Do Bertalanffy's growth curves result from optimal resource allocation?

Bertalanffy's equation is commonly used to model indeterminate growth. Bertalanffy claimed that this growth pattern results from growth potential decreasing with age. An alternative approach provided by life history theory predicts that indeterminate growth is optimal for organisms in a seasonal environment and results not from decreasing growth potential but from allocating increasingly less energy with age into growth, and more into reproduction. Bertalanffy's curves are the result of evolutionary optimization and should not be used in optimization models as an assumption, but they can be used as a tool to describe the indeterminate growth pattern phenomenologically.     

Directionality theory: an empirical study of an entropic principle in life-history evolution

Understanding the relationship between ecological constraints and life-history properties constitutes a central problem in evolutionary ecology. Directionality theory, a model of the evolutionary process based on demographic entropy, a measure of the uncertainty in the age of the mother of a randomly chosen newborn, provides an analytical framework for addressing this problem. The theory predicts that in populations that spend the greater part of their evolutionary history in the stationary growth phase (equilibrium species), entropy will increase. Equilibrium species will be characterized by high iteroparity and strong demographic stability. In populations that spend the greater part of their evolutionary history in the exponential growth phase (opportunistic species), entropy will decrease when population size is large, and will undergo random variation when population size is small. Opportunistic species will be characterized by weak iteroparity and weak demographic stability when population size is large, and random variations in these attributes when population size is small. This paper assesses the validity of these predictions by employing a demographic dataset of 66 species of perennial plants. This empirical analysis is consistent with directionality theory and provides support for its significance as an explanatory and predictive model of life-history evolution.     

On the optimal policies of cancer screening.

Some problems of optimal screening are considered. A screening strategy is allowed to be nonperiodic. Two approaches to screening optimization are used: the minimum delay time approach and the minimum cost approach. Both approaches are applied to the analysis of an optimization problem when the natural history of the disease is known and when it is unknown (a minimax problem). The structure of optimal screening policies is investigated as well as the benefit they can provide compared to the periodic screening policy. The detection probability is assumed to depend only on the stage of the disease, though it may not be constant throughout each stage. It is shown that periodic screening appears to be optimal when one has no information on the natural history of the disease, the minimum delay time criterion being used for optimization. Some applications to lung cancer screening are presented.     

Life histories as adaptive strategies

Optimal control theory is used to produce a general model of life history evolution in a stationary environment. Several disparate trends in current theorizing on life histories are thereby unified. An optimal life history (OLH) is defined as one which maximizes individual fitness (the Malthusian parameter in density-independent populations, the carrying capacity in density-dependent ones). Since the components of fitness depend on the phenotype, the search for an OLH is accomplished in phenotypic space. The optimization is controlled by apportioning the energy obtained at any age between conflicting processes of growth, survival and reproduction. The methods of dynamic optimization which pertain to this problem are reviewed briefly, and its results interpreted biologically. Of these, Pontryagin's method is selected and used to examine some simple models. This method leads one to define a dual variable matched to each phenotypic variable, the prospective value. This provides an indicator of the selective pressures acting at any age on a phenotypic feature to push it towards coincidence with the OLH. This also suggests that at ages in which these dual variables are low (i.e. late ages) there will be greater phenotypic variability around the OLH in any population. The problem of the optimal distribution of reproductive effort over the life history is discussed as well.     

Formation of evolutionary approach as an explanatory principle for ecology

Although the connection of ecology with evolutionary idea and specifically with Darwinism was proclaimed for a long time it seems that Herbert Spencer's approach with its emphasize on natural equilibrium was much more often used as its real theoretical base. Elements of Darwinian approach appeared only in 1920-30s in works of those few researchers who studying the distribution and population dynamics of different species tried to understand general mechanisms providing their continuing existence. Later, in the middle of 1950s the first attempts were undertaken to consider the population life history (primarily the age specific schedule of death and reproduction) as a result of natural selection aimed to maintain the necessary level of fitness. A special attention in these studies that burgeoned in 1980-90s was paid to looking for various trade-offs between particular parameters of life history, e.g., between the survival of juveniles and fecundity of adults. The problem of life history optimization became central for the whole branch of science named "evolutionary ecology". Though traditionally this branch is connected with Darwinism, it is rooted rather in Spencer's ideas on moving equilibrium and deals more with static than dynamic. Disproportionately less attention was paid to the evolution of communities since these formations could be hardly interpreted as units of Darwinian selection. Moreover, the ecologists dealing with biosphere as a unified biogeochemical system began insist on "nondarwinian" nature of its evolution. The author considers this opinion as not sufficiently grounded. Darwin's ideas about unavoidable exponential growth, intrinsic for any population, consequent deficiency of resources, and differential survival and reproduction of individuals are still useful while studying the evolution of living organisms (phylogenetics) or the development of biosphere as a global ecosystem.     

Modelling spatial dynamics of fish

Our ability to model spatial distributions of fish populations is reviewed by describing the available modelling tools. Ultimate models of the individual's motivation for behavioural decisions are derived from evolutionary ecology. Mechanistic models for how fish sense and may respond to their surroundings are presented for vision, olfaction, hearing, the lateral line and other sensory organs. Models for learning and memory are presented, based both upon evolutionary optimization premises and upon neurological information processing and decision making. Functional tools for modelling behaviour and life histories can be categorized as belonging to an optimization or an adaptation approach. Among optimization tools, optimal foraging theory, life history theory, ideal free distribution, game theory and stochastic dynamic programming are presented. Among adaptation tools, genetic algorithms and the combination with artificial neural networks are described. The review advocates the combination of evolutionary and neurological approaches to modelling spatial dynamics of fish.     

Heuristic optimization of the general life history problem: A novel approach

Summary The general life history problem concerns the optimal allocation of resources to growth, survival and reproduction. We analysed this problem for a perennial model organism that decides once each year to switch from growth to reproduction. As a fitness measure we used the Malthusian parameterr, which we calculated from the Euler-Lotka equation. Trade-offs were incorporated by assuming that fecundity is size dependent, so that increased fecundity could only be gained by devoting more time to growth and less time to reproduction. To calculate numerically the optimalr for different growth dynamics and mortality regimes, we used a simplified version of the simulated annealing method. The major differences among optimal life histories resulted from different accumulation patterns of intrinsic mortalities resulting from reproductive costs. If these mortalities were accumulated throughout life, i.e. if they were senescent, a bangbang strategy was optimal, in which there was a single switch from growth to reproduction: after the age at maturity all resources were allocated to reproduction. If reproductive costs did not carry over from year to year, i.e. if they were not senescent, the optimal resource allocation resulted in a graded switch strategy and growth became indeterminate. Our numerical approach brings two major advantages for solving optimization problems in life history theory. First, its implementation is very simple, even for complex models that are analytically intractable. Such intractability emerged in our model when we introduced reproductive costs representing an intrinsic mortality. Second, it is not a backward algorithm. This means that lifespan does not have to be fixed at the begining of the computation. Instead, lifespan itself is a trait that can evolve. We suggest that heuristic algorithms are good tools for solving complex optimality problems in life history theory, in particular questions concerning the evolution of lifespan and senescence.     

A parameter optimization approach for the optimal control of large-scale musculoskeletal systems.

This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).     

The evolution of fledging age in songbirds

In birds with altricial young an important stage in the life history is the age at fledging. In this paper we use an approach proven successful in the prediction of the optimal age at maturity in fish and reptiles to predict the optimal age of fledging in passerines. Integrating the effects of growth on future fecundity and survival leads to the prediction that the optimal age at fledging is given by a function that comprises survival to maturity, the exponent of the fecundity-body size relationship and nestling growth. Growth is described by the logistic equation with parameters, A, K and t(i). Assuming that the transitional mortality curve can be approximated by the nestling mortality, M(n), the optimal fledging age, t(f), is given by a simple formula involving the three growth parameters, nestling mortality (M(n)) and the exponent (d) of the fecundity-body size relationship. Predictions of this equation underestimate the true values by 11-16%, which is expected as a consequence of the transitional mortality function approximation. A transitional mortality function in which mortality is approximately 0.3-0.4 of nesting mortality (i.e. mortality declines rapidly after fledging) produces predictions which, on average, equal the observed values. Data are presented showing that mortality does indeed decline rapidly upon fledging.     

Target size and optimal life history when individual growth and energy budget are stochastic

I extend my previous work on life history optimization when body mass is divided into reserves and structure components. Two important innovations are: (1) effect of finite target size on optimal structural growth; (2) incorporating reproduction in the optimization objective. I derive optimal growth trajectories and life histories, given that the individual is subject to both starvation mortality and exogenous hazards (e.g., predation). Because of overhead costs in building structural mass, it is optimal to stop structural growth close to the target size, and to proceed only by accumulating reserves. Higher overhead costs cause earlier cessation of structural growth and smaller final structures. Semelparous reproduction also promotes early cessation of structural growth, compared to when only survival to target size is maximized. In contrast, iteroparous reproduction can prolong structural growth, resulting in larger final structures than in either the survival or the semelparous scenarios. Increasing the noise in individual growth lowers final structural mass at small target sizes, but the effect is reversed for large target sizes. My results provide predictions for comparative studies. I outline important consequences of my results to additional important evolutionary questions: evolution of sexual dimorphism, optimization of clutch size and evolution of progeny and adult sizes.     

Improved Hidden Markov Model training for multiple sequence alignment by a particle swarm optimization-evolutionary algorithm hybrid

Multiple sequence alignment (MSA) is one of the basic problems in computational biology. Realistic problem instances of MSA are computationally intractable for exact algorithms. One way to tackle MSA is to use Hidden Markov Models (HMMs), which are known to be very powerful in the related problem domain of speech recognition. However, the training of HMMs is computationally hard and there is no known exact method that can guarantee optimal training within reasonable computing time. Perhaps the most powerful training method is the Baum-Welch algorithm, which is fast, but bears the problem of stagnation at local optima. In the study reported in this paper, we used a hybrid algorithm combining particle swarm optimization with evolutionary algorithms to train HMMs for the alignment of protein sequences. Our experiments show that our approach yields better alignments for a set of benchmark protein sequences than the most commonly applied HMM training methods, such as Baum-Welch and Simulated Annealing.     

Estimation of optimal feeding strategies for fed-batch bioprocesses

A generic methodology for feeding strategy optimization is presented. This approach uses a genetic algorithm to search for optimal feeding profiles represented by means of artificial neural networks (ANN). Exemplified on a fed-batch hybridoma cell cultivation, the approach has proven to be able to cope with complex optimization tasks handling intricate constraints and objective functions. Furthermore, the performance of the method is compared with other previously reported standard techniques like: (1) optimal control theory, (2) first order conjugate gradient, (3) dynamical programming, (4) extended evolutionary strategies. The methodology presents no restrictions concerning the number or complexity of the state variables and therefore constitutes a remarkable alternative for process development and optimization. This revised version was published online in June 2005 with corrections to the Appendix.     

The history of development of evolutionary methods in St. Petersburg school of computer simulation in biology

The history of rise and development of evolutionary methods in Saint Petersburg school of biological modelling is traced and analyzed. Some pioneering works in simulation of ecological and evolutionary processes, performed in St.-Petersburg school became an exemplary ones for many followers in Russia and abroad. The individual-based approach became the crucial point in the history of the school as an adequate instrument for construction of models of biological evolution. This approach is natural for simulation of the evolution of life-history parameters and adaptive processes in populations and communities. In some cases simulated evolutionary process was used for solving a reverse problem, i. e., for estimation of uncertain life-history parameters of population. Evolutionary computations is one more aspect of this approach application in great many fields. The problems and vistas of ecological and evolutionary modelling in general are discussed.     

Evolutionary game theoretic strategy for optimal drug delivery to influence selection pressure in treatment of HIV-1

Cytotoxic T-lymphocyte (CTL) escape mutation is associated with long-term behaviors of human immunodeficiency virus type 1 (HIV-1). Recent studies indicate heterogeneous behaviors of reversible and conservative mutants while the selection pressure changes. The purpose of this study is to optimize the selection pressure to minimize the long-term virus load. The results can be used to assist in delivery of highly loaded cognate peptide-pulsed dendritic cells (DC) into lymph nodes that could change the selection pressure. This mechanism may be employed for controlled drug delivery. A mathematical model is proposed in this paper to describe the evolutionary dynamics involving viruses and T cells. We formulate the optimization problem into the framework of evolutionary game theory, and solve for the optimal control of the selection pressure as a neighborhood invader strategy. The strategy dynamics can be obtained to evolve the immune system to the best controlled state. The study may shed light on optimal design of HIV-1 therapy based on optimization of adaptive CTL immune response.     

Modeling forest stand dynamics from optimal balances of carbon and nitrogen

We formulate a dynamic evolutionary optimization problem to predict the optimal pattern by which carbon (C) and nitrogen (N) are co-allocated to fine-root, leaf, and wood production, with the objective of maximizing height growth rate, year by year, in an even-aged stand. Height growth is maximized with respect to two adaptive traits, leaf N concentration and the ratio of fine-root mass to sapwood cross-sectional area. Constraints on the optimization include pipe-model structure, the C cost of N acquisition, and agreement between the C and N balances. The latter is determined by two models of height growth rate, one derived from the C balance and the other from the N balance; agreement is defined by identical growth rates. Predicted time-courses of maximized height growth rate accord with general observations. Across an N gradient, higher N availability leads to greater N utilization and net primary productivity, larger trees, and greater stocks of leaf and live wood biomass, with declining gains as a result of saturation effects at high N availability. Fine-root biomass is greatest at intermediate N availability. Predicted leaf and fine-root stocks agree with data from coniferous stands across Finland. Optimal C-allocation patterns agree with published observations and model analyses.     

We age because we grow

Why do we age? Since ageing is a near-universal feature of complex organisms, a convincing theory must provide a robust evolutionary explanation for its ubiquity. This theory should be compatible with the physiological evidence that ageing is largely due to deterioration, which is, in principle, reversible through repair. Moreover, this theory should also explain why natural selection has favoured organisms that first improve with age (mortality rates decrease) and then deteriorate with age (mortality rates rise). We present a candidate for such a theory of life history, applied initially to a species with determinate growth. The model features both the quantity and the quality of somatic capital, where it is optimal to initially build up quantity, but to allow quality to deteriorate. The main theoretical result of the paper is that a life history where mortality decreases early in life and then increases late in life is evolutionarily optimal. In order to apply the model to humans, in particular, we include a budget constraint to allow intergenerational transfers. The resultant theory then accounts for all our basic demographic characteristics, including menopause with extended survival after reproduction has ceased.     

Clustering in non-stationary environments using a clan-based evolutionary approach

 Clustering techniques are used to discover structure in data by optimizing a defined criterion function. Most of these methods assume that the data are stationary, and these techniques are based on gradient descent which converge to a locally optimal clustering. There are many potential applications that require clustering to be performed in non-stationary temporal environments. In this paper, we investigate the applicability of a clan-based evolutionary optimization method for clustering data in non-stationary environments. Due to the stochastic nature of the technique, the problem of becoming entrapped in local optima is avoided, and the method can converge to (nearly) optimal clusters. Different cases are considered in the experiments, and the results demonstrate the efficacy of the evolutionary approach for clustering time-varying data. Received: 7 September 1994/Accepted in revised form: 28 March 1995     

Probabilistic maturation reaction norms assessed from mark–recaptures of wild fish in their natural habitat

Reaction norms are a valuable tool in evolutionary biology. Lately, the probabilistic maturation reaction norm approach, describing probabilities of maturing at combinations of age and body size, has been much applied for testing whether phenotypic changes in exploited populations of fish are mainly plastic or involving an evolutionary component. However, due to typical field data limitations, with imperfect knowledge about individual life histories, this demographic method still needs to be assessed. Using 13 years of direct mark–recapture observations on individual growth and maturation in an intensively sampled population of brown trout (Salmo trutta), we show that the probabilistic maturation reaction norm approach may perform well even if the assumption of equal survival of juvenile and maturing fish does not hold. Earlier studies have pointed out that growth effects may confound the interpretation of shifts in maturation reaction norms, because this method in its basic form deals with body size rather than growth. In our case, however, we found that juvenile body size, rather than annual growth, was more strongly associated with maturation. Viewed against earlier studies, our results also underscore the challenges of generalizing life‐history patterns among species and populations.     

Metabolic flux estimation--a self-adaptive evolutionary algorithm with singular value decomposition

Metabolic flux analysis is important for metabolic system regulation and intracellular pathway identification. A popular approach for intracellular flux estimation involves using ¹³C tracer experiments to label states that can be measured by nuclear magnetic resonance spectrometry or gas chromatography mass spectrometry. However, the bilinear balance equations derived from ¹³C tracer experiments and the noisy measurements require a nonlinear optimization approach to obtain the optimal solution. In this paper, the flux quantification problem is formulated as an error-minimization problem with equality and inequality constraints through the ¹³C balance and stoichiometric equations. The stoichiometric constraints are transformed to a null space by singular value decomposition. Self-adaptive evolutionary algorithms are then introduced for flux quantification. The performance of the evolutionary algorithm is compared with ordinary least squares estimation by the simulation of the central pentose phosphate pathway. The proposed algorithm is also applied to the central metabolism of Corynebacterium glutamicum under lysine-producing conditions. A comparison between the results from the proposed algorithm and data from the literature is given. The complexity of a metabolic system with bidirectional reactions is also investigated by analyzing the fluctuations in the flux estimates when available measurements are varied.     

Physiological dynamics,reproduction‐maintenance allocations,and life history evolution

Allocation of resources to competing processes of growth, maintenance, or reproduction is arguably a key process driving the physiology of life history trade‐offs and has been shown to affect immune defenses, the evolution of aging, and the evolutionary ecology of offspring quality. Here, we develop a framework to investigate the evolutionary consequences of physiological dynamics by developing theory linking reproductive cell dynamics and components of fitness associated with costly resource allocation decisions to broader life history consequences. We scale these reproductive cell allocation decisions to population‐level survival and fecundity using a life history approach and explore the effects of investment in reproduction or tissue‐specific repair (somatic or reproductive) on the force of selection, reproductive effort, and resource allocation decisions. At the cellular level, we show that investment in protecting reproductive cells increases fitness when reproductive cell maturation rate is high or reproductive cell death is high. At the population level, life history fitness measures show that cellular protection increases reproductive value by differential investment in somatic or reproductive cells and the optimal allocation of resources to reproduction is moulded by this level of investment. Our model provides a framework to understand the evolutionary consequences of physiological processes underlying trade‐offs and highlights the insights to be gained from considering fitness at multiple levels, from cell dynamics through to population growth.