On a mathematical model describing the aggregation of amoebae

In this paper an extension of a mathematical model of Keller and Segel (1970) describing the aggregation of amoebae is presented. In their paper (Keller and Segel, 1970) they showed that the onset of the aggregation could be viewed as a spatial instability. Their instability condition involved diffusion constants of the cyclic AMP and of the amoebae as well as a constant describing the chemotactic behavior of the amoebae. In our case we consider a temporal instability that depends only on the kinetics of cyclic AMP production, degradation and transport through the cell wall. Our model then explains the oscillatory behavior of the cyclic AMP in well-stirred suspensions of amoebae. In addition we discuss existence and non-existence of nonuniform steady states of the nonlinear parabolic system involved.     

Modeling the Role of Diffusion Coefficients on Turing Instability in a Reaction-diffusion Prey-predator System

The paper is concerned with the effect of variable dispersal rates on Turing instability of a non-Lotka-Volterra reaction-diffusion system. In ecological applications, the dispersal rates of different species tends to oscillate in time. This oscillation is modeled by temporal variation in the diffusion coefficient with large as well as small periodicity. The case of large periodicity is analyzed using the theory of Floquet multipliers and that of the small periodicity by using Hill's equation. The effect of such variation on the resulting Turing space is studied. A comparative analysis of the Turing spaces with constant diffusivity and variable diffusivities is performed. Numerical simulations are carried out to support analytical findings.     

On a population pathogen model incorporating species dispersal with temporal variation in dispersal rate

In the present paper, we consider a mathematical model of ecosystem population interaction where the population suffers from a susceptible–infectious–susceptible disease. Dispersal of both the susceptible and the infective is incorporated using reaction–diffusion equations. We first study the stability criteria of the basic (non-spatial) model around the disease-free and the infected steady states. We find that the loss rate of the infective species controls disease prevalence. Also without predation pressure, the disease will continue to exist among the population. Then we analyze the spatial model with species dispersal in constant as well as in time-varying form. It is observed that though constant dispersal is unable to generate diffusion-driven instability, dispersal with sinusoidal variation in dispersion rate can generate diffusive instability when the wave number of the perturbation lies within a given range. Numerical simulations are performed to illustrate analytical studies.     

Enzyme reactions at the surface of living cells. I. Electric repulsion of charged ligands and recognition of signals from the external milieu

The dynamic behaviour of a polyelectrolyte-bound enzyme is studied when diffusion of substrate or diffusion of product is coupled to electric repulsion and to Michaelis-Menten enzyme reaction. The definition of the classical concepts of electric partition coefficients and Donnan potential of a polyelectrolyte membrane has been extended under global non-equilibrium conditions. This extension is permissible when a strong repulsion exists of substrate and product by the fixed negative charges of the membrane. Coupling between product diffusion, electric repulsion and enzyme reaction at constant advancement may result in a hysteresis loop of the partition coefficient as the product concentration is increased in the reservoir. This hysteresis loop vanishes as the rate of product diffusion increases. No hysteresis loop may occur when electric repulsion effects are coupled to substrate diffusion and reaction. The existence of multiple values of the partition coefficient for a fixed concentration of product implies that the membrane may store short-term memory of the former product concentration present in the external milieu. The occurrence of hysteresis generated by coupling enzyme reaction, product diffusion, electric partition effects at constant advancement of the reaction may be viewed as a sensing device of product concentration in the external milieu. Surprisingly, non-linearities required to generate this sensing device come from electrostatic effects and not from enzyme kinetics.     

Bet‐hedging as a complex interaction among developmental instability,environmental heterogeneity,dispersal, and life‐history strategy

One potential evolutionary response to environmental heterogeneity is the production of randomly variable offspring through developmental instability, a type of bet‐hedging. I used an individual‐based, genetically explicit model to examine the evolution of developmental instability. The model considered both temporal and spatial heterogeneity alone and in combination, the effect of migration pattern (stepping stone vs. island), and life‐history strategy. I confirmed that temporal heterogeneity alone requires a threshold amount of variation to select for a substantial amount of developmental instability. For spatial heterogeneity only, the response to selection on developmental instability depended on the life‐history strategy and the form and pattern of dispersal with the greatest response for island migration when selection occurred before dispersal. Both spatial and temporal variation alone select for similar amounts of instability, but in combination resulted in substantially more instability than either alone. Local adaptation traded off against bet‐hedging, but not in a simple linear fashion. I found higher‐order interactions between life‐history patterns, dispersal rates, dispersal patterns, and environmental heterogeneity that are not explainable by simple intuition. We need additional modeling efforts to understand these interactions and empirical tests that explicitly account for all of these factors.     

Single-image diffusion coefficient measurements of proteins in free solution

Diffusion coefficient measurements are important for many biological and material investigations, such as studies of particle dynamics and kinetics, and size determinations. Among current measurement methods, single particle tracking (SPT) offers the unique ability to simultaneously obtain location and diffusion information about a molecule while using only femtomoles of sample. However, the temporal resolution of SPT is limited to seconds for single-color-labeled samples. By directly imaging three-dimensional diffusing fluorescent proteins and studying the widths of their intensity profiles, we were able to determine the proteins' diffusion coefficients using single protein images of submillisecond exposure times. This simple method improves the temporal resolution of diffusion coefficient measurements to submilliseconds, and can be readily applied to a range of particle sizes in SPT investigations and applications in which diffusion coefficient measurements are needed, such as reaction kinetics and particle size determinations.     

Molecular motors: thermodynamics and the random walk

The biochemical cycle of a molecular motor provides the essential link between its thermodynamics and kinetics. The thermodynamics of the cycle determine the motor's ability to perform mechanical work, whilst the kinetics of the cycle govern its stochastic behaviour. We concentrate here on tightly coupled, processive molecular motors, such as kinesin and myosin V, which hydrolyse one molecule of ATP per forward step. Thermodynamics require that, when such a motor pulls against a constant load f, the ratio of the forward and backward products of the rate constants for its cycle is exp [-(DeltaG + u(0)f)/kT], where -DeltaG is the free energy available from ATP hydrolysis and u(0) is the motor's step size. A hypothetical one-state motor can therefore act as a chemically driven ratchet executing a biased random walk. Treating this random walk as a diffusion problem, we calculate the forward velocity v and the diffusion coefficient D and we find that its randomness parameter r is determined solely by thermodynamics. However, real molecular motors pass through several states at each attachment site. They satisfy a modified diffusion equation that follows directly from the rate equations for the biochemical cycle and their effective diffusion coefficient is reduced to D-v(2)tau, where tau is the time-constant for the motor to reach the steady state. Hence, the randomness of multistate motors is reduced compared with the one-state case and can be used for determining tau. Our analysis therefore demonstrates the intimate relationship between the biochemical cycle, the force-velocity relation and the random motion of molecular motors.     

Diffusion driven instability in an inhomogeneous domain

Diffusion driven instability in reaction-diffusion systems has been proposed as a mechanism for pattern formation in numerous embryological and ecological contexts. However, the possible effects of environmental inhomogeneities has received relatively little attention. We consider a general two species reaction-diffusion model in one space dimension, with one diffusion coefficient a step function of the spatial coordinate. We derive the dispersion relation and the solution of the linearized system. We apply our results to Turing-type models for both embryogenesis and predator-prey interactions. In the former case we derive conditions for pattern to be isolated in one part of the domain, and in the latter we introduce the concept of “environmental instability”. Our results suggest that environmental inhomogeneity could be an important regulator of biological pattern formation.     

Impacts of dispersal on rapid adaptation and dynamic stability of Daphnia in fluctuating environments

Prior ecological research has shown that spatial processes can enhance the temporal stability of populations in fluctuating environments. Less explored is the effect of dispersal on rapid adaptation and its concomitant impact on population dynamics. For asexually reproducing populations, theory predicts that dispersal in fluctuating environments can facilitate asynchrony among clones and enhance stability by reducing temporal variability of total population abundance. This effect is predicted when clones exhibit heritable variation in environmental optima and when fluctuations occur asynchronously among patches. We tested this in the field using artificial ponds and metapopulations composed of a diverse assemblage of Daphnia pulex clones. We directly manipulated dispersal presence/absence and environmental fluctuations in the form of nutrient pulses. Consistent with predictions, dispersal enhanced temporal asynchrony among clones in the presence of nutrient pulses; this in turn stabilized population dynamics. This effect only emerged when patches experienced spatially asynchronous nutrient pulses (dispersal had no effect when patches were synchronously pulsed). Clonal asynchrony was driven by strong positive selection for a single clone that exhibited a performance advantage under conditions of low resource availability. Our work highlights the importance of dispersal as a driver of eco-evolutionary dynamics and population stability in variable environments.     

Transient response of encapsulated enzymes in hollow-fiber reactor

A mathematical model for the transient response of encapsulated enzymes is developed showing the effects of the outer boundary layer, the encapsulating membrane, the partition coefficient, and diffusion with reaction within the encapsulating medium. The model incorporates both first-order kinetics and Michaelis-Menten kinetics for the reaction rate. Using typical hollow-fiber or microcapsule parameters, the model shows that (a) the partition coefficient affects the overall rate only when the rate-limiting step is diffusion through the membrane, (b) the transient overall effectiveness factor rises sharply with time and approaches an asymptotic value for most situations, and (c) the first-order approximation to Michaelis-Menten kinetics is not valid when the initial outside bulk concentration is higher than the Michaelis constant and the overall rate is reaction limited. The model is compared with experimental data using uricase in a hollow-fiber enzyme reactor configuration. Batch assay and CSTUER (continuous-stirred ultrafiltration enzyme reactor) studies were conducted on the free enzyme to provide some of the parameters used in the model. The CSTUER data fit the case of substrate inhibition kinetics with the apparent Michaelis constant approaching zero. The hollow-fiber reactor was conducted with uricase dissolved in both a buffer solution and a concentrated hemoglobin solution. Diffusivities of the solute were measured in both solutions as was the osmotic pressure of the hemoglobin solution. While experimental data for uricase in buffer solution could easily be matched by the model, that in the concentrated hemoglobin solution could not.     

Monte Carlo Simulation of Buffered Diffusion into and out of a Model Synapse

Buffered diffusion occurs when ligands enter or leave a restricted space, such as a chemical synapse, containing a high density of binding sites. This study used Monte Carlo simulations to determine the time and spatial dependences of buffered diffusion without a priori assumptions about kinetics. The synapse was modeled as a box with receptors on one inner face. The exterior was clamped to some ligand concentration and ligands diffused through two sides. Onset and recovery simulations were carried out and the effects of receptor density, ligand properties and synapse geometry were investigated. This study determined equilibration times for binding and the spatial gradient of unliganded receptors. Onset was characterized by a high spatial gradient; equilibration was limited by the time needed for sufficient ligands to enter the synapse. Recovery showed a low spatial gradient with receptor equilibration limited by ligand rebinding. Decreasing ligand association rate or increasing ligand diffusion coefficient reduced the role of buffered diffusion and decreased the spatial gradient. Simulations with irreversible ligands showed larger, persistent spatial gradients. These simulations identify characteristics that can be used to test whether a synaptic process is governed by buffered diffusion. They also indicate that fundamental differences in synapse function may occur with irreversible ligands.     

The effect of temporal variability on persistence conditions in rivers

There has been great interest in the invasion and persistence of algal and insect populations in rivers. Recent modeling approaches assume that the flow speed of the river is constant. In reality, however, flow speeds in rivers change significantly on various temporal scales due to seasonality, weather conditions, or many human activities such as hydroelectric dams. In this paper, we study persistence conditions by deriving the upstream invasion speed in simple reaction-advection-diffusion equations with coefficients chosen to be periodic step functions. The key methodological idea to determine the spreading speed is to use the exponential transform in order to obtain a moment generating function. In a temporally periodic environment, the averages of each coefficient function determine the minimal upstream and downstream propagation speeds for a single-compartment model. For a two-compartment model, the temporal variation can enhance population persistence.     

Statistical Analysis of Lateral Diffusion and Multistate Kinetics in Single-Molecule Imaging

Single-molecule trajectories of molecules on the membrane of living cells have indicated the possibility that the lateral mobility of individual molecules is variable with time. Such temporal variation in mobility may indicate intrinsic kinetics of multiple molecular states. To clarify the mechanisms of signal processing on the membrane, quantitative characterizations of such temporal variations are necessary. Here we propose a method to analyze and characterize the multiple states in lateral mobility and their transition kinetics from single-molecule trajectories based on a displacement probability density function and an autocorrelation function of squared displacements. We performed our method for three cases: a molecule with a single diffusion coefficient (D), a mixture of molecules in two states with different D-values, and a molecule switching between two states with different D-values. Our analysis of numerically generated trajectories successfully distinguished the three cases and estimated the characteristic parameters for mobility and the kinetics of state transitions. This method is applicable to single-molecule tracking analysis of molecules in multiple functional states with different lateral mobility on the membrane of living cells.     

Finding the one: optimal choosiness under sequential mate choice

When mates are encountered sequentially, each encounter involves a decision whether to reject the current suitor and risk not finding a better mate, or to accept them despite their flaws. I provide a flexible framework for modelling optimal choosiness when mate encounters occur unpredictably in time. The model allows for temporal variation in the fitness benefits of mating, including seasonal breeding conditions, accrual of mate search costs, survival of the choosing individual or senescence of gametes. The basic optimality framework can be applied iteratively to obtain mate choice equilibria in dynamically evolving populations. My model predicts that individuals should be choosier when the average rate of mate encounters is high, but that choosiness should decline over time as the likelihood of future mate encounters decreases. When mate encounters are uncertain, there is a trade‐off between reproductive timing and mate choice (the ‘when’ and the ‘who’). Mate choice may be selected against when reproductive timing is highly important (e.g. when breeding conditions show a narrow peak in time). This can even lead to step‐shaped mate choice functions, where individuals abruptly switch from rejecting to accepting all suitors as peak breeding conditions approach. The model contributes to our understanding of why individuals may not express mate preferences, even when there is substantial variation in mate quality.     

Influence of temporal correlation of synaptic input on the rate and variability of firing in neurons

The spike trains that transmit information between neurons are stochastic. We used the theory of random point processes and simulation methods to investigate the influence of temporal correlation of synaptic input current on firing statistics. The theory accounts for two sources for temporal correlation: synchrony between spikes in presynaptic input trains and the unitary synaptic current time course. Simulations show that slow temporal correlation of synaptic input leads to high variability in firing. In a leaky integrate-and-fire neuron model with spike afterhyperpolarization the theory accurately predicts the firing rate when the spike threshold is higher than two standard deviations of the membrane potential fluctuations. For lower thresholds the spike afterhyperpolarization reduces the firing rate below the theory's predicted level when the synaptic correlation decays rapidly. If the synaptic correlation decays slower than the spike afterhyperpolarization, spike bursts can occur during single broad peaks of input fluctuations, increasing the firing rate over the prediction. Spike bursts lead to a coefficient of variation for the interspike intervals that can exceed one, suggesting an explanation of high coefficient of variation for interspike intervals observed in vivo.     

Diffusion coefficients of metabolites in active biofilms

Diffusion coefficients of actual metabolites in completely active biofilms can be determined by applying a new concept that is based on a constant local activity in the entire biofilm. In that case, a concentration step will be transmitted unattenuated. Subsequently, the diffusion coefficient can be calculated from the response monitored with a microelectrode positioned in the biofilm without quantitative knowledge of the local microbial kinetics. The conditions required for such a constant microbial biofilm activity were formulated in terms of the Thiele modulus and the substrate concentration in the bulk liquid. This proposed method was successfully applied to determine diffusion coefficients of oxygen and glucose in agar gels containing various fractions of active immobilized microorganisms. The values obtained were compared to experimental results from well-defined inert systems. The transient response of oxygen was far more affected by the presence of the immobilized cells than glucose. This can be explained by partition of the diffusing solute between the microbial cells and the aqueous phase.     

Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model

In this paper we consider a modified spatiotemporal ecological system originating from the temporal Holling-Tanner model, by incorporating diffusion terms. The original ODE system is studied for the stability of coexisting homogeneous steady-states. The modified PDE system is investigated in detail with both numerical and analytical approaches. Both the Turing and non-Turing patterns are examined for some fixed parametric values and some interesting results have been obtained for the prey and predator populations. Numerical simulation shows that either prey or predator population do not converge to any stationary state at any future time when parameter values are taken in the Turing-Hopf domain. Prey and predator populations exhibit spatiotemporal chaos resulting from temporal oscillation of both the population and spatial instability. With help of numerical simulations we have shown that Turing-Hopf bifurcation leads to onset of spatio-temporal chaos when predator's diffusivity is much higher compared to prey population. Our investigation reveals the fact that Hopf-bifurcation is essential for the onset of spatiotemporal chaos.     

Dramatic phenotypic plasticity in barnacle legs (Balanus glandula Darwin): magnitude,age dependence,and speed of response

Abstract.— The precise dependence of barnacle leg form on flow suggests the wave-swept environment imposes strong selection on suspension feeding limbs. I conducted three experiments to determine the mechanism, age dependence, and response time of cirrus variation in the acorn barnacle Balanus glandula . (1) To test whether cirrus variation arises via genetic or environmental mechanisms, I transplanted juvenile barnacles from one wave-exposed and one protected population into high and low flow conditions. Both populations exhibited similar abilities to modify cirri in response to experimental velocities: transplanted barnacles grew legs up to 84% longer in low flow. A small (up to 24%), but significant difference between source populations suggested slight genetic divergence in leg form. (2) Because flow is heterogeneous over space and time, I tested whether cirrus plasticity was limited to juveniles by transplanting both juveniles and adults from exposed and protected shores into quiet water. Remarkably, both juveniles and adults from the wave-exposed population produced legs over 100% longer than the original population, whereas protected barnacles remained unchanged. (3) A third transplant of adults into quiet water demonstrated that wave-exposed B. glandula modified cirrus form very quickly-within 18 days, or one to two molts. Results from these experiments suggest that variation in cirrus form is largely environmentally induced, but genetic differences may account for some variation observed among field populations; spatial and temporal flow heterogeneity appear to have selected for extreme flexibility of feeding form throughout a barnacle's life; and flow heterogeneity in the wave-swept environment appears to have selected for rapid ecophenotypic responses in the form of feeding structures.     

A minimum on the mean number of steps taken in adaptive walks

I consider the adaptation of a DNA sequence when mutant fitnesses are drawn randomly from a probability distribution. I focus on "gradient" adaptation in which the population jumps to the best mutant sequence available at each substitution. Given a random starting point, I derive the distribution of the number of substitutions that occur during adaptive walks to a locally optimal sequence. I show that the mean walk length is a constant:L = e-1, where e approximately 2.72. I argue that this result represents a limit on what is possible under any form of adaptation. No adaptive algorithm on any fitness landscape can arrive at a local optimum in fewer than a mean of L = e-1 steps when starting from a random sequence. Put differently, evolution must try out at least e wild-type sequences during an average bout of adaptation.