Equilibrium Theory and Geometrical Constraint Equation for Two-Component Lipid Bilayer Vesicles

This paper aims at the general mathematical framework for the equilibrium theory of two-component lipid bilayer vesicles. To take into account the influences of the local compositions together with the mean curvature and Gaussian curvature of the membrane surface, a general potential functional is constructed. We introduce two kinds of virtual displacement modes: the normal one and the tangential one. By minimizing the potential functional, the equilibrium differential equations and the boundary conditions of two-component lipid vesicles are derived. Additionally, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are presented. The physical, mathematical, and biological meanings of the equilibrium differential equations and the geometrical constraint equations are discussed. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. Numerical results can be used to explain recent experiments.     

Shape equations and curvature bifurcations induced by inhomogeneous rigidities in cell membranes

This article aims at two objectives: one is the shape equation for the equilibrium configurations of biomembranes with heterogeneous rigidities; another is the possible mechanism for curvature bifurcations in various biomembranes such as human red blood cells (RBC). The shape equation is established by treating the inhomogeneous biomembrane as a lipid bilayer vesicle containing inclusions or impurities. After careful investigation of the equation, the rigidity gradient is found to be an initial "driving force" that may destabilize the biomembrane and stimulate shape transitions, and the concept (or mechanism) termed "curvature bifurcations induced by rigidity gradients" is suggested. Various post-bifurcation modes recording the new equilibrium configurations are disclosed. A few post-bifurcation modes are found to coincide well with some practical shape transitions in cells such as the cup-like shape (stomatocyte) transition and spiculated shape (echinocyte) transition in RBC.     

含扩散和时滞的偏微分方程解的振动性

研究一类含扩散和时滞的偏微分方程解的振动性,利用平均法,通过使用偏泛函微分方程上、下解思想和泛函微分方程振动性理论,获得了其解的非负性和关于正平衡态振动的充分条件．     

Equilibrium Shape Equation and Geometrically Permissible Condition for Two-Component Lipid Bilayer Vesicles

Equilibrium shapes of vesicles composed of a mixture of partially miscible amphiphiles are investigated. To take into account the influences of the composition, a simple phenomenological coupling between the co mposition and the curvatures, including the mean curvature and the Gauss curvature of the membrane surface, is suggested. By minimizing the potential functional, the general shape equation is obtained and solved analytically for vesicles with simple shapes. Besides, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are put forward. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. The close relations between the predictions and existing experimental phenomena published recently are shown.     

Non-equilibrium theory of the allele frequency spectrum

A forward diffusion equation describing the evolution of the allele frequency spectrum is presented. The influx of mutations is accounted for by imposing a suitable boundary condition. For a Wright-Fisher diffusion with or without selection and varying population size, the boundary condition is lim(x downward arrow0)xf(x,t)=thetarho(t), where f(.,t) is the frequency spectrum of derived alleles at independent loci at time t and rho(t) is the relative population size at time t. When population size and selection intensity are independent of time, the forward equation is equivalent to the backwards diffusion usually used to derive the frequency spectrum, but this approach allows computation of the time dependence of the spectrum both before an equilibrium is attained and when population size and selection intensity vary with time. From the diffusion equation, a set of ordinary differential equations for the moments of f(.,t) is derived and the expected spectrum of a finite sample is expressed in terms of those moments. The use of the forward equation is illustrated by considering neutral and selected alleles in a highly simplified model of human history. For example, it is shown that approximately 30% of the expected total heterozygosity of neutral loci is attributable to mutations that arose since the onset of population growth in roughly the last 150,000 years.     

Markov State Models Reveal a Two-Step Mechanism of miRNA Loading into the Human Argonaute Protein: Selective Binding followed by Structural Re-arrangement

Argonaute (Ago) proteins and microRNAs (miRNAs) are central components in RNA interference, which is a key cellular mechanism for sequence-specific gene silencing. Despite intensive studies, molecular mechanisms of how Ago recognizes miRNA remain largely elusive. In this study, we propose a two-step mechanism for this molecular recognition: selective binding followed by structural re-arrangement. Our model is based on the results of a combination of Markov State Models (MSMs), large-scale protein-RNA docking, and molecular dynamics (MD) simulations. Using MSMs, we identify an open state of apo human Ago-2 in fast equilibrium with partially open and closed states. Conformations in this open state are distinguished by their largely exposed binding grooves that can geometrically accommodate miRNA as indicated in our protein-RNA docking studies. miRNA may then selectively bind to these open conformations. Upon the initial binding, the complex may perform further structural re-arrangement as shown in our MD simulations and eventually reach the stable binary complex structure. Our results provide novel insights in Ago-miRNA recognition mechanisms and our methodology holds great potential to be widely applied in the studies of other important molecular recognition systems.     

'Steady/equilibrium approximation' in relaxation and fluctuation. II. Mathematical theory of approximations in first-order reaction

A mathematical theory of the steady/equilibrium approximation for first-order reactions is presented. This gives the theoretical basis for the methods of simplifying the complex first-order reactions described in the preceding work The steady/equilibrium relation holds on every fast component after a proper inducation period T degrees T degrees is either of O(1) or less, or nearly of O(1/epsilon) depending on the reaction scheme and on the initial condition but is always less than O(1/epsilon) (as in the preceding paper [1], we use the symbol O(1) to denote a positive number of the order of unity). In the open group, the determinant of the submatrix M(p), representing the interconversion between the fast components in the group and their dissipation, is of O(1). The concentration of the fast components in the open group can thus be expressed as a linear combination of those components neighboring the group after the establishment of a steady/equilibrium relation, and can be eliminated from the reaction scheme leaving the pathway through them. On the other hand, in the closed group the determinant of Mp is of O(epsilon) or less and the components in the group are in quasi equilibrium with each other after T degrees . They are eliminated from the reaction scheme leaving the sum of the components in the closed group as a slow component.     

Stochastic models for an open biochemical system

This paper uses the theory of Markov processes to derive stochastic models for a single open biochemical system at st?ady state under 3 sets of assumptions. The system is a one substrate, one product reaction. Each set of assumptions results in a separate solution for the probability functions. A system of linear equations in the probability function as well as an equivalent differential equation in its generating function are derived. The assumption of no flux leads to the first (exact) solution of the linear equations. The form agrees with that of the closed systems. Making assumptions that simplify the system to model active transport results in the second (exact) solution to the linear equations. Assuming the presence of a large number of molecules in the system facilitates obtaining the third (approximate) solution to the differential equations.     

Steady-state kinetics of solitary batrachotoxin-treated sodium channels. Kinetics on a bounded continuum of polymer conformations.

The underlying principles of the kinetics and equilibrium of a solitary sodium channel in the steady state are examined. Both the open and closed kinetics are postulated to result from round-trip excursions from a transition region that separates the openable and closed forms. Exponential behavior of the kinetics can have origins different from small-molecule systems. These differences suggest that the probability density functions (PDFs) that describe the time dependences of the open and closed forms arise from a distribution of rate constants. The distribution is likely to arise from a thermal modulation of the channel structure, and this provides a physical basis for the following three-variable equation: [formula; see text] Here, A0 is a scaling term, k is the mean rate constant, and sigma quantifies the Gaussian spread for the contributions of a range of effective rate constants. The maximum contribution is made by k, with rates faster and slower contributing less. (When sigma, the standard deviation of the spread, goes to zero, then p(f) = A0 e-kt.) The equation is applied to the single-channel steady-state probability density functions for batrachotoxin-treated sodium channels (1986. Keller et al. J. Gen. Physiol. 88: 1-23). The following characteristics are found: (a) The data for both open and closed forms of the channel are fit well with the above equation, which represents a Gaussian distribution of first-order rate processes. (b) The simple relationship [formula; see text] holds for the mean effective rat constants. Or, equivalently stated, the values of P open calculated from the k values closely agree with the P open values found directly from the PDF data. (c) In agreement with the known behavior of voltage-dependent rate constants, the voltage dependences of the mean effective rate constants for the opening and closing of the channel are equal and opposite over the voltage range studied. That is, [formula; see text] "Bursts" are related to the well-known cage effect of solution chemistry.     

Models of growth, self-reproduction and autoregulation based on consideration of reaction vessels with distensible, semipermeable walls]

We constructed non-equilibrium thermodynamics of the open physical-chemical irreversible processes in reactors with the strain semipermeable walls. This thermodynamics does not use the reciprocal relations of Onsager, so it may be applied when the stability stationary state is far from equilibrium. One of a general consequences of this thermodynamics is the statement: coordinated growth and self-reproduction are possible near the absolute equilibrium of the dissolvent and near the absolute stability stationary state of all chemicals with the absolute conservation of the differential equations of chemical kinetics. The supposition of ideal mixing is unnecessary; this condition is fulfilled automatically with diffusion. Growth and self-reproduction are not connected with positive eigenvalue of the differential equation of chemical kinetics. It is possible to construct a model of autoregulation and differentiation with this thermodynamics. The uniquness of such autoregulation follows from the mathematical theory [1]. The mathematical foundation of this thermodynamics is given in [1].     

Quality and irreversibility: constraints on ecosystem development.

We discuss one of the most general mathematical tools for analysing dynamical systems: the master equation (ME). The ME is used to derive models for entropy production in closed and open systems. Due to dissipation in open systems, the direction of evolution of important characteristics can be opposite to those imposed on closed systems. When applying these models to soil organic matter it can be shown that the principle of minimum entropy production necessitates that more and more recalcitrant organic matter is produced the further the decomposition proceeds. The necessity to dissipate entropy can also impose a limit on the degree to which litters can decompose, but interaction between litters of differing ages can remove this constraint. This is an example of the 'priming' effect.     

A dynamical approach to chemical kinetics: Mass-action laws as generalized Riccati equations

It is shown how the fundamental laws of chemical kinetics for either open or closed systems with an arbitrarily large number of reactants can be represented as a system of Riccati-like differential equations. Through the use of a concise tensor notation, it is shown when and how the differential system is exactly reducible to linear form, a reduction without approximation that parallels the well-known similar reduction of a single simle Riccati equation. An example is worked out to show how open kinetics can lead to oscillatory chemical concentrations of the Change-Higgins type. The biologically central problem of great chemical speciation is discussed from the viewpoint of Gibbs ensemble theory within the linearized kinetics and, approximately, within the starting nonlinear kinetics where it is shown roughly how to estimate, from an overall temperature-like parameter characterizing the whole system, mean chemical levels and mean frequencies of oscillation, and where a gross oscillation of the total mass is estimated in terms of an anharmonic oscillator whose general structure is fixed from the structure of the chemical kinetic laws.     

A modified theory of the potential and inhomogeneous ion distribution near charged surfaces

A new theory is proposed to calculate the surface potential. The potential is derived from a modified force equilibrium resulting in an algebraic equation. Special boundary conditions are introduced to handle the modified forces. Numeric examples are given based on erythrocyte data and are compared with the Gouy-Chapman theory.     

Hydrostatic pressure affects the conformational equilibrium of Salmonella typhimurium tryptophan synthase

The effect of hydrostatic pressure on the tryptophan (Trp) synthase alpha2beta2 complex from Salmonella typhimurium has been investigated. Trp synthase has been shown previously to exhibit low-activity (open) and high-activity (closed) conformations. The equilibrium between the open and closed conformations of Trp synthase has been found to be affected by a wide range of variables, including alpha-subunit ligands, monovalent cations, organic solvents, pH, and temperature. The absorption spectrum of the Trp synthase-L-Ser complex shows an increase in absorption of the 423 nm band of the external aldimine, which is a characteristic of the open conformation, as hydrostatic pressure is increased from 1 to 2000 bar. The deltaV(o) and K(o) for the equilibrium between the closed and open conformations of the Trp synthase-L-Ser complex are -126 mL/mol and 0.12 for the Na+ form and -171 mL/mol and 2.3 x 10(-4) for the NH4+ form. When the Trp synthase-L-Ser complex is subjected to pressure jumps of 100-400 bar, relaxations are observed, exhibiting an increase in fluorescence emission at wavelengths greater than 455 nm, with 405 nm excitation. The relaxation to the new equilibrium position requires two exponentials to fit the data in the presence of 0.1 M Na+ and three exponentials to obtain a reasonable fit in the absence of cations and with 0.1 M NH4+. Fluorescence emission at 325 nm, with excitation at 280 nm, also increases when the Trp synthase-L-Ser complex is subjected to pressure jump. These data demonstrate that the open conformation of Trp synthase is favored by higher pressure. Thus, the open conformation has a smaller apparent net system volume than the closed conformation. We estimate that there are 35-47 more waters in the solvation shell of the open conformation than in that of the closed conformation.     

Voltage-Gated Lipid Ion Channels

Synthetic lipid membranes can display channel-like ion conduction events even in the absence of proteins. We show here that these events are voltage-gated with a quadratic voltage dependence as expected from electrostatic theory of capacitors. To this end, we recorded channel traces and current histograms in patch-experiments on lipid membranes. We derived a theoretical current-voltage relationship for pores in lipid membranes that describes the experimental data very well when assuming an asymmetric membrane. We determined the equilibrium constant between closed and open state and the open probability as a function of voltage. The voltage-dependence of the lipid pores is found comparable to that of protein channels. Lifetime distributions of open and closed events indicate that the channel open distribution does not follow exponential statistics but rather power law behavior for long open times.     

二阶脉冲微分方程边值问题解的存在性

利用Leray-Schauder不动点理论研究了一类二阶脉冲微分方程三点边值问题,将以往所研究的方程的边界条件和脉冲项做了推广,得到了其解的存在性的新结果,最后通过实例说明了结论的应用.     

Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure

We describe a mathematically exact method for the analysis of spatially structured Markov processes. The method is based on a systematic perturbation expansion around the deterministic, non-spatial mean-field theory, using the theory of distributions to account for space and the underlying stochastic differential equations to account for stochasticity. As an example, we consider a spatial version of the Levins metapopulation model, in which the habitat patches are distributed in the d-dimensional landscape Rd in a random (but possibly correlated) manner. Assuming that the dispersal kernel is characterized by a length scale L, we examine how the behavior of the metapopulation deviates from the mean-field model for a finite but large L. For example, we show that the equilibrium fraction of occupied patches is given by p(0)+c/L(d)+O(L(-3d/2)), where p(0) is the equilibrium state of the Levins model and the constant c depends on p(0), the dispersal kernel, and the structure of the landscape. We show that patch occupancy can be increased or decreased by spatial structure, but is always decreased by stochasticity. Comparison with simulations show that the analytical results are not only asymptotically exact (as L-->infinity), but a good approximation also when L is relatively small.     

Plasmastatic models of galathea traps with magnetically transparent boundaries

Mathematical models and results of calculation of plasma equilibrium in a circular cylinder with three helical or straight imbedded current-carrying conductors (i.e., in a straightened analog of a toroidal Galathea trap) are presented. The equilibrium is described in the framework of two-dimensional boundary value problems with plane and helical analogs of the Grad-Shafranov equation for the scalar magnetic flux function. Problems with first-kind boundary conditions corresponding to a magnetically transparent boundary of the cylinder and problems with second-kind boundary conditions and a given value of the electric current flowing in plasma (in addition to those flowing in the conductors) are considered. Deformations of magnetoplasma configurations in the cylinder for different formulations of the above-specified problems are investigated numerically.     

Modeling relapse in infectious diseases

An integro-differential equation is proposed to model a general relapse phenomenon in infectious diseases including herpes. The basic reproduction number R(0) for the model is identified and the threshold property of R(0) established. For the case of a constant relapse period (giving a delay differential equation), this is achieved by conducting a linear stability analysis of the model, and employing the Lyapunov-Razumikhin technique and monotone dynamical systems theory for global results. Numerical simulations, with parameters relevant for herpes, are presented to complement the theoretical results, and no evidence of sustained oscillatory solutions is found.