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.     

An approximate solution to the Lamm equation

An approximate solution to the Lamm equation subject to the initial and boundary conditions for conventional sedimentation velocity experiments is derived and compared with the approximate solution of Fujita and MacCosham. Calculations with this solution demonstrate that the half-height method of estimating sedimentation coefficients yields correct values for epsilon < 0.02.     

An improved approximate solution of the Lamm equation for the simultaneous estimation of sedimentation and diffusion coefficients from sedimentation velocity experiments

Sedimentation and diffusion coefficients are important parameters to describe size and shape of macromolecules in solution. The data can be obtained from sedimentation velocity experiments by a nonlinear fitting procedure using approximate solutions for the Lamm equation. Here, we present a modification of such a model function that was originally proposed by Fujita [H. Fujita, Mathematical Theory of Sedimentation Analysis, Wiley, New York, 1962]. The extended model function is well suitable to study low molecular mass compounds. The improvement of this solution given here is based on using an adjustable value for the explicit integration variable, z, the reduced radius. This modification leads to more accurate sedimentation and diffusion coefficients compared to using a constant value of 0.5 as used by Fujita. The advantage of our modification was demonstrated by the analysis of noise-free curves calculated using the finite element method, as well as experimental curves obtained for the peptides angiotensin I and II. The relatively low sedimentation and diffusion coefficients found for both substances indicate that the peptides exist as extended chains of about 3.65 nm (angiotensin I) or 3.04 nm length (angiotensin II) in solution. The lack of higher-order structure of the peptides that was derived also from CD spectra might facilitate receptor binding, and could be one reason for the fast proteolytic digestion of the free peptides.     

Effect of orientational order on the decay of the fluorescence anisotropy in membrane suspensions. A new approximate solution of the rotational diffusion equation

We discussed the time-dependence of fluorescent emission anisotropy of a cylindrical probe in membrane vesicles. We showed that, if the motion of the probe were described as diffusion in an anisotropic environment, it would be possible to determine not only the second-rank but also the fourth-rank orientational order parameter from the decay of the fluorescence anisotropy. The approximations involved were based on an interpolation of short-time and long-time behavior of the relevant correlation functions. A general expression was derived for the time dependence of the fluorescence anisotropy in closed form, which applies to any particular distribution model. It was shown to be in good agreement with previously reported results for the cone model and the Gaussian model. Finally, the applicability of the theory to time-resolved and differential phase fluorescence depolarization experiments was discussed.     

Properties of solution of the diffusion-reaction equation. II

Two theorems relating to properties of the solutions of the equations of continuity for the concentrations of the chemical species in a diffusion-reaction system are proved. The theorems concern boundary conditions under which the flux of a specified species can be guaranteed to be directed into the reaction region and the circumstances under which any two of the conditions (i) stationarity, (ii) flux equilibrium, and (iii) chemical equilibrium, imply the third. Application of these theorems to apparent active transport and to the properties of the differential equations for specific activities in a distributed tracer system are noted.     

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.     

Properties of solution of the diffusion-reaction equation

An arbitrary set of chemical reactions is considered to occur among chemical speciesX _i . In a closed uniform reaction system certain linear combinations of the concentrations of theX _i are constants. The general construction of all such linear combinations with non-negative coefficients is given in terms of the molecular formulae for theX _i . It is shown that to each such linear combination there corresponds another which is a harmonic function when the reactions take place in an open spatially distributed stationary reaction system of arbitrary shape. Under the usual boundary conditions these harmonic functions are constants. With some restrictions upon the diffusion and permeability coefficients these constants are evaluated. This evaluation is the basis for relations between the total concentration of a given chemical group (e.g., the sum of the concentrations of a free molecule, or ion, and its various bound forms) in the reaction system, and in the surrounding medium. The bearing of these relations on apparent active transport is noted and illustrated.     

Exact solution of the multi-allelic diffusion model

We give an exact solution to the Kolmogorov equation describing genetic drift for an arbitrary number of alleles at a given locus. This is achieved by finding a change of variable which makes the equation separable, and therefore reduces the problem with an arbitrary number of alleles to the solution of a set of equations that are essentially no more complicated than that found in the two-allele case. The same change of variable also renders the Kolmogorov equation with the effect of mutations added separable, as long as the mutation matrix has equal entries in each row. Thus, this case can also be solved exactly for an arbitrary number of alleles. The general solution, which is in the form of a probability distribution, is in agreement with the previously known results. Results are also given for a wide range of other quantities of interest, such as the probabilities of extinction of various numbers of alleles, mean times to these extinctions, and the means and variances of the allele frequencies. To aid dissemination, these results are presented in two stages: first of all they are given without derivations and too much mathematical detail, and then subsequently derivations and a more technical discussion are provided.     

Spring-mass running: simple approximate solution and application to gait stability

The planar spring-mass model is frequently used to describe bouncing gaits (running, hopping, trotting, galloping) in animal and human locomotion and robotics. Although this model represents a rather simple mechanical system, an analytical solution predicting the center of mass trajectory during stance remains open. We derive an approximate solution in elementary functions assuming a small angular sweep and a small spring compression during stance. The predictive power and quality of this solution is investigated for model parameters relevant to human locomotion. The analysis shows that (i), for spring compressions of up to 20% (angle of attack > or = 60 degree, angular sweep < or = 60 degree) the approximate solution describes the stance dynamics of the center of mass within a 1% tolerance of spring compression and 0.6 degree tolerance of angular motion compared to numerical calculations, and (ii), despite its relative simplicity, the approximate solution accurately predicts stable locomotion well extending into the physiologically reasonable parameter domain. (iii) Furthermore, in a particular case, an explicit parametric dependency required for gait stability can be revealed extending an earlier, empirically found relationship. It is suggested that this approximation of the planar spring-mass dynamics may serve as an analytical tool for application in robotics and further research on legged locomotion.     

On the solution of the non-linear bio-heat equation

With reference to microwave localized hyperthermia, a non-linear model of the thermal behavior of living tissues, where local thermoregulating convective and conducting effects due to blood flow are taken into account, has been assumed. The non-linear operator equation for the space and time temperature distribution, which describes local energy balance (bio-heat equation), has been linearized and solved by using a variant of the Newton iterative method. Numerical calculations for plane stratified structures simulating living bodies, irradiated by plane electromagnetic waves, have been carried out.     

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.     

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 approximate solution to the periodic bidomain equations in one dimension

An approximate, computationally tractable solution is proposed for the potentials in the bidomain model with periodic intracellular junctions (the periodic bidomain model). This new approach is based on the one-dimensional rigorous spectral method described previously by Trayanova and Pilkington (IEEE Trans. Biomed. Eng., May 1993). The total solution to the one-dimensional periodic bidomain problem is decomposed in the spectral domain into solutions to (1) the single-fiber classical bidomain problem in which the intracellular conductivity value incorporates the average contribution from cytoplasm and junction and (2) the “junctional” potential problem due to the presence of junctions at discrete locations alone. Solving for the junctional term rigorously requires most of the numerical effort in the solution for the periodic bidomain potentials. Here the junctional potential is found approximately with little numerical effort. A comparison between the rigorous and the approximate solutions serves as a justification for the proposed approximate solution procedure. The procedure outlined in this paper is applicable to higher spatial dimensions where both tissue anisotropy and junctional inhomogeneities play a role in establishing the transmembrane potential distribution.