Chromatin jets define the properties of cohesin-driven in vivo loop extrusion.

Complex genomes show intricate organization in three-dimensional (3D) nuclear space. Current models posit that cohesin extrudes loops to form self-interacting domains delimited by the DNA binding protein CTCF. Here, we describe and quantitatively characterize cohesin-propelled, jet-like chromatin contacts as landmarks of loop extrusion in quiescent mammalian lymphocytes. Experimental observations and polymer simulations indicate that narrow origins of loop extrusion favor jet formation. Unless constrained by CTCF, jets propagate symmetrically for 1-2 Mb, providing an estimate for the range of in vivo loop extrusion. Asymmetric CTCF binding deflects the angle of jet propagation as experimental evidence that cohesin-mediated loop extrusion can switch from bi- to unidirectional and is controlled independently in both directions. These data offer new insights into the physiological behavior of in vivo cohesin-mediated loop extrusion and further our understanding of the principles that underlie genome organization.

Entropy Production in Exactly Solvable Systems.

The rate of entropy production by a stochastic process quantifies how far it is from thermodynamic equilibrium. Equivalently, entropy production captures the degree to which global detailed balance and time-reversal symmetry are broken. Despite abundant references to entropy production in the literature and its many applications in the study of non-equilibrium stochastic particle systems, a comprehensive list of typical examples illustrating the fundamentals of entropy production is lacking. Here, we present a brief, self-contained review of entropy production and calculate it from first principles in a catalogue of exactly solvable setups, encompassing both discrete- and continuous-state Markov processes, as well as single- and multiple-particle systems. The examples covered in this work provide a stepping stone for further studies on entropy production of more complex systems, such as many-particle active matter, as well as a benchmark for the development of alternative mathematical formalisms.

Time-dependent branching processes: a model of oscillating neuronal avalanches.

Recently, neuronal avalanches have been observed to display oscillations, a phenomenon regarded as the co-existence of a scale-free behaviour (the avalanches close to criticality) and scale-dependent dynamics (the oscillations). Ordinary continuous-time branching processes with constant extinction and branching rates are commonly used as models of neuronal activity, yet they lack any such time-dependence. In the present work, we extend a basic branching process by allowing the extinction rate to oscillate in time as a new model to describe cortical dynamics. By means of a perturbative field theory, we derive relevant observables in closed form. We support our findings by quantitative comparison to numerics and qualitative comparison to available experimental results.

Volume explored by a branching random walk on general graphs.

Branching processes are used to model diverse social and physical scenarios, from extinction of family names to nuclear fission. However, for a better description of natural phenomena, such as viral epidemics in cellular tissues, animal populations and social networks, a spatial embedding-the branching random walk (BRW)-is required. Despite its wide range of applications, the properties of the volume explored by the BRW so far remained elusive, with exact results limited to one dimension. Here we present analytical results, supported by numerical simulations, on the scaling of the volume explored by a BRW in the critical regime, the onset of epidemics, in general environments. Our results characterise the spreading dynamics on regular lattices and general graphs, such as fractals, random trees and scale-free networks, revealing the direct relation between the graphs' dimensionality and the rate of propagation of the viral process. Furthermore, we use the BRW to determine the spectral properties of real social and metabolic networks, where we observe that a lack of information of the network structure can lead to differences in the observed behaviour of the spreading process. Our results provide observables of broad interest for the characterisation of real world lattices, tissues, and networks.

Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects.

We revisit the problem of Brownian diffusion with drift in order to study finite-size effects in the geometric Galton-Watson branching process. This is possible because of an exact mapping between one-dimensional random walks and geometric branching processes, known as the Harris walk. In this way, first-passage times of Brownian particles are equivalent to sizes of trees in the branching process (up to a factor of proportionality). Brownian particles that reach a distant reflecting boundary correspond to percolating trees, and those that do not correspond to nonpercolating trees. In fact, both systems display a second-order phase transition between "conducting" and "insulating" phases, controlled by the drift velocity in the Brownian system. In the limit of large system size, we obtain exact expressions for the Laplace transforms of the probability distributions and their first and second moments. These quantities are also shown to obey finite-size scaling laws.

Exact Derivation of a Finite-Size Scaling Law and Corrections to Scaling in the Geometric Galton-Watson Process.

The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary.

Finite-size scaling of survival probability in branching processes.

Branching processes pervade many models in statistical physics. We investigate the survival probability of a Galton-Watson branching process after a finite number of generations. We derive analytically the existence of finite-size scaling for the survival probability as a function of the control parameter and the maximum number of generations, obtaining the critical exponents as well as the exact scaling function, which is G(y)=2ye(y)/(e(y)-1), with y the rescaled distance to the critical point. Our findings are valid for any branching process of the Galton-Watson type, independently of the distribution of the number of offspring, provided its variance is finite. This proves the universal behavior of the finite-size effects in branching processes, including the universality of the metric factors. The direct relation to mean-field percolation is also discussed.