Solution of the time-dependent ionic diffusion equation

The ionic diffusion problem here solved is that of the voltage clamp experiment. In this, the total voltage across the membrane and double layers is held constant. Since the diffusion of the ions would normally result in a change in the membrane voltage with time, a part of the complexity of the solution of the problem is in the correction procedure required to so modify the electric field as to hold the voltage at a fixed value. The program could, however, be readily modified to conform to other requirements: e.g., current clamp, linearly rising voltages, etc.A significant improvement in speed of convergence is obtained by using the membrane parameters existing at the midpoint of the time step (Crank-Nicolson method), rather than, as has been more customarily done, at the end of the step (“implicit” method).Convergence is further speeded by estimating the new value of the electric field at each station by extrapolation; and convergence of iterations at each time level is aided by a backward-averaging of the computed electric field. So far as the author is aware, these procedures have not previously been described for use in this type of problem.     

Single file ion diffusion and the Schmoluchowski equation

The Schmoluchowski equation is introduced into the problem of single file ion diffusion in a channel. The ions mutually interact due to coulomb repulsion and are also subject to a single ion potential due to the channel. The positions of the ions are represented by a continuous co-ordinate. The problem is reduced to the solution of a pair of transfer integral equations. The resistivity of finite and infinite channels is calculated for various dielectric constants and mean ionic separations. The ionic density for finite channels is also calculated. The results clearly demonstrate that strong coulomb interaction leads to a co-operative motion of the ions across channels.     

An approximation diffusion equation for a long narrow channel with varying cross-sectional area

The diffusion equation for long narrow channels which lie parallel to a rectangular coordinate and have varying cross-sectional areas may be approximated by an equation which involves only one space variable and the average concentration at each value of this space variable. This equation is derived and is discussed along with its assumptions.     

Singular solutions of the diffusion equation of population genetics

The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the "weight" associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval (0,1) that excludes the frequencies of 0 and 1. Previously, the probabilities of the absorbing frequencies of 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range (0,1) were found from the forward diffusion equation. Our unified approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability.     

A continuity equation for cell populations

A continuity equation for cell-number density in a population of cells is derived, and a system of equations for eliminating parameters between the general solution and the initial distribution obtained.     

Comments on the approximate solution of the diffusion equation: I

The approximation method of N. Rashevsky is discussed and reviewed. It is shown that in addition to theexplicit assumptions and approximations there is involved the assumption that the rate of metabolism is the same at every point in the cell and that theaverage rate of metabolism is different from zero. An expression is given for the error in the approximate method when the rate of metabolism is any function of the concentration. It is also shown that a solution in theform of that obtained by the approximate method is not possible if the generalized laws of diffusion are assumed to apply. A portion of this work was performed while the author was a research participant, Oak Ridge Institute of Nuclear Studies, assigned to the Mathematics Panel, Oak Ridge National Laboratory.     

Comments on the approximate solution of the diffusion equation: II

On the basis of a previous general formulation (Bull. Math. Biophysics,15, 21–29, 1953a) a discussion is given of the error in the approximation method of N. Rashevsky. This error, inherent in the method when the metabolic rate is different at each point in the cell, is discussed in detail and numerical values are presented for two particular cases: the rate proportional to the concentration and the rate a prescribed function of the spatial coordinates. It is shown that the formulation for the first case also applies to several other cases, that the error is negligible provided the rate is sufficiently small, and that the error is fairly sensitive to the cell size. If the rate depends upon the coordinatesalone a small rate is not sufficient to insure a negligible error. The relations between the exact method, the standard approximate method, an earlier approximate method (Physics,7 260, 1936), and a more recent refinement (Bull. Math. Biophysics,10, 201, 1948) of the standard method are discussed. This work was performed while the author was a research participant, Oak Ridge Institute of Nuclear Studies, assigned to the Mathematics Panel, Oak Ridge National Laboratory.     

A nonlinear integral equation for visual impedance

The Hartline-Ratliff equation is a linear integral equation of the second kind and is employed in modeling inhibitory networks. Saturation of the inhibiting elements is commonly modeled as a function whose form is sigmoid; however, the resulting integral equation is nonlinear. Whenever the unknown function within the integral is hypothesized to be a nondecreasing nonlinear function, the Hartline-Ratliff equation becomes a nonlinear integral equation of the Hammerstein type. We present existence and uniqueness theorems for a Hammerstein equation which represents a further generalization of the Hartline-Ratliff equation.     

A Fokker-Planck equation for growing cell populations

Closed form solutions are obtained for a Fokker-Planck model for cell growth as a function of maturation velocity and degree of maturation. For reproduction rules where daughter cells inherit their parent's maturation velocity the complete solution is derived in terms of Airy functions. For more complicated reproduction rules partial results are obtained. Emphasis is given to the relationship of these problems to time dependent linear transport theory.     

A rate equation for blood platelet aggregation

A kinetic scheme for agonist-induced aggregation of blood platelets was formulated in terms of the kinetics of agonist interaction with the nonaggregable, discoid platelets, formation of aggregable forms by shape-change reactions, and interactions among shape-changed forms. Taking into account the relative magnitudes of the rate constants of the different steps and assuming aggregation to be by hydrophobic forces, an equation similar in form to the Michaelis-Menten equation was derived to characterize aggregation kinetics. The kinetic formulation could account for several empirical observations and may be used to interpret kinetic effects of antiplatelet drugs more informatively than at present.     

A simple method for estimating cytoplasmic diffusion coefficients

The diffusion coefficients of radioactively labelled substances in cytoplasm or other fluids are determined in vitro. The fluid containing the labelled substance is filled into a cylinder with one open end, through which the labelled substance diffuses out into a stirred outer medium. The diffusion coefficient is calculated by a one-dimensional diffusion equation from the rate of loss from the cylinder, and the length of the cylinder. The diffusion coefficients of tritiated water in several fluids have been determined. The results are in good agreement with those obtained by other methods.     

A delay differential equation model for tumor growth

We present a competition model of tumor growth that includes the immune system response and a cycle-phase-specific drug. The model considers three populations: Immune system, population of tumor cells during interphase and population of tumor during mitosis. Delay differential equations are used to model the system to take into account the phases of the cell cycle. We analyze the stability of the system and prove a theorem based on the argument principle to determine the stability of a fixed point and show that the stability may depend on the delay. We show theoretically and through numerical simulations that periodic solutions may arise through Hopf Bifurcations.Send offprint requests to:Minaya Villasana