### #16 How enzymes propel themselves-VI

We are finally going to finish this series. Today we will talk about the final piece of the puzzle -- the relation between heat of reaction and diffusion constant.

In #13(http://biophys3min.blogspot.tw/2016/09/13-how-enzymes-propel-themselves-iii.html) we have mentioned
Assume that the original diffusion constant of enzyme is D0 while the diffusion constant of catalyzing enzyme is D1. Given the reaction rate V (with unit sec^(-1)) and the duration of enzyme deformation caused by each "pulse of heat" δt, the overall diffusion constant could be estimated as

D1 could be estimated from Langevin's diffusion theory. As we mentioned in #14 (http://biophys3min.blogspot.tw/2016/09/14-how-enzymes-propel-themselves-iv.html), the diffusion constant could be expressed in terms kinetic energy:
If there is no extra energy source,  m<v(0)v(0)>=3kT, and D = kT/ζ, which is simply the Einstein relation. However, given the heat of reaction Q, we assume that the kinetic energy of enzyme becomes γQ. As Vδt approaches 0, the diffusion constant could be written as

Finally we have to deal with δt. δt could be estimated as the relaxation time of each acceleration, or how long it takes a propelled enzyme to return back to its original speed. Since δt = m/ζ, we finally get
It means there is a linear relationship between diffusion constant and reaction rate, which is proven experimentally.

#biophysics #3minBiophysics #生物物理 #三分鐘生物物理
#enzyme #diffusion