Solvable null model for the distribution of word frequencies.

Zipf's law asserts that in all natural languages the frequency of a word is inversely proportional to its rank. The significance, if any, of this result for language remains a mystery. Here we examine a null hypothesis for the distribution of word frequencies, a so-called discourse-triggered word choice model, which is based on the assumption that the more a word is used, the more likely it is to be used again. We argue that this model is equivalent to the neutral infinite-alleles model of population genetics and so the degeneracy of the different words composing a sample of text is given by the celebrated Ewens sampling formula [Theor. Pop. Biol. 3, 87 (1972)]], which we show to produce an exponential distribution of word frequencies.

Mutation accumulation in growing asexual lineages.

The stochastic loss of entire classes of individuals bearing the fewest number of mutations-a process known as Muller's ratchet-is studied in asexual populations growing unconstrained from a single founder. In the neutral regime, where mutations have zero effect on fitness, we derive a recursion equation for the probability distribution of the minimum number of mutations carried by individuals in the least-loaded class, and obtain an explicit condition for the halting of the ratchet. Next, we consider the case of deleterious mutations, and show that weak selection can actually accelerate the ratchet beyond that achieved for the neutral regime. This effect is transitory, however, as our results suggest that even weak purifying selection will eventually lead to the complete cessation of the ratchet. These results may have important implications for problems in biology and the medical sciences.

Evolution of protein synthesis in a lattice model of replicators.

The traditional chemical-kinetics approach to the study of prebiotic evolution cannot explain the evolution of protein synthesis in a homogeneous population of self-replicating molecules, because the invasion of a resident population of simpler, template-directed replicators by mutant protein-assisted replicators is deemed impossible. Approaching this problem in a spatial cellular automaton framework, we argue here that in such a setting evolution of protein synthesis is a likely event. In addition, we show that the onset of invasion may be viewed as a nonequilibrium phase transition, that can be characterized quantitatively by a set of critical exponents.

Complementarity and diversity in a soluble model ecosystem.

Complementarity among species with different traits is one of the basic processes affecting biodiversity, defined as the number of species in the ecosystem. We present here a soluble model of microbial-based ecosystems in which the species are characterized by binary traits and their pairwise interactions follow a complementarity principle. Manipulation of the species composition, and so the study of its effects on the species diversity, is achieved through the introduction of a bias parameter favoring one of the traits. Using statistical mechanics tools we find explicit expressions for the allowed values of the equilibrium species concentrations in terms of the control parameters of the model.

Nonequilibrium phase transitions in a model for the origin of life.

The requisites for the persistence of small colonies of self-replicating molecules living in a two-dimensional lattice are investigated analytically in the infinite diffusion or mean-field limit and through Monte Carlo simulations in the position-fixed or contact process limit. The molecules are modeled by hypercyclic replicators A that are capable of replicating via binary fission A+E-->2A with production rate s, as well as via catalytically assisted replication 2A+E-->3A with rate c. In addition, a molecule can degrade into its source materials E with rate gamma. In the asymptotic regime, the system can be characterized by the presence (active phase) and the absence (empty phase) of replicators in the lattice. In both diffusion regimes, we find that for small values of the ratio c/gamma these phases are separated by a second-order phase transition that is in the universality class of the directed percolation, while for small values of s/gamma the phase transition is of first order. Furthermore, we show the suitability of the dynamic Monte Carlo method, which is based on the analysis of the spreading behavior of a few active cells in the center of an otherwise infinite empty lattice, to address the problem of the emergence of replicators. Rather surprisingly, we show that this method allows an unambiguous identification of the order of the nonequilibrium phase transition.

Extinctions in the random replicator model.

The statistical properties of an ecosystem composed of species interacting via pairwise, random interactions and deterministic, concentration limiting self-interactions are studied analytically with tools of equilibrium statistical mechanics of disordered systems. Emphasis is given to the effects of externally induced extinction of a fixed fraction of species at the outset of the evolutionary process. The manner the ecosystem copes with the initial extinction event depends on the degree of competition among the species as well as on the strength of that event. For instance, in the regime of high competition the ecosystem diversity, given by the fraction of surviving species, is practically insensitive to the strength of the initial extinction provided it is not too large, while in the less competitive regime the diversity decreases linearly with the size of the event. In the case of large extinction events we find that no further biotic extinctions take place and, furthermore, that rare species become very unlikely to be found in the ecosystem at equilibrium. In addition, we show that the reciprocal of the Edwards-Anderson order parameter yields a good measure of the diversity of the model ecosystem.

Soluble model for the accumulation of mutations in asexual populations.

Finite asexual populations can accumulate an increasing number of deleterious mutations by a process known as Muller's ratchet, which consists of successive losses of the fittest or least-loaded classes of individuals in the population. We present here a simplified theoretical framework to describe the serial bottleneck passages setup used in experiments to demonstrate the decrease of the population mean fitness due to the operation of ratchet. In particular, we calculate the expected time between consecutive clicks of the ratchet and derive expressions relating the moments of the mean fitness distribution to the mutation and selection parameters.

Phase transitions in a model for the formation of herpes simplex ulcers.

The critical properties of a cellular automaton model describing the spreading of infection of the herpes simplex virus in corneal tissue are investigated through the dynamic Monte Carlo method. The model takes into account different cell susceptibilities to the viral infection, as suggested by experimental findings. In a two-dimensional square lattice the sites are associated with two distinct types of cells, namely, permissive and resistant to the infection. While a permissive cell becomes infected in the presence of a single infected cell in its neighborhood, a resistant cell needs to be surrounded by at least R>1 infected or dead cells in order to become infected. The infection is followed by the death of the cells resulting in ulcers whose forms may be dendritic (self-limited clusters) or amoeboid (percolating clusters) depending on the degree of resistance R of the resistant cells as well as on the density of permissive cells in the healthy tissue. We show that a phase transition between these two regimes occurs only for R>/=5 and, in addition, that the phase transition is in the universality class of the ordinary percolation.

Group selection models in prebiotic evolution.

The evolution of enzyme production is studied analytically using ideas of the group selection theory for the evolution of altruistic behavior. In particular, we argue that the mathematical formulation of Wilson's structured deme model [The Evolution of Populations and Communities (Benjamin-Cumings, Menlo Park, 1980)] is a mean-field approach in which the actual environment that a particular individual experiences is replaced by an average environment. That formalism is further developed so as to avoid the mean-field approximation and then applied to the problem of enzyme production in the prebiotic context, where the enzyme producer molecules play the altruists role while the molecules that benefit from the catalyst without paying its production cost play the nonaltruists role. The effects of synergism (i.e., division of labor) as well as of mutations are also considered and the results of the equilibrium analysis are summarized in phase diagrams showing the regions of the space of parameters where the altruistic, nonaltruistic, and the coexistence regimes are stable. In general, those regions are delimitated by discontinuous transition lines which end at critical points.

Error propagation in the hypercycle.

We study analytically the steady-state regime of a network of n error-prone self-replicating templates forming an asymmetric hypercycle and its error tail. We show that the existence of a master template with a higher noncatalyzed self-replicative productivity a than the error tail ensures the stability of chains in which m < n-1 templates coexist with the master species. The stability of these chains against the error tail is guaranteed for catalytic coupling strengths K on the order of a. We find that the hypercycle becomes more stable than the chains only if K is on the order of a2. Furthermore, we show that the minimal replication accuracy per template needed to maintain the hypercycle, the so-called error threshold, vanishes as square root of n/K for large K and N < or = 4.