Today we finished the Markov chains and posed your homework for this block. The homework is about a molecular motor where the potential landscapes of a (simply described) Kinesin moving have been discretized. We commented on the different states and transition rates to include non-equilibrium effects. In regards of this homework, I ask you to:
a) Find values of T1 and T2 (and lambda) such that a net current of "particles" appears.
b) Estimate the value of the fluxes. How can you define the current of particles in terms of fluxes?.
c) Is it possible to reverse the current in terms of the parameters T1, T2, and lambda?
Due date will be the day of the exam (whenever that date will be during January and independently of the fact that you have or you haven´t to do the exam).
In addition I introduced the Master equation, the underlying idea of the Gillespie simulation scheme, and the resolution in terms of momenta by means of the characteristic function. In particular we solved the problem of a Brownian particle diffusing in a lattice and also the birth and death processes in a population where we saw that no matter that the average could grow exponentially the fate of the population actually depends on the initial condition since (again) fluctuation can be very important.
2 comments:
Hi!
I have tried to solve the problem and when I obtain the stationary (not equilibrium) probabilities of each state. I find (with Mathematica) very very very very long expressions.
So my question is, do we have to manipulate such horrible expressions? or the aim of the problem is only to find numerically some solutions that show that indeed there is a net particle flux?
Of course, there is the high probability of the result to be mistaken.
Sure the expressions are long....
As said in the blog "a) Find values of T1 and T2 (and lambda) such that a net current of "particles" appears.". That means that you just need to find numerical value as you commented on. In any case what is also very important is that you (all) think about the criteria that (in terms of fluxes) provides information about a net flux of particles in one particular direction. Best,
J
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