From: John Philo <jphilo@earthlink.net>
  To  : rasmb (E-mail) <rasmb@bbri.harvard.edu>
  Date: Tue, 24 Mar 1998 15:33:09 -0800

RE: analysis of heteromeric system

In response to Yujia's comments, and with all due respect, in my view (and 
in my experience) things are generally not quite that easy.

One problem arises from the fact that NONLIN does not really deal with 
concentration units---internally it simply works in instrument units 
(fringes or absorbance).  As long as one is just doing self-association, 
there is only a single type of monomer so this doesn't really matter and 
you can convert the association constants to mg/ml or molar units at the 
end of the analysis.

However, for heteroassociations such as A + B --> AB, the equations only 
hold true in molar units (one mole of A + one mole of B gives one mole of 
AB).  It will generally NOT be true that one fringe or one AU of species A 
combines with one (fringe or AU) of B.  For heteroassociations the 
association equations and fitting functions really need to be written on a 
molar scale.

Thus I believe Yujia's suggestion of using non-integer 'n' values in NONLIN 
will work correctly ONLY with an additional assumption that all types of 
monomer contribute to the experimentally observed signal in a manner 
exactly proportional to their mass.  That is, for interference data all 
species must have the same dn/dc, and for absorbance data all species must 
have the same extinction coefficient (per milligram) in order to use this 
approach.  It has, of course, traditionally been assumed for interference 
data that all non-conjugated proteins have the same dn/dc (and this is 
indeed probably a good assumption within a per cent or two), but it is 
certainly false that all glycoproteins have the same dn/dc, and obviously 
false that all proteins the same extinction coefficient.

In addition, a more subtle, but quite serious, problem arises when you try 
to fit hetero-associations using the NONLIN approach whenever the system 
makes complexes containing two or more of one type of monomer (e.g. an AB2 
or A2B2 complex).  In such systems the overall weight average molecular 
weight of the sample is generally a MULTI-VALUED function of the monomer 
concentrations, i.e. two (or more) sets of monomer concentrations give the 
same overall Mw and therefore virtually indistinguishable concentration 
profiles in the centrifuge, particularly when the system is strongly 
associated and the free monomers contribute very little to the total 
signal.  Consequently what happens when you try to fit such data is that 
for given value(s) of association constant(s), for each data set in the fit 
there are two or more solutions (monomer concentrations at some reference 
radius) that fit the experimental data nearly equivalently.  Thus if you 
are globally fitting, say, 10 data sets, there are generally at least 1024 
(2^10) local minima with nearly equivalent variances and residuals!

This multiple minima problem is not intractable, however, provided we can 
input more information into the analysis.  One approach that has been used 
successfully, particularly by Alan Minton and Marc Lewis at the NIH, is to 
somehow measure the concentration distribution of each type of monomer 
independently, using approaches such as post-run collection of fractions to 
be analyzed via gels, or using a multi-wavelength approach to take 
advantage of a difference in spectral properties among the monomers.

An alternative approach to this multiple minima problem which does NOT 
require the ability to separate the contributions of different species, and 
the one that I use in work, is to use numerical constraints to impose mass 
balance onto the solutions.  It turns out that for each data set only one 
of the possible solutions is physically realistic, based on the amounts of 
each protein that were actually put into the cell.  Thus if you can input 
into the fitter the initial concentration of each species (or at least 
their molar ratios), and then use this information to penalize solutions 
which deviate strongly from mass balance, the fitter can avoid the physi  
cally unrealistic minima and converge on a true and unique solution.

In my view, without these or some alternative approach to solving this 
multiple minima problem, the use of the NONLIN approach on complex 
heteroassociations is, at best, fraught with peril and, at worst, 
impossible.

John Philo, Alliance Protein Laboratories

On Tuesday, March 24, 1998 8:34 AM, yujia xu 
[SMTP:yujia@hxiris.med.upenn.edu] wrote:
>
> For the analysis of heteromeric system using equilibrium sedimentation,
> commercial mathematicl modeling software packages could be usefull, but
> acctually NONLIN can do all the work with little or no modification. The
> parameter 'n' in NONLIN, which stands for 'the degree of association' 
does not
> have to be an interger. For self-associating systems n is set to 2 for
> dimerization, 3 for trimerization etc.. For heteromeric association, the 
n can
> be set to the  ratio of the molecular weight of the two types of monomer
> involved in the interaction, and the analysis can be carried out as one 
would
> for a self-association system.
>
> Yujia
>
>
> --
> Yujia Xu
> Department of Biochemistry and Biophysics
> University of Pennsylvania, School of Medicine
> Philadelphia, PA 19104
> (215) 898-6580
>