A new approach to the problem of bulk-mediated surface diffusion

Diffusion-Controlled Reactions: An Overview

We review the milestones in the century-long development of the theory of diffusion-controlled reactions. Starting from the seminal work by von Smoluchowski, who recognized the importance of diffusion in chemical reactions, we discuss perfect and imperfect surface reactions, their microscopic origins, and the underlying mathematical framework. Single-molecule reaction schemes, anomalous bulk diffusions, reversible binding/unbinding kinetics, and many other extensions are presented. An alternative encounter-based approach to diffusion-controlled reactions is introduced, with emphasis on its advantages and potential applications. Some open problems and future perspectives are outlined.

Diffusion coefficients and MSD measurements on curved membranes and porous media

We study some geometric aspects that influence the transport properties of particles that diffuse on curved surfaces. We compare different approaches to surface diffusion based on the Laplace-Beltrami operator adapted to predict concentration along entire membranes, confined subdomains along surfaces, or within porous media. Our goal is to summarize, firstly, how diffusion in these systems results in different types of diffusion coefficients and mean square displacement measurements, and secondly, how these two factors are affected by the concavity of the surface, the shape of the possible barriers or obstacles that form the available domains, the sinuosity, tortuosity, and constrictions of the trajectories and even how the observation plane affects the measurements of the diffusion. In addition to presenting a critical and organized comparison between different notions of MSD, in this review, we test the correspondence between theoretical predictions and numerical simulations by performing finite element simulations and illustrate some situations where diffusion theory can be applied. We briefly reviewed computational schemes for understanding surface diffusion and finally, discussed how this work contributes to understanding the role of surface diffusion transport properties in porous media and their relationship to other transport processes.

Spatially Resolving Size Effects on Diffusivity in Nanoporous Extracellular Matrix-like Materials with Fluorescence Correlation Spectroscopy Super-Resolution Optical Fluctuation Imaging

It is well documented that the nanoscale structures within porous microenvironments greatly impact the diffusion dynamics of molecules. However, how the interaction between the environment and molecules influences the diffusion dynamics has not been thoroughly explored. Here, we show that fluorescence correlation spectroscopy super-resolution optical fluctuation imaging (fcsSOFI) can be used to accurately measure the diffusion dynamics of molecules within varying matrices such as nanopatterned surfaces and porous agarose hydrogels. Our data demonstrate the robustness of fcsSOFI, where it is possible not only to quantify the diffusion speeds of molecules in heterogeneous media but also to recover the matrix structure with resolution on the order of 100 nm. Using dextran molecules of varying sizes, we show that the diffusion coefficient is sensitive to the change in the molecular hydrodynamic radius. fcsSOFI images further reveal that smaller dextran molecules can freely move through the small pores of the hydrogel and report the detailed porous structure and local diffusion heterogeneities not captured by the average diffusion coefficient. Conversely, bigger dextran molecules are confined and unable to freely move through the hydrogel, highlighting only the larger pore structures. These findings establish fcsSOFI as a powerful tool to characterize spatial and diffusion information of diverse macromolecules within biorelevant matrices.

Enhanced Facilitated Diffusion of Membrane-Associating Proteins under Symmetric Confinement

The facilitated surface diffusion of transiently adsorbing molecules in a planar confined microenvironment (i.e., slit-like confinement) is highly relevant to biological phenomena, such as extracellular signaling, as well as numerous biotechnology systems. Here, we studied the surface diffusion of individual proteins confined between two symmetric lipid bilayer membranes, under a continuum of confinement heights, using single-molecule tracking and convex lens-induced confinement as well as hybrid, kinetic Monte Carlo simulations of a generalized continuous time random walk process. Surface diffusion was observed to vary non-monotonically with confinement height, exhibiting a maximum at a height of ∼750 nm, where diffusion was nearly 40% greater than that for a semi-infinite system. This demonstrated that planar confinement can, in fact, increase surface diffusion, qualitatively validating previous theoretical predictions. Simulations reproduced the experimental results and suggested that confinement enhancement of surface diffusion for symmetric systems is limited to cases where the adsorbate exhibits weak surface sticking.

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.

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.

Anomalous protein kinetics on low-fouling surfaces

In this work, protein-surface interactions were probed in terms of adsorption and desorption of a model protein, bovine serum albumin, on a low-fouling surface with single-molecule localization microscopy. Single-molecule experiments enable precise determination of both adsorption and desorption rates. Strikingly the experimental data show anomalous desorption kinetics, evident as a surface dwell time that exhibits a power-law distribution, i.e. a heavy-tailed rather than the expected exponential distribution. As a direct consequence of this heavy-tailed distribution, the average desorption rate depends upon the time scale of the experiment and the protein surface concentration does not reach equilibrium. Further analysis reveals that the observed anomalous desorption emerges due to the reversible formation of a small fraction of soluble protein multimers (small oligomers), such that each one desorbs from the surface with a different rate. The overall kinetics can be described by a series of elementary reactions, yielding simple scaling relations that predict experimental observations. This work reveals a mechanistic origin for anomalous desorption kinetics that can be employed to interpret observations where low-protein fouling surfaces eventually foul when in long-term contact with protein solutions. The work also provides new insights that can be used to define design principles for non-fouling surfaces and to predict their performance.

Slip of fluid molecules on solid surfaces by surface diffusion

The mechanism of fluid slip on a solid surface has been linked to surface diffusion, by which mobile adsorbed fluid molecules perform hops between adsorption sites. However, slip velocity arising from this surface hopping mechanism has been estimated to be significantly lower than that observed experimentally. In this paper, we propose a re-adsorption mechanism for fluid slip. Slip velocity predictions via this mechanism show the improved agreement with experimental measurements.

Bulk-mediated surface transport in the presence of bias

Surface transport, when the particle is allowed to leave the surface, travel in the bulk for some time, and then return to the surface, is referred to as bulk-mediated surface transport. Recently, we proposed a formalism that significantly simplifies analysis of bulk-mediated surface diffusion [A. M. Berezhkovskii, L. Dagdug, and S. M. Bezrukov, J. Chem. Phys. 143, 084103 (2015)]. Here this formalism is extended to bulk-mediated surface transport in the presence of bias, i.e., when the particle has arbitrary drift velocities on the surface and in the bulk. A key advantage of our approach is that the transport problem reduces to that of a two-state problem of the particle transitions between the surface and the bulk. The latter can be solved with relative ease. The formalism is used to find the Laplace transforms of the first two moments of the particle displacement over the surface in time t at arbitrary values of the particle drift velocities and diffusivities on the surface and in the bulk. This allows us to analyze in detail the time dependence of the effective drift velocity of the particle on the surface, which can be highly nontrivial.