Fluctuation in excitability of nerve fibres — a model study

Summary In this paper a mathematical model for the fluctuation in excitability of nerve fibres is presented. The model is based on the concept of a threshold subject to noise-like variations. Emphasis is laid on the rate of fluctuation of these disturbances.     

Remarks on the Respective Roles of Logical Parameters and Time Delays in Asynchronous Logic: An Homage to El Houssine Snoussi

As a tribute to E.H. Snoussi for his essential mathematical formalization of the logic approach to qualitatively described gene networks, this note presents some thoughts on the relationships between logical parameters and time delays in asynchronous logic.     

On Bartlett's formulation of the Luria-Delbrück mutation model

Current knowledge of microbial mutation rates was accumulated largely by means of fluctuation experiments. A mathematical model describing the cell dynamics in a fluctuation experiment is indispensable to the estimation of mutation rates through fluctuation experiments. In almost six decades the model formulated by Lea and Coulson dominated in research and application, although the model formulated by Bartlett is generally believed to describe the cell dynamics more faithfully. The neglect of the Bartlett formulation was mainly due to mathematical difficulties. The present investigation overcomes some of these difficulties, thereby paving the way for the application of the Bartlett formulation in estimating mutation rates. Specifically, the article offers an algorithm for computing the distribution function of the number of mutants under the Bartlett formulation. The article also provides algorithms for computing point and interval estimates of mutation rates that are based on the maximum-likelihood principle. In addition, the article examines and compares the asymptotic behavior of the distributions of the number of mutants under the two formulations.     

Steady state bifurcation analysis of reaction-diffusion equations-A critique

This note supplies missing details and corrects some mathematical errors in the recent significant paper by Auchmuty and Nicolis (1975).     

Stability considerations in community organization.

We show how mathematical stability considerations (persistence in a fluctuating environmental fluctuation) suggest that competing species should partition the habitat dimension more finely than the food-type dimension.     

Fluctuation Induces Evolutionary Branching in a Mathematical Model of Ecosystems

Ecological systems are always subjected to various environmental fluctuations. They evolve under these fluctuations and the resulting systems are robust against them. The diversity in ecological systems is also acquired through the evolution. How do the fluctuations affect the evolutionary processes? Do the fluctuations have direct impact on the species diversity in ecological systems? In the present paper, we investigate the relation between the environmental fluctuation and the evolution of species diversity with a mathematical model of evolutionary ecology. In the model, individual organisms compete for a single restricted resource and the temporal fluctuation in the resource supply is introduced as the environmental fluctuation. The evolutionary process is represented by the mutational change of genotypes which determines their resource utilization strategies. We found that when the environmental state is switched form static to fluctuating conditions, the initial closely related population distributed around the genotype adapted for the static environment is destabilized and divided into two groups in the genotype space; i.e., the evolutionary branching is induced by the environmental fluctuation. The consequent multiple species structures is evolutionary stable at the presence of the fluctuation. We perform the evolutionary invasion analysis for the phenomena and illustrate the mechanisms of the branchings. The results indicate a novel process of increasing the species diversity via evolutionary branching, and the analysis reveals the mechanisims of the branching preocess as the response to the environmental fluctuation. The robustness of the evolutionary process is also discussed.     

Sightseeing Around the Single Cosinor

Misuses of the single cosinor analysis are numerous. We give three rules of thumb, providing guidelines as to whether results obtained from this analysis are dubious. This short note avoids mathematical development so as to be comprehensible to the nonspecialist. However, a few endnotes, give some technical comments.     

Sightseeing Around the Single Cosinor

Misuses of the single cosinor analysis are numerous. We give three rules of thumb, providing guidelines as to whether results obtained from this analysis are dubious. This short note avoids mathematical development so as to be comprehensible to the nonspecialist. However, a few endnotes, give some technical comments.     

Interpretation of the Ussing flux ratio from the fluctuation theorem

The fluctuation theorem gives a mathematical expression to quantify the probability of observing events violating the second law of thermodynamics in a small system over a short period of time. The theorem predicts the ratio of forward (entropy-producing) runs to the backward (entropy-consuming) runs for a nanometer-sized molecular machine in a nonequilibrium system. However, few experimental verifications of the theorem have been carried out. In this paper, I show that the Ussing flux ratio, the ratio of outward to inward unidirectional ion fluxes across a membrane channel, can be derived from the fluctuation theorem if we consider the ion channel and the contacting solutions as a small nonequilibrium system. The entropy change due to ion electrodiffusion is expressed from the fundamental equation for the entropy change. Thus, the empirical flux ratio equation can be interpreted from the more general fluctuation theorem, and serves as a verification of the theorem.     

Occurrence of planktonic blooms under environmental fluctuations and its possible control mechanism--mathematical models and experimental observations

Planktonic blooms and its control is an intriguing problem in ecology. To investigate the oscillatory successions of blooms, three simple phytoplankton-zooplankton systems are proposed. It is observed that if the uptake function is linear and the process of toxin liberation is instantaneous, the oscillatory nature of blooms is not observed. On the other hand, periodic planktonic blooms are observed when toxin liberation process follows discrete time variation. The bloom phenomena described by this mechanism can be controlled through toxin producing phytoplankton (TPP). Introducing environmental fluctuation in the system, a critical value of time delay in terms of correlation time of the fluctuation is worked out. We observed from our mathematical analysis, numerical simulation and field observation that TPP and control of the rapidity of environmental fluctuation are key factors for the termination of planktonic blooms.     

Efficacy of artesunate-amodiaquine for the treatment of acute uncomplicated falciparum malaria in southern Mauritania

Mathematical analyses and modelling have an important role informing malaria eradication strategies. Simple mathematical approaches can answer many questions, but it is important to investigate their assumptions and to test whether simple assumptions affect the results. In this note, four examples demonstrate both the effects of model structures and assumptions and also the benefits of using a diversity of model approaches. These examples include the time to eradication, the impact of vaccine efficacy and coverage, drug programs and the effects of duration of infections and delays to treatment, and the influence of seasonality and migration coupling on disease fadeout. An excessively simple structure can miss key results, but simple mathematical approaches can still achieve key results for eradication strategy and define areas for investigation by more complex models.     

On the evolution of fluctuating segregation distortion

Previous mathematical models of the genetic control by one locus of the segregation at another have all concluded that alleles causing departures from Mendelian segregation should succeed. In this study the segregation ratios induced at the major locus by the modifier locus fluctuate cyclically. It is shown that if initially there is Mendelian segregation and if the rare modifying allele induces symmetric fluctuation about the Mendelian ratios it cannot succeed. It is further proven that if initially there are symmetric fluctuations about Mendelian segregation then an allele reducing the amplitude of the fluctuation will succeed.     

Microbial Ecological Model of Filamentous Bulking and Mechanisms

The paper proposes a prototype mathematical model for the interactions among filamentous bacteria, floc-forming bacteria and protozoa within completely mixed activated sludge systems, which is capable of describing the practical coexistence of three microorganisms in a generic way. The systems with substrate fluctuation and without substrate fluctuation are investigated, respectively. The results show that filamentous bulking will occur within an interested range of substrate concentrations under unsuitable dilution rates, particularly at low substrate concentrations. When substrate concentration fluctuates, the same-frequency synchronous control parameters are beneficial to prevent filamentous bulking. Thus, the present investigations not only reveal the cause of the problem of filamentous bulking theoretically, but also may provide a crucial theoretical foundation for overcoming it.     

Remarks on the Evolutionary Effect of Natural Selection

The so-called "Fundamental Theorem of Natural Selection", that the mean fitness of a population increases with time under natural selection, is known not to be true, as a mathematical theorem, when fitnesses depend on more than one locus. Although this observation may not have particular biological relevance, (so that mean fitness may well increase in the great majority of interesting situations), it does suggest that it is of interest to find an evolutionary result which is correct as a mathematical theorem, no matter how many loci are involved. The aim of the present note is to prove an evolutionary theorem relating to the variance in fitness, rather than the mean: this theorem is true for an arbitrary number of loci, as well as for arbitrary (fixed) fitness parameters and arbitrary linkage between loci. Connections are briefly discussed between this theorem and the principle of quasi-linkage equilibrium.     

A biokinetic model to describe consequences of inhibition/stimulation in DNA-proofreading and repair-1. Development of the model

A biokinetic model is described which deals with the mathematical consequences of the inhibition or stimulation of DNA proofreading. It demonstrates the development of the number of DNA mismatch-dependent cells (e.g. cells with a malignant phenotype), where such mismatches arise by the in situ interaction of various substances with nucleotides of the DNA. The model can test for consequences by a logic gating on an "if-then" type of analysis in relation to the separate and consecutive processes of proofreading and repair. In particular, the consequences are considered in cases where either (i) the efficacy of proofreading and repair are reduced/prevented (inhibited) or (ii) are increased by some form of stimulation. On the chosen kinetic parameters, the model is accessible to manipulation as new data arising from further investigations become available and are introduced. The model is based on recently published data which show that an increased "mutant fraction" (see note on terms) arises in DNA replication when intracellular nucleotide pools show "asymmetries" (see note on terms). Extraordinarily high mutant fractions can be predicted/have been recorded in the presence of proofreading inhibitors. The model expresses data in mathematical terms of the competition between the development of mismatch-dependent cells and those with authentic genetic information. (Feedback and metastasis-effects and those of wild-type replicates are included.) A computerized (numerical) integration of the corresponding set of differential equations is offered. (A diskette with the program CANCER.xls is available upon request.)     

The differential difference equation for epidemics

In a series of recently published papers E. B. Wilson and his collaborators have investigated some of the mathematical questions connected with epidemics. A differential difference equation was set up and various properties of its solution were obtained. This equation implies that the epidemic has an equilibrium state, and it is the purpose of this note to show that this equilbrium is stable in the sense that any small deviation from it will tend to zero.     

Changes in the fluctuation of interbeat intervals in spontaneously beating cultured cardiac myocytes: experimental and modeling studies

 Isolated and cultured neonatal cardiac myocytes contract spontaneously and cyclically, and have the properties of a non-linear oscillator. In this study, we have analyzed the relationship between the fluctuation of contraction rhythm of spontaneously beating cultured cardiac myocytes, and the coupling strength among them. The coefficient of variation of contraction intervals increased transiently in the early stages of incubation, and then decreased almost monotonically with time. The contraction rhythm of the myocytes became synchronized in the late stage of the culture. The day on which synchronization occurred almost coincided with the day when the coefficient of variation reached its lowest value. In addition, we have performed a mathematical analysis using interacting Bonhoeffer–van der Pol oscillators to clarify the mechanisms underlying the changes in the fluctuation of contraction rhythm with time. As the coupling strength among oscillators increased, the coefficient of variation of oscillation periods increased temporarily, but then decreased rapidly when the oscillators showed synchronization. These results suggest that the changes in the fluctuation of beating rhythm result from the increase in strength of electrical coupling among spontaneously beating cardiac myocytes. Received: 10 August 2000 / Accepted in revised form: 19 August 2001