# User:TheLarry/Notebook/Larrys Notebook/2009/09/19

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## Overall Rate ConstantsSo with Igor's help we (mostly him) came up with an expression that could be how to connect all the intermediate states into 1 rate constant. However, i am not happy with it since it doesn't match simulation results. So back to the drawing board. I took out from an actual library 3 books on markov process but even the simplest is way over my head. I have some print outs on continuous time markov processes to read. Koch also suggests Poisson Process which is a off shoot of markov chains. So I'll do some reading today and tomorrow, and then in the end say FUCKETT. because i have a simulation that works that i can just plug values into and see what the overall k is from that. So I have a method that is based on absorbing markov chains. That was able to predict when all the k's=1 in A↔B↔C→D reaction. It gave an answer of 1/6 = .1667. It was also able to predict k=2 giving 1/3 = .333. It came close to predicting the all k forwards are 3 and all back are 1. It gave a k total of .125 while the day i did it, i got 1.25, but i think i was off by 10 that day because all my values were higher than what i get now. It must be because i was normalizing the integral which would explain it. Or i am faking myself into believing that. Anyways, i'll read about continuous time Markov chain and some Poisson process over the weekend, but I am feeling good about approximating the chain as an absorbing markov chain and calling it quits with that. For the record, what this does is i can find out how many times on average the object is in each step. So how many times the object is in step A, B, and C (it's in D once since that is when it ends). It is in A three times, B 4 times, and C 2 times. With that knowledge I can tell how many times the transition is from A-->B (all the times, 3 times), B-->A, B-->C, and C-->D. Those transitions depend on the probability. I take that probability and multiply it by how many times the object is in each state and that says how many of which transitions it takes. Then i multiply how many transitions by the average number of time each transition takes (inverse rate constant). Add those together should say average time for whole cycle. Inverse that is k total. So hopefully this works. Everything I have done so far is rough. I need to make a better model for the chemical reaction, and write a .vi that does all this math for me. Including calculating the matrix that finds how many times it is in each state. Again this whole process might be a third way to check k total. First being ODE's. Second is the simulation, and third Absorbing Markov Chain. But it is feeling sort of good. And i think it'll look cool in the appendix of the paper. OK I'm out. I printed out a 50 page thing on continuous time markov chain. So i'll read that tonight and that'll at least give me a background of what i can do with this section of probability. I haven't found anything really in depth for Poisson Process, but i can look again when i finish this part. The library also had two books that i can download in .pdf's. So later in the week, if i am feeling energetic i can print them out and read them since the books i took out at the library were way over my head. Note for Koch: Sorry if i seemed cranky in yesterday's notes, it was because the Rockies were losing. And that was pissing me off. So sorry if it seemed like i was taking it out on you. - Steve Koch 05:31, 21 September 2009 (EDT):HA--I didn't think that...Rockies are looking good, but man Tigers lead is dicey. Anyway, I'm reading all this, but not finding much to say constructive--usually I'm thinking I should wait to talk with you in person, which maybe is true. Keep up the great notebook, though, even if I'm not saying anything that helps!
## Continuous Markov ChainsThis stuff is pretty exciting. I can't see how this won't lend itself to our problem. I am trying not to think of this problem while i read it so i can try to fully learn this stuff. However the rule P_{ij}(t+s)=Σ_k P_{ik}(t) P_{kj}(s) might be what i want. It says that the probability to move from state i to j is equal to the summation of the probability of moving from i to k times k to j as a function of time. That can be something like saying A-->D is equal to A-->B * B-->C *...whatever. I haven't worked through the example yet but it is interesting. Also this .pdf started off with talking about Poisson Process which is what a kinesin walking is, but i still don't have a firm grasp on that. http://www.mast.queensu.ca/~stat455/lecturenotes/set5.pdf There is the link to what i am reading Since there are no examples or questions in this text, I am trying to take note of things I wanna come back to and try later. So the first thing is above, and the second thing is that P(t) = exp{t*G}. I think i might be able to find the probability with respect to time to go from state i to j. And say i = A and j = D then i can take the derivative of P(t) and find the maximum probability that should happen. And possibly the inverse of that is the rate constant. Hopefully |