Erik Rauch - Papers

For abstracts, see below.

Topics:


Erik M. Rauch, Mark M. Millonas and Dante R. Chialvo. Pattern Formation and Functionality in Swarm Models. Physics Letters A 207 185-193 (1995). [html] [postscript] [pdf]

Erik M. Rauch and Mark M. Millonas. The Role of Trans-membrane Signal Transduction in Turing-type Cellular Pattern Formation. Journal of Theoretical Biology 226: 401-407 (2004). [pdf]

J. Asikainen, A. Aharony, B.B. Mandelbrot, E.M. Rauch, and J.-P. Hovi. Fractal geometry of critical Potts clusters. European Physical Journal B 34: 479 (2003). [pdf]

Erik M. Rauch. Discrete, Amorphous Physical ModelsInternational Journal of Theoretical Physics 42: 329-348 (2003).

M. A.M. de Aguiar, H. Sayama, E. M. Rauch, Y. Bar-Yam, and M. Baranger. Stability and Instability of Polymorphic Populations and the Role of Multiple Breeding Seasons in Phase III of Wright's Shifting Balance Theory. Physical Review E 65: 031909 (2002). [pdf]

Erik M. Rauch, Hiroki Sayama and Yaneer Bar-Yam. The relationship between measures of fitness and time scale in evolution. Physical Review Letters 88: 228101 (2002). [pdf]

Erik M. Rauch, Hiroki Sayama and Yaneer Bar-Yam. Dynamics and genealogy of strains in spatially extended host-pathogen models. Journal of Theoretical Biology 221: 655-664 (2003). [pdf]

M.A.M. de Aguiar, E. M. Rauch, and Y. Bar-Yam. Mean-field approximation to a spatial host-pathogen model. Physical Review E 67: 047102 (2003). [pdf]

M.A.M. de Aguiar, E. M. Rauch, and Y. Bar-Yam. Invasion and Extinction in the Mean Field Approximation for a Spatial Host-Pathogen Model. Journal of Statistical Physics 114: 1417-1451 (2004). [pdf]

Erik M. Rauch and Yaneer Bar-Yam. Theory predicts the uneven distribution of genetic diversity within species. Nature 431, 449-452 (2004).


Abstracts

Biodiversity

Erik M. Rauch and Yaneer Bar-Yam. Theory predicts the uneven distribution of genetic diversity within species. Nature 431, 449-452 (2004).

Global efforts to conserve species have been strongly influenced by the heterogeneous distribution of species diversity across the Earth. This is manifest in conservation efforts focused on diversity hotspots. The conservation of genetic diversity within an individual species is an important factor in its survival in the face of environmental changes and disease. Here we show that diversity within species is also distributed unevenly. Using simple genealogical models, we show that genetic distinctiveness has a scale-free power-law distribution. This property implies that a disproportionate fraction of the diversity is concentrated in small sub-populations. It holds even when the population is well-mixed. Small groups are of such importance to overall population diversity that even without extrinsic perturbations, there are large fluctuations in diversity due to extinctions of these small groups. We also show that diversity can be geographically non-uniform, potentially including sharp boundaries between distantly related organisms, without extrinsic causes, such as barriers to gene flow or past migration events. We obtain these results by studying the fundamental scaling properties of genealogical trees. Our theoretical results agree with field data from global samples of Pseudomonas bacteria. Contrary to previous studies, our results imply that diversity loss due to severe extinction events is high and focusing conservation efforts on highly distinctive groups can save much of the diversity.

Spatial evolution and host-pathogen systems

Erik M. Rauch, Hiroki Sayama and Yaneer Bar-Yam. The relationship between measures of fitness and time scale in evolution. Physical Review Letters 88: 228101 (2002). [pdf]

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.


Erik M. Rauch, Hiroki Sayama and Yaneer Bar-Yam. Dynamics and genealogy of strains in spatially extended host-pathogen models. Journal of Theoretical Biology 221: 655-664 (2003). [pdf]

We examine the dynamics of evolution in a generic spatial model of a pathogen infecting a population of hosts, or an analogous predator-prey system. Previous studies of this model have found a range of interesting phenomena that differ from the well-mixed version. We extend these studies by examining the spatial and temporal dynamics of strains using genealogical tracing. When transmissibility can evolve by mutation, strains of intermediate transmissibility dominate even though high-transmissibility mutants have a short-term reproductive advantage. Mutant strains continually arise and grow rapidly for many generations but eventually go extinct before dominating the system. We find that, after a number of generations, the mutant pathogen characteristics strongly impact the spatial distribution of their local host environment, even when there are diverse types coexisting. Extinction is due to the depletion of susceptibles in the local environment of these mutant strains. Studies of spatial and genealogical relatedness reveal the self-organized spatial clustering of strains that enables their impact on the local environment. Thus, we find that selection acts against the high-transmissibility strains on long time-scales as a result of the feedback due to environmental change. Our study shows that averages over space or time should not be assumed to adequately describe the evolutionary dynamics of spatially distributed host-pathogen systems.


M.A.M. de Aguiar, E. M. Rauch, and Y. Bar-Yam. Mean-field approximation to a spatial host-pathogen model. Physical Review E 67: 047102 (2003). [pdf]

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.


M.A.M. de Aguiar, E. M. Rauch, and Y. Bar-Yam. Invasion and Extinction in the Mean Field Approximation for a Spatial Host-Pathogen Model. Journal of Statistical Physics 114: 1417-1451 (2004). [pdf].

We derive the mean field equations of a simple spatial host-pathogen, or predator-prey, model that has been shown to display interesting evolutionary properties. We compare these equations, and the equations including pair-correlations, with the low-density approximations derived by other authors. We study the process of invasion by a mutant pathogen, both in the mean field and in the pair approximation, and discuss our results with respect to the spatial model. Both the mean field and pair correlation approximations do not capture the key spatial behaviors|the moderation of exploitation due to local extinctions, preventing the pathogen from causing its own extinction. However, the results provide important hints about the mechanism by which the local extinctions occur.

M. A.M. de Aguiar, H. Sayama, E. M. Rauch, Y. Bar-Yam, and M. Baranger. Stability and Instability of Polymorphic Populations and the Role of Multiple Breeding Seasons in Phase III of Wright's Shifting Balance Theory. Physical Review E 65: 031909 (2002). [pdf]

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.

Pattern formation

Erik M. Rauch, Mark M. Millonas and Dante R. Chialvo. Pattern Formation and Functionality in Swarm Models. Physics Letters A 207 185-193 (1995). [html] [postscript] [pdf]

We explore a simplified class of models we call swarms, which are inspired by the collective behavior of social insects. We perform a mean-field stability analysis and numerical simulations of the model. Several interesting types of behavior emerge in the vicinity of a second-order phase transition in the model, including the formation of stable lines of traffic flow, and memory reconstitution and bootstrapping. Tn addition to providing an understanding of certain classes of biological behavior, these models bear a generic resemblance to a number of pattern formation processes in the physical sciences.

Erik M. Rauch and Mark M. Millonas. The Role of Trans-membrane Signal Transduction in Turing-type Cellular Pattern Formation. Journal of Theoretical Biology 226: 401-407 (2004). [pdf]

The Turing mechanism  for the production of a broken spatial symmetry in an initially homogeneous system of reacting and diffusing substances has attracted much interest as a potential model for certain aspects of morphogenesis such as pre-patterning in the embryo.  The two features necessary for the formation of Turing patterns are  short-range autocatalysis and long-range inhibition  which usually only occur when the diffusion rate of the inhibitor is significantly greater than that of the activator.   This observation has sometimes been used to cast doubt on applicability of the Turing mechanism to cellular patterning since many messenger molecules that diffuse between cells do so at more-or-less similar rates. Here we show that Turing-type patterns will be able to robustly form under a wide variety of realistic physiological conditions though plausible mechanisms of intra-cellular chemical communication without relying on differences in diffusion rates.  In the mechanism we propose, reactions occur within cells. Signal transduction leads to the production of messenger molecules, which diffuse between cells at approximately equal rates, coupling the reactions occurring in different cells.  These mechanisms also suggest how this process can be controlled in a rather precise way by the genetic machinery of the cell.

Fractal geometry

J. Asikainen, A. Aharony, B.B. Mandelbrot, E.M. Rauch, and J.-P. Hovi. Fractal geometry of critical Potts clusters. European Physical Journal B 34: 479 (2003). [pdf]

Numerical simulations on the total mass, the numbers of bonds on the hull, external perimeter, singly connected bonds and gates into large fjords of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality are presented. The data are found consistent with the recently derived corrections-to-scaling theory. However, the approach to the asymptotic region is slow, and the present range of the data does not  allow a unique identification of the exact correction exponents.

Digital physics

Erik M. Rauch. Discrete, Amorphous Physical ModelsInternational Journal of Theoretical Physics 42: 329-348 (2003).

Discrete models of physical phenomena are an attractive alternative to continuous models such as partial differential equations. In discrete models, such as cellular automata, space is treated as having finitely many locations per unit volume, and physical processes are modelled by rules that depend on a small number of nearby locations. Such models depend critically on a regular (crystalline) lattice, as well as the global synchronization of all sites. We should ask, on the grounds of minimalism, whether the global synchronization and crystalline lattice are inherent in any discrete formulation. Is it possible to do without these conditions and still have a useful physical model? Or are they somehow fundamental? We will answer this question by presenting a class of models that are "extremely local" in the sense that the update rule does not depend on synchronization with the other sites, or on knowledge of the lattice geometry. All interactions involve only a single pair of sites. The models have the further advantage that they exactly conserved the analog of quantities such as momentum and energy which are conserved in physics. An example model of waves is given, and evidence is given that it agrees well qualitatively and quantitatively with continuous differential equations.



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