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Absorbing Markov Chains
i wanted to model this question i had yesterday as a Markov chain and see if i can figure out total k from that. So after reading halfway through a fantastic chapter on Markov Chains, I realized that ideally this is an absorbing Markov chain since (for the example of yesterday) once ATP-->ADP the person recorded the rate constant of say 300 1/s. Thus ADP is the finish and there is no turning around. So that is an absorbing Markov chain. The criteria for that is there is a state where it is impossible to leave.
Using some simple matrix algebra, it is possible to find out the probability of being in which state after n steps. So say that the reaction took 3 steps what is the probability that the state is ADP-P. Well that can be figured out fairly easy. Also the number of steps on average it takes to get to the finish. So how many steps does it take to go from ATP-->ADP on average? As well as how many times the reaction is in which state. How many times is there ADP-P or ADP+P or ATP? This is the important one to me since knowing which steps the process took, it might be possible to figure out on average how long it took to do it. That is inverse the rate constant.
So from yesterday I ran a simulation that A↔B↔C→D. Where each rate constant was 1. I got a k total of something like 17. Well i ran it in the absorbing Markov and got 16.6667 as a total rate constant. Of course figuring out the rate constant was made easy because all the rate constants were equal, but that is still kind of close. Andy is going home tonight to try this simplified model out and see what he gets for a total rate constant from the coupled differential equations.
Can absorbing Markov actually give me the answer? I think so, but if it is easier than coupled differential equations, or flat out simming the reaction 10,000 times then fitting the histogram to a decaying exponential, I am not sure. Three ways to check is not a bad idea, and would make for a better apendix. For the record, i believe coupled differential equations is probably better since it might give us a relation between k's and k total.
Here's the link for the chapter i am reading
I guess i should explain what a markov chain is
It is a process where the probability of the next step does not rely on the anything but the present. So my simulation is a markov chain (technically it is an absorbing markov chain since the state of both heads unbound is impossible to leave from). My simulation is a markov chain because the only thing going to the next state in the simulation depends on is what the current state is. The current state provides the rate constants that is used to figure out the next step. For the record I don't think this is a regular markov chain.
Also yea, i really like this chapter i am going through.