A simple model of co-evolutionary dynamics caused by epistatic selection

Epistasis is the dependency of the effect of a mutation on the genetic background in which it occurs. Epistasis has been widely documented and implicated in the evolution of species barriers and the evolution of genetic architecture. Here we propose a simple model to formalize the idea that epistasis can also lead to co-evolutionary patterns in molecular evolution of interacting genes. This model epistasis is represented by the influence of one gene substitution on the fitness rank of the resident allele at another locus. We assume that increasing or decreasing fitness rank occur equally likely. In simulations we show that this form of epistasis leads to co-evolution in the sense that the length of an adaptive walk between interacting loci is highly correlated. This effect is caused by episodes of elevated rate of evolution in both loci simultaneously. We find that the influence of epistasis on these measures of co-evolutionary dynamics is relatively robust to the details of the model. The main factor influencing the correlation in evolutionary rates is the probability that a substitution will have an epistatic effect, but the strength of epistasis or the asymmetry of the initial fitness ranks of the alleles have only a minor effect. We suggest that covariance in rates of evolution among loci could be used to detect epistasis among loci.     

A model of competition

Summary The paper considers a model of competition, based upon the Lotka-Volterra equations, which explicitly considers the effect of density independent mortality upon the outcome of competition. The model's possible application to wild Drosophila species in Europe are considered.     

Traveling waves in a simple population model involving growth and death

The structure of solutions to a simple spatially dependent population model involving growth and death is investigated. Two forms of motility of the population are considered: (1) random motion only modeled by a Fickian law, and (2) a directed component of motion (chemotaxis), included in addition to the random motion. Under certain growth conditions a traveling wave of constant speed is approached. This speed can be increased by the addition of the chemotaxis with a corresponding increase in the asymptotic population. Development of initial conditions into a wave is illustrated numerically.     

Founder control and coexistence in a simple model of asymmetric competition for light

Size asymmetry in plant light acquisition complicates predictions of competitive outcomes in light-limited communities. We present a mathematically tractable model of asymmetric competition for light and discuss its implications for predicting outcomes of competition during establishment in two-, three-, and many-species communities. In contrast to the resource-reduction model of symmetric competition for a single resource, the model we present predicts that outcomes of asymmetric competition for light will sometimes depend on the timing of establishment and the consequent hierarchy among species in canopy position. Competitive outcomes in the model depend on the minimum light requirements (L(c)) and self-shading of species lower in the canopy compared to the light available (L(out)(*)) beneath species higher in the canopy. Succession progresses towards species with decreasing values for L(c), but arrested successions occur when initial dominants have relatively high values for L(c) but low values for L(out)(*), leading to founder control. A theoretically limitless number of species may coexist in competition for light when dominance is founder controlled. These model predictions have implications for an array of applied ecological questions, including methods to control invasive species in light-limited restored ecosystems.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

A population model for limited food competition

In this paper a deterministic differential equation system is proposed to model the population dynamics of a biological community in which two species on the same trophic level compete for a common food, taken to be in limited supply. Food limitation is assumed to be the only inhibition of the growth of the populations and food quantity is assumed to be only affected by consumption. The model is thus designed to mimic a closed experimental situation rather than a natural community.Analytical properties of the solution of the differential equation system are developed and corresponding biological interpretations suggested.Cited laboratory data on the experimental batch community consisting of the marine ciliates Euplotes vannus and Uronema marinum feeding on bacteria motivated the model and supported its analytic properties.     

A multi-locus continuous-time selection model

Summary A continuous time selection model is formulated for a diploid monoecious population with multiple alleles at each of an arbitrary number of loci, incorporating differential fertility and mortality as well as arbitrary mating and age structure. The model is simplified in the case of age-independence and for the case of a stable age distribution. The age-independent model is examined in detail for the special case of multiple alleles at each of two loci. This model is analyzed under the assumptions of random mating and additive fertilities, with close attention given to the behavior of the system with respect to Hardy-Weinberg proportions and linkage equilibrium.M. M. was supported by a U.S. Public Health Service training grant (Grant No. GM780).     

Habitat-specific estimates of competition in stream salmonids: A field test of the isodar model of habitat selection

Summary The population densities of sympatric Atlantic salmon,Salmo salar and brook charr,Salvelinus fontinalis, were measured in riffle and pool stream habitats to test whether non-linear isodars, a multispecific model of habitat selection based on ideal distribution assumptions, could (1) predict the distribution of densities between habitats and (2) reproduce the processes postulated to underlie spatial segregation and species interactions in previous laboratory and field studies. The model provided a good fit to observed density patterns and indicated that habitat suitability declined non-linearly with increased heterospecific competitor densities. Competitive effects in riffles appeared to be due to exploitative resource use, with salmon always emerging as the superior competitor. No evidence was found for interference competition in riffles. In contrast, interspecific competition in pools seemed to occur through exploitation and interference. The specific identity of the superior competitor in pools depended on the density of both species; pools provided the charr with refuge from competition with the salmon, presumably through the adoption by the charr of density-dependent behaviours, such as schooling and group foraging, that mitigated the negative impact of the salmon. Charr were displaced from the riffles toward the pools as the total salmon density increased. The isodar analysis, based on limited density data, successfully reproduced the processes suggested to underlie spatial segregation in previous field and laboratory studies and provided new insights into how changes in competitor densities modify habitat suitability in this system.     

Individual and group selection with competition

Summary The response of a randomly mating population which is expected to follow selection of phenotypic units, comprising individuals or groups whose members have an arbitrary degree of relatedness, was formulated using a model which included additive and dominance competition effects. The derivation involved three steps. Twenty-two quadratic components were defined, six describing individual (direct) and neighbor (associate) effects, and 16 describing direct by associate interactions for different loci, for single loci with different alleles, and for identical alleles. Six covariances between pairs of individual phenotypes and three of individuals with their offspring were defined according to whether or not their direct or associate genotypes are common, and expressed in terms of the quadratic components. Finally, variances of selection units of different types and their covariance with their offspring were expressed as compounds of these individual covariances. Explicit formulations for mass, clonal and full-sib selection show that without constraints on the quadratic components, and hence on the magnitude and type of competition operative, no predictions as to the relative efficiencies of these three methods can be made.     

Structural competition involving G-quadruplex DNA and its complement

Structural competition between the G-quadruplex, the I-motif, and the Watson-Crick duplex has been implicated for repetitive DNA sequences, but the competitive mechanism of these multistranded structures still needs to be elucidated. We investigated the effects of sequence context, cation species, and pH on duplex formation by the G-quadruplex of dG(3)(T(2)AG(3))(3) and its complement the I-motif of d(C(3)TA(2))(3)C(3), using ITC, DSC, PAGE, CD, UV, and CD stopped-flow kinetic techniques. ITC and PAGE experiments confirmed Watson-Crick duplex formation by the complementary strands. The binding constant of the two DNA strands in the presence of 10 mM Mg(2+) at pH 7.0 was shown to be 5.28 x 10(7) M(-1) at 20 degrees C, about 400 times larger than that in the presence of 100 mM Na(+) at pH 5.5. The dynamic transition traces of the duplex formation from the equimolar mixture of G-/C-rich complementary sequences were obtained at both pH 7.0 and pH 5.5. Fitting to a single-exponential function gave an observed rate of 8.06 x 10(-3) s(-1) at 20 degrees C in 10 mM Mg(2+) buffer at pH 7.0, which was about 10 times the observed rate at pH 5.5 under the same conditions. Both of the observed rates increased as temperature rose, implying that the dissociation of the single-stranded structured DNAs is the rate-limiting step for the WC duplex formation. The difference between the apparent activation energy at pH 7.0 and that at pH 5.5 reflects the fact that pH significantly influences the structural competition between the G-quadruplex, the I-motif, and the Watson-Crick duplex, which also implies a possible biological role for I-motifs in biological regulation.     

A simple framework for selection of water quality models

Water quality modeling is no longer just the preserve of specialists who seek to describe water quality processes but also for use by non specialists in everyday water quality management issues. With so many models already developed, it becomes prudent to adapt them to a situation than to develop a completely new model that would probably do the same simulations. The question is: which is the most appropriate model to apply to a situation? The specialist can always draw on past experiences to make a decision. However, this is not the case for the non specialist. A lot of different criteria can be used to decide which model to use for a particular situation based on some important factors. The objectives of modeling exercises differ and each water body is unique so there cannot be hard and fast rules on which is the best criteria for selecting the appropriate model. Furthermore, there is usually hardly any time on the project work plan allocated for model selection. Therefore there is need for a simple procedure to select the appropriate model. The objective of this paper was to develop a simple framework for selecting water quality models to aid the non specialist. The framework was then applied to a case study in order to evaluate its usefulness. The results from the case study show that after a thorough literature review, models can be evaluated against chosen criteria and the most appropriate model singled out. It was concluded that the framework is only effective if the research objective is adequately defined and the models are reviewed thoroughly, but it saves time for the actual modeling exercise.     

A simple model of retina-LGN transmission

To gain a deeper understanding of the transmission of visual signals from retina through the lateral geniculate nucleus (LGN), we have used a simple leaky integrate and-fire model to simulate a relay cell in the LGN. The simplicity of the model was motivated by two questions: (1) Can an LGN model that is driven by a retinal spike train recorded as synaptic (‘S’) potentials, but does not include a diverse array of ion channels, nor feedback inputs from the cortex, brainstem, and thalamic reticular nucleus, accurately simulate the LGN discharge on a spike-for-spike basis? (2) Are any special synaptic mechanisms, beyond simple summation of currents, necessary to model experimental recordings? We recorded cat relay cell responses to spatially homogeneous small or large spots, with luminance that was rapidly modulated in a pseudo-random fashion. Model parameters for each cell were optimized with a Simplex algorithm using a short segment of the recording. The model was then tested on a much longer, distinct data set consisting of responses to numerous repetitions of the noisy stimulus. For LGN cells that spiked in response to a sufficiently large fraction of retinal inputs, we found that this simplified model accurately predicted the firing times of LGN discharges. This suggests that modulations of the efficacy of the retino-geniculate synapse by pre-synaptic facilitation or depression are not necessary in order to account for the LGN responses generated by our stimuli, and that post-synaptic summation is sufficient.     

A simple first-principles model of reduplication

From first principles, a simple model was derived to describe the reduplication of the molecular structure as a sequence of stages whose probabilities can be estimated using the general theory of molecular transformations. It was shown that the genetic information transfer in time (the phenomenon of life) for a long period is impossible without simultaneous “phenomenon of death” and that taking into account the features of transition states in reactions (quantum beats) leads to self-oscillations.     

A simple model of interactive pollutants

This study presents a model of two interactive pollutants in which the natural rate of regeneration of each pollutant is affected by the quantity of the other pollutant present in the atmosphere. The behavior of pollution accumulations in the natural unintervened system is characterized. With human intervention, it is shown that while a sustainable environment can be obtained, a deteriorating environment may also arise. There is a unique continuous boundary beyond which the environment is non-revitalizable.     

A simple stochastic gene substitution model

If the fitnesses of n haploid alleles in a finite population are assigned at random and if the alleles can mutate to one another, and if the population is initially fixed for the kth most fit allele, then the mean number of substitutions that will occur before the most fit allele is fixed is shown to be

$\text{1}\text{2}\text{+}\text{1}\text{k}\text{+}\text{∑}\text{i=2}\text{k?1}\text{(i+3)}\text{(2i(i+1))}$

when selection is strong and mutation is weak. This result is independent of the parameters that went into the model. The result is used to provide a partial explanation for the large variance observed in the rates of molecular evolution.     

A simple model of molecular construction

The microscopic model illustrating the ideas of L.A. Blumenfeld about the role of slow conformational relaxation in the mechanism of action of enzymes is discussed.     

A simple model for gene targeting

Sequence-specific binding to genomic-size DNA sequences by artificial agents is of major interest for the development of gene-targeting strategies, gene-diagnostic applications, and biotechnical tools. The binding of one such agent, peptide nucleic acid (PNA), to a randomized human genome has been modeled with statistical mass action calculations. With the length of the PNA probe, the average per-base binding constant k(0), and the binding affinity loss of a mismatched base pair as main parameters, the specificity was gauged as a "therapeutic ratio" G = maximum safe [PNA](tot)/minimal efficient [PNA](tot). This general, though simple, model suggests that, above a certain threshold length of the PNA, the microscopic binding constant k(0) is the primary determinant for optimal discrimination, and that only a narrow range of rather low k(0) values gives a high therapeutic ratio G. For diagnostic purposes, the value of k(0) could readily be modulated by changing the temperature, due to the substantial Delta H degrees associated with the binding equilibrium. Applied to gene therapy, our results stress the need for appropriate control of the binding constant and added amount of the gene-targeting agent, to meet the varying conditions (ionic strength, presence of competing DNA-binding molecules) found in the cell.     

A simple model for plankton patchiness

Recent work on zooplankton spatial variability shows that thepower spectrum of the wave-number variance is flatter than thatfor chlorophyll, as a consequence of greater fine structurein the herbivores. The relative slopes of the power spectracan result from white noise forcing of simple coupled phytoplankton-herbivoremodels with diffusion.