We discovered a dynamic phase transition induced by sexual reproduction. The dynamics is a pure Darwinian rule with both fundamental ingredients to drive evolution: 1) random mutations and crossings which act in the sense of increasing the entropy (or diversity); and 2) selection which acts in the opposite sense by limiting the entropy explosion. Selection wins this competition if mutations performed at birth are few enough. By slowly increasing the average number m of mutations, however, the population suddenly undergoes a mutational degradation precisely at a transition point mc. Above this point, the "bad" alleles spread over the genetic pool of the population, overcoming the selection pressure. Individuals become selectively alike, and evolution stops. Only below this point, m < mc, evolutionary life is possible. The finite-size-scaling behaviour of this transition is exhibited for large enough "chromosome" lengths L. One important and surprising observation is the L-independence of the transition curves, for large L. They are also independent on the population size. Another is that mc is near unity, i.e. life cannot be stable with much more than one mutation per diploid genome, independent of the chromosome length, in agreement with reality. One possible consequence is that an eventual evolutionary jump towards larger L enabling the storage of more genetic information would demand an improved DNA copying machinery in order to keep the same total number of mutations per offspring.
The theoretical question posed in this work concerns the length-scaling properties of chromosomes. Let's call L the chromosome length, an integer number measuring the number of coding units along the chain, which for simplicity we consider as a bit-string: 0-bits represent the wild alleles, whereas 1-bits correspond to harmful mutations, the "bad" alleles. The larger this length L is, the larger is the space to store more genetic information. Therefore, in principle, evolution should lead to species with larger and larger chromosomes, of course with the same value of L for all individuals belonging to the same species.
Consider first a simple case of haploid individuals which reproduce through cloning. The chromosome of each newborn is copied from an already alive individual, taken at random, plus an average fixed number m of point mutations. Being an average over all newborns, this number m is not necessarily an integer, it can be tuned in a continuously way as explained later. One point mutation means a 0-bit in the parent’s chromosome which is flipped into a 1-bit in the offspring’s, or vice-versa. The position where this mutation is performed is random. The wild genotype corresponds to a bit-string where all bits are set to zero. A mutation in the sense 0 → 1 makes the offspring farther to the wild genotype than its parent, another in the reverse sense makes it closer. A fixed birth rate b defines the probability of each individual to produce an offspring each new time step.
Let’s ignore any kind of correlation along the chromosome, i.e. the fitness of individual i depends only on a single phenotype defined here as N i , the total number of 1-bits in its genome. One individual with phenotype N + 1 is at a disadvantage, when compared to another individual with phenotype N . The disadvantage here corresponds to a smaller survival chance: the probability to survive a new time step is smaller for the former individual by a factor of x , when compared to the latter, where x is a number strictly smaller than 1. Therefore, the survival probability for different individuals decrease for increasing N . This number x measures the overall selection pressure, and can be tuned in order to keep the population size constant, i.e. to keep the death rate equal to the birth rate b . After evolving for many generations the distribution of phenotypes stabilizes. In order to keep the wild genotype (the only for which the phenotype is N = 0 ) inside this equilibrium distribution, the number of mutations m cannot be too high.
Let’s now compare different chromosome lengths. One can follow a simple and intuitive reasoning: 1) the length L is increased; 2) the same ratio m/L is kept; 3) after many generations, the steady-state population presents the same distribution of phenotypes versus N/L , independent of the (large enough) chromosome length. This expected behaviour is exactly what is obtained by simulating this simple haploid, asexual model on a computer [1,2]. Fig. 1 shows an example of this behaviour [3]. The above-mentioned item 2) deserves an important remark: the genetic storing media (the bit-strings) are one-dimensional objects. Therefore, the average number m of mutations should be scaled proportionally to L . As a result, the whole genetic distribution curve and consequently both its average N and its width ∆N also scale proportionally to L (note the collapsed distribution curves in Fig. 1 plotted versus N/L , not N ).
The reasoning and the corresponding simulational results do not cause any surprise. The purpose of this work is to study a similar reasoning for sexual, diploid reproduction. Let’s pose the first question.
Should the same ratio m/L be kept for increasing chromosome lengths?
The answer to this simple question is not so simple. Intuition can betray who thinks about it. Sex deals with half the genetic information inherited from each parent, a nonlinear behaviour which requires prudence to avoid false conclusions. Moreover, a crossing-over performed with homologous chromosomes within each parent’s genome indicates that now the genetic information is no longer stored along strictly one-dimensional objects: we should not trust on the linear reasoning leading to the fixed ratio m/L . Dominance and recessiveness are further sources of doubts. In order to answer this and many other related questions, we present in the next sections the results obtained from computer simulations of a population dynamics. Compared to reality, the model is simplified in order to retain only the fundamental features of sexual, diploid reproduction. It is based on a pure Darwinian evolutionary rule with two basic ingredients: random mutations which tends to increase the entropy (or diversity); and natural selection which acts on the opposite sense by removing from the population many of these mutations and, consequently, preventing entropy explosion.
Fig. 2 shows a computer-simulated result of a model described later on. Fo
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