Mathematical modeling of evolution. Solved and open problems

Evolution is a highly complex multilevel process and mathematical modeling of evolutionary phenomenon requires proper abstraction and radical reduction to essential features. Examples are natural selection, Mendel’s laws of inheritance, optimization by mutation and selection, and neutral evolution. An attempt is made to describe the roots of evolutionary theory in mathematical terms. Evolution can be studied in vitro outside cells with polynucleotide molecules. Replication and mutation are visualized as chemical reactions that can be resolved, analyzed, and modeled at the molecular level, and straightforward extension eventually results in a theory of evolution based upon biochemical kinetics. Error propagation in replication commonly results in an error threshold that provides an upper bound for mutation rates. Appearance and sharpness of the error threshold depend on the fitness landscape, being the distribution of fitness values in genotype or sequence space. In molecular terms, fitness landscapes are the results of two consecutive mappings from sequences into structures and from structures into the (nonnegative) real numbers. Some properties of genotype–phenotype maps are illustrated well by means of sequence–structure relations of RNA molecules. Neutrality in the sense that many RNA sequences form the same (coarse grained) structure is one of these properties, and characteristic for such mappings. Evolution cannot be fully understood without considering fluctuations—each mutant originates form a single copy, after all. The existence of neutral sets of genotypes called neutral networks, in particular, necessitates stochastic modeling, which is introduced here by simulation of molecular evolution in a kind of flowreactor.     

The search for a macroevolutionary theory in German paleontology

Summary Six schools of thought can be detected in the development of evolutionary theory in German paleontology between 1859 and World War II. Most paleontologists were hardly affected in their research by Darwin's Origin of Species. The traditionalists (School 1) accepted evolution within lower taxa (genera and families) but not for organisms in general. They also rejected Darwin's theory of selection. The early Darwinians (School 2) accepted Darwin's theory of transmutation and theory of selection as axioms and applied them fruitfully to the fossil record, thereby laying the foundation for the new research areas of phylogeny and paleo-biology. The enthusiasm of the early Darwinians faded when the fossil record and the problems of its interpretion became more widely known. The pluralists of the turn of the century (School 3) invented and adopted a wealth of hypothetical mechanisms in order to explain individual features of the fossil record. They failed, however, to provide one coherent theory. Dissatisfaction with this situation led to adoption of a dogmatic neo-Lamarckism (School 4), which was regarded as a coherent theory providing a fruitful research program. The rejection of the Lamarckian mechanism early in this century left paleontologists with only one kind of evolutionary mechanism: inner causes.Like many neo-Lamarckians several orthogeneticists (School 5) were highly interested in adaptation and did not see any contradiction between the inner causes of evolution and adaptation. The dominance of stratigraphical research programs in paleontology led in the 1930s and 1940s to a decrease in interest in adaptation. Stratigraphical records of taxa were accepted as meaningful in the context of evolutionary theory. Orthogenesis and the new concepts of saltation and cyclicism were amalgamated into one theory: typostrophism (School 6). This theory dominated German paleontology for decades after the war and only recently has the synthetic theory been seriously considered.Evolution was never very intensively discussed in German paleontology in the hundred years after Darwin's book. Most information used here comes from textbooks or from papers given on special occasions. It has been impossible to summarize how members of one school defended their views or discussed the ideas of competing schools.     

The fundamental units,processes and patterns of evolution,and the Tree of Life conundrum

Background

The elucidation of the dominant role of horizontal gene transfer (HGT) in the evolution of prokaryotes led to a severe crisis of the Tree of Life (TOL) concept and intense debates on this subject.

Concept

Prompted by the crisis of the TOL, we attempt to define the primary units and the fundamental patterns and processes of evolution. We posit that replication of the genetic material is the singular fundamental biological process and that replication with an error rate below a certain threshold both enables and necessitates evolution by drift and selection. Starting from this proposition, we outline a general concept of evolution that consists of three major precepts.1. The primary agency of evolution consists of Fundamental Units of Evolution (FUEs), that is, units of genetic material that possess a substantial degree of evolutionary independence. The FUEs include both bona fide selfish elements such as viruses, viroids, transposons, and plasmids, which encode some of the information required for their own replication, and regular genes that possess quasi-independence owing to their distinct selective value that provides for their transfer between ensembles of FUEs (genomes) and preferential replication along with the rest of the recipient genome.2. The history of replication of a genetic element without recombination is isomorphously represented by a directed tree graph (an arborescence, in the graph theory language). Recombination within a FUE is common between very closely related sequences where homologous recombination is feasible but becomes negligible for longer evolutionary distances. In contrast, shuffling of FUEs occurs at all evolutionary distances. Thus, a tree is a natural representation of the evolution of an individual FUE on the macro scale, but not of an ensemble of FUEs such as a genome.3. The history of life is properly represented by the "forest" of evolutionary trees for individual FUEs (Forest of Life, or FOL). Search for trends and patterns in the FOL is a productive direction of study that leads to the delineation of ensembles of FUEs that evolve coherently for a certain time span owing to a shared history of vertical inheritance or horizontal gene transfer; these ensembles are commonly known as genomes, taxa, or clades, depending on the level of analysis. A small set of genes (the universal genetic core of life) might show a (mostly) coherent evolutionary trend that transcends the entire history of cellular life forms. However, it might not be useful to denote this trend "the tree of life", or organismal, or species tree because neither organisms nor species are fundamental units of life.

Conclusion

A logical analysis of the units and processes of biological evolution suggests that the natural fundamental unit of evolution is a FUE, that is, a genetic element with an independent evolutionary history. Evolution of a FUE on the macro scale is naturally represented by a tree. Only the full compendium of trees for individual FUEs (the FOL) is an adequate depiction of the evolution of life. Coherent evolution of FUEs over extended evolutionary intervals is a crucial aspect of the history of life but a "species" or "organismal" tree is not a fundamental concept.

Reviewers

This articles was reviewed by Valerian Dolja, W. Ford Doolittle, Nicholas Galtier, and William Martin

     

A Mathematical Model for Evolution and SETI

Darwinian evolution theory may be regarded as a part of SETI theory in that the factor f_l in the Drake equation represents the fraction of planets suitable for life on which life actually arose. In this paper we firstly provide a statistical generalization of the Drake equation where the factor f_l is shown to follow the lognormal probability distribution. This lognormal distribution is a consequence of the Central Limit Theorem (CLT) of Statistics, stating that the product of a number of independent random variables whose probability densities are unknown and independent of each other approached the lognormal distribution when the number of factors increased to infinity. In addition we show that the exponential growth of the number of species typical of Darwinian Evolution may be regarded as the geometric locus of the peaks of a one-parameter family of lognormal distributions (b-lognormals) constrained between the time axis and the exponential growth curve. Finally, since each b-lognormal distribution in the family may in turn be regarded as the product of a large number (actually “an infinity”) of independent lognormal probability distributions, the mathematical way is paved to further cast Darwinian Evolution into a mathematical theory in agreement with both its typical exponential growth in the number of living species and the Statistical Drake Equation.     

Noah and the spaceship: evolution for twenty-first century christians

Evolution has increasingly become a topic of conflict between scientists and Christians, but Alexandre Meinesz’s recent book How Life Began aims to provide a reconciliation between the two. Here I review his somewhat unorthodox perspective on major transitions, alien origins and the meaning of life, with a critical focus on his account of the generation of multicellularity.     

Density-Dependent Growth and Life History Evolution of Polycyclic Leaf Pathogens: A Continuous Time Growth Model

A density-dependent growth model of a polycyclic leaf pathogen is analysed. Viability and fecundity of the pathogen are regulated by the current population density. Spores are produced continuously from a sporulating infection until death of the infection, and consequently, all age classes of infections are present at a certain point in time. The leaf area of the host varies with time. Evolution of life history strategies are studied by letting different pathogen genotypes compete with each other. Evolution of life history strategies and evolution of impact of disease are discussed in relation to the ecology of the host-pathogen system. The model is exemplified by Erysiphe graminis f. sp. hordei growing on Hordeum vulgare.     

Alfred Russel Wallace deserves better

During 2009, while we were celebrating Charles Darwin and his The origin of species, sadly, little was said about the critical contribution of Alfred Russel Wallace (1823–1913) to the development of the theory of evolution. Like Darwin, he was a truly remarkable nineteenth century intellect and polymath and, according to a recent book by Roy Davies (The Darwin conspiracy: origins of a scientific crime), he has a stronger claim to the Theory of Evolution by Natural Selection than has Darwin. Here we present a critical comparison between the contributions of the two scientists. Sometimes referred to as ‘The other beetle-hunter’ and largely neglected for many decades, Wallace had a far greater experience of collecting and investigating animals and plants from their native habitats than had Darwin. He was furthermore much more than a pioneer biogeographer and evolutionary theorist, and also made contributions to anthropology, ethnography, geology, land reform and social issues. However, being a more modest, self-deprecating man than Darwin, and lacking the latter’s establishment connections, Wallace’s contribution to the theory of evolution was not given the recognition it deserved and he was undoubtedly shabbily treated at the time. It is time that Wallace’s relationship with Darwin is reconsidered in preparation for 2013, the centenary of Wallace’s death, and he should be recognized as at least an equal in the Wallace-Darwin theory of evolution.     

On the Theory of Evolution Versus the Concept of Evolution: Three Observations

Here we address three misconceptions stated by Rice et al. in their observations of our article Paz-y-Mi?o and Espinosa (Evo Edu Outreach 2:655–675, 2009), published in this journal. The five authors titled their note “The Theory of Evolution is Not an Explanation for the Origin of Life.” First, we argue that it is fallacious to believe that because the formulation of the theory of evolution, as conceived in the 1800s, did not include an explanation for the origin of life, nor of the universe, the concept of evolution would not allow us to hypothesize the possible beginnings of life and its connections to the cosmos. Not only Stanley Miller’s experiments of 1953 led scientists to envision a continuum from the inorganic world to the origin and diversification of life, but also Darwin’s own writings of 1871. Second, to dismiss the notion of Rice et al. that evolution does not provide explanations concerning the universe or the cosmos, we identify compelling scientific discussions on the topics: Zaikowski et al. (Evo Edu Outreach 1:65–73, 2008), Krauss (Evo Edu Outreach 3:193–197, 2010), Peretó et al. (Orig Life Evol Biosph 39:395–406, 2009) and Follmann and Brownson (Naturwissenschaften 96:1265–1292, 2009). Third, although we acknowledge that the term Darwinism may not be inclusive of all new discoveries in evolution, and also that creationists and Intelligent Designers hijack the term to portray evolution as ideology, we demonstrate that there is no statistical evidence suggesting that the word Darwinism interferes with public acceptance of evolution, nor does the inclusion of the origin of life or the universe within the concept of evolution. We examine the epistemological and empirical distinction between the theory of evolution and the concept of evolution and conclude that, although the distinction is important, it should not compromise scientific logic.     

Genetic Constraints on Protein Evolution

ABSTRACT

Evolution requires the generation and optimization of new traits (“adaptation”) and involves the selection of mutations that improve cellular function. These mutations were assumed to arise by selection of neutral mutations present at all times in the population. Here we review recent evidence that indicates that deleterious mutations are more frequent in the population than previously recognized and that these mutations play a significant role in protein evolution through continuous positive selection. Positively selected mutations include adaptive mutations, i.e. mutations that directly affect enzymatic function, and compensatory mutations, which suppress the pleiotropic effects of adaptive mutations. Compensatory mutations are by far the most frequent of the two and would allow potentially adaptive but deleterious mutations to persist long enough in the population to be positively selected during episodes of adaptation. Compensatory mutations are, by definition, context-dependent and thus constrain the paths available for evolution. This provides a mechanistic basis for the examples of highly constrained evolutionary landscapes and parallel evolution reported in natural and experimental populations. The present review article describes these recent advances in the field of protein evolution and discusses their implications for understanding the genetic basis of disease and for protein engineering in vitro.     

A theory of evolution that includes prebiotic self-organization and episodic species formation

A theory has been proposed that encompasses pre-replication changes in RNA synthesis and non-gradual variant formation, in addition to competitive replication. Using a fundamental theorem of natural selection and maximum principle scaled to nucleotide condensation, evolutionin vitro was demonstrated to maximally damp both kinetic and thermodynamic forces driving this reaction, from its pre-replication stage. This led to the finding that evolution follows a path of least action. These principles form the framework for a general theory of evolution, whose scope extends beyond evolution modeled by synthesis of non-interacting RNA molecules. It applies, in particular, to standard processes, such as competitive crystallization. In calculations simulatingde novo formation of self-replicating RNA molecules in the Qβ replicase system, spontaneous changes in strand secondary structure promoted the transition from random copolymerization to template-directed polymerization. This finding indicates selection preceded genome self-propagation. Non-gradual species formation was attributed to the presence of heterogeneous thermodynamic forces. Growth unconstrained by competition follows mutation to a variant able to utilize a free energy source alien to its progenitors. Evolution in a heterogeneous system can, therefore, exhibit discontinuous rates of species formation and spawn new species populations. Natural selection among competing self-propagators thus gives way to a principle of wider scope stating that evolution optimally damps the physicochemical forces causing change within an evolving system.     

Darwin's error? Patrick Matthew and the catastrophic nature of the geologic record

In 1831, the Scottish horticulturalist Patrick Matthew (1790–1874) published a clear statement of the law of natural selection in an Appendix to his book Naval Timber and Arboriculture, which both Darwin and Wallace later acknowledged. Matthew, however, was a catastrophist, and he presented natural selection within the contemporary view that relatively long intervals of environmental stability were episodically punctuated by catastrophic mass extinctions of life. Modern studies support a similar picture of the division of geologic time into long periods of relative evolutionary stability ended by sudden extinction events. Mass extinctions are followed by recovery intervals during which surviving taxa radiate into vacated niches. This modern punctuated view of evolution and speciation is much more in line with Matthew's episodic catastrophism than the classical Lyellian–Darwinian gradualist view.     

DIRECT SELECTION ON LIFE SPAN IN DROSOPHILA MELANOGASTER

An important issue in the study of the evolution of aging in Drosophila melanogaster is whether decreased early fecundity is inextricably coupled with increased life span in selection experiments on age at reproduction. Here, this problem has been tackled using an experimental design in which selection is applied directly to longevity. Selection appeared successful for short and long life, in females as well as males. Progeny production of females selected for long life was lower than for short-lived females throughout their whole life. No increase of late-life reproduction in long-lived females occurred, as has been found in selection experiments on age at reproduction. This discrepancy is explained in terms of the inadequacy of the latter design to separate selection on life span from selection on late-life fecundity. Moreover, starvation resistance and fat content were lower for adults selected for short life. In general, the data support the negative-pleiotropy–disposable-soma theory of aging, and it is hypothesized that the pleiotropic allocation of resources to maintenance versus to reproduction as implicated in the theory might involve lipid metabolism. It is argued that further research on this suggestion is urgent and should certainly comprise observations on male reproduction because these are for the greater part still lacking. In conclusion, the longevity of D. melanogaster can be genetically altered in a direct-selection design, and such an increase is accompanied by a decreased general reproduction and thus early reproduction.     

SEXUAL SELECTION AFFECTS THE EVOLUTION OF LIFESPAN AND AGEING IN THE DECORATED CRICKET GRYLLODES SIGILLATUS

Recent work suggests that sexual selection can influence the evolution of ageing and lifespan by shaping the optimal timing and relative costliness of reproductive effort in the sexes. We used inbred lines of the decorated cricket, Gryllodes sigillatus, to estimate the genetic (co)variance between age‐dependent reproductive effort, lifespan, and ageing within and between the sexes. Sexual selection theory predicts that males should die sooner and age more rapidly than females. However, a reversal of this pattern may be favored if reproductive effort increases with age in males but not in females. We found that male calling effort increased with age, whereas female fecundity decreased, and that males lived longer and aged more slowly than females. These divergent life‐history strategies were underpinned by a positive genetic correlation between early‐life reproductive effort and ageing rate in both sexes, although this relationship was stronger in females. Despite these sex differences in life‐history schedules, age‐dependent reproductive effort, lifespan, and ageing exhibited strong positive intersexual genetic correlations. This should, in theory, constrain the independent evolution of these traits in the sexes and may promote intralocus sexual conflict. Our study highlights the importance of sexual selection to the evolution of sex differences in ageing and lifespan in G. sigillatus.     

Evolution’s past and future: an introduction and partial précis to Stephen Jay Gould’s The Structure of Evolutionary Theory

In The Structure of Evolutionary Theory, his last and largest book on evolution, Stephen Jay Gould conceives of the structure of evolutionary theory since Darwin as comprising three major propositions. First, natural selection is the most important direction-giving force in evolution. Second, it operates at the level of the individual organism. Third, selection can be extrapolated smoothly from its actions on individuals in living species throughout geologic time, to produce the gradual divergence of species and adaptations that characterizes the history of life. Challenges to each of these major propositions, according to Gould, can be of three kinds of severity. The most severe challenges, if true, would nullify one of the major propositions entirely, thus destroying the integration of the theory (and perhaps the logic and support of its other propositions). Other, less severe challenges would revise, modify, and expand the content and scope of one or another of the propositions, but not destroy any of them or the theory in toto. Still other, even less severe challenges add to what is known and extend the scope and possibilities of the theory, but do not call for a revision in its fundamental structure. Gould acknowledges that the theory has withstood all presumptive challenges that would destroy it, and has accommodated those that simply extend and add to it. His principal concern is with those challenges that would revise the theory substantially: for example, if processes other than natural selection were of great importance in evolution; if selection acted in important ways at the level of species and clades, and (or) at the level of the genes, alleles, and chromosomes; and if the extrapolation of what is known from living populations could not by itself explain many patterns of large-scale evolution seen in the fossil record. He thinks that, both through the history of evolution since Darwin and in the present day, challenges that substantially revise these basic propositions are valid, and that the theory needs to integrate them in order to retain the explanatory power that it has had for many decades. To cite this article: K. Padian, C. R. Palevol 2 (2003).     

Experimental evolution of aging in a bacterium

Background  

Aging refers to a decline in reproduction and survival with increasing age. According to evolutionary theory, aging evolves because selection late in life is weak and mutations exist whose deleterious effects manifest only late in life. Whether the assumptions behind this theory are fulfilled in all organisms, and whether all organisms age, has not been clear. We tested the generality of this theory by experimental evolution with Caulobacter crescentus, a bacterium whose asymmetric division allows mother and daughter to be distinguished.     

Evolution of butterfly seasonal plasticity driven by climate change varies across life stages

Photoperiod is a common cue for seasonal plasticity and phenology, but climate change can create cue–environment mismatches for organisms that rely on it. Evolution could potentially correct these mismatches, but phenology often depends on multiple plastic decisions made during different life stages and seasons that may evolve separately. For example, Pararge aegeria (Speckled wood butterfly) has photoperiod-cued seasonal life history plasticity in two different life stages: larval development time and pupal diapause. We tested for climate change-associated evolution of this plasticity by replicating common garden experiments conducted on two Swedish populations 30 years ago. We found evidence for evolutionary change in the contemporary larval reaction norm—although these changes differed between populations—but no evidence for evolution of the pupal reaction norm. This variation in evolution across life stages demonstrates the need to consider how climate change affects the whole life cycle to understand its impacts on phenology.     

Optimal age of maturity in fluctuating environments under r‐ and K‐selection

Optimality models for evolution of life histories have shown that increased environmental stochasticity promotes early age of maturity. Here we argue that if r‐selection for early maturation implies a tradeoff making those phenotypes more sensitive to a change in population size than phenotypes maturing at older ages, K‐selection can favor delayed onset of maturation. We analyze a general stochastic Leslie‐matrix model with a simplified density regulation affecting all survivals equally through a function of the population vector, often called the ‘critical age class’. We show that the outcome of such an age‐dependent r‐ and K‐selection is that the expected value of the ‘critical age class’ is maximized by evolution, a strategy strongly influenced by the magnitude of the environmental stochasticity. We also demonstrate that evolution caused by such density‐dependent selection influences the population dynamics, showing a possible reciprocal effect between ecology and evolution in age‐structured populations. This modeling approach reveals that changes in population size affecting the fitness of phenotypes with different age of maturity may be an important selective agent for variation in onset of reproduction in fluctuating environments. This provides a testable hypothesis for how patterns in the population dynamics should affect life history variation.     

The evolution of plant response to herbivory: simultaneously considering resistance and tolerance in Brassica rapa

Although the evolution of plant response to herbivory can involve either resistance (a decrease in susceptibility to herbivore damage) or tolerance (a decrease in the per unit effect of herbivory on plant fitness), until recently few studies have explicitly incorporated both of these characters. Moreover, theory suggests these characters do not evolve independently, and also that the pattern of natural selection acting on resistance and tolerance depends on their costs and benefits. In a genotypic selection analysis on an experimental population of Brassica rapa (Brassicaceae) I found a complex set of correlational selection gradients acting on resistance and tolerance of damage by flea beetles (Phyllotreta cruciferae: Chrysomelidae) and weevils (Ceutorhynchus assimilis: Curculionidae), as well as directional and stabilizing selection on resistance to attack by weevils. Evolution of response to flea beetle attack is constrained by a strong allocation cost of tolerance, and this allocation cost may be caused by a complex correlation among weevil resistance, weevil tolerance, flea beetle resistance, and flea beetle tolerance. Thus, one important conclusion of this study is that ecological costs may involve complex correlations among multiple characters, and for this reason these costs may not be detectable by simple pairwise correlations between characters. The evolution of response to weevil attack is probably constrained by a series of correlations between weevil resistance, weevil tolerance, and fitness in the absence of weevil damage, and possibly by a cost of tolerance of weevil damage. However, the nature of these constraints is complicated by apparent overcompensation for weevil damage. Because damage by both flea beetles and weevils had non-linear effects on plant fitness, standard measures of tolerance were not appropriate. Thus, a second important contribution of this study is the use of the area under the curve defined by the regression of fitness on damage and damage-squared as a measure of tolerance. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of assay conditions in life history experiments with Drosophila melanogaster

Selection experiments with Drosophila have revealed constraints on the simultaneous evolution of life history traits. However, the responses to selection reported by different research groups have not been consistent. Two possible reasons for these inconsistencies are (i) that different groups used different environments for their experiments and (ii) that the selection environments were not identical to the assay environments in which the life history traits were measured. We tested for the effect of the assay environment in life history experiments by measuring a set of Drosophila selection lines in laboratories working on life history evolution with Drosophila in Basel, Groningen, Irvine and London. The lines measured came from selection experiments from each of these laboratories. In each assay environment, we measured fecundity, longevity, development time and body size. The results show that fecundity measurements were particularly sensitive to the assay environment. Differences between assay and selection environment in the same laboratory or differences between assay environments between laboratories could have contributed to the differences in the published results. The other traits measured were less sensitive to the assay environment. However, for all traits there were cases where the measurements in one laboratory suggested that selection had an effect on the trait, whereas in other laboratories no such conclusion would have been drawn. Moreover, we provide good evidence for local adaptation in early fecundity for lines from two laboratories.