Barrier crossing in a viscoelastic medium under active noise: Predictions of Kramers' flux-over-population method.

The biochemical activity inside a cell has recently been suggested to act as a source of hydrodynamic fluctuations that can speed up or slow down enzyme catalysis [Tripathi et al., Commun. Phys. 5, 101 (2022).] The idea has been tested against and largely corroborated by simulations of activated barrier crossing in a simple fluid in the presence of thermal and athermal noise. The present paper attempts a wholly analytic solution to the same noise-driven barrier crossing problem but generalizes it to include viscoelastic memory effects of the kind likely to be present in cellular interiors. A calculation of the model's barrier crossing rate, using Kramers' flux-over-population formalism, reveals that in relation to the case where athermal noise is absent, athermal noise always accelerates barrier crossing, though the extent of enhancement depends on the duration τ0 over which the noise acts. More importantly, there exists a critical τ0-determined by the properties of the medium-at which Kramers' theory breaks down and, on approach to which, the rate grows significantly. The possibility of such a giant enhancement is potentially open to experimental validation using optically trapped nanoparticles in viscoelastic media that are acted on by externally imposed colored noise.

Internal friction as a factor in the anomalous chain length dependence of DNA transcriptional dynamics.

Recent experiments by Brückner et al. [Science 380, 1357 (2023)] have observed an anomalous chain length dependence of the time of near approach of widely separated pairs of genomic elements on transcriptionally active chromosomal DNA. In this paper, I suggest that the anomaly may have its roots in internal friction between neighboring segments on the DNA backbone. The basis for this proposal is a model of chain dynamics formulated in terms of a continuum scaled Brownian walk (sBw) of polymerization index N. The sBw is an extension of the simple Brownian walk model widely used in path integral calculations of polymer properties, differing from it in containing an additional parameter H (the Hurst index) that can be tuned to produce varying degrees of correlation between adjacent monomers. A calculation using the sBw of the mean time τc for chain closure predicts-under the Wilemski-Fixman approximation for diffusion-controlled reactions-that at early times, τc varies as the 2/3 power of N, in close agreement with the findings of the Brückner et al. study. Other scaling relations of that study, including those related to the probability of loop formation and the mean square displacements of terminal monomers, are also satisfactorily accounted for by the model.

Survival probabilities and first-passage distributions of self-propelled particles in spherical cavities.

A model of self-propelled motion in a closed compartment containing simple or complex fluids is formulated in this paper in terms of the dynamics of a point particle moving in a spherical cavity under the action of random thermal forces and exponentially correlated noise. The particle's time evolution is governed by a generalized Langevin equation (GLE) in which the memory function, connected to the thermal forces by a fluctuation-dissipation relation, is described by Jeffrey's model of viscoelasticity (which reduces to a model of ordinary viscous dynamics in a suitable limit). The GLE is transformed exactly to a Fokker-Planck equation that in spherical polar coordinates is in turn found to admit of an exact solution for the particle's probability density function under absorbing boundary conditions at the surface of the sphere. The solution is used to derive an expression (that is also exact) for the survival probability of the particle in the sphere, starting from its center, which is then used to calculate the distribution of the particle's first-passage times to the boundary. The behavior of these quantities is investigated as a function of the Péclet number and the persistence time of the athermal forces, providing insight into the effects of nonequilibrium fluctuations on confined particle motion in three dimensions.

Effects of Hydrodynamic Backflow on the Transmission Coefficient of a Barrier-Crossing Brownian Particle.

The slow power law decay of the velocity autocorrelation function of a particle moving stochastically in a condensed-phase fluid is widely attributed to the momentum that fluid molecules displaced by the particle transfer back to it during the course of its motion. The forces created by this backflow effect are known as Basset forces, and they have been found in recent analytical work and numerical simulations to be implicated in a number of interesting dynamical phenomena, including boosted particle mobility in tilted washboard potentials. Motivated by these findings, the present paper is an investigation of the role of backflow in thermally activated barrier crossing, the governing process in essentially all condensed-phase chemical reactions. More specifically, it is an exact analytical calculation, carried out within the framework of the reactive-flux formalism, of the transmission coefficient κ(t) of a Brownian particle that crosses an inverted parabola under the influence of a colored noise process originating in the Basset force and a Markovian time-local friction. The calculation establishes that κ(t) is significantly enhanced over its backflow-free limit.

Statistical Dynamics of Flow-Driven Globular Polymers.

The response of collapsed polymers to the effects of linear mixed flow is studied theoretically in this paper using a model of a self-interacting finitely extensible Gaussian chain that evolves stochastically in the presence of random thermal fluctuations and an external fluid velocity gradient. The interactions that produce compact chain configurations are described by a harmonic pair potential of strength κ that acts between nonbonded sites on the chain backbone. Several chain properties are calculated analytically from this model as a function of κ for elongational and shear flows, including the dependence of the chain's steady-state mean-square end-to-end distance on the Weissenberg number of the flow, the time-dependence of the chain's relaxation to equilibrium from a steady-state of given chain extension, and the nature of the force-extension curves that are obtained from the free energy change between unperturbed and flow-stretched states of the chain. For both elongational and shear flows (but to different degrees), it is found that the greater the value of κ (and the more compact the chain), the more difficult it is, in general, for the imposed flow to induce a transition between compact and extended states, in broad agreement with available data from numerical simulations. For the relaxation process, the differences between the two flow types are more marked. The characteristic decay time for relaxation from a state prepared by elongational flow is essentially independent of κ, whereas in the case of a state prepared by shear flow, it is distinctly κ-dependent, the relaxation becoming faster at larger κ.

The relaxation dynamics of single flow-stretched polymers in semidilute to concentrated solutions.

Recent experiments on the return to equilibrium of solutions of entangled polymers stretched by extensional flows [Zhou and Schroeder, Phys. Rev. Lett. 120, 267801 (2018)] have highlighted the possible role of the tube model's two-step mechanism in the process of chain relaxation. In this paper, motivated by these findings, we use a generalized Langevin equation (GLE) to study the time evolution, under linear mixed flow, of the linear dimensions of a single finitely extensible Rouse polymer in a solution of other polymers. Approximating the memory function of the GLE, which contains the details of the interactions of the Rouse polymer with its surroundings, by a power law defined by two parameters, we show that the decay of the chain's fractional extension in the steady state can be expressed in terms of a linear combination of Mittag-Leffler and generalized Mittag-Leffler functions. For the special cases of elongational flow and steady shear flow, and after adjustment of the parameters in the memory function, our calculated decay curves provide satisfactory fits to the experimental decay curves from the work of Zhou and Schroeder and earlier work of Teixeira et al. [Macromolecules 40, 2461 (2007)]. The non-exponential character of the Mittag-Leffler functions and the consequent absence of characteristic decay constants suggest that melt relaxation may proceed by a sequence of steps with an essentially continuous, rather than discrete, spectrum of timescales.

Particle dynamics in viscoelastic media: Effects of non-thermal white noise on barrier crossing rates.

The growing interest in the dynamics of self-driven particle motion has brought increased attention to the effects of non-thermal noise on condensed phase diffusion. Thanks to data recently collected by Ferrer et al. on activated dynamics in the presence of memory [Phys. Rev. Lett. 126, 108001 (2021)], some of these effects can now be characterized quantitatively. In the present paper, the data collected by Ferrer et al. are used to calculate the extent to which non-thermal white noise alters the time taken by single micron-sized silica particles in a viscoelastic medium to cross the barrier separating the two wells of an optically created bistable potential. The calculation-based on a generalized version of Kramers's flux-over-population approach-indicates that the added noise causes the barrier crossing rate (compared to the noise-free case) to first increase as a function of the noise strength and then to plateau to a constant value. The precise degree of rate enhancement may depend on how the data from the experiments conducted by Ferrer et al. are used in the flux-over-population approach. As claimed by Ferrer et al., this approach predicts barrier crossing times for the original silica-fluid system that agree almost perfectly with their experimental counterparts. However, this near-perfect agreement between theory and experiment is only achieved if the theoretical crossing times are obtained from the most probable values of a crossing time distribution constructed from the distributions of various parameters in Kramers's rate expression. If the mean values of these parameters are used in the expression instead, as would be commonly done, the theoretical crossing times are found to be as much as 1.5 times higher than the experimental values. However, these times turn out to be consistent with an alternative model of viscoelastic barrier crossing based on a mean first passage time formalism, which also uses mean parameter values in its rate expression. The rate enhancements predicted for barrier crossing under non-thermal noise are based on these mean parameter values and are open to experimental verification.

Stochastic thermodynamics of a harmonically trapped colloid in linear mixed flow.

In this paper, motivated by a general interest in the stochastic thermodynamics of small systems, we derive an exact expression-via path integrals-for the conditional probability density of a two-dimensional harmonically confined Brownian particle acted on by linear mixed flow. This expression is a generalization of the expression derived earlier by Foister and Van De Ven [J. Fluid Mech. 96, 105 (1980)10.1017/S0022112080002042] for the case of the corresponding free Brownian particle, and reduces to it in the appropriate unconfined limit. By considering the long-time limit of our calculated probability density function, we show that the flow-driven Brownian oscillator attains a well-defined steady state. We also show that, during the course of a transition from an initial flow-free thermal equilibrium state to the flow-driven steady state, the integral fluctuation theorem, the Jarzynski equality, and the Bochkov-Kuzovlev relation are all rigorously satisfied. Additionally, for the special cases of pure rotational flow we derive an exact expression for the distribution of the heat dissipated by the particle into the medium, and for the special case of pure elongational flow we derive an exact expression for the distribution of the total entropy change. Finally, by examining the system's stochastic thermodynamics along a reverse trajectory, we also demonstrate that in elongational flow the total entropy change satisfies a detailed fluctuation theorem.

The effects of slit-like confinement on flow-induced polymer deformation.

This paper is broadly concerned with the dynamics of a polymer confined to a rectangular slit of width D and deformed by a planar elongational flow of strength γ̇. It is interested, more specifically, in the nature of the coil-stretch transition that such polymers undergo when the flow strength γ̇ is varied, and in the degree to which this transition is affected by the presence of restrictive boundaries. These issues are explored within the framework of a finitely extensible Rouse model that includes pre-averaged surface-mediated hydrodynamic interactions. Calculations of the chain's steady-state fractional extension x using this model suggest that different modes of relaxation (which are characterized by an integer p) exert different levels of control on the coil-stretch transition. In particular, the location of the transition (as identified from the graph of x versus the Weissenberg number Wi, a dimensionless parameter defined by the product of γ̇ and the time constant τp of a relaxation mode p) is found to vary with the choice of τp. In particular, when τ1 is used in the definition of Wi, the x vs. Wi data for different D lie on a single curve, but when τ3 is used instead (with τ3 > τ1) the corresponding data lie on distinct curves. These findings are in close qualitative agreement with a number of experimental results on confinement effects on DNA stretching in electric fields. Similar D-dependent trends are seen in our calculated force vs. Wi data, but force vs. x data are essentially D-independent and lie on a single curve.

Non-Gaussian Brownian Diffusion in Dynamically Disordered Thermal Environments.

In this article, we suggest simple alternatives to the methods recently used by Jain and Sebastian [ J. Phys. Chem. B 2016 , 120 , 3988 ] and Chechkin et al. [ Phys. Rev. X 2017 , 7 , 021002 ] to treat a model of non-Gaussian Brownian diffusion based on the dynamics of a particle governed by Ornstein-Uhlenbeck modulated white noise. In addition to substantiating these authors' earlier findings (which show that a particle can execute a simple random walk even when the distribution of its displacements deviates from Gaussianity), our approach identifies another process, two-state white noise, that exhibits the same "anomalous" Brownian behavior. Indeed, we find that the modulation of white noise by any stochastic process whose time correlation function decays exponentially is likely to behave similarly, suggesting that the occurrence of such behavior can be widespread and commonplace.

Polymer extension under flow: Some statistical properties of the work distribution function.

In an extension of earlier studies from this group on the application of the Jarzynski equality to the determination of the elastic properties of a finitely extensible Rouse model of polymers under flow [A. Ghosal and B. J. Cherayil, J. Chem. Phys. 144, 214902 (2016)], we derive several new theoretical results in this paper on the nature of the distribution function P(w) that governs the long-time limit t>>1 of the fluctuations in the work w performed by the polymer during flow-induced stretching. In particular, we show that an expression for the average of the nth power of the work, ⟨wn(t)⟩, can be obtained in closed form in this limit, making it possible to exactly calculate three important statistical measures of P(w): the mean µ, the skewness γ1, and the kurtosis γ2 (apart from the variance σ2). We find, for instance, that to leading order in t, the mean grows linearly with t at a constant value of the dimensionless flow rate Wi and that the slope of the µ-t curve increases with increasing Wi. These observations are in complete qualitative agreement with data from Brownian dynamics simulations of flow-driven double-stranded DNA by Latinwo and Schroeder [Macromolecules 46, 8345 (2013)]. We also find that the skewness γ1 exhibits an interesting inversion of sign as a function of Wi, starting off at positive values at low Wi and changing to negative values at larger Wi. The inversion takes place in the vicinity of what we interpret as a coil-stretch transition. Again, the finding exactly reproduces behavior seen in other numerical and experimental work by the above group Latinwo et al. [J. Chem. Phys. 141, 174903 (2014)]. Additionally, at essentially the same value of Wi at which this sign inversion takes place, we observe that the kurtosis reaches a minimum, close to 1, providing further evidence of the existence of a coil-stretch transition at this location. Our calculations reproduce another numerical finding: a power law dependence on Wi of the rate of work production that is characterized by two distinct regimes, one lying below the putative coil-stretch transition, where the exponent assumes one value, and the other above, where it assumes a second.

Polymer extension under flow: A path integral evaluation of the free energy change using the Jarzynski relation.

The Jarzynski relation (and its variants) has provided a route to the experimental evaluation of equilibrium free energy changes based on measurements conducted under arbitrary non-equilibrium conditions. Schroeder and co-workers [Soft Matter 10, 2178 (2014) and J. Chem. Phys. 141, 174903 (2014)] have recently exploited this fact to determine the elastic properties of model DNA from simulations and experiments of chain extension under elongational flow, bypassing the need to make these measurements mechanically using sophisticated optical trapping techniques. In this paper, motivated by these observations, we investigate chain elasticity analytically, using the Jarzynski relation and a finitely extensible nonlinear elastic-type Rouse model within a path integral formalism to calculate (essentially exactly) both the flow-induced free energy change between chain conformations of definite average end-to-end distance, as well as the force-extension curve that follows from it. This curve, based on a new analytic expression, matches the trends in the corresponding curve obtained from a model of chain stretching developed by Marko and Siggia [Macromolecules 28, 8759 (1995)], which itself is in very satisfactory agreement with the numerical and experimental data from the work of Schroeder et al.

Dynamics of the reaction between the free end of a tethered self-avoiding polymer and a flat penetrable surface: a renormalization group study.

The average time τr for one end of a long, self-avoiding polymer to interact for the first time with a flat penetrable surface to which it is attached at the other end is shown here to scale essentially as the square of the chain's contour length N. This result is obtained within the framework of the Wilemski-Fixman approximation to diffusion-limited reactions, in which the reaction time is expressed as a time correlation function of a "sink" term. In the present work, this sink-sink correlation function is calculated using perturbation expansions in the excluded volume and the polymer-surface interactions, with renormalization group methods being used to resum the expansion into a power law form. The quadratic dependence of τr on N mirrors the behavior of the average time τc of a free random walk to cyclize, but contrasts with the cyclization time of a free self-avoiding walk (SAW), for which τr ∼ N(2.2). A simulation study by Cheng and Makarov [J. Phys. Chem. B 114, 3321 (2010)] of the chain-end reaction time of an SAW on a flat impenetrable surface leads to the same N(2.2) behavior, which is surprising given the reduced conformational space a tethered polymer has to explore in order to react.

The diffusion and relaxation of Gaussian chains in narrow rectangular slits.

The confinement of a polymer to volumes whose characteristic linear dimensions are comparable to or smaller than its bulk radius of gyration R(G,bulk) can produce significant changes in its static and dynamic properties, with important implications for the understanding of single-molecule processes in biology and chemistry. In this paper, we present calculations of the effects of a narrow rectangular slit of thickness d on the scaling behavior of the diffusivity D and relaxation time τr of a Gaussian chain of polymerization index N and persistence length l0. The calculations are based on the Rouse-Zimm model of chain dynamics, with the pre-averaged hydrodynamic interaction being obtained from the solutions to Stokes equations for an incompressible fluid in a parallel plate geometry in the limit of small d. They go beyond de Gennes' purely phenomenological analysis of the problem based on blobs, which has so far been the only analytical route to the determination of chain scaling behavior for this particular geometry. The present model predicts that D ∼ dN(-1)ln (N∕d(2)) and τr ∼ N(2)d(-1)[ln (N∕d(2))](-1) in the regime of moderate confinement, where l0 ≪ d < R(G,bulk). The corresponding results for the blob model have exactly the same power law behavior, but contain no logarithmic corrections; the difference suggests that segments within a blob may actually be partially draining and not non-draining as generally assumed.

Interfacial growth as a model of tube-width heterogeneities in concentrated solutions of stiff polymers.

Recent experimental measurements of the distribution P(w) of transverse chain fluctuations w in concentrated solutions of F-actin filaments [B. Wang, J Guan, S. M. Anthony, S. C. Bae, K. S. Schweizer, and S. Granick, Phys. Rev. Lett. 104, 118301 (2010); J. Glaser, D. Chakraborty, K. Kroy, I. Lauter, M. Degawa, N. Kirchgessner, B. Hoffmann, R. Merkel, and M. Giesen, Phys. Rev. Lett. 105, 037801 (2010)] are shown to be well-fit to an expression derived from a model of the conformations of a single harmonically confined weakly bendable rod. The calculation of P(w) is carried out essentially exactly within a path integral approach that was originally applied to the study of one-dimensional randomly growing interfaces. Our results are generally as successful in reproducing experimental trends as earlier approximate results obtained from more elaborate many-chain treatments of the confining tube potential.

Subdiffusion in hair bundle dynamics: the role of protein conformational fluctuations.

The detection of sound signals in vertebrates involves a complex network of different mechano-sensory elements in the inner ear. An especially important element in this network is the hair bundle, an antenna-like array of stereocilia containing gated ion channels that operate under the control of one or more adaptation motors. Deflections of the hair bundle by sound vibrations or thermal fluctuations transiently open the ion channels, allowing the flow of ions through them, and producing an electrical signal in the process, eventually causing the sensation of hearing. Recent high frequency (0.1-10 kHz) measurements by Kozlov et al. [Proc. Natl. Acad. Sci. U.S.A. 109, 2896 (2012)] of the power spectrum and the mean square displacement of the thermal fluctuations of the hair bundle suggest that in this regime the dynamics of the hair bundle are subdiffusive. This finding has been explained in terms of the simple Brownian motion of a filament connecting neighboring stereocilia (the tip link), which is modeled as a viscoelastic spring. In the present paper, the diffusive anomalies of the hair bundle are ascribed to tip link fluctuations that evolve by fractional Brownian motion, which originates in fractional Gaussian noise and is characterized by a power law memory. The predictions of this model for the power spectrum of the hair bundle and its mean square displacement are consistent with the experimental data and the known properties of the tip link.

Chain extension of a confined polymer in steady shear flow.

The growing importance of microfluidic and nanofluidic devices to the study of biological processes has highlighted the need to better understand how confinement affects the behavior of polymers in flow. In this paper we explore one aspect of this question by calculating the steady-state extension of a long polymer chain in a narrow capillary tube in the presence of simple shear. The calculation is carried out within the framework of the Rouse-Zimm approach to chain dynamics, using a variant of a nonlinear elastic model to enforce finite extensibility of the chain, and assuming that the only effect of the confining surface is to modify the pre-averaged hydrodynamic interaction. The results, along with results from the corresponding calculations of finitely extensible versions of both the Rouse and Rouse-Zimm models, are compared with data from experiments on the flow-induced stretching of λ-phage DNA near a non-adsorbing glass surface [L. Fang, H. Hu, and R. G. Larson, J. Rheol. 49, 127 (2005)]. The comparison suggests that close to a surface hydrodynamic screening is significant, and causes the chains to become effectively free-draining.

Confinement and viscoelastic effects on chain closure dynamics.

Chemical reactions inside cells are typically subject to the effects both of the cell's confining surfaces and of the viscoelastic behavior of its contents. In this paper, we show how the outcome of one particular reaction of relevance to cellular biochemistry--the diffusion-limited cyclization of long chain polymers--is influenced by such confinement and crowding effects. More specifically, starting from the Rouse model of polymer dynamics, and invoking the Wilemski-Fixman approximation, we determine the scaling relationship between the mean closure time t(c) of a flexible chain (no excluded volume or hydrodynamic interactions) and the length N of its contour under the following separate conditions: (a) confinement of the chain to a sphere of radius d and (b) modulation of its dynamics by colored Gaussian noise. Among other results, we find that in case (a) when d is much smaller than the size of the chain, t(c) ~ Nd(2), and that in case (b), t(c) ~ N(2/(2-2H)), H being a number between 1/2 and 1 that characterizes the decay of the noise correlations. H is not known a priori, but values of about 0.7 have been used in the successful characterization of protein conformational dynamics. At this value of H (selected for purposes of illustration), t(c) ~ N(3.4), the high scaling exponent reflecting the slow relaxation of the chain in a viscoelastic medium.

Distribution of transverse chain fluctuations in harmonically confined semiflexible polymers.

Two different experimental studies of polymer dynamics based on single-molecule fluorescence imaging have recently found evidence of heterogeneities in the widths of the putative tubes that surround filaments of F-actin during their motion in concentrated solution. In one [J. Glaser, D. Chakraborty, K. Kroy, I. Lauter, M. Degawa, N. Kirchesner, B. Hoffmann, R. Merkel, and M. Giesen, Phys. Rev. Lett. 105, 037801 (2010)], the observations were explained in terms of the statistics of a worm-like chain confined to a potential determined self-consistently by a binary collision approximation, and in the other [B. Wang, J. Guan, S. M. Anthony, S. C. Bae, K. S. Schweizer, and S. Granick, Phys. Rev. Lett. 104, 118301 (2010)], they were explained in terms of the scaling properties of a random fluid of thin rods. In this paper, we show, using an exact path integral calculation, that the distribution of the length-averaged transverse fluctuations of a harmonically confined weakly bendable rod (one possible realization of a semiflexible chain in a tube), is in good qualitative agreement with the experimental data, although it is qualitatively different in analytic structure from the earlier theoretical predictions. We also show that similar path integral techniques can be used to obtain an exact expression for the time correlation function of fluctuations in the tube cross section.