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Register a new userA stochastic model for the evolution of species with random fitness
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We generalize the evolution model introduced by Guiol, Machado and Schinazi (2010). In our model at odd times a random number X of species is created. Each species is endowed with a random fitness with arbitrary distribution on $[0, 1]$. At even times a random number Y of species is removed, killing the species with lower fitness. We show that there is a critical fitness $f_c$ below which the number of species hits zero i.o. and above of which this number goes to infinity. We prove uniform convergence for the distribution of surviving species and describe the phenomena which could not be observed in previous works with uniformly distributed fitness.
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We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $mu_N$, and each beneficial mutation increases the individuals fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individuals fitness to give birth. Under certain conditions on the parameters $mu_N$ and $s_N$, we obtain rigorous results for the rate at which mutations accumulate in the population and the distribution of the fitnesses of individuals in the population at a given time. Our results confirm predictions of Desai and Fisher (2007).
Most human tumors result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Mutations that confer a fitness advantage to the cell are known as driver mutations and are causally related to tumorigenesis. Other mutations, however, do not change the phenotype of the cell or even decrease cellular fitness. While much experimental effort is being devoted to the identification of the different functional effects of individual mutations, mathematical modeling of tumor progression generally considers constant fitness increments as mutations are accumulated. In this paper we study a mathematical model of tumor progression with random fitness increments. We analyze a multi-type branching process in which cells accumulate mutations whose fitness effects are chosen from a distribution. We determine the effect of the fitness distribution on the growth kinetics of the tumor. This work contributes to a quantitative understanding of the accumulation of mutations leading to cancer phenotypes.
Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction network theory. In our previous paper textit{Prevalence of deficiency zero reaction networks in an ErdH os-Renyi framework}, we provided a framework to quantify the prevalence of deficiency zero among randomly generated reaction networks. Specifically, given a randomly generated binary reaction network with $n$ species, with an edge between two arbitrary vertices occurring independently with probability $p_n$, we established the threshold function $r(n)=frac{1}{n^3}$ such that the probability of the random network being deficiency zero converges to 1 if $frac{p_n}{r(n)}to 0$ and converges to 0 if $frac{p_n}{r(n)}toinfty$, as $n to infty$. With the base ErdH os-Renyi framework as a starting point, the current paper provides a significantly more flexible framework by weighting the edge probabilities via control parameters $alpha_{i,j}$, with $i,jin {0,1,2}$ enumerating the types of possible vertices (zeroth, first, or second order). The control parameters can be chosen to generate random reaction networks with a specific underlying structure, such as closed networks with very few inflow and outflow reactions, or open networks with abundant inflow and outflow. Under this new framework, for each choice of control parameters ${alpha_{i,j}}$, we establish a threshold function $r(n,{alpha_{i,j}})$ such that the probability of the random network being deficiency zero converges to 1 if $frac{p_n}{r(n,{alpha_{i,j}})}to 0$ and converges to 0 if $frac{p_n}{r(n,{alpha_{i,j}})}to infty$.
In the mean field integrate-and-fire model, the dynamics of a typical neuron within a large network is modeled as a diffusion-jump stochastic process whose jump takes place once the voltage reaches a threshold. In this work, the main goal is to establish the convergence relationship between the regularized process and the original one where in the regularized process, the jump mechanism is replaced by a Poisson dynamic, and jump intensity within the classically forbidden domain goes to infinity as the regularization parameter vanishes. On the macroscopic level, the Fokker-Planck equation for the process with random discharges (i.e. Poisson jumps) are defined on the whole space, while the equation for the limit process is on the half space. However, with the iteration scheme, the difficulty due to the domain differences has been greatly mitigated and the convergence for the stochastic process and the firing rates can be established. Moreover, we find a polynomial-order convergence for the distribution by a re-normalization argument in probability theory. Finally, by numerical experiments, we quantitatively explore the rate and the asymptotic behavior of the convergence for both linear and nonlinear models.
In the mean field integrate-and-fire model, the dynamics of a typical neuron within a large network is modeled as a diffusion-jump stochastic process whose jump takes place once the voltage reaches a threshold. In this work, the main goal is to establish the convergence relationship between the regularized process and the original one where in the regularized process, the jump mechanism is replaced by a Poisson dynamic, and jump intensity within the classically forbidden domain goes to infinity as the regularization parameter vanishes. On the macroscopic level, the Fokker-Planck equation for the process with random discharges (i.e. Poisson jumps) are defined on the whole space, while the equation for the limit process is on the half space. However, with the iteration scheme, the difficulty due to the domain differences has been greatly mitigated and the convergence for the stochastic process and the firing rates can be established. Moreover, we find a polynomial-order convergence for the distribution by a re-normalization argument in probability theory. Finally, by numerical experiments, we quantitatively explore the rate and the asymptotic behavior of the convergence for both linear and nonlinear models.
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