The formation of protein patterns inside cells is generically described by reaction-diffusion models. The study of such systems goes back to Turing, who showed how patterns can emerge from a homogenous steady state when two reactive components have different diffusivities (e.g., membrane-bound and cytosolic states). However, in nature, systems typically develop in a heterogeneous environment, where upstream protein patterns affect the formation of protein patterns downstream. Examples for this are the polarization of Cdc42 adjacent to the previous bud site in budding yeast and the formation of an actin-recruiter ring that forms around a PIP3 domain in macropinocytosis. This suggests that previously established protein patterns can serve as a template for downstream proteins and that these downstream proteins can “sense” the edge of the template. A mechanism for how this edge sensing may work remains elusive. Here we demonstrate and analyze a generic and robust edge-sensing mechanism, based on a two-component mass-conserving reaction-diffusion (McRD) model. Our analysis is rooted in a recently developed theoretical framework for McRD systems, termed local equilibria theory. We extend this framework to capture the spatially heterogeneous reaction kinetics due to the template. This enables us to graphically construct the stationary patterns in the phase space of the reaction kinetics. Furthermore, we show that the protein template can trigger a regional mass-redistribution instability near the template edge, leading to the accumulation of protein mass, which eventually results in a stationary peak at the template edge. We show that simple geometric criteria on the reactive nullcline's shape predict when this edge-sensing mechanism is operational. Thus, our results provide guidance for future studies of biological systems and for the design of synthetic pattern forming systems.

We investigate the use of artificial neural networks (NNs) as an alternative tool to current analytical methods for recognizing knots in a given polymer conformation. The motivation is twofold. First, it is of interest to examine whether NNs are effective at learning the global and sequential properties that uniquely define a knot. Second, knot classification is an important and unsolved problem in mathematical and physical sciences, and NNs may provide insights into this problem. Motivated by these points, we generate millions of polymer conformations for five knot types: 0, $3_1, 4_1, 5_1$, and $5_2$, and we design various NN models for classification. Our best model achieves a five-class classification accuracy of above $99\%$ on a polymer of 100 monomers. We find that the sequential modeling ability of recurrent NNs is crucial for this result, as it outperforms feed-forward NNs and successfully generalizes to differently sized conformations as well. We present our methods and suggest that deep learning may be used in specific applications of knot detection where some error is permissible. Hopefully, with further development, NNs can offer an alternative computational method for knot identification and facilitate knot research in mathematical and physical sciences.

About three-quarters of eukaryotic DNA is wrapped into nucleosomes; DNA spools with a protein core. The affinity of a given DNA stretch to be incorporated into a nucleosome is known to depend on the base-pair sequence-dependent geometry and elasticity of the DNA double helix. This causes the rotational and translational positioning of nucleosomes. In this study we ask the question whether the latter can be predicted by a simple coarse-grained DNA model with sequence-dependent elasticity, the rigid base-pair model. Whereas this model is known to be rather robust in predicting rotational nucleosome positioning, we show that the translational positioning is a rather subtle effect that is dominated by the guanine-cytosine content dependence of entropy rather than energy. A correct qualitative prediction within the rigid base-pair framework can only be achieved by assuming that DNA elasticity effectively changes on complexation into the nucleosome complex. With that extra assumption we arrive at a model which gives an excellent quantitative agreement to experimental in vitro nucleosome maps, under the additional assumption that nucleosomes equilibrate their positions only locally.

We present an energy conserving lattice Boltzmann model based on a crystallographic lattice for simulation of weakly compressible flows. The theoretical requirements and the methodology to construct such a model are discussed. We demonstrate that the model recovers the isentropic sound speed in addition to the effects of viscous heating and heat flux dynamics. Several test cases for acoustics and thermal and thermoacoustic flows are simulated to show the accuracy of the proposed model.

Adsorption on a boundary line confining a monolayer of particles self-assembling into clusters is studied by Monte Carlo simulations. We focus on a system of particles interacting via competing interaction potential in which effectively short-range attraction is followed by long-range repulsion. For the chemical potential values below the order-disorder phase transition the adsorption isotherms were shown to undergo nonstandard behavior, i.e., the adsorption exhibits a maximum on structural transition between structureless and disordered cluster fluid. In particular, we have found that the adsorption decreases for increasing chemical potential when (i) clusters dominate over monomers in the bulk, (ii) the density profile in the direction perpendicular to the confining line exhibits an oscillatory decay, and (iii) the correlation function in the layer near the adsorption wall exhibits an oscillatory decay in the direction parallel to this wall. Our report indicates striking differences between simple and complex fluid adsorption processes.

In spite of their enormous applications as alternative energy storage devices and lubricants, room-temperature ionic liquids (ILs) still pose many challenges from a pure scientific viewpoint. We develop an IL microscopic theory in terms of ionic clusters, which describes the IL behavior close to charged interfaces. The full structure factor of finite-size clusters is considered and allows us to retain fine and essential details of the system as a whole. Beside the reduction in the screening, it is shown that ionic clusters cause the charge density to oscillate near charged boundaries, with alternating ion-size thick layers, in agreement with experiments. We distinguish between short-range oscillations that persist for a few ionic layers close to the boundary, as opposed to long-range damped oscillations that hold throughout the bulk. The former can be captured by finite-size ion pairs, while the latter is associated with larger clusters with a pronounced quadrupole (or higher) moment. The long-wavelength limit of our theory recovers the well-known Bazant-Storey-Kornyshev (BSK) equation in the linear regime, and elucidates the microscopic origin of the BSK phenomenological parameters.

In this article, we propose a minimal model for the cooking-induced deformation of spaghetti and related food products. Our approach has parallels to the use of rod theories for the mechanics of slender bodies undergoing growth and is inspired by a wealth of experimental data from the food science literature. We use our model to investigate the cooking of a single strand of spaghetti confined to a pot and reproduce a curious three-stage deformation sequence that arises in the cooking process.

We study airplane boarding in the limit of a large number of passengers using geometric optics in a Lorentzian metric. The airplane boarding problem is naturally embedded in a ($1+1$)-dimensional space-time with a flat Lorentzian metric. The duration of the boarding process can be calculated based on a representation of the one-dimensional queue of passengers attempting to reach their seats in a two-dimensional space-time diagram. The ability of a passenger to delay other passengers depends on their queue positions and row designations. This is equivalent to the causal relationship between two events in space-time, whereas two passengers are timelike separated if one is blocking the other and spacelike if both can be seated simultaneously. Geodesics in this geometry can be utilized to compute the asymptotic boarding time, since space-time geometry is the many-particle (passengers) limit of airplane boarding. Our approach naturally leads to the introduction of an effective refractive index that enables an analytical calculation of the average boarding time for groups of passengers with different aisle-clearing time distribution. In the past, airline companies attempted to shorten the boarding times by trying boarding policies that allow either slow or fast passengers to board first. Our analytical calculations, backed by discrete-event simulations, support the counterintuitive result that the total boarding time is shorter with the slow passengers boarding before the fast passengers. This is a universal result, valid for any combination of the parameters that characterize the problem: the percentage of slow passengers, the ratio between aisle-clearing times of the fast and the slow group, and the density of passengers along the aisle. We find an improvement of up to 28% compared with the fast-first boarding policy. Our approach opens up the possibility to unify numerous simulation-based case studies under one framework.

Among the family of hard convex lens-shaped particles (lenses), the one with aspect ratio equal to 2/3 is “optimal” in the sense that the maximally random jammed (MRJ) packings of such lenses achieve the highest packing fraction ${\varphi }_{\mathrm{MRJ}}\simeq 0.73$ [G. Cinacchi and S. Torquato, Soft Matter 14, 8205 (2018)]. This value is only a few percent lower than ${\varphi }_{\mathrm{DKP}}=0.76210\cdots$, the packing fraction of the corresponding densest-known crystalline (degenerate) packings [G. Cinacchi and S. Torquato, J. Chem. Phys. 143, 224506 (2015)]. By exploiting the appreciably reduced propensity that a system of such optimal lenses has to positionally and orientationally order, disordered packings of them are progressively generated by a Monte Carlo method–based procedure from the dilute equilibrium isotropic fluid phase to the dense nonequilibrium MRJ state. This allows us to closely monitor how the (micro)structure of these packings changes in the process of formation of the MRJ state. The gradual changes undergone by the many structural descriptors calculated here can coherently and consistently be traced back to the gradual increase in contacts between the hard particles until the isostatic mean value of ten contact neighbors per lens is reached at the effectively hyperuniform MRJ state. Compared to the MRJ state of hard spheres, the MRJ state of such optimal lenses is denser (less porous), more disordered, and rattler-free. This set of characteristics makes them good glass formers. It is possible that this conclusion may also hold for other hard convex uniaxial particles with a correspondingly similar aspect ratio, be they oblate or prolate, and that, by using suitable biaxial variants of them, that set of characteristics might further improve.

Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of link percolation in nonamenable two-dimensional branching simplicial and cell complexes, i.e., simplicial and cell complexes in which the boundary scales like the volume. We establish the relation between the equations determining the percolation probability in random branching cell complexes and the equation for interdependent percolation in multiplex networks with interlayer degree correlation equal to one. By using this relation we show that branching cell complexes can display more than two percolation phase transitions: the upper percolation transition, the lower percolation transition, and one or more intermediate phase transitions. At these additional transitions the percolation probability and the fractal exponent both feature a discontinuity. Furthermore, by using the renormalization group theory we show that the upper percolation transition can belong to various universality classes including the Berezinskii-Kosterlitz-Thouless (BKT) transition, the discontinuous percolation transition, and continuous transitions with anomalous singular behavior that generalize the BKT transition.

Using modulation theory, we develop a simple $\mathrm{[}(2+1)$-dimensional] model to describe the synergy between the thermo-optical and reorientational responses of nematic liquid crystals to light beams to describe the routing of spatial optical solitary waves (nematicons) in such a uniaxial environment. Introducing several approximations based on the nonlocal physics of the material, we are able to predict the trajectories of nematicons and their angular steering with temperature, accounting for the energy exchange between the input beam and the medium through one-photon absorption. The theoretical results are then compared to experimental data from previous studies, showing excellent agreement.

We study properties of Markov chain Monte Carlo simulations of classical spin models with local updates. We derive analytic expressions for the mean value of the acceptance rate of single-spin-flip algorithms for the one-dimensional Ising model. We find that for the Metropolis algorithm the average acceptance rate is a linear function of energy. We further provide numerical results for the energy dependence of the average acceptance rate for the three- and four-state Potts model, and the $XY$ model in one and two spatial dimensions. In all cases, the acceptance rate is an almost linear function of the energy in the critical region. The variance of the acceptance rate is studied as a function of the specific heat. While the specific heat develops a singularity in the vicinity of a phase transition, the variance of the acceptance rate stays finite.

A high-precision test of the Kolmogorov-Avrami (KA) model of “geometric” interactions between growing nuclei is performed for an Ising ferromagnet driven by Metropolis dynamics. As long as the interface boundaries of interacting nuclei are rough, the KA model is adequate for arbitrary large densities of the new phase, well beyond the percolation transition, as in the case of homogeneous nucleation. For a prenucleated system the KA description breaks down at sufficiently low temperature, when the length of a kinetically smooth interface exceeds the distance between seeds.

We describe a simple form of importance sampling designed to bound and compute large-deviation rate functions for time-extensive dynamical observables in continuous-time Markov chains. We start with a model, defined by a set of rates, and a time-extensive dynamical observable. We construct a reference model, a variational ansatz for the behavior of the original model conditioned on atypical values of the observable. Direct simulation of the reference model provides an upper bound on the large-deviation rate function associated with the original model, an estimate of the tightness of the bound, and, if the ansatz is chosen well, the exact rate function. The exact rare behavior of the original model does not need to be known in advance. We use this method to calculate rate functions for currents and counting observables in a set of network- and lattice models taken from the literature. Straightforward ansätze yield bounds that are tighter than bounds obtained from Level 2.5 of large deviations via approximations that involve uniform scalings of rates. We show how to correct these bounds in order to recover the rate functions exactly. Our approach is complementary to more specialized methods and offers a physically transparent framework for approximating and calculating the likelihood of dynamical large deviations.

Motivated by the idea that intrinsically disordered proteins (IDPs) condense into liquidlike droplets within cells, we carry out Monte Carlo simulations of a polymer lattice model to study the relationship between charge patterning and phase separation. Polymer chains containing neutral, positively charged, and negatively charged monomers are placed on a cubic lattice. Only nearest-neighbor interactions between charges are considered. We determine the phase diagram for a systematically varied set of sequences. We observe homogeneous fluids, liquid condensation, cluster phases, filaments, and crystal states. Of the six sequences we study, three form crystals at low temperatures. The other three sequences, which have lower charge densities, instead collapse into gel-like networks or unconnected finite clusters. Longer neutral patches along the sequence sterically limit the size and shape of low-energy structures, which is analogous to the effect of charge or limited valence in attractive colloids. Only one sequence clearly exhibits liquid behavior; this sequence has a reduced tendency to individually fold and crystallize compared to others of similar charge density and draws parallels to real IDP behavior.

Ordinary nematic liquid crystals are characterized by having a uniform director field as ground state. In such a state, the director is the same everywhere and no distortion is to be seen at all. We give a definition of uniform distortion which makes precise the intuitive notion of seeing everywhere the same director landscape. We characterize all such distortions and prove that they fall into two families, each described by two scalar parameters. Uniform distortions exhaust R. Meyer's heliconical structures, which, as it has recently been recognized, include the ground state of twist-bend nematics. The generalized elasticity of these new phases is treated with a simple free-energy density, which can be minimized by both uniform and nonuniform distortions, the latter injecting a germ of elastic frustration.

The transport of nanoparticles in confined geometries plays a crucial role in several technological applications ranging from nanopore sensors to filtration membranes. Here we describe a Brownian approach to simulate the motion of a rigid-body nanoparticle of an arbitrary shape under confinement. A quaternion formulation is used for the nanoparticle orientation, and the corresponding overdamped Langevin equation, completed by the proper fluctuation-dissipation relation, is derived. The hydrodynamic mobility matrix is obtained via dissipative particle dynamics simulation equipped with a new method for enforcing the no-slip boundary condition for curved moving solid-liquid interfaces. As an application, we analyzed the motion of a nanoparticle in a cylindrical channel under the action of external fields. We show that both axial effective diffusion and rotational diffusion decrease with confinement.

It is well known that flagellated bacteria swim in circles near surfaces. However, recent experiments have shown that a sulfide-oxidizing bacterium named Thiovulum majus can transition from swimming in circles to a surface bound state where it stops swimming while remaining free to move laterally along the surface. In this bound state, the cell rotates perpendicular to the surface with its flagella pointing away from it. Using numerical simulations and theoretical analysis, we demonstrate the existence of a fluid-structure interaction instability that causes cells with relatively short flagella to become surface bound.

We study the dynamic properties of a thermal autonomous machine made up of two quantum Brownian particles, each of which is in contact with an environment at different temperature and moves on a periodic sinusoidal track. When such tracks are shifted, the center of mass of the system exhibits a nonvanishing velocity, for which we provide an exact expression in the limit of small track undulations. We discuss the role of the broken spatial symmetry in the emergence of directed motion in thermal machines. We then consider the case in which external deterministic forces are applied to the system, and we characterize its steady-state velocity. If the applied external force opposes the system motion, work can be extracted from such a steady-state thermal machine, without any external cyclic protocol. When the two particles are not interacting, our results reduce to those of Fisher and Zwerger [Phys. Rev. B 32, 6190 (1985)] and Aslangul, Pottier, and Saint-James [J. Phys. France 48, 1093 (1987)] for a single particle moving in a periodic tilted potential. We finally use our results for the motor velocity to check the validity of the quantum molecular dynamics algorithm in the nonlinear, nonequilibrium regime.

In many biological systems, motile agents exhibit random motion with short-term directional persistence, together with crowding effects arising from spatial exclusion. We formulate and study a class of lattice-based models for multiple walkers with motion persistence and spatial exclusion in one and two dimensions, and use a mean-field approximation to investigate relevant population-level partial differential equations in the continuum limit. We show that this model of a persistent exclusion process is in general well described by a nonlinear diffusion equation. With reference to results presented in the current literature, our results reveal that the nonlinearity arises from the combination of motion persistence and volume exclusion, with linearity in terms of the canonical diffusion or heat equation being recovered in either the case of persistence without spatial exclusion, or spatial exclusion without persistence. We generalize our results to include systems of multiple species of interacting, motion-persistent walkers, as well as to incorporate a global drift in addition to persistence. These models are shown to be governed approximately by systems of nonlinear advection-diffusion equations. By comparing the prediction of the mean-field approximation to stochastic simulation results, we assess the performance of our results. Finally, we also address the problem of inferring the presence of persistence from simulation results, with a view to application to experimental cell-imaging data.