[RASMB] still 2 ligands but a couple of equations

[Holger Strauss]

Thanks a lot to Arthur Rowe, Claus Urbanke, Dmitry Veptintsev and Allen
Minton for their very, very helpful suggestions, which are summarised
below. With this, we can get now. 

Cheers!

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Multiple equilibria are indeed horrible to solve
explicitely. There is one trick that is used in such a
situation:
Set up the kinetic equations e.g. d[AX]/dt =
k(1)[A][X]-k(-1)[AX] and d[AY]/dt = k(2)[A][Y]-k(-2)[AY] and let any
numerical solver for such differential equations run until equilibrium is
reached (There are simulation programs around although I tend to use some
own stuff. If you do not have anything, ask a chemical kineticist nearby).
This should always give you physically meaningful and therefor correct
solutions.

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As regards your problem, it seems to me that doing maths on dynamic  
processes is boring - and hardly necessary when there is
plenty of software which "is designed to facilitate the mapping, modeling and
simulating of dynamic processes". To quote from the web site of those
who make and market my own favoured product (STELLA, from High Performance
Systems, Inc:
http://www.hps-inc.com/index.htm, click on STELLA in the LH menu).

Of course there are other programs out there, but STELLA
is long established, is both Mac and PC compatible (and models can
be transferred across the platforms), and does not take lots of learning
to get output.
Although a lot of its applications are social science
based, a rate is a rate, and it's easy to set a model* up for a
multi-component interacting system.

e.g. Alistair McGregor, Andrew D Blanchard, Arthur J Rowe
and David R Critchley (1994) Identification of the vinculin binding
site in the cytoskeletal protein a-actinin  Biochem  J  301  225-233

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The answer is no to look for an analitical solution, which is horredos and
non-existent as it is at least a cubic equation,  but for
a numerical one, in which case it is faily trivial. 
I can recommend "Numerical recepes in C" by Teukolsky at
all by Academic Press. You will need to solve a set of equasions: 
                  A+B<K1>AB; 
                  A+C<K2>AC; 

                  A+AB+AC=A0; 
                  B+AB=B0; 
                  C+AC=C0; 
                  K1=AB/(A*B);
                  K=AC/(A*C); 

This could either be done by numerically solving the
system of equasions or you could use  substitution and
express it in terms of one unknown (which will look horrifying), and solve
this using root finding procedures.  
You shold get one root withing a range of (0 to A0) if you
express everithing in terms of A. 
for known A0,B0,C0,K1,K2 

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Let the two ligands be A and B and the protein be C.  Write conservation of 
mass of all three species;  Atot = [A] + [AC]; 
Btot = [B] + [BC]; Ctot = [C] + [AC] + [BC].  
Substitute into these equations in the
equilibrium relations: [AB] = K1 [A][B] and [AC] = K2 [A][C] (K1 and
K2 being equilibrium association constants).  Now you have 3
equations in 3 unknowns, [A], [B], and [C] as functions of Atot, Btot,
Ctot, K1 and K2.  With a little straightforward algebra you can reduce
the 3 equations into a single cubic equation in [C] which is easy to solve
numerically for the single real value of [C] between 0 and Ctot.



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Holger Strauss

Forschungsinstitut fuer Molekulare Pharmakologie (FMP)
Robert-Roessle Strasse 10

13125 Berlin

Tel: +49 (0)30 94793 - 223 (office)
                     - 316 (lab)

Fax: +49 (0)30 94793 - 169


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Science is spectrum analysis; art is photosynthesis.

                                                    Karl Kraus