User:TheLarry/Notebook/Larrys Notebook/2009/10/11

Fixing the Rate Constants

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Continuing from yesterday's idea Yesterday i wiped the slate clean and decided to look at the core cycle first to get that to have a k effective above 100 possibly something like 120. Then I'll add in the side cycles. A side cycle has two options end the run or reenter the core cycle. And the probability to do either should depend on how often it enters the side cycle. Say entering a side cycle at all has a probability of 1:100 so entering one means basically 100% chance of removal from the microtubule. But this is all something I have to think about after i get the core cycle to have the right k effective. OK i got the core cycle that is on the white board to have a k effective of 121.969. Again this is calculated through the Markov method. Anyway, i could only do it by increasing the rate constant of releasing inorganic phosphate. The literature says it is a limiting step with rate constant of ~100, but i had to bump the step up to 450. It is still the rate limiting step however. A decrease of 50 from 450-400 in this step changes the k from 121 to 117. for comparison changing the back step from 10,000 to 500 changes the k effective by 4 as well. And i checked with other steps as well. I don't like a three fold increase but i am not sure what else to do without changing a lot of the other steps. It doesn't make sense that a cycle that has a k effective of 100 would have a step that takes 100. That means every other step is instantaneous since the k effective can only be lower than each sub steps forward rate constant. So thinking of it like that it is impossible that inorganic phosphate could be released with a rate constant of 100. Also literature disagrees if this is the limiting step. Pointing to a waiting step for instance. If the waiting step isn't this step than the longest step is somewhere else. So that means there is a step lower than 100 or this is higher and another step is the rate limiting step. So I prattled on for long enough about this. So i am keeping it. The question Andy and I had a while back when making this core cycle was what happened first ATP hydrolysis or ADP release when the head binds in front of the other head which moves the stalk 8 nm. So i added the ability for either path to occur. This raised the k effective to 123. This isn't a side cycle since this still leads forward and isn't a path to unbinding from the microtubule. I still have a question: that is how often does ADP bind to a head? This comes down to the question of how much ADP is in solution in a normal gliding motility assay? I am not sure the answer to this yet. After putting in this new option to reach the goal i was able to lower the inorganic phosphate rate constant to 300 and still keep an effective rate constant of ~120. The thing i don't understand, and i might have a mistake somewhere is that the Markov predicts 114 while running this 1000 times gives me 120. I don't get the disparity and i searched for an error somewhere. I don't like this, and i don't want it to get too complicated if i am building on top of an error somewhere but i am just not sure the deal. and i searched for a mistake quite a bit. After looking through it again i found two errors, and fixed them. so now Markov says 114 and the 1000 sims says 118. I found another error and it changed the sim k effective to 121 and again the Markov is 114. So i don't know what is causing this problem, but i am getting a bit frustrated by it. I am sure i'll find it in time, but for right now i want to play around with a side chain. I am a bit less upset about this error right now because if i look at the expected time to finish: for Markov it is .0087 and for 1000 sim it is .0083. These numbers are pretty close to each other and it is these numbers that i am calculating then inverting to get k effective. So the fact that .0004 is creating this disparity is a bit easier to swallow. Right now i have a core cycle that takes a reasonable amount of time to complete. Now i want to add on side cycles, and not disrupt the time it takes to complete a cycle. These side cycles are what is going to lead to the kinesin coming off. So let's look at μ-ADP/μ-Φ where Φ is empty. in the core cycle this state ADP should unbind from the microtubule at 600/s. this means there are 5 different options to divert from the core cycle. so i should look at how many times on average this state is entered and figure out how many times i want to leave the core cycle. If it enters this state 100 times then i want it to divert possibly once. i don't know i have to think about it. But to figure out how often this state is entered i may be able to look at the discrete Markov chain like i did way early on. this should tell me how many times on average this state is entered per cycle. Then I can say if i want to divert once per 100 times entered i might be able to set up the probabilities correctly. The probability of a transition is [math]\displaystyle{ P_{ij}=\frac{q_{ij}}{\sum{q_{ij}}} }[/math]. And i can look at the probability of everything but the core cycle for each state. And hopefully set those to cumulatively equal .01. It sounds so crazy it might just work. Of course this'll be a lot of work. After i figure out each state on the core cycle i then can set the side chains to either go back to the core or to detach from the microtubule. This will depend on how often i want to leave. Right now i want to divert the core once. So if it leaves it has a large chance to come off the microtubule. My reasoning is that if it comes off the microtubule, and ADP concentration is low in the solution like i am currently assuming, then the best way it can get back on the cycle is through ATP binding and since it is basically 1 ATP per step, coming off the core cycle doesn't happen too often. I found my notes on the discrete Markov chains, but I am not thinking right and I am getting a bit hungry. So I think I'll call it a day and work on this part tomorrow. But I think I am on the right track. It feels that way at least. I also found a paper by William Schief that looks interesting so I'll read that tonight. It is titled "Inhibition of kinesin motility by ADP and phosphate supports a hand-over-hand mechanism." It looks like a quick read, so I'll stay busy tonight and come in tomorrow ready to analyze some side chains. I want to get these rate constants right and see how well this works. It is kind of exciting. Alright i found the times in each state. I am note 100% confident in the answers but I don't know a way to check it yet. I could run the 1000 sims and keep track of each time in each state. Damn that is a way to do it. Alright...I'll take care of that. Or I could do it tomorrow. hmmmm...i like that idea. Yeah i am going to save this for tomorrow because checking these numbers is annoying since the sim looks at each head individually and I'll need to combine them since the discrete markov looks at the combined heads as a state. So i'll need a 8 option case structure. not impossible just annoying. The case structure will actually have to be double. One embedded in the other. And I'll have to add 1 to it each time I am in the state. So in the end of the 1000 sims I'll have 8 1,000 arrays which I can then average and see if they match these numbers. If they do that'll make my life easier in the future. If they don't I'll have to check the time spent in each state through this sim method every time and it'll be annoying when I get much more complicated than the core cycle. So I'll keep my fingers crossed when I do this tomorrow.