User:TheLarry/Notebook/Larrys Notebook/2009/09/06

Force Dependence

Main project page Previous entry      Next entry

Evans annual review letter This is an extremely complicated and complex paper. It seems like every sentence in this thing is as full of information as it physically can be. a lot of this has to do with the fact that there aren't a lot of written equations. most of the time they are just described in words. also evans likes to put in real numbers to give the reader a sense of size for these equations. like kT is approximately 4 pN nm. stuff like that. anyway, i read this paper, and then reread what i consider the most useful parts for this project. since this is a single simple bond. and if i combine the parts i read with Koch's Dynamic Force Spectroscopy of Protein-DNA Interactions by Unzipping DNA i can glisten some information out of this. but if i can jump to the punchline... Koch has an equation [math]\displaystyle{ F^*=\frac{k_B T}{d}\ln(\frac{t_{off}d}{k_B T}) }[/math] which he got from finding the maximum from the probability density as a function of F and r. r being the link between time and force. [math]\displaystyle{ r=\tfrac{dF}{dt} }[/math]. anyways that maximum is written above. this can be rearranged to solve for 1/t_off. [math]\displaystyle{ \tfrac{1}{t_{off}}=\tfrac{rd}{k_BT}e^{\frac{-dF^*}{k_BT}} }[/math]. so evans says "The off rate of 1/toff obtained from extrapolation of lipid pullout to zero force is..." I think that means that 1/toff is what i was calling k the problem being i don't know F*...damn i confused myself again. i can go back to the probability distribution and plug in time for force through r...damn aghhh confused i want an equation where i can plug in at the very least F and get out 1/time. ok lemme think. if F maps to t...and F* is the most likely force then that should map to t* the most likely time? so that means i have an equation that looks like this [math]\displaystyle{ t^*=\tfrac{k_BT}{rd}\ln(\tfrac{t_{off}rd}{k_BT}) }[/math]. so t* is a just a function of r. which is ok since that encodes in force. and it has t_off which is just 1/k when there is no force. the more i think about it the more i like that equation. that is until i get confused again. so would i want to take 1/t* to get the new off rate and what if the force applied is constant? then r = 0? no no...ahhh...no--yep I'm confused again. Learnding is hard. and i haven't even looked at the rebinding section I could also take the probability distribution switch it to becoming a function of time not force and r. then i can put this into the monte carlo simulation instead of the off rate. this should tell time. i'd have to integrate the function. this will be annoying since some change in the code will be needed. right now the code takes in an array of k's and takes that to exponential and finds time. but if i did this then anytime the foot binds/rebinds i'd need a new probability function. namely this one. [math]\displaystyle{ p(F;r)=\tfrac{1}{t_{off}r}e^{\tfrac{k_BT}{t_{off}rd}}e^{\tfrac{Fd}{k_BT}-\tfrac{k_BT}{t_{off}rd}e^{\tfrac{Fd}{k_BT}}} }[/math]. yeah that's scary looking anyways. i can send in a number in the array of k's in the code. if the number pops up i can go through a sub.vi that calculates the time from this probability density. i'd need to integrate this. i might need to switch from F to t using F=rt. then roll the dice on the t. actually that doesn't matter since i can roll the dice on the F on the integrated function. but i would need time in the end so i'll need to switch it at some point. is there a dimensional problem with this equation. the exponentials shouldn't have a dimension but the coefficient has units of 1/N? oh but if i integrate it'll be unitless so that's cool nevermind. i definitely think i gotta talk to koch about this stuff. but i'll do that tuesday. i'll just write a .vi that'll integrate this PDF for now. ok i got a .vi that'll integrate that there equation. it does what i expect for higher r there is a higher chance it'll break at higher force, and for lower it'll break at a lower force. and higher r breaks at a lower time while a lower r breaks at a higher time. so if i knew what i was doing i might be able to do something tomorrow. but if i don't by the time i come back i'll just fix up the security cam software. I like when there are pictures in my notebook. Here is a picture with high r compared to a low r in the previous image