Transmembrane electrical potential of excitable membranes: a pore analysis influence of surface charges and surface dipoles

A treatment is proposed in order to establish the general expression of the zero-current transmembrane potential of excitable membranes. The membrane model considered here is that of a hydrocarbon layer which is impermeable to ions and which represents the lipid bilayer matrix. In this matrix are incorporated ionic channels. The ion transport process through the channels is described by the absolute-rate theory applied to pores which are seen as chains of potential energy maxima and minima. Only one of the energy barriers corresponds to the gate step, and it is strongly dependent on the transmembrane potential. The kinetic equation is related to the zero current, to electrostatic boundary conditions and to the Gouy-Chapman equation for the aqueous diffuse layer.     

The reliability of relative anion-cation permeabilities deduced from reversal (Dilution) potential measurements in ion channel studies

Measurements of anion-cation permeability ratios (e.g., P _Cl/P _Na) are most readily made by measuring changes in zero-current reversal potential when the salt concentration on one side of the membrane (e.g., external NaCl) is decreased. This is particularly useful for measuring changes in ion selectivity in wild-type and mutant channels, such as those of the ligand-gated ion channel superfamily, and has shown that many of these channels have a significant permeability to counter-ions. One Brownian dynamics study of ion permeation through such narrow ion channels failed to observe such counter-ion movement, although later, another Brownian dynamics study did observe counter-ion movement through simulations of the same channels. The question has been raised as to the reliability of such reversal potential measurements for determining permeability ratios, particularly given the use of an equation such as the Goldman-Hodgkin-Katz (GHK) equation, which is often used to calculate such ratios. A new derivation of the GHK equation in terms of activity coefficients is also included. The application of irreversible thermodynamics will be shown to qualitatively support the reliability of such experimental anion-cation permeability values derived from reversal potential measurements. It will then be shown that for such zero-current situations, different electrodiffusion models, with very different underlying assumptions, produce almost identical relative permeabilities (and reversal potentials). Finally, the results of the two Brownian dynamics simulation studies and the relationship between reversal potentials and relative permeability will be discussed.     

The conductance of single cardiac sodium channels from guinea pig depends on the intracellular sodium concentration.

Currents through DPI 201-106 modified single sodium channels have been measured in cell-free inside-out patches from guinea-pig ventricular myocytes. Single-channel conductance and reversal potential of the sodium channel have been calculated at different intracellular sodium concentrations [( Na+]i) from microscopic I-V curves, which were obtained by application of linear voltage ramps. The relation between the reversal potential and [Na+]i could be fitted with a modified Goldman-Hodgkin-Katz equation with a relative permeability for K+ over Na+ ions of 0.054. The zero-current conductance of the Na channel as a function of [Na+]i shows a plateau value at low Na concentrations, and increases in a sigmoidal manner at higher concentrations. It is concluded that the Na channel can carry outward currents and that its conductance depends on [Na+]i.     

Anomalous permeabilities of the egg cell membrane of a starfish in K(+)-Tl(+) mixtures

The electrical properties of “inward” rectifying egg cell membranes of the starfish mediastera aequalis have been studied in the presence of K(+)-Tl(+) mixtures. When the ratio of the external concentrations of these ions is changed while their sum is kept constant, both the conductance and the zero-current membrane potential go through a minimum, showing clear discrepancies from theoretical results based on conventional electrodiffusion models (E.g., Goldman’s equation). By contrast, when the ration of the two concentrations is fixed and their sum varied, the potential follows an ideal Nernst slope, consistent with Goldman’s equation. The membrane conductance which, according to previous studies on similar membranes, is to be viewed as a function of the displacement of the membrane potential from its resting value δV, shows marked differences between the cases in which K(+) or Tl(+) are the predominant ions: when K(+) is the predominant permeant ion in solution, the addition of small amounts of Tl(+) inhibits the current, while corresponding blocking effects of K(+) on the current are not observed when Tl(+) is the predominant permeant ion. Also, the time course of the conductance during voltage clamp is different in the two cases, being much faster in Tl(+) than in K(+) solution for comparable values of δV. Most of the above features are accounted for by a model in which it is assumed that the ionic channels have external binding sites for cations and that their permeability properties depend on the species of the cation bound (K(+)or Tl(+) in the present experiments).     

A self-consistent kinetic model of the stratification of plane and spherical low-pressure discharges in argon

A self-consistent kinetic model is used to describe the effect of stratification of the positive column of a plane and a spherical gas discharge in argon at low pressure. The model is based on solving the Boltzmann kinetic equation for the electron energy distribution function, the time-dependent ion continuity equation, and Poisson’s equation for the self-consistent electric field. The spatial distributions of the electron and ion densities and of the electric field in the positive column of a stratified discharge are determined. The kinetic mechanism for discharge stratification in noble gases at low pressures is explained in terms of the proposed model. The model makes it possible to describe the moving strata and to confirm the validity of the experimentally obtained dependence of the radii of the strata on their numbers in a spherical discharge.     

Unidirectional sodium and potassium fluxes through the sodium channel of squid giant axons.

Unidirectional 22Na-traced sodium influx or 42K-traced potassium efflux across the membranes of voltage-clamped squid giant axons was measured at various membrane potentials under bi-ionic conditions. Tetrodotoxin almost entirely eliminated the extra K+ efflux induced by short repetitive depolarizations in the presence of tetraethylammonium or 3,4-diaminopyridine. A method of determining the voltage dependence of the unidirectional flux through voltage-gated channels is described. This technique was used to obtain the unidirectional flux-voltage relation for the sodium channel in bi-ionic and single-ion conditions. It allows the determination of the unidirectional flux at the zero-current potential which, for influx, was found to be approximately 20% of the value measured 80 mV negative to the zero-current potential. The unidirectional flux ratio under bi-ionic conditions was also measured and the flux ratio exponent found to average 1.15 with an external sodium and an internal potassium solution. A three-barrier, two-site, multi-occupancy model previously obtained for other conditions was found to predict a similar non-unity average for the flux ratio exponent. It is also shown that some single-occupancy models can predict non-unity values for the flux ratio exponent in bi-ionic conditions.     

Density-dependence in single-species populations of plants

The general form of yield-density relationships in plant populations is discussed with reference to reciprocal equations and the $\text{3}\text{2}$ power law, which describes the concomitant changes in plant weight and density during self-thinning. A model to describe the pattern of mortality in high density populations is also discussed with particular reference to the nature of intraspecific competition within plant populations.A reparameterized version of a reciprocal equation proposed by Bleasdale & Nelder is used to describe the relationship between individual plant weight and surviving plant density. The biological interpretation of the parameters is discussed in relation to the dry matter production of isolated plants, the density at which mutual interference between neighbours becomes appreciable, and the efficiency of resource utilization at high densities.The reparameterized equation is then used together with an equation which describes mortality during self-thinning as the basis for a new model to describe the relation between total plant yield and sowing density. The law of allometry is used in conjunction with the model to describe the relationship between the weight of a plant part and density, and this then forms the basis for a model of the population dynamics of annual plants with effectively discrete generations. Finally the dynamical behaviour of plant populations is discussed. It is concluded that most plant populations will show neighbourhood stability with exponential or perhaps oscillatory damping towards an equilibrium.     

Potassium transport through lipid bilayer membranes facilitated by tentoxin dimers: A new mechanism of ion carrier transport?

The cyclic tetrapeptide tentoxin at concentrations greater than 5 X 10(-7) M selectively increases the ion conductivity for potassium of lipid bilayer membranes, while the naturally occurring derivative dihydrotentoxin has no influence on this property. Current-voltage curves, zero-current potential and charge-pulse measurements were used to characterize the action of tentoxin. The results suggest that a new mechanism of facilitated ion transport operates. The model of tentoxin dimerization and tentoxin-K+ association developed is in contradiction to the model of tentoxin pore formation described recently by Heitz et al. (Biophys. Chem. 23 (1986) 245).     

Behavior of microorganisms and internal state variables

It is shown that the dependence of cell density on the swimming parameters of microorganisms—the average speed, the average turning frequency and the motility—obtained by Oosawa and Nakaoka using a theory with internal state variables to describe the behavior of individual cells, may be derived using only macroscopic quantities. These results are compared with the predictions of a model in which a partial differential equation is used to describe the behavior of microbial populations.     

Effects of Hydraulic Soil Properties on Vegetation Pattern Formation in Sloping Landscapes

Current models of vegetation pattern formation rely on a system of weakly nonlinear reaction–diffusion equations that are coupled by their source terms. While these equations, which are used to describe a spatiotemporal planar evolution of biomass and soil water, qualitatively capture the emergence of various types of vegetation patterns in arid environments, they are phenomenological and have a limited predictive power. We ameliorate these limitations by deriving the vertically averaged Richards’ equation to describe flow (as opposed to “diffusion”) of water in partially saturated soils. This establishes conditions under which this nonlinear equation reduces to its weakly nonlinear reaction–diffusion counterpart used in the previous models, thus relating their unphysical parameters (e.g., diffusion coefficient) to the measurable soil properties (e.g., hydraulic conductivity) used to parameterize the Richards equation. Our model is valid for both flat and sloping landscapes and can handle arbitrary topography and boundary conditions. The result is a model that relates the environmental conditions (e.g., precipitation rate, runoff and soil properties) to formation of multiple patterns observed in nature (such as stripes, labyrinth and spots).     

副溶血性弧菌温度-盐度双因素预测模型的建立

本文以副溶血性弧菌VP BJ1.1997为研究对象, 采用均匀设计试验方法, 建立并验证了温度范围为7°C~43°C, 盐度范围为0.5%~9.5%NaCl的生长动力学模型。结果表明, 所选一级模型的拟合效果优劣依次为Logistic方程>Gompertz方程>Linear方程, 以Logistic方程为一级模型计算生长参数; 二级模型采用平方根模型进行拟合, 得到模型相关系数r为0.9863, 最低生长温度T min为9.0506°C, 最高生长盐度为5.93%NaCl(对应最低生长水分活度Aw min     

A critical review of the use and performance of different function types for modeling temperature-dependent development of arthropod larvae

Temperature-dependent development influences production rates of arthropods, including crustaceans important to fisheries and agricultural pests. Numerous candidate equation types (development functions) exist to describe the effect of temperature on development time, yet most studies use only a single type of equation and there is no consensus as to which, if any model predicts development rates better than the others, nor what the consequences of selecting a potentially incorrect model equation are on predicted development times. In this study, a literature search was performed of studies fitting development functions to development data of arthropod larvae (99 species). The published data of most (79) of these species were then fit with 33 commonly-used development functions. Overall performance of each function type and consequences of using a function other than the best one to model data were assessed. Performance was also related to taxonomy and the range of temperatures examined. The majority (91.1%) of studies were found to not use the best function out of those tested. Using the incorrect model lead to significantly less accurate (e.g., mean difference±SE 85.9±27.4%, range: −1.7 to 1725.5%) predictions of development times than the best function. Overall, more complex functions performed poorly relative to simpler ones. However, performance of some complex functions improved when wide temperature ranges were tested, which tended to be confined to studies of insects or arachnids compared with those of crustaceans. Results indicate the biological significance of choosing the best-fitting model to describe temperature-dependent development time data.     

Ionic selectivity, saturation, and block in sodium channels. A four- barrier model

Ionic fluxes in Na channels of myelinated axons show ionic competition, block, and deviations from simple flux independence. These phenomena are particularly evident when external Na+ ions are replaced by other permeant or impermeant ions. The observed currents require new flux equations not based on the concepts of free diffusion. A specific permeability model for the Na channel is developed from Eyring rate theory applied to a chain of saturable binding sites. There are four energy barriers in the pore and only one ion is allowed inside at a time. Deviations from independence arise from saturation. The model shows that ionic permeability ratios measured from zero-current potentials can differ from those measured from relative current amplitudes or conductances. The model can be fitted to experiments with various external sodium substitutes by varying only two parameters: For each ion the height of the major energy barrier (the selectivity filter) determines the biionic zero-current potential and the depth of the energy well (binding site) just external to that barrier then determines the current amplitudes. Voltage clamp measurements with myelinated nerve fibers are given showing numerous examples of deviations from independence in ionic fluxes. Strong blocks of ionic currents by guanidinium compounds and Tl+ ions are fitted by binding within the channel with apparent dissociation constants in the range 50-122 mM. A small block with high Na+ concentrations can be fitted by Na+ ion binding with a dissociation constant of 368 mM. The barrier model is given a molecular interpretation that includes stepwise dehydration of the permeating ion as it interacts with an ionized carboxylic acid.     

Mathematical modeling of induced foreign protein production by recombinant bacteria

A new generalized mathematical model for recombinant bacteria which includes inducer effects on cell growth and foreign protein production is developed. The model equation set was applied to a host-vector system, Escherichi coli D1210 and plasmid pSD8. Batch experiments were designed and performed in shake flasks to verify the model. A parameter estimation method was developed and proven to be efficient. Although simple, the model can effectively describe the dynamics of the production of foreign protein in recombinant bacteria and can be used for optimization and control studies to maximize foreign protein production.     

Analytical solution for a hybrid Logistic-Monod cell growth model in batch and continuous stirred tank reactor culture

Monod and Logistic growth models have been widely used as basic equations to describe cell growth in bioprocess engineering. In the case of the Monod equation, the specific growth rate is governed by a limiting nutrient, with the mathematical form similar to the Michaelis–Menten equation. In the case of the Logistic equation, the specific growth rate is determined by the carrying capacity of the system, which could be growth-inhibiting factors (i.e., toxic chemical accumulation) other than the nutrient level. Both equations have been found valuable to guide us build unstructured kinetic models to analyze the fermentation process and understand cell physiology. In this work, we present a hybrid Logistic-Monod growth model, which accounts for multiple growth-dependent factors including both the limiting nutrient and the carrying capacity of the system. Coupled with substrate consumption and yield coefficient, we present the analytical solutions for this hybrid Logistic-Monod model in both batch and continuous stirred tank reactor (CSTR) culture. Under high biomass yield (Y_x/s) conditions, the analytical solution for this hybrid model is approaching to the Logistic equation; under low biomass yield condition, the analytical solution for this hybrid model converges to the Monod equation. This hybrid Logistic-Monod equation represents the cell growth transition from substrate-limiting condition to growth-inhibiting condition, which could be adopted to accurately describe the multi-phases of cell growth and may facilitate kinetic model construction, bioprocess optimization, and scale-up in industrial biotechnology.     

Calculation of viscoelastic bead–rod flow mediated by a homogenised kinetic scale with holonomic constraints

We present a new multiscale model for complex fluids based on three scales: microscopic, kinetic and continuum. We choose the microscopic level as Kramers' bead–rod model for polymers, which we describe as a system of stochastic differential equations with an implicit constraint formulation. The associated Fokker–Planck equation is then derived, and adiabatic elimination removes the fast momentum coordinates. Approached in this way, the kinetic level reduces to a dispersive drift equation. The continuum level is modelled with a finite volume Godunov-projection algorithm. We demonstrate the computation of viscoelastic stress divergence using this multiscale approach.     

PCR扩增试验的动力学数学模型

PCR技术已日趋成熟,但因为影响因素较多、反应过程比较复杂,直到目前PCR技术已创立近二十年,尚未能给出较好的描述PCR 反应的数学方法。我们根据它的基本原理提出了能够描述其反应过程的动力学方程：Wamp=［Ntarg×(1+P)n1+0.5×Cenz×U×P×Ceactiv×(n-n1)-Ntarg× (1+n×P)］×Cu×M,准确地描述了PCR反应的产物积累规律,建立了PCR反应的动力学数学模型。用动力学数学模型预测的PE 7700仪器的CT值与仪器的实际数值一致。动力学数学模型配合适当的监测设备可以构成自动化的PCR 定量仪器。PE 7700 仪器使用本动力学模型处理、分析数据,定量结果的准确性会更好。各实验室可根据各自的实验条件,由模型估算PCR产物数量,为PCR后产物继续处理提供较准确的数量信息。本模型阐明了PCR反应在多次循环后必然由指数扩增转变为线性扩增的分子基础,为定量PCR 提供了准确的计算方法。 Abstract:The PCR technique has been set up for nearly twenty years and is becoming more and more ripe.But because of the multiple influencing factors and complicated reaction procedures,no mathematical method that can describe the PCR reaction has been given.On the basis of its elementary principle,we suggested a kinetic equation to describe the reaction procedure,Wamp=［Ntarg×(1+P)n1+0.5×Cenz×U×P×Ceactive×(n-nl)-Ntarg×(1+n×P)］×Cu×M.This equation can describe correctly the accumulation rule of PCR product and thus build up the kinetic-mathematical model of PCR reaction.The predicted CT value of PE 7700 by the kinetic-mathematical model was in accordance with the real value detected by the machine.This kinetic-mathematical model accompanied by proper detecting equipment and computer could make an automatic PCR instrument,which would produce much better result.A laboratory can predict the amount of PCR product by this model and provide accurate information for further handling of PCR product according to its own condition.In this model,the molecular basis that PCR reaction is doomed to change from exponential amplification to linear amplification had been clarified.     

When growth models are not universal: evidence from marine invertebrates

The accumulation of body mass, as growth, is fundamental to all organisms. Being able to understand which model(s) best describe this growth trajectory, both empirically and ultimately mechanistically, is an important challenge. A variety of equations have been proposed to describe growth during ontogeny. Recently, the West Brown Enquist (WBE) equation, formulated as part of the metabolic theory of ecology, has been proposed as a universal model of growth. This equation has the advantage of having a biological basis, but its ability to describe invertebrate growth patterns has not been well tested against other, more simple models. In this study, we collected data for 58 species of marine invertebrate from 15 different taxa. The data were fitted to three growth models (power, exponential and WBE), and their abilities were examined using an information theoretic approach. Using Akaike information criteria, we found changes in mass through time to fit an exponential equation form best (in approx. 73% of cases). The WBE model predominantly overestimates body size in early ontogeny and underestimates it in later ontogeny; it was the best fit in approximately 14% of cases. The exponential model described growth well in nine taxa, whereas the WBE described growth well in one of the 15 taxa, the Amphipoda. Although the WBE has the advantage of being developed with an underlying proximate mechanism, it provides a poor fit to the majority of marine invertebrates examined here, including species with determinate and indeterminate growth types. In the original formulation of the WBE model, it was tested almost exclusively against vertebrates, to which it fitted well; the model does not however appear to be universal given its poor ability to describe growth in benthic or pelagic marine invertebrates.     

Kinetic analysis of the chemotaxis of bacteria

On the basis of a kinetic model of bacterial chemotactic movement the system of differential equations was reduced to describe the phenomenon of bacterial bonds migration. It follows that Keller-Segel equation is a private case of a more general "diffusion approximation" of the kinetic model. The functional parameters of the reduced equation for E. coli K-12 are estimated.