Diffusional boundary spreading in the ultracentrifuge

A common approximation for deriving solutions to the Lamm equation is to neglect diffusion. This paper presents a singular perturbation technique that allows one to estimate the band spreading due to nonzero diffusion coefficient. We illustrate the general mathematical technique by its application to sedimentation when pressure effects are important. Comparison of the approximate solution with accurate numerical solutions shows that the relative errors are of the order of 1% both for concentration and concentration gradient for parameters of chemical interest.     

Distribution of the Fittest Individuals and the Rate of Muller's Ratchet in a Model with Overlapping Generations

Muller''s ratchet is a paradigmatic model for the accumulation of deleterious mutations in a population of finite size. A click of the ratchet occurs when all individuals with the least number of deleterious mutations are lost irreversibly due to a stochastic fluctuation. In spite of the simplicity of the model, a quantitative understanding of the process remains an open challenge. In contrast to previous works, we here study a Moran model of the ratchet with overlapping generations. Employing an approximation which describes the fittest individuals as one class and the rest as a second class, we obtain closed analytical expressions of the ratchet rate in the rare clicking regime. As a click in this regime is caused by a rare, large fluctuation from a metastable state, we do not resort to a diffusion approximation but apply an approximation scheme which is especially well suited to describe extinction events from metastable states. This method also allows for a derivation of expressions for the quasi-stationary distribution of the fittest class. Additionally, we confirm numerically that the formulation with overlapping generations leads to the same results as the diffusion approximation and the corresponding Wright-Fisher model with non-overlapping generations.     

Accuracy of neuronal interspike times calculated from a diffusion approximation

The theory of neuronal firing in Stein's model is outlined as well as the corresponding theory for a diffusion approximation which has the same first two infinitesimal moments. The diffusion approximation is derived from the discontinuous model in the limit of large input frequencies and small postsynaptic potential amplitudes. A comparison of the calculated mean interspike intervals is made for various values of the threshold for firing and various input frequencies. The diffusion approximation can underestimate the interspike interval by up to 100% or severely overestimate it, depending on the input frequencies and the threshold. A general relation between the predictions of the two models is deduced.     

Mathematical biophysics of the cell with reference to the contractility of tissues and amoeboid movements

Using a recently developed approximation method, the problem of the deformation of a cell under the action of diffusion forces is studied for the general case when the rate of deformation is so rapid that the diffusion state is not quasistationary. The possibilities of configurations with several equilibria, as well as of periodical contractions and expansions of the cell around one of such configurations are shown. The bearing of these results on the theory of tissue contractility and amoeboid movements is discussed.     

A master equation for a spatial population model with pair interactions

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approximation, the spatial logistic equation. In the limit of strong diffusion the spatial logistic equation is a good approximation to the model. However, in the limit of weak diffusion the spatial logistic equation is inaccurate because of spontaneous clustering driven by reproduction. The weak-diffusion limit can be partially analyzed using an exact solution of the master equation applicable to a competition kernel with infinite range. This analysis shows that in the case of a top-hat kernel, reducing the diffusion can increase the total population. For a Gaussian kernel, reduced diffusion invariably reduces the total population. These theoretical results are confirmed by simulation.     

Translational Diffusion in Lipid Membranes beyond the Saffman-Delbrück Approximation

The Saffman-Delbrück approximation is commonly used in biophysics to relate the membrane inclusion size to its translational diffusion coefficient and membrane viscosity. However, this approximation has a restricted validity range, and its application to determination of inclusion sizes from diffusion data may in certain cases lead to unreliable results. At the same time, the model by Hughes et al. (Hughes, B. D., B. A. Pailthorpe, and C. R. White. 1981. J. Fluid Mech. 110:349-372.), providing diffusion coefficients of membrane inclusions for arbitrary inclusion sizes and viscosities of the membrane and surrounding fluids, involves substantial computational efforts, which prevents its use in practical data analysis. We develop a simple and accurate analytical approximation to the Hughes et al. model and demonstrate its performance and utility by applying it to the recently published experimental data on translational diffusion of micrometer-sized membrane domains.     

Translational diffusion in lipid membranes beyond the Saffman-Delbruck approximation

The Saffman-Delbrück approximation is commonly used in biophysics to relate the membrane inclusion size to its translational diffusion coefficient and membrane viscosity. However, this approximation has a restricted validity range, and its application to determination of inclusion sizes from diffusion data may in certain cases lead to unreliable results. At the same time, the model by Hughes et al. (Hughes, B. D., B. A. Pailthorpe, and C. R. White. 1981. J. Fluid Mech. 110:349-372.), providing diffusion coefficients of membrane inclusions for arbitrary inclusion sizes and viscosities of the membrane and surrounding fluids, involves substantial computational efforts, which prevents its use in practical data analysis. We develop a simple and accurate analytical approximation to the Hughes et al. model and demonstrate its performance and utility by applying it to the recently published experimental data on translational diffusion of micrometer-sized membrane domains.     

Computational studies of ligand diffusion in globins: I. Leghemoglobin

The thermally assisted diffusion of a small ligand (carbon monoxide) through a protein matrix (lupine leghemoglobin) is investigated computationally. The diffusion paths are calculated by a variant of the time-dependent Hartree approximation which we call LES (locally enhanced sampling). The variant which was recently introduced by Elber and Karplus is based on the classical TDSCF approximation of Gerber et al. The simulation enables more significant search for diffusion pathways than was possible before. This is done by increasing the number of ligand trajectories using a single trajectory for the protein. We compare qualitatively diffusion rates in leghemoglobin and in myoglobin. The calculation shows that the diffusion in leghemoglobin is much faster than the diffusion in myoglobin, in agreement with experiment. The gate in leghemoglobin is opened by fluctuations at a close contact between the B/C and the G helices. The most relevant fluctuation is the rigid shift of the C helix with respect to the G helix. This path is not observed in a comparable calculation for myoglobin. This finding is rationalized by the lack of the D helix in leghemoglobin and a significantly more flexible CE loop. Supporting experimental evidence for the importance of the CE loop in leghemoglobin can be found in the kinetics studies of Gibson et al.     

Diffusion approximation of the neuronal model with synaptic reversal potentials

The stochastic neuronal model with reversal potentials is approximated. For the model with constant postsynaptic potential amplitudes, a deterministic approximation is the only one which can be applied. The diffusion approximations are performed under the conditions of random postsynaptic potential amplitudes. New diffusion models of nerve membrane potential are devised in this way. These new models are more convenient for an analytical treatment than the original model with discontinuous trajectories.     

The probability of fixation of a single mutant in an exchangeable selection model

The Cannings exchangeable model for a finite population in discrete time is extended to incorporate selection. The probability of fixation of a mutant type is studied under the assumption of weak selection. An exact formula for the derivative of this probability with respect to the intensity of selection is deduced, and developed in the case of a single mutant. This formula is expressed in terms of mean coalescence times under neutrality assuming that the coefficient of selection for the mutant type has a derivative with respect to the intensity of selection that takes a polynomial form with respect to the frequency of the mutant type. An approximation is obtained in the case where this derivative is a continuous function of the mutant frequency and the population size is large. This approximation is consistent with a diffusion approximation under moment conditions on the number of descendants of a single individual in one time step. Applications to evolutionary game theory in finite populations are presented.      

Diffusion models for chemotaxis: a statistical analysis of noninteractive unicellular movement.

A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell physiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.     

Population Genetics Inference for Longitudinally-Sampled Mutants Under Strong Selection

Longitudinal allele frequency data are becoming increasingly prevalent. Such samples permit statistical inference of the population genetics parameters that influence the fate of mutant variants. To infer these parameters by maximum likelihood, the mutant frequency is often assumed to evolve according to the Wright–Fisher model. For computational reasons, this discrete model is commonly approximated by a diffusion process that requires the assumption that the forces of natural selection and mutation are weak. This assumption is not always appropriate. For example, mutations that impart drug resistance in pathogens may evolve under strong selective pressure. Here, we present an alternative approximation to the mutant-frequency distribution that does not make any assumptions about the magnitude of selection or mutation and is much more computationally efficient than the standard diffusion approximation. Simulation studies are used to compare the performance of our method to that of the Wright–Fisher and Gaussian diffusion approximations. For large populations, our method is found to provide a much better approximation to the mutant-frequency distribution when selection is strong, while all three methods perform comparably when selection is weak. Importantly, maximum-likelihood estimates of the selection coefficient are severely attenuated when selection is strong under the two diffusion models, but not when our method is used. This is further demonstrated with an application to mutant-frequency data from an experimental study of bacteriophage evolution. We therefore recommend our method for estimating the selection coefficient when the effective population size is too large to utilize the discrete Wright–Fisher model.     

The approximation method,relational biology and organismic sets

It is pointed out that the approximation method for diffusion problems, developed by N. Rashevsky in 1937 and successfully used since then by many authors, was in a sense a precursor of relational biology. The connection between the approximation method, relational biology, and the theory of organismic sets, developed in a series of recent papers by N. Rashevsky, is discussed. A number of conclusions known to hold experimentally, are then derived from relational considerations and some of them are applied to organismic sets.     

Analytical solution to the initial-value problem for traveling bands of chemotactic bacteria

The governing parabolic partial differential equations for the diffusion and chemotactic transport of a distribution of bacteria and for the diffusion and bacterial degradation of a distribution of chemotactic agent are supplemented with boundary and initial conditions that model the recent capillary tube experiments on the formation and propagation of traveling bands of chemotactic bacteria. An iteration procedure that takes the exact solution to the “diffusionless” problem as a first approximation is applied to solve the equations of the complete theoretical model. It is shown that satisfactory agreement with experiment obtains for the analytical results of the first approximation which relate the velocity of propagation and total number of bacteria cells per unit cross-sectional area in a traveling band to the constant parameters in the governing equations and supplementary conditions. The second approximation is shown to yield approximate analytical expressions for the solution functions which are in close correspondence with previously derived traveling band solutions for values of time after the initial period of formation.     

An introduction to the mathematical structure of the Wright–Fisher model of population genetics

In this paper, we develop the mathematical structure of the Wright–Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection. We establish a new concept of a global solution for the diffusion approximation (Fokker–Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution. Thus, our solution turns out to be superior to the local solution constructed by Kimura.     

A stability analysis of the power-law steady state of marine size spectra

This paper investigates the stability of the power-law steady state often observed in marine ecosystems. Three dynamical systems are considered, describing the abundance of organisms as a function of body mass and time: a “jump-growth” equation, a first order approximation which is the widely used McKendrick–von Foerster equation, and a second order approximation which is the McKendrick–von Foerster equation with a diffusion term. All of these yield a power-law steady state. We derive, for the first time, the eigenvalue spectrum for the linearised evolution operator, under certain constraints on the parameters. This provides new knowledge of the stability properties of the power-law steady state. It is shown analytically that the steady state of the McKendrick–von Foerster equation without the diffusion term is always unstable. Furthermore, numerical plots show that eigenvalue spectra of the McKendrick–von Foerster equation with diffusion give a good approximation to those of the jump-growth equation. The steady state is more likely to be stable with a low preferred predator:prey mass ratio, a large diet breadth and a high feeding efficiency. The effects of demographic stochasticity are also investigated and it is concluded that these are likely to be small in real systems.     

An Efficient Wavelet Analysis Method to Film–Pore Diffusion Model Arising in Mathematical Chemistry

In this paper, we have established an efficient Legendre wavelet based approximation method to solve film–pore diffusion model arising in engineering. Film–pore diffusion model is widely used to determine study the kinetics of adsorption systems. The use of Legendre wavelet based approximation method is found to be accurate, simple, fast, flexible, convenient, and computationally attractive. It is shown that film–pore diffusion model satisfactorily describe kinetics of methylene blue adsorption onto the three low-cost adsorbents, Guava, teak and gulmohar plant leaf powders, used in this study.     

Instantaneous Non-Linear Processing by Pulse-Coupled Threshold Units

Contemporary theory of spiking neuronal networks is based on the linear response of the integrate-and-fire neuron model derived in the diffusion limit. We find that for non-zero synaptic weights, the response to transient inputs differs qualitatively from this approximation. The response is instantaneous rather than exhibiting low-pass characteristics, non-linearly dependent on the input amplitude, asymmetric for excitation and inhibition, and is promoted by a characteristic level of synaptic background noise. We show that at threshold the probability density of the potential drops to zero within the range of one synaptic weight and explain how this shapes the response. The novel mechanism is exhibited on the network level and is a generic property of pulse-coupled networks of threshold units.