Smoluchowski Diffusion Equation Research Articles

Probability distribution of Brownian motion in periodic potentials

We calculate the probability distribution function (PDF) of an overdamped Brownian particle moving in a periodic potential energy landscape $U(x)$. The PDF is found by solving the corresponding Smoluchowski diffusion equation. We derive the solution for any periodic even function $U(x)$, and demonstrate that it is asymptotically (at large time $t$) correct up to terms decaying faster than $\sim t^{-3/2}$. As part of the derivation, we also recover the Lifson-Jackson formula for the effective diffusion coefficient of the dynamics. The derived solution exhibits agreement with Langevin dynamics simulations when (i) the periodic length is much larger than the ballistic length of the dynamics, and (ii) when the potential barrier $\Delta U=\max(U(x))-\min(U(x))$ is not much larger than the thermal energy $k_BT$.

Position-Dependent Diffusion Tensors in Anisotropic Media from Simulation: Oxygen Transport in and through Membranes.

A Bayesian-based methodology is developed to estimate diffusion tensors from molecular dynamics simulations of permeants in anisotropic media, and is applied to oxygen in lipid bilayers. By a separation of variables in the Smoluchowski diffusion equation, the multidimensional diffusion is reduced to coupled one-dimensional diffusion problems that are treated by discretization. The resulting diffusivity profiles characterize the membrane transport dynamics as a function of the position across the membrane, discriminating between diffusion normal and parallel to the membrane. The methodology is first validated with neat water, neat hexadecane, and a hexadecane slab surrounded by water, the latter being a simple model for a lipid membrane. Next, a bilayer consisting of pure 1-palmitoyl 2-oleoylphosphatidylcholine (POPC), and a bilayer mimicking the lipid composition of the inner mitochondrial membrane, including cardiolipin, are investigated. We analyze the detailed time evolution of oxygen molecules, in terms of both normal diffusion through and radial diffusion inside the membrane. Diffusion is fast in the more loosely packed interleaflet region, and anisotropic, with oxygen spreading more rapidly in the membrane plane than normal to it. Visualization of the propagator shows that oxygen enters the membrane rapidly, reaching its thermodynamically favored center in about 1 ns, despite the free energy barrier at the headgroup region. Oxygen transport is quantified by computing the oxygen permeability of the membranes and the average radial diffusivity, which confirm the anisotropy of the diffusion. The position-dependent diffusion constants and free energies are used to construct compartmental models and test assumptions used in estimating permeability, including Overton's rule. In particular, a hexadecane slab surrounded by water is found to be a poor model of oxygen transport in membranes because the relevant energy barriers differ substantially.

Subdiffusion in Membrane Permeation of Small Molecules.

Within the solubility–diffusion model of passive membrane permeation of small molecules, translocation of the permeant across the biological membrane is traditionally assumed to obey the Smoluchowski diffusion equation, which is germane for classical diffusion on an inhomogeneous free-energy and diffusivity landscape. This equation, however, cannot accommodate subdiffusive regimes, which have long been recognized in lipid bilayer dynamics, notably in the lateral diffusion of individual lipids. Through extensive biased and unbiased molecular dynamics simulations, we show that one-dimensional translocation of methanol across a pure lipid membrane remains subdiffusive on timescales approaching typical permeation times. Analysis of permeant motion within the lipid bilayer reveals that, in the absence of a net force, the mean squared displacement depends on time as t0.7, in stark contrast with the conventional model, which assumes a strictly linear dependence. We further show that an alternate model using a fractional-derivative generalization of the Smoluchowski equation provides a rigorous framework for describing the motion of the permeant molecule on the pico- to nanosecond timescale. The observed subdiffusive behavior appears to emerge from a crossover between small-scale rattling of the permeant around its present position in the membrane and larger-scale displacements precipitated by the formation of transient voids.

Wetting Driven by Thermal Fluctuations on Terraced Nanostructures.

Theoretical analysis and fully atomistic molecular dynamics simulations reveal a Brownian ratchet mechanism by which thermal fluctuations drive the net displacement of immiscible liquids confined in channels or pores with micro- or nanoscale dimensions. The thermally driven displacement is induced by surface nanostructures with directional asymmetry and can occur against the direction of action of wetting or capillary forces. Mean displacement rates in molecular dynamics simulations are predicted via analytical solution of a Smoluchowski diffusion equation for the position probability density. The proposed physical mechanisms and derived analytical expressions can be applied to engineer surface nanostructures for controlling the dynamics of diverse wetting processes such as capillary filling, wicking, and imbibition in micro- or nanoscale systems.

Novel Single-Molecule Resolution Method for Spatio-Temporal Simulations of Protein Binding and Recruitment on the Membrane

In the early stages of clathrin-mediated endocytosis (CME), a variety of distinct proteins can bind to the membrane and engage in further interactions with proteins on the membrane and in solution. Understanding the dynamics of this process requires correctly accounting for the behavior of protein interactions while restricted to the 2D membrane surface, as it is fundamentally distinct from binding in solution due to changes in the dynamics of the proteins. Here we introduce the 2D Free-Propagator Re-weighting (2D-FPR) method that accurately models the spatial and temporal dynamics of proteins as they are recruited to the membrane surface and as they interact with one another while bound to the membrane. In this method the position of each diffusing protein is tracked, and reactions between binding partners can occur upon collisions. Reaction probabilities are determined by the solution to the 2D Smoluchowski diffusion equation with reactive boundary conditions, allowing us to take large time steps. Molecule positions are propagated by free diffusion, but by using a trajectory reweighting approach we can recover the exact association rates for all reactive pairs. This approach is uniquely able to capture the changes in protein binding dynamics that can occur upon membrane binding because it accounts for both the diffusional motion of proteins and their binding reactions. These important details are absent from models that lack spatial resolution. We present our simulation results on modeling adaptor protein interaction dynamics, and discuss the effects of varying local protein concentration on both recruitment to the membrane and complex formation in the confined 2D geometry.

SIGEST

This issue's SIGEST paper, “Chemical Reactions as $\Gamma$-Limit of Diffusion,” by Mark A. Peletier, Giuseppe Savaré, and Marco Veneroni, is an update of their paper “From Diffusion to Reaction via $\Gamma$-Convergence” from the SIAM Journal on Mathematical Analysis. The work uses advanced ideas from measure theory to connect macroscale chemical reactions, the ones you see in a test tube, to a limit of a molecular (micro)scale diffusion processes. The macroscale reactions of interest in this paper are “unimolecular,” $A \rightleftharpoons B$. The meaning of this is that A and B are two forms of the same molecule and one can convert to the other. One can express the reaction between the two forms with a reaction diffusion equation, which you may have seen in an elementary course in applied partial differential equations. The microscale or molecular problem is very different. Here we think of A and B as the endpoints of the domain of an continuous “chemical variable.” The chemical variable is the independent variable in the microscale equation for a probability distribution. This microscale equation is the Kramers–Smoluchowski diffusion equation, which is parabolic. The authors show that for reactions like $A \rightleftharpoons B$, the classical reaction-diffusion equations are a singular limit of the microscale equations. The authors have completely rewritten the introduction for SIAM Review, and the new introduction is a delight. In four pages they walk the reader through much of the technical background and preview the analysis to come. With this background, one can appreciate the discussion of the models and the statements of the two main theorems. This paper is a fine example of how hard analysis contributes to our understanding of multiscale phenomena.

Nucleus-nucleus cold fusion reactions analyzed with thel-dependent “fusion by diffusion” model

We present a modified version of the Fusion by Diffusion (FBD) model aimed at describing the synthesis of superheavy nuclei in cold fusion reactions, in which a low excited compound nucleus emits only one neutron. The modified FBD model accounts for the angular momentum dependence of three basic factors determining the evaporation residue cross section: the capture cross section ${\ensuremath{\sigma}}_{\text{cap}}(l)$, the fusion probability ${P}_{\text{fus}}(l)$, and the survival probability ${P}_{\text{surv}}(l)$. The fusion hindrance factor, the inverse of ${P}_{\text{fus}}(l)$, is treated in terms of thermal fluctuations in the shape degrees of freedom and is expressed as a solution of the Smoluchowski diffusion equation. The $l$ dependence of ${P}_{\text{fus}}(l)$ results from the $l$-dependent potential energy surface of the colliding system. A new parametrization of the distance of starting point of the diffusion process is introduced. An analysis of a complete set of 27 excitation functions for production of superheavy nuclei in cold fusion reactions, studied in experiments at GSI Darmstadt, RIKEN Tokyo, and LBNL Berkeley, is presented. The FBD model satisfactorily reproduces shapes and absolute cross sections of all the cold fusion excitation functions. It is shown that the peak position of the excitation function for a given 1$n$ reaction is determined by the $Q$ value of the reaction and the height of the fission barrier of the final nucleus. This fact could possibly be used in future experiments (with well-defined beam energy) for experimental determination of the fission barrier heights.

Single-file diffusion of externally driven particles

We study one-dimensional diffusion of N hard-core interacting Brownian particles driven by the space- and time-dependent external force. We give the exact solution of the N-particle Smoluchowski diffusion equation. In particular, we investigate the nonequilibrium energetics of two interacting particles under the time-periodic driving. The hard-core interaction induces entropic repulsion which differentiates the energetics of the two particles. We present exact time-asymptotic results which describe the mean energy, the accepted work and heat, and the entropy production of interacting particles, and we contrast these quantities against the corresponding ones for the noninteracting particles.

Two-particle random walk simulation of outer-sphere nuclear relaxation

We present a two-particle Monte Carlo method for computing the outer-sphere (OS) dipolar time correlation function (DTCF) of the relative position of a nuclear spin I on a diamagnetic molecule M(I) with respect to a nuclear or electronic spin S on a molecule M(S) when both molecules are anisotropic and undergo translational and rotational diffusion. As a first application, we question the validity of the appealing interspin procedure [L. P. Hwang, Mol. Phys. 51, 1235 (1984); A. Borel et al., Chem. Eur. J. 7, 600 (2001)] based on the solutions of a Smoluchowski diffusion equation, which conserve the interspin radial distribution function in the course of time. We show that the true random spatial motion of the interspin vector obtained by simulation can be very different from that given by the Smoluchowski solutions and lead to notable retardation of the time decay of the OS-DTCF. Then, we explore the influence of the solvation properties of M(S) on the decay rate of the DTCF. When M(S) is significantly larger than M(I), its rotation accelerates the decay only weakly, even if M(I) follows M(S) in its Brownian tumbling. By contrast, viscous solvation layers in OS pockets of M(S) can yield an important local slowdown of the relative translational diffusion of M(I), leading to a decay retardation of the DTCF, which adds to that due to the shape anisotropy of M(S). When M(S) is a Gd(3+)-based contrast agent, this retardation leads to a notable increase of the OS contribution to relaxivity even at rather high imaging field.

A method for the analysis of thermal tweezers for manipulation and trapping of nanoparticles and adatoms on crystalline surfaces

We develop a computationally efficient method for the theoretical analysis of thermophoresis of nanoparticles and adatoms on crystalline surfaces (thermal tweezers) for efficient parallel nanofabrication. The analysis of surface diffusion of particles or adatoms in the presence of strong temperature gradients is conducted through the direct determination of probability distributions for diffusing particles, using the numerical solution of the Smoluchowski diffusion equation with varying (temperature-dependent) diffusion constant. The local values of the diffusion constant are determined from the Fokker–Planck equation for the considered crystalline potential of the substrate and local temperature. Steady-state and nonsteady-state particle distributions on the surface are obtained and analyzed in the presence of optically-induced strong temperature gradients. Detailed comparison of this approach with the previously obtained results from the Monte Carlo simulations of the Langevin equation is conducted, demonstrating high computational efficiency, and accuracy of the new method in the high-friction regime. Applicability conditions for the developed method are also determined and discussed.

Characterization of Internal Reorientation of Methyl Group in 2,6-Dichlorotoluene

The two correlation times previously obtained in our coupled relaxation measurement for the methyl group in 2,6-dichlorotoluene may be used as a criterion for evaluating the reorientation dynamics of an internal rotor. We numerically tested an extended diffusion model and the Smoluchowski diffusion equation to see how the rotational inertial effect and jump character contribute to the internal correlation time ratio of the internal rotor. We also analytically solved the general jump model with three different rate constants in a sixfold symmetric potential barrier. By assuming that the internal rotation of the methyl group in 2,6-dichlorotoluene can be described in terms of jumps among sixfold harmonic potential wells, we can conclude that the jump model satisfactorily reproduce the experimental data and the rate for sixfold jump is at least 1.53 times as great as that of a threefold jump.

Length and Time-Dependent Rates in Diffusion-Controlled Reactions with Conjugated Polymers

Rate constants for diffusion-controlled reactions of solvated electrons with conjugated fluorene oligomers (oF) and polymers (pF) were measured in liquid tetrahydrofuran (THF). Preparative gel permeation chromatography (GPC) was used to separate the polyfluorenes into fractions having narrowed distributions of lengths. Both oF and pF's were used in determinations of the attachment rate constants k(inf) as a function of length, where k(inf) refers to the rate coefficients at long times where they are indeed constant. The results find that in going from oF(1) to pF(133), k(inf) increases by a factor of 16, which is much smaller than that of the 133-fold increase in length. The extent of this increase and its change with length are in excellent agreement with published theoretical models that describe diffusion to long thin objects as either prolate spheroids or one-dimensional arrays of spheres. As the concentration of polymer was increased, the effects of large transient terms in the rate constant were observed. As predicted by the Smoluchowski diffusion equation, with modifications by more contemporary theorists, these transient effects are larger and persist to longer times for the larger molecules. For the longest molecule, pF(133), k(t) increases by more than a decade at short times. In that case, the "transient term" becomes dominant and the rate coefficient is approximately proportional to the square of the effective reaction radius in contrast to the linear dependence usual for diffusional reactions. The size of these transient effects and their quantitative confirmation are unprecedented.

Diffusional Channeling in the Sulfate-Activating Complex: Combined Continuum Modeling and Coarse-Grained Brownian Dynamics Studies

Enzymes required for sulfur metabolism have been suggested to gain efficiency by restricted diffusion (i.e., channeling) of an intermediate APS 2– between active sites. This article describes modeling of the whole channeling process by numerical solution of the Smoluchowski diffusion equation, as well as by coarse-grained Brownian dynamics. The results suggest that electrostatics plays an essential role in the APS 2– channeling. Furthermore, with coarse-grained Brownian dynamics, the substrate channeling process has been studied with reactions in multiple active sites. Our simulations provide a bridge for numerical modeling with Brownian dynamics to simulate the complicated reaction and diffusion and raise important questions relating to the electrostatically mediated substrate channeling in vitro, in situ, and in vivo.

ROLE OF MASS ASYMMETRY IN FUSION OF SUPER-HEAVY NUCLEI

By using well tested standard statistical model for calculating survival probabilities of super-heavy compound nuclei, Psurv, and reliably predicted capture cross sections σcap, "empirical" values of the fusion hindrance Pfus = σsyn/(σcap · Psurv) have been determined from the formation cross sections σsyn for production of super-heavy nuclei of 102 ≤ Z ≤ 113 measured at GSI Darmstadt and RIKEN. So determined fusion hindrance can be well reproduced with a simple model based on the Smoluchowski diffusion equation applied to describe thermal shape fluctuations of the fusing system. An applicability of this model for a wider class of fusing systems could be verified on data from an interesting experiment on the synthesis of isotopes of Hs (Z = 108) in the 136 Xe + 136 Xe reaction, scheduled to be carried out in Dubna this year. Synthesis cross sections for 269-271 Hs isotopes, strongly reduced by the fusion hindrance effect in this symmetric fusion reaction, are predicted.

Colloidal electrolyte friction: the effect of finite-sized electrolyte ions

The electro-hydrodynamic coupling of electrolyte ions and surface-dissociated counterions,i.e., microions, to the motion of a suspended colloidal macroion leads to an additionalcontribution to the colloidal friction coefficient. On the basis of the primitive model and thegeneralized Smoluchowski diffusion equation, a simplified mode–mode coupling scheme(MCS) is developed for quantifying the effect of electrolyte friction on the tracer diffusionof a macroion. In this scheme, far-field hydrodynamic interactions between all ionic speciesare considered. The influence of the finite size of the microions is accounted for by usingmean spherical approximation expressions of static pair correlation functions forunequal sizes. The present paper extends earlier work of one of the authors toinclude the effects of finite-sized and hydrodynamically interacting microions. Ourtheoretical results are used to test the relevance of finite size effects in suspensions ofnano-sized particles such as charged globular micelles. Significant finite size effectsare only observed for macroion–microion size ratios typically smaller than 10.

Rapid demulsification of dense oil-in-water emulsion by low external electric field.: II. Theory

Rapid demulsification of dense oil-in-water emulsions under external low electric fields was theoretically analyzed in the framework of the DLVO theory by solving the Smoluchowski diffusion equation for ions and the Poisson’s equation for electrostatic fields. The theory revealed that the applied field induces the demulsification according to the following mechanisms. (1) The applied field induces a steady-state current of ions in an aqueous phase, which in turn generates an electric field in the aqueous phase. (2) The electric field in the aqueous phase rearranges surface charges on oil droplets to compensate the field gradient on the surface of each droplet, which causes the polarization of the electrostatic surface potential. (3) The polarization lowers repulsive osmotic pressure in the overlapped diffuse electric double layers for the droplets, so that the demulsification is accelerated.

Molecular dynamics of C-peptide of ribonuclease A studied by replica-exchange Monte Carlo method and diffusion theory

Generalized-ensemble algorithm and diffusion theory have been combined in order to compute the dynamical properties monitored by nuclear magnetic resonance (NMR) experiments from efficient and reliable evaluation of statistical averages. Replica-exchange Monte Carlo simulations have been performed with a C-peptide analogue of ribonuclease A, and the Smoluchowski diffusion equation has been applied. A fairly good agreement between the calculated and measured 1H-NOESY NMR cross peaks has been obtained. The combination of these advanced and continuously improving statistical tools allows the calculation of a wide variety of dynamical properties routinely obtained by experiments.

Modeling the dynamics of the solvated SL1 domain of HIV-1 genomic RNA.

A mode-coupling solution of the Smoluchowski diffusion equation (MCD theory), designed to describe the dynamics of wobbling macromolecules in water, is applied to a macromolecular bead model including water beads in the nearest layers. The necessary statistical averages are evaluated by time averaging along a molecular dynamics (MD) trajectory where both solute and water are introduced as atomistic models. The cross peaks in (1)H nuclear Overhauser effect spectroscopy (NOESY) NMR spectra that are routinely measured to determine biological structures are here calculated for the mutated 23 nucleotides stem-loop fragment of the SL1 domain in the HIV-1(Lai) genomic RNA. The calculations are in acceptable agreement with experiments without requiring any screening of the hydrodynamic interactions. The screening of hydrodynamics was necessary in previous MCD calculations obtained by using the same full atomistic MD trajectory, but a nonsolvated frictional model.

Silylation and sulfonation of structured supported catalysts active in the decoloration of azo-dyes under visible light

Structured silica woven fabrics have been derivatized with functional groups able to anchor by exchange of Fe3+-ions and TiO2 showing a stable performance during the visible light induced decoloration of the Orange II azo-dye. The kinetics and efficiency of the decoloration mediated by the catalytic loaded silica fabrics with Fe3+-ions were seen to be much higher than found with homogeneous Fenton reagents (Fe3+/H2O2) with the equivalent Fe3+ content. The same was observed for derivatized membranes where TiO2 has been anchored as the active catalyst surface species. In the case of the silica Fe3+-ions loaded fabrics, the decoloration was studied as a function of the amount of H2O2 oxidant added in solution, the intensity of the applied visible light and the concentration of the initial Orange II. In the case of the silica-TiO2 fabrics the decoloration kinetics was observed to be a function of the O2 present in solution. In the case of the derivatized Fe3+ and TiO2 loaded silica fabrics, the decoloration process presented three common features: (a) the decoloration process was observed only in the presence of light pointing to a photo-induced process in both cases, (b) the decoloration was also observed to be truly catalytical following repetitive cycles for Orange II, and finally (c) the decoloration processes were limited by the mass transfer kinetics taking place at the surface of both derivatized fabric catalyst and proceeded with about the same kinetics in both cases. The numerical values for the diffusion distance of the radicals species OH and HO2 as well as the decrease in the concentration of radicals away from the silica fabric during the photodegradation of Orange II is estimated by the Smoluchowski diffusion equation.

Electronic energy transfer in polymers labeled at both ends with fluorescent groups

The energy transfer between chromophores attached at the ends of an isolated polymer chain was studied. Chains labeled with different or equal fluorophores at the ends were considered. Interpolated analytical expressions dependent on a unique parameter, were derived for the fluorescence decay curves taking into account the distribution of end-to-end separations in both θ and good solvents for the polymer. The donor decay curves for chains with dissimilar chromophores were compared with the exact solution obtained by numerical integration. The derived expressions reproduce the decay for all times with a precision better than 1% for pseudo-ideal chains ( θ solvents) and 3% for chains in good solvents. The inhomogeneous broadening of chromophore spectra is important when its width is larger than 20% of the homogeneous broadening width. The fluorescence decay curves in the presence of Brownian motion were also calculated after solving the Smoluchowski diffusion equation for the end-to-end separation distance distribution function. Chain dynamics is important for donors with long lifetimes, high diffusion coefficients and donor–acceptor pairs with small Förster radius.