D. Speed selection and length of an experiment Equilibrium Experiments: ● The speed in equilibrium experiments determines the steepness of the equilibrium gradient. By performing equilibrium experiments at multiple speeds, it is possible to generate multiple representations of the identical sample which can be fitted globally for better precision. ● Speed selection is dictated by the molecular weight of the sample, and the range of possible and reasonable speeds is given by the reduced molecular weight, sigma. The speeds should be chosen such that the sigma value is between 1 and 4 for the smallest component in the system (generally the monomer molecular weight, or the smallest associate in the system) Experimental Design and Data Collection

D. Speed selection and length of an experiment Equilibrium Experiments: ● Equally space 4-5 speeds between sigma limits of 1-4 ● Use the “Equilibrium:Suggest Best Speed” module to estimate the speed appropriate for an experiment DEMO Equilibrium Speed Selection Experimental Design and Data Collection

D. Speed selection and length of an experiment Equilibrium Experiments: ● The length of an equilibrium experiment depends on several factors: column height (discussed below), speed in the approach to equilibrium, and the diffusion properties of the molecule. To predict the approximate time for reaching equilibrium at multiple speeds use the “Simulation:Model s, D and f from Molecular Weight for 4 basic shapes” module to first estimate the molecular parameters for your sample, and then use the “Simulation:Estimate Equilibrium Times” module to estimate the time it takes to reach equilibrium at each speed. Always estimate the largest component in the system, since this component will have the smallest diffusion coefficient. You can also overestimate the axial ratio to be on the safe side. To be certain that equilibrium has been reached, take multiple scans spaced 4-6 hours apart and see if they remain unchanged. DEMO Equilibrium Time Simulation Experimental Design and Data Collection

E. Column height ● For velocity experiments, the full column length should always be used. Obviously, the longer the sample column, the more datapoints are available for fitting. Naturally, more data is always better than less data. Therefore, the maximum column height should always be used. When total sample is limiting, a 1.4 cm column experiment with a diluted sample at 0.45 OD is much more informative than a concentrated sample at 0.9 OD with a 0.7 cm column length. ● Column height has a large effect on the time it takes to reach equilibrium. The longer the column, the longer the experiment. Too short a column doesn't provide enough datapoints. A good compromise is to use a mm column height, which translates into a loading volumne of approximately μl. Equilibrium experiments should be performed at multiple concentrations to obtain additional equilibrium profiles and to span a better signal range for all associated multimers in the sample. Experimental Design and Data Collection

F. Concentration selection ● For velocity experiments, a loading concentration between OD is recommended. ● For interference experiments, at least 1 mg/ml is desirable to reduce time- invariant noise from refractive index in-homogeneities in the cell windows. ● For equilibrium experiments, loading concentrations of 0.3, 0.5 and 0.7 OD at each wavelength are recommended. ● In order to maximize the signal from all species in a reversibly self- associating sample, it is recommended that the concentration range be made as large as possible to assure that sufficient signal from each species is present in the data. ● In order to provide maximum concentration spread in the analysis, it is recommended to combine data from 210, 230 and 280 nm, as well as interference in a global fit. Experimental Design and Data Collection

G. Temperature considerations Velocity experiments require a constant velocity. Therefore it is critically important to temperature-equilibrate the rotor before acceleration. This is best accomplished by letting the rotor with loaded sample cells sit in vacuum at the temperature at which the run is to be performed for at least 1 hour before the experiment is started. Other than thermal stability of a sample, there are no considerations for temperature. There is plenty of time for the rotor to temperature-equilibrate before the first equilibrium scan is taken. Experimental Design and Data Collection

H. Wavelength scans Whenever data at multiple wavelengths are measured, it is important to obtain extinction information for each wavelength, so appropriate corrections can be made when concentration dependent parameters like the association constant are determined. It is recommended to obtain wavelength scans at multiple concentrations spanning all wavelengths plus nm on either side. The wavelength scans are then fitted globally to obtain an intrinsic extinction profile which can be normalized with the known extinction at 280 nm. Use the “Utilities:Global Extinction Fit” module to determine the extinction profile. The wavelength measurements should be performed as follows: measure 3 scans of each concentration with 1 nm resolution and zero averages. DEMO: Wavelength Fitter Experimental Design and Data Collection

I. Instrument settings For UV absorbance experiments it is important that optimal data acquisition settings are selected. Options include radial resolution, the number of repeat measurements, and for wavelength measurements the wavelength resolution. Settings for each experiment: ● Wavelength measurements: Wavelength measurements should be performed as follows: measure 3 scans of each concentration with 1 nm resolution and zero averages, continuous mode. Measurement should include 20 nm above and below the desired wavelengths. ● Velocity experiments: cm resolution, no averaging, no scan delays, continuous mode ● Equilibrium experiments: cm resolution, 20 averages, step mode. Optional: repeat scans 4-6 hours apart to determine if equilibrium has been reached. ● Interference data collection for velocity experiments should be performed in continuous mode without delay between scans. Superfluous scans can be discarded later. Experimental Design and Data Collection

Time =  low rotorspeed Equilibrium

Sedimentation Equilibrium At Equilibrium, the total flow is zero, and diffusion and sedimentation cancel out: Therefore: Svedberg's Law: After Integration:

Self-Association Models Monomer+Monomer  Dimer Monomer+Dimer  Trimer Dimer+Dimer  Tetramer etc...

Monomer-Dimer Equilibrium: Adding Constraints

Linearized Models: Fixed Molecular Weight Distribution In a Fixed Molecular Weight Distribution Model all species are independent, the molecular weight is held constant and the concentration constraints are released. The only constraints are now on the linear coefficients of each molecular weight term, which is imposed by the NNLS fitting method, which doesn't allow negative concentration contributions. The model can be represented as: The zeroth-order term is simply the baseline ( M 0 = 0 ). This model is very degenerate, and equivalent to the single ideal species model if n=1 or two component model if n=2, and if the molecular weight is fixed. It is OK to fit 100 or more species. This model will give the lowest variance of all models, provided the correct molecular weight range is chosen.

Sedimentation Equilibrium Summary Sedimentation Equilibrium can be useful in the following cases: ● Molecular weight ● Self-associating systems ● Equilibrium constants ● Stoichiometry determinations It is not useful for assessing heterogeneity. Composition analysis should be performed with sedimentation velocity experiments

Global Fitting Nonlinear Least Squares Take-Home Messages: ● Reducing the number of fitted Parameters improves the confidence in the fitting results. ● Constraining Parameters with known relationships ● Constraining Parameters with other experiments ● If possible, measure a parameter rather than fitting it. ● Use many datasets from different conditions to improve confidence ● Combine multiple experiments to improve confidence ● Only use high-quality data

Nonlinear Least Squares Fitting Basics Nonlinear least squares fitting is a difficult art if done correctly. Several assumptions need to be satisfied in order to obtain reliable results: ● Experimental Uncertainties (Noise): In order for least squares fitting to succeed, the experimental uncertainties need to only distributed along the y-axis, and they have to be distributed normally, i.e., follow a Gaussian distribution. ● Residuals: Residuals HAVE to be distributed randomly, otherwise the resulting parameters are ABSOLUTELY meaningless. If the residuals are not randomly distributed about the mean, the model needs to be changed to assure that the model is appropriate for the data under investigation. ● Number of Parameters: The larger the number of parameters, the higher is the uncertainty associated with each parameter. Try to fit the simplest model first, and use as much data from different experimental conditions as possible.

Nonlinear Least Squares Fitting Basics ● Confidence: The confidence of each parameter is best ascertained with the Monte Carlo method. Simple fit variances do not reflect the confidence of each individual parameter. ● Model: Attempt to determine the correct model by first doing a different, independent experiment, which is analyzed with a model-independent method such as the van Holde - Weischet analysis. This is true for both equilibrium and finite element fitting. The following rule of thumb should be observed: The more independent experimental data are available, the more complex the model can be by maintaining the same level of confidence in the determined parameters. At the same token, a reduction in the number of parameters in the model with the same number of experimental data will generally increase the statistical confidence in each parameter.