Peter Schuster, Institut für Theoretische Chemie der Universität Wien, Austria Darwin s Optimization in the looking glasses of physics and chemistry

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1 Peter Schuster, Institut für Theoretische Chemie der Universität Wien, Austria Darwin s Optimization in the looking glasses of physics and chemistry Prologue The traditions of scientific research in physics and biology are rather different. Theories and mathematical formulations are the backbone of physical thought. Mere data collections without a unifying or generalizing concept are considered of low value and have little prestige. In biology observations and experimental findings without an interpretation by theory are quite often considered as having the same rank as fundamental insights into the mechanisms operating in the living world. Mathematical and theoretical biology were thought to be dispensable for a real biologist until the recent data explosion in molecular genetics required new strategies for the extraction of relevant information. Charles Darwin s centennial theory of biological evolution is an exception: A clearly stated formal concept, natural selection, is subjected to proof not by mathematics but by a wealth of empirical observations with careful interpretations. The great success of the Darwinian explanation for the driving force of optimization and adaptaion in nature is based on the radical abstraction of fitness. What counts in evolution is only fertility over generations, the head count of reproductive progeny. Neither the details of reproduction play a role, nor the mechanisms creating variability. Otherwise, Darwin would have failed badly because he had totally wrong thoughts about in heritance and variation. a The different viewpoints of physicists and biologists date back to the beginning of modern science, to the famous statement of Galileo Galilei: 1 Philosophy (science) is written in this grand book, the universe. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures;. At the beginning of science stands the marriage between physics and mathematics that turned out to be an incredibly stable liaison until present time. Mathematics provided the frame for quantitative thinking and physics has fertilized mathematics. In order to make mathematical modeling feasible the complexity of observations has to be reduced to the essentials. This is expressed in the statement: Make things as simple as possible, but not simpler. b Reduction is an indispensible part of the methods of physics. It turned out to be highly fruitful in biology too as the well known example of Mendelian genetics illustrates. Gregor Mendel made an enormous reductionist s assumption by postulating atoms of inheritance and succeded to provide evidence for his concept by analyzing a large amount of exprimental data by means of mathematical statistics. Most biologists, on the other hand, were and are primarily engaged in observations, collecting data from nature, and ordering them according to macroscopic features. Therefore most biologists prefer holistic descriptions of phenomena, theory is considered with skepticism, and a It is an historical irony that Gregor Mendel had the correct solution to inheritance and genetics but his works were completely ignored by evolutionary biologists. b This statement is often called Einstein s razor in analogy to Occam s razor, but the precise reference to Albert Einstein is unknown.

2 - - most of the great books of biology do not contain any mathematics at all. Examples are Charles Darwin s seminal book Origin of Species 3 or Ernst Mayr s remarkable historical treatise The Growth of Biological Thought. 4 The development of molecular biology and molecular genetics within the second half of the twentieth century completely changed the situation: Current biology is flooded by data, which cannot be analyzed without a solid theoretical and mathematical background, and massive assistance by computation is indispensible. Present day molecular biology is closing the gap between chemistry and biology and in the new discipline of systems biology the reductionists bottom up approach aims at reaching the description level of whole cells and organisms. 5 This contribution deals with Darwins s principle of optimization, not in the conventional view but rather seen with the eyes of a mathematician or a physicist. We shall try to aswer the question, how Darwin might have formulated his theory if he were a mathematician. Darwin s principle tells how selection operates and chooses variants according to their effect on progeny: The better adapted variant can produce more progeny and will outgrow the less efficient ones in future generations. Darwin s principle tells how selection operates but says nothing about the spectrum of variants that is available for choosing. The present-day knowledge on possible forms of organisms is still rudimentary but some basic information of shapes and patterns is already available. Accordingly pattern formation is discussed first. The special role of genetic information is illustrated by means of a comparison between pattern formation in physics and biology. Then, natural selection will be discussed and analyzed as a pure concept and as an implementation in molecular systems. Molecular evolution will be modeled as an exercise in chemical kinetics and a brief account is given on evolution experiments under controlled conditions. Folding of biopolymer sequences into structures is considered as a simple example of genetically encoded pattern formation. Neutrality with respect to selection is shown to be an integral part of evolution in real systems. Finally, we illustrate the role of stochasticity be means of a model system accessible by computer simulation. Pattern formation in physics and biology In previous centuries pattern formation has been considered as a phenomenon that is unique to biology. Almost all people are, for example, impressed by the intriguing beauties of flowers and animal skins. The metamorphosis of insects from caterpillars to butterflies and insect mimicry by orchids may serve as illustrative examples. Seeming purposefulness of biological pattern has attracted the interest of biologists since long. Some biological patterns are interpreted as the results of optimization processes that aim at hiding e.g., stripes and dots similar to shadows in a forest or white fur colors in winter. In mimicry, in contrary, patterns resemble the appearance of other pattern carriers, which are dangerous and have advantageous properties for survival. Mimicry found in numerous natural examples provides clear evidence for optimization. Harmless species, for example, resemble poisonous species in order to avoid being eaten by predators. Two properties of pattern formation are required for successful mimicry: (i) The pattern must be inherited, that is reproducible in the progeny, and (ii) the patterns need to be

3 - 3 - easily changeable in order to provide enough variability to resemble patterns of other species. Patterns determine not only the externally visible features of organisms they are likewise responsible for body plans: somites rips and vertebrae in later development or other periodic and non-periodic body segments that are formed though the developmental structuring process. Patterns do exist in the inanimate world as well but the lack of apparent usefulness has delayed the attention of scientists. Patterns in rock formations were of characterized as jokes of nature or ludus naturae. Although spontaneous formation of regular patterns in crystallization 6 and oscillatory concentrations in heterogeneous chemical reactions 7 were known already in the late nineteenth century, systematic research on pattern formation did not start before the nineteen fifties. Alan Turing s path-breaking work on the spontaneous formation of spatial structures in an isothermal homogeneous chemical reaction systems 8 provided a chemical model that pointed at necessary requirements for pattern formation: (i) thermodynamic conditions far away from equilibrium, (ii) autocatalysis or self-enhancement caused by a chemical reaction network, and (iii) different spatial migration rates as expressed by different diffusion coefficients c of the molecular species. The Turing model can be cast into a simple two-component system with two molecular species U and V whose concentrations are variables in space and time denoted by u(x,y,z,t) and v(x,y,z,t), respectively. Spatial spreading of the two components described by a partial differential equation of the reaction-diffusion type: u t = D v = D t The partial derivatives with respect to time, u u + v v + t f ( u, v) g( u, v), on the l.h.s. of the equation describe the local change in the concentration of u and v, respectively. These changes consist of two contributions: (i) the change as a result of spatial diffusion expressed by the sum of the second derivatives with respect to the spatial coordinates, = x + y + z, multiplied by the diffusion coefficients of the two molecular species, D u and D v, respectively, and (ii) the change resulting from chemical reactions that is encapsulated in the two functions f(u,v) and g(u,v). Provided the diffusion coefficients, D u and D v, are sufficiently different and the two functions are nonlinear and autocatalytic, d spontaneous breaking of spatial symmetry occurs at certain concentration values and a pattern consisting of stripes, checker board like or other distributions occurs. c The diffusion coefficient is a measure for the spreading of a peak of concentration in space. The higher the diffusion coefficient, the faster the substance approaches the uniform distribution representing thermodynamic equilibrium in an unperturbed system. d Autocatalysis implies that either f(u,v) or g(u,v) or both increase in response to an increase in u or y, and this at least in certain ranges of concentrations. The increase in the functions results in a larger positive time derivative of the concentration that in turn causes an increase in the variable. The self-enhancement in the positive feedback loop continues until an external concentration limit or constraint prevents further increase.

4 - 4 - Spontaneous pattern formation became a central issue in physics of the second half of twentieth century. Two approaches turned out to be of general applicability: (i) the theory of nonequilibrium or dissipative structures occurring at conditions far from equilibrium introduced by Ilya Progogine and his coworkers in Brussels 9 and (ii) the theory of synergetics in nonlinear dynamical systems conceived by Hermann Haken and his group in Stuttgart. 10 Pattern formation found widespread interest after the first oscillatory chemical reaction in homogeneous solution had been discovered by Boris Belousov 11 and was taken up again five years later by Anatol Zhabotinsky. 1 A great variety of detailed investigations revealed the kinetic mechanisms of several reactions that have some features in common. 13 Autocatalysis is commonly encapsulated by a multi-step mechanism involving a chemical element like chlorine, bromine, or iodine in multiple valence states. Although oscillations in chemical reactions as well as spatiotemporal structures like concentric or target waves and coupled spirals e were well understood, no truly stationary Turing patterns were known until the elegant works of the nonlinear chemistry group in Bordeaux. 14,15 The trick applied to stabilize the Turing pattern is to use an acrylamide gel in order to avoid undesired convective motion that obscures pure diffusion. Patterns in the inanimate world are characterized by limited predictability. The wave length of spatiotemporal patterns or the period of oscillations are reproducible and can be calculated with fair accuracy. The positioning of patterns, however, depends on microscopic fluctuations, which escape even the most precise measurements at least so far and cannot be predicted therefore. The same is true for atmospheric patterns: The theory of hurricanes is well developed and necessary conditions like the minimal surface temperature of the ocean can be given but there is no chance to predict when and where a hurricane will form. Patterns on the skins of genetically identical or very closely related animals are often remarkably constant (see, e.g., figure 1): Genetic control of development allows for precise positioning of patterns, f which is particularly remarkable, since organisms of very different sizes carry certain features at precisely the same relative position. Embryonic development has been studied now for more than 150 years and these investigations revealed a true wealth of interesting details. Molecular biology opened new doors for the study of cell differentiation and development but still many important features wait to be discovered. About forty years ago new impacts were given to the field through the application of methods from molecular genetics. 16,17 A pre-pattern is formed first followed by cell differentiation that leads eventually to the adult organism. Lewis Wolpert postulated that the cell in an organism receives information on its position from neighboring cells or from signals in the intercellular medium. 18 A pre-pattern consisting of a gradient of a substance called morphogen initiates development. Thresholds in morphogen concentration separate cells that differentiate into different cell lines with different properties, since different chemical reactions dominate within the cells lying above or below threshold (see e A pair of coupled spirals in a plane consists of two spirals being mirror images of each other that rotate in opposite directions. f Developmental biology and developmental genetics are very broad disciplines and therefore an updated textbook and a collective volume are recommended to the interested reader here. 16,17

5 - 5 - the French-flag model illustrated in figure ). The precision of relative positioning is determined by the sharpness of the response to the morphogen concentrations. Further investigations have shown that the initial gradients and thresholds are rather fuzzy but sharpen at later time. 19 Direct insight in early embryonic development was provided by molecular genetics studies of larval development in the fruit fly drosophila. Proteins derived from maternal genes and genes of the individual itself form gradients in the early periods of embryonic development and the gradients are translated into a pre-pattern of further gene activities that determine the fate of cells. 0,1 Work in the following decades revealed a whole cascade of decisions in gene activities, which lay down the protein expression pattern and determine the ultimate properties of the fully differentiated cells. The Turing model was adapted for the description of biological pattern formation by Alfred Gierer and Hans Meinhardt.,3 The basic idea is to postulate two morphogens, an activator and an inhibitor, which form a pre-pattern of Turing type and initiate cell differentiation. Stable primary patterns are obtained under the assumption of slow activator and fast inhibitor diffusion that gives rise to short-range activation and long-range inhibition. In terms of the Turing model, where A is the activator and H the inhibitor, respectively, these properties are encoded in the diffusion coefficients, D A á D H, and in the functions representing chemical kinetics: 4 a F( a, h) = ρ ρ0 + µ a h G( a, h) = η ( η + a ) υ h 0 Both morphogens, activator and inhibitor, are produced at a low constant rates, 0 and 0, and degraded through first order reactions, - a and - h. Uncatalyzed production and degradation alone would give rise to a homogeneous stationary state. Pattern formation is introduced by the nonlinear autocatalytic terms containing the factor a. g The production of both morphogens is strongly catalyzed by the activator A but the growth in activator concentration a is reduced by the inhibitor H as the factor 1/h shows. The model reproduces well many features of pattern formation in biology. Despite the apparent beauty of the formalism the model suffers from the fact that the individual terms were introduced ad hoc without reference to a defined kinetic mechanism. Nevertheless, the Turing-Gierer-Meinhardt model provides an excellent study case describing how pattern formation sets the stage for the forthcoming Darwinian evolution (see, e.g., figure 3). Many other models, which are based on the Turing mechanism and yield similar spatial patterns can be found in the literature (For an excellent review see, e.g., the corresponding chapters in the two volume treatise of mathematical biology by Jim Murray 5 ). The Gierer-Meinhardt model in its original form has a parameter set with six parameter values P(, 0,,, 0, ), which represents one point in an abstract six-dimensional parameter space. The g Systematic modeling and experimental studies have shown that simple linear autocatalysis of the form dx/dt = k x is not sufficient for complex dynamics consisting of oscillations or deterministic chaos and for spatial pattern formation. It is, however, the basis of Darwinian selection.

6 - 6 - set of all points P in parameter space is the set of all patterns that are accessible by the sixparameter version of the model. It is worth to illustrate by means of the Gierer-Meinhardt model that often the dimension of parameter space can be reduced without losing generality. Transformation to dimensionless variables yields in this case 6 u = γ f ( u, v) + u t and u f ( u, v) = α β u + v v = γ g( u, v) + ϑ u t and g( u, v) = u v, and reduces the dimensionality of parameter space by two: P(,,, ). The model parameters are functions or properties of biological macromolecules and their interactions. In simple cases these functions can be directly derived from molecular structures but development is an exceedingly complex process and we are miles away from a complete understanding. Hence, predictions of how changes in molecular properties are transformed into the changes in the phenotypes of organisms are not possible at present. Nevertheless, development sets the stage for evolution since it translates the variations in the genetic molecules into differences in patterns and eventually different adult organisms. How does evolution come into play? Given we cannot predict phenotypic changes of organisms from molecular variations yet, how can we then study the mechanism of evolutionary optimization and adaptation? Two strategies are possible: (i) The molecular mechanism of phenotype development is treated as black box and insight into mechanistic details is replaced by a wealth of observations that support an empirical approach or (ii) the system is reduced to such an extent that the simplified system allows for a study of the molecular mechanism. Here, we shall adopt the second approach that is the conventional viewpoint of physicists and chemists. Evolution is reduced to the core of Darwin s principle based on reproduction, variation and selection. Darwin s principle seen with the eyes of a mathematician Darwin s principle of evolution predicts changes in the distribution of variants in a population. Variants with more progeny these are fitter variants or variants with higher fitness values f i will be present at higher frequencies, x i (t), in future generations on the expense of less fit variants whose frequencies decrease until they finally die out. At the same time the mean fitness of the population, expressed by n i= 1 i i= 1 i t n φ ( t) = f ( t) = x ( t) fi with x ( ) = 1, increases until the population becomes homogeneous and contain only the fittest variant. The variables x i (t) describe the population as an ensemble of variants. Normalization of the sum of frequencies to one facilitates the description. Reduction of Darwin s principle to the simplest meaningful model leads to three indispensible prerequisites: (i) multiplication with inheritance with

7 - 7 - offspring resembles the parents, (ii) variation as a necessary byproduct of correct reproduction, and (iii) selection as a consequence of finite population size. First we consider the growth of a population without limitation: The number of individuals is denoted by N(t) and the increase in population size due to multiplication at instant t, dn/dt, is assumed to be proportional to N(t). By elementary calculus we find dn dt rt = r N and N( t) = N(0) e, where r is the so-called Malthusian parameter named after the famous English economist Robert Malthus 7 and N(0) the number of individuals at time t = 0. The equation illustrates a simple kind of autocatalysis: Increase in N leads to an increase in dn/dt, and this increase results in further increase in N completing positive feedback. The population size N(t) exhibits exponential growth and this growth behavior turned out to be essential for Darwinian selection. h The consideration of finite resources is straightforward and has been suggested already by Pierre-François Verhulst, 8 a contemporary mathematician of Charles Darwin: An ecosystem has a limited carrying capacity and can sustain only a population of C individuals. This leads to a negative quadratic term in the differential equation describing the growth of the population dn dt N N(0) C = r N 1 and N( t) = rt C N(0) + ( C N(0 ). ) e As shown in figure 4 initial exponential growth is turned into saturation and the population size converges to the carrying capacity for sufficiently long time, lim N( t C. i t ) = The next steps, the introduction of n different variants X i in the population and the normalization of variables as indicated above, lead directly to an equation describing selection in a population as predicted by Darwin s principle: dxi dt = x i n ( fi x j f j ) = xi ( fi f ) = xi ( fi φ( t) ) j = 1 x ( t) = i x (0) e i n j = 1 j f i t x (0) e The interpretation of the solution curve is straightforward: The term containing the largest fitness value, f m = max {f 1, f,, f n }, dominates the sum in the denominator and after sufficiently long time the population becomes homogeneous and contains only the fittest variant: n j = 1 x (0) e j f j t x m (0) e f m t f j t for large t and with x ( t) 1, m h Subexponential growth may lead to coexistence of variants with different fitness and hyper exponential or hyperbolic growth is characterized by survival of the first rather than survival of the fittest. In other words the variant that appears first in the population is very likely the one to be selected. i The Verhulst equation is identical with the often used logistic equation, dx/dt = r x (1-x), which is obtained by the substitution N(t) = x(t) C or x(t) = N(t) / C.

8 - 8 - selection of the fittest has occurred. Optimization of the mean fitness takes place in the population and the fittest variant, X m, is selected. Selection can be verified also without reference to the explicit solution. For this goal we consider the mean fitness of the population, (t), and its time derivative: n n n ( fi xi xi f j x j ) = fi xi ( fi xi ) dφ = dx dt dt n i n f = 1 f i= i i= 1 i j = 1 i= 1 i= 1 = f ( f ) = var{ f } 0. The change of mean fitness with time is the variance of the distribution of fitness values and hence always nonnegative. In other words (t) is non-decreasing, it increases as long as the population contains more than one variant and becomes constant after the fittest variant has been selected. Direct application of Darwin s principle in the sense that the parameters of the kinetic equations, the fitness values f i, are determined independently of the selection experiment has been carried out for in vitro RNA replication catalyzed by a simple enzyme isolated from Escherichia coli bacteria infected by the bacteriophage Q. RNA-replication by this enzyme, Q replicase, is a multi-step mechanism based on complementary synthesis: 9 The plus strand is the template for the synthesis of the minus strand and vice versa, individual reaction steps involve among other processes binding of template and activated nucleotide monomers to the enzyme, nucleation and propagation of RNA synthesis as well as dissociation of the newly synthesized strand and the template strand from the enzyme. Despite various complications in detail the over-all reproduction of the plus-minus ensemble of RNA follows a perfect exponential growth law as long as activated monomers and enzyme are present in excess or, in other words at sufficiently low RNA concentration. Fitness values were calculated and competition between RNA species leading to selection has been studied experimentally. 30 Darwin s natural selection studied in cellfree systems is a law of nature in the same rank as laws in physics like gravity, Maxwell s equations of electrodynamics or others. It has a clear mathematical formulation and the range of accurate application can be predicted by chemical kinetics and checked by experiment. The mathematical derivation of Darwin s principle has been chosen here in order to show that elementary mathematics is sufficient to demonstrate the universal predictive power of natural selection. It is worth noticing that the neither the mechanism of inheritance nor the detailed properties of phenotypes appear in the formal description of selection. Otherwise, Darwin s approach would have been doomed to fail. Darwin s concept of inheritance was completely wrong the Austrian monk Gregor Mendel had the correct solution but, unfortunately, it had been completely ignored by evolutionary biologists until the rediscovery of Mendel s work around the turn of the century. In addition, nothing has been said so far about the origin of variation and Darwin had no mechanism for the alteration of phenotypic properties either. What is needed for successful application of Darwin s concept is only the count of fertile progeny, the fitness, and the fact that variants appear once in a while no matter whether they are the result of a well understood mechanism or the product of a deus ex machina process. Simplicity or =

9 - 9 - complexity of evolution is not a result of its dynamics, it is entirely encapsulated in the process that unfolds the phenotype. This process may as complex as the development of an adult organism of a multi-cellular species or as simple as the folding of a biological macromolecule. Mutations, error thresholds, and in vitro evolution The discovery of deoxyribonucleic acid (DNA) and its identification with the carrier of genetic information opened the door for a mechanistic understanding of multiplication and variation. 31 Multiplication is always initiated by copying the genetic information of the cell. DNA consists of a backbone and digits (A, T, G, C) j attached to it. The molecular structure of DNA two complementary strands in a double helical arrangement bound together by base pairs A = T and G C provides the basis for digital replication starting from one end and proceeding digit by digit until the other end is reached. Stereochemistry and thermodynamics of base pairing guarantee high copying fidelity that is further increased in nature by enzyme catalysis and proofreading. Two major classes of variations occur in nature, mutation and recombination. Mutations were identified as copying errors and range from single digit errors, to deletions of DNA stretches and insertions through double or multiple copying of sequence parts, and further to duplications of genes and whole genomes. k Recombination occurs with sexual reproduction of higher organisms: The two parental genomes are split into parts and recombined anew whereby genetic information from mother and father are mixed. In order to avoid too much complication we shall refer here only to the simplest case of variation, to point mutations. Chemical kinetics of evolution through reproduction, mutation, and selection provides a frame to handle correct replication and mutation as parallel reaction channels (figure 5). 3 Fitness values f i are a measure of the number of copies synthesized on template molecules of class X i. Expressed as the number of events per time and concentration unit, f i is the conventional replication rate constant used in reaction kinetics. Mutation is accounted for by the probability Q ji to obtain X j as an error copy from X i. Accordingly, Q ji f i counts the number of replications producing X j on a template molecule X i and Q ji is the normalized rate for the mutation channel X i X j. The sum of n all rates for channels starting from X i fulfils = Q = 1, since replications are either correct or incorrect. The kinetic equations for the mutation-selection system are then of the form: dx dt j n n Q i ji fi xi x j φ ( t) with ( t) = = 1 i= 1 = φ j 1 ji x ( t) f Not unexpectedly, the mean production rate φ(t) is independent of the mutation terms. The mutation-selection equation can be solved exactly in terms of an eigenvalue problem 33 but we dispense here from all details and mention only the most relevant general results: (i) Provided i i j In ribonucleic acid (RNA) the nucleotide T is replaced by the closely related nucleotide U. k A gene is a DNA stretch, which is transcribed into RNA and translated into protein. The entire genetic information of an organism is called the genome or the genotype. Here, we shall also use the word sequence as an abbreviation for DNA- or RNA-sequence.

10 every variant can be reached from every variant by means of a finite number of consecutive mutations, the population approaches an asymptotically stable stationary state, this long-time solution is unique, and it is obtained as the largest eigenvector of the matrix W = W = Q f }, { ji ji i and (ii) the stationary concentrations of variants are given by the components of this eigenvector and they are strictly positive: lim x ( t) = x > 0. In biological terms this results implies that no t i i variant is vanishing. The long-time solution of the mutation-selection equation has been denoted as quasispecies, 34 because it represents the genetic reservoir of an asexually reproducing species similarly as a species is the genetic reservoir in case of sexual reproduction. A quasispecies consists of a fittest genotype, the master sequence X m and its mutants. Point mutations as we assume here for the purpose of illustration occur independently and with the same rate per site and replication event with a probability p, which is called the local mutation rate. Under this assumption the total mutation rate from one sequence X i to X j depends only on the number of positions in which the two sequences differ. This number is called the Hamming distance of the two sequences, d H (X i, X j ). For two sequences of chain length l we obtain (figure 6) Q ji = (1 p) p = (1 p) 1 p l d H ( X i, X j ) d H ( X i, X j ) l The diagonal elements of the mutation matrix Q, p = ( 1 p) dh ( Xi, X j ), describe the accuracy of replication in form of the probability to obtain a correct copy of Xi on the template X i. The stationary frequency of the master sequence is visualized as a function of the local mutation rate, x m ( p). Exact calculation is possible by solving the corresponding eigenvalue problem (figure 7). In order to illustrate the most relevant property of quasispecies at high error rates a simple analytical approximation that neglects mutational backflow is sufficient. Mutational backflow considers the production of the master genotype through mutation of mutants, these are the mutations X X for all k m. In other words, neglect of mutational backflow considers only mutations x m (0) k m X m X k Qmm σ = 1 σ 1 m 1 m Q ii, and then the stationary concentration of the master sequence is obtained as 1 = σ 1 m l fm 1 n ( σ m(1 p) 1) with σ m = and f m = i = 1, i The quantity m is the superiority of the master sequence and fulfils the condition m > 1, since by definition f m is the largest fitness value. The stationary concentration of the master sequence decreases with increasing local mutation rate p and vanishes at a critical value f m l 1 x m cr 1 m x i f i.. This point is called the error threshold of error prone replication because error propagation at pcr and higher values of p prevents the formation of quasispecies in the form of a stationary mutant distribution with a dominant master sequence. The insert in figure 7 shows also the exact solution x m ( p), which according to the exact result reported above does not vanish but p σ 1/ l m

11 approaches the (very small) value l x m ( p) 1 κ that becomes exact at p = -1, where correct replication and mutation have equal probabilities is the number of digits in the nucleotide alphabet; = 4 in the natural (A, T, G, C) alphabet. In figure 7 the occurrence of quasispecies with a dominant master sequence is shown to be confined between the accuracy limit of replication, which is determined by the replication mechanism, and the error threshold. The error threshold provides a relation between local mutation rate p and chain length of the replicating nucleic acid molecule l. The interpretation of this relation is straightforward: At constant local mutation rate the relation defines a maximum chain length l max lnσ m p, and accordingly, the genetic information that can be faithfully transmitted from one generation to the next is limited to. At constant chain length the relation defines a maximum mutation rate pmax lnσ m l l max l and this gives hints for the development of antiviral strategies based on an increase of p beyond p max by means of pharmaceutical drugs. 35 The error threshold relation can be interpreted in a third way too: The product of chain length and local mutation rate is proportional to the logarithm of the superiority: µ = l p = lnσ. If the superiority is approximately constant under similar evolutionary conditions what can be assumed to be approximately fulfilled for related species this product should be constant and this is indeed found in nature: The mutation rate per genome,, amounts to approximately one mutation per replication in RNA viruses, one mutation per ten replications in retroviruses and retrotransposons, and one mutation per 300 replications in bacteria and higher organisms. 36 The three prerequisites for Darwin s principle are not only fulfilled by cellular organisms but also by polynucleotides, in particular RNA, in suitable cell-free environments. This fact was mentioned in the discussion of the selection equation and it is used in the design of laboratory evolution experiments with RNA. 37 Other applications are the evolutionary design of molecules tailored for predefined purposes, for example RNA molecules binding to predefined targets called aptamers 38 and enzymes optimized for catalysis of non-natural reactions or reactions under non-natural conditions. 39 One selection technique based on exponential growth of molecules became standard in laboratory practice: Systematic Evolution of Ligands by Exponential Enrichment (SELEX). 40,41 Because of its general importance the SELEX method has been modeled and analyzed by mathematics. 4 Evolutionary design of catalytic RNA molecules called ribozymes became a central issue of artificial evolution. 43 It is worth noticing that RNA catalysts were also developed for reactions that have no analogue or occur only very rarely in nature like, for example, the Diels-Alder cycloaddition. 44 Catalytic activity is not restricted to RNA molecules as the successful preparation of deoxyribozymes has shown. 45 Genotypes and phenotypes in RNA evolution Why are most systems in biology so complex? The solution of the mutation-selection equation discussed here or of the equations of population dynamics is often complicated, because of nonlinear terms and large numbers of variables corresponding to independent degrees of m

12 - 1 - freedom. In case of the mutation-selection equation the eigenvalue problems are highdimensional and to find numerical solutions without approximations is commonly very hard, but it is not complex. l The source of biological complexity is indeed the relation between genotypes and phenotypes. The equations modeling evolution in the preceding two sections were exclusively dealing with genotypes, being polynucleotide sequences undergoing evolution in vitro, virus genomes or cellular DNA. The same is true for the differential equations in population genetics. Phenotypes come into play only through the fitness values or other parameters of the equations describing population dynamics. Understanding of formation and properties of phenotypes, however, requires detailed knowledge on unfolding genotypes to yield phenotypes. Solid data on genotype-phenotype relations, however, are scarce and often the currently available information is too complex for constructing models that are suitable for studying evolution. Retaining the focus on genetics, mutation is tantamount to taking a step in an abstract space of genotypes commonly called sequence space. Single point mutations represent the smallest possible or elementary steps in this context. The metric or the distance between sequences is given by the Hamming distance d H. Phenotypic properties, fitness values for example, are represented in sequence space by assigning a fitness value to every sequence, and the resulting object is called a fitness landscape or fitness (hyper)surface m The most frequently used concept for assigning function to a polynucleotide or protein considers two consecutive mappings: S j = (X i ) f j = (S j ) sequence structure function The first mapping considers the structure that is formed by the sequence under specific conditions, for example thermodynamic equilibrium at given temperature T, pressure p, phvalue, ionic strength, etc. The second map assigns a quantity, commonly a real number, to the structure. The rationale behind splitting fitness assignment to genotypes into two sub-problems is the common belief that each of the two mappings is easier to understand, analyze and predict than the combination. Prediction of structure from sequence is an old and at least partially understood task. Structures, in particular nucleic acid and protein structures, are often telling function. Nevertheless, the only case of a Darwinian system at present where landscape theory can meet experiment is in vitro evolution. l Complexity, in essence, arises from two properties: (i) Typically nonlinear interactions in networks with highconnectivity and (ii) high-dimensionality in the sense of large numbers of players on the stage. In biology both features are very pronounced. Genetic and metabolic networks are characterized, for example, by regulation with highly nonlinear switches and high connectivity. In addition, we are commonly dealing with thousands of genes and even more metabolites. m A geographic landscape results from plotting a quantity, altitude, upon a two-dimensional support, latitude and longitude, or in general z(x,y). Hypersurface indicates higher dimensionality of the support.

13 A fairly well understood biopolymer folding problem is the formation of thermodynamically stable RNA secondary structures from unfolded sequences (figure 8). Mapping RNA sequences into structures is also an experimentally accessible system that can serve as a toy model for more complicated cases. 46,47 A typical example is sketched in figure 9. Two general features are eminent in sequence-structure mappings of RNA molecules: (i) ruggedness and (ii) neutrality. Ruggedness means that sequences, which are close neighbors in sequence space, may (but need not) have very different properties. This fact is illustrated, for example, by compensatory mutations n of nucleotide pairs in double helical regions of the structure (figure 8). If in a Watson- Crick base pair A = U the U is mutated, U C, then no pairing is possible for A C and the structure is destabilized. A consecutive mutation A G compensates the destabilization, because another Watson-Crick base pair G C can be formed now. Other types of compensatory mutations are well known in nature, for example in protein structures, in protein-protein interaction and in the structure of viruses. Nonlocal interactions are responsible for structure formation and hence, a single point mutation can lead to an entirely different structure with very different properties. Most conventional model (fitness) landscapes are smooth, in particular those used in population genetics like the additive and the multiplicative landscapes, where it is assumed that every mutation contributes the same reduction to the fitness being either a constant term or a constant factor. Obviously, the results obtained by these models are likely to be very unrealistic. Many sequences form the same structure and this case of structural neutrality was found to be important for the success of evolutionary optimization. The set of sequences folding into one structure space is denoted as neutral network o and represents the pre-image of the structure in sequence space, for example G k is the neutral network of the structure S k (figure 9). Depending on the degree of neutrality, (G k ), a network may consist of one component all points of G k are connected or of several components. Populations on neutral networks migrate by neutral drift (see next section) 48 in the same way as the neutral theory of evolution predicts, 49 and as a rule jumping from one component of a neutral network to another is a rare event. Therefore, connectedness of networks is relevant for evolution. Neutrality of biopolymers, nucleic acids or proteins, is common and this implies that a relatively high percentage of mutations has no or very little influence on the molecular properties, which are relevant for evolution. A particularly illustrative example is the existence of synonymous codons: Several nucleotide triplets code for the same amino acid and mutations leading from one of the codons to another one within such a class yield the same protein on translation. The consequence for evolution is that several genotypes cannot be distinguished by selection, they n A compensatory mutation is a second (point) mutation, which compensates the effect of the first mutation, without restoring the original genotype. o The neutral network is a graph, which is constructed from the set of points folding into the same structure. Individual points or sequences represent the nodes of the graph and all pairs of sequences with Hamming distance one, i.e. all pairs of nearest neighbors, form the edges (figure 9).

14 are neutral. 47 The frequent case of small differences in fitness is handled in the so-called nearlyneutral theory of evolution. 50 Population sizes and stochastic simulation of evolution Modeling by differential equations is common in biology but there are important restrictions that have to be considered. Differential equations make the implicit assumption of infinite population size, which is well justified for conventional systems in chemistry. The number of molecules is commonly 10 0 and more, and the number of chemical species in a reaction network hardly exceeds one hundred. In general, we have many orders of magnitude more molecules than species even at very low concentrations. In biology very often the opposite situation is true: We are dealing with orders of magnitude more genotypes than individuals in a population. An example will illustrate this fact: We assume a population size of individuals, which is about the maximal number in test tube experiments with RNA molecules, and ask for the size of a clone in sequence space that contains all possible mutants. The total number of mutants with k errors is k l N( l, κ, k) = ( κ 1), k with l being the sequence length of the genotype and the size of the nucleotide alphabet. For l=300, which is the smallest length of a genome found with so-called viroids p the radius of a completely populated clone is d H = 7; for very small viruses with a genome length of 3000 this radius is d H = 5 and thus only mutants up to five point mutations away from the master sequence are completely covered by the population. Accordingly, stochastic effects are non-negligible and introduce a random element into the evolutionary process. The mathematics of stochastic processes is rather involved and not so well developed as modeling by differential equations. Here we shall restrict the considerations to a computer model for the simulation of RNA structure optimization. 51 The simulation has the advantage allowing also for direct incorporation of an RNA sequence-structure map into the simulation algorithm. In the example presented in figure 10 an initial population of a homogeneous population with a random structure evolved towards the predefined target structure, the structure of phenyl-alanyltransfer RNA from yeast. The setup of the simulation mimics correctly replication and mutation according to figure 5. The underlying stochastic process is conceived to have two absorbing states: (i) the state of extinction and (ii) the target state. Individual trajectories resulting from independent calculations are obtained by means of an algorithm developed and analyzed by Daniel Gillespie, 5 which has the advantage to be fully consistent with stochastic chemical reaction kinetics in form of the master equation. 53 Individual trajectories are sampled in order to yield a statistically consistent description of the evolutionary process with the following general p Viroids are the smallest known plant pathogens. Outside plants they survive as unprotected RNA molecules, which enter the plants through injuries in the cell wall. Inside the plant cell they are multiplied and released to the environment after cell death.

15 features: (i) A minimal population size is required in order to guarantee both, survival of the population and approach towards the target structure, (ii) the individual trajectories show large differences with respect to the intermediate structures of the optimization process no two trajectories were the same and enormous scatter in the time required to reach the target, and (iii) the optimization does not occur gradually, instead the target is approached in a stepwise or punctuated manner (figure 10). On the quasi-stationary plateaus no progress is made in the approach towards target but genotypes change and the population migrates in sequence space until a position in found from where further optimization is possible. The role of neutrality in evolution is sketches in figure 11. Because of the ruggedness of fitness landscapes evolution in a world without neutrality would soon come to an end on some minor local optima. Whenever the optimization process came to a temporary end on a landscape with a sufficient degree of neutrality, a random drift process begins on some neutral network that extends in a direction perpendicular to previous optimization. Random drift continues until a point is reached in sequence space from where further optimization is possible. Alternating periods of optimization and random drift are continued until, eventually, the target is reached. Accordingly, efficient evolution on rugged landscapes requires neutral networks. Conclusion and perspectives Biology developed differently from physics because it refrained from using mathematics as the primary tool to analyze and unfold theoretical concepts. Application of mathematics enforces clear definitions and reduction of observations to problems that can be managed quantitatively. Over the years physics became the science of abstractions and generalizations, biology the science of encyclopedias of special cases with all their beauties and peculiarities. Among others there is one eminent exception of the rule: Charles Darwin presented a grand generalization derived from a wealth of personal observations and reports by others together with knowledge from economics concerning population dynamics. In the second half of the twentieth century the appearance of molecular biology on the stage changed the situation substantially. A bridge was built from physics and chemistry to biology and mathematical models from biochemical kinetics or population genetics became presentable in biology. Nevertheless, the vast majority of biologists still smiled and smiles at the works of theorists. By the end of the twentieth century, however, molecular genetics began to produce such a wealth of data that almost everybody feels nowadays that progress cannot be made without a comprehensive theoretical foundation and a rich box of suitable mathematical, in particular computational tools. Nothing like this is at hand but indications for attempts in the right direction are already visible. Biology is going to enter the grand union of science that started with physics and chemistry and is progressing fast. Molecular biology started out with biological macromolecules in isolation and deals now with cells, organs, and organisms. Hopefully, this spectacular success will end the so far fruitless reductionism versus holism debate. Stochasticity is still an unsolved problem in molecular evolution. The mathematics of stochastic processes encounters difficulties in handling the equations of evolution in sufficient detail. A

16 comprehensive stochastic theory is still not at hand and the simulations are lacking more systematic approaches, since computer simulations in chemical kinetics of evolution are in an early state too. Another fundamental problem concerns the spatial dimensions: Almost all mathematical treatments of evolutionary processes are assuming spatial homogeneity but we saw the prominent role of patterns in development. In addition there is ample evidence for the role of spatial structures in biochemical catalysis, for example the solid particle like structures of the chemical factories in the cell. In the future, any comprehensive theory of the cell will have to deal with these supra-molecular structures that are rich in optimal design, which is a result of evolution over billions of years. Insight into the mechanisms of evolution reduced to the conceivably simplest systems was presented here. These systems deal with evolvable molecules in cell-free assays and are accessible to rigorous mathematical analysis as well as experimentation. An extension to asexual species, in particular viruses and bacteria, is within reach. The molecular approach based on sequence space considerations provides many advantages. It offers, for example, a simple explanation why we have species formed by asexually reproducing organisms despite the fact that there is neither unrestricted recombination nor reproductive isolation. The sequence spaces are so large that populations, colonies or clones can migrate for time spans of the age of the universe without coming close to another. We can give an answer to the question of the origin of complexity: Complexity in evolution results primarily from genotype-phenotype relations and from the influences of the environment. Evolutionary dynamics alone may be complicated in some cases but it is not complex at all. This has been reflected already by the sequence-structure map of our toy example. Conformation spaces depending on the internal folding kinetics as well as on environmental conditions and peculiarities of structures are metaphors for more complex features in evolution proper. Progress in understanding development in quantitative terms will encourage model in evodevo evolution and development. 1 Galilei, G Il Saggiatore. Edition Nationale, Bd.6, Florenz 1896, p.3. Translation into English is taken from The Assayer, S. Drake and C.D. O Malley in The Controversy of the Comets of University of Pennsylvania Press, Philadelphia, PA. Mendel, G Versuche über Pflanzen-Hybriden. Verhandlungen des naturforschenden Vereins in Brünn 4: Darwin, C On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. John Murray. London. 4 Mayr, E The Growth of Biological Thought. Diversity, Evolution and Inheritance. The Belknap Press of Harvard University Press, Cambridge, MA. 5 Schuster, P The beginning of the end of the holism versus reductionism debate? Molecular biology goes cellular and organismic. Complexity 13(1): Ostwald, W A-Linien von R.E. Liesegang. Zeitschrift für physikalische Chemie 3:365 7 Ostwald, W Periodisch veränderliche Rektionsgeschwindigkeiten. Physikalische Zeitschrift 1:87-88, und Periodische Erscheinungen bei der Auflösung des Chroms in Säuren. Erste und zweite Mitteilung. Zeitschrift für physikalische Chemie 35:33-76 und 35:04-56

17 Turing, A.M The chemical basis of morphogenesis. Philosophical Transactions of the Royal Socienty (London) B 37: Nicolis, G., Prigogine, I Self-Organization in Non-Equilibrium Systems. From Dissipative Structures to Order Through Fluctuations. John Wiley & Sons, Hoboken, NJ. 10 Haken, H Synergetics. Introduction and advanced topics. Springer-Verlag, Berlin. 11 Belousov, B.P Sbornik Referator po Radiatsioni Medizin 1958, Medgiz, Moscow, p Zhabotinsky, A.M Periodic kinetics of oxidation of malonic acid in solution (study of the Belousov reaction kinetics)..biofizika 9: Sagués, F., Epstein, I.R Nonlinear chemical dynamics. Dalton Transactions 003: Castets, V., Dulos, E., Boissonade, J., De Kepper, P Experimental Evidence of a Sustained Standing Turing_Type Nonequilibrium Chemical Pattern. Physical Review Letters 64: Rudovics, B., Barillot, E., Davies, P.W., Dulos, E., Boissonade, J., De Kepper, P Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction. Journal of Physical Chemistry A 103: Wolpert, L., Smith, J., Jessell, T., Lawrence, P., Robertson, E., Meyerowitz, E Principles of Development. Third edition. Oxford University Press, New York. 17 Moody, S.A., Editor Principles of Developmental Genetics. Academic Press, Burlington, MA. 18 Wolpert, L Positional information and the spatial pattern of cellular differentiation. Journal of Theoretical Biology 5: Jaeger, J., Martinez-Arlas, A Getting the Measure of Positional Information. PLoS Biology 7:e Driever W., Nüsslein-Volhard, C A Gradient of Bicoid Protein in Drosophila embryos. Cell 54: Driever W., Nüsslein-Volhard, C The Bicoid Protein Determines Position in the Drosophila Embryo in a Concentration-Dependent Manner. Cell 54: Gierer, A., Meinhardt, H A Theory of Biological Pattern Formation. Kybernetik 1: Meinhardt, H Pattern formation in biology: A comparison of models and experiments. Reports on Progress in Physics. 55: Meinhardt, H.198. Models of Biological Pattern Formation. Academic Press, London. 5 Murray, J.D. 00 and 003. Mathematical Biology I and II. Springer-Verlag, New York. 6 Murray, J.D Mathematical Biology II: Spatial Models and Biomedical Applications, pp Springer- Verlag, New York. 7 Malthus, T.R An Essay of the Principle of Population as it Affects the Future Improvement of Society. J. Johnson, London. 8 Verhulst, P Notice sur la loi que la population pursuit dans son accroisement. Correspondance Mathématique et Physique 10: Biebricher, C.K., Eigen, M., Gardiner jr., W.C Kinetics of RNA replication. Biochemistry : Biebricher, C.K., Eigen, M., Gardiner jr., W.C Kinetics of RNA replication: Competition and Selection among Self-Replicating RNA Species. Biochemistry 4: Judson, H.F The Eighth Day of Creation. The Makers of the Revolution in Biology. Jonathan Cape, London. 3 Eigen, M Selforganization of Matter and the Evolution of Biological Macromolecules. Naturwissenschaften 58:

18 Thompson, C.J., McBride, J.L On Eigen s Theory of the Self-Organization of Matter and the Evolution of Biological Macromolecules. Mathematical Biosciences 1:17-14 and Jones, B.L., Enns, R.H., Ragnekar, S.S On the Theory of Selection of Coupled Macromolecular Systems. Bulletin of Mathematical Biology 38: Eigen, M., Schuster, P The Hypercycle. A Principle of Natural Self-Organization. Part A: Emergence of the Hypercycle. Naturwissenschaften 64: Domingo, E., ed Quasispecies: Concepts and Implications for Virology. Springer-Verlag, Berlin. 36 Drake, J.W., Charlesworth, R., Charlesworth, D., Crow, J.F Rates of spontaneous mutation. Genetics 148: Joyce, G.F Forty Years of in vitro Evolution. Angew. Chem. Int. Ed. 46: Klussmann, S The Aptamer Handbook. Fuctional Oligonucleotides and Their Applications. Wiley-VCh, Weinheim (Bergstraße), DE. 39 Brakmann, S., Johnsson, K. 00. Directed Molecular Evolution of Proteins or how to Improve Enzymes for Biocatalysis. Wiley-VCh, Weinheim (Bergstraße), DE 40 Ellington, A.D., Szostak, J.W In vitro Selection of RNA Molecules That Bind Specific Ligands. Nature 346: Tuerk, C., Gold, L Systematic Evolution of Ligands by Exponential Enrichment: RNA Ligands to Bacteriophage T4 DNA Polymerase. Science 49: Levine, H.A., Nielsen-Hamilton, M A Mathematical Analysis of SELEX. Computational Biology and Chemistry 31: Joyce, G.F Directed Evolution of Nucleic Acid Enzymes. Annual Reviews of Biochemistry 73: Serganow, A., Keiper, S., Malina, L., Tereshko, V., Skripkin, E., Höbartner, C., Polonskaia, A., Phan, A.T., Wombacher, R., Micura, R., Dauter, Z., Jäschka, A., Patel, D.J Structural Basis for Diels-Alder Ribozyme-Catalyzed Carbon-Carbon Bond Formation. Nature Structural & Molecular Biology 1: Breaker, R.R., Joyce, G.F A DNA Enzyme that Cleaves RNA. Chemistry & Biology 1: Schuster, P., Fontana, W., Stadler, P.F., Hofacker, I.L From sequences to shapes and back A case study in RNA secondary structures. Proceedings of the Royal Society. London B 55:79-84, and Reidys, C.M., Stadler, P.F., Schuster, P Generic properties of combinatory landscapes: Neutral networks of RNA secondary structures. Bulletin of Mathematical Biology59: Schuster, P Prediction of RNA secondary structures: From theory to models and real molecules. Reports on Progress in Physics 69: Huynen, M.A., Stadler, P.F., Fontana, W Smoothness within ruggedness. The role of neutrality in adaptation. Proceedings of the National Academy of Sciences USA 99: Kimura, M The neutral theory of evolution. Cambridge University Press, Cambridge, UK. 50 Ohta, T. 00. Near-neutrality in evolution of genes and gene regulation. Proceedings of the National Academy of Sciences USA 99: Fontana, W., Schuster, P Continuity in evolution. On the nature of transitions. Science 80: Gillespie, D.T Stochastic simulation of chemical kinetics. Annual Review of Physical Chemistry 58: Gardiner, C.W Stochastic methods. A handbook for the natural and social sciences. Springer-Verlag, Berlin.

19 Figure captions: Figure 1: Skin patterns of inbred cats. The figure shows a family of inbred cats. The mother and the child are assigned with certainty. The male is the presumed father. He was around before and during the time of pregnancy. The similarity of the patter on the forehead is remarkable and almost identical on all three animals (Photographs by Dr. Inge Schuster). Figure : French flag model of gradient evaluation in biological pre-patterns. Gradients in morphogen concentration are evaluated by means of thresholds. Every cell that sensors a morphogen concentration below threshold develops into a red cell, a value between threshold 1 and threshold triggers formation of a white cell and a morphogen concentration above threshold 1 gives rise to the development of a blue cell. This model converting a continuous concentration gradient into a discrete developmental field is called a French flag model. 18 Figure 3: Turing patterns in the Gierer-Meinhardt model. The four 3D-plots show specific integrations of the partial differential equation for the Turing model in the Gierer- Mainhardt implementation. Space-time development is shown for the activator A (top row) and the inhibitor H (bottom row). The inhibitor follows the activator in peak formation but reaches a wider range because of faster diffusion. Three of the four parameters of the model were identical in all simulations: = 0.01, = 0.5, and = 5. Two values were chosen for the fourth parameter : = gives rise to the formation of a single peak (l.h.s. plots) and = 0.01 was used for the formation of a three peak solution on the r.h.s. plots. The change in a single parameter produces a completely different pattern (one versus three band). As diffusion becomes slower, more separated activator-inhibitor peaks are sustained within the morphogenetic field. Figure 4: Exponential growth and the Verhulst equation. The solution of the differential equation dy/dt = r t is the exponential function y(t) = y(0) e r t (black). It is compared with the solution curve of the Verhulst equation, x(t) = x(0)/(x(0)+(1-x(0)) e -r t ) (red). At short times the solution of the Verhulst equation behaves like the exponential growth curve but levels off at later times and goes into saturation when the carrying capacity of the ecosystem (here x = 1) is reached. Figure 5: Chemical kinetics of replication and mutation as parallel reaction channels. Correct replication and mutation are considered as parallel chemical reactions. Replication is initiated by binding of the template RNA to a free replicase molecule, then the polymer molecule is synthesizes nucleotide by nucleotide until the copy is completed Mutations are copying errors leading to sequences that differ from the template.

20 - 0 - Figure 6: Mutation rates from local replication accuracies. Two sequences of chain length l = 6, X i and X j, are shown in end to end alignment. They differ at four positions and thus their Hamming distance is d H (X i,x j ) = 4. The mutation rate for the (error prone) replication X i X j, Q ji = q p 4, is computed in the third row: Every correct digit incorporation contributes a factor q = 1-3p, every local or single point mutation a factor p. The local replication accuracy q plus all local mutation rates p sum up to q + 3p = 1. Figure 7: The error threshold. The stationary frequency of the master sequence X m is shown as a function of the local mutation rate p. In the approximation neglecting mutational backflow the function x m ( p) is almost linear in the particular example shown here. In the insert the approximation (black) is shown together with the exact solution (red). The error rate p has two natural limitations: (i) the physical accuracy limit of the replication process provides a lower bound for the mutation rate and (ii) the error threshold defines a minimum accuracy of replication that is required to sustain inheritance and sets an upper bound for the mutation rate. Parameters used in the calculations: binary sequences, l = 6, = Figure 8: From RNA sequence to secondary structure. The folding of RNA sequences into secondary structures is a string matching problem. Double helical stretches are formed whenever there is a thermodynamically favorable possibility. The structure in the figure contains four such stretches, which are color coded in the sequence and in the structure. In the double helices the two stretches run locally in opposite directions and nucleotides within a pair must either fulfill the Watson-Crick rules, A=U, U=A, G C, C G, or be one of the pairs G-U or U-G. The RNA molecule has two chemically different ends; by convention the 5 end of the sequence is at the left hand beginning of the string, whereas the 3 -end is always on the right hand side. Commonly, a sequence can from many structures, one of them is the thermodynamically minimum free energy structure and the others are suboptimal conformations, which may also play a role under certain circumstances.

21 - 1 - Figure 9: Mapping sequences into structures and structures into fitness values. In the upper part the figure presents two consecutive mappings: (i) from sequence space into structure space, S j = (I i ), and (ii) from structure space into the real numbers representing a quantitative measure for function, for example the fitness values, f k = (S j ). In one structure is uniquely assigned to every sequence. The mapping is commonly many-to-one and non-invertible: Many sequences fold into the same secondary structure and form a neutral network as shown in the lower part. The neutral network is the pre-image of the structure S k in sequence space. Both sequence space and shape space are highdimensional. The two-dimensional representation is used here exclusively for the purpose of illustration. Figure 10: A trajectory of evolutionary optimization. The topmost plot presents the mean distance to the target structure of a population of N = 1000 molecules (black). The plot in the middle (blue) shows the width of the population in Hamming distance between sequences and the plot at the bottom (green) is a measure of the velocity with which the center of the population migrates through sequence space. Diffusion on neutral networks causes spreading on the population in the sense of neutral evolution. A remarkable synchronization is observed: At the end of each quasi-stationary plateau a new adaptive phase in the approach towards the target is initiated, which is accompanied by a drastic reduction in the population width and a jump of the population center (The top of the peak at the end of the second long plateau is marked by a black arrow). The chain length of the molecules was l = 76, the mutation rate was chosen to be p = 0.001, the replication rate parameter was defined by f = 1 ( a d ( S, ) / l) where S k is the structure of the k + H template molecule and S the target structure (a is an empirical parameter common chosen in the range of 0.1). Figure 11: The role of neutrality in evolutionary optimization. Darwinian optimization on rugged landscapes is very often caught in local optima from which populations cannot escape (upper part). Provided the mapping creating the landscape has a sufficiently high degree of neutrality, the local traps are parts of neutral networks that allow for random drift in a direction perpendicular to the past adaptive walk. The random walk of the population on the neutral network is continued until the population reaches a point from which further optimization is possible. The alternation of fast adaptive phases and slow quasi-stationary epochs results in a stepwise approach towards the optimum (lower part). k S τ

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