Growth of cell populations with arbitrarily distributed cycle durations. I. basic model

A discrete model is proposed describing the growth of cell populations with arbitrary frequency distributions of cycle durations. The model assumes that each cell divides into two cells at the end of its cycle, and that each new cell is assigned an individual cycle duration according to a probability distribution that can be arbitrarily defined. The increase in the cell number is calculated, either from the numbers of cells at earlier time points or from the initial conditions of the population, by a recurrence formula; it is also approximated by the optimal exponential function, whose parameters are determined by the initial conditions. The appropriate average cycle duration is shown not to be the arithmetic or geometric mean, but rather the solution to a more complex equation. Age distributions are calculated and compared with those found in the literature. The results of the model calculations are compared with computer simulations and with observed data on populations of the ciliate Tetrahymena geleii.     

The dispersal of an initial concentration of motile bacteria.

The dispersal of an initial concentration of identical Brownian particles is accurately described by the solution of the conventional diffusion equation, and a diffusion coefficient can be assigned to the assembly of particles. However, the dispersal of an initial concentration of motile bacteria is not well described by the same solution, in spite of the similarity between the random motion of a bacterium and a Brownian particle. Reasons for the failure of the Gaussian solution of the diffusion equation to describe the dispersal of Escherichia coli are discussed. An equation is formulated which gives the concentration of dispersing organisms as a function of space and time if the speed distribution function of the assembly of organism is known and reproduction is suppressed. For three assumed speed distributions the results are compared with concentrations measured by previous authors.     

Electrostatic potentials in membrane systems

A membrane with an arbitrary distribution of fixed charges inside and on its surfaces is considered. A procedure for calculating the local electrostatic potential at an arbitrary point of the system is described and its validity discussed. This procedure is based on the linearization of the 3-dimensional Poisson-Boltzmann equation around an exact 1-dimensional solution.     

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.     

On the spread of epidemics in a closed heterogeneous population

Heterogeneity is an important property of any population experiencing a disease. Here we apply general methods of the theory of heterogeneous populations to the simplest mathematical models in epidemiology. In particular, an SIR (susceptible-infective-removed) model is formulated and analyzed when susceptibility to or infectivity of a particular disease is distributed. It is shown that a heterogeneous model can be reduced to a homogeneous model with a nonlinear transmission function, which is given in explicit form. The widely used power transmission function is deduced from the model with distributed susceptibility and infectivity with the initial gamma-distribution of the disease parameters. Therefore, a mechanistic derivation of the phenomenological model, which is believed to mimic reality with high accuracy, is provided. The equation for the final size of an epidemic for an arbitrary initial distribution of susceptibility is found. The implications of population heterogeneity are discussed, in particular, it is pointed out that usual moment-closure methods can lead to erroneous conclusions if applied for the study of the long-term behavior of the models.     

On the fisher and the cubic-polynomial equations for the propagation of species properties

For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)^1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.     

Dissipation of Energy of an AC Electric Field in a Semi-Infinite Electron Plasma with Mirror Boundary Conditions

Absorption of the electromagnetic energy in a semi-infinite electron plasma is calculated for an arbitrary degree of the electron gas degeneracy. Absorption is determined by solving the boundary-value problem on the oscillations of electron plasma in a half-space with mirror boundary conditions for electrons. The Vlasov?Boltzmann kinetic equation with the Bhatnagar–Gross–Krook collision integral for the electron distribution function and Maxwell’s equation for the electric field are employed. The electron distribution function and the electric field inside plasma are searched for in the form of expansions in the eigenfunctions of the initial set of equations. The expansion coefficients are found for the case of mirror boundary conditions. The contribution of the plasma surface to absorption is analyzed. Cases with different degrees of electron gas degeneracy are considered. It is shown that absorption of the electromagnetic energy near the surface depends substantially on the ratio between the electric field frequency and the volumetric electron collision frequency.     

Transient electric birefringence of colloidal particles immersed in shear flow. Part II. The initial response under the action of a rectangular electric pulse and the behavior at a low alternating electric field

The time-dependent rotational diffusion equation for rigid macromolecules in solution has been approximately solved for two cases in order to extend the electric birefringence technique to streaming-electric birefringence. One is for the initial period through the application of a rectangular electric pulse to the solution immersed in a low shear flow. The purpose of this is expansion of the distribution function into a function series made by the product of the powers of reduced time (= Thetat) and hydrodynamic field alpha (= G Theta , G: velocity gradient, Theta: rotary diffusion constant) and a surface harmonic P(i)(j)cos jphi. The solution for the build-up process at arbitrary electric field strength is found, but is limited to low hydrodynamic fields. The other is for the response when an alternating electric field is applied to the solution in a shear flow. Here, instead of reduced time, the maximum electric field E(0) is chosen as a parameter for the expansion. The expressions for the intensity of the transmitted light through crossed Nicols are derived in two optical systems where the polarizer is set at an angle of 45 degrees and 0 degrees to the direction of the electric field. The results in the former case show that we can determine four parameters, the ratio of velocity gradient to rotary diffusion constant, the axial ratio of a particle, the anisotropy of electric polarizability, and the optical anisotropy factor, from four values observed in two optical systems, namely, two light intensities before applying an electric field and two initial slopes of the build-up after applying an electric field. On the other hand, when a low alternating electric field with extremely high frequency is applied, the build-up of the light intensity in the former case is the same as that of electric birefringence for pure induced dipole orientation. The build-up for the latter optical system is the same as the expression for pure induced dipole orientation of Eq. (38) shown in a previous work.     

Mortality rates from size distributions

Summary A population model explicitly describing the dynamics of an arbitrary population size distribution is presented. One consequence of the model is an equation for the exact shape of the size distribution of a stationary or steady-state population. The shape is expressed as a function of sizespecific mortality and growth rates. From the equation, various mortality estimation formulas can be derived, two of which are discussed in detail. One of the methods permits estimation of size-specific mortality rates without the assumption of a theoretical growth model.Supported by a National Research Council Research Associateship     

Exact fundamental solutions of linear parabolic equations with spatially varying coefficients

The exact general solution is obtained to a linear second order ordinary differential equation which has quite general coefficients depending on an arbitrary function of the independent variable. From this, the exact fundamental solution is derived for the corresponding linear parabolic partial differential equation with coefficients depending on the single space coordinate. In a special case this latter equation reduces to one of the Fokker-Planck type. These coefficients are then generalised and the appropriate fundamental solution is obtained. Extensions are given to linear parabolic equations in two andn space dimensions. The paper provides a collection of basic examples which illustrate and develop the theory for the generation of the exact fundamental solutions. Reduction to, and the corresponding fundamental solutions of the Fokker-Planck equations is presented, where appropriate.     

Calculation of the Electrophoretic Mobility of a Particle Bearing Bound Polyelectrolyte Using the Nonlinear Poisson-Boltzmann Equation

A numerical method for determining the electrophoretic mobility of a polyelectrolyte-coated particle is presented. The particle surface is modeled as having a permeable layer of polyelectrolyte molecules anchored to its surface. Fluid flow within the polyelectrolyte layer is subject to Stokes drag arising from the polyelectrolyte segments. The method allows arbitrary distribution of polymer segments and charge density normal to the surface to be used. The hydrodynamic plane of shear may also be varied. The potential profile is determined by a numerical solution to the nonlinearized Poisson-Boltzmann equation. The potential profile is then used in a numerical solution to the Navier-Stokes equation to give the required mobility. The use of the nonlinearized Poisson-Boltzmann equation extends the results to higher charge density/lower ionic strength conditions than previous treatments. The surface potentials and mobilities for three limiting charge distributions are compared for both the linear and nonlinear treatments to delimit the range of validity of the linear treatment. The utility of the numerical, nonlinear treatment is demonstrated by an improved fit to the electrophoretic mobility of human erythrocytes as a function of ionic strength in the range 10 to 150 mM.     

A mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment

A situation is considered in which a fluid containing a substance flows through a vessel at a constant rate, the substance being permeable to the vessel wall. In the region outside the vessel there is supposed to be rapid mixing in the direction perpendicular to the axis of the vessel but no mixing longitudinally. The solution for the spatial distribution at any time is given for the case of an arbitrary initial distribution along the vessel length in the absence of an input. The solution is also given for the case of a single impulsive input, the concentration being initially zero everywhere. Work performed under Contract W-7405-eng-26 for the Atomic Energy Commission.     

Required ultracentrifugal time for condensing solution by ultracentrifuge

Required ultracentrifugal time for condensing a solution by ultracentrifuge with an angle rotor or a swing rotor was estimated by considering the calculated distribution of concentration in an ultracentrifugal tube. From the calculated distribution of the concentration a free boundary between solvent and solution was found in the ultracentrifugal tube under the condition that t^*>0.25. A condense constant was newly defined as the ratio of the mean concentration in the ultracentrifugal tube after removing the solvent to the concentration of initial solution. The equation determining the required ultracentrifugal time for condensing a solution up to a given condense constant is derived and solved numerically. The required ultracentrifugal time for an angle rotor was shorter than that for a swing rotor.     

Electromagnetic van Kampen waves

The theory of van Kampen waves in plasma with an arbitrary anisotropic distribution function is developed. The obtained solutions are explicitly expressed in terms of the permittivity tensor. There are three types of perturbations, one of which is characterized by the frequency dependence on the wave vector, while for the other two, the dispersion relation is lacking. Solutions to the conjugate equations allowing one to solve the initial value problem are analyzed.     

A general method for the evaluation of diffusion constants,dilute-solution viscoelasticity,and the complex dielectric constant of a rigid macromolecule with an arbitrary configuration. II

As a continuation of a previous paper [Biopolymers 16 , (1977)], in which we described a general method for the evaluation of the diffusion constants of a rigid macromolecule with an arbitrary configuration, this paper presents a theory for evaluating the relaxational behavior of the molecule in solution. The diffusion equation of a molecule subjected to hydrodynamic or electric forces is solved and both a complex viscosity and a complex dielectric constant are obtained by assuming that the molecule is composed of Tokes spheres and, in the dielectric property, the molecule possesses a fixed charge distribution. The viscoelasticity of a rigid rod is calculated and compared with the analytical results.     

A revised approach for an exact analytical solution for thermal response in biological tissues significant in therapeutic treatments

The genesis of the present research paper is to develop a revised exact analytical solution of thermal profile of 1-D Pennes’ bioheat equation (PBHE) for living tissues influenced in thermal therapeutic treatments. In order to illustrate the temperature distribution in living tissue both Fourier and non-Fourier model of 1-D PBHE has been solved by ‘Separation of variables’ technique. Till date most of the research works have been carried out with the constant initial steady temperature of tissue which is not at all relevant for the biological body due to its nonhomogeneous living cells. There should be a temperature variation in the body before the therapeutic treatment. Therefore, a coupled heat transfer in skin surface before therapeutic heating must be taken account for establishment of exact temperature propagation. This approach has not yet been considered in any research work. In this work, an initial condition for solving governing differential equation of heat conduction in biological tissues has been represented as a function of spatial coordinate. In a few research work, initial temperature distribution with PBHE has been coupled in such a way that it eliminates metabolic heat generation. The study has been devoted to establish the comparison of thermal profile between present approach and published theoretical approach for particular initial and boundary conditions inflicted in this investigation. It has been studied that maximum temperature difference of existing approach for Fourier temperature distribution is 19.6% while in case of non-Fourier, it is 52.8%. We have validated our present analysis with experimental results and it has been observed that the temperature response based on the spatial dependent variable initial condition matches more accurately than other approaches.     

Cable Theory for Finite Length Dendritic Cylinders with Initial and Boundary Conditions

The cable equation is solved in the Laplace transform domain for arbitrary initial and boundary conditions. The cable potential is expressed directly in terms of the impedance of the terminations and the cable electrotonic length. A computer program is given to invert the transform. Numerical solutions may be obtained for any particular model by inserting expressions describing the terminations and parameter values into the program, without further computation by the modeler. For a finite length cable, sealed at one end, the solution is expressed in terms of the ratio of the termination impedance to the impedance of the finite length cable, a generalization of the steady-state conductance ratio. Analysis of a model of a soma with several primary dendrites shows that the dendrites may be lumped into one equivalent cylinder if they have the same electrotonic length, even though they may vary in diameter. Responses obtained under voltage clamp are conceptually predictable from measurements made under current clamp, and vice versa. The equalizing time constants of an infinite series expression of the solution are the negative reciprocals of the roots of the characteristic equation. Examination of computed solutions shows that solutions which differ theoretically may be indistinguishable experimentally.     

An Analysis and Comparison of Convergence and Uniqueness of Time-Independent Bone Adaptation Models

Several stimuli are proposed in the bone remodeling theory. It is not clear, if a unique solution exists and if the result is convergent using a certain stimulus. In this study, the strain stimulus, strain energy stimulus and the von Mises stress stimulus for bone remodeling are compared and applied to a square plate model using the finite element method. In the plane stress state, the remodeling equilibrium equations are transformed into functions of only the principal strains and the graphs of these functions are drawn in a diagram using the principal strains as the variables of two coordinate axes. The equation of the sum of principal strain squared equal to a constant is a circle in the diagram. The remodeling equilibrium equation of the strain stimulus is a quadrangle fitting into the circle, the remodeling equilibrium equation of the strain energy stimulus is an ellipse and the remodeling equilibrium equation of the von Mises stress stimulus is also an ellipse close to the principal strains circle when we take the same constants in the above equations. Using the finite element method, two models are performed with the uniform initial elastic properties and with the semi-random initial distribution of the elastic properties. The principal strains as the final finite element results converge within 2% of the objective constant for all the different stimuli. The obtained Young's moduli of two models as the adaptation object are different but in equilibrium, i.e. the equilibrium solution of adaptation model is not unique. The principal strains can not be used to examine the uniqueness of solution, since two different solutions can have the same results of principal strains. Using a certain stimulus, certain initial properties and a certain iterative equation, the solution is unique in equilibrium. The results using the model in this study show also that the same results can be obtained using any of the three stimuli when a proper constant in each remodeling equilibrium equation is chosen.     

The SMM model as a boundary value problem using the discrete diffusion equation

A generalized single-step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.     

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.