I am wondering if there is any reported attempt to mathematically model the interaction of two multiple-sites molecules?
I know that there is a lot of literature on how to mathematically model the binding of single molecules on multiple-sites molecules. For example, the Monod-Wyman-Changeux model of allostery can be used to model the concerted binding of N individual ligands on a single protein, and was expanded to many contexts.
I am interested in a case in which protein A contains, say, N binding sites X, and protein B contains M binding sites Y, and any site X can potentially interact with any site Y. That would be somewhat equivalent to how polymers interact in a hydrogel, but at a smaller scale (i.e. with smaller molecules so that there is no "immobilization" of the system and molecules can still diffuse freely). This recent paper (Hnisz, Cell 2017) reports a simulation that assumes that each A and B can only form one bond, which is obviously simplistic since each one of the molecules potentially has other sites nearby, ready for additional parallel binding events.
I am interested in both stochastic models and deterministic approximations.
EDIT: A more detailed explanation of the problem:
Let's imagine that a protein A (concentration [A]) is mixed with a protein B (concentration [B]) in solution. Each protein has a single binding site. At any time the reversible reaction A+B=C can be described by the evolution of [C] for example:
$\frac{d[C]}{dt} = k_a[A][B]-k_d[C]$
With $k_a$ and $k_d$ being rate constants.
Now, if I have two molecules, A and B, and A has $n$ independent sites for binding (let's call them $a_1$, $a_2$, etc.). Obvisouly, the problem gets more complex as there can be multiple binding events, and each binding of a molecule of B reduces potential for future interactions (at some point, A saturates). There are models for that, which get pretty simple assuming that sites do not otherwise interact. The reaction can be modelled as a series of single-binding events:
$A + nB = AB + (n-1)B = ... = AB_n$
If B also has multiple sites, then I think I cannot simply build a matrix of all the possible reactions, because of the resulting branched structures. Let's compare two cases:
1) two molecules of A are bound by a B "bridge", and each one carries, say, two other B molecules (the structure is something like B=A-A=B)
2) two molecules of A are bound by two different B "bridges" and each is also bound to single B molecules (the structure is more like B-A=A-B).
In that case, breaking a "A bridge" has fundamentally different consequences depending on the topology; in the first case it results in two free molecules which can bind different pertners, whereas in the second case there is still one big molecule.
I am looking for a way to mathematically simulate this type of multiple reaction and keep track of the average "size" of the clusters, the multiplicity of topologies becomes a huge issue which (I think) cannot be recapitulated by usual sequential models. I am looking for a way to approach that problem mathematically.