Manual

Finite Element Analysis - Tutorial:

Selecting the proper Model:

The first step in the finite element analysis of sedimentation velocity or approach-to-equilibrium data is the definition of an appropriate model. Unfortunately, the most difficult step in this procedure is to decide what exactly the appropriate model is. To assist you in this decision, you should analyze the corresponding velocity data with the van Holde - Weischet Analysis method first. A good description for interpreting experimental velocity data by this method can be found in Demeler et al., 1998.

As with all methods of fitting imperfect data to a model, the solution space is degenerate and often many different models, though similar, can result in the same chi-square value. There are several ways to check if a particular model is appropriate, the most obvious method is to check the residuals of the fit. Unless the residuals are randomly scattered about zero and don't show any systematic drift or runs, you should not use the results, since they will be meaningless.

The other important rule to keep in mind is that the simplest model that fits the data is generally more correct than an overdetermined complex model. The fewer parameters necessary to fit a dataset, the better. In order to improve the confidence in the results, it is strongly recommended to utilize scans from the early beginning of the experiment until late into the run when the material is almost pelleted. Future versions of UltraScan will include the ability to fit multiple experiments under different conditions to enhance the information content of the fitted data.

Even if a model results in randomly scattered residuals, this is not a guarantee that the model is representative of the data. For example, a gaussian distribution of s-values due to a gaussian distribution of particle sizes (a common occurence) may be approximated by an artificially high diffusion coefficient, since the boundary spreading due to heterogeneity will have a similar effect on the shape of the boundary as the spreading of the boundary due to diffusion. On the other hand, an overdetermined model with too many parameters may explain the experimental data equally well, but of course is totally meaningless. For a single velocity experiment, three separate components may be the maximum of what can be reliably fitted. Always, the results obtained with the finite element analysis need to be carefully cross-checked against other methods before taken for granted.

Improving the confidence of the Results:

There are several ways to improve the confidence of your results obtained from finite element fitting. The most obvious method of improving the confidence of individually fitted parameters is to reduce the number of parameters fitted simultaneously. For example, if you have measured the molecular weight of a sample by mass spectroscopy or gel electrophoresis, or if the molecular weight is known from the sequence of the protein or nucleic acid, the molecular weight could be fixed during the fit, which will also fix the diffusion coefficient and constrain the fit. Another method for improving the confidence of the fit is to utilize only clean and noise-free data. The higher the signal to noise ratio, the better the confidence in the fitted results. The same holds true for data obtained from a wider range of experimental conditions. A particular model needs to hold for all conditions (such as varying the speed or loading concentration) and should ideally result in the same result. If the results differ from run to run, the model is most likely inappropriate.

Fitting Strategies:

In order to succeed in fitting sedimentation data to finite element solutions, it helps to follow the steps outlined below:

• It is generally best to start with a simple model, such as a single ideal, non-interacting system.
• Obtain initializing guesses from the van Holde - Weischet analysis
• Start with a large time discretization for an initial approximation of the the data (it goes faster), then refine the model with a smaller stepsize once the fitted data is close to the final answer. Use the unconstrained method for unknown data and refine the fit with the constrained method. Make sure the constraints are not too close to the final answer, or the solution matrix may turn singular.
• Float as few parameters as possible to assure a high confidence in each
• Try different starting guesses to make sure the solution converges always to the same answer
• If at first the fit doesn't succeed, try a different model and/or different initialization parameters.
• Utilize information known from other experiments to supplement the information gained from finite element. For example, measure the partial specific volume of the sample separately to constrain the molecular weight.