Inversion of Markov processes to determine rate constants from single-channel data.

The determination of rate constants from single-channel data can be very difficult, in part because the single-channel lifetime distributions commonly analyzed by experimenters often have a complicated mathematical relation to the channel gating mechanism. The standard treatment of channel gating as a Markov process leads to the prediction that lifetime distributions are exponential functions. As the number of states of a channel gating scheme increases, the number of exponential terms in the lifetime distribution increases, and the weights and decay constants of the lifetime distributions become progressively more complicated functions of the underlying rate constants. In the present study a mathematical strategy for inverting these functions is introduced in order to determine rate constants from single-channel lifetime distributions. This inversion is easy for channel gating schemes with two or fewer states of a given conductance, so the present study focuses on schemes with more states. The procedure is to derive explicit equations relating the parameters of the lifetime distribution to the rate constants of the scheme. Such equations can be derived using the equality between symmetric functions of eigenvalues of a matrix and sums over principle minors, as well as expressions for the moments, derivatives, and weights of a lifetime distribution. The rate constants are then obtained as roots to this system of equations. For a gating scheme with three sequential closed states and a single gateway state, exact analytical expressions were found for each rate constant in terms of the parameters of the three-exponential closed-time distribution. For several other gating schemes, systems of equations were found that could be solved numerically to obtain the rate constants. Lifetime distributions were shown to specify a unique set of real rate constants in sequential gating schemes with up to five closed or five open states. For kinetic schemes with multiple gating pathways, the analysis of simulated data revealed multiple solutions. These multiple solutions could be distinguished by examining two-dimensional probability density functions. The utility of the methods introduced here are demonstrated by analyzing published data on nicotinic acetylcholine receptors, GABA(A) receptors, and NMDA receptors.     

The attractiveness of the Droop equations.

The Droop equations are a system of three coupled, nonlinear ordinary differential equations describing the growth of a microorganism in a chemostat. The growth rate of the organism is limited by the availability of a single nutrient. In contrast to the better known Monod equations, the nutrient is divided into external and internal cellular pools. Only the internal pool can catalyze growth. This paper proves that the Droop equations are globally stable. Based on a single combination of parameters, either the chemostat organism goes extinct or it tends to a fixed, positive concentration.     

Deviations from Michaelis-Menten kinetics. Computation of the probabilities of obtaining complex curves from simple kinetic schemes.

1. It is possible to calculate the intrinsic probability associated with any curve shape that is allowed for rational functions of given degree when the coefficients are independent or dependent random variables with known probability distributions. 2. Computations of such probabilities are described when the coefficients of the rational function are generated according to several probability distribution functions and in particular when rate constants are varied randomly for several simple model mechanisms. 3. It is concluded that each molecular mechanism is associated with a specific set of curve-shape probabilities, and this could be of value in discriminating between model mechanisms. 4. It is shown how a computer program can be used to estimate the probability of new complexities such as extra inflexions and turning points as the degree of rate equations increases. 5. The probability of 3 : 3 rate equations giving 2 : 2 curve shapes is discussed for unrestricted coefficients and also for the substrate-modifier mechanisms. 6. The probability associated with the numerical values of coefficients in rate equations is also calculated for this mechanism, and a possible method for determining the approximate magnitude of product-release steps is given. 7. The computer programs used in the computations have been deposited as Supplement SUP 50113 (21 pages) with the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem, J. (1978) 169, 5.     

Cell Growth and Division: IV. Determination of Volume Growth Rate and Division Probability

Volume growth rate and division probability functions for mammalian cells have been determined as functions of cell volume with good reproducibility and statistical precision using Coulter volume spectrometry and the equations of the Bell model. Results are compared with independent measurements on synchronous cultures. The slow rate of volume dispersion requires that the growth rate F(tau, V) be closely proportional to volume for cells of a given age. However, when F(tau, V) is averaged over the age distribution of a population in balanced exponential growth to give the growth rate function f(V), the latter may rise more steeply than V.     

Tubular fluid flow and distal NaCl delivery mediated by tubuloglomerular feedback in the rat kidney

The glomerular filtration rate in the kidney is controlled, in part, by the tubuloglomerular feedback (TGF) system, which is a negative feedback loop that mediates oscillations in tubular fluid flow and in fluid NaCl concentration of the loop of Henle. In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a short loop of Henle with compliant walls. The proximal tubule and the outer-stripe segment of the descending limb are assumed to be highly water permeable; the thick ascending limb (TAL) is assumed to be water impermeable and have active NaCl transport. A bifurcation analysis of the TGF model equations was performed by computing parameter boundaries, as functions of TGF gain and delay, that separate differing model behaviors. The analysis revealed a complex parameter region that allows a variety of qualitatively different model equations: a regime having one stable, time-independent steady-state solution and regimes having stable oscillatory solutions of different frequencies. A comparison with a previous model, which represents only the TAL explicitly and other segments using phenomenological relations, indicates that explicit representation of the proximal tubule and descending limb of the loop of Henle lowers the stability of the TGF system. Model simulations also suggest that the onset of limit-cycle oscillations results in increases in the time-averaged distal NaCl delivery, whereas distal fluid delivery is not much affected.     

Ribonucleic acid synthesis in rabbit erythroid cells.

Kinetic studies on the synthesis of RNA in mature bone-marrow erythroid cells from rabbits were made by measuring the incorporation of [2-3H]adenosine into the ATP pool and RNA over periods up to 8h. By use of equations to fit the pool specific radioactivity and an equation using the same type of pool to generate the rate of linear DNA synthesis, good agreement between the pool parameters is found, provided that the ATP pool is measured in whole cell extracts, and assuming that the dATP and ATP pools equilibrate rapidly. RNA-synthesis rates were measured by using curve fits to equations developed by using the pool specific-radioactivity curves. The rate of synthesis of poly(A)-containing RNA varied in three experiments from 90 to 220mol/min per cell, with half-life of nuclear processing of 12-22 min with a mean of 16 min. Ribosomal RNA is synthesized at a rate of 70-200 mol/min per cell with an average half-life of nuclear processing of 37 min for the 18S RNA and 214 min for the 28S RNA. When the stable rRNA components are subtracted from the nRNA synthesis, the rate of nRNA synthesis is between 2 and 6fg/min per cell with an average half-life of degradation of 27 min. The rate of synthesis of poly(A)-containing RNA is 1.5-3.5% of the RNA-synthesis rates. These rates are compared with the RNA-synthesis rates found in L cells and concentrations of globin mRNA found in various erythroid-cell preparations.     

Turing structures in an enzyme-induction system with gap junction-mediated non-linear diffusion

Two cells, each containing a reaction system modeling genetic induction, are coupled by diffusion. The substrate is moving through gap junctions, the number of which is regulated by the adjacent cells. This leads to a non-linear substrate diffusion term in the rate equations. Stability analysis reveals the conditions for the emergence of stable asymmetric solutions (dissipative structures). Due to non-linear diffusion rigid restrictions on the ratio of the two diffusion constants no longer exist. We demonstrate that substances operating as regulators of intercellular communication and participating in cellular metabolism may exhibit morphogenetic functions.     

Combined influences of ambient temperature and graded work on cardiopulmonary functions.

Several cardiopulmonary items were measured in eight adult females who had performed stationary ergometer cycling of 75-450 kgm/min in air temperature between 14 degrees and 35 degreesC. The experiments were designed on the basis of the Latin square method and the results were analyzed by computing the analysis of variance and multiple regression equations for each item linked with work rate and air temperature. In view of the degree of affinity of the effect of work rate and that of air temperature, the items could be divided into three groups. The first group consisted of items of pulmonary functions closely related with work rate but independent of air temperature, such as pulmonary ventilation, oxygen intake, carbon dioxide production, respiratory exchange ratio, and ratio of oxygen removal. The second group characterized by linear dependency on air temperature included mean skin temperature and mean innermost air temperature. The third group consisting of heart rate, pulse sum during work, and work pulse sum was intermediate. In spite of the confusion in the literature about the attitude of oxygen intake or mean skin temperature during work in heat, the former was the most stable in relation to change in air temperature and the latter was independent of work intensity.     

Thermodynamic and kinetic studies of a stoichiometric model of energetic metabolism under starvation conditions

Abstract A stoichiometric model of anaerobic glycolysis is presented and the influence on its dynamics by the ATP-consuming membrane transport processes and substrate input rate are studied. The model is represented by a system of four ODE (ordinary differential equations), mass conservation equations and functions of state variables, such as thermodynamic efficiency. A low substrate input rate provokes damped oscillations while a high enrgy load determines sustained oscillations in all the metabolites and in thermodynamic efficiency. Due to the lack of linearity between fluxes and forces in the oscillatory region it may be stated that oscillations appear when the system is kinetically controlled.     

On the localization of the origin of DNA replication in E. coli.

A theory is presented to describe the behavior of micro-organisms, bacteria and protozoa. Individual cells are regarded as particles having internal state variables. The change of each variable with time depends on the environmental condition. The velocity and the frequency of direction change of swimming cells are determined by the values of these variables. With this framework, the theory gives a method to connect the behaviour in a spatial gradient of the environment and the behaviour upon a change of the environment with time. Observed behaviors of bacteria and protozoa are understandable on the basis of simple rate equations for internal state variables and the product expressions for the velocity and the frequency of direction change as functions of these variables. Experimental data on the thermotaxis of paramecium are shown for comparison with the theoretical results.     

一类具时滞的禽流感模型

针对具扩散和时滞的SI-SIR传染病模型,用特征分析和Lyapunov泛函方法研究了相应的具齐次Neumann边界条件反应扩散方程组解的渐近性质.最后给出数值模拟来说明如果染病鸟类的接触率和染病人类的接触率小,那么全系统的无病平衡点是全局渐近稳定的;但当染病鸟类的接触率大或者和染病人类的接触率大时,变异的禽流感将在人类中扩散.     

Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods

In recent years, three related methods have been used to model the phenotypic dynamics of traits under the influence of natural selection. The first is based on an approximation to quantitative genetic recursion equations for sexual populations. The second is based on evolution in asexual lineages with mutation-generated variation. The third method finds an evolutionarily stable set of phenotypes for species characterized by a given set of fitness functions, assuming that the mode of reproduction places no constraints on the number of distinct types that can be maintained in the population. The three methods share the property that the rate of change of a trait within a homogeneous population is approximately proportional to the individual fitness gradient. The methods differ in assumptions about the potential magnitude of phenotypic differences in mutant forms, and in their assumptions about the probability that invasion or speciation occurs when a species has a stable, yet invadable phenotype. Determining the range of applicability of the different methods is important for assessing the validity of optimization methods in predicting the evolutionary outcome of ecological interactions. Methods based on quantitative genetic models predict that fitness minimizing traits will often be evolutionarily stable over significant time periods, while other approaches suggest this is likely to be rare. A more detailed study of cases of disruptive selection might reveal whether fitness-minimizing traits occur frequently in natural communities.     

An unconditionally stable numerical method for the Luo-Rudy 1 model used in simulations of defibrillation

Numerical simulations of defibrillation using the Bidomain model coupled to a model of membrane kinetics represent a serious numerical challenge. This is because very high voltages close to defibrillation electrodes demand that extreme time step restrictions be placed on standard numerical schemes, e.g. the forward Euler scheme. A common solution to this problem is to modify the cell model by simple if-tests applied to several equations and rate functions. These changes are motivated by numerical problems rather than physiology, and should therefore be avoided whenever possible. The purpose of this paper is to present a numerical scheme that handles the original model without modifications and which is unconditionally stable for the Luo-Rudy phase 1 model. This also shows that the cell model is mathematically well-behaved, even in the presence of very high voltages. Our theoretical results are illustrated by numerical computations.     

A flexible sigmoid function of determinate growth

A new empirical equation for the sigmoid pattern of determinate growth, 'the beta growth function', is presented. It calculates weight (w) in dependence of time, using the following three parameters: t(m), the time at which the maximum growth rate is obtained; t(e), the time at the end of growth; and w(max), the maximal value for w, which is achieved at t(e). The beta growth function was compared with four classical (logistic, Richards, Gompertz and Weibull) growth equations, and two expolinear equations. All equations described successfully the sigmoid dynamics of seed filling, plant growth and crop biomass production. However, differences were found in estimating w(max). Features of the beta function are: (1) like the Richards equation it is flexible in describing various asymmetrical sigmoid patterns (its symmetrical form is a cubic polynomial); (2) like the logistic and the Gompertz equations its parameters are numerically stable in statistical estimation; (3) like the Weibull function it predicts zero mass at time zero, but its extension to deal with various initial conditions can be easily obtained; (4) relative to the truncated expolinear equation it provides more reasonable estimates of final quantity and duration of a growth process. In addition, the new function predicts a zero growth rate at both the start and end of a precisely defined growth period. Therefore, it is unique for dealing with determinate growth, and is more suitable than other functions for embedding in process-based crop simulation models to describe the dynamics of organs as sinks to absorb assimilates. Because its parameters correspond to growth traits of interest to crop scientists, the beta growth function is suitable for characterization of environmental and genotypic influences on growth processes. However, it is not suitable for estimating maximum relative growth rate to characterize early growth that is expected to be close to exponential.     

A Lyapunov function for piecewise-independent differential equations: stability of the ideal free distribution in two patch environments

In this article we construct Lyapunov functions for models described by piecewise-continuous and independent differential equations. Because these models are described by discontinuous differential equations, the theory of Lyapunov functions for smooth dynamical systems is not applicable. Instead, we use a geometrical approach to construct a Lyapunov function. Then we apply the general approach to analyze population dynamics describing exploitative competition of two species in a two-patch environment. We prove that for any biologically meaningful parameter combination the model has a globally stable equilibrium and we analyze this equilibrium with respect to parameters.      

A mathematical approach to spatial distribution and temporal succession in plant communities

The Volterra equations which represent competitions between two species are utilized to examine the phenomenon of boundary formation between two species of plants. The set of stable stationary points for these equations is determined and is illustrated in a product space of parameters and dynamical variables. The stages of boundary appearance and succession are visualized by considering slow changes of the parameters as functions of time and space.     

Modeling and analysis of a predator-prey model with disease in the prey

A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator k=k(0) is constant (independent of delay tau;, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay tau; is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If k=k(0)e(-dtau;) (d is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.     

Bend propagation in flagella. I. Derivation of equations of motion and their simulation.

A set of nonlinear differential equations describing flagellar motion in an external viscous medium is derived. Because of the local nature of these equations and the use of a Crank-Nicolson-type forward time step, which is stable for large deltat, numerical solution of these equations on a digital computer is relatively fast. Stable bend initiation and propagation, without internal viscous resistance, is demonstrated for a flagellum containing a linear elastic bending resistance and an elastic shear resistance that depends on sliding. The elastic shear resistance is derived from a plausible structural model of the radial link system. The active shear force for the dynein system is specified by a history-dependent functional of curvature characterized by the parameters m0, a proportionality constant between the maximum active shear moment and curvature, and tau, a relaxation time which essentially determines the delay between curvature and active moment.     

Lotka-Volterra equations: Decomposition,stability, and structure

Summary The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.Research supported by U.S. Department of Energy under the Contract EC-77-S-03-1493.On leave from Kobe University, Kobe, Japan.     

Feedback-mediated dynamics in a model of a compliant thick ascending limb

The tubuloglomerular feedback (TGF) system in the kidney, which is a key regulator of filtration rate, has been shown in physiologic experiments in rats to mediate oscillations in tubular fluid pressure and flow, and in NaCl concentration in the tubular fluid of the thick ascending limb (TAL). In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a TAL with compliant walls. The model was used to investigate the dynamic behaviors of the TGF system. A bifurcation analysis of the TGF model equations was performed by deriving and finding roots of the characteristic equation, which arises from a linearization of the model equations. Numerical simulations of the full model equations were conducted to assist in the interpretation of the bifurcation analysis. These techniques revealed a complex parameter region that allows a variety of qualitatively different model solutions: a regime having one stable, time-independent steady-state solution; regimes having one stable oscillatory solution only; and regimes having multiple possible stable oscillatory solutions. Model results suggest that the compliance of the TAL walls increases the tendency of the model TGF system to oscillate.