Deviation from the Einstein relation of the single-file diffusion coefficient

The deviation of the diffusion coefficient D of a one-dimensional fluid from its tracer diffusion coefficient D^★ is calculated from Gürsey's equation of State, using a square-well interaction potential. Enormous departures from the Einstein relation D = D^* are noted.     

Kinetic models and phenomenological analysis of passive lipid translocation in single-file

Passive movement of lipids through a membrane-embedded pore is analysed with kinetic equations of transport in single-file. The number of lipids arranged along the translocation coordinate in the pore is not limited in the calculations. The assumption is made that the energetic state of a pore is independent of the sequence of lipids contained in it. The results are valid for an arbitrary number of species with identical kinetic constants. It is shown that infinitely fast diffusion of one vacant site is equivalent to the filled pore approximation, which has been used here. We introduce the concept of non-strict single-file, which allows also for exchanges of neighbouring lipids inside the pore at specified rates. The model successfully simulates the redistribution of lipids between the monolayers of red blood cell plasma membranes under operation of an active aminophospholipid translocase. Kinetic equations are related to linear flux force relations. Phenomenological coefficients are expressed and analysed in terms of kinetic constants. Plausible kinetic pore model parameters are derived from comparison with a reference simulation of a thermodynamic model of the erythrocyte transmembrane lipid distribution. Mechanical forces due to differences in compressions of the lipid molecules between the monolayers are incorporated in kinetic rate constants. It is seen how the active inward transport of aminophospholipids causes an unsymmetrical passive redistribution of the other components due to mechanical effects and cross-coupling of fluxes.     

The relation between osmotic flow and tracer solvent diffusion for single-file transport.

It is demonstrated that a reasonably general model for single-file passage of solvent through an ultra-narrow pore implies the equality of tracer diffusion and osmotic flow. This result is not trival, but follows from the exactly compensating effects of solvent-solvent interaction on the paritioning of bulk solvent into the pore and on the diffusion rate within the pore. A previous calculation of Longuet-Higgins and Austin is seen to be valid only in the absence of interactions among solvent molecules in the pore.     

Theory of single-file noise

A general theoretical approach to the analysis of electric fluctuations generated by the so-called single-file diffusion through narrow channels is presented. The formalism is a slight extension of an approach to electric fluctuations in discrete transport systems with negligible interactions between the particles recently developed by one of the authors. In the single-file transport mechanism interactions between the particles must be taken into account. Three main results of principal interest are: (a) the electric fluctuations around stationary states (at equilibrium and non-equilibrium) are determined by the time-dependent solutions of the macroscopic single-file transport equations, (b) as a direct consequence of the interactions between the ions in the single-file transport the macroscopic time-dependent current and the autocorrelation function of the microscopic current fluctuations can exhibit damped oscillatory behavior, and the current noise spectrum can show peaking, (c) the number of binding sites for the ions within the pores seems to have a strong influence on the oscillatory behavior: with increasing number of binding sites the damping of the oscillations decreases and the peaking of the spectrum becomes stronger.     

Computer simulation of single-file transport

A stochastic model of single-file transport was developed as the Markov process in continuous time technique. The model was constructed using an EC-1060 computer. Unidirectional fluxes were investigated and populations of channels were correlated with flux fluctuations. The profiles of channel populations were shown to have nonlinear shapes even with the transport of nonelectrolyte (the classical diffusion approach gives linear profiles). The relationship between the paired correlation function F(AB) and the concentration of transported particles was examined. The F(AB) profile was shown to become flattened (or exponential for asymmetrical cases) at high concentrations. The concentration dependence jA/jA0 ratio were analyzed, where jA is a single-file unidirectional flux, jA0 is unidirectional flux for the case of free diffusion. An interesting "stack" phenomenon was observed for abnormal time correlations of single-file fluxes.     

Theory of single-file noise.

A general theoretical approach to the analysis of electric fluctuations generated by the so-called single-file diffusion through narrow channels is presented. The formalism is a slight extension of an approach to electric fluctuations in discrete transport systems with negligible interactions between the particles recently developed by one of the authors. In the single-file transport mechanism interactions between the particles must be taken into account. Three main results of principal interest are: (a) the electric fluctuations around stationary states (at equilibrium and non-equilibrium) are determined by the time-dependent solutions of the macroscopic single-file transport equations, (b) as a direct consquence of the interactions between the ions in the single-file transport the macroscopic time-dependent current and the autocorrelation function of the microscopic current fluctuations can exhibit damped oscillatory behavior, and the current noise spectrum can show peaking, (c) the number of binding sites for the ions within the pores seems to have a strong influence on the oscillatory behavior: with increasing number of binding sites the damping of the oscillations decreases and the peaking of the spectrum becomes stronger.     

Potassium channels as multi-ion single-file pores

A literature review reveals many lines of evidence that both delayed rectifier and inward rectifier potassium channels are multi-ion pores. These include unidirectional flux ratios given by the 2--2.5 power of the electrochemical activity ratio, very steeply voltage-dependent block with monovalent blocking ions, relief of block by permeant ions added to the side opposite from the blocking ion, rectification depending on E--EK, and a minimum in the reversal potential or conductance as external K+ ions are replaced by an equivalent concentration of T1+ ions. We consider a channel with a linear sequence of energy barriers and binding sites. The channel can be occupied by more than one ion at a time, and ions hop in single file into vacant sites with rate constants that depend on barrier heights, membrane potential, and interionic repulsion. Such multi-ion models reproduce qualitatively the special flux properties of potassium channels when the barriers for hopping out of the pore are larger than for hopping between sites within the pore and when there is repulsion between ions. These conditions also produce multiple maxima in the conductance-ion activity relationship. In agreement with Armstrong's hypothesis (1969. J. Gen. Physiol. 54:553--575), inward rectification may be understood in terms of block by an internal blocking cation. Potassium channels must have at least three sites and often contain at least two ions at a time.     

An analysis of neutral-alleles and variable-environment diffusion models

Three diffusion models are formulated for the evolution of a diploid population with K alleles at one locus with completely symmetric mutation and random genetic drift, a variable-environment, and all the above mechanisms. For the diallelic case, the transient behavior is studied by solving the corresponding diffusion equations by an asymptotic method valid for short time intervals. The transient behavior of the three models is compared for the case when their stationary distributions are identical. The expected amount of heterozygosity is computed using the asymptotic solution and is compared to an exact result. The asymptotic results are extended to the general case with K alleles at the locus for the symmetric mutation and variable-environment models.Research supported by the National Science Foundation under Grant MCS 79-01718     

A multi-substrate single-file model for ion-coupled transporters.

Ion-coupled transporters are simulated by a model that differs from contemporary alternating-access schemes. Beginning with concepts derived from multi-ion pores, the model assumes that substrates (both inorganic ions and small organic molecules) hop a) between the solutions and binding sites and b) between binding sites within a single-file pore. No two substrates can simultaneously occupy the same site. Rate constants for hopping can be increased both a) when substrates in two sites attract each other into a vacant site between them and b) when substrates in adjacent sites repel each other. Hopping rate constants for charged substrates are also modified by the membrane field. For a three-site model, simulated annealing yields parameters to fit steady-state measurements of flux coupling, transport-associated currents, and charge movements for the GABA transporter GAT1. The model then accounts for some GAT1 kinetic data as well. The model also yields parameters that describe the available data for the rat 5-HT transporter and for the rabbit Na(+)-glucose transporter. The simulations show that coupled fluxes and other aspects of ion transport can be explained by a model that includes local substrate-substrate interactions but no explicit global conformational changes.     

A study of some diffusion models of population growth

The stochastic differential equations of many diffusion processes which arise in studies of population growth in random environments can be transformed, if the Stratonovich stochastic calculus is employed, to the equation of the Wiener process. If the transformation function has certain properties then the transition probability density function and quantities relating to the time to first attain a given population size can be obtained from the known results for the Wiener process. Some other random growth processes can be derived from the Ornstein-Uhlenbeck process. These transformation methods are applied to the random processes of Malthusian growth, Pearl-Verhulst logistic growth and a recent model of density independent growth due to Levins.     

Transport properties of single-file pores with two conformational states.

Complex facilitative membrane transporters of specific ligands may operate via inner channels subject to conformational transitions. To describe some properties of these systems, we introduce here a kinetic model of coupled transport of two species, L and w, through a two-conformational pore. The basic assumptions of the model are: a) single-file of, at most, n molecules inside the channel; b) each pore state is open to one of the compartments only; c) there is at most only one vacancy per pore; d) inside the channel, a molecule of L occupies the same positions as a molecule of w; and e) there is at most only one molecule of L per pore. We develop a general representation of the kinetic diagram of the model that is formally similar to the one used to describe one-vacancy transport through a one-conformational single-file pore. In many cases of biological importance, L could be a hydrophilic (ionic or nonionic) ligand and w could be water. The model also finds application to describe solute (w) transport under saturation conditions. In this latter case, L would be another solute, or a tracer of w. We derive steady-state expressions for the fluxes of L and w, and for the permeability coefficients. The main results obtained from the analysis of the model are the following. 1) Under the condition of equilibrium of w, the expression derived for the flux of L is formally indistinguishable from the one obtainable from a standard four-state model of ligand transport mediated by a two-conformational transporter. 2) When L is a tracer of w, we can derive an expression for the ratio between the main isotope and tracer permeability coefficients (Pw/Pd). We find that the near-equilibrium permeability ratio satisfies (n - 1) < or = (Pw/Pd)eq < or = n, a result previously derived for the one-conformational, single-file pore for the case that n > or = 2. 3) The kinetic model studied here represents a generalization of the carrier concept. In fact, for the case that n = 1 (corresponding to the classical single-occupancy carrier), the near-equilibrium permeability ratio satisfies 0 < or = (Pw/Pd)eq < or = 1, which is characteristic of a carrier performing exchange-diffusion.     

Transition densities for neutral multi-allele diffusion models.

Kimura [1955, 1956] partially solved the problem of finding the transition density function which describes the behavior of a diffusion model for evolution at one genetic locus with many neutral alleles. We complete the solution using a system of polynomials biorthogonal to polynomials suggested by Appell [1881] as generalizations of Jacobi polynomials.     

Reaction-diffusion models of development with state-dependent chemical diffusion coefficients

Reaction-diffusion models are widely used to model developmental processes. The great majority of current models invoke constant diffusion coefficients. However, the diffusion of metabolites or signals through tissues is frequently such that this assumption may reasonably be questioned. We consider several different physical mechanisms leading to effective diffusion coefficients in biological tissues which vary with the local conditions, including models in which juxtacrine signaling results in the diffusion of a signal in the absence of material transport. We develop a mathematical formalism for transforming local transport laws into diffusive terms. This procedure is appropriate when the typical length scale over which the concentrations change significantly is much greater than the dimensions of a cell. We review previous developmental models which considered the possibility of state-dependent diffusion coefficients. We also provide a few new motivating examples.     

Accounting for diffusion in agent based models of reaction-diffusion systems with application to cytoskeletal diffusion

Diffusion plays a key role in many biochemical reaction systems seen in nature. Scenarios where diffusion behavior is critical can be seen in the cell and subcellular compartments where molecular crowding limits the interaction between particles. We investigate the application of a computational method for modeling the diffusion of molecules and macromolecules in three-dimensional solutions using agent based modeling. This method allows for realistic modeling of a system of particles with different properties such as size, diffusion coefficients, and affinity as well as the environment properties such as viscosity and geometry. Simulations using these movement probabilities yield behavior that mimics natural diffusion. Using this modeling framework, we simulate the effects of molecular crowding on effective diffusion and have validated the results of our model using Langevin dynamics simulations and note that they are in good agreement with previous experimental data. Furthermore, we investigate an extension of this framework where single discrete cells can contain multiple particles of varying size in an effort to highlight errors that can arise from discretization that lead to the unnatural behavior of particles undergoing diffusion. Subsequently, we explore various algorithms that differ in how they handle the movement of multiple particles per cell and suggest an algorithm that properly accommodates multiple particles of various sizes per cell that can replicate the natural behavior of these particles diffusing. Finally, we use the present modeling framework to investigate the effect of structural geometry on the directionality of diffusion in the cell cytoskeleton with the observation that parallel orientation in the structural geometry of actin filaments of filopodia and the branched structure of lamellipodia can give directionality to diffusion at the filopodia-lamellipodia interface.     

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.