We propose a mathematical analysis of an integro-differential model arising in population genetics. The model describes the dynamics of fitness distribution in an asexual population under the effect of mutation and selection. These two processes are represented by two nonlocal terms. First, we prove the existence and uniqueness of the solution, and we derive asymptotic estimates of the distribution as the fitness tends to ±∞. Based on these asymptotic estimates, we show that the cumulant generating function of the distribution is well-defined and satisfies a linear nonlocal transport equation that we solve explicitly. This explicit formula allows us to characterize the dependence of the long time behavior of the distribution with respect to the mutation kernel. On the one hand, if the kernel contains some beneficial mutations, the distribution diverges, which is reminiscent of the results of Alfaro and Carles (2014) who analysed a mutator-replicator equation with a diffusive mutation term. On the other hand, if the initial fitness distribution admits some upper bound, purely deleterious kernels lead to the convergence of the distribution towards an equilibrium. The shape of the equilibrium distribution strongly depends on the kernel through its harmonic mean −s H : the distribution admits a positive mass at the best initial fitness class if and only if s H = 0.

We present an original method to estimate the conductivity of a single molecule anchored to nanometric-sized metallic electrodes, using a Mechanically Controlled Break Junction (MCBJ) operated at room temperature in liquid. We record the conductance through the metal / molecules / metal nanocontact while keeping the metallic electrodes at a fixed distance. Taking advantage of thermal diffusion and electromigration, we let the contact naturally explore the more stable configurations around a chosen conductance value. The conductance of a single molecule is estimated from a statistical analysis of raw conductance and conductance standard deviation data for molecular contacts containing up to 14 molecules. The single molecule conductance values are interpreted as time-averaged conductance of an ensemble of conformers at thermal equilibrium. arXiv:1706.09818v2 [cond-mat.mes-hall]

We propose an integro-differential description of the dynamics of the fitness distribution in an asexual population under mutation and selection, in the presence of a phenotype optimum. Due to the presence of this optimum, the distribution of mutation effects on fitness depends on the parent's fitness, leading to a non-standard equation with "contextdependent" mutation kernels.Under general assumptions on the mutation kernels, which encompass the standard n−dimensional Gaussian Fisher's geometrical model (FGM), we prove that the equation admits a unique time-global solution. Furthermore, we derive a nonlocal nonlinear transport equation satisfied by the cumulant generating function of the fitness distribution. As this equation is the same as the equation derived by Martin and Roques (2016) while studying stochastic Wright-Fisher-type models, this shows that the solution of the main integro-differential equation can be interpreted as the expected distribution of fitness corresponding to this type of microscopic models, in a deterministic limit. Additionally, we give simple sufficient conditions for the existence/non-existence of a concentration phenomenon at the optimal fitness value, i.e, of a Dirac mass at the optimum in the stationary fitness distribution. We show how it determines a phase transition, as mutation rates increase, in the value of the equilibrium mean fitness at mutation-selection balance. In the particular case of the FGM, consistently with previous studies based on other formalisms Peck, 1998, 2006), the condition for the existence of the concentration phenomenon simply requires that the dimension n of the phenotype space be larger than or equal to 3 and the mutation rate U be smaller than some explicit threshold.The accuracy of these deterministic approximations are further checked by stochastic individual-based simulations.

We propose an integro-differential description of the dynamics of the fitness distribution in an asexual population under mutation and selection, in the presence of a phenotype optimum. Due to the presence of this optimum, the distribution of mutation effects on fitness depends on the parent's fitness, leading to a non-standard equation with 'context-dependent' mutation kernels. Under general assumptions on the mutation kernels, which encompass the standard n − dimensional Gaussian Fisher's geometrical model (FGM), we prove that the equation admits a unique time-global solution. Furthermore, we derive a nonlocal nonlinear transport equation satisfied by the cumulant generating function of the fitness distribution. As this equation is the same as the equation derived by Martin and Roques (2016) while studying stochastic Wright-Fisher-type models, this shows that the solution of the main integro-differential equation can be interpreted as the expected distribution of fitness corresponding to this type of microscopic models, in a deterministic limit. Additionally, we give simple sufficient conditions for the existence/non-existence of a concentration phenomenon at the optimal fitness value, i.e. of a Dirac mass at the optimum in the stationary fitness distribution. We show how it determines a phase transition, as mutation London Mathematical Society * This work has been carried out in the framework of Archimède Labex of Aix-Marseille University. The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University-A*MIDEX, a French 'Investissements d'Avenir' programme, from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) ERC Grant Agreement n. 321186-ReaDi-Reaction-Diffusion Equations, Propagation and Modelling and from the ANR NONLOCAL project (ANR-14-CE25-0013).