Persistent random walk with exclusion

Modelling the propagation of a pulse in a dense {\em milieu} poses fundamental challenges at the theoretical and applied levels. To this aim, in this paper we generalize the telegraph equation to non-ideal conditions by extending the concept of persistent random walk to account for spatial exclusion effects. This is achieved by introducing an explicit constraint in the hopping rates, that weights the occupancy of the target sites. We derive the mean-field equations, which display nonlinear terms that are important at high density. We compute the evolution of the mean square displacement (MSD) for pulses belonging to a specific class of spatially symmetric initial conditions. The MSD still displays a transition from ballistic to diffusive behaviour. We derive an analytical formula for the effective velocity of the ballistic stage, which is shown to depend in a nontrivial fashion upon both the density (area) and the shape of the initial pulse. After a density-dependent crossover time, nonlinear terms become negligible and normal diffusive behaviour is recovered at long times.Comment: Revised version accepted for publication in Europ. Phys. J.

Diffusion of tagged particles in a crowded medium

The influence of crowding on the diffusion of tagged particles in a dense medium is investigated in the framework of a mean-field model, derived in the continuum limit from a microscopic stochastic process with exclusion. The probability distribution function of the tagged particles obeys to a nonlinear Fokker-Planck equation, where the drift and diffusion terms are determined self-consistently by the concentration of crowders in the medium. Transient sub-diffusive or super-diffusive behaviours are observed, depending on the selected initial conditions, that bridge normal diffusion regimes characterized by different diffusion coefficients. These anomalous crossovers originate from the microscopic competition for space and reflect the peculiar form of the non-homogeneous advection term in the governing Fokker-Planck equation. Our results strongly warn against the overly simplistic identification of crowding with anomalous transport {\em tout court}.Comment: 10 pages, 3 figure

Constraining the high redshift formation of black hole seeds in nuclear star clusters with gas inflows

In this paper we explore a possible route of black hole seed formation that appeal to a model by Davies, Miller & Bellovary who considered the case of the dynamical collapse of a dense cluster of stellar black holes subjected to an inflow of gas. Here, we explore this case in a broad cosmological context. The working hypotheses are that (i) nuclear star clusters form at high redshifts in pre-galactic discs hosted in dark matter halos, providing a suitable environment for the formation of stellar black holes in their cores, (ii) major central inflows of gas occur onto these clusters due to instabilities seeded in the growing discs and/or to mergers with other gas-rich halos, and that (iii) following the inflow, stellar black holes in the core avoid ejection due to the steepening to the potential well, leading to core collapse and the formation of a massive seed of $<~ 1000\, \rm M_\odot$. We simulate a cosmological box tracing the build up of the dark matter halos and there embedded baryons, and explore cluster evolution with a semi-analytical model. We show that this route is feasible, peaks at redshifts $z <~ 10$ and occurs in concomitance with the formation of seeds from other channels. The channel is competitive relative to others, and is independent of the metal content of the parent cluster. This mechanism of gas driven core collapse requires inflows with masses at least ten times larger than the mass of the parent star cluster, occurring on timescales shorter than the evaporation/ejection time of the stellar black holes from the core. In this respect, the results provide upper limit to the frequency of this process

Macroscopic Transport Equations in Many-Body Systems from Microscopic Exclusion Processes in Disordered Media: A Review

Describing particle transport at the macroscopic or mesoscopic level in non-ideal environments poses fundamental theoretical challenges in domains ranging from inter and intra-cellular transport in biology to diffusion in porous media. Yet, often the nature of the constraints coming from many-body interactions or reflecting a complex and confining environment are better understood and modeled at the microscopic level.In this paper we review the subtle link between microscopic exclusion processes and the mean-field equations that ensue from them in the continuum limit. We show that in an inhomogeneous medium, i.e. when jumps are controlled by site-dependent hopping rates, one can obtain three different nonlinear advection-diffusion equations in the continuum limit, suitable for describing transport in the presence of quenched disorder and external fields, depending on the particular rule embodying site inequivalence at the microscopic level. In a situation that might be termed point-like scenario, when particles are treated as point-like objects, the effect of crowding as imposed at the microscopic level manifests in the mean-field equations only if some degree of inhomogeneity is enforced into the model. Conversely, when interacting agents are assigned a finite size, under the more realistic extended crowding framework, exclusion constraints persist in the unbiased macroscopic representation

Conformation-controlled binding kinetics of antibodies

International audienceAntibodies are large, extremely flexible molecules, whose internal dynamics is certainly key to their astounding ability to bind antigens of all sizes, from small hormones to giant viruses. In this paper, we build a shape-based coarse-grained model of IgG molecules and show that it can be used to generate 3D conformations in agreement with single-molecule Cryo-Electron Tomography data. Furthermore, we elaborate a theoretical model that can be solved exactly to compute the binding rate constant of a small antigen to an IgG in a prescribed 3D conformation. Our model shows that the antigen binding process is tightly related to the internal dynamics of the IgG. Our findings pave the way for further investigation of the subtle connection between the dynamics and the function of large, flexible multi-valent molecular machines

Reaction rate of a composite core-shell nanoreactor with multiple, spatially distributed embedded nano-catalysts

We present a detailed theory for the total reaction rate constant of a composite core-shell nanoreactor, consisting of a central solid core surrounded by a hydrogel layer of variable thickness, where a given number of small catalytic nanoparticles are embedded at prescribed positions and are endowed with a prescribed surface reaction rate constant. Besides the precise geometry of the assembly, our theory accounts explicitly for the diffusion coefficients of the reactants in the hydrogel and in the bulk as well as for their transfer free energy jump upon entering the hydrogel shell. Moreover, we work out an approximate analytical formula for the overall rate constant, which is valid in the physically relevant range of geometrical and chemical parameters. We discuss in depth how the diffusion-controlled part of the rate depends on the essential variables, including the size of the central core. In particular, we derive some simple rules for estimating the number of nanocatalysts per nanoreactor for an efficient catalytic performance in the case of small to intermediate core sizes. Our theoretical treatment promises to provide a very useful and flexible tool for the design of superior performing nanoreactor geometries and with optimized nanoparticle load.Comment: 12 pages, 3 figures, Physical Chemistry Chemical Physics, 201

Diffusion of small ligands in complex confining and reactive landscapes: The geometry of chemoreception

The rate constant that describes the diffusive encounter/reaction between a particle and a large sphere can be computed easily by solving the stationary diffusion (i.e. Laplace) equation for the particle density with appropriate boundary conditions imposed on the surface of the sphere. In one classic, textbook example, this calculation is used to estimate the binding rate constant of a ligand to a receptor-covered cell. But what happens if the particles are diffusing in the presence of many reactive boundaries of different strength (intrinsic reaction rate), which compete for the same ligands and amidst a landscape of inert obstacles? In spite of the apparent overwhelming complexity, the same mathematical framework as the two-body problem can be used to solve the N-body problem exactly, by resorting to addition theorems for the appropriate fundamental solutions of the Laplace equation. Please click Additional Files below to see the full abstract