Feedback control for a metabolic pathway

[arunma]

I'm having a problem setting up a certain differential equation which I need to solve a problem. Here's the problem: I'm given a feedback control for a certain metabolic pathway. I've attached a picture of the network below.

Now what I can't figure out is how to write the differential equation which governs the amount that accumulates at any particular node. It says that the reactions, except for the first one, obey a first-order rate law, and that all the rate constants are proprtional. But is the time derivative of S1 equal to f(Sn), or is it equal to f(Sn) - kS2? Also, in general, should I find the time derivative of each Sn by subtracting the incoming and outgoing rates?

I'd appreciate any help that anyone can give me. Thanks.

 

[lippyka]

The differential equation governing the amount of substance at any particular node can be written as follows: dSn/dt = f(Sn) - kSn-1, where Sn is the amount of substance at node n, and f(Sn) is the rate of inflow of substance to node n. The rate constant k is proportional to all rate constants in the system. The time derivative of S1 is equal to f(S1), since it does not have an incoming rate (i.e., there is no Sn-1). For all other nodes, the time derivative should be found by subtracting the incoming and outgoing rates.