Denis S. Grebenkov

Geometrical Structure of Laplacian Eigenfunctions

We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is put onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.

Anomalous, non-Gaussian tracer diffusion in crowded two-dimensional environments

A topic of intense current investigation pursues the question of how the highly crowded environment of biological cells affects the dynamic properties of passively diffusing particles. Motivated by recent experiments we report results of extensive simulations of the motion of a finite sized tracer particle in a heterogeneously crowded environment made up of quenched distributions of monodisperse crowders of varying sizes in finite circular two-dimensional domains. For given spatial distributions of monodisperse crowders we demonstrate how anomalous diffusion with strongly non-Gaussian features arises in this model system. We investigate both biologically relevant situations of particles released either at the surface of an inner domain or at the outer boundary, exhibiting distinctly different features of the observed anomalous diffusion for heterogeneous distributions of crowders. Specifically we reveal an asymmetric spreading of tracers even at moderate crowding. In addition to the mean squared displacement (MSD) and local diffusion exponent we investigate the magnitude and the amplitude scatter of the time averaged MSD of individual tracer trajectories, the non-Gaussianity parameter, and the van Hove correlation function. We also quantify how the average tracer diffusivity varies with the position in the domain with a heterogeneous radial distribution of crowders and examine the behaviour of the survival probability and the dynamics of the tracer survival probability. Inter alia, the systems we investigate are related to the passive transport of lipid molecules and proteins in two-dimensional crowded membranes or the motion in colloidal solutions or emulsions in effectively two-dimensional geometries, as well as inside supercrowded, surface adhered cells.

First exit times of harmonically trapped particles: a didactic review

We revise the classical problem of characterizing first exit times of a harmonically trapped particle whose motion is described by a one- or multidimensional Ornstein–Uhlenbeck process. We start by recalling the main derivation steps of a propagator using Langevin and Fokker–Planck equations. The mean exit time, the moment-generating function and the survival probability are then expressed through confluent hypergeometric functions and thoroughly analyzed. We also present a rapidly converging series representation of confluent hypergeometric functions that is particularly well suited for numerical computation of eigenvalues and eigenfunctions of the governing Fokker–Planck operator. We discuss several applications of first exit times, such as the detection of time intervals during which motor proteins exert a constant force onto a tracer in optical tweezers single-particle tracking experiments; adhesion bond dissociation under mechanical stress; characterization of active periods of trend-following and mean-reverting strategies in algorithmic trading on stock markets; relation to the distribution of first crossing times of a moving boundary by Brownian motion. Some extensions are described, including diffusion under quadratic double-well potential and anomalous diffusion.

Optimal reaction time for surface-mediated diffusion.

We present an exact calculation of the mean first-passage time to a small target on the surface of a 2D or 3D spherical domain, for a molecule performing surface-mediated diffusion. This minimal model of interfacial reactions, which explicitly takes into account the combination of surface and bulk diffusion, shows the importance of correlations induced by the coupling of the switching dynamics to the geometry of the confinement, ignored so far. Our results show that, in the context of interfacial systems in confinement, the reaction time can be minimized as a function of the desorption rate from the surface, which puts forward a general mechanism of enhancement and regulation of chemical and biological reactivity.

Analytical solution for restricted diffusion in circular and spherical layers under inhomogeneous magnetic fields

We propose an analytical solution for restricted diffusion of spin-bearing particles in circular and spherical layers in inhomogeneous magnetic fields. More precisely, we derive exact and explicit formulas for the matrix representing an applied magnetic field in the Laplacian eigenbasis and governing the magnetization evolution. For thin layers, a significant difference between two geometrical length scales (thickness and overall size) allows for accurate perturbative calculations. In these two-scale geometries, apparent diffusion coefficient (ADC) as a function of diffusion time exhibits a new region with a reduced but constant value. The emergence of this intermediate diffusion regime, which is analogous to the tortuosity regime in porous media, is explained in terms of the underlying Laplace operator eigenvalues. In general, regions with constant ADCs would be reminiscent of multiscale geometries, and their observation can potentially be used in experiments to detect the length scales by varying diffusion time.

Pulsed-gradient spin-echo monitoring of restricted diffusion in multilayered structures

A general mathematical basis is developed for computation of the pulsed-gradient spin-echo signal attenuated due to restricted diffusion in multilayered structures (e.g., multiple slabs, cylindrical or spherical shells). Individual layers are characterized by (different) diffusion coefficients and relaxation times, while boundaries between adjacent layers are characterized by (different) permeabilities. Arbitrary temporal profile of the applied magnetic field can be incorporated. The signal is represented in a compact matrix form and the explicit analytical formulas for the elements of the underlying matrices are derived. The implemented algorithm is faster and much more accurate than classical techniques such as Monte Carlo simulations or numerical resolutions of the Bloch-Torrey equation. The algorithm can be applied for studying restricted diffusion in biological systems which exhibit a multilayered structure such as composite tissues, axons and living cells.

Spectral properties of the Brownian self-transport operator

Applying the technique of characteristic functions developped for one-dimensional regular surfaces (curves) with compact support, we obtain the distribution of hitting probabilities for a wide class of finite membranes on square lattice. Then we generalize it to multi-dimensional finite membranes on hypercubic lattice. Basing on these distributions, we explicitly construct the Brownian self-transport operator which governs the Laplacian transfer. In order to verify the accuracy of the distribution of hitting probabilities, numerical analysis is carried out for some particular membranes.

A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging

Diffusion magnetic resonance imaging (dMRI) is a non-invasive imaging technique that gives a measure of the diffusion characteristics of water in biological tissues, notably, in the brain. The hindrances that the microscopic cellular structure poses to water diffusion are statistically aggregated into the measurable macroscopic dMRI signal. Inferring the microscopic structure of the tissue from the dMRI signal allows one to detect pathological regions and to monitor functional properties of the brain. For this purpose, one needs a clearer understanding of the relation between the tissue microstructure and the dMRI signal. This requires novel numerical tools capable of simulating the dMRI signal arising from complex microscopic geometrical models of tissues. We propose such a numerical method based on linear finite elements that allows for a more accurate description of complex geometries. The finite elements discretization is coupled to the adaptive Runge-Kutta Chebyshev time stepping method. This method, which leads to the second order convergence in both time and space, is implemented on FeniCS C++ platform. We also use the mesh generator Salome to work efficiently with multiple-compartment and periodic geometries. Four applications of the method for studying the dMRI signal inside multi-compartment models are considered. In the first application, we investigate the long-time asymptotic behavior of the dMRI signal and show the convergence of the apparent diffusion coefficient to the effective diffusion tensor computed by homogenization. The second application aims to numerically verify that a two-compartment model of cells accurately approximates the three-compartment model, in which the interior cellular compartment and the extracellular space are separated by a finite thickness membrane compartment. The third application consists in validating the macroscopic Karger model of dMRI signals that takes into account compartmental exchange. The last application focuses on the dMRI signal arising from isolated neurons. We propose an efficient one-dimensional model for accurately computing the dMRI signal inside neurite networks in which the neurites may have different radii. We also test the validity of a semi-analytical expression for the dMRI signal arising from neurite networks.

Mean First-Passage Time of Surface-Mediated Diffusion in Spherical Domains

We present an exact calculation of the mean first-passage time to a target on the surface of a 2D or 3D spherical domain, for a molecule alternating phases of surface diffusion on the domain boundary and phases of bulk diffusion. The presented approach is based on an integral equation which can be solved analytically. Numerically validated approximation schemes, which provide more tractable expressions of the mean first-passage time are also proposed. In the framework of this minimal model of surface-mediated reactions, we show analytically that the mean reaction time can be minimized as a function of the desorption rate from the surface.

Optical trapping microrheology in cultured human cells

We present the microrheological study of the two close human epithelial cell lines: non-cancerous HCV29 and cancerous T24. The optical tweezers tracking was applied to extract the several seconds long trajectories of endogenous lipid granules at time step of 1μs. They were analyzed using a recently proposed equation for mean square displacement (MSD) in the case of subdiffusion influenced by an optical trap. This equation leads to an explicit form for viscoelastic moduli. The moduli of the two cell lines were found to be the same within the experimental accuracy for frequencies 102 – 105 Hz. For both cell lines subdiffusion was observed with the exponent close to 3/4, the value predicted by the theory of semiflexible polymers. For times longer than 0.1s the MSD of cancerous cells exceeds the MSD of non-cancerous cells for all values of the trapping force. Such behavior can be interpreted as a signature of the active processes and prevents the extraction of the low-frequency viscoelastic moduli for the living cells by passive microrheology.