# Biological modelling math question?

I am trying to write a biological model that models protein interaction. I am having an issue with one aspect. Lets say protein A and protein B interact with eachother to form complex AB. Now every A molecule and B molecule will come together to form AB. The amount of AB formed is described by a constant k where $$k=\frac{(free A)*(free B)}{(complexed AB)}$$ When there are just two components that interact in a one to one manner, it is easy to calculate the amount of free A, free B, and complexed AB when the total amount of A and B, and k are known. However, lets say there is a mixture of containing A1, A2, B1, and B2. Now you can have either A1B1, A1B2, A2B1, and A2B2 forming, each with their own k. This gives a set of equations:

$$k_{11} = \frac{(free A_1)(free B_1)}{(complexed A_1B_1)}$$ $$k_{12} = \frac{(free A_1)(free B_2)}{(complexed A_1B_2)}$$ $$k_{21} = \frac{(free A_2)(free B_1)}{(complexed A_2B_1)}$$ $$k_{22} = \frac{(free A_2)(free B_2)}{(complexed A_2B_2)}$$

Which are are connected by the concentration of free A and B. I cannot come up with a formula to find the free As, free Bs, and complexed ABs in this scenario with the total concentration of the As and Bs, and each k. I may be asking a bit much, and this may be a bit ambiguous, but I would be happy to provide more information if needed, and I would be more than happy if someone could so much as point me in the right direction to figure this out.

• You need to tell us (I am assuming each phrase denotes a single real number) which ones are known, or will be known after running an experiment, and which ones are unknown. Put another way, you need enough symbols so that there are no words used, and carefully describe all relationships. For example, you say "total amount of A and B, and k are known." With the relationships you typed, that phrase is completely meaningless. Mar 28 '15 at 23:23
• What does the total amount of A and B mean here? Do you have some relation like $$\text{Total} A+\text{Total} B= \text{Free} A+\text{Free} B + \text{Complexed} AB$$? Mar 28 '15 at 23:35
• In the one to one case, total A = free A + complexed AB and total B = free B + complexed AB. In the 2nd case, total A1 = free A1 + complexed A1B1 + complexed A1B2. The knowns are all of the total concentrations and all of the ks Mar 29 '15 at 1:52