Evolution of linkage disequilibrium of the founders in exponentially growing populations

Evolution of linkage disequilibrium of the founders in exponentially growing populations was studied using a time-inhomogeneous It? process model. The model is an extension of the diffusion approximation of the Wright-Fisher model. As a measure of linkage disequilibrium, the squared standard linkage deviation, which is defined by a ratio of the moments, was considered. A system of ordinary differential equations that these moments obey was obtained. This system can be solved numerically. By simulations, it was shown that the squared standard linkage deviation gives a good approximation of the expectation of the squared correlation coefficient of gamete frequencies. In addition, a perturbative solution was obtained when the growth rate is not large. By using the perturbation, an asymptotic formula for the squared standard linkage deviation after a large number of generations was obtained. According to the formula, the squared standard linkage deviation tends to be 1/(4Nc), where N is the current size of the population and c is the recombination fraction between two loci. It is dependent on neither the initial effective size, the growth rate, nor the mutation rate. In exponentially growing populations, linkage disequilibrium will be asymptotically the same as that in a constant size population, the effective size of which is the current effective size.     

A spatial stochastic neuronal model with Ornstein–Uhlenbeck input current

 We consider a spatial neuron model in which the membrane potential satisfies a linear cable equation with an input current which is a dynamical random process of the Ornstein–Uhlenbeck (OU) type. This form of current may represent an approximation to that resulting from the random opening and closing of ion channels on a neuron's surface or to randomly occurring synaptic input currents with exponential decay. We compare the results for the case of an OU input with those for a purely white-noise-driven cable model. The statistical properties, including mean, variance and covariance of the voltage response to an OU process input in the absence of a threshold are determined analytically. The mean and the variance are calculated as a function of time for various synaptic input locations and for values of the ratio of the time constant of decay of the input current to the time constant of decay of the membrane voltage in the physiological range for real neurons. The limiting case of a white-noise input current is obtained as the correlation time of the OU process approaches zero. The results obtained with an OU input current can be substantially different from those in the white-noise case. Using simulation of the terms in the series representation for the solution, we estimate the interspike interval distribution for various parameter values, and determine the effects of the introduction of correlation in the synaptic input stochastic process. Received: 5 March 2001 / Accepted in revised form: 7 August 2001     

Perturbation approximation of solutions of a nonlinear inverse problem arising in olfaction experimentation

In this paper, a mathematical model of the diffusion of cAMP into olfactory cilia and the resulting electrical activity is presented. The model, which consists of two nonlinear differential equations, is studied using perturbation techniques. The unknowns in the problem are the concentration of cAMP, the membrane potential, and the quantity of most interest in this work: the distribution of CNG channels along the length of a cilium. Experimental measurements of the total current during this diffusion process provide an extra boundary condition which helps determine the unknown distribution function. A simple perturbation approximation is derived and used to solve this inverse problem and thus obtain estimates of the spatial distribution of CNG ion channels along the length of a cilium. A one-dimensional computer minimization and a special delay iteration are used with the perturbation formulas to obtain approximate channel distributions in the cases of simulated and experimental data.      

Mathematical modelling of dynamics and control in metabolic networks. I. On Michaelis-Menten kinetics

As a starting point for modeling of metabolic networks this paper considers the simple Michaelis-Menten reaction mechanism. After the elimination of diffusional effects a mathematically intractable mass action kinetic model is obtained. The properties of this model are explored via scaling and linearization. The scaling is carried out such that kinetic properties, concentration parameters and external influences are clearly separated. We then try to obtain reasonable estimates for values of the dimensionless groups and examine the dynamic properties of the model over this part of the parameter space. Linear analysis is found to give excellent insight into reaction dynamics and it also gives a forum for understanding and justifying the two commonly used quasi-stationary and quasi-equilibrium analyses. The first finding is that there are two separate time scales inherent in the model existing over most of the parameter space, and in particular over the regions of importance here. Full modal analysis gives a new interpretation of quasi-stationary analysis, and its extension via singular perturbation theory, and a rationalization of the quasi-equilibrium approximation. The new interpretation of the quasi-steady state assumption is that the applicability is intimately related to dynamic interactions between the concentration variables rather than the traditional notion that a quasi-stationary state is reached, after a short transient period, where the rates of formation and decomposition of the enzyme intermediate are approximately equal. The modal analysis reveals that the generally used criterion for the applicability of quasi-stationary analysis that total enzyme concentration must be much less than total substrate concentration, et much less than St, is incomplete and that the criterion et much less than Km much less than St (Km is the well known Michaelis constant) is the appropriate one. The first inequality (et much less than Km) guarantees agreement over the longer time scale leading to quasi-stationary behavior or the applicability of the zeroth order outer singular perturbation solution but the second half of the criterion (Km much less than St) justifies zeroth order inner singular perturbation solution where the substrate concentration is assumed to be invariant. Furthermore linear analysis shows that when a fast mode representing the binding of substrate to the enzyme is fast it can be relaxed leading to the quasi-equilibrium assumption. The influence of the dimensionless groups is ascertained by integrating the equations numerically, and the predictions made by the linear analysis are found to be accurate.(ABSTRACT TRUNCATED AT 400 WORDS)     

An illness-death process with time-dependent covariates

A general model for the illness-death stochastic process with covariates has been developed for the analysis of survival data. This model incorporates important baseline and time-dependent covariates in order to make an appropriate adjustment for the transition and survival probabilities. The follow-up period is subdivided into small intervals and a constant hazard is assumed for each interval. An approximation formula is derived to estimate the transition parameters when the exact transition time is unknown. The method developed is illustrated with data from a study on the prevention of the recurrence of a myocardial infarction and subsequent mortality, the Beta-Blocker Heart Attack Trial (BHAT). This method provides an analytical approach with which the effectiveness of the treatment can be compared between the placebo and propranolol treatment groups with respect to fatal and nonfatal events simultaneously.     

Enzyme localization as a mechanism of apparent active transport

A model based on enzyme localization is developed which gives rise to an apparent active transport of a metabolite into or out of cells. The model is applied to three simple situations, using Fick's equation and the Rashevsky approximation. It is shown that the apparent efficiency can be made as large as desired if, for constant reaction, the outer cell region is made sufficiently small, or, for autocatalytic reaction, if the metabolite concentration in the outer region is sufficiently small. The physical limitations imposed by this mechanism are developed for all three situations.     

Metastable behavior in Markov processes with internal states

A perturbation framework is developed to analyze metastable behavior in stochastic processes with random internal and external states. The process is assumed to be under weak noise conditions, and the case where the deterministic limit is bistable is considered. A general analytical approximation is derived for the stationary probability density and the mean switching time between metastable states, which includes the pre exponential factor. The results are illustrated with a model of gene expression that displays bistable switching. In this model, the external state represents the number of protein molecules produced by a hypothetical gene. Once produced, a protein is eventually degraded. The internal state represents the activated or unactivated state of the gene; in the activated state the gene produces protein more rapidly than the unactivated state. The gene is activated by a dimer of the protein it produces so that the activation rate depends on the current protein level. This is a well studied model, and several model reductions and diffusion approximation methods are available to analyze its behavior. However, it is unclear if these methods accurately approximate long-time metastable behavior (i.e., mean switching time between metastable states of the bistable system). Diffusion approximations are generally known to fail in this regard.     

The singularly perturbed Hodgkin-Huxley equations as a tool for the analysis of repetitive nerve activity

A qualitative analysis of the Hodgkin-Huxley model (Hodgkin and Huxley 1952), which closely mimics the ionic processes at a real nerve membrane, is performed by means of a singular perturbation theory. This was achieved by introducing a perturbation parameter that, if decreased, speeds up the fast variables of the Hodgkin-Huxley equations (membrane potential and sodium activation), whereas it does not affect the slow variables (sodium inactivation and potassium activation). In the most extreme case, if the perturbation parameter is set to zero, the original four-dimensional system degenerates to a system with only two differential equations. This degenerate system is easier to analyze and much more intuitive than the original Hodgkin-Huxley equations. It shows, like the original model, an infinite train of action potentials if stimulated by an input current in a suitable range. Additionally, explanations for the increased sensitivity to depolarizing current steps that precedes an action potential can be found by analysis of the degenerate system. Using the theory of Mishchenko and Rozov (1980) it is shown that the degenerate system does not only represent a simplification of the original Hodgkin-Huxley equations but also gives a valid approximation of the original model at least for stimulating currents that are constant within a suitable range.     

A stochastic model for disease transmission in a managed herd, motivated by Neospora caninum amongst dairy cattle

A stochastic model for the spread of Neospora caninum infection within a herd of dairy cattle is studied, in particular the long-term (equilibrium) behaviour of the model. The model incorporates the interesting feature that total herd size is constrained to lie within a fairly small interval, but not held exactly constant. Approximations for the joint distribution of numbers of susceptible and infected individuals present in equilibrium are derived based upon a diffusion approximation to the infection process. The effect of both 'typical herd size' and 'the amount of permitted variation in herd size' upon disease prevalence in equilibrium are considered using both the exact equilibrium distribution of the process and our approximations.     

Prediction of the power of detection of marker-quantitative trait locus linkages using analysis of variance

Analysis of variance can be used to detect the linkage of segregating quantitative trait loci (QTLs) to molecular markers in outbred populations. Using independent full-sib families and assuming linkage equilibrium, equations to predict the power of detection of a QTL are described. These equations are based on an hierarchical analysis of variance assuming either a completely random model or a mixed model, in which the QTL effect is fixed. A simple prediction of power from the mean squares is used that assumes a random model so that in the mixed-model situation this is an approximation. Simulation is used to illustrate the failure of the random model to predict mean squares and, hence, the power. The mixed model is shown to provide accurate prediction of the mean squares and, using the approximation, of power.     

On the quasi-stationary distribution of the Ross malaria model.

Approximations are derived for the quasi-stationary distribution of the fully stochastic version of the classical Ross malaria model. The approximations are developed in two stages. In the first stage, the Ross process is approximated with a bivariate Markov chain without an absorbing state. The second stage of the approximation uses ideas from perturbation theory to derive explicit expressions that serve as approximations of the joint stationary distribution of the approximating process. Numerical comparisons are made between the approximations and the quasi-stationary distribution.     

Kinetic analysis of the attachment of a biological particle to a surface by macromolecular binding

Motivated by the problem of microbial deposition, a dynamic model is developed for the attachment of a Brownian particle to a surface mediated by colloidal forces as well as macromolecular bridging. The model predicts the attachment probability of the particle to the surface based upon the free energy as a function of fluctuating bond number and separation distance from the surface. From this model, the mean first-passage time approach is used to predict the mean time required for the particle moving from the unattached state to the attached state based on the properties of the binding macromolecules. This approach provides an analytical approximation for mean transition time from the secondary energy minimum as well as the attachment rate constant for the general case where neither binding nor particle diffusion are necessarily rate-limiting.     

Cell cycle progression

In this paper we consider cell cycle models for which the transition operator for the evolution of birth mass density is a simple, linear dynamical system with a stochastic perturbation. The convolution model for a birth mass distribution is presented. Density functions of birth mass and tail probabilities in n-th generation are calculated by a saddle-point approximation method. With these probabilities, representing the probability of exceeding an acceptable mass value, we have more control over pathological growth. A computer simulation is presented for cell proliferation in the age-dependent cell cycle model. The simulation takes into account the fact that the age-dependent model with a linear growth is a simple linear dynamical system with an additive stochastic perturbation. The simulated data as well as the experimental data (generation times for mouse L) are fitted by the proposed convolution model.     

Accuracy of neuronal interspike times calculated from a diffusion approximation

The theory of neuronal firing in Stein's model is outlined as well as the corresponding theory for a diffusion approximation which has the same first two infinitesimal moments. The diffusion approximation is derived from the discontinuous model in the limit of large input frequencies and small postsynaptic potential amplitudes. A comparison of the calculated mean interspike intervals is made for various values of the threshold for firing and various input frequencies. The diffusion approximation can underestimate the interspike interval by up to 100% or severely overestimate it, depending on the input frequencies and the threshold. A general relation between the predictions of the two models is deduced.     

Nonlinear control of HIV-1 infection with a singular perturbation model

Using a singular perturbation approximation, a nonlinear state-space model of HIV-1 infection, having as state variables the number of healthy and infected CD4+T cells and the number of virion particles, is simplified and used to design a control law. The control law comprises an inner block that performs feedback linearizing of the virus dynamics and an outer block implementing an LQ regulator that drives the number of virion particles to a number below the specification. A sensitivity analysis of the resulting law is performed with respect to the model parameter to the infection rate, showing that the controlled system remains stable in the presence of significant changes of this parameter with respect to the nominal value.     

Kinetics and thermodynamics of isothermal seed germination.

We have been successful in building a mathematical model that fits both the germination rate and the total number of seeds that germinate as a function of time. This mathematical model is the same autocatalytic reaction model that describes biochemical reactions in which enzymes play an important role. The model gives values for the initial concentration of two enzymes. From these initial enzyme concentrations an equilibrium constant is calculated and the thermodynamic model gives the change in enthalpy, entropy, free energy and the activation energy. A plot of the natural logarithm of the equilibrium constant as a function of the reciprocal of the absolute temperature gives two straight lines. The change of enthalpy for the process below 33 °C differs considerably to the change above 33 °C. The free energy as a function of the absolute temperature gives a straight line from which the change in entropy is calculated. The activation energy is determined from the slope of the natural logarithm of the rate constant as a function of the reciprocal of the absolute temperature.     

Multipoint linkage analysis using sib pairs: an interval mapping approach for dichotomous outcomes.

I propose an interval mapping approach suitable for a dichotomous outcome, with emphasis on samples of affected sib pairs. The method computes a lod score for each of a set of locations in the interval between two flanking markers and takes as its estimate of trait-locus location the maximum lod score in the interval, provided it exceeds the prespecified critical value. Use of the method depends on prior knowledge of the genetic model for the disease only through available estimates of recurrence risk to relatives of affected individuals. The method gives an unbiased estimate of location, provided the recurrence risk are correctly specified and provided the marker identity-by-descent probabilities are jointly, rather than individually, estimated. I also discuss use of the method for traits determined by two loci and give an approximation that has good power for a wide range of two-locus models.     

A new concept to reveal protein dynamics based on energy dissipation

Protein dynamics is essential for its function, especially for intramolecular signal transduction. In this work we propose a new concept, energy dissipation model, to systematically reveal protein dynamics upon effector binding and energy perturbation. The concept is applied to better understand the intramolecular signal transduction during allostery of enzymes. The E. coli allosteric enzyme, aspartokinase III, is used as a model system and special molecular dynamics simulations are designed and carried out. Computational results indicate that the number of residues affected by external energy perturbation (i.e. caused by a ligand binding) during the energy dissipation process shows a sigmoid pattern. Using the two-state Boltzmann equation, we define two parameters, the half response time and the dissipation rate constant, which can be used to well characterize the energy dissipation process. For the allostery of aspartokinase III, the residue response time indicates that besides the ACT2 signal transduction pathway, there is another pathway between the regulatory site and the catalytic site, which is suggested to be the β15-αK loop of ACT1. We further introduce the term "protein dynamical modules" based on the residue response time. Different from the protein structural modules which merely provide information about the structural stability of proteins, protein dynamical modules could reveal protein characteristics from the perspective of dynamics. Finally, the energy dissipation model is applied to investigate E. coli aspartokinase III mutations to better understand the desensitization of product feedback inhibition via allostery. In conclusion, the new concept proposed in this paper gives a novel holistic view of protein dynamics, a key question in biology with high impacts for both biotechnology and biomedicine.     

A neuronal model for the discharge patterns produced by cyclic inputs

To understand the patterns of nerve impulses produced by sinusoidal stimuli, a simple model is considered which integrates input currents with a finite time constant until a threshold voltage is reached, whereupon an output pulse is produced and the process is restarted. We show here that (a) a general analytic solution exists for this model driven by sinusoidal stimuli, determining the interval between every member of the pulse train, (b) for all values of the parameters of the model a pattern exists which repeats periodically after a finite number of pulses in the absence of noise, (c) the system will approach a stable pattern which, if perturbed, will be recovered once the perturbation is removed, (d) the linear integrator or relaxation oscillator and the curren multiplier are limiting cases of this model.