Special issue on Marian Smoluchowski's 1916 paper—a century of inspiration

We investigate the statistics of the first detected passage time of a quantum walk. The postulates of quantum theory, in particular the collapse of the wave function upon measurement, reveal an intimate connection between the wave function of a process free of measurements, i.e. the solution of the Schrödinger equation, and the statistics of first detection events on a site. For stroboscopic measurements a quantum renewal equation yields basic properties of quantum walks. For example, for a tight binding model on a ring we discover critical sampling times, diverging quantities such as the mean time for first detection, and an optimal detection rate. For a quantum walk on an infinite line the probability of first detection decays like ${{\left(\text{time}\right)}^{-3}}$ with a superimposed oscillation, critical behavior for a specific choice of sampling time, and vanishing amplitude when the sampling time approaches zero due to the quantum Zeno effect.

100 years after Smoluchowski introduced his approach to stochastic processes, they are now at the basis of mathematical and physical modeling in cellular biology: they are used for example to analyse and to extract features from a large number (tens of thousands) of single molecular trajectories or to study the diffusive motion of molecules, proteins or receptors. Stochastic modeling is a new step in large data analysis that serves extracting cell biology concepts. We review here Smoluchowski's approach to stochastic processes and provide several applications for coarse-graining diffusion, studying polymer models for understanding nuclear organization and finally, we discuss the stochastic jump dynamics of telomeres across cell division and stochastic gene regulation.

We study the current $J_m(\beta U_0)$ of particles to an immobile perfect trap attached to an antenna—a linear array of m partially adsorbing sites with a barrier against desorption U₀ and β being the inverse temperature. Supposing that particles perform standard random walks, in discrete time n, between the nearest-neighbouring sites of an infinite simple cubic lattice, we calculate the current analytically in the limit $n \to \infty$ as a function of m and $\beta U_0$. We find that for each $\beta U_0$, there exists some effective length m^* of the antenna, such that $J_m(\beta U_0)$ is an increasing function of m for m  <  m^*, $J_m(\beta U_0) \sim m/\ln(m)$, and saturates at an m-independent value for m  >  m^*, meaning that only a portion m^*/m of the antenna (which otherwise can be arbitrarily long) effectively enhances the reaction rate. Our analysis is relevant to such practically important situations as, e.g. reactions with the so-called antenna molecules or protein binding to specific sites on a stretched DNA.

In this paper, we present the stochastic foundations of fractional dynamics driven by the fractional material derivative of distributed-order type. Before stating our main result, we present the stochastic scenario which underlies the dynamics given by the fractional material derivative. Then we introduce the Lévy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with the material derivative of distributed-order type.

We consider super-diffusive Lévy walks in $d\geqslant 2$ dimensions when the duration of a single step, i.e. a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d-dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of the central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have a universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called 'infinite density', describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of the PDF of velocity in one step. A typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d-dimensional Lévy distribution) interspersed with long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distributions the d-dimensional process can be reduced to a one-dimensional Lévy walk. We briefly discuss the consequences of non-universality for the d  >  1 dimensional fractional diffusion equation, in particular the non-uniqueness of the fractional Laplacian.

We study a mesoscopic model of a chemically active colloidal particle which on certain parts of its surface promotes chemical reactions in the surrounding solution. For reasons of simplicity and conceptual clarity, we focus on the case in which only electrically neutral species are present in the solution and on chemical reactions which are described by first order kinetics. Within a self-consistent approach we explicitly determine the steady state product and reactant number density fields around the colloid as functionals of the interaction potentials of the various molecular species in solution with the colloid. By using a reciprocal theorem, this allows us to compute and to interpret—in a transparent way in terms of the classical Smoluchowski theory of chemical kinetics—the external force needed to keep such a catalytically active colloid at rest (stall force) or, equivalently, the corresponding velocity of the colloid if it is free to move. We use the particular case of triangular-well interaction potentials as a benchmark example for applying the general theoretical framework developed here. For this latter case, we derive explicit expressions for the dependences of the quantities of interest on the diffusion coefficients of the chemical species, the reaction rate constant, the coverage by catalyst, the size of the colloid, as well as on the parameters of the interaction potentials. These expressions provide a detailed picture of the phenomenology associated with catalytically-active colloids and self-diffusiophoresis.

Diffusion is a ubiquitous phenomenon that impacts virtually all processes that involve random fluctuations, and as such, the foundational work of Smoluchowski has proven to be instrumental in addressing innumerable problems. Here, we focus on a critical biological problem that relies on diffusive transport and is analyzed using a probabilistic treatment originally developed by Smoluchowski. The search of a DNA binding protein for its specific target site is believed to rely on non-specific binding to DNA with transient hops along the chain. In this work, we address the impact of protein crowding along the DNA on the transport of a DNA-binding protein. The crowders dramatically alter the dynamics of the protein while bound to the DNA, resulting in single-file transport that is subdiffusive in nature. However, transient unbinding and hopping results in a long-time behavior (shown to be superdiffusive) that is qualitatively unaffected by the crowding on the DNA. Thus, hopping along the chain mitigates the role that protein crowding has in restricting the translocation dynamics along the chain. The superdiffusion coefficient is influenced by the quantitative values of the effective binding rate, which is influenced by protein crowding. We show that vacancy fraction and superdiffusion coefficient exhibits a non-monotonic relationship under many circumstances. We leverage analytical theory and dynamic Monte Carlo simulations to address this problem. With several additional contributions, the core of our modeling work adopts a reaction-diffusion framework that is based on Smoluchowski's original work.

The reactive and diffusive dynamics of a single chemically powered Janus motor in a crowded medium of moving but passive obstacles is investigated using molecular simulation. It is found that the reaction rate of the catalytic motor reaction decreases in a crowded medium as the volume fraction of obstacles increases as a result of a reduction in the Smoluchowski diffusion-controlled reaction rate coefficient that contributes to the overall reaction rate. A continuum model is constructed and analyzed to interpret the dependence of the steady-state reaction rate observed in simulations on the volume fraction of obstacles in the system. The steady-state concentration fields of reactant and product are shown to be sensitive to the local structure of obstacles around the Janus motor. It is demonstrated that the active motor exhibits enhanced diffusive motion at long times with a diffusion constant that decreases as the volume fraction of crowding species increases. In addition, the dynamical properties of a passive tracer particle in a system containing many active Janus motors is studied to investigate how an active environment influences the transport of non-active species. The diffusivity of a passive tracer particle in an active medium is found to be enhanced in systems with forward-moving Janus motors due to the cooperative dynamics of these motors.

The collectivity in the simultaneous diffusion of many particles, i.e. the interdependence of stochastic forces affecting different particles in the same solution, is a largely overlooked phenomenon with no well-established theory. Recently, we have proposed a novel type of thermodynamically consistent Langevin dynamics driven by spatially correlated noise (SCN) that can contribute to the understanding of this problem. This model draws a link between the theory of effective interactions in binary colloidal mixtures and the properties of SCN. In the current article, we review this model from the perspective of collective diffusion and generalize it to the case of multiple (N  >  2) particles. Since our theory of SCN-driven Langevin dynamics has certain issues that could not be resolved within this framework, in this article we also provide another approach to the problem of collectivity. We discuss the multi-particle Mori–Zwanzig model, which is fully microscopically consistent. Indeed, we show that this model supplies a lot of information, complementary to the SCN-based approach, e.g. it predicts the deterministic dynamics of the relative distance between the particles, it provides an approximation for non-equilibrium effective interactions and predicts the collective sub-diffusion of tracers in the group. These results provide the short-range, inertial limit of the earlier model and agree with its predictions under some general conditions. In this article we also review the origin of SCN and its consequences for a variety of physical systems, with emphasis on the colloids.

Fractional Brownian motions (FBMs) have been observed recently in the measured trajectories of individual molecules or small particles in the cytoplasm of living cells and in other dense composite systems, among others. Various types of FBMs differ in a number of ways, including the strength, range and type of damping of the memory encoded in their definitions, but share several basic characteristics: distributions, non-ergodic properties, and scaling of the second moment, which makes it difficult to determine which type of Brownian motion (fractional or normal) the measured trajectory belongs to. Here, we show, by introducing FBMs with regulated range and strength of memory, that it is the structure of memory which determines their physical properties, including mean velocity of diffusion; therefore, the course and kinetics of several processes (including coagulation and some chemical reactions). We also show that autocorrelation functions possess characteristic features which enable identification of an observed FBM, and of the type of memory governing its trajectory.

We investigate the stochastic dynamics of an active particle moving at a constant speed under the influence of a fluctuating torque. In our model the angular velocity is generated by a constant torque and random fluctuations described as a Lévy-stable noise. Two situations are investigated. First, we study white Lévy noise where the constant speed and the angular noise generate a persistent motion characterized by the persistence time ${\tau }_{D}$. At this time scale the crossover from ballistic to normal diffusive behavior is observed. The corresponding diffusion coefficient can be obtained analytically for the whole class of symmetric α-stable noises. As typical for models with noise-driven angular dynamics, the diffusion coefficient depends non-monotonously on the angular noise intensity. As second example, we study angular noise as described by an Ornstein–Uhlenbeck process with correlation time ${\tau }_{c}$ driven by the Cauchy white noise. We discuss the asymptotic diffusive properties of this model and obtain the same analytical expression for the diffusion coefficient as in the first case which is thus independent on ${\tau }_{c}$. Remarkably, for ${\tau }_{c}\gt {\tau }_{D}$ the crossover from a non-Gaussian to a Gaussian distribution of displacements takes place at a time ${\tau }_{G}$ which can be considerably larger than the persistence time ${\tau }_{D}$.

The continuous time random walk model plays an important role in modelling of the so-called anomalous diffusion behaviour. One of the specific properties of such model is the appearance of constant time periods in the trajectory. In the continuous time random walk approach they are realizations of the sequence called waiting times. In this work we focus on the analysis of waiting time distribution by introducing novel methods of parameter estimation and statistical investigation of such a distribution. These methods are based on the modified cumulative distribution function. In this paper we consider three special cases of waiting time distributions, namely α-stable, tempered stable and gamma. However, the proposed methodology can be applied to broad set of distributions—in general it may serve as a method of fitting any distribution function if the observations are rounded. The new statistical techniques are applied to the simulated data as well as to the real data of $\text{C}{{\text{O}}_{2}}$ concentration in indoor air.

Several experiments on tagged molecules or particles in living systems suggest that they move anomalously slow—their mean squared displacement (MSD) increase slower than linearly with time. Leading models aimed at understanding these experiments predict that the MSD grows as a power law with a growth exponent that is smaller than unity. However, in some experiments the growth is so slow (fitted exponent  ∼0.1–0.2) that they hint towards other mechanisms at play. In this paper, we theoretically demonstrate how in-plane collective modes excited by thermal fluctuations in a two dimensional membrane lead to logarithmic time dependence for the the tracer particle's MSD.

We consider an anisotropic needle-like Brownian particle with nematic symmetry confined in a 2D domain. For this system, the coupling of translational and rotational diffusion makes the process $\mathbf{x}(t)$ of the positions of the particle non Markovian. Using scaling arguments, a Gaussian approximation and numerical methods, we determine the mean first passage time $\langle \mathbf{T}\rangle$ of the particle to a target of radius a and show in particular that $\langle \mathbf{T}\rangle \sim {{a}^{-1/2}}$ for $a\to 0$, in contrast with the classical logarithmic divergence obtained in the case of an isotropic 2D Brownian particle.

We study diffusion-controlled single-species annihilation with sparse initial conditions. In this random process, particles undergo Brownian motion, and when two particles meet, both disappear. We focus on sparse initial conditions where particles occupy a subspace of dimension δ that is embedded in a larger space of dimension d. We find that the co-dimension Δ = d − δ governs the behavior. All particles disappear when the co-dimension is sufficiently small, Δ ≤ 2; otherwise, a finite fraction of particles indefinitely survive. We establish the asymptotic behavior of the probability S(t) that a test particle survives until time t. When the subspace is a line, δ = 1, we find inverse logarithmic decay, $S\sim {(\mathrm{ln}t)}^{-1}$, in three dimensions, and a modified power-law decay, $S\sim (\mathrm{ln}t){t}^{-1/2}$, in two dimensions. In general, the survival probability decays algebraically when Δ < 2, and there is an inverse logarithmic decay at the critical co-dimension Δ = 2.

We study diffusion-controlled single-species annihilation with a finite number of particles. In this reaction-diffusion process, each particle undergoes ordinary diffusion, and when two particles meet, they annihilate. We focus on spatial dimensions $d\gt 2$ where a finite number of particles typically survive the annihilation process. Using scaling techniques we investigate the average number of surviving particles, M, as a function of the initial number of particles, N. In three dimensions, for instance, we find the scaling law $M\sim {N}^{1/3}$ in the asymptotic regime $N\gg 1$. We show that two time scales govern the reaction kinetics: the diffusion time scale, $T\sim {N}^{2/3}$, and the escape time scale, $\tau \sim {N}^{4/3}$. The vast majority of annihilation events occur on the diffusion time scale, while no annihilation events occur beyond the escape time scale.

The term 'Lévy flights' was coined by Benoit Mandelbrot, who thus poeticized α-stable Lévy random motion, a Markovian process with stationary independent increments distributed according to the α-stable Lévy probability law. Contrary to the Brownian motion, the trajectories of the α-stable Lévy motion are discontinous, that is exhibit jumps. This feature implies that the process of first passage through the boundary of a given space domain, or the first escape, is different from the process of first arrival (hit) at the boundary. Here we investigate the properties of first escapes and first arrivals for Lévy flights and explore how the asymptotic behavior of the corresponding (passage and hit) probabilities is sensitive to the size of the domain. In particular, we find that the survival probability to stay in a large enough, finite domain has a universal Sparre Andersen temporal scaling ${t}^{-1/2}$, which is transient and changes to an exponential non-universal decay at longer times. Also, the probability to arrive at a finite domain possesses a similar transient Sparre Andersen universality that turns into a non-universal and slower power-law decay in course of time. Finally, we demonstrate that the probability density of the leapover length ℓ over the boundary, related to overshooting events, has an intermediate asymptotics ${{\ell }}^{-(1+\alpha /2)}$ ($0\lt \alpha \lt 2$) which is inherent for the escape from a semi-infinite domain. However, for larger leapovers the probability density decays faster according to the ${{\ell }}^{-(1+\alpha )}$ law. Thus, we find that the laws derived for the α-stable processes on the semi-infinite domain, manifest themselves as transients for Lévy flights on the finite domain.

Rapid recognition by a protein of its DNA target site is achieved through a combination of one- and three-dimensional (1D and 3D) diffusion, which allows efficient scanning of the many alternative sites. This facilitated diffusion mechanism is expected to be affected by cellular conditions, particularly crowding, given that up to 40% of the total cellular volume may by occupied by macromolecules. Both experimental and theoretical studies showed that crowding particles can enhance facilitated diffusion and accelerate search kinetics. This effect may originate from crowding forcing a trade-off between 3D and 1D diffusion. In this study, using coarse-grained molecular dynamic simulations, we investigate how the molecular properties of the crowders may modulate the effect exerted by crowding on a searcher protein. We show that crowders with an affinity to the DNA are less effective search facilitators than particles whose contribution is solely entropic. Crowders that have affinity to DNA may occupy DNA sites and thereby function as obstacles or roadblocks that slow down the searcher protein, and they may also produce a smaller excluded volume effect and so reduce usage of the hopping searching mode in favor of less-effective 3D diffusion in the bulk. We discuss how strong repulsive interactions between the crowding particles themselves may affect the overall dynamics of the crowders and their excluded volume effect. Our study shows that search kinetics and its mechanism are modulated not only by salt concentration and crowding occupancy, but also by the properties of the crowding particles.

We perform the thermodynamic analysis of an engine consisting of a Brownian particle in a space-periodic and time-periodic potential, including the issues of power, efficiency and dissipation. We derive the explicit expressions for the Onsager coefficients characterizing the linear response regime.

Protein search and association to specific sequences on DNA is a starting point for all fundamental biological processes. It has been intensively studied in recent years by a variety of experimental and theoretical methods. However, many features of these complex biological phenomena are still not resolved at the molecular level. Experiments indicate that proteins can be bound non-specifically to DNA in multiple configurations. But the role of conformational fluctuations in the protein search dynamics remains not well understood. Here we develop a theoretical method to analyze how the conformational transitions affect the process of finding the specific targets on DNA. Our approach is based on discrete-state stochastic calculations that take into account the most relevant physical–chemical processes. This allows us to explicitly evaluate the protein search for the targets on DNA at different conditions. Our calculations suggest that conformational fluctuations might strongly affect the protein search dynamics. We explain how the shift in the conformational equilibrium influences the target search kinetics. Theoretical predictions are supported by Monte Carlo computer simulations.

Particle diffusion in the presence of sparse obstacles may be considered as a sequence of relatively long intervals, during which the particle diffuses in regions free of obstacles, separated by relatively short intervals during which the particle suffers multiple collisions with an obstacle. The present paper focuses on the mean duration of the short intervals, called mean encounter time, assuming that the obstacles are identical spheres. Based on scaling arguments, one can deduce that this time is proportional to the ratio of the square of the obstacle radius and the particle diffusivity. We derive an expression for the mean encounter time, which shows that the proportionality coefficient is 1/6.

We study the dynamics of a starving random walk in general spatial dimension d. This model represents an idealized description for the fate of an unaware forager whose motion is not affected by the presence or absence of resources. The forager depletes its environment by consuming resources and dies if it wanders too long without finding food. In the exactly solvable case of one dimension, we explicitly derive the average lifetime of the walk and the distribution for the number of distinct sites visited by the walk at the instant of starvation. We also give a heuristic derivation for the averages of these two quantities. We tackle the complex but ecologically relevant case of two dimensions by an approximation in which the depleted zone is assumed to always be circular and which grows incrementally each time the walk reaches the edge of this zone. Within this framework, we derive a lower bound for the scaling of the average lifetime and number of distinct sites visited at starvation. We also determine the asymptotic distribution of the number of distinct sites visited at starvation. Finally, we solve the case of high spatial dimensions within a mean-field approach.

A combined dynamics consisting of Brownian motion and Lévy flights is exhibited by a variety of biological systems performing search processes. Assessing the search reliability of ever locating the target and the search efficiency of doing so economically of such dynamics thus poses an important problem. Here we model this dynamics by a one-dimensional fractional Fokker–Planck equation combining unbiased Brownian motion and Lévy flights. By solving this equation both analytically and numerically we show that the superposition of recurrent Brownian motion and Lévy flights with stable exponent $\alpha \lt 1$, by itself implying zero probability of hitting a point on a line, leads to transient motion with finite probability of hitting any point on the line. We present results for the exact dependence of the values of both the search reliability and the search efficiency on the distance between the starting and target positions as well as the choice of the scaling exponent α of the Lévy flight component.

We construct an equivalent probability description of linear multi-delay Langevin equations subject to additive Gaussian white noise. By exploiting the time-convolutionless transform and a time variable transformation we are able to write a Fokker–Planck equation (FPE) for the 1-time and for the 2-time probability distributions valid irrespective of the regime of stability of the Langevin equations. We solve exactly the derived FPEs and analyze the aging dynamics by studying analytically the conditional probability distribution. We discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non-Markov process as both preparation and observation times have bearing on its dynamics. As our analytic procedure can also be applied to linear Langevin equations with memory kernels, we compare the non-Markov dynamics of a one-delay system with that of a generalized Langevin equation with an exponential as well as a power law memory. Application to a generalization of the Green–Kubo formula is also presented.

The helical structure of DNA imposes constraints on the rate of diffusion-limited protein binding. Here we solve the reaction–diffusion equations for DNA-like geometries and extend with simulations when necessary. We find that the helical structure can make binding to the DNA more than twice as fast compared to a case where DNA would be reactive only along one side. We also find that this rate advantage remains when the contributions from steric constraints and rotational diffusion of the DNA-binding protein are included. Furthermore, we find that the association rate is insensitive to changes in the steric constraints on the DNA in the helix geometry, while it is much more dependent on the steric constraints on the DNA-binding protein. We conclude that the helical structure of DNA facilitates the nonspecific binding of transcription factors and structural DNA-binding proteins in general.

Molecular signalling in living cells occurs at low copy numbers and is thereby inherently limited by the noise imposed by thermal diffusion. The precision at which biochemical receptors can count signalling molecules is intimately related to the noise correlation time. In addition to passive thermal diffusion, messenger RNA and vesicle-engulfed signalling molecules can transiently bind to molecular motors and are actively transported across biological cells. Active transport is most beneficial when trafficking occurs over large distances, for instance up to the order of 1 metre in neurons. Here we explain how intermittent active transport allows for faster equilibration upon a change in concentration triggered by biochemical stimuli. Moreover, we show how intermittent active excursions induce qualitative changes in the noise in effectively one-dimensional systems such as dendrites. Thereby they allow for significantly improved signalling precision in the sense of a smaller relative deviation in the concentration read-out by the receptor. On the basis of linear response theory we derive the exact mean field precision limit for counting actively transported molecules. We explain how intermittent active excursions disrupt the recurrence in the molecular motion, thereby facilitating improved signalling accuracy. Our results provide a deeper understanding of how recurrence affects molecular signalling precision in biological cells and novel medical-diagnostic devices.