A general diffusion model for analyzing the efficacy of synaptic input to threshold neurons

We describe a general diffusion model for analyzing the efficacy of individual synaptic inputs to threshold neurons. A formal expression is obtained for the system propagator which, when given an arbitrary initial state for the cell, yields the conditional probability distribution for the state at all later times. The propagator for a cell with a finite threshold is written as a series expansion, such that each term in the series depends only on the infinite threshold propagator, which in the diffusion limit reduces to a Gaussian form. This procedure admits a graphical representation in terms of an infinite sequence of diagrams. To connect the theory to experiment, we construct an analytical expression for the primary correlation kernel (PCK) which profiles the change in the instantaneous firing rate produced by a single postsynaptic potential (PSP). Explicit solutions are obtained in the diffusion limit to first order in perturbation theory. Our approximate expression resembles the PCK obtained by computer simulation, with the accuracy depending strongly on the mode of firing. The theory is most accurate when the synaptic input drives the membrane potential to a mean level more than one standard deviation below the firing threshold, making such cells highly sensitive to synchronous synaptic input.     

Rare events out of equilibrium: mobility through the liquid–liquid interface

Transition state theory provides a well established means to compute the rate at which rare events occur; however, this is strictly an equilibrium approach. Here we consider a nonequilibrium problem of this nature in the form of transport through a liquid–liquid interface. When two immiscible liquids are coexisting in equilibrium, there will be a certain amount of mixing between the two phases, resulting in a finite linear mobility across the liquid–liquid interface. We derive an exact relationship between the mobility and the local diffusion in the direction perpendicular to the interface. We compute the mobility using both nonequilibrium molecular dynamics and a variety of linear response type approaches, with accurate agreement being obtained for the best of these. Our analysis makes it clear how the local diffusion is influenced by the inhomogeneities of the interface, even when at a distance from it. This nonlocal character to the mobility has not been appreciated before and results in a strong variation in the local diffusion, which is formally coupled to the variation in the potential of mean force. The nonlocal aspect of the diffusion requires the velocity autocorrelation function to be integrated out to far longer times than is the case for homogeneous liquids, and requires special care with regard to the choice of numerical approach.     

A method for defining steady-state unidirectional fluxes through branched chemical, osmotic and chemiosmotic reactions

A general mathematical technique is described for deriving analytical expressions and obtaining numerical solutions for the steady-state unidirectional fluxes between two chemical states via any set of intermediate states present within any hypothetical system of unbranched or branched and overlapping elementary processes. The technique is a restricted application of the theory of Markov processes with conditional probabilities being assigned to the chemical state transitions constituting the system of reactions. While, in principle, the technique requires the summation of an infinite power series of a matrix defining the conditional probabilities of single state transitions, the power series is evaluated by means of the Taylor series expansion for matrices. As this technique allows isotopic exchange velocity equations to be derived from systems of reactions in which no distinction between the labelled and unlabelled species is required it provides a distinct and independent alternative to previously proposed methods. The technique is illustrated by application to a mechanism for second-order carrier-mediated transport.     

Molecular genetics and epigenetics in the morphogenesis mechanisms

The progress of molecular genetics principally changed the views on heredity and, in the long run, wrecked the synthetic theory of evolution, designed for the microevolutionary processes in populations only. Molecular genetics as a whole is sufficient for analyzing evolutionary processes in viruses and prokaryotes. But in multicellular organisms, with the advent of more complicated morphogenesis, epigenetic processes took effect. Appealing exclusively to the integrity of the organism in ontogeny is insufficient for the understanding of these processes; further studies of the molecular basis for this integrity are required. The discovery of homeobox genes was an important step on this path. The theory of evolution should include not only the molecular processes but also the laws of ecosystem and biosphere processes, which study requires handling many problems of ecology, parasitology, palaeontology, and geology. All these fields together comprise an enormous area of knowledge for which the development of a unified theory is scarcely possible.     

A density functional theory parameterised neural network model of zirconia

We have developed a Behler–Parrinello Neural Network (BPNN) that can be employed to calculate energies and forces of zirconia bulk structures with oxygen vacancies with similar accuracy as that of the density functional theory (DFT) calculations that were used to train the BPNN. In this work, we have trained the BPNN potential with a reference set of 2178 DFT calculations and validated it against a dataset of untrained data. We have shown that the bulk structural parameters, equation of states, oxygen vacancy formation energies and diffusion barriers predicted by the BPNN potential are in good agreement with the reference DFT data. The transferability of the BPNN potential has also been benchmarked with the prediction of structures that were not included in the reference set. The evaluation time of the BPNN is orders of magnitude less than corresponding DFT calculations, although the training process of the BPNN potential requires non-negligible amount of computational resources to prepare the dataset. The computational efficiency of the NN enabled it to be used in molecular dynamics simulations of the temperature-dependent diffusion of an oxygen vacancy and the corresponding diffusion activation energy.     

A mathematical model of the P-glycoprotein pump as a mediator of multidrug resistance

Cells displaying the classic multidrug resistant (MDR) phenotype possess a transmembrane protein (p170 or P-glycoprotein) which can actively extrude cytotoxic agents from the cytoplasm. A mathematical model of this drug efflux pump has been developed. Outward transport is modeled as a facilitated diffusion process. Since energy-dependent efflux of cytotoxic agents requires that ATP also bind to p170, the model includes a dynamic calculation for efflux rate which considers Michaelis-Menten kinetics for both the substrate agent and ATP. The final system consists of one partial differential equation (PDE) for the facilitated diffusion of substrate agents out of the cell a 2×2 ordinary differential equation (ODE) system for the dynamic calculation of the ATP-ADP pool, and a dynamic algebraic calculation of the efflux rate given substrate levels at the interior cell membrane interface and ATP levels in the cell. A stability analysis of the ATP-ADP pool distribution and a simplistic closed form solution of the linearized PDE are included. Numerical simulations are also provided.     

Curve-crossing problem for Gaussian stochastic processes and its application to neural modelling

A first time crossing problem for Gaussian stochastic process and monotonic time curve is considered and results are discussed with application to neural modelling. Using diffusion approximation of the stochastic process, integral equation for probability density function of the first time crossing has been obtained. Exact solution of the equation is given for two kinds of stochastic processes which have correspondingly infinitesimal and infinitely large correlation time; approximation methods are constructed for processes characterized by intermediate values of this parameter.     

Diffusion-based learning theory for organizing visuo-motor coordination

A diffusion-based learning theory is presented and applied to organize the visuomotor coordination of an eye-hand system which has redundant motion degree of freedom (dof). This theory considers the spatial optimality of the coordination: to minimize the end-effector position error of the eye-hand system as well as the differentiation of the joint angles with respect to the end-effector positions over all the bounded work space. By introducing variational methods with respect to the space, we derive a partial differential equation (PDE) of the joint angles with respect to the work space. The equation includes a diffusion term. For the given boundary conditions and the initial conditions, it can be solved uniquely, and the solution is a well organized map. From the motor learning point of view, our approach contains both the aspects of supervised learning as well as self-organization. Firstly, we assume that the forward relation from the hand system's joint angles to its end-effector positions can be obtained using supervised learning, and at the boundary of the work space, the supervisor can provide correct joint information. Then, by evolving the diffusion equation, we organize the visuomotor coordination. We show the effectiveness of this approach using a 3-dof scale manipulator. The problems of how to realize the visuomotor map; how to utilize the resultant map in several motions; and what are the influences of the initial conditions on the map formation and the relation to the boundary conditions are also discussed using computer simulations. Our approach has three advantages: (1) it does not require too many trial motions for the eye-hand system; (2) during the map formation process, it requires only the local interactions between each node; and (3) it guarantees the final map's spatial optimality over all the bounded work space. Received: 8 May 1997 / Accepted in revised form: 12 June 1998     

Filtration,diffusion, and molecular sieving through porous cellulose membranes

1. A study has been made of the diffusion and filtration of a graded series of molecules (including tritium-labelled water, urea, glucose, antipyrine, sucrose, raffinose, and hemoglobin) in aqueous solution through porous cellulose membranes of three degrees of porosity. 2. Experimental results were in close agreement with predictions based on the membrane pore theory of Pappenheimer et al. (1,2). Restriction to molecular diffusion is a function of pore radius and molecular radius described by equation (11) in the text. Molecular sieving during ultrafiltration is a function of total pore area per unit path length, pore radius, molecular radius, and filtration rate given by equations (16) and (19). 3. Estimates of average pore radius made by means of this theory were considerably larger than estimates made by the method of Elford and Ferry (3) (Table II). Sources of error in the latter method are discussed and a new method of membrane calibration is proposed in which the total cross-sectional area of the pores is measured by direct diffusion of isotope-labelled water. 4. Steady-state osmotic pressures of solutions of sucrose and raffinose measured during molecular sieving through cellulose membranes were found to be close to the "ideal" osmotic pressures calculated by van't Hoff's law. Thus the present experimental data support the methods used by Pappenheimer et al. in their studies on living capillary walls as well as their theory of membrane pore permeability.     

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].     

Random genetic drift and gamete frequency

An analytic expression of conditional expectation of transient gamete frequency, given that one of the two loci remains polymorphic, is obtained in terms of the diffusion process by calculating the moments of the distribution. Using this expression, a model where linkage disequilibrium is introduced by a single mutation is considered. The conditional expectation of the gamete frequency given that the locus with the mutant allele remains polymorphic is presented. The behavior is significantly different from the monotonic decrease observed in the deterministic model without random genetic drift.     

On the theory of diffusion of electrolytes

In the theory of diffusion of electrolytes the following assumptions are frequently made: (i) the electrolytic solution is electrically neutral everywhere, (ii) the ionic concentrations and the electric potential all depend on a single Cartesian coordinate as the only space variable. Often the electric potential of the solution is determined on the basis of the Poisson equation alone, disregarding any other relation between this potential and the ionic concentrations. Since the Poisson equation only represents a condition which the potential fulfills, the use of this equation alone may lead to error unless the explicit relation for the potential involving a space integration of ionic concentrations is also taken into account. But if this relation is used the Poisson equation becomes redundant and, more important, assumptions (i) and (ii) appear unacceptable, the former because it leads to a zero electric potential everywhere, the latter because it is mathematically incorrect. The present paper is based on general equations of diffusion of ions, excluding the Poisson equation. These equations form a system of nonlinear integrodifferential quations whose number equals the number of ionic species present in the solution. It appears that when all ions are distributed symmetrically around a point all functions related to the above system of equations can be made dependent on a single space coordinate: the distance from the center of symmetry. Two methods of successive approximations are given for the solution of the equations in the case of spherical symmetry with limitation to the steady state. These methods are then applied to the study of the distribution of ionic concentrations and electrical potentials inside a cell of spherical shape in equilibrium with its surroundings. These methods are rapidly convergent; exact theoretical values of the electric potential are calculable on the boundary of the cell. It appears that the potential at the center of the cell is not more than ∼50% higher than at its boundary and that variation of concentration inside the cell is not very large. For instance, with 100 mV on the boundary the ionic concentration there is about four times higher than at the center. Calculations show that extremely small amounts of electricity are sufficient to account for the electric potentials currently observed. In a cell of 100 micra diameter an average concentration of only 10⁻¹⁴ mole/cm³ of a monovalent ion would be sufficient to give 1 millivolt on the boundary. This concentration is directly proportional to the voltage and inversely proportional to the square of the cell diameter. Most of the numerical results given above are obtained by considering only those ions whose electrical charge is not compensated for by ions of an opposite sign. The total concentrations may be much higher than those quoted. The theory does not take into account possible effects of structural heterogeneities which may exist in the cell, particularly of various phase boundaries. An incidental result shows that the Boltzmann distribution function in the form employed in modern theory of electrolytes is fundamentally a consequence of the mathematical theory of diffusion alone. It is pointed out, however, that Boltzmann distribution is not always compatible with the definition of the electric potential.     

M5 mesoscopic and macroscopic models for mesenchymal motion

In this paper mesoscopic (individual based) and macroscopic (population based) models for mesenchymal motion of cells in fibre networks are developed. Mesenchymal motion is a form of cellular movement that occurs in three-dimensions through tissues formed from fibre networks, for example the invasion of tumor metastases through collagen networks. The movement of cells is guided by the directionality of the network and in addition, the network is degraded by proteases. The main results of this paper are derivations of mesoscopic and macroscopic models for mesenchymal motion in a timely varying network tissue. The mesoscopic model is based on a transport equation for correlated random walk and the macroscopic model has the form of a drift-diffusion equation where the mean drift velocity is given by the mean orientation of the tissue and the diffusion tensor is given by the variance-covariance matrix of the tissue orientations. The transport equation as well as the drift-diffusion limit are coupled to a differential equation that describes the tissue changes explicitly, where we distinguish the cases of directed and undirected tissues. As a result the drift velocity and the diffusion tensor are timely varying. We discuss relations to existing models and possible applications.Dedicated to K.P. Hadeler, a great scientist, teacher, and friend.     

标记基因型中QTL基因型条件概率分布

随着分子数量遗传学的发展,人们提出了很多统计模型用于QTL定位分析。在这些模型中,首先得确定QTL在标记基因型中的条件概率分布,然后利用适当的统计方法对QTL在基因组中所处的位置进行估计。本文讨论了常见作图群体（如F2和回交群体）中在给定标记基因型下QTL的条件概率分布,提出了用Mathematics软件推导QTL基因型条件概率分布的方法。用该方法能够快速地、准确地推导出比较复杂的标记基因型中QTL基因型的条件概率分布。     

Orientational exchange approach to fluorescence anisotropy decay.

Fluorescence depolarization is a powerful technique in resolving dynamics of molecular systems. Data obtained in fluorescence depolarization experiments are highly complex. Mathematical models for analyzing data from depolarization due to rotational motion have been largely based on the rotational diffusion equation. These results have been verified by Monte Carlo simulations. It has been implicitly stated that a 90 degrees jump model between predefined orientations such as presented by G. Weber (1971. J. Chem. Phys. 55:2399-2411) should, for the specific case of fluorescence depolarization, give the same answer as the diffusion equation. Since the highly symmetric cases considered by G. Weber gave the same result as the diffusion equation, it has been desirable to use this method in cases where depolarization arises from both discrete processes and rotational diffusion. We have derived, in a compartmental formalism, the general result for excitation and emission dipoles not necessarily coincident with any of the principal rotational axes of the fluorophore from this exchange model, and have found it to be different from that of the diffusion equation approach. We have also verified this difference with a Monte Carlo simulation of our exchange model. This derivation allows us to define the limits of validity of the 90 degrees exchanges to model rotational diffusion. Also, for systems where movements may be jumps between a few preferred orientations, the actual physical mechanism of depolarization may not be accurately represented by continuous diffusion. The compartmental formalism developed here can be used to easily combine rotational motions with discrete position jumps or other level kinetics.(ABSTRACT TRUNCATED AT 250 WORDS)     

Concentration-dependent isomerization of bacteriophage T2L

We have studied the concentration and ionic strength dependence of the fiber-reorientation transition in T2L bacteriophage, using sedimentation velocity and quasielastic light scattering diffusion measurements. High-salt and low-phage concentration favor the form with fibers extended. The equilibrium between the two forms is dependent on the phage Concentration. This behavior is unexpected for a reaction that is apparently unimolecular with respect to phage, and attests to the strong increase in excluded volume and other interactions accompanying fiber extension. The fiber-extended form has anomalously high diffusion coefficient and low molecular weight according to the Svedberg equation.     

Microscopic diffusion-reaction coupling in steady-state enzyme kinetics

The theory of diffusion-controlled processes is applied to describe the steady state of a reversible enzymatic reaction with special emphasis on the effects of enzyme saturation. A standard macroscopic steady-state treatment requires only that the average diffusional influx of substrate equals the net reaction flux as well as the average diffusional efflux of product. In contrast, the microscopic diffusion-reaction coupling used here takes properly into account the conditional concentration distributions of substrate and product: Only when the enzyme is unoccupied will there be a diffusional association flux; when the enzyme is occupied, the concentration distributions will relax towards their homogeneous bulk values. In this way the relaxation effects of the non-steady state will be constantly reoccurring as the enzyme shifts between unoccupied and occupied states. Thus, one is forced to describe the steady state as the weighted sum of properly time-averaged non-stationary conditional distributions. The consequences of the theory for an appropriate assessment of the parameters obtained in Lineweaver-Burk plots are discussed. In general, our results serve to justify the simpler macroscopic coupling scheme. However, considerable deviations between the standard treatment and our analysis can occur for fast enzymes with an essentially irreversible product release.     

Temporal generalization and diffusion in forgetting

Two processes may contribute to the decrement in discriminability with increasing temporal distance between the occasioning event and later choice. One is the length of the interval and the other is generalization decrement. In the model described by White and Wixted [J. Exp. Anal. Behav. 71 (1999) 91-113], choice was predicted by the relative payoff for correct delayed matching responses, conditional on the current value of the stimulus sampled from Thurstone-like probability distributions of the effect of the sample stimuli. In the model, discriminability decreased with increasing temporal distance because the variance of the distributions increased with time. However, White and Wixted did not specify the function relating variance to temporal distance. If a diffusion process is assumed, and if diffusion increases exponentially with time, the resulting forgetting function is a negative exponential. An additional process involves exponential generalization of remembering from one time to other times. Alternative diffusion functions result in hyperbolic or power forgetting functions. The combination of two exponential processes yields forgetting functions that are double exponential in form and which appear consistent with a wide range of data.