SEDANAL

USERS' MANUAL

Walter F. Stafford

and

Peter J. Sherwood

12 Francis Avenue

Cambridge, MA 02138

stafford@sedanal.org

sherwood@computer.org

21 May 2024

SEDANAL v7.91

[A4]

This will print as-is on A4 paper.

(8.27″ x 11.69″ -- 21.0 x 29.7 cm)

Reduce to 94% for 8.5" x 11" paper size

TABLE of CONTENTS

1!!!!INTRODUCTION*AND*SETUP*

1.1!!!!BACKGROUND*

1.2!!!!THE,SEDANAL,SOFTWARE,PACKAGE*

1.3!!!!INSTALLATION:,PROGRAM,AND,DATA,FILES,–,PATHS*

1.3.1!!!!THE#DEFAULT#INSTALLATION:&

1.3.2!!!!ALTERNATE#INSTALLATION#METHODS&

1.3.3!!!!DEFAULT#PATH#STRUCTURE#FOR#SEDANAL&

1.3.3.1!!!!!In&summary,&for&getting&started,&the&default&directory&structure&should&look&like:&

1.3.4!!!!A#GRAPHICAL#REPRESENTATION#OF#THIS#PATH#STRUCTURE:&

2!!!!MAIN*MENU*

2.1.1!!!!PREPROCESSOR#SCREEN&

2.1.2!!!!MODELEDITOR&

2.1.3!!!!DCDT#AND#WDA#SCREENS&

2.1.4!!!!FITTER:#CONTROL#PANEL#SCREEN#AND#CONTROL#FILE#CREATION&

2.1.5!!!!BIOSPIN&

2.1.6!!!!GENERAL#PURPOSE#KINETICS#SIMULA T OR&

2.1.7!!!!NOTE:#THE#MAIN#MENU#WINDOW#CAN#PLACED#IN#THE#S C R E E N#WITH#THE #F O L L O WING#KEYBO A RD#SHORT-

CUTS:&

3!!!!MODULES*OF*SEDANAL*

3.1!!!!THE!MODELEDITOR*

3.1.1!!!!THE#NAMING#TAB&

3.1.2!!!!THE#SPECIES#TAB&

3.1.3!!!!THE#MOLECULAR#PARAMETERS#TAB&

3.1.4!!!!REACTIONS#TAB&

3.1.5!!!!MODELS&

3.1.5.1!!!!!One&Component&-&Single&Ideal&Species&Model&

3.1.5.2!!!!!One&Component&-&Two&Species&Ideal&Model&(e.g.&Monomer-dimer,&rapidly&reversible)&

3.1.5.3!!!!!Two&Component&-&Two&Species&Ideal&Model&

3.1.5.4!!!!!Three&Component&-&Three&Species&Ideal&Model&

3.1.5.5!!!!!Two&Component&-&Three&Species&Interacting&Model&

3.1.5.6!!!!!Two&Component&-&Four&Species&Interacting&Model&

3.1.5.7!!!!!Three&Component&-&Four&Species&Interacting&Model&

3.1.6!!!!MODELS#INCLUDING#NON-IDEALITY&

3.1.6.1!!!!!Hydrodynamic&Concentration&Dependence&and&Thermodynamic&Non-ideality&

3.1.6.2!!!!!Single&species&with&non-ideality&

3.1.6.3!!!!!Multiple&Species&with&Non-ideality&

3.1.6.4!!!!!Cross&Term&Non-ideality:&

3.1.6.4.1!!!!!Setting&limits&to&avoid&numerical&instability&in&the&finite&element&solutions&for&non-ideal&

systems.&

3.1.7!!!!INDEFINITE#SELF-ASSOCIATION#MODELS&

3.1.8!!!!NON-IDEAL#ISODESMIC#SYSTEM S:&

3.1.9!!!!NON-INTERACTING#SYSTEMS#OF#FIXED#KNOWN#STOICHIOMETRY&

3.1.10!!!!EDITING#MODELS#FOR#THE#SEDEQ#FITTER&

3.1.11!!!!USER#AUXILIARY#PARAMETERS#FOR#FITTING:&

3.2!!!!MAIN,MENU,-,OTHER,MODULES:*,ACCESS,BY,MAIN,MENU*

3.3!!!!PREPROCESS,ULTRACENTRIFUGE,DATA*

3.3.1!!!!LOADING#DATA&

3.3.2!!!!ADJUSTING#DATA#FOR#OFF-SETS#AND#JITTER&

3.3.3!!!!SELECTION#OF#SCANS#TO#BE#DISPLAYED.&

3.3.4!!!!SELECTION#OF#SCANS#TO#BE#FITTED.&

3.3.5!!!!PROCESSING#INTENSITY#DATA:&

3.3.5.1!!!!!Beckman&Optima&data&

3.3.6!!!!SUMMARY#OF#THE#PREPROCESSING#PROCEDURE:&

3.3.6.1!!!!Zoom.&

3.3.6.2!!!!!Interference&Data.&

3.3.6.2.1!!!!Remove&optical&jitter:&

3.3.6.2.2!!!!!Remove&integral&fringe&shifts:&

3.3.6.3!!!!!Absorbance&and&Interference&Data:&

3.3.6.4!!!!!Fluorescence&and&Intensity&(pseudo-absorbance)&data.&

3.3.6.5!!!!!Wavelength&Data&from&XL-A.&

3.3.6.6!!!!!Very&large&multi-wavelength&(MWL)&datasets&containing&several&hundred&wavelengths&-&

3.3.6.7!!!!!Deconvolution&of&Concentration&Profiles&from&Multi-Wavelength&Data&

3.3.6.8!!!!!Preprocessing&multi-speed&datasets&for&Wide&Distribution&Analysis&(WDA):&

3.3.6.9!!!!!Preprocessing&Multiwavelngth&data&from&the&Optima&

3.3.6.10!!!!!Preprocessing&Sedimentation&Equilibrium&data&

3.3.6.10.1!!!!!Testing&for&Equilibrium&

DISPLAYING,A PP R O A C H ,TO ,EQ U IL IB R IU M&

1.!!!!IF*THE*USER*HAS*ENTERED*RADII,*THOSE*ARE*USED.*

2.!!!!ELSE*IF*THE*MENISCUS*AND/OR*CELL*BASE*HAS*BEEN*CHOSEN,*IT*WILL*BE*USED*AFTER*

ADJUSTMENT*.*THE*ADJUSTMENT*IS*TO*ADD*0.02*CM*TO*THE*MENISCUS,*AND*SUBTRACT*0.002*

CM*FROM*THE*CELL*BASE.*

3.!!!!ELSE*IF*THE*RANGE*TO*FIT*HAS*BEEN*CHOSEN,*IT*WILL*BE*USED*

4.!!!!OTHERWISE,*THE*CENTER*96%*OF*THE*RANGE*OF*RADII*FOR*ALL*SCANS*WILL*BE*USED*AS*

AN*ESTIMATE.*

3.3.6.10.2!!!!Equilbrium run in standard double sector centerpiece.&

3.3.6.10.3!!!!!Multi-channel&data&with&6&channel&centerpieces&

3.4!!!!FIT,PREPROCESSED,DATA*

3.4.1!!!!THE#CONTROL#SCREEN&

3.4.1.1!!!!!Create&a&New&Control&File&

3.4.1.2!!!!!Choose&between&Analyze&Data&and&Simulate&Data&

3.4.1.3!!!!!How&to&Save&the&Control&file&if&you&haven't&started&with&the&"New"&button.&

3.4.1.4!!!!!Reload&a&previous&control&file&

3.4.1.5!!!!!Loading&cell&data&("abr")&files:&

3.4.1.6!!!!!The&control&screen&is&divided&into&several&regions.&

3.4.1.6.1!!!!!TOP&LEFT:&

3.4.1.6.2!!!!!MIDDLE-LEFT:&

3.4.1.6.3!!!!!MIDDLE-RIGHT:&

3.4.1.6.4!!!!!LOWER:&

3.4.1.7!!!!!The&Equation&Editor&

3.4.1.8!!!!!Start&the&fitting:&

3.4.1.9!!!!!Advanced&Parameters&Button&

3.4.1.9.1!!!!!Error&Estimation&Control&

3.4.1.9.2!!!!!Lamm&Equation&Solutions&(Claverie&control):&

3.4.1.9.3!!!!!Kinetic&integrator&control&

3.4.1.9.4!!!!!Fitting&Tab&

3.4.1.9.5!!!!!Plot&color&and&symbol&size&

3.4.1.9.6!!!!!Select&output&for&Simulations&

3.4.1.9.7!!!!!Set&limits&on&s&and&D&under&non-ideal&conditions.&

3.4.1.10!!!!!Concentration&dependence&of&s&and&D&without&cross&terms.&

3.4.1.11!!!!!Pressure&dependence&of&density&and&viscosity:&

3.4.1.11.1!!!!!Compressibility&

3.4.1.11.2!!!!!Pressure&dependence&of&viscosity&

3.4.1.12!!!!!Selecting&datasets&to&be&fitted:&

3.4.1.12.1!!!!!Removing&a&dataset&from&the&fit.&

3.4.1.12.2!!!!!Weighting&Factors&

3.4.1.13!!!!!Multi-Wavelength&Weighting&Factors&by&Wavelength&

3.4.1.13.1!!!!!Standard&deviation&by&wavelength&file&format&

3.4.1.14!!!!!Selecting&parameters&for&the&finite&element&solutions&to&the&Lamm&equation.&

3.4.1.15!!!!!Extinction&coefficients:&Global&fitting&with&multiple&optical&systems&-&

3.4.1.16!!!!!Local&vs.&global&parameters&-&changing&

3.4.1.17!!!!!Indefinite&Self-association&

3.4.1.17.1!!!!!Isodesmic&case:&

3.4.1.17.2!!!!!Isoenthalpic&indefinite&self-association:&

3.4.1.18!!!!!Constraining&the&range&of&parameter&values&during&fitting.&

3.4.2!!!!EXITING#THE#CONTROL#SCREEN#TO#START#FITTING&

3.4.2.1!!!!!Fitting&Screen&

3.4.2.2!!!!!Screen&Dumps&

3.4.2.2.1!!!!!Toggling&Plots&During&Fitting&

3.4.2.2.2!!!!!Displaying&the&Residuals&Bitmap&

3.4.2.3!!!!!Fitting&to&Sedimentation&Equilibrium&Data:&

3.4.2.3.1!!!!!Loading&the&dataset&-&load&equilibrium&datasets&before&choosing&the&model&

3.4.2.3.2!!!!!Choose&a&model&

3.4.2.3.3!!!!!Enter&initial&guesses&and&other&parameters&

3.4.2.3.4!!!!!Start&the&fit&

3.4.2.3.5!!!!!Global&fit&to&a&dilution&series&with&linked&cells&

3.4.2.4!!!!!Since&the&molar&masses&and&density&increments&are&known&for&the&two&components,&only&the&

loading&concentrations,&y–offsets&and&global&equilibrium&constant&were&fit.&

3.4.2.4.1!!!!!Global&fit&with&un-linked&cells&

3.4.2.5!!!!!Global&Fitting&of&Multi-wavelength&Data&

3.4.2.6!!!!!Fitting&Multiwavelength&Data&at&a&Single&Wavelength&

3.4.2.7!!!!!Error&Analysis:&&&Boot-strap&with&Replacement,&Monte&Carlo&and&F-statistics&

3.4.2.7.1!!!!!Error&Estimation:&Parameter&standard&deviations.&

3.4.2.7.1.1!!!!!Bootstrap&with&replacement&

3.4.2.7.1.2!!!!!Monte&Carlo&Analysis&

3.4.2.7.2!!!!!Error&Estimation:&Parameter&Confidence&Limits&

3.4.2.7.2.1!!!!!F-Statistics.&

3.4.2.7.2.1.1!!!!!General&F-stat&preferences&

3.4.2.7.2.1.2!!!!!Turning&on&F-stats&for&a&particular&parameter&

3.4.2.7.2.1.3!!!!!Displaying&F-stat&progress&

3.4.2.8!!!!!Fitting&for&Extinction&Coefficients&

3.4.2.9!!!!!Fitting&flotation&data&

3.5!!!!DCDT,AND,WDA*

3.5.1!!!!DCDT&

3.5.1.1!!!!!Switching&between&g(s*)&vs&s*&and&s*.g(s*)&vs&ln(s*)&

3.5.1.2!!!!!Smoothing&of&g(s*)&vs&s*&for&presentation&purposes&

3.5.2!!!!WIDE#DISTRIBU T ION#ANALYSIS#(WDA)&

3.5.2.1!!!!!Choice&of&radii&for&WD&Analysis&

3.5.2.2!!!!!Bad&Scans&

3.5.2.3!!!!!Smoothing&after&numerical&differentiation&

3.5.2.4!!!!!Adding&additional&radii&for&WD&Analysis:&

3.5.2.5!!!!!Averaging&overlayed&WDA&curves.&

3.5.3!!!!FLOTATION&

3.5.4!!!!THE#ADV#BUTTON:&

3.5.5!!!!TIME#DERIVATIVE#ANALYSIS#OF#MULTI-WAVELENGTH#DATA:#DCDT#AND#WD.&

3.5.5.1!!!!!WDA:&Extracting&Spectra.&

3.5.5.2!!!!!Least&Squares&deconvolution&of&component&(constituent)&concentration&profiles.&

3.5.5.3!!!!!Multi-wavelength&Fluoresence&Intensity&Data&

3.5.5.4!!!!!Deconvoluting&Multi-wavelength&Fluoresence&Intensity&Data&

3.6!!!!SYNTHETIC,BOUNDARY,AND,BAND,SEDIMENTATION*

3.6.1!!!!FITTING#OR#SIM ULATING#SYN T HETIC#BOUN D ARY#EXPERIM E N TS&

3.6.1.1!!!!!Fitting&

3.6.1.2!!!!!Simulating&

3.6.1.2.1!!!!!Concentration&file&format&

3.7!!!!PREFERENCES*

3.7.1!!!!GENERAL#PREFERENCES&

3.7.1.1!!!!!Location&of&Preferences.txt&files&

3.7.1.2!!!!!Multiple&users&and&multiple&Preference&files:&

3.7.1.3!!!!!Developer&Options&

3.7.2!!!!PREPROCESSOR#PREFERENCES&

3.7.3!!!!DCDT#AND#WDA#PREFERENCES&

3.7.4!!!!WEIGHTING#FACTOR#PREFERENCES#-#"CONTROL"#TAB&

3.7.5!!!!CONFIDENCE#LIMITS#--#ERROR#ESTIMATION#CONTROL:&#ADVANCED,#CONTROL#EXTENDED#TAB&

3.7.6!!!!"CLAVERIE#CONTROL".&

3.7.7!!!!KINETICS/EQUILIBRIUM#CONTROLS.&

3.7.8!!!!THE#FITTING#CONTROL#TAB&

3.7.9!!!!SIMULATING #OUTPUT#CHOICES&

3.7.10!!!!REPORTS:&#OUTPUT,#CONTROL#EXTENDED#TAB&

3.7.11!!!!THE#OTHER#“OUTPUT#…”#TABS&

3.7.11.1!!!!!Concentration&as&a&function&of&radius&output.&

3.7.11.2!!!!!Concentration&as&a&function&of&time&for&the&initial&equilibration&step&

3.7.11.3!!!!!Disposition&of&various&log&files&

3.7.12!!!!FITTING#PREF E R E NCES&

3.7.13!!!!SIMULATION #PREFERENCES&

3.8!!!!BIOSPIN*

3.9!!!!SIMULATING,DATA*

3.9.1!!!!SAVING#A#“PACKAGE”.&

3.9.2!!!!OUTPUT#OF#INITIAL#REACTION#TIME#COURSE.&

3.9.3!!!!OUTPUTTING#THE#TIME#COURSE#OF#THE#INITIAL#EQUILIBRATION#STEP&

3.10!!!!KINETICS,SIMUL A TO R *

3.11!!!!SCRIPTING:,FITTING,AND,SIMULATION*

3.11.1!!!!SCRIPTING#COMMANDS:&

3.11.2!!!!SPECIFYING#CONTROL#FILES#IN#SCRIPTS&

3.11.3!!!!STORING#SCRIPTING#CONTROL#FILES&

3.12!!!!KEYBOARD,SHORTCUTS*

3.13!!!!HELP*

3.14!!!!EXIT*

4!!!!BACKGROUND*THEORY*

4.1!!!!SEDIMENTATION,VELOCITY,THEORY:*

4.1.1!!!!CONCENTRATION#TIME-DIFFERENCE#CURVES.&

4.1.3!!!!PROCEDURE#USED#BY#SEDANAL#FOR#FITTING#TIME#DIFFERNCE#DATA.&

4.2!!!!SEDIMENTATION,EQUILIBRIUM,THEORY*

4.2.1!!!!IDEAL#CASE:&

4.2.1.1!!!!!Monomer-dimer:&

4.2.1.2!!!!!Hetero-Association:&

4.2.2!!!!NON-IDEAL#CASE:&

5!!!!APPENDIX*A:***HELP*FILE*AND*CHANGE*LOG.*

6!!!!INDEX*

7!!!!LIST*OF*FIGURES*

2 Introduction and Setup

SEDANAL is a computer program for analysis sedimentation velocity and sedimentation

equilibrium data that are produced by the BeckmanCoulter Optima Series XLA/I

ProteomeLab analytical ultracentrifuge system or the Beckman Optima AUC, or the Open-

Multiwavelength machines available in Germany. SEDANAL also has a kinetics simulator that can

simulate almost any reaction scheme by choosing a reaction scheme in the Model Editor.

⚠

Note: This is free software. There is NO warranty; not even for MERCHANTABILITY or

FITNESS FOR A PARTICULAR PURPOSE.

2.1 Background

Starting from the raw data files produced by intensity, absorbance, interference, fluorescence,

or multi-wavelength intensity, absorbance and fluorescence experiments, the SEDANAL

sedimentation velocity module (SedVel) uses numerical solutions to the Lamm equation

along with chemical kinetics and equilibrium models to fit parameters to the experimental

data. SEDANAL can also produce simulated data in the same format as the raw data files

produced by the XLA/I and Optima optical systems. SEDANAL SedVel fits to time

difference curves to completely remove time invariant systematic errors from the data. It also

allows preprocessing of the data to essentially eliminate time dependent, radially independent

systematic instrumental errors characteristic of the interference optics as well as integral

fringe jumps that can occur at the meniscus.

The SEDANAL sedimentation equilibrium module (SedEq) fits sums of exponentials to

sedimentation equilibrium data to obtain estimates of molecular masses and equilibrium

constants for interacting and relative amounts for non-interacting systems. Non-ideality

(second and third virial coefficients) can also be estimated.

The output of the SEDANAL SedVel curve fitter is a report file containing the root mean

squared deviation (rmsd) and best-fit values for the parameters, as well as the experimental

and calculated difference curves and the residuals of the fit for each of the difference curves.

The user may specify which parameters are to be fitted and which to be held constant on the

Control Screen

The output of the SEDANAL SedEq curve fitter is a file containing the root mean squared

deviation (rmsd) and best-fit values for the parameters, as well as the experimental and

calculated equilibrium concentration curves and the residuals of the fit for each of the curves.

The user may specify which parameters are to be fitted and which to be held constant.

SEDANAL also writes a detailed Report File containing all the input parameters, model fitted

and parameters returned by the fits.

2.2 The SEDANAL software package

The software package is CPU-intensive: it runs best on the fastest possible multi-core CPU

with lots of memory and at least 100 Mb of the hard drive space for installation. SEDANAL

runs under Windows 7 64 bit, Windows 10 64 bit, and Windows 11. This version does many

of the computations in parallel. To take full advantage of parallel computing, the machine

should have as many cores as datasets being fit at any one time.

2.3 INSTALLATION: Program and Data Files – paths

2.3.1 The default installation:

SEDANAL expects to find a specific path structure for files, (This can be changed to

some extent in the Preferences menu and is not a "hard and fast" requirement). Most

screens now have a "Browse" function, as well.

The top level directory (also called a "folder") may be named anything you wish; we'll

call it "SEDANAL". In the "SEDANAL" directory, create three subdirectories called

"ModelEditor", "Program" and "User_data".

The file "ModelInfo.txt", if it already exists, must be placed in the "ModelEditor"

directory. If it doesn’t exist, for example in a new installation, SEDANAL will create an

empty ModelInfo.txt file for you in the "ModelEditor" directory. The application

“SEDANAL.exe” must be placed in the "Program" directory along with

“Preferences.txt” If preferences.txt is omitted, SEDANAL will create a default

preferences file. There is also a DLL named "libiomp5md.dll" (for all versions after

6.01) that must be placed along with SEDANAL.exe in the Program folder. A help file

for the on-line help also goes in the Program folder. It is named "SEDANAL.chm". If

you downloaded the SEDANAL package from the internet, the SEDANAL.chm file may

be blocked. It must be un-blocked by changing its Properties. Both the Help file and

Change Log contain detailed descriptions of various functions only briefly described in

this manual. The Help File is searchable.

Each "User_data" directory will contain sub-directories in 3 levels. At the highest level

within “User_data”, any number of "experiment" directories can be created. Each

experiment directory, in turn, must contain the data directories in the standard

BeckmanCoulter XLA/I date/time format. See Figure 2-1 for an example. Usually one

will store control files (which have extension *.abc) in an experiment folder. In this case

(see example below) it would be the "experiment 1" folder. The run files (with extension

*.abr) are automatically stored at the level of the XLA/I data files.

2.3.2 Alternate installation methods

The location of both the User_data directory and the ModelEditor directory can be

specified in the Preferences:

Figure 2-1 Setting up paths to the data files and to the model file.

The “User_data” folder can have any name and be located where-ever you want--like a network

server over a VPN, for example (be careful; this doesn't always work well.). The “ModelEditor”

folder also can have any name and be stored anywhere. You might have multiple “User_data”

folders, each with multiple experiment folders and multiple “ModelEditor” folders depending on

the analysis du jour. The SEDANAL executable must be located on the local machine, however.

2.3.3 Default path structure for SEDANAL

2.3.3.1 In summary, for getting started, the default directory structure should look like:

C:\SEDANAL\ModelEditor\ModelInfo.txt

C:\ SEDANAL\Program\ SEDANAL.exe

C:\ SEDANAL\Program\Preferences.txt

C:\SEDANAL\Program\

C:\ SEDANAL\User_data\my_experiment#1\mmddyy\hhmmss\*.IPn

C:\ SEDANAL\User_data\my_experiment#2\mmddyy\hhmmss\*.IPn

C: \SEDANAL\User_data\my_experiment#3\mmddyy\hhmmss\*.RAn

C: \SEDANAL\User_data\my_experiment#4\mmddyy\hhmmss\*.FLn

2.3.4 A graphical representation of this path structure:

3 MAIN MENU

Figure 3-1 MAIN MENU

3.1.1 Preprocessor Screen

• Read in a dataset of scan files

• Designate whether it is a SedVel or a SedEq run.

• Eliminate systematic error

• Choose meniscus and base

• Choose radial fitting range

• Interpolated optical blank subtraction can be carried out on equilibrium data

• Save preprocessed data in a “cell data file” (*.abr) for fitting, or DCDT and

wide distribution analysis (WDA).

• Experiment Information page

• Approach to EQ

3.1.2 ModelEditor

• Enumerate the species in the model

• Create reaction schemes

• Choose parameters to be fit

• Set default limits on parameter values

3.1.3 DCDT and WDA Screens

• Standard time derivative g(s*) analysis (DCDT)

• Multi-speed and single-speed, wide distribution analysis (WDA)

• Both DC/DT and WDA have multi-wavelength capability

• Measure extinction spectra from multi-wavelength data with WDA

3.1.4 Fitter: Control Panel Screen and control file creation

• Read either an existing Control file or Select "New"

• Select either "Analyze data" or "Simulate data"

• Chose a model, enter the parameter values

• Choose datasets to be fit and scans to be included in the fits.

• Enter fitting/simulating parameters ( # of points, time increment,...)

• Enter initial guesses

• Fit the model to the data

• The Fitter can be used to fit multiwavelength data, given the

extinction spectra of the components.

3.1.5 BIOSPIN

• Computes number, weight, and z-average molar mass as a function of local

cell concentration and position in a sedimentation equilibrium run. Also

computes M

and M

, molar mass moments that are independent of the

second virial coefficient. Based on BIOSPIN by Roark and Yphantis (1968)

3.1.6 General Purpose Kinetics Simulator

• A general purpose kinetics simulator can be accessed from the Main Menu

and can be used to simulate any reaction scheme that can be represented in

the ModelEditor.

3.1.7

⚠

NOTE: The Main Menu Window can placed in the screen with the following

keyboard short-cuts:

The commands are

• Ctrl-C: Center the main menu both vertically and horizontally in the current monitor.

• Ctrl-H: Center the main menu horizontally in the current monitor, leaving the vertical

position unchanged.

• Ctrl-V: Center the main menu vertically in the current monitor, leaving the horizontal

position unchanged.

• Ctrl-R: Restore the default position (upper left of the primary monitor) and size.

4 Modules of SEDANAL

4.1 The ModelEditor

The Model Editor allows one to create and edit various models and reaction schemes. It is also where

the default fitting parameters and their default limits are chosen. (The defaults can be adjusted later

if necessary on the Control Screen.)

Models may consist of any combination of reaction schemes ranging from a single species to

combinations of non-interacting species, and hetero- and self- associations. It will allow up to 28

thermodynamic components and up to 28 species related by up-to 27 chemical reactions.

When the ModelEditor is first invoked, it starts up with an empty box into which the model name or

its aliases may be entered as a text string. The drop-down window will show the other models that

have been entered previously. The new model will be added to this list when the "STORE" button is

clicked or the "EXIT" button is selected. The user will select one of these models from the Control

Screen when fitting data.

4.1.1 THE NAMING TAB

Figure 4-1 The Model Naming tab.

4.1.2 THE SPECIES TAB

Figure 4-2 The species tab

In this example, we have chosen 3 species, A, B, and C. And since we have not indicated any

reactions between them, the Model Editor has designated them as components by selecting the "C"

button next to their names. Examples will be give below for several types of interacting and non-

interacting systems.

4.1.3 THE MOLECULAR PARAMETERS TAB

Figure 4-3 Molecular Parameters

The parameters include:

• molar mass

• sedimentation coefficient

• density increment, (dr/dc)

which for most practical purposes is just (1-vr),

• or partial specific volume (actually the product of vr as in (1-vr), since the density is set

to 1.000 internally)

• mass extinction coefficient, either A.U.-(mg/mL)

-1

or fringes-(mg/mL)

-1

for a the optical path

length of the centerpiece used.

• concentration dependence of the frictional coefficient.,

• k

[s(c)=s

/(1 + k

*c)] or [D(c)=D

/(1 + k

*c)]; where k

, in this case, applies to both s and D,

and second order coefficients

• thermodynamic non-ideality expressed through the second virial coefficient, BM

, as in the

expression (1 + 2BM

*c)

• thermodynamic non-ideality expressed through the third virial coefficient, CM

, as in the

expression (1 + 2BM

*c + 3M

)

• User defined parameters

The simplest model is a single ideal species. In this case the unknown parameters of interest usually

are the sedimentation coefficient and molar mass (or diffusion coefficient). The Model Editor species

tab for a single species is shown below. One species has been selected in the species box.

In the list of molecular parameters, the desired boxes are checked, their scale factors indicated and

their default limits set.

• In this example, the first parameter is the molar mass and its scale factor has been set to unity.

Therefore, future references to this parameter on the control screen must be in units of daltons

or grams/mole. If one had entered 1000 in the scale factor field, future references to the molar

mass would be in kilodaltons or kg/mole.

• The second parameter is the sedimentation coefficient and has been scaled to 1x10

-13

so that

future references to it must be in units of svedbergs. At the same time it would also be a good

idea to rename the parameter to reflect the new units; for example, you might change it to

"Sedimentation coefficient, (S)" as shown below.

• The third parameter is either the density increment or partial specific volume times the

density, and the fourth, is the extinction coefficient on the mass concentration scale, (g/L).

For absorbance optics this is the specific extinction coefficient multiplied by either 1.2 for

the 12 mm path length of the centrifuge cell (i.e. one has to multiply the usual value

(normally corresponding to a 1 cm optical path) by 1.2 for a 12 mm centerpiece before

entering it into the box or by 0.3 for a 3 mm centerpiece, for example, etc ...). For

interference optics, it is the number of fringes produced by a 1 mg/mL solution and is

approximately 3.29 in a 12 mm centerpiece for typical proteins, but varies depending on

whether the macromolecule is a protein, carbohydrate or nucleic acid. Each species may have

a different extinction coefficient. SEDANAL will allow data from different optical

systems and wavelengths and different centerpieces to be combined for global fitting with a

different extinction coefficient to be entered for each species in each dataset (or cell) obtained

with a different optical system or wavelength or with a different optical path length.

4.1.4 REACTIONS TAB

The reactions tab (see below) allows the user to specify the reaction scheme relating the several

species present in the model.

4.1.5 MODELS

4.1.5.1 One Component - Single Ideal Species Model

The simplest model is a single species with no non-ideality. The necessary parameters are entered

into the ModelEditor control screen as shown in Figure 4-4.

Figure 4-4. Screen showing default fitting parameters for a One Component - Single Ideal

Species Model.

In this example, (Figure 4-4), the model is named "1 comp 1 species". The number of species is "1",

the number of reactions is "0", and the number of parameters is “4”. One might want to fit for the

molar mass and sedimentation coefficient by default. So, these buttons would be checked. They will

appear with a light gray background on the control screen. If they aren't checked here, they will have

a blue background on the control screen indicating that the default is to hold them constant. However,

their status can be changed later from "hold" to "fit" by right-clicking on the corresponding box on

the control screen.

The species having a selected button in the "C" column next to their names are the components as

determined by the reaction scheme.

One may also select the allowable range for each parameter to be checked during entry of values on

the control screen. Allowed ranges to be used during fitting are set by right clicking on the parameter

value on the control screen.

⚠

NOTE: Pay special attention to these limiting values. Many systems may require that these be

changed, especially those that involve organic polymers in non-aqueous solvents and those that

involve inorganic compounds or nearly neutrally buoyant or negatively buoyant compounds. They

will have to be changed for compounds present in your sample that do not contribute to the optical

signal—like background proteins in a fluorescent tracer experiment. Although some of these can be

changed on the Control Screen, it is better to change them in the model.

4.1.5.2 One Component - Two Species Ideal Model (e.g. Monomer-dimer, rapidly reversible)

Stoichiometric relationships are established on the Reactions Tab, after the species are indicated on

the Species Tab

Figure 4-5 One component, two species model.

4.1.5.3 Two Component - Two Species Ideal Model

The next example, "2 comp 2 species", comprises two independent ideal species (i.e. two

components) with no interaction between them. Necessary parameters in the ModelEditor control

box are shown in Figure 4-6.

Figure 4-6 Screen showing parameters for the Two Independent Ideal Species Model.

Since we are fitting for A and B as independent species in this example, the model tab for species 2

would be the same as for species 1. If a stoichiometric or other relation between the two independent

species is to be specified, it can be done on the control screen using the Equation Editor (The EQN

button)

4.1.5.4 Three Component - Three Species Ideal Model

Similarly, 3 component, non-interacting 3 species system, let’s call it “3 comp, 3 species”, would be

designated by choosing 3 species and no reactions. The Model Editor screen would look like Figure

3-4-7 with each of species tabs the same having molecular mass and sedimentation

coefficient checked. As above, if a stoichiometric or other relation between the two independent

species is to be specified, it can be done on the control screen using the Equation Editor (The EQN

button)

Figure 3-4-7. Model Editor Species tab showing parameters for the Three Independent Ideal

Species Model.

4.1.5.5 Two Component - Three Species Interacting Model

In this case, there is simple bimolecular complex formation between species A and species B:

A + B = C K

= k

Where K

is the association constant, k

is the forward rate constant and k

is the reverse

rate constant.

In this case, one would have studied A and B separately, and therefore, would not have to fit for their

properties. So none of the “Default is fit” boxes are checked for either A or B (Figure 3-4-8). In

general, in this case, we would be fitting for the sedimentation coefficient of C as well as the

association equilibrium constant for complex formation. Note that “Species 3” is not highlighted

because it is not an independent component; the two components are A and B on the "species" tabs.

(Note: the two components could have been chosen as A and C but that is not convenient in this

case.) The species tab for A and B would be filled in the same way (Figure 3-4-8).

Figure 3-4-8. Screen showing parameters for the Interacting Species Model

For Species C, we have (by clicking in the “Species 3” window) the parameters as shown in

Figure 3-4-9.

Figure 3-4-9. Species tab showing parameters for Species 3 (“C”).

We have checked the “mass-wtg avg” boxes for Density Increment and Mass extinction

coefficient to indicate that these parameters are calculated as mass averages of those for species A

and B.

In this example, on the row labeled "Reaction 1", the “fit” box for Keq is checked to indicate

that we are going to fit for the equilibrium constant by default. For this particular model, SEDANAL

uses an analytical solution to the mass action equations. SEDANAL uses analytical solutions for

monomer-dimer and A+B=C.

4.1.5.6 Two Component - Four Species Interacting Model

A more complicated example is a 2 component system, comprising of 4 species, and 2 reactions.

This might represent an antigen-antibody reaction.

A + B = C K

C + D = D K

Selection of parameters is shown in Figure 3-4-10.

Figure 3-4-10 Selection of species

Figure 3-4-11. Parameters for 2 component, 4 species interacting model: A + B = AB, AB +

B = AB

When the 4 species are selected on the species tab (Figure 3-4-10), all the buttons are automatically

selected as components until the reaction is specified on the reactions tab. After selecting the reaction

(Figure 3-4-11, top), the Model Editor decides which species represent independent components and

which are related by a chemical reaction (Figure 3-4-11, bottom). The number of selected buttons

in the "C" column is the number of independent components. SEDANAL fits for the loading

concentration of each independent component.

⚠

NOTE: Setting allowable ranges for the parameters shown in the Model

Editor affects only the range allowed be inputted on the Control Screen. One may also

select the allowable range for each parameter to be used during the fitting procedure

by left-shift clicking in the parameter’s box on the control screen. If one of the limits

is hit, the function evaluator will return a large value for the r.m.s. deviation for that

iteration.

4.1.5.7 Three Component - Four Species Interacting Model

An example of a 3 component, 4 species system would be a system such as A+B=C with an

impurity or aggregate, D. A and B are the first 2 components and D will be the third. The

ModelEditor will correctly understand that D is not participating in a reaction and is, therefore, an

additional component.

When this model is chosen (see below), the Control Screen, will show the three

components with their appropriate labels. (The species’ names can be edited to reflect the three

components. For example species C could be renamed, “AB”. So the model editor screen would

show the third species as AB, as in the figure below.) The figure also shows the tab for species 4 now

labeled as D. The Model Editor can determine which species represent independent components and

which are not independent.

Figure 4-12. Selection of parameters in the control box for three component system

Selection of parameters in the control box for three component system - two interacting species

(A+B=C) plus one non-interacting species (D) model (Figure 4-12).

4.1.6 MODELS INCLUDING NON-IDEALITY

4.1.6.1 Hydrodynamic Concentration Dependence and Thermodynamic Non-ideality

Non-ideality is treated to first order in concentration through two parameters, k

, the

concentration dependence of the frictional coefficient, and BM

, the second virial coefficient term.

These nonideality parameters are included in the modeling in the following way for sedimentation

and diffusion coefficients: see below: (Cross Term Non-ideality:)

The factor (1+k

c) represents the concentration dependence of the frictional coefficient:

The factor (1+2BM

c) represents the contribution from thermodynamic non-ideality through the

second virial coefficient.

It is possible, to add a second order non-ideality terms to allow fitting to data at very high

concentrations.

Figure 4-13 Higher order non-ideality terms

4.1.6.2 Single species with non-ideality

Figure 4-14 Selection of parameters in the control box for single species with non-ideality.

The “Species 1” tab has both Ks and BM1 checked.

4.1.6.3 Multiple Species with Non-ideality

Species "A” and Species "B” would both look the same with the ks and BM1 both checked.

Figure 4-15. Selection of parameters for two species with non-ideality.

If you suspected that the non-ideality was the same for both species A and B, you would check the

“copy” box for “Species B” and uncheck the “Default is fit” boxes. This would allow for fitting for

a single value of Ks and BM1 for both species Figure 4-16. This might be appropriate on a mass

concentration scale.

Figure 4-16 “Copying” values for “Species 2” from “Species 1”

If one uses only the BM1 and Ks entry boxes on the control screen for each species,

SEDANAL assigns the non-ideality as follows:

Use either "Fit" or "Hold constant" if fitting for non-ideality with cross term non-ideality

represented as follows:

This amounts to requiring that the elements of each column of the matrix be equal. This is a

reasonable approximation for initial fits to get Ks and BM1 values that are close to the correct ones.

This is equivalent to saying that each species has the same effect on all other species. For example

it says that the backflow due to species 1 creates a backflow that slows down all other species in

proportion to its effective stokes radius and local concentration. The same argument applies to the

excluded volume and charge effects (Donnan effects) that species 1 has on the other species in its

vicinity in proportion to its local concentration.

⚠

NOTE: In most cases, BM1 and Ks are of roughly the same magnitude (in units of cc/g) and

both should usually be included in a fit or a simulation. See section below on setting limits to D for

the sake of numerical stability. This limit is usually not necessary and generally should be left

blank as long as both Ks and BM1 are included in the fit or simulation and are of similar order of

magnitude.

4.1.6.4 Cross Term Non-ideality:

(In versions 6.79 and later.)

In a multispecies system, each species can affect each of the others through cross non-ideality

coefficients requiring both Ks and BM1 matrices to accommodate the cross interactions.

The factor in the denominator for both s and D is attributed to the back flow and is expressed as the

concentration dependence of the frictional coefficient.

and the factor in the numerator represents the contribution of the thermodynamic non-ideality

expressed through the second virial coefficients. The thermodynamic factor is

These equations imply the following matrices that SEDANAL will use to express the non-ideality

and whose elements are entered on the Control screen:

So for a three species system we would have the following matrices (the order of subscripts is

"row, column"):

The matrix elements can be entered on the control screen for a non-ideal model by right-clicking on

either the Ks or BM1 window:

Click on "Matrix..." to see the matrix entry form. For example, for a three species system, the

following matrix window opens to allow entry of the self (diagonal) elements and the cross (off-

diagonal) elements of the Ks matrix.

or entry of the BM1 matrix. (where B'

=BM1

above)

⚠

NOTE: These parameters are read-only and must be determined from a series of

separate, pairwise experiments.

4.1.6.4.1 Setting limits to avoid numerical instability in the finite element solutions for non-

ideal systems.

Not all systems will require setting of limits, but the more highly non-ideal systems might.

On the Fitting Control Screen, choose Advanced > Non-ideality > and enter 2.0 for D/Do (you

may have to adjust this if you get "CHECK GRID" messages).

Leave (Min s/s

) blank.

You may also have to set an upper limit on the concentrations near the base:

Advanced > Claverie control > Base conc params

Click on

to get to:

Figure 4-17 Base Concentration Parameters

And be sure to set the “Concentration maximum” to a value several times the loading concentration

in g/L. (this limits the concentration and non-ideality contribution at the base of the cell and

improves the numerical stability of the finite element calculations. DO NOT check the “Fit

exponentials to all points” or the Smooth every “n” points boxes. These are experimental and don’t

work very well.

4.1.7 INDEFINITE SELF-ASSOCIATION MODELS

The model editor can create isodesmic and isoenthalpic models for use with

SEDANAL. Click on the “Number of reactions” to indicate 1 reaction and the type the letters “sa”

in the window below the “Default name” for “species 1”, name the reaction either “isodesmic” or

"isoenthalpic", click the corresponding button to the right of the row and click the “Store” button.

Figure 4-18. Setup for indefinite self-association.

Figure 4-19. Blow-up of upper left of the reactions tab.

Figure 4-20. Isodesmic case

For the isoenthalpic case (Figure 4-20), the ModelEditor window looks the same except that the

“Isoenthalpic” button should be clicked to indicate to the fitter that it should use the equations for

that case.

4.1.8 Non-ideal Isodesmic systems:

Non-ideality for isodesmic systems is handled that same way as for discreet systems except that

SEDANAL uses the Fujita-Adams approximation by assuming the non-ideality is a linear function

of the total local concentration. Therefore, the Ks and BM1 matrices are not used.

4.1.9 NON-INTERACTING SYSTEMS OF FIXED KNOWN STOICHIOMETRY

Models for non-interacting (i.e. not controlled by mass action) systems composed of components

with known molar mass ratios can be treated using the Equation Editor on the control screen. Fixed

relationships between parameters can be established by typing in FORTRAN-like equations in the

Equation Editor Page (“Eqn” button”). - More on this later when we get to the control screen.

4.1.10 EDITING MODELS FOR THE SEDEQ FITTER

Figure 4-21. SEDANAL will determine whether the data are from SedVel or SedEq

SEDANAL will determine whether the data are from a SedVel or SedEq run from information in

the cell data file (*.abr) and display the appropriate boxes on the control screen for a particular

model after reading in the cell data file. The type of run is indicated when preprocessing the

data. (This shows the right half of the model editor species tab with the "Equilibrium" button

selected.) The model editor has a page for SedVel and a separate page for SedEq in each

model selected by clicking the appropriate button above.

4.1.11 User Auxiliary Parameters for fitting:

The fitting process for sedimentation velocity experiments always uses the molecular parameters

molar mass, sedimentation coefficient, density increment (or partial specific volume), and extinction

coefficient. For equilibrium experiments, sedimentation coefficients are not used. Other molecular

parameters, such as virial coefficients, are optional.

SEDANAL v4.86 introduced a new type of molecular parameter, the user auxiliary parameter,

which is not used directly for fitting. Instead, it is used indirectly via the Equation Editor. For

example, suppose we want to fit for the stoichiometric ratios, n, of species relative to a "monomer"

species rather than fitting for their molar masses. For a three species system, the relationships might

be M

and M

First, we create a model in the MODEL EDITOR with a user auxiliary parameter, which we will name

N entering the symbol nu for them. (Figure 4-22) Now N(i) can be used in the Equation Editor.

Figure 4-22 Auxiliary parameters—user defined

Next, create a control file, specifying the new model.

The symbol for the auxiliary and other molecular parameters used in the equation editor, N in this

example, is given in the model (in the MODEL EDITOR). Symbol names are made up of letters, digits,

and underscores (“_”), with the initial character not a digit. Symbol names are case-sensitive.

The symbol names must be unique, and cannot conflict with the symbol names used for other fitting

parameters; currently these are K, kf, kr, L, r, rw, m, b, y (for equilibrium runs only), p, v and T.

Versions after version 5.78, now include the Frictional Ratio, F, as a fitting parameter which allows

one to set limits for f/fo requiring it to be in any specified range, like 1.0 to 5.0, and not to allow it to

drift below 1.0, the theoretical lower limit of a perfect, unhydrated sphere.

⚠

NOTE: When storing a model, the name and aliases (if any) are checked, and storing is

disallowed if any other models has the same name or alias. Names are considered the same if they

differ only in spacing. For example, "A+B=C" and "A + B = C" are the same

4.2 MAIN MENU - Other Modules: Access by Main Menu

When SEDANAL is launched, the Main Menu appears with 11 choices, see Figure 4-23.

Figure 4-23 The Main Menu

4.3 Preprocess Ultracentrifuge Data

The preprocessor screen presents the following window:

Figure 4-24 PreProcessor Screen

The "View" menu in the upper left corner of the Preprocessor window provides several choices for

the display of data: the default values can be set in the Preprocessor Preferences tab.

Figure 4-25 View choices

The first step is to select the experiment folder by clicking on the top upper left box to reveal a

list of experiments: in the top drop-down window, one must select the experiment folder

(Figure 4-26). Then, once the experiment folder has been selected, in the bottom drop-down

window, one will see the available datasets and select the dataset to be preocessed (Figure

4-27).

Processing is different for absorance data and interference data. For both absorbance data and

interference data, one must select the meniscus, the base and range of data to be fitted.

Additionally, for interference data, one must remove optical jitter (time dependent, radially

independent noise) by aligning the fringes in the air-air space to the left of (centripetal to) the

meniscus and then remove any integral fringe jumps by selecting a spot--either in the plateau,

near the meniscus, or near the hinge point depending on the dataset--where the data changes the

least from scan to scan.

In addition to absorbance and interference data, the preprocessor recognizes both intensity data

from the XL-A and the Optima AUC and fluorescence data from the Aviv FDS. The

preprocessor also recognizes data from the Cölfen type Multi-wavelength (Open MWL)

absorbance optical systems. Each set of intensity data is presented in the preprocessor as two

separate datasets, one from the sample side and one from the reference side. The intensity data

are converted to “pseudo-absorbance” data by taking their logarithm before being processed.

Pseudo-absorbance data are linearly related to absorbance data but differ from the corresponding

absorbance data by an arbitrary additive constant offset.

4.3.1 Loading Data

Figure 4-26. The Experiment drop-down window

The Experiment drop-down window displays a list of folders in the User_data

directory. Folders with names in bold format contain scan files. Folders in

regular font, contain subfolders that contain scan files.

Figure 4-27 List of folders

In this example, we’ll select “AGSG171736” which contains XL datasets stored in the standard

format. After expanding the folder “AGSG171736” select a set of files such as "15KCELL1",

and now click on the “Centrifuge” drop-down window. To reveal the scan files available for

fitting.

(N.B: The standard Beckman folder format is

…\User_data\experiment 1\”date”\”time”\*.IPn ).

Figure 4-28 The datasets are shown with alternating yellow and white backgrounds.

The time and date shown in the “Centrifuge” drop down window are derived from the folder

names (in this case from a 4 cell run).

When one of the datasets is selected, the entire dataset will be read in and will be used for fitting.

The actual files (i.e. scans) to be used in fitting will be chosen from the control screen by right-

clicking on the run file name window (see below)

⚠

NOTE: If a file or files are missing from a dataset, SEDANAL will break the set up into two

sets. For example, if the run generated scans 00001.IP1 thru 00099.IP1 but scan 00050.IP1 was

missing for some reason, SEDANAL would show two batches of scans from that cell, one from

00001.IP1 thru 00049.IP1 and another from 00051.IP1 thru 00099.IP1.

Figure 4-29 missing scan file

In order to rejoin the two parts of the run, a dummy file would be need to be created with the name

00050.IP1. Now the dataset will appear as one contiguous set of files and would be treated as a

single dataset. Later, on the control screen one would have to exclude that dummy file from the

analysis. The dummy file must at least contain the two header lines at the top. The resulting series

of scans (showing only those from 00040.IP1 to 00060.IP1) would look as shown in Figure 4-30 and

now the data can be treated as a single set.

Figure 4-30 Missing scan

Occasionally, the XL-A/I will write an empty file and the data will appear as shown above. The

name (i.e. number) of the missing file must be noted and that file must be excluded from the fitting

or from the DCDT/WD analysis. Versions 6.01 and later will automatically flag a scan as bad if the

file is not complete.

After a dataset has been selected and read in, the screen will look something like this for interference

data:

Figure 4-31 Raw Interference data

Initially when the data (in this case, interference data) are read in, they will look like above

Figure 4-31 and will require alignment in the air-air space (jitter adjust) and removal of integer

fringe jumps in the solution column that may occur at the meniscus.

Now indicate what type of run it is:

Figure 4-32 Click the appropriate button for this run type.

Next, we take care of vertical “jitter” by aligning the fringes in the air-air space. Under the

“Jitter adjust” label click on “Go” to activate the cursor and click in the air-air space. The number

“10” indicates that a range of plus and minus 10 pixels will be used for the alignment. If you need

to undo what you have done, click on “reset”.

4.3.2 Adjusting Data for off-sets and jitter

Figure 4-33 The “Go” button: Jitter adjust.

When the "Go" button is clicked the cursor is activated to indicate the spot at which the jitter

adjustment will be carried out. the number "10" in the window next to the "Go" button indicates the

number of pixels around the clicked position (i.e. +/- 10 points) that will be used in the jitter

correction process. Adjust this number to make the range narrower, as one might for absorbance

data.

Figure 4-34 After the jitter adjustments. The scans are now all aligned in the air-air space.

Once the data have been pre-processed, they will be written into a “run” file (with the extention

“.abr”) and will be available for re-editing by selecting them in the middle window labeled “Cell

Data”. All the parameters, meniscus, base etc are stored in the run file and are used by the fitter to

perform the fits. Run files can be re-editted at any time, for example, for re-fitting with a different

range or better estimate of the base or meniscus positions.

The preprocessing procedure is explained in detail below.

The fringes are aligned first on the air-air space to remove vertical variations due to time dependent

instrumental noise. Under "Jitter adjust" click on the “Go” button and then click in the air-air space.

The fringe patterns will all be aligned as shown in Figure 4-34.

Now we take care of the integral fringe adjustment:

1. à 2. à 3.

Figure 4-35 Re-adjust the data for integral fringe shifts.

The next step is to re-adjust the data for integral fringe shifts introduced by the data acquistion

software at the meniscus. Next (1) click on the “Locate” button and select a spot – usually

someplace near the bottom in the plateau region – by clicking there. Then (2) click the

“Subtract” button. The result will be as shown in Figure 4-36, and the buttons will look like (3)

above (Figure 4-35). Clicking the “undo” button will undo the last adjustment.

After manually re-scaling the plot, click on “Meniscus” and ‘click and drag and unclick’ at the

meniscus position. Repeat the process to choose the base position by clicking on the “cell base”

button and then ‘click and drag and unclick’ at the base position. Note that the base of the normal

double sector centerpieces is very close to 7.20 cm and so the actual base postion will probably be

at a higher radius than you are at first inclined to choose. (You can verify this by running some

simulations: You’ll be surprised.)

Figure 4-36 Scans adjusted for integral fringe shifts

After the fringe adjustments have been completed and after the meniscus, base and range to be

fitted have been selected, the screen will look like one presented in with all five windows at the

bottom filled in their corresponding values (Figure 4-36).

The plot can be rescaled by manually by selecting the y-axis and x-axis ranges in the four windows

at the top of the plot. Click on the “Plot” button to refresh the screen.Figure 4-37

Figure 4-37 Set min and max x and y values manually

The “View” button (Figure 4-38) in the upper left corner of the preprocessor screen allows one to

choose various charcteristics of the plot, as well as to show how many of the loaded scans are

currently being plotted, which are shown in the upper right corner of the plot.

Figure 4-38 Plot format

Figure 4-39 Number of scans plotted

The number of scans being plotted is shown in the upper-right corner of the plot.(Figure 4-39)

4.3.3 Selection of Scans to be displayed.

Click on the “Select scans to be plotted” button to select which scans will be displayed on the screen.

Figure 4-40 Select scans to be plotted

Note that all scans will be stored in the “run” file and can be used in the fitting process. Check or

uncheck each scan as shown below:Figure 4-42..

A range of scans and the increment can be selected by typing in the small boxes in the middle right

of the window (Figure 4-43) as shown in Figure 4-42..below.

4.3.4 Selection of scans to be fitted.

After the meniscus, base, etc. have been selected, it’s a good idea to choose an initial range of scans

to be fitted at this point, while you have them on the screen. Although these can be selected later in

the fitting control window by right clicking on the file name for the cell whose scans are to be chosen

(see below), it’s easier to do it at this stage to see what is being chosen. For example, you might

decide to fit the following scans after seeing them in the preprocessor (i.e. scans 400 to 499 by 1):

Later, on the fitting screen, this range of scans can be recalled by right-clicking on the data file name

and reloading them as they were displayed in the preprocessor. This also gives you opportunity to

select the appropriate radial “range-to-fit” for the chosen set. Bad scans can be excluded from fitting

and plotting by right clicking on the scan box as descibed in the green box in Figure 4-42. If you

want one of the scans to be excluded from analysis, right-click on the corresponding box. A red "X"

will indicate which scans have been labeled as "bad" scans that will be excluded from

analyses.(Figure 4-41) Right clicking it again will undo the "bad" scan indication.

Figure 4-41 Bad scans

Figure 4-42. Designate scans as "bad scans"

Scan selection for Fitting (Figure 4-43)

Figure 4-43 Select scans for plotting or fitting later

Figure 4-44 Scans plotted are from 400 to 499.

You might want to limit the fitting to just this subset and range because of an aggregate you want to

ignore, for example (Figure 4-44).

4.3.5 Processing intensity data:

4.3.5.1 Beckman Optima data

Optima intensity data appears as two sets of data, one from the "sample" sector and one from the

"reference" sector. If here is a smple is each of those sectors, each dta set can be treated as a smple

(doubling the thoughput). I that case the internsity data is conveted to abrsorbance units by taking

the negative logarithm of the scan intenity data. These are then treated as pseudo-absorbance data.

If there is a sample in the "sam" sector and a reference buffer in the "ref" sector, then these

intensity scans can be combined and after taking the ratio of the intensities, into absobance scans.

First the "sam" sector is loaded,

then the "ref" button is clicked to indicatet hat you intend to load reference intensity scans

The you select the "ref" scans dataset and the referene scans are displayed:

After clikcing on the "+" button, the scans are combined and converted to absorbance data.

Optima Multiwavelength intensity data are preocessed the same way.

4.3.6 Summary of the preprocessing procedure:

4.3.6.1 Zoom.

The image on the plot can be expanded by clicking at the point around which you would like the

picture to expand. There are three levels of scale expansion, the fourth click will return the image

to its original size.

4.3.6.2 Interference Data.

4.3.6.2.1 Remove optical jitter:

To remove optical jitter, click on the button labeled “Jitter adjust” and then click in a

relatively flat region to the left of the meniscus. The fringes will be aligned to each other by least

squares fitting the curves to each other over a small region (default is +/- 10 points).

4.3.6.2.2 Remove integral fringe shifts:

Integral fringe shifts can sometime occur at the meniscus. Because there may be a horizontal

region of several pixels where the are no fringes, the fringe tracing algorithm can get confused and

loose count by one or more integer jumps. These shifts are eliminted by comparing fringe pattterns

in a region where there is a shallow gradient like the plateau. To remove integral shifts, click on the

“Locate” button under Integral fringe alignment. Then click on a region of the pattern to the right of

the mencscus where the pattern changes least between scans; the plateau region is usually a good

spot. A green band will appear. Then click on “Subtract” to eliminate the shifts. This is done simply

by subtracting or adding integers to the data as required.

4.3.6.3 Absorbance and Interference Data:

To set the meniscus position, first click on the mensicus region to expand the image. Then click on

the button labeled “Meniscus” and then click and while holding the mouse down move the black

vertical line until it coincides with the meniscus, and release the mouse button. Now a red vertical

line will appear at the meniscus position and the radius of the meniscus will appear in a box at the

bottom of the screen. The value in the box at the bottom of the screen will track the mouse while the

button is held down. You can also type a vlue for th emeniscus postion into this box.

To set the base position, repeat the same process for the location of the base of the cell by clicking

on “Cell base” first.

To select the range to be fittted: The range to be fitted involves a “click-and-drag-from-left-to-

right-and-release” operation. Click on the “Range to fit” button, then click on the centripetal point

and drag the pointer to the centrifugal point. This selects the range of the data to which the model will

be fitted.

4.3.6.4 Fluorescence and Intensity (pseudo-absorbance) data.

Since neither of these types of data have a reference solution, they are processed a little differently.

Fluorescence data are processed in the same way as absorbance data; however no background is

removed from the scans at this point. The delta-c procedure in DCDT/WDA and the Fitter will

remove the time independent background signal when these data are rad in for these types of analysis.

Intensity data are converted to pseudo-absorbance (PsAbs) data by taking the negative logarithm of

the intensity signal. Since the PsAbs data have no reference background signal they will contain all

the time independent background systematic noise from the optical system. Similarly to the

fluorescence data, the PsAba data will have the time independent background signal removed by the

delta-c procedure in DCDT/WDA and the Fitter.

Save Preprocessed data.

Once the data have been preprocessed, the data are ready to be fitted. The preprocessed data are

stored in a “run” file with the extension “.abr”.

Figure 4-45 Choose cell file name.

As shown above, the file name will be of the form “yyyymmdd_<user_suffix>.abr”, where

<user_suffix> is supplied by the user when either the “OK” or the “Write adjusted data file” button

is clicked. See Figure 4-45, Top: before the user adds the <user_suffix>. Bottom: After the user has

added a <user_suffix>. the suffix will be instered between the inintial name of the abr file and the

".abr" extension.

4.3.6.5 Wavelength Data from XL-A.

The advantages of using absorbance optics on the ultracentrifuge is the high sensitivity and

specificity; by choosing an appropriate wavelength, you can often observe one species

independently of others. This is not the case with interference optics, where all species contribute to

the signal.

However, multiwavelength analysis is not recommended on the XL-A: it's slow to scan multiple

wavelengths and inaccurate because the monochromator does not always return to the nominal

wavelengths chosen. And even when it does, it doesn't always return to the exact true wavelength

because of mechanical jitter in the wavelength selection mechanism.

Nevertheless, it can be useful if the wavelengths chosen for eth analysis correspond to either a

maximum or minimum in the spectrum of each component. In that case the jitter will have the

smallest effect on the inaccuracy of the results.

To differentiate among species, it is convenient to use the multi-wavelength feature of the XL-A or

XL-I. This allows you to have every nth scan be done at a different wavelength (n can be up to 3).

This is accomplished by moving the monochromator grating between scans. The nominal wavelength

is recorded in the file header. Beckman’s control software allows you to choose among a single

wavelength for all scans, or alternate scans at two or three different wavelengths. On the XL-I, you

can also do an interference scan, as well as absorbance at up to 3 different wavelengths, in the same

run.

When doing scans at multiple wavelengths, the monochromator reproducibility is about 2-4 nm; the

reported wavelength accuracy is unknown. Therefore, it is desirable to avoid wavelengths where the

extinction coefficient is changing rapidly with wavelength, although some compromise may be

necessary to take advantage of the peak intensities of the Xenon flash lamp.

When preprocessing the scan files produced by the centrifuge, SEDANAL will allow you to separate

the scans for each wavelength. This is necessary because the extinction coefficients vary. Normally,

the choice of wavelengths is designed to maximize the difference between extinction coefficients at

each wavelength for the species of interest.

There are two ways to do the separation using SEDANAL. Remember that the first step in

analyzing scans is to store the data in a run file (.abr). As an example, suppose you have collected

100 scans of absorbance data at 280 and 350 nm in cell 2. The XL-A will use 280 nm for the odd-

numbered scans (00001.RA2, 00003.RA2, ...), and 350 nm for even-numbered (00002.RA2,

00004.RA2, ...). (Yikes! What in heaven's name were they thinking when they thought up this

scheme??)

(1) Create a single run file containing all the scans for the cell. During preprocessing you choose

a single meniscus, cell base, and range to fit. On the fitting control screen, you will choose the run

file n times, selecting sets of scans at each wavelength. For the example, we would analyze two

instances of the run file; for the first, we select “scans to be fitted” of 1, 3, ..., 99, and for the second,

2, 4, ..., 100. On the control screen, the two cells will be linked (see below) indicating they are

physically identical. For the extinction coefficients of the various species, you will put in different

values for the two cells, corresponding to the different wavelengths (see below).

(2) Create multiple run files containing only the scans at a particular wavelength. Each wavelength

is treated independently. During preprocessing you choose a meniscus, cell base, and range to fit for

each wavelength. When fitting the data, you use the n run files, and no scan selection is needed,

although you may want to do so to speed processing or restrict the part of the run being analyzed.

For the example, we have SEDANAL create two separate run files; the first containing the 280 nm

data, the second, 350 nm data. These cells will also be linked (see below), because they have identical

loading concentrations. For the extinction coefficients of the various species, you will put in different

values for each of the two cells (see below), corresponding to the different wavelengths.

The difference between these choices is only how many run files are created from the original scans,

whether the scans need to be separated by wavelength, and whether different menisci, etc, can be

used for different wavelengths.

The XL-A monochromator does not reposition itself to the exact specified wavelength for successive

scans. In addition, the measurement of the actual wavelength, stored in the header of the scan file, is

subject to error. SEDANAL has a tolerance for wavelength matching: two scans are considered to be

at the same wavelength if they are within this tolerance. The value of this tolerance is set in

Preferences->Preprocessor, by checking the box “Preprocessor should offer the option to separate

scans...” and entering a tolerance below. The default tolerance is 4.0 nm, with the option enabled.

When the preprocessor is reading in a set of scans, it looks at the wavelengths, and if it detects more

than one, it will offer the choice to create separate run files, assuming this option has been enabled

in Preferences, as described above.

If the wavelength tolerance is too small, there may be more different wavelengths than

intended. In this case, SEDANAL will list the first six wavelengths seen, and suggest increasing

the wavelength tolerance.

The wavelength tolerance can be set in the Preferences under Preprocessor:

Figure 4-46. Main menu > Preferences > Preprocessor

4.3.6.6 Very large multi-wavelength (MWL) datasets containing several hundred

wavelengths -

When loading multi-wavelength data from either Helmut Cölfen's 2D detector, or from

Kristian Schilling’s 2D detector, a new window will open allowing the user to select which of the

many possible wavelengths (up to 2048) to process in the Preprocessor. (J. Walter et al. Anal. Chem.

87(6):3396-403, 2015).

Figure 4-47 Scans from several wavelengths

Scans from multiple wavelengths can by plotted here but usually only one is necessary to choose the

meniscus, base and range-to-fit, and take care of vertical jitter. After adjusting the jitter, only the

meniscus position is necessary for either DCDT or WD analysis. But base radius and range-to-fit are

required for the Fitter

4.3.6.7 Deconvolution of Concentration Profiles from Multi-Wavelength Data

If the user has the extinction spectra of the components that compose a mixture, the Preprocessor can

read those data in and then can deconvolute the MWL data into component concentration profiles

that are stored as separate abr files for further processing by either WDA or the Fitter.

4.3.6.8 Preprocessing multi-speed datasets for Wide Distribution Analysis (WDA):

When loading a multi-speed dataset, the preprocessor will stop after loading the scans from each

speed and allow you to choose a radius at which to align the fringes in the air-air space and to choose

a meniscus position at that speed. The integral fringe subtraction will be done after the scans from

the last speed have been loaded.

After the scans from the first speed have been loaded the boxes at the top of the screen will look

something like Figure 4-48.

Figure 4-48 load speed set

One clicks on the little black arrow to advance the preprocessor to the next speed’s dataset where the

next meniscus and vertical alignment in the air-air space are carried out. Neither the “”Range to

fit” nor “Cell Base” needs to be set for DCDT or WDA data.

Figure 4-49 Load more speed sets

After all the speeds have been processed, carry out the integral fringe alignment and click “OK” to

save the abr file. Next proceed to the DCDT/WDA button on the Main menu.

If it has not been set in the preferences as the default (which is recommended), click the “Wide

distribution” button (left side above the dcdt window) to enter the WDA mode. Then load the abr

file. Enter 1.0 for the smoothing window. The data from each speed will appear in a different color

(Figure 4-50). See below for more on WDA.

Figure 4-50 Multi-speed Wide Distribution Analysis (WDA).

4.3.6.9 Preprocessing Multiwavelngth data from the Optima

Preprocessing multiwavelength data from the Optima is essentially the same as for the OpenMWL

machines but usually with many fewer wavelengths. Components can be deconvoluted in the same

way when a spectrum for each component has been measured with by WDA or in a

spectrophotometer.

4.3.6.10 Preprocessing Sedimentation Equilibrium data

The scan or scans to be precessed for SedEq analysis should be put in a folder on the same “date/time”

path as for sedimentation velocity. Only the scan(s) to be precessed should be made visible in that

folder. SEDANAL will allow the user to load an equilbrium scan and an optical baseline scan (a

background scan taken usually with water vs water to be used to correct for systematic errors that

arise from irregularities in the optical system). SEDANAL will subtract the optical background scan

from the equilibiurm data scan. This is mandatory for interfernce data and often helpful for

absorbance data if there is a scratch on a window or dirt on the optics. SEDANAL will perform an

interpolated baseline subtraction using the optical basline scan to correct the run scan at

corresponding radial points.

4.3.6.10.1 Testing for Equilibrium

Displaying approach to equilibrium

The Preprocessor displays the approach to equilibrium when either a set of centrifuge files, or a cell

data (.abr) file is loaded. It does this by comparing each scan to a reference scan (normally the last

one). The comparison is displayed numerically in the scan summary window, and graphically in a

new Scans' approach to equilibrium window.

Two plots can be shown: either (1) the root mean squared deviation (RMSD) between the reference

scan and each earlier scan is computed as the average of the squared deviations, corrected for

baseline shifts (i.e. vertical jitter).

or (2) the user can choose to plot the "angle" between the last scan and the others by taking the dot

product between each earlier scan and the last scan.(Ninety degrees means the scan do not match

at all and zero degrees means they are the same.)

The left plot shows the entire span of the run, while the right window show, in this case, scans 200

to 400. It is clear that equilibrium has been reached effectively by about scan 220.

Figure 4-51 Displaying approach to equilibrium

The "r from" and "to" radii are chosen automatically, in this priority:

1. If the user has entered radii, those are used.

2. Else if the meniscus and/or cell base has been chosen, it will be used after adjustment . The

adjustment is to add 0.02 cm to the meniscus, and subtract 0.002 cm from the cell base.

3. Else if the range to fit has been chosen, it will be used

4. Otherwise, the center 96% of the range of radii for all scans will be used as an estimate.

The reference scan and starting scan for comparison are taken to be the initial and final scan,

respectively. The initial parameters and appearance of the graph are controlled by eight preference

variables, but at present (v7.55), these are not editable by users.

The Scans' approach to equilibrium window can be resized by dragging on its edges.

For multi-speed data, the approach to equilibrium is calculated separately for each speed:

This feature is useful. for example, when using a multispeed equilibrium method to verify whether

or not equilibrium had been achieved before each of the speed changes.

4.3.6.10.2 Equilbrium run in standard double sector centerpiece.

Here is an example from a run performed in a standard double sector cell with a loading volume of

0.25 mL (Figure 4-52).

Figure 4-52 Equilibrium scan

After selecting the meniscus, base and range-to-be-fitted, the screen will look like in Figure 4-53,

(Do not forget to click the “Equilibrium” check box).

Figure 4-53 Scan with meniscus, base, and range to fit selected

If an optical blank is to be subtracted from the data run, select it now in the Centrifuge data file

drop-down window. Three curves will apear, the equilibrium data, the blank, and the corrected

data.

4.3.6.10.3 Multi-channel data with 6 channel centerpieces

After loading a dataset from a 6 channel centerpiece, we select the meniscus, base and range to fit

for each channel and repeat until three cell data files are produced one for each channel pair.

Figure 4-54 Data from 6 channel Yphantis type equilibrium centerpeice.

Figure 4-55 Selecting channel A. Repeat for remaining channels

After the meniscus, base, and range-to-fit have been chosen for the first channel, the processed data

can be saved with a name like 20040314_60K_CHA, for example. And then these data must be

reloaded and the process is repeated for the other channels and each is saved as a separate cell data

(*.abr) file using similar names (eg. 20040314_60K_CHB, and 20040314_60K_CHC, etc). Rather

than reloading the file three times, use the "Save cell data as..." feature of the File menu in the upper

left corner of the menu bar:

4.4 Fit Preprocessed Data

4.4.1 The Control Screen

When "Fit Preprocessed Data" is selected from the Main Menu, the default control screen will

appear.

Figure 4-56 The main control screen for the fitting preprocessed data

4.4.1.1 Create a New Control File

To start a new fit, click on “New” and a dialog box asking you to specify an “experiment” directory

will appear. The directory you specify next will be the location in which the control file will be

stored with extension “.abc” (Figure 4-57).

Figure 4-57 Dialog box for output directory

Figure 4-58 Select experiment folder

Click on “Browse” and select the experiment directory in which the control file will be stored,

here: “simulate single species’

The control file name will appear in the Control file box. It will be named “New_Control_Filename”

by default and should be renamed immediately at this point before storing it.

Figure 4-59 The main control screen for the fitting preprocessed data

4.4.1.2 Choose between Analyze Data and Simulate Data

Figure 4-60. Now indicate whether you are going to “Analyze data” (the default for fitting) or

“Simulate data” by clicking the appropriate button, and select the model from the “Model to be

fitted” dropdown window.

Figure 4-61 Select model

Now the Control Screen will change to correspond to the model chosen Figure 4-60.

Figure 4-62. The main control screen for the fitting preprocessed data. As the model is selected

the appropriate boxes will appear on the top portion of the control screen.

4.4.1.3 How to Save the Control file if you haven't started with the "New" button.

If you have filled in all the parameters but had not pressed "New" before you started, you will see

this message:

You can still save the new control file by typing the full path to the folder, into which you want to

store the new control file, into the path window just to the left of the "Browse" button.

At that point, if you make a mistake in the path, you will see this message:

After you have entered the correct path, SEDANAL will save the new control file.

4.4.1.4 Reload a previous control file

There are two ways to load an existing control file: (1) To reload the last control file for the fit you

just finished; click on the "Last" button. The previous abc file will be loaded onto the screen and to

update the best fit parameters, click on the "B" button and the Best Fit values of the parameters that

were allowed to float in the last fit will be loaded in RED font. (2) To load a previous abc file from

a fit performed at some other time, click on the button with the down arrow to see a list of abc files

that were processed previously (The number of files in this list can be set in the Preferences.) The

Best Fit for this abc file can be reloaded as described above with the "B" button.

When you click on the down arrow you will see a list like this:

The icon at the start of each line identifies the type of fit: SV, SE, or SIMulation

4.4.1.5 Loading cell data ("abr") files:

Figure 4-63 Select cell data file

Figure 4-63. Select the cell data (*.abr) file from the “Cell data” drop-down window. When the

Cell data file is selected, the type of run (velocity or equilibrium) will be recognized and the

parameter boxes on the control screen updated accordingly.

⚠

NOTE: The default screen

assumes velocity run until the abr file is loaded.

If the cell data files are NOT stored on the default path, you will have to "Browse" for the files. This

could happen, for example, if the runs were done at different times and/or stored in different folders.

4.4.1.6 The control screen is divided into several regions.

4.4.1.6.1 TOP LEFT:

In the top left region of the control screen, one supplies a name for the Control file. Below that

window is a drop-down window from which a model may be chosen. Below that is a box for entering

the number of points to be used between the meniscus and the base of the cell by the Lamm equation

solver:. (Other parameters used by the Lamm equation solver can be set under the Advanced >

Fitting tab (or from the Preferences > Control extended > Advanced > Fitting); these include the

time interval to be used by the ODE (Ordinary Differential Equation) solver, and the maximum

number of iterations to be performed by the non-linear least squares function minimizer ... more on

this later.) The Equation Editor can be accessed by clicking on the "Eqn" button. And the Model

Editor can be accessed by clicking on the "Model Editor" button.

4.4.1.6.2 MIDDLE-LEFT:

In the mid-left region, in the box labeled "Kinetic Parameters", kinetic rate constants (k

, k

) or

equilibrium constant (Keq) may be entered by the user. If an equilibrium constant is chosen, the

internal routine uses 0.01 sec

-1

for the reverse rate constant and fits for the forward rate constant only.

The default value for the reverse rate constant may be changed on the "Advanced >

Kinetics/equilibrium control" tab. If nothing is entered in the box labeled Keq, one must supply

an initial guess for both the forward and reverse rate constants. Other parameters used by the

Kinetics ODE solver also may be adjusted in the "Advanced > Kinetics/equilibrium control" tab.

4.4.1.6.3 MIDDLE-RIGHT:

In the middle right region, in the box labeled "Molecular parameters", one enters the molecular

parameters for each of the species participating in the reaction scheme, as well as those which might

be extra components. These parameters include molar mass or diffusion coefficient or frictional

coefficient, sedimentation coefficient, density increment, and mass scale extinction coefficient for

the optical path length corresponding to the centerpiece in use.

4.4.1.6.4 LOWER:

The Lower region allows one to select the datasets that will be fitted either singly (if only one is

chosen) or globally, if several (up to thirty-two datasets in v6.14) are chosen. The datasets are chosen

from the drop-down windows on the left and will be found in the Experiment folder in which one

had preprocessed the datasets under consideration. The dataset (*abr) files live at the “time” directory

level i.e. the lowest level along with the scan files.

The best guesses for the loading concentrations in molar or mass units are entered in the boxes

at the right end of the dataset line of boxes. These values should usually be allowed to float (you may

think you know what they are but you really don't). One may either float or hold a common value for

the ratio of [B]

/[A]

or c

B,o

A,o

. In most cases in which a dilution series is being analyzed, this ratio

should be the same for all cells. To constrain the ratio to be the same value for all cells, left click on

the text above the ratio boxes.

To switch between mass and molar concentrations click on the little red/blue squares to the bottom

right of the column headers. To switch between molar or mass concentrations and molar or mass

ratios, click on the yellow square on the bottom left of the column header. (⚠ Note: The yellow

square at the bottom left of the first column header for species A does nothing.)

Figure 4-64. Switching between molar and mass concentrations

Figure 4-65. Switching between mole ratios and weight (i.e. mass) ratios.

4.4.1.7 The Equation Editor

Other relationships between parameters not established by the Model Editor can be

established with the Equation Editor.

The equation editor window can be opened by clicking on the button labeled "Eqn".

Relationships established by the ModelEditor will appear in green font. Relationships established

by the Control screen will appear in blue font. Custom relationships that are entered by the user

will appear in grey font and labeled as "Custom". (see below).

Figure 4-66. Equation Editor

In this example, for the model A+B=C, the equations are generated by the ModelEditor and show

the relationships established by clicking on the boxed on the tab for species 3, “C”. That is, the molar

mass is the sum of those of species A and B. The density increment and the extinction coefficient

of “C” are both expressed as mass weighted averages of those of species A and B.

Other relationships between the parameters can be entered manually. For example, consider the case

of a 2 component, 2 species system comprising a non-interacting monomer and dimer. One would

choose the model “2 comp 2 species” and then enter the relationship relating the molar mass of the

two species.

Relationships in Green are established by the ModelEditor; those in Blue by the control screen and

those in Grey by you. So, for the system A+B=AB we might have a case in which we had performed

an experiment in which we had done a dilution series (4 cells) at a constant ratio of [B]

/[A]

concentrations and combined those data with another experiment in which we had varied those ratios

(2 more cells). Moreover, we’ve used both absorbance and interference optics for 2 of the cells. That

gives us a total of 8 run files to combine in a global fit. Assuming we know the extinction coefficients

accurately (both in units of A.U.-mL-mg and of fringes-mL-mg for a 12 mm optical path length),

we can require that the loading concentrations in the cells in which both optics were used be the

same by linking the cells on the control screen. Or we can use the equation editor to establish those

constraints if the 2 cells in question were not adjacent to each other in the list of run files on the

screen. If we wanted to link dataset 1 (absorbance data) to dataset 4 (interference data), because they

are the same cell and therefore must have the same loading concentrations, we would enter L(4,1)

= L(1,1) in the Equation editor.

Figure 4-67. Using the equation editor to link parameters in different cell datasets.

4.4.1.8 Start the fitting:

To start the fitting after all the initial guesses and other constraints have been entered, the fitting is

started by clicking the “Store control file and start fit” button.

4.4.1.9 Advanced Parameters Button

4.4.1.9.1 Error Estimation Control

The first set of advanced parameters is under the “Error estimates” tab and allows choice of either a

Bootstrap with replacement, a Monte Carlo type of error analysis, or search for confidence limits by

computing F-statistics. (F-statistics is selected in Figure 4-68.)

Figure 4-68. Advanced window.

4.4.1.9.2 Lamm Equation Solutions (Claverie control):

The second set of advanced parameters under the “Claverie control” tab allow adjustment of the

performance of the finite element procedure for solving the Lamm equation and concerns the

adjustable time increment used for each step and the distribution of grid points along the radial

axis. The example below says essentially, if the fractional change in concentration does not

exceed 0.02 at any point in the cell over that last time interval, then increase the time interval by

a factor of 1.02. A smaller rate of change can be set for the first few steps in the box

labeled "Sedimentation time step initial fractional change".

The boxes at the bottom of this tab allow termination of the finite element algorithm in the event

of numerical instability :viz:

Figure 4-69 criterion for stopping F.E.M.

Figure 4-70. Control of the finite element calculations.

The grid spacing can be controlled as in the following example which shows the cell divided up

into three regions spanning 0.1, 0.7 and 0.2 of the distance from the meniscus to the base. The

point density and actual number of points used are shown in the designated boxes.

Figure 4-71 Specify grid spacing: three zones.

Limit concentrations near the base

Figure 4-72 Limit concentrations near base

4.4.1.9.3 Kinetic integrator control

The third set of advanced parameters under the “Kinetic integrator control” tab allows one to

adjust the performance of the ODE (Ordinary Differential Equation) solver. The current version

of SEDANAL uses either the Bulirsch-Stoer algorithm (with Richardson rational polynomial

extrapolation) or the Euler extrapolation method to solve the differential equations describing the

kinetics. For those cases in which instantaneous equilibrium can be assumed, the equilibrium

equations may be solved directly using the Newton-Raphson successive approximation iterative

method. One method, kinetic or equilibrium solution, could be faster than the other depending on

the model.

Figure 4-73. Control of the kinetic integrator.

After each time step of the Lamm equation solution, the updated concentrations of all the species are

recomputed according to the kinetic rate constants. If the kinetic methods are too slow and you are

sure that your system is not kinetically limited, use the Newton-Raphson method; it may run faster.

You will have to determine this on a case-by-case basis. For fitting equilibrium runs, use only

Newton-Raphson.

4.4.1.9.4 Fitting Tab

First choose the fitting method, either simplex or Levenberg-Marquardt minimization. For simplex,

the mode for setting up the initial simplex is controlled by selecting the forth tab “Fitting”. In the

upper right-hand area, The top button is to vary only the diagonal elements of the simplex array in

the positive direction. The middle selection is experimental – don’t use it. The last one (bottom

button) varies all the elements by small positive and negative amounts. It seems to work best and is

recommended. The convergence criteria are entered in the bottom set of windows. Whether the fit

is stopped, or perturbed and restarted is indicated in the "Perturb fit and re-do" section of this screen.

Other settings here affect the display.

Figure 4-74. Choosing fitting parameters.

4.4.1.9.5 Plot color and symbol size

Plot color and symbol size are set in the Main Menu > Preferences under the control Extended tab:

4.4.1.9.6 Select output for Simulations

4.4.1.9.7 Set limits on s and D under non-ideal conditions.

Figure 4-75. Set limits on D and s for non-ideal systems

Leave s/so blank and set D/Do to 2.0. If needed, s/so should be set to a fraction like 0.5.

4.4.1.10 Concentration dependence of s and D without cross terms.

Hydrodynamic concentration dependence of both s and D through the frictional coefficient is

expressed through the coefficient k

defined as follows: f=f

(c=0)

(1+k

c).

Thermodynamic concentration dependence of D is expressed through the second viral coefficient,

, as follows. D=D

(c=0)

(1+2BM

c)/(1+k

c) One may either fit for these parameters or hold them

constant at preset, "known" values.

Therefore, the hydrodynamic concentration dependence for s is given by

while the total concentration dependence for D is given by

This approach has been superseded by Cross term fitting using the Ks and BM1

matrices (cf Cross Term Non-ideality:)

4.4.1.11 Pressure dependence of density and viscosity:

4.4.1.11.1 Compressibility

When dealing with compressible solutions, one can enter the isothermal compressibility value in

the Experiment information window in the preprocessor: [the units of compressibility are

/dyne (a.k.a. Ba

= bayre

-1

).]

⚠ Note that the compressibility can also be expressed in terms the bulk modulus 𝐵 The SI unit

for bulk modulus is the pascal. (10 bayre=1 pascal). Therefore, the bulk modulus is just the

reciprocal of the compressibility.

Figure 4-76 Setting Compressibility for each cell.

Load the abr file and click on Experiment information and enter the compressibility as indicated.

We have generally followed the equations described by: Schuck,P., "A model for sedimentation in

inhomogeneous media. II. Compressibility of aqueous and organic solvents Bioph. Chem. 108

(2004) 201–214.

See also, in terms of bulk modulus, Stoutjesdyk, M et al., European Biophysics Journal (2020)

49:711–718.

A default value of compressibility can also be entered in the Preferences page:

4.4.1.11.2 Pressure dependence of viscosity

If the pressure dependence of solvent viscosity is to be taken into account. The values for the

pressure-viscosity coefficient can be entered as a developer option: select option 024 and enter the

viscosity-pressure coefficient under Parameter 1. (Version 7.05: this will be moved to experiment

info in later versions)

Figure 4-77 Pressure dependent viscosity

4.4.1.12 Selecting datasets to be fitted:

Figure 4-78. Preprocessed datasets.

Once the experiment directory has been chosen, all the available preprocessed run files (*.abr) will

be visible in the dropdown windows.

When the cells are chosen, one must enter guesses for the loading concentration of each component.

Selected scans within a cell may be fitted. The default is all scans; to change this, right-click on the

run file name (lower left), a small window appears that allows you to either choose the scans to fit

or to enter a weighting factor for that dataset.

Figure 4-79 Specifying scans to fit

If "Select scans to fit" is chosen, a window similar to the one for selecting scans to be plotted in the

preprocessor will appear. You may select the scans to be used for fitting in that window. The result

is written into the control file. There is an option in the lower right of the scan-select dialog box to

use the same scans for fitting that had been chosen for plotting in the preprocessor (Figure 4-80).

Figure 4-80 select scans to be fitting or copy those used in the preprocessor

4.4.1.12.1 Removing a dataset from the fit.

To remove dataset from the fit, highlight the name in the “Cell data file” window (requires two left

clicks) and press the “Delete” (not the “Backspace”) key.

4.4.1.12.2 Weighting Factors

In the Fitter the use of weighting factors and user settable values for weighting factors can be set

for each cell data file by right-clicking on the cell data file name on the control screen. To enter

weighting factors for a particular dataset, right-click on the cell data file name

(the *.abr file) and select "weights for this cell"

(

⚠

NOTE: THIS HAS BEEN UPDATED TO USE AS INPUT, STANDARD

DEVIATIONS, instead of inverse variance, as of version 6.92. The inverse variance is

calculated and used internally.)

Figure 4-81. Selecting "Weights for this cell” brings up the following window.

Figure 4-82 Select weighting for each cell

If datasets from different optical systems are to be combined in a global fit, they must be weighted

according to the magnitude of the data and the noise on the data. If the inverse of the variance

(1/s

) of the data is used as the weighting factor for the squared residuals, then data from different

datasets can be combined. In the example above, the value of 27777.8 is the square of the inverse

of 0.006, the typical order of the standard deviation of either interference (0.004) or absorbance data

(0.006) from the XL-A/I. When weighting factors derived this way are used, the reduced chi-

squared values instead of sums of squares of the residuals are minimized. The results are

equivalent: in both cases the procedure is called Maximum Likelihood Estimation (MLE).

⚠

NOTE:: SEDANAL displays the reduced chi-squared values on the fitting screen instead

of the rms deviation when inverse variances are used as weighting factors and “Chi-square”

is selected at the top of control screen.

Default values can be set in the Preferences under the Control tab:

These can be changed on the control screen for each cell.

4.4.1.13 Multi-Wavelength Weighting Factors by Wavelength

When a multiwavelength dataset has been loaded, the following weight option becomes available:

allowing the user to load a file containing the standard deviations of the data as a function of

wavelength.

4.4.1.13.1 Standard deviation by wavelength file format

The standard deviation files may have any extension (e g, .txt or .csv), but the contents will be

interpreted as ASCII text. SEDANAL looks for lines like

wavelength stdDev

The stdDev is the standard deviation to be used for all points at that wavelength, in the same units

as the scans contained in the corresponding cell data (.abr) file. The standard deviation is used to

weight scan points at the given wavelength.

The format is identical to that for extinction coefficient spectrum files.

The wavelength may contain a decimal point, so either 253 or 253.0 is valid, and has the same

meaning. The unit for wavelength is nanometer (nm), and the values are rounded to the nearest 0.1

nm.

Lines beginning with // are comments, and will be ignored.

The wavelengths may appear in any order, and need not be consecutive. Within the range of

wavelengths contained in the standard deviation file, the standard deviation will be linearly

interpolated to the scan data wavelength. Any scan data with wavelength outside that range will be

ignored during fitting (weight = 0). Only the first 2500 wavelengths will be included. Wavelengths

should not be < 200 nm.

4.4.1.14 Selecting parameters for the finite element solutions to the Lamm equation.

One must select the grid density, i.e. the number of points between the meniscus and the base, to be

used for the fitting. The data are interpolated onto this grid and the finite element solutions to the

Lamm equation are generated on this grid. The time interval is the initial time increment used by

the finite element procedure. This initial time increment can be adjusted according to the

parameters enter under the “Advanced” menu.

Figure 4-83 Select number of points for F.E.M for fitting

For initial fits to a new system, it is recommended to choose 200 points between the meniscus and

the base. This will make the exploratory fits relatively fast. The number of points should be increased

to 400 or 800 later to refine the fits for the most accurate estimation of fitted parameters. Higher

numbers of points may be necessary if the sample generates steep gradients. The maximum is 10,000

points

4.4.1.15 Extinction coefficients: Global fitting with multiple optical systems -

If several optical systems and/or wavelengths have been used, different mass extinction coefficients

can be used for each cell. (

⚠

Note: the extinction coefficients to be entered must correspond

to a 12 mm path length: i.e. one must multiply the usual (mg/mL for 1 cm) UV extinction

coefficients by 1.2). Or if 1 or 3 mm centerpieces were used, the usual 1 cm value must be

multiplied by either 0.1 or 0.3, respectively, to correspond to actual path length used.

If all cells were run with interference optics, then the extinction coefficients would be entered by

choosing the "ALL CELLS" option which is the default:

Figure 4-84 ALL CELLS

Left-clicking on the text “ALL CELLS” will allow one to enter different extinction coefficients for

species for each cell.

Figure 4-85. Left-clicking on the text “ALL CELLS”

Each click advances to the next cell. If interference optics have been used for cell 1, one would input

the number of fringes per 1 mg/mL:

Figure 4-86. Set extinction coefficients for each cell

⚠ NOTE: Starting with version 7.40, SEDANAL allows a matrix of extinction

coefficients to be filled in for each species in each cell. For example, (cf Figure 4-87)

for the system 2A=A

; 2A

and two cells, with, say, a 12mm and a 3 mm

centerpiece, respectively, we would have the following matrix that would allow us to

either fit or hold the extinction coefficients. (In this case, for example, we have

assumed that the extinction coefficients do not change upon association.)

Figure 4-87 Extinction Coefficient Matrix

If absorbance at 280 nm had been used, one would click again on the text and enter the mass

extinction coefficients of each species at 280 nm: where the last box is computed as a weight average

based, in this case, on the model A+B=C. If the data were taken with different optical systems on

the same cell, then the loading concentrations for those data will be the same. SEDANAL allows one

to constrain the loading concentration to be identical for those datasets.

Figure 4-88 data from “cell data files” 1 and 2

For example, if the data from “cell data files” 1 and 2 were taken from the same cell, their

loading concentration will be identical. One may click on the Number “2” on the far left of the

row to “slave” that dataset to dataset # “1”.

If cell “3” had been taken at another wavelength on the same cell, then cell data files ”2” and “3”

would both be “slaved” to cell data file “1” by clicking on the number “3” at the left (not shown).

⚠

NOTE: Many of the relationships discussed in the following section can now be established

in the Equation Editor which is especially useful for cells not adjacent in the list shown in

Figure 4-89 and for more complicated relationships between cells.

Figure 4-89. Linking cells that have the same sample but different optical system.

For example, if cell data files “1” and “2” are from one cell and cell data files “3” and “4”, then click

on the row for cell data file “2” to slave it to cell data file “1”. Then click on the row for cell data file

“4” to slave it to cell data file “3”. Cell dataset that are slaved are highlighted in the same color.

Figure 4-90 The main control screen for the fitting preprocessed data with all parameters filled in.

When all the parameters have been filled in, the screen will look as shown above (Figure 4-90).

⚠

NOTE: The loading concentrations can be expressed in moles per liter. The loading

concentration of component 2 can be entered in terms of the molar ratio of B to A: [B]

/[A]

. For

the 3 component system, the last column will be the ratio of [C]

/[A]

. This will allow one to specify

that the ratio be fit as a global value for all runs, and is used in cases in which the runs are part of a

dilution series for which these ratios might not be known but which are usually expected to be the

same for all cells.

4.4.1.16 Local vs. global parameters - changing

To change the ratio from a local to a global parameter left-click on the text “Loading conc ratio

B/A” or “Loading conc ratio C/A” above the column of boxes. The first click will fill all the boxes

with the value entered into the top box and propagate that value to the other cells and change to

background color to yellow to indicate that these are not being fit separately.

All cells will use the same ratio in the fitting process.

Figure 4-91 The loading concentration ratio is entered as 1.0 in this example.

Figure 4-92. A single left-click on the text “Loading conc ratio B/A” propagates the top 1 into the

boxes below with a yellow background to indicate that these values are derived from cell #1 and

will be kept equal to and vary with the value for cell #1 as a single global parameter.

Figure 4-93. A second click on the text “Loading conc ratio B/A” turns the backgrounds back to

gray to indicate that these values will be now be allowed to float independently for all cells:

Figure 4-94. Right clicking on one of the gray boxes will turn its background blue and now its

value would be held constant during the fit, if that were desired:

4.4.1.17 Indefinite Self-association

4.4.1.17.1 Isodesmic case:

Figure 4-95 Isodesmic model selection.

On the control screen, if an isodesmic model has been chosen (Figure 4-95), a button labeled

“Indefinite self-ass’n…” will become active. Click the “Indefinite self-ass’n…” button to enter the

coefficients for a polynomial representing the relationship between the sedimentation coefficient of

the oligomers and the degree of polymerization (Figure 4-96).

Figure 4-96. This example is using the s(i-mer) = s(monomer)* (i)

2/3

Other coefficients, for end-to-end polymerization for example, can be entered from bead modeling

or other theoretical considerations. The coefficients can be entered as rationals in the numerator box

or as fractions by specifying both numerator and denominator.

The coefficients, a0, a1, etc… represent the coefficients of the following polynomial

ln(s(i-mer)) = a0 + a1*ln(i) + a2*( ln(i))**2 + a3*(ln(i))**3.

The coefficients are obtained by fitting a polynomial to the values of ln(s(i)) as a function of ln(i).

Usually a0 will be set to ln(s(monomer)). The values of s(i-mer) can be obtained from the analysis

of bead modeling, for example.

4.4.1.17.2 Isoenthalpic indefinite self-association:

Figure 4-97 Isoenthalpic indefinite self-association

On the control screen (Figure 4-97), if an isoenthalpic model has been chosen, a button labeled

“Indefinite self-ass’n…” will become active:

Click the “Indefinite self-ass’n” button to enter the coefficients for a polynomial representing the

relationship between the sedimentation coefficient of the oligomers and the degree of

polymerization. Then select the polymerization method according to how the entropy terms will be

handled. [For details, see Ronald Chatelier, “Indefinite isoenthalpic self-association of solute

molecules”, Biophysical Chemistry, 28, 121-128 (1987)]

Figure 4-98. Entering parameters for indefinite self-association

4.4.1.18 Constraining the range of parameter values during fitting.

Figure 4-99. The range of allowable values

The range of allowable values for a parameter may be set by shift-left-clicking (Figure 4-99) on

the value in the control screen (Limits set by the ModelEditor can be changed here.):

4.4.2 Exiting the Control Screen to start fitting

Figure 4-100. Start the fit

After all the appropriate boxes are filled in, the fit can be started by clicking the button

labeled “Store control file and start fit” or if you just want to store this control file and maybe

open up another one, click the button labeled (you guessed it) “Store control file only”.

4.4.2.1 Fitting Screen

Figure 4-101. Fitting screen display for 4 cell global fit to the system A+B=C

Here (Figure 4-101) is an example of what the fitting screen looks like before convergence, after a

few iterations, when performing a 4 cell global fit to the system A+B=C:

During the fitting process, the screen is updated every 10 seconds or each time a new minimum is

reached. SEDANAL will print at the bottom of the screen 7 lines of text: (1) column titles, (2) the

initial guesses, (3-6) the last 4 best minima, (7) the current guess being processed. The middle region

of the screen displays 2 sets of three plots for each cell. The screen above shows 4 cells. In the

upper left panel, the first set of 3 plots is from the first difference curve and the second set is from

the last difference curve of the dataset. The intermediate difference curves are not displayed; these

can be found in the output file (with extension *.min). Within the window for each cell are plotted

(1) the difference data being fitted (red dots), (2) the best fit to that difference data (green line) and

(3) the residuals for the current best fit (blue dots).

After the fitting process, files are written that contain the data, the fit and the residuals. These files

are named “<control_filename_prefix>__0nMin00m.txt” where "n" is the cell number and “m”

is the sequence number corresponding to the number of times that control file has been used to start

a fit.

The first few lines of a “Min” file look like this:

The “Min” file header contains information about the fit. Most of the lines are self-explanatory.

The global root mean square deviation for the fit is labeled ”Std dev =” The fitted

parameters corresponding to the last minimum are listed after the word ”Params:” These will be

labeled more clearly in a future version.

After the first two columns, “#” and “Radius”, the min file contains, in groups of three columns, the

entire set of difference curves from the experimental data, “delta-C(obs)”, as well as all the

difference curves generated by the fitting procedure, “delta-C(calc)” and then the

deviations between the data and the fit for each difference curve, “DEV”.

“#” is the point number

“Radius” is the radius

“delta-C(obs)” is the first difference curve from the experimental data,

“delta-C(calc)” is the first difference curve generated for the same times

“DEV” is the residual at that point for the first difference curve

From this point on, the last three columns are repeated for each difference curve:

delta-C(obs) delta-C(calc) DEV delta-C(obs) delta-C(calc) DEV …etc…

These data can be imported into the plotting program of your choice.

4.4.2.2 Screen Dumps

Figure 4-102. Screen Dumps

The screen can either be dumped directly to a printer at any time during the fitting process by right

clicking on the “Stop/Close” button (Figure 4-102): Doing so will cause a printer dialog window to

open. The screen can be dumped as a bit map to a file by shift-right-clicking on the Stop/Close button.

This is done silently and no dialog window will appear. The bit map will be found in a bmp file with

same name and sequence number as the report file.

4.4.2.2.1 Toggling Plots During Fitting

Figure 4-103 Toggling plots during fitting

As long we have this little picture (Figure 4-103) handy … One can turn on or off any of the plots

on the fitting screen by clicking the little red, green or blue boxes sporting the text “Exp”, “Calc” or

“Diff” corresponding to the plot of the experimental data, the best fit, or the residuals, respectively.

4.4.2.2.2 Displaying the Residuals Bitmap

Residuals can be displayed in either grey scale or on a color scale by clicking on the "R"

button on the upper right hand corner of the fitting window (Figure 4-104):

Figure 4-104 Displaying residuals

The residuals window shows the difference between observed and calculated difference curves (the

residuals). The colors can be scaled either relative to the minimum and maximum values or to

absolute minimum and maximum user specified values by clicking on the "abs" button.

Figure 4-105 Residual plots

The plot in the residuals window has x = the radial point, y = the scan number, and the color = the

residual value. Only one cell is shown at a time. The range of x is the user-selected range to fit. The

color mapping is determined in one of three different ways, and scaled to the range of data, z, in

the run:

Gray scale: intensity = 255 ((z – z

min

) / (z

max

–z

min

)), so z

min

gives black and z

max

gives white.

Color: A 16-bit look-up table is used (65,535 colors) to convert p = 2

((z – z

min

) / (z

max

–z

min

)) to a

color. The table is shown below. Again, z

min

gives black and z

max

gives white.

Custom: Same as color, but the look-up table is user-specified, read in from the file

ResidualColors.txt. The format of ResidualColors.txt is

// Optional comments

0 0 0 0

1 0 1 0

2 1 0 0

...

65534

65535

Each line is p Red Green Blue, where the color values are an 8-bit intensity 0–255.

Horizontal and/or vertical scroll bars appear if the plot window is not large enough to accommodate

the data (current size is 930 radial points and 132 scan pairs). Clicking the scroll bar arrows moves

one pixel (i.e. one radial point or one scan pair), while clicking the bar outside the thumb moves

half the size of the plot window.

The run to be shown is selected with a thumbwheel. Only a single run’s residuals are visible.

Normally, every scan is displayed (one scan per row of pixels). This can be changed with the nth

scan thumbwheel to show only every n

scan in adjacent rows.

4.4.2.3 Fitting to Sedimentation Equilibrium Data:

4.4.2.3.1 Loading the dataset - load equilibrium datasets before choosing the model

Open the control screen and click “New’ and set the path and create a new control file. Select one

of the cell data files. (Selecting a cell data file before a model is selected is not mandatory but is

less confusing than the other way around). Selecting the cell data file tells the Fitter whether this is

SedEq or SedVel data and arranges the Molecular Parameters boxes appropriately when the model

is chosen (Figure 4-106).

Figure 4-106 The main control screen for the fitting preprocessed data

4.4.2.3.2 Choose a model

Choose a cell data file first (Figure 4-107) and then choose a model (in this case for a single species)

and now the control screen changes to the appropriate sedimentation equilibrium model. (i.e. no

window for s is shown under Molecular parameters). The fitting screen reads the abr file to see

whether this is a sedimentation velocity or an equilibrium run before loading the model.

Figure 4-107. The main control screen for the fitting of preprocessed data for a

sedimentation equilibrium run.

For an A+B=C model the screen would change to (Figure 4-108). In addition to the molar mass,

density increments, extinction coefficients, an additional “local” cell parameter, y-offset, has been

added to account for vertical offsets in the data. It is absolutely necessary to include this parameter

in fit of interference data since the fringe displacements are known only to within an arbitrary

additive constant. Often it is necessary to include a y-offset to fit absorbance data if there are buffer

mismatches or optical system calibration issues.

Figure 4-108. The control screen for the fitting an equilibrium run for the model A + B =C.

4.4.2.3.3 Enter initial guesses and other parameters

After filling in guesses for the parameters for a single species fit, the screen looks like the one

below (Figure 4-109).

Figure 4-109. The main control screen for the fitting of preprocessed data for a sedimentation

equilibrium run. Here we will float the molar mass, the loading concentration and the y-

offset.

4.4.2.3.4 Start the fit

Now, after pressing “Store … and start fit”, we get to the following fitting screen at convergence

(Figure 4-110).

Figure 4-110. Final fit to equilibrium data after convergence.

For a multi-cell fit of a 3 component model to 3 datasets spanning 3 loading concentrations at one

speed we might see (Figure 4-111).

Figure 4-111. Three-cell global fit to sedimentation equilibrium data

4.4.2.3.5 Global fit to a dilution series with linked cells

Global fit to 9 datasets from a run at 3 speeds and 3 loading concentrations to the Model

A+B=C:

The control screen was set up as follows with linked cells (note colored bars on the left side) and,

since this is a dilution series, with fixed molar ratios of [B]

/[A]

Figure 4-112. The main control screen for the fitting preprocessed data for sedimentation

equilibrium run.

This constrained fit (i.e. requiring the cell loading concentration to be the same for the same cell at

different speeds) resulted in a fairly good fit but having some systematic errors in a few of the

datasets.

Figure 4-113 Fitting with linked cells

4.4.2.4 Since the molar masses and density increments are known for the two components,

only the loading concentrations, y–offsets and global equilibrium constant were fit.

4.4.2.4.1 Global fit with un-linked cells

Now, not linking the cells (i.e. not linking the cell loading concentrations) gives a better fit. This

because the cell loading concentration depends quite strongly on having estimated the correct

position of the base of the cell. In this fit we are still holding the molar ratio of the loading

concentrations of B/A to be the same for all cells with the overall value allowed to float

Figure 4-114. The main control screen for the fitting preprocessed data for sedimentation

equilibrium run.

Figure 4-115. Fit with unlinked cells.

Returning to the case of linked cells but allowing the radius of the base of each cell to be a fitting

parameter: we get a better linked-cell fit than the first one but still not as good letting all the

loading concentrations float.

Figure 4-116 Fitting for the base radius

Figure 4-117 Fitting for R

. with linked cells

Since a better fit was obtained without linking the cells, it may be that some material had pelleted

or gelled at the bottom at the higher speeds.

Allowing separate ratios of B/A for each cell gives an even better fit in the unlinked case.

Figure 4-118. Allowing the ratio B/A to vary

Note that although this fit is better, the equilibrium constant found in this case is about 1 order of

magnitude higher than that value found with the more constrained fits.

4.4.2.5 Global Fitting of Multi-wavelength Data

The large, 4-dimensional, MWL datasets can be fit directly if you supply the extinction

spectra for each of the components in the mixture. After loading an abr file that was written

by the PreProcessor, a model can be selected, and parameter guesses entered as usual. The

extinction spectra files, and wavelengths to be used for fitting, must be specified by right-

clicking on the abr filename window and selecting “Extinction Coeff file” from the drop-

down list:

The following window appears that allows you to browse to choose the path to the

extinction coefficient files to be used.

Figure 4-119 Select paths to the extinction coefficient spectra

Wavelengths to be used for fitting are selected by clicking on the intersections in the

wavelength selection window:

Figure 4-120. Select wavelengths to be used in the fit

When we start the fit, we see a curve corresponding to the lowest wavelength (in this case 250.1

nm)

For monitoring the fitting process, any particular wavelength can be displayed by clicking on the

wavelength, "lambda", button in the upper right corner of the screen and clicking on the

wavelength matrix at the desired intersection. All the wavelengths selected on the control screen

are used in the global fitting procedure.

Figure 4-121 Select wavelength whose residuals are to be displayed

4.4.2.6 Fitting Multiwavelength Data at a Single Wavelength

Regarding the Extinction Coefficient file:

Weight extinction coefficients are adjusted to the 1.2 cm path--or whatever path, like 3mm, that

you used.

The file can be named anything you want with ".txt" extension

The spectrum files can be stored anywhere you want. It is convenient to store them in a folder one

level up from the data folders in case I need them for other experiments on the same material.

To load the spectrum file, right-click on the cell data file window in the Fitter to reveal the

dropdown menu:

In the dropdown window, click on "Extinction coeff file" and then click on the "..." button to

browse for the file.

My extinction coefficient files are in the “spectra” folder in this example: like so

Select the file you want and click "open":

A spectrum file can look like this:

Spectra output from dc/dt File version 1

Peak 1 (0.4798-0.7797)

Input file: C:\SEDANAL\User_data Optima AUC\20170905.abr

WL,nm Area

220.0 0.0593433

260.0 0.0326965

280.0 0.0632744

The important part is the two columns of numbers, wavelength and extinction coefficient; all

other text is ignored (This particular file is spectral output directly from DCDT/WDA i.e. it hasn't

been converted to actual extinction coefficients yet; it needs to normalized at some wavelength at

which the extinction coefficient is known, and multiplied by 1.2 cm or other pathlength like 0.3

cm).

For single wavelength, it would look like

WL,nm abs

280.0 1.200

I.e. only one wavelength need be entered into the file.

If you have done the experiment at, say, 280 nm but have only extinction coefficient values at, say,

279 nm and 282nm, SEDANAL will interpolate to get the value you need at 280 nm. Therefore, the

entries in the file do not have to correspond exactly to the wavelengths being fitted.

SEDANAL will obtain the value at 280 nm by interpolating the value from these data:

(or you can do it yourself)

WL,nm abs

279.0 1.210

281.0 1.230

However, with only one or a few wavelengths, exact values are relatively easy to generate.

You must select the wavelengths at which you want to fit:

In this particular example, that sample had only two wavelengths, 260 nm and 280 nm, and was to

be used to analyze a mixture of protein and RNA at 260 and 280 nm. To fit the pure protein in this

cell, only 280 nm is selected.

4.4.2.7 Error Analysis: Boot-strap with Replacement, Monte Carlo and F-statistics

4.4.2.7.1 Error Estimation: Parameter standard deviations.

4.4.2.7.1.1 Bootstrap with replacement

SEDANAL will carry out a specified number of so called “boot-strap with replacement” or

fitting operations to estimate the standard deviations of the estimates of the fitted parameter values.

Essentially the way it works is that the original data, consisting of N data points, are randomly

sampled, with replacement, N times to produce a new dataset that is then fit using the best fit

parameters values as the starting guesses. Each fit is carried out until convergence is reached and

the new fitted parameters are written into a table. After a specified number of boot-strap fits, the

standard deviation of each parameter is computed. In the case of SEDANAL, for which the fitting

times can be very long, it is practical to carry out only a limited number of boot-strap fits. Ideally

one would prefer to repeat the boot-strap operation 500 to 1000 times and generate a distribution

function for each parameter from which true confidence limits can be computed. However, it is

feasible to carry out only a limited number of boot-strap fits and so an abbreviated boot-strap

procedure has been implemented to compute only the standard deviation of each parameter. The

ability to compute a standard deviation from a smaller number of samples of a potentially non-

normal distribution relies on the Central Limit Theorem which basically states that the distribution

of repeated estimates of the parameters will be normally distributed no matter what the shape of the

actual distribution of the parameters. This will be an approximation that will give one a reasonably

good idea of the degree to which any given parameter is determined by the data being fitted.

To specify the number of boot-strap operations to be performed, one enters the number in the boot-

strap tab under the Advanced button of the Control Screen (Figure 4-122).

Figure 4-122. Error estimation control

4.4.2.7.1.2 Monte Carlo Analysis

The Monte Carlo analysis works similarly except that the best fit model is simulated first and then

multiple different noise sets are added and the simulated data with noise are refit each time.

For a Monte Carlo analysis, the Monte Carlo button is clicked the number of simulations is

entered and the standard deviation of the normally distributed random noise that will be added to

the data is entered:

Figure 4-123. Error estimation - Monte Carlo

N.B.: Monte Carlo analysis is also useful when designing experiments. One can simulate

particular cases using the values of the expected parameters to see the effect of noise on the

ability to obtain reliable values of the fitted parameters.

The convergence level of each fit, including each boot-strap fit, can be set under the “Fitting” tab

under the Advanced button.

4.4.2.7.2 Error Estimation: Parameter Confidence Limits

4.4.2.7.2.1 F-Statistics.

4.4.2.7.2.1.1 General F-stat preferences

Figure 4-124. Error estimation: F-statistics.

To turn on the ability to compute F-Statistics for any parameter, under the advanced menu, error

estimation tab, click the “Calculate F-statistics” button. Indicate the desired default confidence

level, the maximum number of steps to take in each direction and the tolerance, i.e. allowed error

in the confidence level, to stop the search of parameter space. In the boxes along the bottom of this

window, you can tailor the size of the search steps to be taken for each type of variable.

4.4.2.7.2.1.2 Turning on F-stats for a particular parameter

To indicate that F-statistics are to be calculated for a particular parameter, left-shift-click on the

parameter window on the control screen. A window will appear; check the F-statistics box and

enter the desired confidence level for that parameter.

Figure 4-125 Turning on F-stats and choosing the CL for an individual parameter

Confidence limits can be determined by computing F-statistics in the following way (cf. M.L.

Johnston and M. Straume, Meth. Enz. 1994). After the main fitting has finished, confidence

intervals of any desired magnitude can be computed for any of the parameters by shift-left-clicking

on the parameter box on the control screen, checking the "F-statistics" box and inputting the

confidence level (CL), e.g. 0.95. F-Statistics are computed by stepping each parameter, holding it

constant and refitting the dataset. The new rmsd is used to compute F = [rmsd(new

fit)/rmsd(minimum)]

. When F increases to the value corresponding to the CL and number of

degrees of freedom, that value of the parameter is the 95% confidence level for the parameter. The

parameter is varied in both directions to yield the (usually) asymmetric confidence intervals. When

a fit has finished, SEDANAL will compute the confidence intervals at the level indicated (95% in

this case) for those variables for which F-Stats have been enabled. F-Stats must also be enabled in

the Error Estimation control tab under the "Advanced ..." button on the control screen or under the

"Control Extended tab" on the preferences screen that can be accessed from the Main Menu.

The results of the F-Stat can be written to a text file for further analysis. Under the "Outputs" button

on the control screen, select the "Log files" tab and click the button for the F-Stat log (Figure 4-126):

Figure 4-126 Output the F-Stat log file

4.4.2.7.2.1.3 Displaying F-stat progress

To display the F-stat progress, click on the "F" button on the upper right corner of the fitting

window:

The following floating window (Figure 4-127) will open allowing you to monitor the F-Stat

progress.

Figure 4-127. F-statistics monitoring window.

4.4.2.8 Fitting for Extinction Coefficients

By combining data from interference optics and absorbance optics, we can determine extinction

coefficients of pure components. Two datasets can be combined from a single run such that the

interference optics provides an estimate of the concentration, while the concentrations of the two

cells are linked (i.e. requiring them to be the same) while the extinction coefficient is allowed to

float as a fitting parameter. The most important requirement is that the “extinction coefficient” (i.e.

numbers of fringes per g/L) for the interference signal is known or calculated from its composition,

taking into account the amino acid composition, glycosylation, and nucleotide content, and any

other absorbing moieties or adducts.

4.4.2.9 Fitting flotation data

Flotation data can be fit if one enters a negative value for the sedimentation coefficient of a floating

species and also sets its density increment to a negative value. Models can also be constructed that

allow for different species to sediment and float in the same sample. While DCDT and WDA can

handle flotation data, they cannot be used to analyze sample which contain both sedimenting and

floating species.

4.5 DCDT and WDA

Time derivative analysis, g(s*), and multi-speed wide distribution analysis (WDA) can be carried

out by selecting the “dc/dt” button on the main menu (Figure 4-23). The “Concentration profile

time-derivative analysis” window will then appear.

4.5.1 DCDT

A run file is loaded by clicking in the “Experiment” window to specify which experiment is to be

analyzed followed by clicking in the “Cell data” window to select a particular cell data (.abr) file

(Figure 4-128). When the cell data file is selected, all the data in the “abr” file will be presented.

Usually the run will span a large time interval and the data plotted may look rather strange. The

scans to be analyzed are selected in the same way they are in the Fitting control screen, by right

clicking on the Cell data file window. The numbers of scans to be used should be selected consistent

with the molar mass of the species present. If too many scans are included, the g(s*) peaks will be

artificially broadened. After the desired scans have been selected, the plots should look similar to

Figure 4-129.

Figure 4-128 DCDT header from v7.43

Figure 4-129 (left) dc/dt. (right) g(s*)

The left window contains a plot of “dc/dt vs s*” for every scan pair used in the analysis. A range

of data can be selected in the left window to remove undesirable stray data points that might interfere

with the analysis. The range chosen at this point will be used for the final “g(s*) vs s*” plot displayed

in the right window, as well as in the computation of the various averages of the sedimentation

coefficient, which will be displayed in the top window after the “Compute weight averages” box is

checked.

The s*-axis scaling is selected in the s*-grid window. There are a thousand s* points plotted across

the window. For example, if a value of 0.01 is selected for the s*-grid spacing, the value of s* will

span from 0-10S. After the value of s*-grid has been edited, you must right click the Cell data

window and then click on the OK button to refresh the screen. Left clicking on the Cell data window

will reload the entire dataset.

The plots may be re-scaled by clicking on the “Select zoom region” button and using the mouse to

select a rectangular region to be re-plotted. The previous zoom levels are stored in a stack. A

previous zoom level can be regained by typing CTRL-Z and a later zoom level, by typing CTRL-

Y .

Figure 4-130. The weight average sedimentation coefficient

The weight average sedimentation coefficient may be computed by selecting a range over which to

do the average by first clicking on the button labeled “Range for averages”, selecting a range on the

dc/dt plot by clicking and dragging and un-clicking.

Figure 4-131 The corresponding averages appear highlighted in yellow in the adjacent box.

⚠

Note: When averages are computed, the report will also contain the normalized values

of g(s*) and s*g(s*).

The temporal range of the data is displayed in a window at the bottom of the screen.

Also displayed is the maximum molar mass of a macromolecule whose diffusion coefficient could

be reliably determined from the dataset (i.e. the set of scans) chosen by fitting the g(s*) vs. s*pattern

to a gaussian.

4.5.1.1 Switching between g(s*) vs s* and s*.g(s*) vs ln(s*)

It is possible to change the y and x-axis scales. On the s*g(s*) vs ln(s*) scale, the vertical off-set is

a constant and independent of the value of ln(s*).

4.5.1.2 Smoothing of g(s*) vs s* for presentation purposes

Some smoothing can be applied to g(s*) curves.

With no smoothing a plot might look noisy; like:

No smoothing and with 4% smoothing

Figure 4-133 Smoothing DCDT analysis

Adding 4% (i.e. using 4% of the total data span in the sliding fit) smoothing gives the plot on the

right. The unsmoothed individual dc/dt plots are shown as dots in the background. This degree of

smoothing distorts the plot very little while removing most of the noise.

4.5.2 Wide Distribution Analysis (WDA)

Data from a either a single speed run or a multi-speed experiment is processed as described above.

Before loading the run file (.abr file), click on the “Wide distribution” button

Dialog box

The following two windows (Figure 4-134) will appear.

Figure 4-134 The radius control box for performing wide-distribution analysis.

The bottom window shows which radii will be used for WD analysis and allows the user to change

the selection of radii. This box may be just below the bottom of the WD Window and not visible on

some monitors if the resolution is set too low. The minimum recommended resolution is 800

vertical, 100%, and at this setting the tool bar can be only one layer. At this resolution the title bar

will be just visible at the bottom of the screen.

4.5.2.1 Choice of radii for WD Analysis

The choice of which radii to use for WDA depends on whether or not the run was allowed to clear

the slowest sedimenting material is allowed to clear the point of observation.

First, we'll explore how WDA is performed. Given a set of scans, first we choose a radius at which

to "sit" and observe the passage of sedimenting material. The figure below shows the passage of the

boundary at each radius.

Figure 4-135 A set of scans from a sedimentation velocity experiment.

Separation is roughly proportional to the first power of the time (distance traveled) while the

spreading due to diffusion is roughly proportional to the square root of the time (distance traveled).

Therefore, one can expect that the overall resolution to increases as the square root of the distance

traveled. See Figure 4-136 below showing the increase in resolution as the radius at which the

observations are made is increased from 6.1 cm to 7.1 cm. If one were to run until the slowest

species is just at the bottom, a radius in the middle of the cell--say, around 6.5 cm--would allow

essentially all the material to pass that point by the time the run is stopped. It would be best for

most types of analysis to run nearly twice as long as that to assure that the slowest material has

cleared the cell completely. In this case, a radius closer to bottom--say, 6.9 -7.1 cm-- would be

appropriate allowing high resolution.

Figure 4-136 WDA resolution as a function of radius.

After choosing a run file, and selecting a smoothing range (see below), a plot will appear from

data taken at a set of radii selected for plotting in the preferences: in this case a single radius

of 6.6 cm was selected (Figure 4-137).

Figure 4-137. Results of WD analysis.

This was a multispeed run at the speeds shown above: 6000, 12000, 18000, 25000, 35000, and

50000 RPM indicated by the different colors on the plot. For this run a range in s of 2S to 20,000S

is displayed in the plot. (This was a mixture of hemacyanin and polystyrene beads.)

If more than one radius is selected, WDA curves from all radii are plotted and the average of the

plots from all radii is shown as a solid black line. By clicking on the little “O”, one can toggle

between, (1) just the individual plots, (2) just the average, or (3) both sets of plots. The s*g(s*) vs

ln(s*) are interpolated onto an equally spaced grid of ln(s*) before being smoothed or averaged.

Clicking on the "E" will add error bars to the averaged curve (solid black line). ⚠ NOTE: The

error bars reflect the standard error of estimate of the average from each radius at each value of s*

and are not influenced by the smoothing.

If one unchecks the Interpolate button, only the individual, WDA plots can be displayed by

clicking on the little “O”.

Figure 4-138. Control box for picking bad scans in WDA.

4.5.2.2 Bad Scans

Right-click on the numbers of any bad scans (Figure 4-138) (There are 4 in this example: #’s 3, 39,

63, & 495-8) to eliminate them from the analysis. Bad scans can be identified in the preprocessor.

4.5.2.3 Smoothing after numerical differentiation

Some smoothing must be performed on the data since the derivative is computed by taking simple

central differences of the c(r,t) vs s*(r,t) data. Smoothing is performed with three passes of a moving

box-car filter using a window (i.e. number of points) expressed as a percentage of the total span of

the data. This algorithm was chosen because it has very small leakage of high frequency components

that are common to many other smoothing algorithms, and it’s frequency response can be easily

controlled by varying the size of the window (Stafford, 1994).

Control dialog box for choosing the size of the smoothing window is expressed as a percentage of

the entire span of the data. In this case, 1% of 1000 s values on the x-axis, uses 10 points on the x-

axis.

⚠ Note: after entering the percent (i.e. “1” in this case), press enter to have it take effect.

The example given above (Figure 4-133 and Figure 4-137) is for a mixture of hemacyanin and

polystyrene beads. The data are plotted as a differential distribution function, dc/dln(s*) vs ln(s*),

on a logarithmic scale of s*. (N.B. dc/dln(s*) = s*g(s*))

Figure 98. WD analysis for the mixture of hemacyanin with polystyrene beads (Stafford and

Braswell, 2004).

Here, plots from radii 6.53, 6.54, 6.55, 6.56, 6.57, 6.58, 6.59, and 6.60 cm are superimposed. The

black line is the average of the curves from those individual radii. By clicking on the "E" button, the

error bars representing the standard error of the mean are plotted at each averaged point.

Plots from higher radii will have higher resolution than those from a lower radius.

The range of these data is from 2.7 S to over 20,000 S.

The color of the points in the plots corresponds to the speed at which the data were acquired. The

color code is given below the plot. and the list of radii are displayed in the radial points window.

4.5.2.4 Adding additional radii for WD Analysis:

Figure 4-139 Radii available for WD analysis

These values can be changed here and/or in the Preferences -> dcdt/wda.

4.5.2.5 Averaging overlayed WDA curves.

The particular WDA curves that are displayed will be averaged over the radii represented by the

blue dots and are written to a separate *_WDA-ave.txt output file when the "Write report file" button

is clicked. The values of radius corresponding to the green dots will be included in the normal

*_WDA.txt output file. The output file names will have the form "Name_of_abr_file_WDA.txt"

and "Name_of_abr_file_WDA-ave.txt", respectively.

4.5.3 Flotation

Figure 4-140. Flotation button in DCDT/WD

4.5.4 The Adv button:

Figure 4-141 The Adv Button

Under the Adv button, we can add correction factors for the sedimentation coefficient axis of the

WD plot to convert the values to s

(20,w)

. Also, we can specify the range of radial values at each

chosen radial point to average. In the example, shown in Figure 4-141, at each radius selected, WD

will select 5 points in either direction and average them with the central point and use that averaged

value for the signal at the central point. For example, if the user entered 0 and 0, only the central

point would be used. The number of points to se to be used depends on the point density of the

scans data, and the amount of noise that is acceptable. The values of 5/5 are suitable for the

interreference optical system.

4.5.5 Time Derivative Analysis of Multi-wavelength Data: DCDT and WD.

After a MWL (multi-wavelength) run has been processed in the Preprocessor and stored in a run file

(*.abr file), it can be analyzed by either DCDT or WDA. There are several ways in which these large

datasets consisting of several hundred scans from several hundred wavelengths can be treated: (1)

Scans from individual wavelengths may be analyzed separately by either DCDT or WDA. (2)

Spectra of unknown components corresponding to well separated peaks in the DCDT or WDA

patterns can be extracted from each peak, (3) Given the extinction spectra for each of the components

in a mixture, the data can be deconvoluted in the Preprocessor into a set (abr files) of concentration

profiles for each constituent component (Walters, J. et al., 2015 Anal. Chem., 87, 3396-3403).

4.5.5.1 WDA: Extracting Spectra.

After preprocessing and saving the abr file, load it into WDA. At first, select only a few

wavelengths to speed the process up. Next select each peak whose spectrum you want to extract by

clicking on the "Peaks" button as shown in Figure 4-142. This a click-and-drag procedure (Figure

4-143) much like selecting the range-to-fit in the preprocessor. Repeat this process for each peak

whose spectrum you want to extract. Then click on "Select all" in the Wavelength window (Figure

4-144). Now all the absorbance vs wavelength values for each peak will be listed in the

"Integration" window. The spectra can be written to spectrum files as shown in Figure 4-147.

Figure 4-142. Selecting peaks for spectrum extraction:

Click on the "Peaks" button, and click-and drag to select the peak.

Figure 4-143 Three peaks selected for only three wavelengths.

Figure 4-144 Click the "Select all" button to plot all the WD curves for all wavelengths.

Figure 4-145. WD curves for all wavelengths.

At this point, you will want to save these spectra to a file for each peak:

Figure 4-146 Spectra gathered from the WD plot

Click on "Write to files ..."; the following window will appear

Figure 4-147 Spectrum output dialog box

The default file names can be renamed:(Figure 4-148)

Figure 4-148 Specifying a filename for spectrum files

After renaming the files, click on "Save spectra" to write out the spectra to individual files

Figure 4-149. Spectra of the three peaks selected from WD analysis. Plotted in a separate plotting

program.

4.5.5.2 Least Squares deconvolution of component (constituent) concentration profiles.

In the Preprocessor, after loading a multi-wavelength dataset file, the individual constituent

concentration profiles can be deconvoluted into a separate run file for each constituent component.

After loading the data and selecting the meniscus, base, and range-to-fit, etc. We can enter paths to

the extinction spectra and click on "Deconvolve" button and see the green or blue progress bar

indicating that the deconvolution is proceeding (Figure 4-150):

Figure 4-150 Deconvolution of constituent concentrations.

4.5.5.3 Multi-wavelength Fluoresence Intensity Data

When Multi-wavelength intensity data are loaded into the preprocessor, the scans will appear as

inverted pseudo-absorbance. The sign of the data can be changed by clicking on the "F" button that

is located in the upper righthand region of the Preprocessor window - as shown:

Figure 4-151 Location of the "F" button

From this point on, the scans are treated as pseudo-absorbance scans and saved in the abr file as

positive scans.

4.5.5.4 Deconvoluting Multi-wavelength Fluoresence Intensity Data

After the data have been "flipped", the deconvolution process is the same as for ordinary

absorbance data. See above in paragraph 4.5.5.2

4.6 Synthetic Boundary and Band Sedimentation

4.6.1 Fitting or simulating synthetic boundary experiments

4.6.1.1 Fitting

Each run may be treated as a synthetic boundary experiment or a normal sedimentation velocity fit.

From your set of scans, choose the one that is to be used for the initial concentrations (the I-scan).

Typically, it will be the first scan, but it may be later. This must be done separately for each run.

Several species can be accommodated, since the concentration is calculated from the signal, the

extinction coefficient and the signal weighted ratios of the species.

To identify a cell as a synthetic boundary or band sedimentation run, on the control screen, right

click on the cell data file to get the context menu. Click "Synthetic boundary"... to bring up the

Synthetic boundary parameters window.

Check the box, and select one of the two locations from which the initial concentration profile will

be obtained, either from a scan from the cell data, or from a file containing a concentration profile.

For fitting cell data (normal sedimentation velocity fit), you can choose the number of the I-scan. For

simulating, this option is not available. For an I-scan that is not in the set being analyzed, use the

third option, and browse to the I-scan file.

The coloration of the B/A field shows that the ratio is to be fitted, and 1.0 is just the initial guess.

The format of the file containing a concentration profile is the same for single-species and multi-

component SB simulations or fits, and is given below.

Remove the I-scan and any prior scans from scans to fit. This must be done separately for each run.

Each time the grid or parameters change, the I-scan is interpolated onto the simulation grid, and the

concentrations for species 1 calculated as the interpolated signal / extinction coefficient for species

1 for the run. These calculated concentrations are used as the initial concentration profile for species

1 for the Claverie FEM simulation of scans.

The “simulation clock” is started at the time of the I-scan, rather than at 0, to make the simulated

scans appear at the same time relative to the I-scan. For example, if the I-scan was taken at 40 sec,

and the next scan was taken at 60 sec, the Claverie simulation will run for 60−40 = 20 sec to produce

the simulated scan to be compared with the second experimental scan.

Because the loading concentration does not affect the fit directly, you should not attempt to fit for it.

The report will show [SB] after the load concentration in the Parameters section for a synthetic

boundary cell.

4.6.1.2 Simulating

To identify a simulation as a synthetic boundary run, on the control screen click Synthetic boundary...

Check the box, and identify a file containing a either an I-scan, or a concentration profile (the Scan

to be used choice will be grayed, as it is not available for simulations).

The coloration of the B/A ratio shows that the ratio is fixed at 0.5.

You can either type or paste the path into the box, or browse using the button.

4.6.1.2.1 Concentration file format

A concentration file contains the mass concentrations of all nc component species at each of nr radii.

The number of radii and their values need not correspond to either the experimental data or the

simulation grid; values will be linearly interpolated as needed. This means, for example, that a step

function at r=6 cm could be described by points at meniscus, 5.999, 6.001, and base.

SEDANAL concentrations v1 // [optional comment]

radius

... c

radius

... c

radius

... c

The initial line in the file must be exactly as shown, except that the “//” is only needed if there is a

comment following. Any number of spaces may separate values.

Concentrations of the species which are not component species (if any) will be computed from the

mass-action equilibria (reactions). Note that the component concentrations are not the “total

component”, but the equilibrium concentrations of the species which have been chosen as

components.

Here is an example of a concentration file which has a linear gradient of component 1 from 0 g/L at

meniscus=5.9 cm to 5 g/L at the base=7.2 cm, and a 1.5 g/L pulse of component 2 between 6.0 and

6.1 cm.

SEDANAL concentrations v1 // illustration of gradient + pulse

5.9 0.0 0.0

5.9999 0.3846 0.0

6.0001 0.3846 1.5

6.0999 0.7692 1.5

6.1001 0.7692 0

7.2 5.0 0.0

4.7 PREFERENCES

The Preferences screen allows one to choose the names of the directories used by SEDANAL, the

default number of scans to show in the initial plot in the Preprocessor, and the maximum number of

control files (*.abc) that are shown in the “Last” drop-down list on the Control Screen (Figure 4-152,

and Figure 4-153). Also, various default and other parameters can be set for other SEDANAL

functions like DCDT, BIOSPIN, the control screen, etc. …

4.7.1 General Preferences

Figure 4-152 Setting User_data and ModelEditor files paths.

Figure 4-153. Browse to set new paths.

4.7.1.1 Location of Preferences.txt files

When SEDANAL is started, the preferences file is read, and unrecognized lines pop up a

message ("The preference file line nn begins with the unrecognized symbol..."). This message has

been expanded to show the complete path to the file being read. Unrecognized lines usually come

from reading a file using an earlier version of SEDANAL than the version that created it.

Users now have more flexibility in setting the location of their preference files by choosing the path

to the preferences file to be used:

4.7.1.2 Multiple users and multiple Preference files:

4.7.1.3 Developer Options

Developer Options can be activated by entering “432” in the “Key: should be blank except for

developers” window.

On the Fitting screen, the following button will appear if Developer Options are activated:

When the Developer Options button is pressed, the following rather daunting window will appear.

You are now looking at many of the secret, deep inner workings of SEDANAL. Many of the items

on this list are experimental, some are defunct but several are useful and after a little more testing

will be available on the other screens.

For example, items 012 and 013 are useful for doing the positive and negative sides of an F-stat

search separately. Also item 019 allows SEDANAL to handle dynamic density distributions, r(r,t),

calculated by adding up the density contributions from each of the species present in a model.

cf.#Schuck#P#(2004)#A#model#for#sedimentation#in#inhomogeneous#media.#I.#Dynamic#density#gradients#from#sedimenting#co-

solutes.#Biophys#Chem#108(1–3):187–200.

4.7.2 Preprocessor preferences

Figure 4-154. After you have used SEDANAL a few times, the top three items on the left hand side

should be unchecked for genreal use. They can be checked for beginners to force the work flow in

the PP.

The middle section:

allows the user to separate scans based on wavelength. The XL-A has some variation in the

wavelength when a single wavelength has been chosen. In the example above, if the variation is

less than or equal to 4 nm, SEDANAL will treat the scans as if they were from a single wavelength.

And if the difference is greater than 4 nm, the scans are treated as coming from different

wavelengths as a separate set of scans.

4.7.3 DCDT and WDA preferences

Figure 4-155. Setting parameters for dc/dt and WDA.

Setting parameters for dc/dt and WDA. It is recommended that a default value of smoothing percent

= 2 be used for the initial look at WDA data. Smoothing of WDA is required because the derivative

is computed using simple centered divided first differences. For DCDT, set it to zero on the DCDT

screen for the initial look. Smoothing can obscure outliers. You should be aware that a single outlier

(i.e. a delta function), with enough smoothing, will be transformed into a gaussian shape (review

your course on convolution.) You don’t want that to happen.

The bottom window with green and blues dots is for selecting the radii to be used in the WD analysis.

The three little windows in the upper right corner are used to establish the radial grid that will be

used by WDA. Those values determine which of the radii in the lower window are represented by

crosses and are those radii that can be used for WD analysis. The green dots, which are selected by

right-click-n-dragging over the intersection crosses, represent those radii that are used in the

computation. Those results are written to the report file ending in "_WDA.txt"; similarly, the blue

dots represent those radii that are used for the average WDA curve and are displayed in the screen.

The blue dot results are written a report file ending in "_WDA-avg.txt"

BIOSPIN preferences

Figure 4-156. Preferred settings for BIOSPIN

4.7.4 Weighting Factor preferences - "Control" Tab

Figure 4-157. Parameters for the control screen: setting defaults for use of weighting factors.

Parameters for the control screen (Figure 4-157): setting defaults for use of weighting factors. In the

preference be sure to un-click the box labeled "Use errors from scan if available" as a default.

This can be selected later after you have loaded a run file in the Fitter and you want to use the errors

in the third column of the absorbance scan file. The values shown above correspond to values of the

inverse variance (1/s

) typically observed for the various types of data: you should change these

values according to your data type and quality. (Do not use values from the "plateau region" that

was an idea that was never implemented.)

In the case shown here, using fluorescence data from the FDS machine, the noise (i.e. standard

deviation of the noise) on the data was +\- 0.5397 fluorescence units. The inverse square of that is

3.433 and so that value is used as a weighting factor for that dataset. If these data had been

combined with absorbance data whose noise is about +/- 0.006, a value of 27778

(=1.0/(0.006*0.006) would be entered for that dataset.

⚠

NOTE:

But since version 6.92, the user can enter the standard deviation and SEDANAL will compute

the inverse variance for you.

4.7.5 Confidence Limits -- Error estimation control: Advanced, Control Extended tab

⚠ NOTE: The Advanced tab can be accessed also from the Fitting Control screen. Choices made

there will apply only to the control file under consideration. Choices made here in Preferences will

become defaults but may be changed from the control screen for a particular run.

Figure 4-158. "Control extended" parameters allow setting of the default parameters found

under the “Advanced…” button on the control screen.

4.7.6 "Claverie Control".

Figure 4-159 Setting Finite Element controls.

4.7.7 Kinetics/equilibrium Controls.

Figure 4-160. Setting for kinetics and slowly equilibrating systems.

Settings for a system whose re-equilibration is kinetically limited, choose either Kinetic integrator:

BulSt or SEulEx, may be used. For any particular problem, one will be faster than the other: try both.

Figure 4-161 Settings for a system that is in instantaneous equilibrium.

4.7.8 The Fitting Control tab

Figure 4-162. Choice if default fitting parameters.

The choice between the default fitting method: Simplex vs Levenberg-Marquardt is made here.

⚠ NOTE: The Advanced tab can be accessed also from the Fitting Control screen. Choices made

there will apply only to the control file under consideration. Choices made here in Preferences will

become defaults but may be changed from the control screen for a particular run.

4.7.9 Simulating Output Choices

Figure 4-163 Choice of outputs when simulating (choose them all)

4.7.10 Reports: Output, Control Extended tab

Figure 4-164. Control extended parameters also allow setting of the default parameters found under

the “Outputs…” button on the control screen.

4.7.11 The other “Output …” tabs

Control Extended-> Outputs->Minima:

Minimum files: "Minima" tab

Figure 4-165. Outputs: choose the “min” file formats

At each new minimum during fitting, SEDANAL will write a "minimum" file that contains the last

best fit, and the data with the residuals etc... for plotting the results. The number of scans to be

included in the “min” files can be indicated in several ways by selecting the output format in the

lower right section of the Figure 4-165. This format can be chosen to accommodate your plotting

program. Data from all delta-C scan pairs or Selected scan pairs, or first and last and evenly spaced,

or a specified number evenly spaced.

4.7.11.1 Concentration as a function of radius output.

Figure 4-166. Ancillary output files

Ancillary output files can be written during simulation that will contain the concentration of each

species as a function of radius at each time point.

4.7.11.2 Concentration as a function of time for the initial equilibration step

Figure 4-167. Kinetics data for the model under consideration.

This has been superseded by the Kinetics function accessible from the Main Menu.

4.7.11.3 Disposition of various log files

Figure 4-168. Disposition of various log files

In this example, the "Output F-statistics log" is checked. Check this. The other two are for debugging

purposes.

Figure 4-169. Setting the default parameters for indefinite self-association reactions.

Setting the default parameters for indefinite self-association reactions. The coefficient for a1 is 2/3

in this example and signifies that the values of the sedimentation coefficient of each of the

oligomer are based on the s

2/3

relationship for oligomers that all have the same frictional ratio:

this has been used traditionally. However, the other coefficients allow one to select any relationship

between the sedimentation coefficients of the oligomers and can be determined using bead models,

for example, or by other appropriate means.

4.7.12 Fitting preferences

Figure 4-170. Fitting preferences.

The “MOD+INT power of 1.01” button causes each element of the initial simplex to be filled with

a different number. The “Positive increment …” button alters only the diagonal elements of the

initial simplex. Don’t use the “Alternating sign …” button; we don’t remember what it does.

4.7.13 Simulation Preferences

Figure 4-171. Fitting and Simulating Preferences - choosing output files: check them all.

4.8 BIOSPIN

BIOSPIN is the program originally developed by Dennis Roark and David Yphantis to compute

point-by-point molar mass averages as a function of the local cell concentration from sedimentation

equilibrium data (D.E. Roark & D.A. Yphantis, 1969). It has been implemented as a subprogram of

SEDANAL. The reader is referred to the original literature for further information. In addition to

the point-wise number, weight, Z- and Z+1 molar mass averages, it also computes several of the so-

called “charge-independent” molar mass averages that are independent of the second and higher

virial coefficients: the Y1, Y2, Y3, etc… averages. (Yphantis & Roark, 1972). The charge

independent averages are useful for analyzing systems under non-ideal conditions. The original

BIOSPIN user manual can be downloaded from https://sedanal.org/biospin_manual.pdf.

4.9 SIMULATING DATA

To simulate XLA/I or Multi-wavelength data, select "Fit preprocessed data" from the Main Menu.

When the Control Screen appears, click the "new" button and either create a new "experiment"

folder or select an existing “experiment” folder in which the simulated data files will be stored. Then

click the button labeled "Simulate data"; the screen will change to the simulator screen. Now select

a model from the drop-down "Model to be fitted" list. Enter the number of points to be used for the

grid between meniscus and base. Enter the molecular parameters in the "Molecular parameters"

boxes.

In the bottom panel, select the optical system; indicate whether it is a velocity or equilibrium

experiment: enter a comment; specify the meniscus, speed, base and loading concentration. For a

velocity run enter the total number of scans and the time between scans. Now enter the magnitude

of normally distributed random noise to add to the simulated data.

The window labeled "Bottom time, sec" will display the total time for the slowest species to reach

the bottom of the cell. You can adjust the total number of scans and time between scans as you

desire.

When all the boxes have been filled, click "Store control file and start fit" to start the simulation and

to save the simulation control file. The simulated data files will have names of the form 000nnn.IPs

or 00nnn.RAs and will be in XLA/I standard format. The simulated data, *.IPs or *.RAs files, will

be stored in standard time and date subdirectories of the experiment directory which was selected

when you clicked on the "New" button. The time and date used for the path name will be the same

as the time and date of the simulation run. The simulated XLA/I data files are recognized and can

be processed like any other XLA/I data files by SEDANAL or other programs.

Simulation can generate a cell data (run, or .abr) file, as well as the usual scan files. The run file's

name is YYYYMMDD_simHHMMSS.abr, and it has the meniscus and cell base from the

simulation, and the range to fit is (meniscus+0.05) - (base-0.1) (i e, 5.95-7.1 for m=5.9 and b=7.2).

It can also write a prototype control file that can be used to fit the simulated data. To turn on either

of these features go to the "Advanced …" button and click on the "Simulating" tab and select the

appropriate buttons. A prototype fitting control file can also be produced by clicking the appropriate

button below:

Figure 4-172. Choosing output files: Preferences -> Control extended -> Simulating.

Choose them all.

Figure 4-173 Control screen after selecting "simulate"

Other features:

4.9.1 Saving a “package”.

A package consists of all the files required to reproduce the current fit. It is useful for debugging

purposes if an “anomaly” occurs and SEDANAL is misbehaving. The package is a zip file

containing the control file, the run files, the Model file needed to reproduce the fit, and several

other relevant files. A package is also a convenient vehicle for sharing a dataset with another

researcher. The package zip file is stored in the curent "experiment" folder. A Package can be

generated by clicking on the “Package” button on the Fitting Control Screen:

Figure 4-174. Generate a Package

4.9.2 Output of initial reaction time course.

(See paragraph 2.10- “Kinetics Simulator” for a more user-friendly and

comprehensive way of computing the kinetics.)

Kinetics: To write a file of the time course of the initial equilibration of a particular model, click on

the “Output files button and select the “conc(t)” tab and check the “Output concentrations as a

function of time” box. Then select the units for the concentrations

Figure 4-175. Output files

4.9.3 Outputting the time course of the initial equilibration step

When the “Store control file and …” button is clicked, only the kinetics of the model are

calculated and a file named after the abc and report files is written. No fitting will be done in

this case.

If the abc file is named xxxxxx.abc, the report file will be named xxxxxx_Report001.rtf

and the kinetic time course will be found in a filed named: xxxxxx _Equilibration.txt.

Remember to unclick the “Output concentrations as a function of time” box when you are

done.

Figure 4-176. This feature has been replaced by the Kinetics Simulator described in Section

4.10

4.10 Kinetics Simulator

A general purpose kinetics simulator has been added to SEDANAL and is accessed from

the Main Menu (Figure 4-23) by clicking the Kinetics button. The kinetics simulator can handle

any model that can be represented in the ModelEditor. When the kinetic simulator screen is first

opened, a blank screen showing the possible combinations of 28 species and 27 reactions. One

should click on "New" to create a new kinetics control file, or select a previously used control file

from either the drop down menu or by browsing for it.

Figure 4-177 Kinetics Control Screen

Figure 4-178 Kinetics control screen for the Model 2A=A2; 2A2=A4; 2A4=A8

For example, for the model A+B=AB, B+AB=AB2 molar masses of A and B are 20,000 and

50,000 g/mol, respectively. The forward and reverse rate constants for the first and second

reactions are 2 x 10

-1

, 0.01 s

-1

, and 0.5 x 10

-1

, 0.01 s

-1

, respectively. The initial

concentrations of A and B are 1 x 10

-6

and 2 x 10

-6

M, respectively.

Make sure you select one of the two methods to solve the kinetic differential equations under the

"Advanced > Kinetics/equilibrium control" tab:

Clicking "Store control file and do kinetic calculation" results in the following display.

Figure 4-179 Screen dump of kinetic simulation

The point density and/or time span, and other aspects of the display can be adjusted:

Clicking on the black dot next to the species name will toggle that plot on or off

These data will have been written to a file whose file name will have the following form:

New_Control_Filename_Equilibration.txt.

The text "New_Control_Filename" will be replaced by the name you gave the control file after you

pressed "New" or changed the control file name after loading an existing abc file with either the

"Last" or "Browse" buttons.

These simulated data can be read into your favorite plotting program to produce a plot more to your

liking::

Figure 4-180 Plot of molar concentrations as a function of time.

Plot of molar concentrations as a function of time for the system A+B=AB; AB+B=AB2 (Figure

4-180). The forward and reverse rate constants for both the first and second reactions are 1 x 10

-1

, 0.01 s

-1

, respectively. The initial concentrations of A and B are 1 x 10

-6

and 2 x 10

-6

respectively.

Here is another example with the same equilibrium constants but the second reaction forward and

reverse rate constants are one tenth those the previous example.

Figure 4-181 Plot with same equilibrium constants; but rate constants for the second step are

1/10 those of the first example above (Figure 4-180).

Another plot (Figure 4-181) with same equilibrium constants but forward and reverse rate

constants for the second step are 1/10 those of the first example above. The forward and reverse

rate constants for the first and second reactions are 1 x 10

-1

, 0.01 s

-1

, and 1 x 10

-1

, 0.001

-1

, respectively. The initial concentrations of A and B are 1 x 10

-6

and 2 x 10

-6

, respectively

4.11 Scripting: fitting and simulation

The script functionality is rudimentary at present, to give us a chance to experiment with it and find

out what is useful and what isn’t. Not all the error-checking or UI features are present.

The script files must be in the User_data directory, and have an extension of .abs. Script files are

created and modified with a text editor.

4.11.1 Scripting commands:

The initial line of the script must be exactly

SEDANAL script 1

The following commands may be included in a script:

Command

Meaning

Example

MENU FIT

Equivalent to clicking Fit preprocessed

data on the main menu.

MENU FIT

LOAD

controlFile

Specify a control file to be loaded;

controlFile can be either the name of a

control file (see below), or $LAST.

LOAD

MyExperiment\ABCD_Fit

MODIFY

expression

Change the value for a kinetic, molecular, or

cell parameter from the value in the control

file, and as previously modified. The

symbols and format of the expression are the

same as in the equation editor.

MODIFY K(1)=K(1)/10

FIT

Equivalent to clicking Store control file

and start fit on the control screen. See note

below.

FIT

blank lines

These lines are ignored.

[this line intentionally left blank]

comment

lines

These lines are ignored, except for checking

spelling, grammar, and clarity of style.

// This is a comment

A typical script file might look like:

Figure 4-182. typical script

This particular script was used to simulate a dilution series of a monomer-tetramer system creating

three abr files for 1 to 3 serial dilutions of the monomer tetramer system that was specified in the

"SIM_mon-tet.abc" file. This script will load the original abc file and perform the simulation; then

reload it and simulate again with the loading concentration divided by 3; and then repeat with the

loading concentrations divided by 9. This is very useful for generating large datasets for

exploratory studies of interacting systems to help in designing experiments.

4.11.2 Specifying control files in scripts

The control file is identified by the following rules:

1) If it’s just a name, the complete path is completed by trying the directory of the most

recent control file loaded, then the most recent control file stored for this instance of

SEDANAL, then the most recent control file stored for any instance of SEDANAL. The idea

is that SEDANAL tries to guess what you mean if you just say, e g, ABCD_Fit.

2) If it is a directory followed by a name (e g, MyExperiment\ABCD_Fit), the directory

is assumed to be in the user data directory.

3) Anything else is a complete path (e g,

C:\SEDANAL\User_data\MyExperiment\ABCD_Fit).

4) If the control file name does not end in .abc, the extension is added. Thus, ABCD_Fit

and ABCD_Fit.abc mean the same.

4.11.3 Storing scripting control files

To avoid modifying the original control file, when you have FIT in a script, the control file is

stored with the suffix “_script001”. For example, if the script in the example column

above is run, the name of the control file with K

a factor of 10 smaller will be

ABCD_Fit_script001.abc, and ABCD_Fit.abc will not be changed.

When a script file has been loaded, the following screen will appear:

Figure 4-183. Script control

The script is initiated by clicking on "Run script".

4.12 Keyboard Shortcuts

There are several keyboard shortcuts that work from the Main Menu.

Key

Action

p or P

Preprocess centrifuge data

d or D

dc/dt

f or F

Fit preprocessed data

l or L

Reload the previous fit (same as L + LAST)

b or B

BIOSPIN

e or E

Equilibrium calculations

q or Q

Preferences

k or K

Kinetics

s or S

Script

h or H

Help

m or M

Model editor

x or X

Exit

4.13 Help

The help files can be accessed from the main menu. By either selecting "Help" from the main

menu or by clicking on the "yellow-question-mark-on-a-red-background" found on most

screens.

You will see a window like this:

Figure 4-184 SEDANAL On-line Help

Be sure to peruse the Change Logs for important information:

Figure 4-185 Change Logs

4.14 Exit

Self-explanatory … bye…

5 Background Theory

The curve fitting portion of SEDANAL is designed to analyze sedimentation velocity and

sedimentation equilibrium data from interacting systems composed of multiple macromolecular

components.

First, a few words about components, species and reactions:

In the context of this discussion, a component is an electroneutral macromolecule or several

molecules related by chemical reaction that can be added or removed, at least conceptually, from a

solution independently of other molecules. A component may be composed of several

macromolecular species. Species interact through chemical reactions. In general, the number of

components is equal to the number of species minus the number of chemical reactions. At constant

temperature and pressure (Actually, we are assuming here that the solution is incompressible.), the

number of degrees of freedom is equal to the number of components. For any given values of the

equilibrium constants, the composition of the solution (i.e. the amount of each species) is determined

by the total concentration of each component at each position in the centrifuge cell.

An example of a single component, multi-species system is a monomer-dimer, rapidly

reversible self-associating system. The monomer and dimer are in equilibrium with each other and,

therefore, cannot be added to or removed from the solution independently. This is why they cannot

be separated on a gel filtration column, for example. As soon as some dimer and monomer become

separated either the monomer reassociates to form more dimer or the dimer dissociates to form more

monomer. At any given temperature and pressure, the amount of monomer and dimer present is

determined completely by the total concentration of the component and the equilibrium constant for

the dimerization reaction. This is a one component system composed of two species with one

chemical reaction between them. If this were a monomer-dimer-tetramer system, it would be a one

component, three species system with two chemical reactions between them. It would have one

degree of freedom such that the amounts of monomer, dimer and tetramer at each point in the

boundary would be uniquely determined by the total macromolecular concentration at that point.

Similarly, consider a system composed of two components, A and B that interact to form a complex,

C. Let C interact with another molecule of B to form D. This is a two component system which is

composed of four species that are related by 2 chemical reactions. The system has two degrees of

freedom, the total concentrations of A and B, respectively.

A + B = C K1=k

C + B = D K2=k

However, if D were composed of 1 mole of A and 2 moles of B but was not in equilibrium with C

and B, it would be a third component. This system would have 3 degrees of freedom, the total

concentrations of A, B that participate in the reactions and the total concentration of D. In principal,

D could be removed from the system by gel filtration without being reformed from C and B. D

might be a covalently cross-linked aggregate, for example. So …

Number-of-components = number-of-species - number-of-reactions

5.1 Sedimentation Velocity Theory:

5.1.1 Concentration time-difference curves.

Consider the signal obtained from the centrifuge. Whether it is fringe displacements, absorbance or

fluorescence, it is a function of both time and radius, call it S(r,t). After S(r,t) has been preprocessed

to remove optical jitter and integral fringe shifts, it is composed of the contribution from the true

concentration distribution, C(r,t), which is also a function of time and radius, as well as a background

optical (systematic error) component, B(r), that is time independent and a function of radius only.

The signal, S(r,t), also has stochastic noise included with it. We have

where a is the conversion factor (also known as the extinction coefficient) between concentration

(g/L) and either fringes, absorbance or fluorescence.

We can remove completely the time independent background component, B(r), of the signal by

subtracting any two experimental curves, say at times t

and t

The time difference curves, DS(r,t

), are proportional to the concentration difference curves,

DC(r,t

) but have no time independent systematic error. The irregular, time dependent, systematic,

error from the optics (often referred to as jitter) will have been removed at the preprocessing stage.

The data to be fitted, DS(r,t

), have only stochastic errors which makes them suitable for least

squares fitting.

SEDANAL fits to the time difference curves DS(r,t

) for any set of parameter guesses by

generating concentration curves, C(r,t

) and C(r,t

), corresponding to times t

and t

, and subtracting

them to form aDC(r,t

). The root mean square residual is computed as the triple sum over all the

points, scans and cells.

where k is the cell number, L is the number of cells, j is the difference curve index, M is the number

of difference curves and 2M is the total number of scans, j and j+M are the indices of the scans

being subtracted, N is the number of radial points in each difference curve, and i is the radial point

index.

SEDANAL can also minimize reduced Chi-Square for cases in which datasets are combined from

different optical systems that have different relative signal strengths and noise levels. In this case

we, minimize the sum of the squares of the weighted residuals, as normalized by the variance of

the noise on the data. For example, noise levels on interference data are typically on the order of

+/- 0.005 fringes, while the noise on absorbance data is in the range of +/- 0.004-0.006 A.U. and

noise on fluorescence data might be +/- 200 arbitrary signal units. So when data from, say,

absorbance and fluorescence optics are combined, we will calculate the normalized residual where

s is the standard deviation of the raw data.

Summed over points, scans and runs.

5.1.2

5.1.3 Procedure used by SEDANAL for fitting time differnce data.

The figure above (Figure 5-1) schematically outlines the basic procedure used by SEDANAL to fit

the time difference data, Dc vs. radius.

The rmsd or chi-square is minimized with respect to the model parameters using the simplex

directed search method of Nelder and Mead (1965), or the Levenberg-Marquardt (1944, 1963)

method.

The numerical solutions to the Lamm equation are generated using the finite element method as

described by Claverie, (1975, 1976), with corrections and enhancements introduced by Todd and

Haschemeyer (1981, 1983).

The general fitting approach, using numerical solutions to Lamm equation and the

Claverie procedure, is based on the method of Todd and Haschemeyer, (1981)

The kinetic differential equations are solved using the Bulirsch-Stoer (BulSt) algorithm with

Richardson extrapolation. (Numerical Recipes in FORTRAN), or by the Semi-implicit Euler

extrapolation (SEulEx). (Numerical Recipes in FORTRAN)

Bulirsch, R. and Stoer, J. §2.2 in Introduction to Numerical Analysis. New York: Springer-Verlag,

1991.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Richardson

Extrapolation and the Bulirsch-Stoer Method.'' §16.4 in Numerical Recipes in FORTRAN: The Art

of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 718-725,

1992.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Stiff Sets of Equations,

Semi-implicit Euler extrapolation.'' §16.6 in Numerical Recipes in FORTRAN: The Art

of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 718-725,

1992.

Todd, G. P. and Haschemeyer, R. H. ", "General solution to the inverse problem of the differential

equation of the ultracentrifuge", Proc. Natl. Acad. Sci. U.S.A., 78(11) 6739--6743, 1981.

Todd, G.P. and Haschemeyer, R. H. "GENERALIZED FINITE ELEMENT SOLUTION TO ONE-

DIMENSIONAL FLUX PROBLEMS" Biophysical Chemistry 17 (1983) 321-336

5.2 Sedimentation Equilibrium Theory

5.2.1 Ideal Case:

For a thermodynamically ideal, incompressible system we have for each species that

where s is defined as , x=r

/2 and x

ref

/2 where r

ref

is an arbitrary reference

radius usually chosen as the first data point. The value of c

ref

can be related to the loading

concentration by invoking conservation of mass and noting that the concentration at the beginning

of the run is uniform and equal to the loading concentration, C

. Invoking conservation of mass, the

total amount of macromolecule, T, in the cell is given by multiplying the concentration by the

volume:

since this quantity doesn’t change with time the following must also be true at equilibrium:

Now we can equate the two relationships, and after dividing both sides by h

, write:

and

and so c

ref

as a function of the loading concentration c

is given by

can be computed from C

ref

if the position of the meniscus and the base of the cell can be

determined with satisfactory accuracy.

For non-interacting systems each species’ loading concentration can be related to its value of c

ref

in this way, independently of the other species. (In this case, each species is also an independent

component.)

For reversibly interacting systems, on the other hand, the law of mass action has to be taken into

account. This leads to somewhat more complicated, but still easily solvable relationships between

the loading concentrations each component and the values of c

ref

for each species in equilibrium.

We will consider two reversible systems as examples: a monomer-dimer self-association and a

simple one-to-one heterodimer association.

5.2.1.1 Monomer-dimer:

Now, invoking the Law of Mass Action, we can substitute for c

2,ref

and write:

This can be solved easily for c

1,ref

with the quadratic equation for the monomer-dimer system. For

higher oligomers and polydisperse self-association, the equations must be solved by successive

approximation (e.g. Newton-Raphson iteration).

5.2.1.2 Hetero-Association:

Now for the hetero-association,

Conservation of mass for each component, A and B, requires

This pair of equations can be solved for c

A,ref

and c

B,ref

, given values of c

A,o

and c

B,o

, easily by

rearrangement into an analytical expression. More complicated stoichiometries can be solved easily

by successive approximation (The Newton-Raphson method works nicely in the general case).

Success in fitting multi-component systems globally to multiple datasets requires accurate

knowledge of the meniscus and base positions (especially the base position) in order to estimate the

correct values of c

ref

from the conservation of mass relationships.

5.2.2 Non-ideal Case:

For a thermodynamically, non-ideal, incompressible system we have :

This has been updated- see above.

6 APPENDIX A: HELP FILE and CHANGE LOG.

See the built-in Help file that accompanies SEDANAL. It can be accessed either from the Main

menu or from any screen that has the yellow-question-mark-on-a-red-background symbol:

7 INDEX

!"#$%&#!'(%)**+,-----------------------------------------------------------------(./0(

1.23

45-----------------------------------------------------------------------------------------(67(

#%%89:;#*9&(%++*<=*8#>(>8+49&)89-----------------------------(..?(

#%=+8%#,49---------------------------------------------------------(6'(.@'(AB'(.6/(

C&:(%)**+,------------------------------------------------------------------------------(.A/(

C&:#,49&(D#8#$9*98=---------------------------------------------------------(6.(

#EE89E#*9-------------------------------------------------------------------------(FF'(.B0(

#;8<#;8(=>#49----------------------------------------------------------------------(AB'(A0(

alignment---------------------------------------------------------------------------------(A0(

CGG(HIGGJ---------------------------------------------------------------------------------(7?(

Analyze data----------------------------------------------------------------------(.F'(BA(

#,*;E9,<#,*;%+&K-------------------------------------------------------------------(F.(

Approach to EQ-----------------------------------------------------------------------(..(

approach.to.equilibrium------------------------------------------------(@@(

#==+4;#*;+,(4+,=*#,*-------------------------------------------------------------(.0(

%#&(=4#,=----------------------------------------------------------------------------(A6'(A7(

L#&(=4#,=-------------------------------------------------------------------------(??'(.F6(

L#,&(J9&;$9,*#*;+,----------------------------------------------------------(.A@(

base-----------------------------------------------(..'(AB'(?F'(?0'(B 6'(7?'(.@@(

%9=*(M;*-----------------------------------------------------------------------------------------(0?(

LNOJDNP--------------------------------------------------------------(.A6'(.?A'(.@@(

BIOSPIN.preference=------------------------------------------------------(.?A(

%")9--------------------------------------------------------------------------------(.B'(70'(0?(

BM1---------------------------------------------------------------(.@'(FA'(F@'(FB'(67(

boot-strap-------------------------------------------------------------------(..?'(..B(

%++*<=*8#>(Q;*R(89>"#49$9,*----------------------------------------(..?(

LJC---------------------------------------------------------------------------------------------------(7(

L)";8=4R<J*+98-----------------------------------------------------(6?'(.6F'(.6A(

HSCPTI(GOT-------------------------------------------------------------------------(.6B(

HR#,E9(G+E=---------------------------------------------------------------------------(.B7 (

Claverie-------------------------------------------------------------------------------------(.6F(

H"#:98;9(4+,*8+"----------------------------------------------------------------------(6.(

4+$>"9U----------------------------------------------------------------------------(.0'(.B0(

4+$>+,9,*(.A'(.B'(.6'(.7'(.0'(F.'(FF'(FA'(B0'(7/'(76'(./.'(.B0'(.6/'(.6?'(.6@(

4+$>+,9,*=------------------------------------------------------------------------------(.?(

4+$>89==;%;";*K-----------------------------------------------------------------------(60(

concentration dependence-----------------------------(.@'(FA'(F?'(67(

4+,49,*8#*;+,(&;=*8;%)*;+,---------------------------------------------(.6/(

4+,M;&9,49("9:9"---------------------------------------------------------(..6'(..7(

Control.Extended:------------------------------------------------------------(.@/(

4+,*8+"(M;"9----------------------------------------(.F'(B.'(BA'(7.'(0A'(.@@(

4+,*8+"(=4899,(.B'(FF'(AF'(A6'(A7'(B.'(BA'(B?'(BB'(B6'(B7'(B0'(76'(70'(0F'(06'(07'(00'(.//'(./F'(./?'(..6'(..0'(.A6'(

.??'(.?@'(.?0(

H+,*8+"(J4899,---(.F'(.A'(FF'(FA'(B?'(0A'(..?'(.A6'(.@@(

H8+==(V98$(P+,<;&9#";*K-------------------------------------(F?'(F6'(67(

WHWV------------------------------------------------------------------(.F'(A7'(@A'(..0(

WHWV(>89M989,49=---------------------------------------------------------------(.?.(

&94+,:+")*;+,-----------------------------------------------------------------------(.A?(

&9E899=(+M(M899&+$------------------------------------------------------------(.B0(

density increment-------------------------------------------------------------(.?'(B6(

W9:9"+>98(O>*;+,=--------------------------------------------------------------(.A0(

&9:;#*;+,=----------------------------------------------------------------------------------(0?(

&;MM989,49(4)8:9-----------------------------------------------------------(0?'(.6/(

&;MM989,49(4)8:9=----------------------------------------------------(6'(0?'(.6/(

&;MM989,*;#"(9X)#*;+,=------------------------------------------------(6?'(.6F(

&;MM)=;+,(4+9MM;4;9,*--------------------------------------------------------------(F?(

&;$98-----------------------------------------------------------------------------------------(.B0(

&;894*+8K(=*8)4*)89-------------------------------------------------------------(7'(0(

&)$$K(M;"9--------------------------------------------------------------------------------(A7(

&K,#$;4(&9,=;*K(&;=*8;%)*;+,=--------------------------------------(.?/(

Eqn--------------------------------------------------------------------------------(AF'(B6'(B7(

IYP(%)**+,------------------------------------------------------------------------(.6'(.7(

9X)#*;+,(9&;*+8--------------------------------------(AA'(B7'(B0'(6/'(.B?(

IX)#*;+,(I&;*+8--------------------------------(.6'(.7'(AF'(B6'(B7'(7B(

9X);";%8;)$------------------------------------------------------(.0'(F.'(B6'(.B0(

9X);";%8;)$(4+,=*#,*----------------------------------(.0'(F.'(B6'(.B0(

9X);";%8;)$(4+,=*#,*=-------------------------------------------------------(.B0(

988+8(%#8=---------------------------------------------------------------------(.FB'(.F7(

I88+8(I=*;$#*;+,(H+,*8+"---------------------------------------------------(6.(

9U>#,&-----------------------------------------------------------------------------------------(?0(

9U>98;$9,*-------------------------------------(7'(0'(AB'(B.'(BA'(7/'(.@@(

9U>98;$9,*(M+"&98---------------------------------------------------(7'(AB'(.@@(

extinction coefficient------------------------------------(.@'(F.'(B6'(.6/(

M;,;*9(9"9$9,*(=+")*;+,=------------------------------------------------------(7?(

Z;*(D89>8+49==9&(W#*#---------------------------------------------------------(B.(

Z;**;,E(M"+*#*;+,(&#*#----------------------------------------------------------(..0(

Z;**;,E(M+8(IU*;,4*;+,(H+9MM;4;9,*=-------------------------------(..0(

Z;**;,E(>89M989,49=-------------------------------------------------------------(.@?(

Z;**;,E(V#%---------------------------------------------------------------------------------(6@(

M")+89=49,49(&#*#--------------------------------------------------------(AB'(.??(

M+8Q#8&(8#*9----------------------------------------------------------------------(.0'(B6(

M+8Q#8&(8#*9(4+,=*#,*--------------------------------------------------(.0'(B6(

M8#4*;+,#"(4R#,E9-------------------------------------------------------------------(6F(

frictional coefficient----------------------------------------(.@'(FA'(F ?'(67(

M8;,E9(&;=>"#49$9,*=---------------------------------------------------------(.6/(

fringes---------------------------------------------------------(.@'(AB'(?7'(? 0'(.6/(

Z<=*#*(>8+E89==----------------------------------------------------------------------(..7(

Z<=*#*(=9#84R--------------------------------------------------------------------------(.?/(

Z<=*#*;=*;4=--------------------------------------------------(6.'(..A'(..6'(..7(

F-statistics.monitoring.window------------------------------(..0(

E9"(M;"*8#*;+,---------------------------------------------------------------------------(.B0(

T9,98#"(D89M989,49=-----------------------------------------------------------(.A7(

T"+%#"(M;**;,E-----------------------------------------------------------------------------(7?(

T"+%#"(Z;**;,E(+M([)"*;<Q#:9"9,E*R(W#*#------------------(./6(

E"+%#"(>#8#$9*98--------------------------------------------------------------------(76(

T8#"\,------------------------------------------------------------------------------------------(67(

E8#K---------------------------------------------------------------------------------------(.B'(70(

E8;&(&9,=;*K-------------------------------------------------------------------------------(7?(

E8;&(=>#4;,E'(:#8;#%"9-----------------------------------------------------------(6A(

Haschemeyer, Rudy-------------------------------------------------------------(.6F(

S9">(M;"9-----------------------------------------------------------------------------------------(7(

SIGD(M;"9----------------------------------------------------------------------------------(.6B(

S9*98+<C==+4;#*;+,-------------------------------------------------------------(.6?(

R;,E9(>+;,*--------------------------------------------------------------------------------(AB(

;&9#"--------------------------------------------------------------------------------------(.@'(.6(

N,&9M;,;*9(=9"M<#==+4;#*;+,-------------------------------------------------(A/(

;,;*;#"(9X);";%8#*;+,(=*9>--------------------------------------------------(.@7(

initial guesses--------------------------------------------------------------------(.F'(0?(

NPJVCGGCVNOP --------------------------------------------------------------------------(7(

#"*98,#*9($9*R+&=--------------------------------------------------------------(7(

&9M#)"*--------------------------------------------------------------------------------------(7(

integer.fringe.jumps---------------------------------------------------------(A0(

;,*9E8#"(M8;,E9(#&])=*$9,*------------------------------------------------(?.(

;,*9E8#"(M8;,E9(=R;M*=-------------------------------------------(?.'(?0'(.6/(

N,*9E8#"(M8;,E9(=R;M*=-------------------------------------------------------------(?0(

;,*9,=;*K(&#*#---------------------------------------------------------------------------(AB(

;,*98#4*;,E(=K=*9$=------------------------------------------------------------(.B0(

;,*98M989,49----------------------------------------------------------(6'(.@'(AB'(A0(

;,*98>+"#*9&------------------------------------------------------------------------------(7?(

;=+&9=$;4---------------------------------------------------------------------------(A/'(70(

;=+9,*R#">;4---------------------------------------------------------------(A/'(A.'(0F(

iterations----------------------------------------------------------------------------------(B6(

jitter---------------------------------------------------------------------------(AB'(?7'(.6/(

^;**98(#&])=*-------------------------------------------------------------------------------(A0(

39K%+#8&(=R+8*<4)*=-------------------------------------------------------------(.F(

_9K%+#8&(JR+8*4)*=-----------------------------------------------------------(.BB(

3;,9*;4------------------------------------------------------------------------(B6'(6@'(.6F(

_;,9*;4(;,*9E8#*+8(4+,*8+"--------------------------------------------------(6?(

3;,9*;4=-------------------------------------------------------------------------(6'(6?'(.@7(

Kinetics---------------------------------------------------------------------------(B6'(.@7(

3;,9*;4=(=;$)"#*+8---------------------------------------------------(6'(.F'(.@0(

_;,9*;4=(J;$)"#*+8------------------------------------------------------(.F'(.@0(

_=(FA'(F@'(FB(

Ks.and.BM1-----------------------------------------------------------------------------(F6(

G#$$-----------------------------------------------------(6'(B6'(6.'(6@'(7?'(.6F(

"9#=9(=X)#89=----------------------------------------------------------------------------(?7(

"9#=*(=X)#89=(M;* * ;,E------------------------------------------------------------(.6/(

G9:9,%98E<[#8X)#8&*---------------------------------------------------------(6@(

"+#&;,E(4+,49,*8#*;+,------------------------(B6'(7/ '(76'(77'(.@@(

G+4#*9($9,;=4)=---------------------------------------------------------------------(?0(

[#;,([9,)--------------------------------------------------------------(A?'(B.'(.@@ (

$#*8;U-------------------------------------------------------------------(F6'(F7'(AF'(67(

[#*8;U------------------------------------------------------------------------------------------(F7(

$9,;=4)=-----------------------(6'(..'(AB'(?F'(?7'(?0'(B6'(7?'(.@@(

[9,;=4)=------------------------------------------------------------------------------------(?.(

$;,------------------------------------------------------------------------------------------------(0?(

[;,;$)$(M;"9=`----------------------------------------------------------------------(.@/(

model--(.F'(.A'(.@'(.B'(.6'(FA '(?0'(BA'(B?'(B6'(.@@'(.6F(

[+&9"(I&;*+8--------------------------------------(.?'(.7'(FF'(FA'(B6'(B7(

[+&9"(*+(%9(M;**9&--------------------------------------------------------(BA'(.@@(

ModelEditor(7'(0'(..'(.F'(.A'(.@'(.6'(.7'(FF'(A.'(B7'(B0'(0F'(.A7'(.@0(

$+"#8($#==--------------------------------------------------------------------------------(B6(

molecular weight------------------------------------------------------(.?'(.@'(.B(

$+,+$98---------------------------------------------------------------------------------(.B0(

$+,+$98<&;$98------------------------------------------------------------------(.B0(

[+,*9(H#8"+-----------------------------------------------(6.'(..A'(..@'(..B(

$)"*;>"9(+>*;4#"(=K=*9$=----------------------------------------------------(7?(

Multi-wavelength----------------------------------------------------------------(AB(

[)"*;<Q#:9"9,E*R(W#*#-----------------------------------------------------(.A/(

[)"*;<Q#:9"9,E*R(Z")+899,49(;,*9,=;*K(&#*#---------(.A@(

P+,<;&9#"---------------------------------------------------------------------------------(.6@(

non-ideality-----------------------------------------------------------------(.@'(F@'(FB(

P+,;&9#";*K([#*8;49=------------------------------------------------------------(F7(

P+,<;,*98#4*;,E(=K=*9$=----------------------------------------------------(AF(

,+,<";,9#8("9#=*(=X )#89=-----------------------------------------------------(B6(

normalization.of.g(s*)---------------------------------------------------(.F.(

,)$98;4#"(&;MM989,*;#*;+,------------------------------------------------(.F6(

Numerical Recipes in FORTRAN-----------------------(.6F'(.6A(

OWI---------------------------------------------------------------------------------------(B6'(6?(

optical system-----------------------------------------------------------------(.@'(.@@(

O>*;$#-------------------------------------------------------------------------------------------(6(

O8&;,#8K(W;MM989,*;#"(IX)#*;+,------(J99(OWI'(J99(OWI(

+)*>)*(M;"9----------------------------------------------------------------------------------(0?(

>#43#E9------------------------------------------------------------------------------------(.@7(

parameter---------------------------------------------------(.@'(.B'(FF'(?/'(.6/(

>#8#$9*98=(6'(..'(.F'(.A'(.?'(.@'(.B'(F.'(FA'(B6'(6.'(6?'(67'(7?'(76'(0?'(.@@'(.6F(

partial specific volume---------------------------------------------------(.@'(F.(

D98*)8%(M;*(#,&(89<&+------------------------------------------------------------(6B(

>"#*9#)---------------------------------------------------------------------------------(AB'(?0(

D89M989,49=--------------------------------------------------------(7'(0'(.A6'(.?/(

preprocessed data--------------------------------------------------(..'(?0'(.@@(

preprocessor-------------------------------------------------------------------------(7.(

D89>8+49==+8-------------------------------------------------------------------------(.A?(

>89==)89(&9>9,&9,49(+M(:;=4+=;*K---------------------------------(7/(

D8+*9+$9G#%------------------------------------------------------------------------------(6(

>=9)&+<#%=+8%#,49--------------------------------------------------------------(AB(

range-------------------------------------------------------(..'(.B'(FF'(AB'(?F'(?0(

8#>;&"K(89:98=;%"9----------------------------------------------------------------(.B0(

8#*9(4+,=*#,*=------------------------------------------------------------------(B6'(6@(

reaction-------------------------------------------------------(..'(.A'(F.'(B6'(.B0(

reduced.chi-squared---------------------------------------------------------(7F(

89$+:9(&#*#(=9*----------------------------------------------------------------------(7.(

89=4#"9&--------------------------------------------------------------------------------------(?F(

89=;&)#"=--------------------------------------------------------------------------------(6'(0?(

89:98=9(8#*9-----------------------------------------------------------------------(.0'(B6(

89:98=9(8#*9(4+,=*#,*---------------------------------------------------(.0'(B6(

Richardson extrapolation-----------------------------------------(.6F'(.6A(

a;4R#8&=+,(8#*;+,#"(>+"K,+$;#"(9U*8#>+"#*;+,-------(6?(

8$=(&9:;#*;+,---------------------------------------------------------------------------(FF(

8$=&--------------------------------------------------------------------------------------(6'(.6F(

a+#83-----------------------------------------------------------------------------------------(.@@(

8++*($9#,(=X)#89(&9:;#*;+,---------------------------------------------(0?(

root mean square residual--------------------------------------------------(.6/(

8),(M;"9=----------------------------------------------------------------------------------(7'(7/(

a),(=48;>*--------------------------------------------------------------------------------(.BB(

J#:;,E(#(b>#43#E9c--------------------------------------------------------------(.@7(

=4#,=------------------------------------------------(?F'(?A'(?0'(7.'(.@@'(.6/(

=4#,=(*+(%9(M;**9&--------------------------------------------------------------------(?A(

J4899,(W)$>=--------------------------------------------------------------------------(0@(

=48;>*--------------------------------------------------------------------(.B?'(.B@'(.BB(

J48;>*;,E----------------------------------------------------------------------------------(.B?(

JIWCPCG----(6'(7'(0'(.@'(A?'(6?'(0?'(.@B'(.B0'(.6/'(.6F(

sedimentation coefficient---------------(.?'(.@'(. B'(.7'(.0'(B6(

=9&;$9,*#*;+,(9X);";%8;)$(6'(07'(00'(.//'(./F'(./?'(.@@(

J9"94*(=4#,=(*+(%9(>"+**9&---------------------------------------------------(?A(

=9"94*(d#:9"9,E*R=(M+*(M;**;,E----------------------------------------(./7(

=9"M(#==+4;#*;+,------------------(.A'(A/'(0.'(0F'(.@A'(.B0'(.6?(

Simulate data---------------------------------------------------------------------(.F'(BA(

=;$)"#*;,E(&#*#---------------------------------------------------------------------(.@@(

="#:9&(49""(&#*#(=9*=--------------------------------------------------------------(7B(

J$++*R;,E-------------------------------------------------------------------------------(.F6(

=>94;9=(.A'(.@'(.B'(.6'(.7'(.0'(F.'(FF'(FA'(F@'(FB'(B6'(6@'(.@@'(.B0(

=>94*8#--------------------------------------------------------------------(@F'(./6'(.A/(

J>94*8#--------------------------------------------------------------------------------------(.A/(

=*+4R#=*;4--------------------------------------------------------------------------------(.6/(

=*+89------------------------------------------------------------------------------(7'(0A'(.@@(

JK,*R9*;4(L+),&#8K------------------------------------------------------------(.A@(

systematic error--------------------------------------------------------------(..'(.6/(

V9=*;,E(M+8(IX);";%8;)$------------------------------------------------------(@@(

*9*8#$98-----------------------------------------------------------------------------------(.B0(

*;$9(4+)8=9-----------------------------------------------------------------------------(.@7(

time increment-----------------------------------------------------------(.F'(6 .'(7?(

*;$9(;,&9>9,&9,*----------------------------------------------------------------(.6/(

Todd, Peter------------------------------------------------------------------------------(.6F(

)=98e=)MM;U---------------------------------------------------------------------------------(@/(

virial coefficient---------------------------------------------------------------(.@'(FA(

:;8;#"(4+9MM;4;9,*=----------------------------------------------------------------------(6(

Q#88#,*K---------------------------------------------------------------------------------------(6(

wavelengths-------------------------------------------------------------------------------(.@(

dWC---------------------------------------------------------------------------(.F'(@A'(..0(

dWC`(IU*8#4*;,E(J>94*8#-------------------------------------------------(.A/(

Q9;ER*;,E(M#4*+8--------------------------------------------------------------(7.'(7F(

d9;ER*;,E(Z#4*+8=-----------------------------------------------------------------(7.(

d;&9(W;=*8;%)*;+,(C,#"K=;=------------------------------------(@A'(.FF(

d8;*9(#&])=*9&(&#*#(M;"9------------------------------------------------------(@/(

Yphantis-------------------------------------------------------------------------(@0'(.@@(

f++$----------------------------------------------------------------------------------(?7'(.F/(

8 LIST OF FIGURES

)<=>?@"ABA"5@CC<D=">E"EFCGH"CI"CG@"JFCF"K<L@H"FDJ"CI"CG@"MIJ@L"K<L@N"

Figure'1-2'Default'path'structure'for'SEDANAL'initial'default'installation"

)<=>?@"OBA"0$,1"0(17"

)<=>?@"PBA"6G@"0IJ@L"1FM<D="CFQN"

)<=>?@"PBO"6G@"HE@R<@H"CFQ"

Figure'3-3'Molecular'Parameters"

)<=>?@"PBSN"Screen'showing'default'fitting'parameters'for'a'One'Component'-'Single'Ideal'Species'Model."

Figure'3-5'One'component,'two'species'model."

Figure'3-6''Screen'showing'parameters'for'the'Two'Independent'Ideal'Species'Model."

Figure'3-3-7.'Model'Editor'Species'tab'showing'parameters'for'the'Three'Independent'Ideal'Species'Model."

Figure'3-3-8.''Screen'showing'parameters'for'the'Interacting'Species'Model"

Figure'3-3-9.''Species'tab'showing'parameters'for'Species'3'(“C”)."

Figure'3-3-10'Selection'of'species"

Figure'3-3-11.''Parameters'for'2'component,'4'species'interacting'model:'A'+'B'='AB,''AB'+'B'='AB

Figure'3-12.'Selection'of'parameters'in'the'control'box'for'three'component'system"

Figure'3-13'Higher'order'non-ideality'terms"

)<=>?@"PBAS"5@L@RC<ID"IK"EF?FM@C@?H"<D"CG@"RIDC?IL"QIT"KI?"H<D=L@"HE@R<@H"U<CG"DIDB<J@FL<CVN"

)<=>?@"PBAWN"""5@L@RC<ID"IK"EF?FM@C@?H"KI?"CUI"HE@R<@H"U<CG"DIDB<J@FL<CVN"

)<=>?@"PBAX''“Copying”'values'for'“Species'2”'from'“Species'1”"

Figure'3-17'Base'Concentration'Parameters"

Figure'3-18.''Setup'for'indefinite'self-association."

Figure'3-19.'Blow-up'of'upper'left'of'the'reactions'tab."

Figure'3-20.'Isodesmic'case"

Figure'3-21.'SEDANAL'will'determine'whether'the'data'are'from'SedVel'or'SedEq"

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)<=>?@"PBSZ""5RFDH"K?IM"H@_@?FL"UF_@L@D=CGH"

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)<=>?@"PBZP.'Control'of'the'ki netic'integrator."

)<=>?@"PBZS.''Choosing'fitting'parameters."

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Figure'3-76'Setting'Compressibility'for'each'cell."

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Figure'3-91"6G@"LIFJ<D="RIDR@DC?FC<ID"?FC<I"<H"@DC@?@J"FH"AN]"<D"CG<H"@TFMEL@N"

Figure'3-92."$"H<D=L@"L@KCBRL<Rf"ID"CG@"C@TC"g/IFJ<D="RIDR"?FC<I"%h$e"E?IEF=FC@H"CG@"CIE"A"<DCI"CG@"QIT@H"Q@LIU"U<CG"

F"V@LLIU"QFRf=?I>DJ"CI"<DJ<RFC@"CGFC"CG@H@"_FL>@H"F?@"J@?<_@J"K?IM"R@LL"iA"FDJ"U<LL"Q@"f@EC"@a>FL"CI"FDJ"_F?V"U<CG"

CG@"_FL>@"KI?"R@LL"iA"FH"F"H<D=L@"=LIQFL"EF?FM@C@?N"

Figure'3-93."$"H@RIDJ"RL<Rf"ID"CG@"C@TC"g/IFJ<D="RIDR"?FC<I"%h$e"C>?DH"CG@"QFRf=?I>DJH"QFRf"CI"=?FV"CI"<DJ<RFC@"

CGFC"CG@H@"_FL>@H"U<LL"Q@"DIU"Q @"F LLI U@J"CI"KLIFC"<DJ@E@DJ@DCLV"KI?"FLL"R@LLHd"

Figure'3-94."4<=GC"RL<Rf<D="ID"ID@"IK"CG@"=?FV"QIT@H"U<LL"C>?D"<CH"QFRf=?I>DJ"QL>@"FDJ"DIU"<CH"_FL>@"UI>LJ"Q@"G@LJ"

RIDHCFDC"J>?<D="CG@"K<Cb"<K"CGFC"U@?@"J@H<?@Jd"

)<=>?@"PB\W",HIJ@HM<R"MIJ@L"H@L@RC<IDN"

)<=>?@"PB\X.'"6G<H"@TFMEL@"<H">H<D="CG@"H#<BM@?`"j"H#MIDIM@?`k"#<`

2/3

)<=>?@"PB\Z",HI@DCGFLE<R"<DJ@K<D<C@"H@LKBFHHIR<FC<ID"

)<=>?@"PB\[N""Entering'parameters'for'indefinite'self-association"

)<=>?@"PB\\N""6G@"?FD=@"IK"FLLIUFQL@"_FL>@H"

Figure'3-100.'Start'the'fit"

)<=>?@"PBA]AN"")<CC<D="HR?@@D"J<HELFV"KI?'S"R@LL"=LIQFL"K<C"CI"CG@"HVHC@M"$l%j&"

)<=>?@"PBA]ON"5R?@@D"'>MEH"

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)<=>?@"PBA]W"4@H<J>FL"ELICH"

Figure'3-106''The'main'control'screen'for'the'fitting'preprocessed'data"

Figure'3-107.''The'main'control'screen'for'the'fitting'of'preprocessed'data'for'a'sedimentation'equilibrium'

run."

)<=>?@"PBA][N'The'control'screen'for'the'fitting'an'equilibrium'run'for'the'model'A'+'B'=C."

Figure'3-109.'The'main'control'screen'for'the'fitting'of'preprocessed'data'for'a'sedimentation'equilibrium'

run.'Here'we'will'float'the'molar'mass,'the'loading'concentration'and'the'y-offset."

Figure'3-110.''Final'fit'to'equilibrium'data'after'convergenceN"

)<=>?@"PBAAA.'Three-cell'global'fit'to'sedimentation'equilibrium'data"

)<=>?@"PBAAO.'The'main'cont rol'screen'for'the'fitting'preprocessed'data'for'sedimentation'equilibrium'run."

Figure'3-113'Fitting'with'linked'cells"

)<=>?@"PBAAS.'6G@"MF<D"RIDC?IL"HR?@@D"KI?"CG@"K<CC<D="E?@E?IR@HH@J"JFCF"KI?"H@J<M@DCFC<ID"@a><L<Q?<>M"?>DN"

Figure'3-115.'Fit'with'unlinked'cells."

Figure'3-116''Fitting'for'the'base'radius"

Figure'3-117'Fitting'for'R

.'with'linked'cells"

Figure'3-118.'Allowing'the'ratio'B/A'to'vary"

)<=>?@"PBAA\"5@L@RC"EFCGH"CI"CG@"@TC<DRC<ID"RI@KK<R<@DC"HE@RC?F"

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)<=>?@"PBAOO.'Error'estimatio n'control"

)<=>?@"PBAOP.'Error'estimatio n'-'Monte'Carlo"

)<=>?@"PBAOS.'Error'estimatio n:'F-statistics."

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Figure'3-127.'F-statistics'monitoring'window."

Figure'3-128'DCDT'header'from'v7.43"

Figure'3-129'(left)'dc/dt.'(right)'g(s*)"

Figure'3-130.'The'weight'average'sedimentation'coefficient"

Figure'3-131'The'corresponding'averages'appear'highlighted'in'yellow'in'the'adjacent'boxN"

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Figure'3-137.'Results'of'WD'analysisN"

Figure'3-138.''Control'box'for'picking'bad'scans'in'WDAN"

Figure'3-139'Radii'available'for'WD'analysis"

Figure'3-140.'Flotation'button'in''DCDT/WD"

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Figure'3-142.'Selecting'peaks'for'spectrum'extraction:"

Figure'3-143'Three'peaks'selected'for'only'three'wavelengthsN"

Figure'3-144'Click'the'"Select'all"'button'to'plot'all'the'WD'curves'for'all'wavelengths."

Figure'3-145.'WD'curves'for'all'wavelengthsN"

Figure'3-146'Spectra'gathered'from'the'WD'plot"

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Figure'3-156.''Preferred'settings'for'BIOSPIN"

Figure'3-157.'Parameters'for'the'control'screen:'setting'defaults'for'use'of'weighting'factorsN"

Figure'3-158.''"Control'extended"'param e te rs'a llow'setting'of'the'default'parameters'found 'un d e r'th e '

“Advanced…”'button'on'the'control'screen."

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Figure'3-162.'Choice'if'default'fitting'parametersN"

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Figure'3-165.'Outputs:'choose'the'“min”'file'formats"

Figure"PBAXX.'Ancillary'output'files"

Figure'3-167.'Kinetics'data'for'the'model'under'considerationN"

Figure'3-168.'Disposition'of'various'log'files"

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)<=>?@"PBAZS.'Generate'a'Package"

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Figure'3-181'Plot'with'same'equilibrium'constants;'but'rate'constants'for'the'second'step'are'1/10'those'of'

the'first'example'above'(Figure'3-175)."

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Figure'3-183.'Script'control"

Figure'3-184'SEDANAL'On-line'Help"

Figure'3-185'Change'Log"

Figure'4-1'Schematic'of'the'procedure'used'by'SEDANAL'for'fitting'time'difference'curves"