# Biological/chemical mathematical modeling - two partners with multiple binding site interactions

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.

• Perhaps make it clearer that you are interested in mathematical models of the mentioned phenomenon. I suspect the close vote is due to someone thinking this is off-topic. Also, I'd suggest you add more relevant literature (i.e. relevant lit you're aware of) and maybe formalize the problem a bit more. Incidentally, I wonder how common the scenario you speak of is? Perhaps it's been analyzed for a single case, like e.g. cytoskeletal subunits. But for globular monomers it seems somewhat uncommon. Are you thinking of a particular case? Maybe consider asking on the bio SE as well. Jan 16 '19 at 4:12
• Sure, I'll change that - I assumed that I was obvious since we're on Math SE but, well, better safe than sorry. To answer your question, it may be relevant in the context of dynamic phase separation in the cytosol, and more generally in the context of engineered systems (synthetic biology etc). Precise mechanisms of cytosolic phase separation are poorly understood, hence the need to better formalization and models. Jan 16 '19 at 19:07