Dear Alessandro
It was wonderful talking to you yesterday.
I think the project we discussed could lead to significant increases
in understanding of both ionic solutions and ion channels and
proteins.
I was thrilled to hear from you that the change of variables involved
are feasible and may have been done in another context.
Below is a more formal (but superficial) description of the problem
we discussed and then some specific references to the form of the
problem we have worked on. (PLEASE FEEL FREE TO USE ANY
OTHER FORM OF THE PROBLEM AND TO CRITICIZE OUR FORMULATION
OR TREATMENT OF THE PROBLEM. I want to solve it, not to defend
what we have already done)
Description
The problem I am interested in is central to the
treatment of ions (hard charged spheres, which
to begin with are approximated as points) in water
and in proteins. This is a central system in biology,
chemistry, physics, and stochastics, the latter
since it is the system that Brown originally studied
and is described by Einstein's treatment of Brownian
motion.
The problem is best introduced by considering
Na (positive hard spheres) and Cl (negative hard
spheres with similar but not equal diameters and
diffusion coefficients) in a dielectric background.
Many many chemists (including my friends Stuart
Rice, Doug Henderson, and Jean-Pierre Hansen) have
shown that a thermodynamic analysis of this system
is remarkably successful (but of course not totally
successful) in understanding the properties of ionic
solutions over a wide range of compositions.
We (see CV) have shown that a similar approach
works surprisingly well in biology, for ionic channels
(proteins with a hole down their middle, of enormous
importance in health and disease).
The problem starts by describing the motion of Na^+
as a Langevin equation (in the high friction limit if you wish)
in an electric field. The electric field MUST itself be calculated
from the charges present in the system (i.e., the other Na ions,
the Cl ions, and dielectric charges, and the boundary conditions).
The potential can NOT be independently prescribed because in
the physical world there are no sources to maintain that potential
fixed (in either space or time or both) as charges move in thermal
motion.
The Cl^- ion is described the same way.
The electric field is described by Poisson's equation ( i.e., for the
electric potential) and boundary conditions.
The standard problem is to find a description for the number density
of Na, the number density of Cl as a partial differential equation in
the spirit of a Fokker Planck equation.
The problem I pose is to find a similar description for the
probability of the
1) number density of charge
Q defined as the difference of the density of Na and Cl
and
2) for the total number density of both ions C
C defined as the sum of the density of Na and Cl.
The charge fluctuates, the 'concentration' C fluctuate.
A simulation can clearly keep track of the charge and the total
number density.
The question is can we write equations for these.
The reason this issue is so central is that it is an experimental
fact (verified by electrochemists since around 1890) that in many
domains the system of Na and Cl in water behaves according
to "Ohm's law" (i.e., one need be concerned only with charge,
to the first order, except perhaps one scaling constant) and
in other domains the system of Na and Cl in water behaves
according to "Fick's law" ( i.e., one need only be concerned
with the concentration, to first order).
Those domains are the domains that determine a large
fraction of biological behavior of nerve cells, muscle cells,
etc etc.
The mathematical question is can we derive these equations
and the domains in which they are valid.
There are then many things that could be done.
If one develops a mean field description, changing variables
from number density of each ion, to number density of
charge, and of all ions, gives large simplifications.
I imagine that the equivalent stochastic change of variable
(which is what I seek) would be a least as successful.
I suspect I have written a confusing description and fear that this may
discourage you, indeed if you have read this far.
Specifics:
In the attached paper eq. 2.1 -2.9 are meant to specify the problem
precisely. In the case you and I were talking about, these reduce
to two Langevin equations for the location of Na (+) and Cl (-) ions
with forces between them generated by Coulomb's law and a dielectric
boundary nearby. The dielectric boundary is the channel and can be
ignored to begin with (let's understand free space first!)
We then go on to derive the Fokker Planck equations for the density
of Na and Cl in a very high dimensional space, dimension equal to
the number of particles.
The good news is we can make such a derivation including nonequilibrium
boundary conditions.
The good news is that very few assumpotions are needed.
The bad news is that we cannot deal with the correlations between particles
and are left with the famous closure problem.
The good news is that the closure problem is the same as that of equil stat mech
(except for different boundary conditions).
My long term HOPE is that one could "change variables' or construct a similar
derivation for the CHARGE (difference in numbers of Na and Cl) and the MASS
(thFie sume of the numbers of Na and Cl) and that these variables would be
so uncorrelated (in many domains of interest, namely those where Ohm's
law OR Fick's law are know to work) that the closure problem is easy or
irrelevant.
My dream is that quasi particles might e defined for charge and for mass that
follow simple Langevin like equations. Then we could simulate these particles
instead of ions and make a fantastic reduction in complexity, as the definition
of quasiparticles ("holes" and "electrons") does in solid state physics.
I hope this all makes sense and I am thrilled that you are interested in working
on this with me.
See you soon I hope, here or there,
As ever
Bob
PS I am sending copies of this to Prof Sreenivasan and Paolo Carloni who
may be interested in our work, at least I hope so.
--
========================
Return Address for email: beisenbe@rush.edu
Bob aka RS Eisenberg
Bard Professor and Chairman
Dept of Molecular Biophysics & Physiology
Rush Medical Center
1653 West Congress Parkway
Chicago IL 60612 USA
Office Location: Room 1291 of
Jelke Building at 1750 West Harrison
Email: beisenbe@rush.edu
Voice: +312-942-6467
FAX: +312-942-8711
FAX to Email: +801-504-8665
Department WebSite: http://www.rushu.rush.edu/molbio/
Personal WebSite: http://www2.phys.rush.edu/RSEisenberg/physioeis.html
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