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

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Forcing in Force

Andy's questions in yesterday's notebook is what i have been worried about while researching this. I don't know if a force chagnes binding/unbinding of nucleotides. However no one knows this so i am gonna go happily thinking it doesn't. So the other consideration is if the force affects the diffusive step. Now my thinking is that it isn't diffusion at all but a directed motion forward. That some how the back foot is thrown forward. I haven't done the math but it takes the step forward in 1 μS according to Block. So i gotta check this against the diffusion equation to find out. Then if it is like a directed motion then i can use the directed motion equation to add in the force. For the record the diffusion equation is:
[math]\displaystyle{ \lt r^2\gt =4Dt }[/math] normal diffusion
[math]\displaystyle{ \lt r^2\gt =4Dt+(Vt)^2 }[/math] directed motion with diffusion
[math]\displaystyle{ \lt r^2\gt =\lt r_c^2\gt [1-A_1e^{\tfrac{-4A_2Dt}{\lt r_c^2\gt }}] }[/math] corralled motion
[math]\displaystyle{ \lt r^2\gt =4Dt^\alpha }[/math]

Don't let the size fool you i think it is directed motion i am just putting the other ones there for completeness. So lemme find out what D is for a kinesin foot. I have to do more reading on backstepping and stall force to see if it does burn through ATPs or if it comes off quicker. Or any other important information that helps me understand what's going down.

Koch Set Me Straight

After having a long talk with Koch and Andy which was basically dissecting Guydosh/Block paper of 2009, we came up with the idea that force only affects diffusion and unbinding rates. For unbinding no matter what the direction the force is in it will raise the unbinding rate constant SJK 23:19, 9 September 2009 (EDT)
23:19, 9 September 2009 (EDT)
I think this was a really good realization for us today...and I still think it's correct, at least for the connection geometries we were considering. For the record, the two shoelace model was helping me out.
. So for this i had to take the magnitude of the force and then negate it since the equation i am using is [math]\displaystyle{ k(F)=k_0e^{\tfrac{-Fd}{k_BT}} }[/math]. So for a rate constant to increase the force should be negative. but i want to have a force helping motion be positive and a force retarding motion to be negative. Well i could just take out the negative sign in the equation, or take the magnitude of the force and then negate it so for unbinding the exponent is always positive thus increasing the rate constant. However it gets a bit more complicated for diffusion. This should have a high rate constant if say the foot is behind but is being pulled forward or vice versa. There are four different options. Front/Back position and Postive/Negative force. So I needed an embedded case structure so the force is negative anytime i want the rate constant to increase and positive whenever i wanted the rate constant to decrease. Basically it went like
back/retard is negative
front/retard is positive
back/help is negative
front/help is positive SJK 23:25, 9 September 2009 (EDT)
23:25, 9 September 2009 (EDT)
So, I can't follow everything here, but just to clarify, I think we decided:
(1) any value of force, positive or negative will increase foot off rates
(2) we don't expect force-increased off rates to really matter ever as far as regular kinesin motility goes
(3) force does affect the "flip-flop" diffusion step: opposing force slows forward flip and speed backward flop; vice-versa for assisting force
(4) because we're modeling a discrete "flip-flop" diffusion step, force doesn't affect foot on-rates ... even though we though maybe it should, especially given how an assisting force can pull the head to far and torque it.

hopefully i did everything right. Anyways i now have a force dependence in the simulation. And 8 more constants--the d's. WEll with everything in there I get to start the fun job figuring out all the rate constants again, but this time i am gonna make Andy help me with this.

So i can't get graph that looks like Block's. This might not be a problem with the program or the force concentration (though i should double check everything), but this could be a problem with the constants i have in. I am just not sure. Yeah I have no idea what's wrong here. I am really hoping it is just a problem that none of the rate constants including d now are correct. Damn i really want that boltzmann model. Could I not be getting this relation because i am missing a force dependence on another state. RIght now i only have binding and diffusing...nah i gotta be looking for a stupid mistake first like something is wired wrong or i forgot something

OK I took out that stupid negative sign that i was talking about above. So nowa positive force increases the rate constant while a negative force decreases. I also found a bug in the back/retard thing, but that is fixed. I still don't have a boltzmann but it is looking a bit better. It has more of a chance to be a rate constant problem.

Now that is a Boltzmann curve.

Look at this motherfucker.
SJK 23:28, 9 September 2009 (EDT)
23:28, 9 September 2009 (EDT)
Dude, that is fucking awesome! Great work!
SJK 23:30, 9 September 2009 (EDT)
23:30, 9 September 2009 (EDT)
I love how real things fell out of it without tweaking: an apparent stall force; a limit to the assistance of positive force; backward steps. One feature of kinesin is assistive forces and small opposing forces don't change the velocity at "saturating" ATP concentration. Does your model show that if you have a high ATP concentration?

OK i was looking at this stuff before and it looked like a straight line. Apparently I was looking at the wrong scale. Because I was looking at -10 to 10 and then i grew to -100 to 0. I never looked at it from -1 to 1. So I'll buy it. Looks like it is working. HAha Time to fuck around with rate constants..tomorrow.

So to recapitulate the simulation gives us the correct form of concentration versus velocity (Michaelis-Menten), force versus velocity (Boltzmann), and run length versus number (exponential decay). Those are the three major guys I was looking for. So i think we have a working simulation for kinesin walking. Of course the real work is going to be in adjusting these rate constants. I am sure we'll have to do it by hand, but probably in the future a program will be nice. But having a program change 150 rate constants will make it near impossible to find values that are correct and not some that are insanely far away. Whatever, i'll worry about that in the future for now i'll do it by hand.

TheLarry 23:50, 9 September 2009 (EDT):Yeah I was pumped to get this graph. I'll check the saturating ATP tomorrow. This graph above has low ATP concentration. So we'll see. And don't worry about the part that you didn't quite get above because that was for me to keep myself straight while I made sure i had the right force sign (positive or negative).

Andy Maloney 23:58, 9 September 2009 (EDT): This is really awesome. I can't wait to work on the rate constants.