Global patterns of speciation and diversity.

In recent years, strikingly consistent patterns of biodiversity have been identified over space, time, organism type and geographical region. A neutral theory (assuming no environmental selection or organismal interactions) has been shown to predict many patterns of ecological biodiversity. This theory is based on a mechanism by which new species arise similarly to point mutations in a population without sexual reproduction. Here we report the simulation of populations with sexual reproduction, mutation and dispersal. We found simulated time dependence of speciation rates, species-area relationships and species abundance distributions consistent with the behaviours found in nature. From our results, we predict steady speciation rates, more species in one-dimensional environments than two-dimensional environments, three scaling regimes of species-area relationships and lognormal distributions of species abundance with an excess of rare species and a tail that may be approximated by Fisher's logarithmic series. These are consistent with dependences reported for, among others, global birds and flowering plants, marine invertebrate fossils, ray-finned fishes, British birds and moths, North American songbirds, mammal fossils from Kansas and Panamanian shrubs. Quantitative comparisons of specific cases are remarkably successful. Our biodiversity results provide additional evidence that species diversity arises without specific physical barriers. This is similar to heavy traffic flows, where traffic jams can form even without accidents or barriers.

Spectral analysis and the dynamic response of complex networks.

The eigenvalues and eigenvectors of the connectivity matrix of complex networks contain information about its topology and its collective behavior. In particular, the spectral density rho(lambda) of this matrix reveals important network characteristics: random networks follow Wigner's semicircular law whereas scale-free networks exhibit a triangular distribution. In this paper we show that the spectral density of hierarchical networks follows a very different pattern, which can be used as a fingerprint of modularity. Of particular importance is the value rho(0), related to the homeostatic response of the network: it is maximum for random and scale-free networks but very small for hierarchical modular networks. It is also large for an actual biological protein-protein interaction network, demonstrating that the current leading model for such networks is not adequate.

Multiscale analysis of information correlations in an infinite-range, ferromagnetic Ising system.

The scale-specific information content of the infinite-range, ferromagnetic Ising model is examined by means of information-theoretic measures of high-order correlations in finite-sized systems. The order-disorder transition region can be identified through the appearance of collective order in the ferromagnetic phase. In addition, it is found that near the transition, the ferromagnetic phase is marked by characteristic information oscillations at scales comparable to the system size. The amplitude of these oscillations increases with the total number of spins, so that large-scale information measures of correlations are nonanalytic in the thermodynamic limit. In contrast, correlations at scales small relative to the system size have a monotonic behavior both above and below the transition point, and a well-defined thermodynamic limit.

Mean-field approximation to a spatial host-pathogen model.

We study the mean-field approximation to a simple spatial host-pathogen model that has been shown to display interesting evolutionary properties. We show that previous derivations of the mean-field equations for this model are actually only low-density approximations to the true mean-field limit. We derive the correct equations and the corresponding equations including pair correlations. The process of invasion by a mutant type of pathogen is also discussed.

Relationship between measures of fitness and time scale in evolution.

The notion of fitness is central in evolutionary biology. We use a simple spatially extended predator-prey or host-pathogen model to show a generic case where the average number of offspring of an individual as a measure of fitness fails to characterize the evolutionary dynamics. Mutants with high initial reproduction ratios have lineages that eventually go extinct due to local overexploitation. We propose general quantitative measures of fitness that reflect the importance of time scale in evolutionary processes.

Stability and instability of polymorphic populations and the role of multiple breeding seasons in phase III of Wright's shifting balance theory.

It is generally difficult for a large population at a fitness peak to acquire the genotypes of a higher peak, because the intermediates produced by allelic recombination between types at different peaks are of lower fitness. In his shifting-balance theory, Wright proposed that fitter genotypes could, however, become fixed in small isolated demes by means of random genetic fluctuations. These demes would then try to spread their genome to nearby demes by migration of their individuals. The resulting polymorphism, the coexistence of individuals with different genotypes, would give the invaded demes a chance to move up to a higher fitness peak. This last step of the process, namely, the invasion of lower fitness demes by higher fitness genotypes, is known as phase III of Wright's theory. Here we study the invasion process from the point of view of the stability of polymorphic populations. Invasion occurs when the polymorphic equilibrium, established at low migration rates, becomes unstable. We show that the instability threshold depends sensitively on the average number of breeding seasons of individuals. Iteroparous species (with many breeding seasons) have lower thresholds than semelparous species (with a single breeding season). By studying a particular simple model, we are able to provide analytical estimates of the migration threshold as a function of the number of breeding seasons. Once the threshold is crossed and polymorphism becomes unstable, any imbalance between the different demes is sufficient for invasion to occur. The outcome of the invasion, however, depends on many parameters, not only on fitness. Differences in fitness, site capacities, relative migration rates, and initial conditions, all contribute to determine which genotype invades successfully. Contrary to the original perspective of Wright's theory for continuous fitness improvement, our results show that both upgrading to higher fitness peaks and downgrading to lower peaks are possible.

Symmetry breaking and coarsening in spatially distributed evolutionary processes including sexual reproduction and disruptive selection.

Sexual reproduction presents significant challenges to formal treatment of evolutionary processes. A starting point for systematic treatments of ecological and evolutionary phenomena has been provided by the gene-centered view of evolution which assigns effective fitness to each allele instead of each organism. The gene-centered view can be formalized as a dynamic mean-field approximation applied to genes in reproduction and selection dynamics. We show that the gene-centered view breaks down for symmetry breaking and pattern formation within a population and show that spatial distributions of organisms with local mating neighborhoods in the presence of disruptive selection give rise to such symmetry breaking and pattern formation in the genetic composition of local populations. Global dynamics follows conventional coarsening of systems with nonconserved order parameters. The results have significant implications for the ecology of genetic diversity and species formation.

Mesostructure of polymer collapse and fractal smoothing.

We investigate the internal structure of a polymer during collapse from an expanded coil to a compact globule. Collapse is more probable in local regions of high curvature, so a smoothing of the fractal polymer structure occurs that proceeds systematically from the shortest to the longest length scales. A proposed universal scaling relationship is tested by comparison with Monte Carlo simulations. We speculate that the universal form applies to various fractal systems with local processes that promote smoothness over time. The results complement earlier work showing that on the macroscale polymer collapse proceeds by driven diffusion of the polymer ends.

Somatic evolution in the immune system: the need for germinal centers for efficient affinity maturation.

In the course of a humoral immune response, the average affinity of antibody for the immunizing antigen can increase in time. This process of affinity maturation is due to antigen-driven selection of higher affinity B cell clones and somatic hypermutation of the genes that code for the antibody variable region. Through the use of simulation models we show that the efficiency of affinity maturation is substantially improved if mutation and selection occur in germinal centers, specialized regions of lymphoid tissues, rather than in the body as a whole. We show that confining mutation and selection to germinal centers decouples the problem of affinity maturation from the problem of antigen elimination. In the germinal centers, stored antigen, high rates of B cell proliferation and apoptosis combine to create an environment where effective maturation occurs even after the primary response has eliminated much or all of the free antigen. Kepler and Perelson suggested that if hypermutation were turned on and off in a phasic manner, affinity maturation could be made more efficient under circumstances when affinity-improving mutations have a low probability of occurrence. We confirm this in the context of a stochastic model. However, even in the absence of phasic mutation, we show that affinity maturation is significantly improved when proliferation, mutation, and selection are restricted to germinal centers as opposed to occurring systemically.

Motion of polymer ends in homopolymer and heteropolymer collapse.

To investigate the polymer coil-to-globule transition we performed simulations for the kinetics of homopolymer and heteropolymer collapse. Our stimulations made use of abstract models of long flexible polymers to obtain extensive statistical sampling. For a variety of these models, the simulations suggest that collapse of long polymers is dominated by diffusion of the polymer ends, which accrete monomers and small aggregates. The growth of the end aggregate was found to be nearly linear in time for homopolymers and largely unaffected by variations in microstructure. In contrast, for heteropolymers the presence of non-aggregating (hydrophilic) monomers dramatically slows and alters the growth of the end mass. In models simulated, the end mass grows roughly as the cube root of time, but still dominates aggregation along the contour. In a model where only pairwise bonding is allowed, the collapse is uniform since more flexible end motion does not result in continued end accretion. The possible significance of our results for biopolymer kinetics is discussed.

Applications of parallel computing to biological problems.

Parallel computers should provide the greatest processing power and memory for scientific simulations in the coming decades. This review discusses general strategies and specific algorithms for the use of various parallel architectures in simulations of biological and artificial polymers. General strategies include space partitioning (domain decomposition cell methods) and distributed independent simulations. Specific algorithms include cellular automata for efficient abstract polymer simulation. One algorithm, the two-space algorithm, is particularly efficient both for parallel and serial computation. Three applications, 2D melts, gel electrophoresis, and polymer collapse, are described. Simulations of high-density melts in 2D show that contrary to expectations, polymers do not completely segregate at the highest densities; instead, polymer interpenetration is significant. Preliminary simulations of gel electrophoresis show its behavior in the diffusive regimen and demonstrate the use of Cellular Automaton Machines (CAMs). Polymer collapse is studied in the regime of large departures from good solvent conditions. In this regime, kinetics plays a significant role. Collapse is dominated (nucleated) by migration of the chain ends.

Cellular automaton simulation of pulsed field gel electrophoresis.

We describe simulation techniques well suited to detailed investigation of the microscopic behavior of DNA during electrophoretic separation in the diffusive regime. Long polymers moving diffusively in a medium are simulated using microscopic Monte-Carlo steps. Simulations rely upon a recently introduced two-space abstract polymer that enables fine-grained massively parallel simulation. Tests of the two-space polymer dynamics are reviewed. The scaling with polymer length of the size and relaxation time of isolated polymers are shown to agree with universal scaling relations. The relaxation time is found to be significantly faster than the alternative bond-fluctuation method. Simplicity of implementation enables simulation on cellular automaton machines (CAM) including CAM-6, and a prototype of the new CAM-8, as well as other massively parallel architectures. Preliminary simulations of polymers migrating under an external field through a random medium of obstacles in two dimensions are described. Two sequences of simulations are performed, with different obstacle densities corresponding to pore sizes larger and smaller than the polymer radius of gyration. In the dilute medium polymers are characteristically draped on single obstacles. In the denser medium draping across multiple obstacles results in reduced orientation in the field direction. A demonstration of rapid 90 degrees field direction switching results in polymer motion toward the expected intermediate direction.