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Date: Fri, 15 Apr 2005 14:12:31 EDT

Here I explore a possibility with a stochastic model, namely the Markov
chain. Assume we have a protein complex, may be able to bind one or more

subassemblies or ion. The simplest case is one initial state A and one final
state B. In
each small time interval dT there is some probablity P(A,B) for a molecule in
state A to  shift to B. In the same way, a molecule in state B may revert to
A with a probability P (B,A). These probabilities rest on the nature of the
molecule and the environment, nothing from an anterior state has any effect on
them. For example, a molecule in state A could be there for tens of time
intervals dT or just one, the probability to go to B would be the same. Such a
system has no memory.

A simple ionic channel could have many intermediate states, may be half a

dozen or more(1). The simplest interesting case is : A goes to B and B to either
C or D. C may goes back to B or link to D. To be sure, there are back

probabilities from B to A for example. How is this process implemented on a
computer?
Assume the probability A to B, P(A,B) is 30 percent or .3 in the time domain
dT. Implicit in this statement is the probability P(A,A) = .7. P(A,A) is the
probability that the system remains in state A. The sum of all probabilities

must be 1, indeed here P(A,B) + P(A,A) = 1. In a simulation, the computer draw a
random number between 0 and 1, if this number is smaller or equal to .3 the

state moves from A to B, if the number is larger than .3, P(A,A) is selected and
nothing happen.

If B can evolve back to A, stay the same or move to C or D, each probability
will get a slice between 0 and 1 and their sum will be 1 as before. This
system is computer intensive, because an array of states and probabilities is
ascribed to each molecular complex.

This is one more tool in the mathematical modeling box, now assume we want to
simulate a small unbranched dendrite section, what are the possibilities ?

The first possibility would be to use the cable theory, a differential class
model. Depending on the boundaries limit conditions, one or another chunk of
the cable equation will be used, this is a grammar model. Next is the Hodgkin
and Huxley differential model or best an advanced simplified by-product: the
Hindmarsh-Rose model. Next could be a simplified Markov model of the 20 to 40

channels in the membrane as a stochastic supplement to the differential element,
we are here in the Langevin's domain. One step beyond, the full Markov model
would be used. This is the stochastic domain, a taste of it can be found with
StochSim at: http://www.anat.cam.ac.uk/~comp-cell//StochSim.html for

could be
modeled at the molecular dynamics scale. What to choose for uploading ? The
surprising answer may be: All of them.

The molecular Dynamics could be used to validate the probability tables of
the full Markov models. These would be used to find simplified but relevant
models. Used with the Hindmarsh-Rose model, a set of generic dendrite segment

could be computed. the same would be done with cable theory, so that a
correction
factor would be established. In a given neuron, a segment would be computed
may be with the cable theory and then corrected by looking in a table to the
nearest similar predefined segments.

There may be trillions of dendrite segments in a brain, but only some

millions at most are different. This is not a new solution: There are billions
of
people on Earth and only twenty shoe sizes or so.

(1)Sakmann, B., and Neher, E., (eds) (1995). Single-Channel Recording, 2nd
ed. Plenum Press,New-York.

Yvan Bozzonetti.

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