>From Ian Murray's fwd:
``...A mathematical formula known as the Poisson-Boltzmann equation is vital to creating accurate electrostatic models. Calculating this equation conventionally requires a great deal of computer power, even for a simple model.
...But mathematicians Mike Holst and Randolph Bank at the University of California discovered in 2000 that equations of this sort can be broken up into independent parts.
...The innovation has already yielded some useful results. The researchers used the technique to model the electrostatic charges on microtubules, which are part of cell's structure and transport system, as well as ribosomes, which are responsible for making protein within cells....''
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For some more explicit background, microtubules are self-assembling struts that form something like a geodesic armature on the interior of the cell. These struts are in constant assembly and dissassembly as part of the physical machinery of metabolic activity, and if the cell is freely mobil, then these struts also work as a rigid cytoskeleton, against which their attached actin filaments elastically contract and relax, essentially pulling or pushing the membrane around the cytoplasma in a something like a rotatary tank thread motion. Microtubules have many other applications including cila and axion extensions, and chromosome duplication in cell division.
The microtubules are rigid coils that are formed or polymerized tubulin in a cylindrically turned spiral. There is a dynamic equilibruim between the existing pool of soluble tubulin molecules and their polymerized chains. Tubulin is a globular polypeptide composed of two dimer (1alpha + 1beta) molecules that align in a staggarded fashion to each other via the electrostatic forces on their molecular surface and are bound together by GTP through nucleotide hydrolysis. This polymerization occurrs spontaneously, so the organization of these transient formations is controlled through nucleated centers in the cell that act something like star shaped intersections, where microtubule bundles radiate outward. Permanent microtulule structures are maintained through a system of various accessory proteins that link these bundles together in a variety of geometrical arrangements which have a radial symmetry about the bundle as in the case of cilium.
So the general spatio-temperal aspect (the geometry) of these formations can be modeled by using the electrostatic features of the dimer formation. This (guessing) is where the Poisson-Boltzmann equation comes in, used in something like the mathematical modeling of the symmetric formation of a crystal lattice.
The polymerization and dissociation of tubulin takes place in the presence of calicum (Ca2+) and magnesium (Mg2+) ions in some electrolytic dynamic equilibrium (chaos?). By changing the balance of these ions, the tubulin will either polymerize or dissociate. The cell nucleus through intermediaries performs this function on the mucleated centers which in turn either generate or dissociate their tubules.
So this must be the process that is being modeled by the Poisson-Boltzmann equation. What is being modeled (guessing) are the changes in the electric charge distribution along a dynamically changing microtubule which has fixed source charges that are embedded in the array. This fixed source array is modified by mobil positive or negative charges (electrolyte ions) that surround it (as in batteries). These are assumed to be in thermal equilibrium at some temperature T. The potential energy of the charge q located at position r on the array can be altered by the bulk number densities of quantities n(+,-) or cations.
The complications or physically realistic features include considering that the Coulomb force has a long range and is non-linear (varies with square of the distance). This introduces the Boltzmann effect or exponentiation, hence the name Poisson-Boltzmann.
The simplest form of this model is a battery plate in an electrolytic solution with a surface charge density O, omega surrounded by a 1:1 electrolyte with bulk density n+ = n-. The plate is located at x = 0, and has infinite extension along the y and z directions, the dielectric constant e, epsilon is uniform through out space. Thus the electric potential is dimensionless and a dimensionless distance u is given as kappa, kx where k^-1 is a length known as the Debye screening length (a statistical fiction or analog that stands in for the range of the Coulomb force). In other words, bulk ion charge density can be neglected at some vanishing distance away from the plate---something like a thinning atomsphere that gets thinner with distance.
Returning to the spiral form of tubulin polymerization, instead of a simple plane, one can consider only the exposed end beads or dimers as the active site of assembly or dissembly, ie the location r in the array. This constantly changing location is in effect a winding vector (radius, angle) that turns either clockwise or counterclock wise and whose speed of rotation and axial translation is governed by the changing balance of the surrounding ionic charges.
So much for my guessing. Go here for very gory details:
http://bessie.che.uc.edu/tlb/rctb6/node19.html
I blew off the evening looking this stuff up and trying to understand it. I needed something to do since I came back early from a climbing trip to Tuolumne after taking a very stupid fall and banging up my left leg, where hopefully this vary process of tubulin formation is repairing the damage. Tuolumne was a mess. It was overcast all day with heavy smoke from surrounding forest fires and looked like something out of Dante's Inferno. The sunsets were blood red and the sun was an huge ugly orange glob.
Chuck Grimes