arXiv subjects

Bahram Houchmandzadeh

Publications and source records attributed to Bahram Houchmandzadeh.

16 recordsLinked to original sources

arXiv2025

A master equation approach to the n-coalescent problem

Given an evolutionary model, such as Wright--Fisher (WF) or Moran, the n-coalescent problem consists of going backward in time to find for example the time to the most recent common ancestor (MRCA) and the topology of the tree. In the literature, this problem is tackled mostly by computing directly the random variable t, time to reach the MRCA. I show here that by shifting the focus from the random variable t to the joined variable (n,t), where n is the number of ancestors at time t, the problem is greatly simplified. Indeed, P(n,t), the probability of this variable, obeys a simpler master equation that can be solved in a straightforward way for the most general model. This probability can then be used to compute relevant information of the n-coalescent, for both random variables $t_{n}$ (random time to reach a given state n) and $n_{t}$ (random number of ancestors at a given time t). The cumulative distribution function for $t_{1}$ for example is $P(1,t)$. I give in this article the general solution for continuous time models such as Moran and discrete time ones such as WF.

q-bio.PE↗

arXiv2025

A geometric derivation of Noether's theorem

Noether's theorem is a cornerstone of analytical mechanics, making the link between symmetries and conserved quantities. In this article, I propose a simple, geometric derivation of this theorem that circumvents the usual difficulties that a student of this field usually encounters. The derivation is based on the direct use of the differential form $pdq -Hdt$, where $p$ is the momentum and $H$ the Hamiltonian, integrated over a simple curve.

physics.class-ph↗

arXiv2019

The Hamilton-Jacobi Equation: an intuitive approach

The Hamilton-Jacobi equation (HJE) is one of the most elegant approach to Lagrangian systems such as geometrical optics and classical mechanics, establishing the duality between trajectories and waves and paving the way naturally for the quantum mechanics. Usually, this formalism is taught at the end of a course on analytical mechanics through its technical aspects and its relation to canonical transformations. I propose that the teaching of this subject be centered on this duality along the lines proposed here, and the canonical transformations be taught only after some familiarity with the HJE has been gained by the students.

physics.class-ph↗

arXiv2019

Large deviation of long time average for a stochastic process : an alternative method

We present here a simple method for computing the large deviation of long time average for stochastic jump processes. We show that the computation of the rate function can be reduced to that of a partial differential equation governing the evolution of the probability generating function. The long time limit of this equation, which in many cases can be easily obtained, leads naturally to the rate function.

cond-mat.stat-mech↗

arXiv2018

Giant fluctuations in logistic growth

We analyze the fluctuation of the number of individuals when two competing species, beginning with a few initial individuals, are submitted to a logistic growth. We show that when the total number of individuals reaches the carrying capacity, the number of each species is subject to giant fluctuations (variance $\sim$ mean 2) if the two species have similar growth rate. We show that the deterministic logistic equation can be used only when the growth rates are significantly different, otherwise such growth has to be investigated by stochastic processes tools. These results generalize to a wide class of growth law.

physics.bio-ph↗

arXiv2017

Fisher Waves: an individual based stochastic model

The propagation of a beneficial mutation in a spatially extended population is usually studied using the phenomenological stochastic Fisher-Kolmogorov (SFKPP) equation. We derive here an individual based, stochastic model founded on the spatial Moran process where fluctuations are treated exactly. At high selection pressure, the results of this model are different from the classical FKPP. At small selection pressure, the front behavior can be mapped into a Brownian motion with drift, the properties of which can be derived from microscopic parameters of the Moran model. Finally, we show that the diffusion coefficient and the noise amplitude of SFKPP are not independent parameters but are both determined by the dispersal kernel of individuals.

physics.bio-ph↗

arXiv2016

Neutral Aggregation in Finite Length Genotype space

The advent of modern genome sequencing techniques allows for a more stringent test of the neutrality hypothesis of Darwinian evolution, where all individuals have the same fitness. Using the individual based model of Wright and Fisher, we compute the amplitude of neutral aggregation in the genome space, i.e., the probability of finding two individuals at genetic (hamming) distance k as a function of genome size L, population size N and mutation probability per base ν. In well mixed populations, we show that for Nν\textless{}1/L, neutral aggregation is the dominant force and most individuals are found at short genetic distances from each other. For Nν\textgreater{}1 on the contrary, individuals are randomly dispersed in genome space. The results are extended to geographically dispersed population, where the controlling parameter is shown to be a combination of mutation and migration probability. The theory we develop can be used to test the neutrality hypothesis in various ecological and evolutionary systems.

q-bio.PE↗

arXiv2016

Nonlinear modification of the laser noise power spectrum induced by a frequency-shifted optical feedback

In this article, we study the non-linear coupling between the stationary (i.e. the beating modulation signal) and transient (i.e. the laser quantum noise) dynamics of a laser subjected to frequency shifted optical feedback. We show how the noise power spectrum and more specifically the relaxation oscillation frequency of the laser are modified under different optical feedback condition. Specifically we study the influence of (i) the amount of light returning to the laser cavity and (ii) the initial detuning between the frequency shift and intrinsic relaxation frequency. The present work shows how the relaxation frequency is related to the strength of the beating signal and the shape of the noise power spectrum gives an image of the Transfer Modulation Function (i.e. of the amplification gain) of the nonlinear-laser dynamics.The theoretical predictions, confirmed by numerical resolutions, are in good agreements with the experimental data.

physics.optics↗

arXiv2016

A simple, general result for the variance of substitution number in molecular evolution

The number of substitutions (of nucleotides, amino acids, ...) that take place during the evolution of a sequence is a stochastic variable of fundamental importance in the field of molecular evolution. Although the mean number of substitutions during molecular evolution of a sequence can be estimated for a given substitution model, no simple solution exists for the variance of this random variable. We show in this article that the computation of the variance is as simple as that of the mean number of substitutions for both short and long times. Apart from its fundamental importance, this result can be used to investigate the dispersion index R , i.e. the ratio of the variance to the mean substitution number, which is of prime importance in the neutral theory of molecular evolution. By investigating large classes of substitution models, we demonstrate that although R\ge1 , to obtain R significantly larger than unity necessitates in general additional hypotheses on the structure of the substitution model.

q-bio.PE↗

arXiv2015

Neutron fluctuations: the importance of being delayed

The neutron population in a nuclear reactor is subject to fluctuations in time and in space due to the competition of diffusion by scattering, births by fission events, and deaths by absorptions. As such, fission chains provide a prototype model for the study of spatial clustering phenomena. In order for the reactor to be operated in stationary conditions at the critical point, the population of prompt neutrons instantaneously emitted at fission must be in equilibrium with the much smaller population of delayed neutrons, emitted after a Poissonian time by nuclear decay of the fissioned nuclei. In this work, we will show that the delayed neutrons, although representing a tiny fraction of the total number of neutrons in the reactor, have actually a key impact on the fluctuations, and their contribution is very effective in quenching the spatial clustering.

cond-mat.stat-mech↗

arXiv2015

General formulation of Luria-Delbr{ü}ck distribution of the number of mutants

The Luria-Delbr{ü}ck experiment is a cornerstone of evolutionary theory, demonstrating the randomness of mutations before selection. The distribution of the number of mutants in this experiment has been the subject of intense investigation during the last 70 years. Despite this considerable effort, most of the results have been obtained under the assumption of constant growth rate, which is far from the experimental condition. We derive here the properties of this distribution for arbitrary growth function, for both the deterministic and stochastic growth of the mutants. The derivation we propose uses the number of wild type bacteria as the independent variable instead of time. The derivation is surprisingly simple and versatile, allowing many generalizations to be taken easily into account.

physics.bio-ph↗

arXiv2015

Exact results for a noise-induced bistable system

A stochastic system where bistability is caused by noise has been recently investigated by Biancalani et al. (PRL 112:038101, 2014). They have computed the mean switching time for such a system using a continuous Fokker-Planck equation derived from the Taylor expansion of the Master equation to estimate the parameter of such a system from experiment. In this article, we provide the exact solution for the full discrete system without resorting to continuous approximation and obtain the expression for the mean switching time. We further extend this investigation by solving exactly the Master equation and obtaining the expression of other quantities of interests such as the dynamics of the moments and the equilibrium time.

cond-mat.stat-mech↗

arXiv2014

Noise driven emergence of cooperative behavior

Cooperative behaviors are defined as the production of common goods benefitting all members of the community at the producer's cost. They could seem to be in contradiction with natural selection, as non-cooperators have an increased fitness compared to cooperators. Understanding the emergence of cooperation has necessitated the development of concepts and models (inclusive fitness, multilevel selection, ...) attributing deterministic advantages to this behavior. In contrast to these models, we show here that cooperative behaviors can emerge by taking into account the stochastic nature of evolutionary dynamics : when cooperative behaviors increase the carrying capacity of the habitat, they also increase the genetic drift against non-cooperators. Using the Wright-Fisher models of population genetics, we compute exactly this increased genetic drift and its consequences on the fixation probability of both types of individuals. This computation leads to a simple criterion: cooperative behavior dominates when the relative increase in carrying capacity of the habitat caused by cooperators is higher than the selection pressure against them. This is a purely stochastic effect with no deterministic interpretation.

q-bio.PE↗

arXiv2013

An alternative to the breeder's and Lande's equations

The breeder's equation is a cornerstone of quantitative genetics and is widely used in evolutionary modeling. The equation which reads R=h^{2}S relates response to selection R (the mean phenotype of the progeny) to the selection differential S (mean phenotype of selected parents) through a simple proportionality relation. The validity of this relation however relies strongly on the normal (Gaussian) distribution of parent's genotype which is an unobservable quantity and cannot be ascertained. In contrast, we show here that if the fitness (or selection) function is Gaussian, an alternative, exact linear equation in the form of R'=j^{2}S' can be derived, regardless of the parental genotype distribution. Here R' and S' stand for the mean phenotypic lag behind the mean of the fitness function in the offspring and selected populations. To demonstrate this relation, we derive the exact functional relation between the mean phenotype in the selected and the offspring population and deduce all cases that lead to a linear relation between these quantities. These computations, which are confirmed by individual based numerical simulations, generalize naturally to the multivariate Lande's equation Δ\mathbf{\bar{z}}=GP^{-1}\mathbf{S} .

q-bio.PE↗

arXiv2013

The remarkable discreteness of being

Life is a discrete, stochastic phenomena : for a biological organism, the time of the two most important events of its life (reproduction and death) is random and these events change the number of individuals of the species by single units. These facts can have surprising, counter-intuitive consequences. I review here three examples where these facts play, or could play, important roles : the spatial distribution of species, the biodiversity and the (Darwinian) evolution of altruistic behavior.

physics.bio-ph↗

arXiv2012

Exact results for fixation probability of bithermal evolutionary graphs

One of the most fundamental concepts of evolutionary dynamics is the "fixation" probability, i.e. the probability that a mutant spreads through the whole population. Most natural communities are geographically structured into habitats exchanging individuals among each other and can be modeled by an evolutionary graph (EG), where directed links weight the probability for the offspring of one individual to replace another individual in the community. Very few exact analytical results are known for EGs. We show here how by using the techniques of the fixed point of Probability Generating Function, we can uncover a large class of of graphs, which we term bithermal, for which the exact fixation probability can be simply computed.

q-bio.PE↗