The Time Dependence of Single File Diffusion

The single file diffusion of particles through a narrow pore membrane separating two media is treated as a stochastic birth and death process. A set of differential-difference equations is derived to describe the probability of finding n particles in the pore at any time whose source is the left-hand medium. Explicit time-dependent solutions for an arbitary number of sites are obtained. These can be used to calculate both one-way and net flux as a function of time. Parameters are estimated from steady state permeability data, and the results of some numerical calculations are presented to illustrate the time required to approach a steady state. In many cases, significant time delays can occur.     

Spatial patterns for an interaction-diffusion equation in morphogenesis

Summary A certain interaction-diffusion equation occurring in morphogenesis is considered. This equation is proposed by Gierer and Meinhardt, which is introduced by Child's gradient theory and Turing's idea about diffusion driven instability. It is shown that slightly asymmetric gradients in the tissue produce stable striking patterns depending on its asymmetry, starting from uniform distribution of morphogens. The tool is the perturbed bifurcation theory. Moreover, from a mathematical point of view, the global existence of steady state solutions with respect to some parameters is discussed.     

Bifurcation analysis of nonlinear reaction-diffusion equations—I. Evolution equations and the steady state solutions

A model nonlinear network involving chemical reactions and diffusion is studied. The time evolution and bounds on the steady state solutions are analyzed. Spatially ordered solutions of the equations of the dissipative structure type are found by bifurcation theory. These solutions are calculated analytically and their qualitative properties are discussed.     

Why the Lysogenic State of Phage λ Is So Stable: A Mathematical Modeling Approach

We develop a mathematical model of the phage λ lysis/lysogeny switch, taking into account recent experimental evidence demonstrating enhanced cooperativity between the left and right operator regions. Model parameters are estimated from available experimental data. The model is shown to have a single stable steady state for these estimated parameter values, and this steady state corresponds to the lysogenic state. When the CI degradation rate (γ_cI) is slightly increased from its normal value (γ_cI 0.0 min⁻¹), two additional steady states appear (through a saddle-node bifurcation) in addition to the lysogenic state. One of these new steady states is stable and corresponds to the lytic state. The other steady state is an (unstable) saddle node. The coexistence these two globally stable steady states (the lytic and lysogenic states) is maintained with further increases of γ_cI until γ_cI 0.35 min⁻¹, when the lysogenic steady state and the saddle node collide and vanish (through a reverse saddle node bifurcation) leaving only the lytic state surviving. These results allow us to understand the high degree of stability of the lysogenic state because, normally, it is the only steady state. Further implications of these results for the stability of the phage λ switch are discussed, as well as possible experimental tests of the model.     

Numerical bifurcation analysis of the pattern formation in a cell based auxin transport model

Transport models of growth hormones can be used to reproduce the hormone accumulations that occur in plant organs. Mostly, these accumulation patterns are calculated using time step methods, even though only the resulting steady state patterns of the model are of interest. We examine the steady state solutions of the hormone transport model of Smith et al. (Proc Natl Acad Sci USA 103(5):1301–1306, 2006) for a one-dimensional row of plant cells. We search for the steady state solutions as a function of three of the model parameters by using numerical continuation methods and bifurcation analysis. These methods are more adequate for solving steady state problems than time step methods. We discuss a trivial solution where the concentrations of hormones are equal in all cells and examine its stability region. We identify two generic bifurcation scenarios through which the trivial solution loses its stability. The trivial solution becomes either a steady state pattern with regular spaced peaks or a pattern where the concentration is periodic in time.     

Non-Linear Bifurcation Analysis of Reaction-Diffusion Activator-Inhibator System

The paper first deals with the linear stability analysis of an activator-inhibitor reaction diffusion system to determine the nature of the bifurcation point of the system. The non-linear bifurcation analysis determining the steady state solution beyond the critical point enables us to determine characteristic features of the spatial inhomogeneous pattern formation arising out of the bifurcation of the state of the system.     

THE THEORY OF DIFFUSION IN CELL MODELS : II. SOLUTION OF THE STEADY STATE FOR THREE DIFFUSING SUBSTANCES

The differential equations describing diffusion in cell models have been extended to include the simultaneous penetration of water and two salts. These equations have been solved for the steady state. Values for the concentrations in the steady state which may be computed from the equations compare favorably with the experimental values obtained by Osterhout, Kamerling, and Stanley. Moreover, it has been shown elsewhere that the solution for the steady state is essential to a discussion of the volume change or "growth" of phase C in the models and, by analogy, in living cells.     

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.     

Calculation of time constants for intracellular diffusion in whole cell patch clamp configuration.

We present a simplified model to identify and analyze the important variables governing the diffusion of substances from pipettes into canine cardiac Purkinje cells in the whole cell patch clamp configuration. We show that diffusion of substances through the pipette is the major barrier for equilibration of the pipette and cellular contents. We solve numerically the one-dimensional diffusion equation for different pipette geometries, and we derive a simple analytic equation which allows one to estimate the time necessary to reach the steady state of intracellular concentration. The time constant of the transient to steady state is given by a pipette geometric factor times the cell volume divided by the diffusion coefficient of the substance of interest. The geometric factor is shown to be given by the ratio of pipette resistance to the resistivity of the filling solution. Additionally from our modeling, we concluded that pipette perfusion at distances greater than 20 microns from the pipette tip would not substantially reduce the time necessary to achieve the steady state.     

Coupled fluid–structure interaction hemodynamics in a zero-pressure state corrected arterial geometry

Hemodynamic conditions in large arteries are significantly affected by the interaction of the pulsatile blood flow with the distensible arterial wall. A numerical procedure for solving the fluid–structure interaction problem encountered in cardiovascular flows is presented. We consider a patient-specific carotid bifurcation geometry, obtained from 3D reconstruction of in vivo acquired tomography images, which yields a geometrical representation of the artery corresponding to its pressurized state. To recover the geometry of the artery in its zero-pressure state which is required for a fluid–structure interaction simulation we utilize inverse finite elastostatics. Time-dependent flow simulations with in vivo measured inflow volume flow rate in the 3D undeformed artery are performed through the finite element method. The coupled-momentum method for fluid–structure interaction is adopted to incorporate the influence of wall compliance in the numerical computation of the time varying flow domain. To demonstrate the importance in recovering the zero-pressure state of the artery in hemodynamic simulations we compute the time varying flow field with compliant walls for the original and the zero-pressure state corrected geometric configurations of the carotid bifurcation. The most important resulting effects in the hemodynamic environment are evaluated. Our results show a significant change in the wall shear stress distribution and the spatiotemporal extent of the recirculation regions.     

An Efficient, Nonlinear Stability Analysis for Detecting Pattern Formation in Reaction Diffusion Systems

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, nonlinear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and nonlinear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have nonlinear patterning regimes not previously detected. The real power of this technique, however, is its simplicity and applicability to larger complex systems where other nonlinear methods become intractable. This is demonstrated through analysis of a chemotaxis regulatory network comprised of interacting proteins and phospholipids. In each case, predictions of this method are verified against results of numerical simulation, linear stability, asymptotic, and/or full PDE bifurcation analyses.     

Toward a multi-membrane model for potassium conduction in squid giant axon.

Possible physical mechanisms are considered which come close to a quantitative explanation for features of the potassium admittance magnitude. At 1–30 Hz there is an elevation of [Y] and positive phase above that obtained from the Hodgkin-Huxley model. Moreover there appears to be a slight negative phase for lower frequencies. An additional important feature for model fitting is the movement of the middle zero-phase crossing to the left with depolarization. Two general classes of subsystems are discussed. (1) Extracellular: potassium accumulation, barriers to diffusion near or adjacent to the excitable membrane, diffusion with volume flow, bulklimited diffusion through the Schwann cell layer and adsorption or absorption by the Schwann cells; (2) processes intrinsic to the excitable membrane: cyclic steady state, co-operative, inactivating and second order. A generalized potassium inactivation is treated in detail which provides fairly quantitative fits to transmembrane transfer data with a voltage-dependent inactivation time constant ranging between 40 and 100 ms. However, potassium accumulation coupled with hypothesized sorptive effects of the greater membrane, particularly the Schwann cell layer, also provide reasonable fits. Based on lack of experimental evidence for an inactivation, the choice is made for a multicompartment model. When an HH membrane element is combined with accumulation-depletion in an extracellular space and with a bulk limited or surface limited diffusion through the Schwann cells good agreement is obtained with measured admittance.     

The logistic equation with a diffusionally coupled delay

The asymptotic behaviour of a logistic equation with diffusion on a bounded region and a diffusionally coupled delay is investigated. An equivelent parabolic system is derived for certain types of delays. Using a Layapunov functional, sufficient conditions for the global asymptotic stability of the constant steady state are obtained. When the global stability is lost, using Hopf's bifurcation theory, existence of travelling waves is shown for ring-like and periodic one dimensional habitats.     

Periodic and quasi-periodic behavior in resource-dependent age structured population models

To describe the dynamics of a resource-dependent age structured population, a general non-linear Leslie type model is derived. The dependence on the resources is introduced through the death rates of the reproductive age classes. The conditions assumed in the derivation of the model are regularity and plausible limiting behaviors of the functions in the model. It is shown that the model dynamics restricted to its ω-limit sets is a diffeomorphism of a compact set, and the period-1 fixed points of the model are structurally stable. The loss of stability of the non-zero steady state occurs by a discrete Hopf bifurcation. Under general conditions, and after the loss of stability of the structurally stable steady states, the time evolution of population numbers is periodic or quasi-periodic. Numerical analysis with prototype functions has been performed, and the conditions leading to chaotic behavior in time are discussed.     

Persistence in reaction diffusion models with weak allee effect

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak Allee effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation. J. S. is partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary summer grants, and a grant from Science Council of Heilongjiang Province, China.     

Aeromonas Distribution and Survival in a Thermally Altered Lake

Par Pond is a thermally enriched monomictic southeastern lake which receives heated effluent from a production nuclear reactor. Fish populations in the lake have lesions of epizooty from which Aeromonas spp. are readily isolated. Distribution and population densities of Aeromonas in the water column were measured along an oxygen and temperature gradient as well as seasonally. Greater population densities of Aeromonas occurred below the oxygen chemocline when the lake was stratified. Survival of Aeromonas hydrophila under in situ conditions in both epilimnetic and hypolimnetic waters was determined through the use of polycarbonate membrane diffusion chambers during two separate reactor operating conditions. Survival levels of pure cultures of A. hydrophila corresponded to the distribution patterns of the naturally occurring Aeromonas-like populations. The greater survival of A. hydrophila during full reactor operation suggests that the fish populations may be exposed to Aeromonas for a longer period of time than when the reactor is not operating.     

Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion

In this paper, we first propose a prey-predator model with prey-stage structure and diffusion. Then we discuss the following three problems: (1) stability of non-negative constant steady states for the reduced ODE system and the corresponding reaction diffusion system with homogeneous Neumann boundary conditions; (2) Hopf bifurcation for the ODE system; (3) Hopf bifurcation created by diffusion.     

Time scale and dimension analysis of a budding yeast cell cycle model

Background  

The progress through the eukaryotic cell division cycle is driven by an underlying molecular regulatory network. Cell cycle progression can be considered as a series of irreversible transitions from one steady state to another in the correct order. Although this view has been put forward some time ago, it has not been quantitatively proven yet. Bifurcation analysis of a model for the budding yeast cell cycle has identified only two different steady states (one for G1 and one for mitosis) using cell mass as a bifurcation parameter. By analyzing the same model, using different methods of dynamical systems theory, we provide evidence for transitions among several different steady states during the budding yeast cell cycle.     

Spatial and spatio-temporal patterns in a cell-haptotaxis model

We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.