Smoluchowski Limit Research Articles

The impact of diffusion on receptor binding during synaptic transmission

Despite the importance of speed in synaptic transmission, in many synapses, neurotransmitters bind to their receptors at rates that appear to be slower than the diffusion limit. This assessment is generally based on a comparison with the Smoluchowski limit rather than an independent experimental analysis. In many synapses, miniature excitatory postsynaptic currents (mEPSCs) are controlled by the interplay between binding to receptors and diffusion of the neurotransmitter out of the synaptic cleft. A model for mEPSCs that incorporates these features was used to evaluate published data showing that elevated viscosity increases mEPSC amplitude. With diffusion-limited binding, the model predicts that raising the viscosity will decrease the amplitude rather than increase it. Diffusion-independent binding predicts an increase that is larger than that observed. To explore the intermediate behavior between the diffusion-limited and diffusion-independent extremes, a general expression for intermolecular rates was used that depends on both collision frequency and intrinsic reactivity. This analysis yielded an estimate for collision frequency that is about an order of magnitude above the measured rate of association and an order of magnitude below the Smoluchowski limit.

Gravitational collision efficiencies of small viscous drops at finite Stokes numbers and low Reynolds numbers

Collision efficiencies are calculated for two sedimenting spherical drops of different radii with internal circulation under conditions such that drop inertia can be significant while the surrounding fluid inertia remains negligible. A trajectory analysis is employed with exact methods for determining the hydrodynamic forces. When the Reynolds number is small, fluid inertia is negligible, and the hydrodynamic forces are linear functions of the velocities of the drops. However, at nonzero Stokes numbers, drop inertia must be taken into account, and the hydrodynamic forces do not balance the applied forces. For drops in close approach, lubrication forces, slip and attractive molecular forces are considered. Comparison is made between the effects of unretarded and retarded van der Waals forces and Maxwell slip on collision efficiencies. Slip is calculated exactly for viscous drops using bispherical coordinates for motion along the drops’ line of centers. For water droplets in air, at drop radii between 10 and 30μm, drop inertia is important while the Reynolds based on the surrounding air is still small. The collision efficiency goes through a minimum and then approaches the Smoluchowski limit of no hydrodynamic interactions as the drop size and Stokes number become increasingly large. Viscous effects are most important at drop radii of about 15μm. For drops with a radius less than 10μm, the most significant factors are unretarded van der Waals forces, Maxwell slip, and retarded van der Waals forces, respectively. When van der Waals forces and slip are combined, the collision efficiencies for drops with radii of 10μm are significantly greater than the results of previous researchers. At an intermediate drop radius of 20μm, surrounding fluid inertia, as considered in earlier work, appears to lead to greater collision efficiencies at intermediate size ratios, while van der Waals forces, incorporated in the present study, cause larger values at smaller and larger size ratios. At a drop radius of 30μm, drop inertia dominates interactions, and similar results occur for both the current and prior investigations. Since temperature affects both density and viscosity, collision efficiencies can differ by a factor of two or more between 0 °C and 100 °C at certain drop radii. Change in the viscosity ratio itself leads to about a 7% difference in the collision efficiency between drops with internal circulation and solid spheres at room temperature.

Facilitating target search in polymer networks: Effects of target size and mixed one-dimensional and three-dimensional diffusion.

We theoretically investigate the problem of diffusive target search and mean first passage times (MFPTs) of a tracer in a three-dimensional (3D) polymer network with a particular focus on the effects of combined one-dimensional (1D) diffusion along the polymer chains and 3D diffusion within the network. For this, we employ computer simulations as well as limiting theories of a single diffusive tracer searching for a spherical target fixed at a cross-link of a homogeneous 3D cubic lattice network. The free parameters are the target size, the ratio of the 1D and 3D friction constants, and the transition probabilities between bound and unbound states. For a very strongly bound tracer on the chains, the expected predominant set of 1D lattice diffusion (LD) is found. The MFPT in the LD process significantly depends on the target size, yielding two distinct scaling behaviors for target sizes smaller and larger than the network mesh size, respectively. In the limit of a pointlike target, the LD search becomes a random walk process on the lattice, which recovers the analytical solution for the MFPT previously reported by S. Condamin, O. Bénichou, and M. Moreau [Phys. Rev. Lett. 95, 260601 (2005)PRLTAO0031-900710.1103/PhysRevLett.95.260601]. For the very weakly bound tracer, the expected 3D free diffusion (FD) dominates, extrapolating to the well-known Smoluchowski limit. A critical target size is found above which the MFPT in the FD process is faster than in the LD process. For intermediate binding, i.e., a combination of LD and FD processes, the target search time can be minimized for an optimal range of target sizes and partitions between FD and LD, for which the MFPTs are substantially faster when compared to the limiting FD or LD processes. Our study may provide a theoretical basis to better understand and predict search and reaction processes in complex structured materials, thereby contributing to practical applications such as designing nanoreactors where catalytic targets are immobilized in polymer networks.

Guided active particles.

To account for the possibility of an externally driven taxis in active systems, we develop a model of a guided active drift which relies on the presence of an external guiding field and a vectorial coupling between the mechanical degrees of freedom and a chemical reaction. To characterize the ability of guided active particles to carry cargo, we generalize the notion of Stokes efficiency extending it to the case of stall conditions. To show the generality of the proposed mechanism, we discuss guided electric circuits capable of turning fluctuations into a directed current without a source of voltage.

Electrokinetics of nanoparticle gel-electrophoresis.

Gel-electrophoresis has been demonstrated in recent decades to successfully sort a great variety of nanoparticles according to their size, charge, surface chemistry, and corona architecture. However, quantitative theoretical interpetations have been limited by the number and complexity of factors that influence particle migration. Theoretical models have been fragmented and incomplete with respect to their counterparts for free-solution electrophoresis. This paper unifies electrokinetic models that address complex nanoparticle corona architectures, corona and gel charge regulation (e.g., by the local pH), multi-component electrolytes, and non-linear electrostatics and relaxation effects. By comprehensively addressing the electrokinetic aspects of the more general gel-electrophoresis problem, in which short-ranged steric interactions are significant, a stage is set to better focus on the physicochemical and steric factors. In this manner, it is envisioned that noparticle gel-electrophoresis may eventually be advanced from a nanoparticle-characterization tool to one that explicitly probes the short-ranged interactions of nanoparticles with soft networks, such as synthetic gels and biological tissues. In this paper, calculations are undertaken that identify a generalized Hückel limit for nanoparticles in low-conductivity gels, and a new Smoluchowski limit for polyelectrolyte-coated particles in high-conductivity gels that is independent of the gel permeability. Also of fundamental interest is a finite, albeit small, electrophoretic mobility for uncharged particles in charged gels. Electrophoretic mobilities and drag coefficients (with electroviscous effects) for nanoparticles bearing non-uniform coronas show that relaxation effects are typically weak for the small nanoparticles (radius ≈3-10 nm) to which gel-electrophoresis has customarily been applied, but are profound for the larger nanoparticles (radius ≳ 40 nm in low conductivity gels) to which passivated gel-electrophoresis experiments have recently been applied. To demonstrate its practical application, the model is applied to (pH charge regulating) carboxylated polystyrene nanospheres in low-density passivated agarose gels (weak steric effects). This furnishes a new theoretical interpretation of literature data for which a finite diffuse-layer-thickness, pH-charge regulation, high charge, and relaxation effects dominate over the steric influences.

Statistical mechanics of holonomic systems as a Brownian motion on smooth manifolds

The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein‐Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology.

Energy conversion by surface-tension-driven charge separation

In this work, the shear-induced electrokinetic streaming potential present in free-surface electrolytic flows subjected to a gradient in surface tension is assessed. Firstly, for a Couette flow with fully resolved electric double layer (EDL), the streaming potential per surface stress as a function of the Debye parameter and surface potential is analyzed. By contrast to the Smoluchowski limit in pressure-driven channel flow, the shear-induced streaming potential vanishes for increasing Debye parameter (infinitely thin EDL), unless the free surface contains (induced) surface charge or the flow at the charged, solid wall is permitted to slip. Secondly, a technical realization of surface-tension induced streaming is proposed, with surface stress acting on the free (slipping) surfaces of a micro-structured, superhydrophobic wall. The streaming potential is analyzed with respect to the slip parameter and surface charge. Finally, the surface tension is assumed to vary with temperature (thermocapillarity) or with surfactant concentration (destillocapillarity). The maximal thermal efficiency is derived and compared to the Carnot efficiency. For large thermal Marangoni number, the efficiency is severely limited by the large heat capacity of aqueous solvents. By contrast, destillocapillary flows may reach conversion efficiencies similar to pressure-driven flow.

Electroosmotic Flow Through a Circular Tube With Slip-Stick Striped Wall

This is an analytical study on electrohydrodynamic flows through a circular tube, of which the wall is micropatterned with a periodic array of longitudinal or transverse slip-stick stripes. One unit of the wall pattern comprises two stripes, one slipping and the other nonslipping, and each with a distinct ζ potential. Using the methods of eigenfunction expansion and point collocation, the electric potential and velocity fields are determined by solving the linearized Poisson–Boltzmann equation and the Stokes equation subject to the mixed electrohydrodynamic boundary conditions. The effective equations for the fluid and current fluxes are deduced as functions of the slipping area fraction of the wall, the intrinsic hydrodynamic slip length, the Debye parameter, and the ζ potentials. The theoretical limits for some particular wall patterns, which are available in the literature only for plane channels, are extended in this paper to the case of a circular channel. We confirm that some remarks made earlier for electroosmotic flow over a plane surface are also applicable to the present problem involving patterns on a circular surface. We pay particular attention to the effects of the pattern pitch on the flow in both the longitudinal and transverse configurations. When the wall is uniformly charged, the adverse effect on the electroosmotic flow enhancement due to a small fraction of area being covered by no-slip slots can be amplified if the pitch decreases. Reducing the pitch will also lead to a greater deviation from the Helmholtz–Smoluchowski limit when the slipping regions are uncharged. With oppositely charged slipping regions, local recirculation or a net reversed flow is possible, even when the wall is on the average electropositive or neutral. The flow morphology is found to be subject to the combined influence of the geometry of the tube and the electrohydrodynamic properties of the wall.

Dielectric-solid polarization at strong fields: Breakdown of Smoluchowski's electrophoresis formula

We investigate the thin-double-layer electrophoretic drift of a uniformly charged dielectric particle, driven by an intense electric field comparable to the transverse Debye-layer field. Under these circumstances, solid polarization affects the leading-order electrokinetic transport in the fluid by inducing a nonuniform zeta-potential distribution. The resulting expression for the particle velocity is accordingly nonlinear in the applied field. The electrophoretic “mobility”—the ratio of this velocity and the applied field—depends upon two parameters, the first quantifying the surface-charge density, and the second constituting the product of the solid-to-liquid permittivity ratio and the scaled applied-field magnitude. At weak values of this product, solid polarization results in field-cubed deviations from Smoluchowski's velocity; at large values of it, the particle velocity is a slowly increasing function of the applied field, essentially varying with its logarithm. The transition between these two limits features a shift from zeta-potential proportionality to a charge-density proportionality. For all values of the two governing parameters solid polarization acts so as to reduce the electrophoretic velocity relative to the Smoluchowski limit.

Enhancement of transport properties of a Brownian particle due to quantum effects: Smoluchowski limit

The transport property of a quantum Brownian particle that interacts strongly with a bath (in which a typical damping constant by far exceeds a characteristic frequency of the isolated system) under the influence of a tilted periodic potential has been studied by solving quantum Smoluchowski equation (QSE). By invoking physically motivated coordinate transformation into QSE, we have presented a transparent treatment for the determination of the effective diffusion coefficient of a quantum Brownian particle and the current (the average stationary velocity). Substantial enhancement in the efficiency of the diffusive transport is envisaged due to the quantum correction effects only if the bath temperature hovers around an appropriate range of intermediate values. Our findings also confirm the results obtained in the classical cases.

Rate, affinity and calcium dependence of nitric oxide synthase isoform binding to the primary physiological regulator calmodulin

Using interferometry-based biosensors the binding and release of endothelial and neuronal nitric oxide synthase (eNOS and nNOS) from calmodulin (CaM) was measured. In both isoforms, binding to CaM is diffusion limited and within approximately three orders of magnitude of the Smoluchowski limit imposed by orientation-independent collisions. This suggests that the orientation of CaM is facilitated by the charge arrays on the CaM-binding site and the complementary surface on CaM. Protein kinase C phosphorylation of eNOS T495, adjacent to the CaM-binding site, abolishes or greatly slows CaM binding. Kinases which increase the activity of eNOS did not stimulate the binding of CaM, which is already diffusion limited. The coupling of Ca(2+) binding and CaM/NOS binding equilibria links the affinity of CaM for NOS to the Ca(2+) dependence of CaM binding. Hence, changes in the Ca(2+) sensitivity of CaM binding always imply changes in the NOS-CaM affinity. It is possible, however, that in some regimes binding and activation are not synonymous, so that Ca(2+) sensitivity need not be tightly linked to CaM sensitivity of activation. This study is being extended using mutants to probe the roles of individual structural elements in binding and release.

Gravity-induced collisions of spherical drops covered with compressible surfactant

Gravity-induced collisions of two spherical drops covered with an insoluble surfactant at low Reynolds numbers are considered. Unlike in previous collision studies, the present work accounts for nonlinear coupling between the surfactant distribution and drop hydrodynamics by solving the full unsteady convective–diffusion equation for the surfactant transport. Our method includes high-order three-dimensional multipole expansions for hydrodynamics and a Galerkin-type approach for the surfactant transport with implicit marching. The efficiency of the algorithm allows for calculating thousands of trajectories to very close contact and determining the collision efficiency (related to the critical initial horizontal offset) by trial and error. The solution is valid for arbitrary surface Péclet (Pes) and Marangoni (Ma) numbers and sets limitations on approximations used in prior work for collision-efficiency calculations. Two limiting cases are observed: at smallPesor largeMa, the variation in surfactant coverage is small, and the results for the incompressible surfactant model are recovered, while for largePesand smallMa, the collision efficiency approaches the clean-interface value. For moderate drop-size ratios (radius ratiok≤ 0.5), the results generally fall between these limits. At larger size ratios, however, the collision efficiency may even exceed the geometrical Smoluchowski limit for both drops and bubbles. Moreover, with even moderate redistribution of the surfactant, equal-sized drops can move relative to one another and collide. These novel effects do not exist for clean drops or drops covered with an incompressible surfactant, and they are due to the nonlinear coupling between surfactant dynamics and flow. This surfactant-enhanced coalescence takes place, for example, in a physical system of air bubbles in water if the surfactant surface concentration is dilute (Γ ≈ 1×10−9mol m−2, much smaller than the typical maximum-packing value of 10−5−10−6mol m−2).

Realization of Lévy walks as Markovian stochastic processes

Based on multivariate Langevin processes we present a realization of Lévy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity-dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Lévy flights. Our procedure is similar to the treatment of the Kramers-Fokker-Planck equation in the Smoluchowski limit. The proposed approach may open a way to treat Lévy flights in inhomogeneous media or systems with boundaries in the future.

Dynamics of Barrierless and Activated Chemical Reactions in a Dispersive Medium within the Fractional Diffusion Equation Approach

Barrierless chemical reactions have often been modeled as a Brownian motion on a one-dimensional harmonic potential energy surface with a position-dependent reaction sink or window located near the minimum of the surface. This simple (but highly successful) description leads to a nonexponential survival probability only at small to intermediate times but exponential decay in the long-time limit. However, in several reactive events involving proteins and glasses, the reactions are found to exhibit a strongly nonexponential (power law) decay kinetics even in the long time. In order to address such reactions, here, we introduce a model of barrierless chemical reaction where the motion along the reaction coordinate sustains dispersive diffusion. A complete analytical solution of the model can be obtained only in the frequency domain, but an asymptotic solution is obtained in the limit of long time. In this case, the asymptotic long-time decay of the survival probability is a power law of the Mittag-Leffler functional form. When the barrier height is increased, the decay of the survival probability still remains nonexponential, in contrast to the ordinary Brownian motion case where the rate is given by the Smoluchowski limit of the well-known Kramers' expression. Interestingly, the reaction under dispersive diffusion is shown to exhibit strong dependence on the initial state of the system, thus predicting a strong dependence on the excitation wavelength for photoisomerization reactions in a dispersive medium. The theory also predicts a fractional viscosity dependence of the rate, which is often observed in the reactions occurring in complex environments.

A novel method of measuring electrophoretic mobility of gas bubbles

Accurate measurement of electrophoretic mobility for gas bubbles is a challenging task as it requires the creation of a desired number of very small air bubbles to ensure negligible rise velocities during the course of the measurement. Here, we report a simple and reliable method for generating stable dispersions of “nano-bubbles.” Preparation of such dispersions relies on the nucleation of nano-bubbles in solutions supersaturated with gas. Electrophoretic mobility of these nano-bubbles is determined by the ZetaPALS technique (Brookhaven Instruments) using Uzgiris electrodes coated with palladium. The Smoluchowski limit is assumed in the calculation of zeta potentials. In regard to reproducibility and reliability, this novel method shows a clear advantage over other existing techniques of zeta potential measurement for bubbles.

Dissipative curve crossing problem. I. High-barrier crossing.

Variational approach based on the diabatic variational transition state theory optimization is extended and applied to the calculation of the radiationless transition rate for asymmetric high-barrier normal and inverted crossings. Scaling argument is employed to derive an analytic expression for the renormalized frequency as a function of the asymmetry of the crossing and of the dissipation strength for the particular case of Ohmic dissipation (Debye relaxation). The effect of the asymmetry of the crossing on the physical parameters and on the radiationless transition rate is explored. The effective adiabaticity parameter increases with the increasing asymmetry of the crossing in the normal region and decreases with the increasing asymmetry in the inverted region. The physical behavior of the radiationless transition rate in the normal and inverted regions is qualitatively different in the strong dissipation (Smoluchowski) limit. In the inverted region the rate exhibits stretched exponential decay as a function of the dissipation strength reflecting adiabatic suppression in the strong dissipation limit. The accuracy of the approach decreases with the increasing (decreasing) asymmetry of the crossing in the normal (inverted) region and breaks down in the activationless case.

Conformational Dynamics of Hyaluronan Oligomers in Solution. 3. Molecular Dynamics from Monte Carlo Replica-Exchange Simulations and Mode-Coupling Diffusion Theory

We present the calculation of dynamic properties of the hyaluronan oligomer (UA)4 based on the second-order solution of the diffusion equation in the Smoluchowski limit. The equilibrium averages necessary for the solution are calculated from replica-exchange Monte Carlo simulations in implicit water models. This simulation technique was shown to avoid the trapping of the configurations in local minima typical of simulations in the canonical statistical ensemble. A simple implicit solvent model was applied to (UA)4 taking into account the hydrophobic effect in water solution and hydrophilic local effects due to efficient hydrogen bonds with the solvent. The correction of the statistics due to a finite difference solution of the Poisson equation, not directly included in the simulation, was included through a perturbative method. Several configurational distributions and dynamical parameters related to nuclear magnetic relaxation, sensitive both to the molecular structure and to mobility, were calculated from the replica- exchange Monte Carlo statistics at different temperatures. These data compare fairly well with experiments and with the results obtained by standard molecular dynamics simulation in the explicit water solvent and in the canonical ensemble at T ) 300 K. Implications in the macromolecular dynamics of hyaluronan were discussed.

Sequence specific recognition of ssDNA by a lupus autoantibody: kinetics and mechanism of binding

11F8 is a pathogenic anti-ssDNA monoclonal autoantibody isolated from a lupus-prone mouse. Previous studies have established that 11F8 is sequence specific. To determine the basis for the observed binding specificity, stopped-flow fluorescence spectroscopy was used to measure the kinetic parameters and establish the mechanisms for the association of 11F8 with its target sequence, noncognate, and nonspecific ssDNA ligands. The data revealed that sequence-specific binding follows a two-step mechanism where the initial association step is second order. Values of k 1 are fast and above the modified Smoluchowski limit for a diffusion limited interaction (10 5–10 6 M −1 s −1). The dependency of k 1 on [salt] and solvent polarity indicates that electrostatic steering is responsible for this rapid association rate. The second association step is rate limiting and is characteristic of an isomerization process during which binding interfaces are optimized. This step apparently is driven by the desolvation of hydrophobic surfaces within the binding interface. The differences in the rate of dissociation for the various DNA ligands suggest that specificity is governed primarily through the dissociation of the final complexes.

Strong friction limit in quantum mechanics: the quantum Smoluchowski equation.

For a quantum system coupled to a heat bath environment the strong friction limit is studied starting from the exact path integral formulation. Generalizing the classical Smoluchowski limit to low temperatures, a time evolution equation for the position distribution is derived and the strong role of quantum fluctuations in this limit is revealed.

Multi-extreme behavior and giant suppression plateau of activation rate in a periodic system driven by correlated noise

An analytical investigation of the activation rate for Brownian particle in a periodic system driven by cross-correlated Gaussian white noises is presented. As used in the literature the activation rate κ is defined as the inverse of the mean first-passage time and the relative activation rate ν= κ/ κ s in which κ s indicates the rate in the Smoluchowski limit, the limit of the multiplicative to additive noise strength ratio R=0. The center idea of this Letter is to construct an exact soluble model in order to exhibit the possible new effect of the cross-correlated Gaussian white noises on the activation processes. The model clearly shows the following novel activation phenomena. (1) There is one and only extreme (minimum) in the ν∼ R curve for the case of positive correlation. However, for independent noises case the ν∼ R curve is monotonously varied. In other words, the positive correlation leads to the suppression of the activation rate. (2) When the two noises are negative correlative, the ν∼ R curves exist as three types of shape, that is, the non-extreme, the two-extreme, and the four-extreme shape. (3) In the intense negative correlation region, the ν∼ R curve experiences a re-entrant transition between the different types of shape. (4) The ν∼ R curve exhibits a giant suppression plateau when the two noises are intense positive correlation. The mechanism for the appearance of this giant suppression plateau is discussed.