Footprints of loop extrusion in statistics of intra-chromosomal distances: An analytically solvable model.

Active loop extrusion-the process of formation of dynamically growing chromatin loops due to the motor activity of DNA-binding protein complexes-is a firmly established mechanism responsible for chromatin spatial organization at different stages of a cell cycle in eukaryotes and bacteria. The theoretical insight into the effect of loop extrusion on the experimentally measured statistics of chromatin conformation can be gained with an appropriately chosen polymer model. Here, we consider the simplest analytically solvable model of an interphase chromosome, which is treated as an ideal chain with disorder of sufficiently sparse random loops whose conformations are sampled from the equilibrium ensemble. This framework allows us to arrive at the closed-form analytical expression for the mean-squared distance between pairs of genomic loci, which is valid beyond the one-loop approximation in diagrammatic representation. In addition, we analyze the loop-induced deviation of chain conformations from the Gaussian statistics by calculating kurtosis of probability density of the pairwise separation vector. The presented results suggest the possible ways of estimating the characteristics of the loop extrusion process based on the experimental data on the scale-dependent statistics of intra-chromosomal pair-wise distances.

Active motion of passive asymmetric dumbbells in a non-equilibrium bath.

Persistent motion of passive asymmetric bodies in non-equilibrium media has been experimentally observed in a variety of settings. However, fundamental constraints on the efficiency of such motion are not fully explored. Understanding such limits, and ways to circumvent them, is important for efficient utilization of energy stored in agitated surroundings for purposes of taxis and transport. Here, we examine such issues in the context of erratic movements of a passive asymmetric dumbbell driven by non-equilibrium noise. For uncorrelated (white) noise, we find a (non-Boltzmann) joint probability distribution for the velocity and orientation, which indicates that the dumbbell preferentially moves along its symmetry axis. The dumbbell thus behaves as an Ornstein-Uhlenbeck walker, a prototype of active matter. Exploring the efficiency of this active motion, we show that in the over-damped limit, the persistence length l of the dumbbell is bound from above by half its mean size, while the propulsion speed v∥ is proportional to its inverse size. The persistence length can be increased by exploiting inertial effects beyond the over-damped regime, but this improvement always comes at the price of smaller propulsion speeds. This limitation is explained by noting that the diffusivity of a dumbbell, related to the product v∥l, is always less than that of its components, thus severely constraining the usefulness of passive dumbbells as active particles.

Conformational statistics of non-equilibrium polymer loops in Rouse model with active loop extrusion.

Motivated by the recent experimental observations of the DNA loop extrusion by protein motors, in this paper, we investigate the statistical properties of the growing polymer loops within the ideal chain model. The loop conformation is characterized statistically by the mean gyration radius and the pairwise contact probabilities. It turns out that a single dimensionless parameter, which is given by the ratio of the loop relaxation time over the time elapsed since the start of extrusion, controls the crossover between near-equilibrium and highly non-equilibrium asymptotics in the statistics of the extruded loop, regardless of the specific time dependence of the extrusion velocity. In addition, we show that two-sided and one-sided loop extruding motors produce the loops with almost identical properties. Our predictions are based on two rigorous semi-analytical methods accompanied by asymptotic analysis of slow and fast extrusion limits.

Pair dispersion in dilute suspension of active swimmers.

Ensembles of biological and artificial microswimmers produce long-range velocity fields with strong nonequilibrium fluctuations, which result in a dramatic increase in diffusivity of embedded particles (tracers). While such enhanced diffusivity may point to enhanced mixing of the fluid, a rigorous quantification of the mixing efficiency requires analysis of pair dispersion of tracers, rather than simple one-particle diffusivity. Here, we calculate analytically the scale-dependent coefficient of relative diffusivity of passive tracers embedded in a dilute suspension of run-and-tumble microswimmers. Although each tracer is subject to strong fluctuations resulting in large absolute diffusivity, the small-scale relative dispersion is suppressed due to the correlations in fluid velocity which are relevant when the inter-tracer separation is below the persistence length of the swimmer's motion. Our results suggest that the reorientation of swimming direction plays an important role in biological mixing and should be accounted in the design of potential active matter devices capable of effective fluid mixing at microscale.

Restart Could Optimize the Probability of Success in a Bernoulli Trial.

The recently noticed ability of restart to reduce the expected completion time of first-passage processes allows appealing opportunities for performance improvement in a variety of settings. However, complex stochastic processes often exhibit several possible scenarios of completion which are not equally desirable in terms of efficiency. Here we show that restart may have profound consequences on the splitting probabilities of a Bernoulli-like first-passage process, i.e., of a process which can end with one of two outcomes. Particularly intriguing, in this respect, is the class of problems where a carefully adjusted restart mechanism maximizes the probability that the process will complete in a desired way. We reveal the universal aspects of this kind of optimal behavior by applying the general approach recently proposed for the problem of first-passage under restart.

Constructing efficient strategies for the process optimization by restart.

Optimization of the mean completion time of random processes by restart is a subject of active theoretical research in statistical physics and has long found practical application in computer science. Meanwhile, one of the key issues remains largely unsolved: how to construct a restart strategy for a process whose detailed statistics are unknown to ensure that the expected completion time will reduce? Addressing this query here we propose several constructive criteria for the effectiveness of various protocols of noninstantaneous restart in the mean completion time problem and in the success probability problem. Being expressed in terms of a small number of easily estimated statistical characteristics of the original process (MAD, median completion time, low-order statistical moments of completion time), these criteria allow informed restart decision based on partial information.

Universal performance bounds of restart.

As has long been known to computer scientists, the performance of probabilistic algorithms characterized by relatively large runtime fluctuations can be improved by applying a restart, i.e., episodic interruption of a randomized computational procedure followed by initialization of its new statistically independent realization. A similar effect of restart-induced process acceleration could potentially be possible in the context of enzymatic reactions, where dissociation of the enzyme-substrate intermediate corresponds to restarting the catalytic step of the reaction. To date, a significant number of analytical results have been obtained in physics and computer science regarding the effect of restart on the completion time statistics in various model problems, however, the fundamental limits of restart efficiency remain unknown. Here we derive a range of universal statistical inequalities that offer constraints on the effect that restart could impose on the completion time of a generic stochastic process. The corresponding bounds are expressed via simple statistical metrics of the original process such as harmonic mean h, median value m, and mode M, and, thus, are remarkably practical. We test our analytical predictions with multiple numerical examples, discuss implications arising from them and important avenues of future work.

Crumpled polymer with loops recapitulates key features of chromosome organization.

Chromosomes are exceedingly long topologically-constrained polymers compacted in a cell nucleus. We recently suggested that chromosomes are organized into loops by an active process of loop extrusion. Yet loops remain elusive to direct observations in living cells; detection and characterization of myriads of such loops is a major challenge. The lack of a tractable physical model of a polymer folded into loops limits our ability to interpret experimental data and detect loops. Here, we introduce a new physical model - a polymer folded into a sequence of loops, and solve it analytically. Our model and a simple geometrical argument show how loops affect statistics of contacts in a polymer across different scales, explaining universally observed shapes of the contact probability. Moreover, we reveal that folding into loops reduces the density of topological entanglements, a novel phenomenon we refer as "the dilution of entanglements". Supported by simulations this finding suggests that up to ~ 1 - 2Mb chromosomes with loops are not topologically constrained, yet become crumpled at larger scales. Our theoretical framework allows inference of loop characteristics, draws a new picture of chromosome organization, and shows how folding into loops affects topological properties of crumpled polymers.