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Denis S. Grebenkov

Publications and source records attributed to Denis S. Grebenkov.

At least 73 records · Page 4Linked to original sources
arXiv2021

A molecular relay race: sequential first-passage events to the terminal reaction centre in a cascade of diffusion controlled processes

We consider a sequential cascade of molecular first-reaction events towards a terminal reaction centre in which each reaction step is controlled by diffusive motion of the particles. The model studied here represents a typical reaction setting encountered in diverse molecular biology systems, in which, e.g., a signal transduction proceeds via a series of consecutive "messengers": the first messenger has to find its respective immobile target site triggering a launch of the second messenger, the second messenger seeks its own target site and provokes a launch of the third messenger and so on, resembling a relay race in human competitions. For such a molecular relay race taking place in infinite one-, two- and three-dimensional systems, we find exact expressions for the probability density function of the time instant of the terminal reaction event, conditioned on preceding successful reaction events on an ordered array of target sites. The obtained expressions pertain to the most general conditions: number of intermediate stages and the corresponding diffusion coefficients, the sizes of the target sites, the distances between them, as well as their reactivities are arbitrary.

physics.bio-ph↗
arXiv2021

A physicist's guide to explicit summation formulas involving zeros of Bessel functions and related spectral sums

We summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains. Such spectral sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier-Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review summarizes many summation formulas for spectral sums that are dispersed in the literature.

math-ph↗
arXiv2020

On the spectral properties of the Bloch-Torrey equation in infinite periodically perforated domains

We investigate spectral and asymptotic properties of the particular Schrödinger operator (also known as the Bloch-Torrey operator), $-Δ+ i g x$, in infinite periodically perforated domains of $\mathbb R^d$. We consider Dirichlet realizations of this operator and formalize a numerical approach proposed in [J. Phys. A: Math. Theor. 53, 325201 (2020)] for studying such operators. In particular, we discuss the existence of the spectrum of this operator and its asymptotic behavior as $g\to \infty$.

math-ph↗
arXiv2020

Statistics of boundary encounters by a particle diffusing outside a compact planar domain

We consider a particle diffusing outside a compact planar set and investigate its boundary local time $\ell_t$, i.e., the rescaled number of encounters between the particle and the boundary up to time $t$. In the case of a disk, this is also the (rescaled) number of encounters of two diffusing circular particles in the plane. For that case, we derive explicit integral representations for the probability density of the boundary local time $\ell_t$ and for the probability density of the first-crossing time of a given threshold by $\ell_t$. The latter density is shown to exhibit a very slow long-time decay due to extremely long diffusive excursions between encounters. We briefly discuss some practical consequences of this behavior for applications in chemical physics and biology.

cond-mat.stat-mech↗
arXiv2020

Influence of density and release properties of UCx targets on the fission product yields at ALTO

To study the influence of the structural properties of UCx targets on their release properties, several types of targets using different precursors (carbon and uranium) were synthesized, characterized, irradiated and heated leading to the determination of the released fractions of eight elements. In this article, the production rates of these targets are estimated under the use conditions at ALTO, i.e. with targets bombarded by an electron beam (10 μA, 50 MeV). We have simulated the fission number produced using the FLUKA code. Then, we have determined the release efficiency as a function of the half-life of the isotopes using average diffusion coefficients deduced for the elements studied previously. Finally, we compare the production rates obtained from the various targets and conclude that the target must be adapted to the element studied. It is crucial to find in each case the best compromise between the target density and the release efficiency.

physics.ins-det↗
arXiv2020

A first-passage approach to diffusion-influenced reversible binding: insights into nanoscale signaling at the presynapse

Synaptic transmission between neurons is governed by a cascade of stochastic reaction-diffusion events that lead to calcium-induced vesicle release of neurotransmitter. Since experimental measurements of such systems are challenging due their nanometer and sub-millisecond scale, numerical simulations remain the principal tool for studying calcium dependent synaptic vesicle fusion, despite limitations of time-consuming calculations. In this paper we develop an analytical solution to rapidly explore dynamical stochastic reaction-diffusion problems, based on first-passage times. This is the first analytical model that accounts simultaneously for relevant statistical features of calcium ion diffusion, buffering, and its binding/unbinding reaction with a vesicular sensor. In particular, unbinding kinetics are shown to have a major impact on the calcium sensor's occupancy probability on a millisecond scale and therefore cannot be neglected. Using Monte Carlo simulations we validated our analytical solution for instantaneous calcium influx and that through voltage-gated calcium channels. Overall we present a fast and rigorous analytical tool to study simplified reaction-diffusion systems that allow a systematic exploration of the biophysical parameters at a molecular scale, while correctly accounting for the statistical nature of molecular interactions within cells, that can also serve as a building block for more general cell signaling simulators.

physics.bio-ph↗
arXiv2020

Mean first-passage time to a small absorbing target in an elongated planar domain

We derive an approximate but fully explicit formula for the mean first-passage time (MFPT) to a small absorbing target of arbitrary shape in a general elongated domain in the plane. Our approximation combines conformal mapping, boundary homogenisation, and Fick-Jacobs equation to express the MFPT in terms of diffusivity and geometric parameters. A systematic comparison with a numerical solution of the original problem validates its accuracy when the starting point is not too close to the target. This is a practical tool for a rapid estimation of the MFPT for various applications in chemical physics and biology.

cond-mat.stat-mech↗
arXiv2020

Joint distribution of multiple boundary local times and related first-passage time problems with multiple targets

We investigate the statistics of encounters of a diffusing particle with different subsets of the boundary of a confining domain. The encounters with each subset are characterized by the boundary local time on that subset. We extend a recently proposed approach to express the joint probability density of the particle position and of its multiple boundary local times via a multi-dimensional Laplace transform of the conventional propagator satisfying the diffusion equation with mixed Robin boundary conditions. In the particular cases of an interval, a circular annulus and a spherical shell, this representation can be explicitly inverted to access the statistics of two boundary local times. We provide the exact solutions and their probabilistic interpretation for the case of an interval and sketch their derivation for two other cases. We also obtain the distributions of various associated first-passage times and discuss their applications.

cond-mat.stat-mech↗
arXiv2020

Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions

Dynamics of a particle diffusing in a confinement can be seen a sequence of bulk-diffusion-mediated hops on the confinement surface. Here, we investigate the surface hopping propagator that describes the position of the diffusing particle after a prescribed number of encounters with that surface. This quantity plays the central role in diffusion-influenced reactions and determines their most common characteristics such as the propagator, the first-passage time distribution, and the reaction rate. We derive explicit formulas for the surface hopping propagator and related quantities for several Euclidean domains: half-space, circular annuli, circular cylinders, and spherical shells. These results provide the theoretical ground for studying diffusion-mediated surface phenomena. The behavior of the surface hopping propagator is investigated for both "immortal" and "mortal" particles.

cond-mat.stat-mech↗
arXiv2020

From single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of $N$ random walkers

We consider the first-passage problem for $N$ identical independent particles that are initially released uniformly in a finite domain $Ω$ and then diffuse toward a reactive area $Γ$, which can be part of the outer boundary of $Ω$ or a reaction centre in the interior of $Ω$. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the $N$ particles reacts with $Γ$. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fastest first-passage time with the particle number $N$, namely, a much stronger dependence ($1/N$ and $1/N^2$ for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.

cond-mat.stat-mech↗
arXiv2020

Pseudo-Darwinian evolution of physical flows in complex networks

The evolution of complex transport networks is investigated under three strategies of link removal: random, intentional attack and "Pseudo-Darwinian" strategy. At each evolution step and regarding the selected strategy, one removes either a randomly chosen link, or the link carrying the strongest flux, or the link with the weakest flux, respectively. We study how the network structure and the total flux between randomly chosen source and drain nodes evolve. We discover a universal power-law decrease of the total flux, followed by an abrupt transport collapse. The time of collapse is shown to be determined by the average number of links per node in the initial network, highlighting the importance of this network property for ensuring safe and robust transport against random failures, intentional attacks and maintenance cost optimizations.

physics.soc-ph↗
arXiv2020

Paradigm shift in diffusion-mediated surface phenomena

Diffusion-mediated surface phenomena are crucial for human life and industry, with examples ranging from oxygen capture by lung alveolar surface to heterogeneous catalysis, gene regulation, membrane permeation and filtration processes. Their current description via diffusion equations with mixed boundary conditions is limited to simple surface reactions with infinite or constant reactivity. In this letter, we propose a probabilistic approach based on the concept of boundary local time to investigate the intricate dynamics of diffusing particles near a reactive surface. Reformulating surface-particle interactions in terms of stopping conditions, we obtain in a unified way major diffusion-reaction characteristics such as the propagator, the survival probability, the first-passage time distribution, and the reaction rate. This general formalism allows us to describe new surface reaction mechanisms such as for instance surface reactivity depending on the number of encounters with the diffusing particle that can model the effects of catalyst fooling or membrane degradation. The disentanglement of the geometric structure of the medium from surface reactivity opens far-reaching perspectives for modeling, optimization and control of diffusion-mediated surface phenomena.

cond-mat.stat-mech↗
arXiv2020

Diffusion toward non-overlapping partially reactive spherical traps: fresh insights onto classic problems

Several classic problems for particles diffusing outside an arbitrary configuration of non-overlapping partially reactive spherical traps in three dimensions are revisited. For this purpose, we describe the generalized method of separation of variables for solving boundary value problems of the associated modified Helmholtz equation. In particular, we derive a semi-analytical solution for the Green function that is the key ingredient to determine various diffusion-reaction characteristics such as the survival probability, the first-passage time distribution, and the reaction rate. We also present modifications of the method to determine numerically or asymptotically the eigenvalues and eigenfunctions of the Laplace operator and of the Dirichlet-to-Neumann operator in such perforated domains. Some potential applications in chemical physics and biophysics are discussed, including diffusion-controlled reactions for mortal particles.

physics.comp-ph↗
arXiv2020

Diffusion NMR in periodic media: efficient computation and spectral properties

The Bloch-Torrey equation governs the evolution of the transverse magnetization in diffusion magnetic resonance imaging, where two mechanisms are at play: diffusion of spins (Laplacian term) and their precession in a magnetic field gradient (imaginary potential term). In this paper, we study this equation in a periodic medium: a unit cell repeated over the nodes of a lattice. Although the gradient term of the equation is not invariant by lattice translations, the equation can be analyzed within a single unit cell by replacing a continuous-time gradient profile by narrow pulses. In this approximation, the effects of precession and diffusion are separated and the problem is reduced to the study of a sequence of diffusion equations with pseudo-periodic boundary conditions. This representation allows for efficient numerical computations as well as new theoretical insights into the formation of the signal in periodic media. In particular, we study the eigenmodes and eigenvalues of the Bloch-Torrey operator. We show how the localization of eigenmodes is related to branching points in the spectrum and we discuss low- and high-gradient asymptotic behaviors. The range of validity of the approximation is discussed; interestingly the method turns out to be more accurate and efficient at high gradient, being thus an important complementary tool to conventional numerical methods that are most accurate at low gradients.

physics.comp-ph↗
arXiv2019

The Kärger vs bi-exponential model: theoretical insights and experimental validations

We revise three common models accounting for water exchange in pulsed-gradient spin-echo measurements: a bi-exponential model with time-dependent water fractions, the Kärger model, and a modified Kärger model designed for restricted diffusion, e.g. inside cells. The three models are compared and applied to experimental data from yeast cell suspensions. The Kärger model and the modified Kärger model yield very close results and accurately fit the data. The bi-exponential model, although less rigorous, has a natural physical interpretation and suggests a new experimental modality to estimate the water exchange time.

physics.med-ph↗
arXiv2019

Probing surface-to-volume ratio of an anisotropic medium by diffusion {NMR} with general gradient encoding

Since the seminal paper by Mitra et al., diffusion MR has been widely used in order to estimate surface-to-volume ratios. In the present work we generalize Mitra's formula for arbitrary diffusion encoding waveforms, including recently developed q-space trajectory encoding sequences. We show that surface-to-volume ratio can be significantly misestimated using the original Mitra's formula without taking into account the applied gradient profile. In order to obtain more accurate estimation in anisotropic samples we propose an efficient and robust optimization algorithm to design diffusion gradient waveforms with prescribed features. Our results are supported by Monte Carlo simulations.

physics.med-ph↗
arXiv2019

Diffusion-limited reactions in dynamic heterogeneous media

Most biochemical reactions in living cells rely on diffusive search for target molecules or regions in a heterogeneous overcrowded cytoplasmic medium. Rapid re-arrangements of the medium constantly change the effective diffusivity felt locally by a diffusing particle and thus impact the distribution of the first-passage time to a reaction event. Here, we investigate the effect of these dynamic spatio-temporal heterogeneities onto diffusion-limited reactions. We describe a general mathematical framework to translate many results for ordinary homogeneous Brownian motion to heterogeneous diffusion. In particular, we derive the probability density of the first-passage time to a reaction event and show how the dynamic disorder broadens the distribution and increases the likelihood of both short and long trajectories to reactive targets. While the disorder slows down reaction kinetics on average, its dynamic character is beneficial for a faster search and realization of an individual reaction event triggered by a single molecule.

cond-mat.stat-mech↗