Fisher's theorems for multivariable, time- and space-dependent systems, with applications in population genetics and chemical kinetics

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A generalized Fisher equation (GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance. The GFE is an exact mathematical result that has been widely used in population dynamics and genetics, where it originated. Here we demonstrate that the GFE can also be useful in other fields, specifically in chemistry, with models of two chemical reaction systems for which the mechanisms and rate coefficients correspond reasonably well to experiments. A bad fit of the GFE can be a sign of high levels of measurement noise; for low or moderate levels of noise, fulfillment of the GFE is not degraded. Hence, the GFE presents a noise threshold that may be used to test the validity of experimental measurements without requiring any additional information. In a different approach information about the system (model) is included in the calculations. In that case, the discrepancy with the GFE can be used as an optimization criterion for the determination of rate coefficients in a given reaction mechanism. The generalized form of Fisher equation (GFE) holds for temporal functions, which are different from zero (for chemical reaction systems this means strictly positive) with continuous second-order derivatives. These Fisher equations are exact results, which are independent of the detailed kinetics of the process: They are valid whether the evolution equations are linear or nonlinear, or local or nonlocal in space and/or time (2). Here we show that the GFE can be useful in chemical kinetics. This is tested with two chemical reaction systems, for which both the reaction mechanism and the rate coefficients are reasonably well known and correspond to experiments. The use of the GFE is new for chemical kinetics, a subject used in many fields other than chemistry, such as biology, biotechnology, chemical engineering, materials science, etc.In Generalized Form of the Fisher Equation we define the notation to write the GFE as in ref. 5. Then, in Use of the GFE for Testing Experimental Measurements we discuss the effect of noise.If the agreement with the GFE is calculated using experimentally measured concentrations-or concentrations generated with a nominal model-and rates of growth analytically calculated with a set of rate coefficients, the fit to the GFE depends on the values of these rates coefficients. For a given reaction system, optimal rate coefficients yield minimum deviation from the GFE. Hence the GFE can be used as a general criterion in optimization

Section: Introductionmentioning

confidence: 96%

“…We consider only temporal not spatial processes (2). To write down the GFE we need to define several terms.…”

Section: Generalized Form Of the Fisher Equation (Gfe)mentioning

confidence: 99%

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A generalized Fisher equation and its utility in chemical kinetics

Ross

Villaverde

Banga

et al. 2010

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“…Generalizing this picture to realistic processes of molecular evolution has been a long-standing problem (2)(3)(4)(5)(6)(7)(8). The solution presented here involves a number of important conceptual steps.…”

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confidence: 99%

Fitness flux and ubiquity of adaptive evolution

Mustonen

Lässig

2010

169

247

Natural selection favors fitter variants in a population, but actual evolutionary processes may decrease fitness by mutations and genetic drift. How is the stochastic evolution of molecular biological systems shaped by natural selection? Here, we derive a theorem on the fitness flux in a population, defined as the selective effect of its genotype frequency changes. The fitnessflux theorem generalizes Fisher's fundamental theorem of natural selection to evolutionary processes including mutations, genetic drift, and time-dependent selection. It shows that a generic state of populations is adaptive evolution: there is a positive fitness flux resulting from a surplus of beneficial over deleterious changes. In particular, stationary nonequilibrium evolution processes are predicted to be adaptive. Under specific nonstationary conditions, notably during a decrease in population size, the average fitness flux can become negative. We show that these predictions are in accordance with experiments in bacteria and bacteriophages and with genomic data in Drosophila. Our analysis establishes fitness flux as a universal measure of adaptation in molecular evolution. A daptive processes have taken center stage in molecular evolutionary biology. Deep sequencing of populations opens unprecedented opportunities to trace the genomic basis of adaptation in population-genetic studies within and across species as well as in time series of evolution experiments. Various methods are used to infer natural selection from such data; however, their results lack a common gauge and are sometimes difficult to compare. This paper develops the concept of fitness flux in a population as a generic measure of adaptation applicable to molecular data. Whereas fitness characterizes the state of a population at a given point in time, fitness flux can be accumulated in a population's history over a period (a precise definition of this quantity will be given below). Fitness flux, not fitness, turns out to be the right variable to show that adaptive evolution is a generic state of natural populations. The notion of fitness flux is already implicitly contained in Fisher's fundamental theorem of natural selection (1), which states that any fitness difference within a population leads to adaptation in an evolution process governed by natural selection alone. The fitness flux of this deterministic process equals the (additive) fitness variance in the population. Hence, the flux is positive when adaptation occurs and zero otherwise.Generalizing this picture to realistic processes of molecular evolution has been a long-standing problem (2-8). The solution presented here involves a number of important conceptual steps. First, molecular processes are always stochastic because of genetic drift and mutations, and we include these forces into a stochastic theory of fitness flux. Second, we extend the observation of this dynamics to the time scales of genomic data, describing populations by histories of genotype composition and demography that may extend beyond their ...

“…5 and 6 have a similar structure as the evolution equations for gene frequencies in population genetics, which suggests the existence of an evolution criterion similar to Fisher's fundamental theorem of natural selection (7)(8)(9). We introduce the distortion factor (q) (W) ϭ exp(Ϫ (q) (W))Ϫ1, which is a measure of the rate of change of the probability density of W from the qth to the (q ϩ 1)th experiment, and is the analog of the intrinsic rate of growth of a genotype from the qth to the (q ϩ 1)th generation.…”

Section: ]mentioning

confidence: 99%

Incremental parameter evaluation from incomplete data with application to the population pharmacology of anticoagulants

Vlad¹,

Corlan²,

Morán³

et al. 2008

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We develop a method for parameter evaluation from incomplete data. Improved estimates of the desired parameters are evaluated step by step, from experiment to experiment by using both Bayesian and informational methods. We make dynamical, improved predictions while the experiments are still going on and keep and interpret information about local fluctuations, which is lost on applying global techniques. The input of information in small packets leads to semi-analytic methods for data processing. An evolution criterion for parameter evaluation, similar to Fisher's theorem of population selection, is derived. We develop direct processing methods, which can be applied to low dimensional systems, semi-analytic methods based on direct or double logarithmic phase expansions, steepest descent approaches, variation and perturbation methods. The techniques are illustrated by developing a method of long-term planning of treatments with oral anticoagulants based on limited clinical data. The efficiency of treatment by oral anticoagulants depends strongly on various anthropometric and genotypic factors, which lead to large variations of the clinical response. We use the clinical data, which accumulates from medical consultations, for extracting improved, incremental information about the statistical properties of the kinetic and anthropometric parameters for a given patient, which in turn is used for making repeated, improved clinical predictions as the treatment proceeds.