Effects of diffusion on the stability of the equilibrium in multi-species ecological systems

Continuous population distributions that undergo self-diffusion, migrational cross-diffusion and interaction in a region of (1-, 2- or 3-dimensional) space are described dynamically by a governing system of nonlinear reaction-diffusion equations. It is shown that the constants associated with migrational cross-diffusion are ordinarily nonnegative or nonpositive, contingent on the type of species interaction. A simple sign relationship obtains between the latter diffusivity constants and the rate constants for species interaction in the neighborhood of a spatially uniform equilibrium state, and this relationship of signs serves to simplify the general stability theory for the growth or decay of arbitrary perturbations on a spatially uniform equilibrium state. The stability of the equilibrium state is analyzed and discussed in detail for the case of a generic two-species model, where the self-diffusion and migrational cross-diffusion of species act to either stabilize or destabilize the equilibrium, depending essentially on the character of the species interaction and also on the magnitude of the Helmholtz eigenvalues associated with the region and boundary conditions. In particular, for a prey-predator or host-parasite model, self-diffusion usually helps to stabilize the equilibrium state and migrational cross-diffusion can only act as an additional stabilizing influence, as evidenced generally by the experiments on such two-species systems. Sufficient conditions are derived for stability of the equilibrium state in the general case for an arbitrarily large number of interacting species. It is shown that the equilibrium state is always stable if all species undergo significant self-diffusion and the Helmholtz eigenvalues are suitably large.     

Further characterization of the long-run population distribution of a deleterious gene

In a population of constant size, there is an equilibrium distribution for every deleterious autosomal dominant gene. This equilibrium represents the balance between selection and mutation. The purpose of this paper is to describe an approximate method of computing the equilibrium distribution and an exact method of computing its cumulants. If the surrounding population has experienced prolonged growth or decline, then an equilibrium does not develop. However, one can show that the variance of the number of carriers divided by the current population size does stabilize; this quantity is an increasing function of the growth rate.     

Differential selectivity of the Escherichia coli cell membrane shifts the equilibrium for the enzyme-catalyzed isomerization of galactose to tagatose

An Escherichia coli galactose kinase gene knockout (DeltagalK) strain, which contains the l-arabinose isomerase gene (araA) to isomerize d-galactose to d-tagatose, showed a high conversion yield of tagatose compared with the original galK strain because galactose was not metabolized by endogenous galactose kinase. In whole cells of the DeltagalK strain, the isomerase-catalyzed reaction exhibited an equilibrium shift toward tagatose, producing a tagatose fraction of 68% at 37 degrees C, whereas the purified l-arabinose isomerase gave a tagatose equilibrium fraction of 36%. These equilibrium fractions are close to those predicted from the measured equilibrium constants of the isomerization reaction catalyzed in whole cells and by the purified enzyme. The equilibrium shift in these cells resulted from the higher uptake and lower release rates for galactose, which is a common sugar substrate, than for tagatose, which is a rare sugar product. A DeltamglB mutant had decreased uptake rates for galactose and tagatose, indicating that a methylgalactoside transport system, MglABC, is the primary contributing transporter for the sugars. In the present study, whole-cell conversion using differential selectivity of the cell membrane was proposed as a method for shifting the equilibrium in sugar isomerization reactions.     

Stability analysis of the elbow with a load

We present a model of the human elbow and study the problem of existence and stability of equilibrium states. Our main goal is to demonstrate that stable equilibrium states exist just on grounds of the mechanical properties of the muscles and the skeleton. In particular, additional control mechanisms such as reflexes are not necessary to obtain stability. We assume that the activation of flexor and extensor muscles is constant and such that the right angle is an equilibrium state. We give a complete bifurcation diagram of all equilibrium states in terms of the elbow angle, the activation of the muscles and the mass of a load. Moreover, we define a dimensionless model parameter which allows to determine whether or not there are stable equilibria at an angle of ninety degrees. It turns out that the dependency of the muscle forces on the length of the muscles is the crucial factor for the stability of such an equilibrium.     

Kinetic behavior of associating enzyme systems ofthe type M in equilibrium M2 in equilibrium M3 in equilibrium ... and of the type 2M in equilibrium D in equilibrium D2 in equilibrium D3 in equilibrium ...

A theoretical analysis has been made of dependencies of specific enzymatic activity (a) on the concentration of the enzyme for associating enzyme systems, in which the association of protein molecules leads to the formation of linear associates of an unlimited length (M in equilibrium M2 in equilibrium M3 in equilibrium ...) and is accompanied by steric shielding of active centers, and also for systems of the type 2 M in equilibrium D in equilibrium D2 in equilibrium D3 in equilibrium ... (M is an inactive monomer and D is an active dimer), in which the specific enzymatic activity of the dimer does not depend on the degree of association. For both models an analysis has been made of the S-shape of the curves of the dependence of a on the concentration of the substrate. Experimental data for glutamate dehydrogenase from ox liver and phosphofructokinase from rabbit skeletal muscles have been used as illustrations.     

Pi in equilibrium ATP exchange in the absence of proton gradient by the H+-ATPase from yeast plasma membranes

Purified soluble H+-ATPase from Schizosaccharomyces pombe catalyzes a Pi in equilibrium ATP exchange in the absence of a H+ gradient. When the pH of the assay medium is raised from 5.5 to 8.0 there is a decrease of the ATPase activity and an increase of the rate of Pi in equilibrium ATP exchange. At pH 7.0 the addition of the organic solvent dimethyl sulfoxide (20%, v/v) promotes a decrease of ATPase activity and an increase of the Pi in equilibrium ATP exchange reaction. The effect of the organic solvent on the Pi in equilibrium ATP exchange is related to a decrease of the apparent Km for Pi.     

具有年龄结构的接种流行病模型正平衡解的全局稳定性

研究一个具有年龄结构的接种SIS流行病模型正平衡解的稳定性,先利用等价积分方程给出了正平衡解存在的充分条件,再利用迭代方法及函数的单调性,得到了零平衡解与正平衡解全局稳定的充分条件。     

Rational behavioral response and the transmission of STDs

The susceptible-infected (SI) model is extended by allowing for individual optimal choices of self-protective actions against infection, where agents differ with respect to preferences and costs of self-protection. It is shown that a unique endemic equilibrium prevalence exists when the basic reproductive number of a STD is strictly greater than unity, and that the disease-free equilibrium is the unique steady state equilibrium when the basic reproductive number is less than or equal to one. Unlike in models that take individual behavior as given and fixed, the endemic equilibrium prevalence need not vary monotonically with respect to the basic reproductive number. Specifically, with endogenously determined self-protective behavior, a reduction in the basic reproductive number may in fact increase the endemic equilibrium prevalence. The global stability of the endemic steady state is established for the case of a homogeneous population by showing that, for any non-zero initial disease prevalence, there exists an equilibrium path which converges to the endemic steady state.     

Evolution under the multiallelic Levene model

The evolution of the multiallelic Levene model is investigated. New sufficient conditions for nonexistence of a completely polymorphic equilibrium and for global loss of an allele and information on which allele(s) will be lost are deduced from some new results on multidimensional recursion relations. In the absence of dominance, a more detailed analysis is presented. Sufficient conditions for global fixation of a particular allele are established. When the number of alleles equals the number of demes, necessary and sufficient conditions for the existence of an isolated, globally asymptotically stable, completely polymorphic equilibrium point are derived, and this equilibrium is explicitly determined. Three examples, one with arbitrarily many alleles and two with three alleles, illustrate the theory.     

A Geometrical Perspective for the Bargaining Problem

A new treatment to determine the Pareto-optimal outcome for a non-zero-sum game is presented. An equilibrium point for any game is defined here as a set of strategy choices for the players, such that no change in the choice of any single player will increase the overall payoff of all the players. Determining equilibrium for multi-player games is a complex problem. An intuitive conceptual tool for reducing the complexity, via the idea of spatially representing strategy options in the bargaining problem is proposed. Based on this geometry, an equilibrium condition is established such that the product of their gains over what each receives is maximal. The geometrical analysis of a cooperative bargaining game provides an example for solving multi-player and non-zero-sum games efficiently.     

Test of the Inverse Monte Carlo Method for the Calculation of Interatomic Potential Energies in Atomic Liquids

Abstract

The principle purpose of this paper is to demonstrate the use of the Inverse Monte Carlo technique for calculating pair interaction energies in monoatomic liquids from a given equilibrium property. This method is based on the mathematical relation between transition probability and pair potential given by the fundamental equation of the “importance sampling” Monte Carlo method. In order to have well defined conditions for the test of the Inverse Monte Carlo method a Metropolis Monte Carlo simulation of a Lennard Jones liquid is carried out to give the equilibrium pair correlation function determined by the assumed potential. Because an equilibrium configuration is prerequisite for an Inverse Monte Carlo simulation a model system is generated reproducing the pair correlation function, which has been calculated by the Metropolis Monte Carlo simulation and therefore representing the system in thermal equilibrium. This configuration is used to simulate virtual atom displacements. The resulting changes in atom distribution for each single simulation step are inserted in a set of non-linear equations defining the transition probability for the virtual change of configuration. The solution of the set of equations for pair interaction energies yields the Lennard Jones potential by which the equilibrium configuration has been determined.     

Hamiltonian formalism in the equilibrium problem for a plasma with islands

Hamiltonian formalism is applied to the equilibrium problem for a plasma with islands by using an analogy between the equilibrium problem for a plasma with one island and the nonlinear mechanics of a physical pendulum. A relationship is established between magnetic flux coordinates with straightened magnetic field lines and the action-angle variables. The flux and current representations of a magnetic field with islands are obtained, and the solution to the equilibrium problem for a narrow island is presented.     

Ultracentrifugal studies of the effect of molecular crowding by trimethylamine N-oxide on the self-association of muscle glycogen phosphorylase b.

The suitability of sedimentation equilibrium for characterizing the self-association of muscle glycogen phosphorylase b has been reappraised. Whereas sedimentation equilibrium distributions for phosphorylase b in 40 mM Hepes buffer (pH 6.8) supplemented with 1 mM AMP signify a lack of chemical equilibrium attainment, those in buffer supplemented additionally with potassium sulfate conform with the requirements of a dimerizing system in chemical as well as sedimentation equilibrium. Because the rate of attainment of chemical equilibrium under the former conditions is sufficiently slow to allow resolution of the dimeric and tetrameric enzyme species by sedimentation velocity, this procedure has been used to examine the effects of thermodynamic nonideality arising from molecular crowding by trimethylamine N-oxide on the self-association behaviour of phosphorylase b. In those terms the marginally enhanced extent of phosphorylase b self-association observed in the presence of high concentrations of the cosolute is taken to imply that the effects of thermodynamic nonideality on the dimer-tetramer equilibrium are being countered by those displacing the T<==>R isomerization equilibrium for dimer towards the smaller, nonassociating T state. Because the R state is the enzymically active form, an inhibitory effect is the predicted consequence of molecular crowding by high concentrations of unrelated solutes. Thermodynamic nonideality thus provides an alternative explanation for the inhibitory effects of high concentrations of glycerol, sucrose and ethylene glycol on phosphorylase b activity, phenomena that have been attributed to extremely weak interaction of these cryoprotectants with the T state of the enzyme.     

Generalization of the Monod-Wyman-Changeux model for the case of multisubstrate reactions]

A general equation is derived for the rate of multisubstrate reaction catalyzed by oligomeric enzyme E(R, T) liable to concerted transitions Ro in equilibrium To or Ro in equilibrium 2To. It is shown that with some assumptions about the enzymes the rate equations can be constructed from the rates of corresponding reactions catalyzed by a single active site. These single active site rate equations are known for the majority of catalysis mechanisms, otherwise they can be easily deduced. As an example the rate equation is derived for the reaction S1 + S2 + S3 in equilibrium S4 + S5 catalyzed by an oligomeric enzyme according to the ordered ter-bi mechanism.     

The role of recombination and selection in the modifier theory of sex-ratio distortion

The equilibrium configurations for a two-locus multialle model of sex-linked meiotic drive are studied with regard to the recombination fraction:limit cycles can occur in the case of small recombination while stable equilibrium points associated with linkage equilibrium can exist for an intermediate range of recombination values depending on the equilibrium sex ratio, linkage disequilibrium at nearby equilibrium points taking turn with loser linkage. The evolutionary dynamics in two-locus sex-ratio distortion systems is enlightened: while equilibria with a sex ratio closer to 1/2 are more likely to be stable with respect to perturbations on the frequencies of sex-ratio distorters that are represented at equilibrium, such equilibria are also more vulnerable to the invasion of mutant distorters when there is some degree of linkage with the sex-determining locus. For X-linked multimodifier systems of sex-ratio distortion, differential fertilities and viabilities are incorporated and a maximum principle is suggested.     

Equilibrium and non‐equilibrium phases in the radiation of Hakea and the drivers of diversity in Mediterranean‐type ecosystems

Mediterranean‐type ecosystems (MTEs) contain exceptional plant diversity. Explanations for this diversity are usually classed as either “equilibrium,” with elevated MTE diversity resulting from greater ecological carrying capacities, or “non‐equilibrium,” with MTEs having a greater accumulation of diversity over time than other types of ecosystems. These models have typically been considered as mutually exclusive. Here, we present a trait‐based explanatory framework that incorporates both equilibrium and non‐equilibrium dynamics. Using a large continental Australian plant radiation (Hakea) as a case study, we identify traits associated with niche partitioning in coexisting species (α‐traits) and with environmental filtering (β‐traits), and reconstruct the mode and relative timing of diversification of these traits. Our results point to a radiation with an early non‐equilibrium phase marked by divergence of β‐traits as Hakea diversified exponentially and expanded from the southwest Australian MTE into biomes across the Australian continent. This was followed from seven million years ago by an equilibrium phase, marked by diversification of α‐traits and a slowdown in lineage diversification as MTE‐niches became saturated. These results suggest that processes consistent with both equilibrium and non‐equilibrium models have been important during different stages of the radiation of Hakea, and together they provide a richer explanation of present‐day diversity patterns.     

Kudos to Reviewers for the JGP: You Make Our Science Better

Rate-equilibrium free energy relationship (REFER) analysis provides information on transition-state structures and has been applied to reveal the temporal sequence in which the different regions of an ion channel protein move during a closed–open conformational transition. To date, the theory used to interpret REFER relationships has been developed only for equilibrium mechanisms. Gating of most ion channels is an equilibrium process, but recently several ion channels have been identified to have retained nonequilibrium traits in their gating cycles, inherited from transporter-like ancestors. So far it has not been examined to what extent REFER analysis is applicable to such systems. By deriving the REFER relationships for a simple nonequilibrium mechanism, this paper addresses whether an equilibrium mechanism can be distinguished from a nonequilibrium one by the characteristics of their REFER plots, and whether information on the transition-state structures can be obtained from REFER plots for gating mechanisms that are known to be nonequilibrium cycles. The results show that REFER plots do not carry information on the equilibrium nature of the underlying gating mechanism. Both equilibrium and nonequilibrium mechanisms can result in linear or nonlinear REFER plots, and complementarity of REFER slopes for opening and closing transitions is a trivial feature true for any mechanism. Additionally, REFER analysis provides limited information about the transition-state structures for gating schemes that are known to be nonequilibrium cycles.     

The nature of fatty acid modifies the equilibrium position in the esterification catalyzed by lipase

The equilibrium position in lipase mediated esterification of various fatty acids and butanol was studied. The influence of the chain length and the presence of unsaturations in the fatty acids on the equilibrium position was measured and predicted. To predict equilibrium position the program TREP extended (TREPEX) based on the UNIFAC group contribution method was used. Using an equilibrium constant of 35, calculated on the basis of thermodynamic activities, the equilibrium position between butanol and saturated and/or unsaturated fatty acids with different chain lengths can be predicted. The ester mole fraction at equilibrium increases with the fatty acid chain length, and for fatty acids with the same carbon number, the highest values are found for unsaturated fatty acids. For reaction systems containing two saturated fatty acids, a slightly higher mole fraction is obtained for the fatty acid with the higher chain length, while for mixtures consisting of saturated and unsaturated fatty acids, the mole fractions of the unsaturated esters are lower than those of the saturated ones, regardless the chain length of the fatty acid. These experimental results are in good agreement with the calculations with TREPEX.     

Iodination significantly influences the binding of human transferrin to the transferrin receptor

The human transferrin receptor (TfR) and its ligand, the serum iron carrier transferrin, serve as a model system for endocytic receptors. Although the complete structure of the receptor's ectodomain and a partial structure of the ligand have been published, conflicting results still exist about the magnitude of equilibrium binding constants, possibly due to different labeling techniques. In the present study, we determined the equilibrium binding constant of purified human TfR and transferrin. The results were compared to those obtained with either iodinated TfR or transferrin. Using an enzyme-linked assay for receptor-ligand interactions based on the published direct calibration ELISA technique, we determined an equilibrium constant of Kd=0.22 nM for the binding of unmodified human Tf to surface-immobilized human TfR. In a reciprocal experiment using soluble receptor and surface-bound transferrin, a similar constant of Kd=0.23 nM was measured. In contrast, covalent labeling of either TfR or transferrin with 125I reduced the affinity 3-5-fold to Kd=0.66 nM and Kd=1.01 nM, respectively. The decrease in affinity upon iodination of transferrin is contrasted by an only 1.9-fold decrease in the association rate constant, suggesting that the iodination affects rather the dissociation than the association kinetics. These results indicate that precautions should be taken when interpreting equilibrium and rate constants determined with covalently labeled components.     

An insight into tumor dormancy equilibrium via the analysis of its domain of attraction

The trajectories of the dynamic system which regulates the competition between the populations of malignant cells and immune cells may tend to an asymptotically stable equilibrium in which the sizes of these populations do not vary, which is called tumor dormancy. Especially for lower steady-state sizes of the population of malignant cells, this equilibrium represents a desirable clinical condition since the tumor growth is blocked. In this context, it is of mandatory importance to analyze the robustness of this clinical favorable state of health in the face of perturbations. To this end, the paper presents an optimization technique to determine whether an assigned rectangular region, which surrounds an asymptotically stable equilibrium point of a quadratic systems, is included into the domain of attraction of the equilibrium itself. The biological relevance of the application of this technique to the analysis of tumor growth dynamics is shown on the basis of a recent quadratic model of the tumor–immune system competition dynamics. Indeed the application of the proposed methodology allows to ensure that a given safety region, determined on the basis of clinical considerations, belongs to the domain of attraction of the tumor blocked equilibrium; therefore for the set of perturbed initial conditions which belong to such region, the convergence to the healthy steady state is guaranteed. The proposed methodology can also provide an optimal strategy for cancer treatment.