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allows the selection of the sedimentation analysis model.

Sedfit contains numerous
models and methods for data analysis. Central are the ** direct boundary
models**. They follow the strategy of expressing all sedimentation processes by the
underlying equations, and of directly modeling the raw data by least-squares
regression. When necessary, the noise structure of the data is also
explicitly modeled by least squares. Because this concept makes it necessary to specify the type of sedimentation
process, many different models are needed to accommodate the different
experimental situations encountered. Also, different models can be used
reflecting the different knowledge about the sample (e.g., number of
species).

In the following, a brief overview of the models is given, but reading the more specific information following the links below is highly recommended.

Most of the models are based on numerical solutions of the Lamm
equation^{1}, which is the partial differential
equation describing the evolution of concentration distribution with time (see introduction
to Lamm equation solutions).
The Lamm equation is very general, and can be used to analyze species as small
as salts up to virtually unlimited size, can describe sedimentation and flotation, allows the
incorporation of the rotor acceleration into the model, allows modeling of experiments using layering techniques, and more. The numerical algorithms
for solving the Lamm equation are optimized for speed and precision, and do not
require any user intervention (although all relevant parameters are available
for change for the interested user). They use the finite element technique. For
small s-values, it follows the strategy originally described by Claverie^{2},
but combined with a Crank-Nicholson scheme^{3} for
increased precision and a more efficient adaptive driver of the
time-steps. Details can be found in ref. 4. For higher
precision and speed at large s-values, Sedfit
uses a new finite element approach with a moving frame of reference^{5}
, also combined with Crank-Nicholson scheme. There is an automatic
switchover between the methods. Systematic noise is calculated algebraically^{6}.

Models are available for independent ideal species, species with a continuous size-distribution, species in rapid self-association equilibrium (monomer-dimer, monomer-trimer, monomer-dimer-tetramer, and monomer-tetramer-octamer), or non-ideal concentration-dependent sedimentation of the type s(c) = s0/(1+kc) (including terms for the concentration dependence of D).

For the independent ideal species, either a known discrete number of species
can be used, or distribution functions for an unknown number of species are
available^{7}. There is a large variation of the
distribution functions that can be selected according to the different prior
knowledge available. Such prior can be knowledge of the approximate shape
of the molecules, their diffusion coefficient, partial-specific volume, the
molar mass for molecules that can undergo a conformational change, etc.
Further, the distribution can be either a sedimentation coefficient (or
flotation coefficient) distribution c(s) or a molar mass distribution c(M)^{7}.

In a typical scenario with a previously uncharacterized sample, one would for example do a c(s) analysis first, using some knowledge about the nature of the molecules to obtain information on purity, s-values, number of species and relative amounts. (For a detailed description on how to perform this analysis, see the help-page of this model). Usually, it is very useful to run the same sample at a range of concentrations and at different rotor speeds. This will reveal if the molecules can be described as ideal non-interacting particles, or if there is a concentration dependence for an associating system, and the presence of hydrodynamic nonideality. After that, for non-interacting species, one would use the information on the number of species and switch to the discrete species Lamm equation model. Here, the s-values from the c(s) analysis can be used, and either constraining the molar mass to that known from sequence (or hypothesized for the oligomers) or fitting for the molar mass. (If the molar mass is the quantity of interest, a relatively low rotor speed is most useful because it allows for considerable diffusional spreading, while highest resolution and precision in the sedimentation coefficient (and molecular shape) is usually obtained at the highest possible rotor speed.) Alternatively, if there is knowledge on the molecules, their self-association, conformational change, etc, one would of course choose one of the specific models.

In part for comparison with previous sedimentation analysis strategies, and
to complete the tool-box for velocity analysis, the
g*(s) method and the van Holde-Weischet G(s)^{8} is
available. While I would recommend using direct boundary modeling for
quantitative analysis whenever possible, the g*(s) and G(s) transformations can
be very useful as additional tools, for example, for diagnostic purpose and
semi-quantitative description of the data, in particular where an appropriate
explicit boundary model is not yet available.

The g*(s) method is calculated in Sedfit
by least squares and direct boundary modeling, termed ls-g*(s)^{9}.
This method is not limited to small data sets, as no approximation of dcdt is
involved, and can therefore be used conveniently with both absorbance and
interference data. ls-g*(s) is a very good model for the sedimentation of
very large particles where diffusion can be neglected. The extrapolation
of ls-g*(s) to infinite time^{10} is a link between
g*(s) and G(s), and can be considered an extension of the van Holde-Weischet
method to interference data containing systematic noise components.

Some other methods closely related to conventional sedimentation velocity analysis
are also included in Sedfit.
These are those for analytical zone centrifugation^{11} (sedimentation velocity using
a synthetic boundary layering technique), dynamic light scattering (substituting
the Lamm equation solutions by appropriate decaying exponentials for the
autocorrelation functions), and electrophoresis (obeying a Lamm equation in
linear geometry with constant force). Finally, some special modifications
for time-dependent sedimentation are available.

**References:**

1) O. Lamm. (1929) Die Differentialgleichung der Ultrazentrifugierung. Ark. Mat. Astr. Fys. 21B:1-4

2) J.-M. Claverie, H. Dreux and R. Cohen. (1975) Sedimentation of generalized systems of interacting particles. I. Solutions of systems of complete Lamm equations. Biopolymers 14:1685-1700

3) J. Crank and P. Nicholson. (1947) A practical method for numerical evaluation of solutions of partial differential equations of the heat-conducting type. Proc. Cambridge Philos. Soc. 43:50-67

4)
P. Schuck, C.E. McPhee, and G.J. Howlett. (1998) Determination of
sedimentation coefficients for small peptides. * Biophysical Journal*
74:466-474.

5)
P. Schuck (1998) Sedimentation analysis of non-interacting and
self-associating solutes using numerical solutions to the Lamm equation. * Biophysical
Journal*
75:1503-1512.

6) P. Schuck and B. Demeler (1999) Direct sedimentation analysis of
interference-optical data in analytical ultracentrifugation. * Biophysical
Journal*
76:2288-2296.

7) P. Schuck (2000) Size distribution analysis of macromolecules by
sedimentation velocity ultracentrifugation and Lamm equation modeling. * Biophysical
Journal*
78:1606-1619.

8) K.E. van Holde and W.O. Weischet. (1978) Boundary analysis of sedimentation velocity experiments with monodisperse and paucidisperse solutes. Biopolymers 17:1387-1403.

9)
P. Schuck and P. Rossmanith (2000) Determination of the sedimentation
coefficient distribution g*(s) by least-squares boundary modeling. * Biopolymers*
54:328-341.

10) in preparation

11)
J. Lebowitz, M. Teale, and P.W. Schuck (1998) Analytical band
centrifugation of proteins and protein complexes. * Biochemical Society
Transactions* 26:745-749.