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

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Overall Rate Constants

So 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 Chains

This 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


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