# User:TheLarry/Notebook/Larrys Notebook/2010/08/19

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## Rate Constants for traveling a certain distance with a wlc force

OK after working really hard on solving the smoluchowski equation, it turns out Hanngi did this shit already and has a fancy ass fuck equation ready to go and here it is
$k=\frac{\omega_0^2 }{2 \pi \Gamma }\sqrt{\pi \beta E_b}e^{-\beta E_b}$
$\omega _0^2 = \frac{1}{M}U''(x_0)$
$\Gamma = \frac{k_BT}{MD}$

So combining the equations you get
$k=\frac{D U''(x_0)}{2 \pi k_bT}\sqrt{\pi \beta E_b}e^{-\beta E_b}$
D is diffusion constant, k_bT is boltzmann energy, x_0 is the energy at the bottom of the metastable energy well (in this case the energy from WLC at x=0), beta is 1/k_bT. Cool

this k is from the hanggi paper where he dsecribes the rate constant from a smoluchoski dynamic over a cusp shaped barrier. which is what we are saying our potential energy is

using this equation I have rate constants for all the situations. For this i don't have simple drawings so i'll just go with writing it out. so ATP and ADP-P are docked so i'll write them as D and ADP and empty don't so i'll write them as U for undocked. The first letter is the unbound head and it can step forward or backward. The forward step is the first number and the backward step is the next one

• U/D
• 2.2E+6
• 2.36E-35
• U/U
• 6.44E+3
• 6.44E+3
• D/U
• 5.54
• 5.54
• D/D (same as U/U right now but might want to rethink it.)

The distances change for how far the head has to travel. For U/D since the neck linker moves the head forward 3.5 nm it only has to travel forward 4.5 nm but backwards 11.5 nm. Its contour length is 10.5. For U/U it travels back and forward 8 nm with a contour length of 14 nm. And for D/U it also needs to travel forward 8 nm but it's contour length is 10.5 since it is shortened by the neck linker docking to the unbound head. We said earlier that D/D wouldn't dock so it would be just like U/U except maybe unbound it could dock. I am not sure. I'll talk to Koch about it. Steve Koch 23:41, 19 August 2010 (EDT): Here's some extra notes from me: User:Steven J. Koch/Notebook/Kochlab/2010/08/19/Larry's rate constant work