Signal-processing molecules inside cells are often present at low copy number, which necessitates probabilistic models to account for intrinsic noise. Probability distributions have traditionally been found using simulation-based approaches which then require estimating the distributions from many samples. Here we present in detail an alternative method for directly calculating a probability distribution by expanding in the natural eigenfunctions of the governing equation, which is linear. We apply the resulting spectral method to three general models of stochastic gene expression: a single gene with multiple expression states (often used as a model of bursting in the limit of two states), a gene regulatory cascade, and a combined model of bursting and regulation. In all cases we find either analytic results or numerical prescriptions that greatly outperform simulations in efficiency and accuracy. In the last case, we show that bimodal response in the limit of slow switching is not only possible but optimal in terms of information transmission.

What is the origin of rotational motion? An answer is presented through the study of the dynamics for spatially localized spots near codimension 2 singularity consisting of drift and peanut instabilities. The drift instability causes a head-tail asymmetry in spot shape, and the peanut one implies a deformation from circular to peanut shape. Rotational motion of spots can be produced by combining these instabilities in a class of three-component reaction-diffusion systems. Partial differential equations dynamics can be reduced to a finite-dimensional one by projecting it to slow modes. Such a reduction clarifies the bifurcational origin of rotational motion of traveling spots in two dimensions in close analogy to the normal form of 1:2 mode interactions.

Fronts propagating into an unstable background state are an important class of solutions to the cubic complex Ginzburg-Landau equation. Applications of such solutions include the Taylor-Couette system in the presence of through flow and chemical systems such as the Belousov-Zhabotinskii reaction. Plane waves are the typical behavior behind such fronts. However, when the relevant plane-wave solution is unstable, it occurs only as a spatiotemporal transient before breaking up into turbulence. Previous studies have suggested that the band of plane waves immediately behind the front will grow continually through time. We show that this is in fact a transient phenomenon and that in the longer term there is a fixed-width band of plane waves. Moreover, we show that the phenomenon occurs for a wide range of parameter values on both sides of the Benjamin-Feir-Newell and absolute instability curves. We present a method for accurately calculating the parameter dependence of the width of the plane-wave band facilitating future experimental verification in real systems.

The study of reversible, functional, and controllable adhesives is a matter of considerable practical interest, and academic research. We report the adhesive response of a magnetorheological fluid confined between two parallel plates under a probe-tack test, when it is subjected to an applied magnetic field. Our analytical approach is based on a Darcy-like law formulation which considers a magnetic field-dependent yield stress behavior. The adhesion force is calculated in closed form for two different configurations produced by a Helmholtz coils setup: uniform perpendicular, and nonuniform radial magnetic fields. In both cases, we verify that adhesion force is hugely increased as a result of the field-dependent nature of the yield stress. This provides a versatile way to obtain a shear resistant, tough structural adhesive through magnetic means.

We analyze the occurrence of vibrational resonance in a damped quintic oscillator with three cases of single well of the potential $V\left(x\right)=\frac{1}{2}{\omega }_0^2{x}^2+\frac{1}{4}\beta x^4+\frac{1}{6}\gamma x^6$ driven by both low-frequency force $f\text{cos}\omega t$ and high-frequency force $g\text{cos}\Omega t$ with $\Omega ⪢\omega$. We restrict our analysis to the parametric choices (i) ${\omega }_0^2$, $\beta$, $\gamma >0$ (single well), (ii) ${\omega }_0^2$, $\gamma >0$, $\beta <0$, ${\beta }^2<4{\omega }_0^2\gamma$ (single well), and (iii) ${\omega }_0^2>0$, $\beta$ arbitrary, $\gamma <0$ (double-hump single well). From the approximate theoretical expression of response amplitude $Q$ at the low-frequency $\omega$ we determine the values of $\omega$ and $g$ (denoted as ${\omega }_{\text{VR}}$ and $g_{\text{VR}}$) at which vibrational resonance occurs. We show that for fixed values of the parameters of the system when $\omega$ is varied either resonance does not occur or it occurs only once. When the amplitude $g$ is varied for the case of the potential with the parametric choice (i) at most one resonance occur while for the other two choices (ii) and (iii) multiple resonance occur. Further, $g_{\text{VR}}$ is found to be independent of the damping strength $d$ while ${\omega }_{\text{VR}}$ depends on $d$. The theoretical predictions are found to be in good agreement with the numerical result. We illustrate that the vibrational resonance can be characterized in terms of width of the orbit also.

This work presents the generalization of the concept of universal finite-size scaling functions to continuum percolation. A high-efficiency algorithm for Monte Carlo simulations is developed to investigate, with extensive realizations, the finite-size scaling behavior of stick percolation in large-size systems. The percolation threshold of high precision is determined for isotropic widthless stick systems as $N_c{l}^2=5.63726\pm 0.00002$, with $N_c$ as the critical density and $l$ as the stick length. Simulation results indicate that by introducing a nonuniversal metric factor $A=0.106910\pm 0.000009$, the spanning probability of stick percolation on square systems with free boundary conditions falls on the same universal scaling function as that for lattice percolation.

Many cellular processes require a polarization axis which generally initially emerges as an inhomogeneous distribution of molecular markers in the cell. We present a simple analytical model of a general mechanism of cell polarization taking into account the positive feedback due to the coupled dynamics of molecular markers and cytoskeleton filaments. We find that the geometry of the organization of cytoskeleton filaments, nucleated on the membrane (e.g., cortical actin) or from a center in the cytoplasm (e.g., microtubule asters), dictates whether the system is capable of spontaneous polarization or polarizes only in response to external asymmetric signals. Our model also captures the main features of recent experiments of cell polarization in two considerably different biological systems, namely, mating budding yeast and neuron growth cones.

We exhibit a remarkable equivalence between the dynamics of an intermittent nonlinear map and the electronic transport properties (obtained via the scattering matrix) of a crystal defined on a double Cayley tree. This strict analogy reveals in detail the nature of the mobility edge normally studied near (not at) the metal-insulator transition in electronic systems. We provide an analytical expression for the conductance as a function of the system size that at the transition obeys a $q$-exponential form. This manifests as power-law decay or few and far between large spike oscillations according to different kinds of boundary conditions.

We use a power grid model with $M$ generators and $N$ consumption units to optimize the grid and its control. Each consumer demand is drawn from a predefined finite-size-support distribution, thus simulating the instantaneous load fluctuations. Each generator has a maximum power capability. A generator is not overloaded if the sum of the loads of consumers connected to a generator does not exceed its maximum production. In the standard grid each consumer is connected only to its designated generator, while we consider a more general organization of the grid allowing each consumer to select one generator depending on the load from a predefined consumer dependent and sufficiently small set of generators which can all serve the load. The model grid is interconnected in a graph with loops, drawn from an ensemble of random bipartite graphs, while each allowed configuration of loaded links represent a set of graph covering trees. Losses, the reactive character of the grid and the transmission-level connections between generators (and many other details relevant to realistic power grid) are ignored in this proof-of-principles study. We focus on the asymptotic limit, $N\to \infty$ and $N/M\to D=O\left(1\right)>1$, and we show that the interconnects allow significant expansion of the parameter domains for which the probability of a generator overload is asymptotically zero. Our construction explores the formal relation between the problem of grid optimization and the modern theory of sparse graphical models. We also design heuristic algorithms that achieve the asymptotically optimal selection of loaded links. We conclude discussing the ability of this approach to include other effects such as a more realistic modeling of the power grid and related optimization and control algorithms.

Recently there has been some suggestions that fragmentation of thin brittle sheets is qualitatively different from pure two-dimensional fragmentation. The obvious reason for such a discrepancy is the possibility of the sheet to deform out of plane. There is a generic crack-branching mechanism that creates power-law fragment size distribution in the small fragment range for two-dimensional (2D) and three-dimensional bulk fragmentation with the power exponent $\left(2D-1\right)/D$. For thin sheets, the power exponent seems to be close to 1.2 which differs from the $D=2$ exponent 1.5. In order to make a distinct separation between sheet and 2D fragmentation, high-resolution fragment size distributions are required for fragmentation models with minimal differencies other than dimensionality. Here a very efficient numerical model which can be switched from 2D fragmentation to out-of-plane sheet fragmentation with minimal changes is used to produce high-resolution fragment size distribution for the two cases. The model results cast some doubt on the existence of separate universality classes for sheet and 2D fragmentation.

Many complex networks are described by directed links; in such networks, a link represents, for example, the control of one node over the other node or unidirectional information flows. Some centrality measures are used to determine the relative importance of nodes specifically in directed networks. We analyze such a centrality measure called the influence. The influence represents the importance of nodes in various dynamics such as synchronization, evolutionary dynamics, random walk, and social dynamics. We analytically calculate the influence in various networks, including directed multipartite networks and a directed version of the Watts-Strogatz small-world network. The global properties of networks such as hierarchy and position of shortcuts rather than local properties of the nodes, such as the degree, are shown to be the chief determinants of the influence of nodes in many cases. The developed method is also applicable to the calculation of the PAGERANK. We also numerically show that in a coupled oscillator system, the threshold for entrainment by a pacemaker is low when the pacemaker is placed on influential nodes. For a type of random network, the analytically derived threshold is approximately equal to the inverse of the influence. We numerically show that this relationship also holds true in a random scale-free network and a neural network.

In computational thermodynamics, a sequence of intermediate states is commonly introduced to connect two equilibrium states. We consider two cases where the choice of intermediate states is particularly important: minimizing statistical error in free-energy difference calculations and maximizing average acceptance probabilities in replica-exchange simulations. We derive bounds for these quantities in terms of the thermodynamic distance between the intermediates, and show that in both cases the intermediates should be chosen as equidistant points along a geodesic connecting the end states.

The effective transport properties and percolation of continuum composites have commonly been studied using discrete models, i.e., by mapping the continuum to a lattice or network. In this study we instead directly solve the continuum transport equations for composite microstructures both analytically and numerically, and we extract the continuum percolation threshold and scaling exponents for the two-dimensional square tile system. We especially focus on the role of corner contacts on flux flow and further show that mapping such “random checkerboard” systems to a network leads to a spurious secondary percolation threshold and causes shifts in the critical scaling exponents of the effective transport properties.

Understanding the dependence and interplay between architecture and function in biological networks has great relevance to disease progression, biological fabrication, and biological systems in general. We propose methods to assess the association of various microbe characteristics and phenotypes with the topology of their networks. We adopt an automated approach to characterize metabolic networks of 32 microbial species using 11 topological metrics from complex networks. Clustering allows us to extract the indispensable, independent, and informative metrics. Using hierarchical linear modeling, we identify relevant subgroups of these metrics and establish that they associate with microbial phenotypes surprisingly well. This work can serve as a stepping stone to cataloging biologically relevant topological properties of networks and toward better modeling of phenotypes. The methods we use can also be applied to networks from other disciplines.

The theory of diffusion-mediated reactions is already established for the target problem in the dilute limit, where the immobile target is surrounded by many quenchers. For lattice random walks in the crowded situation, each quencher is surrounded by other quenchers differently. As a result, each quencher migrates differently in the presence of site blocking effects. However, in the conventional theory, such difference is ignored and quenchers are assumed to move independently of each other. In this paper, theory of diffusion-mediated reactions of target problem is developed by taking into account the site blocking effects for quencher migration and the difference in the configuration of quenchers around each quencher. Our result interpolates between those in high and low limits of quencher concentrations and is a lower bound of the survival probability. In the static limit, the exact result is reproduced for a localized sink. In the presence of diffusion, the approximation is better when intrinsic reaction rates are low.

In systems that exhibit deterministic diffusion, the gross parameter dependence of the diffusion coefficient can often be understood in terms of random-walk models. Provided the decay of correlations is fast enough, one can ignore memory effects and approximate the diffusion coefficient according to dimensional arguments. By successively including the effects of one and two steps of memory on this approximation, we examine the effects of “persistence” on the diffusion coefficients of extended two-dimensional billiard tables and show how to properly account for these effects using walks in which a particle undergoes jumps in different directions with probabilities that depend on where they came from.

We study the time-dependent transmitted intensity and the scattered pattern from magnetic nanofluids at constant ramping of uniform external magnetic field. The nanofluid used is the dispersion of magnetite particles with an average diameter of 6.5 nm with a protective surfactant coating. We observe several critical fields at which the transmitted light intensity decreases drastically followed by the formation of a ringlike pattern on a screen placed perpendicular to the field direction. Interestingly, the critical fields occur at a regular interval of 20 G. The observed critical fields are attributed to zippering transitions of the chains due to attractive energy well when the chains are of different lengths or shifted with respect to one another. Interaction energy calculations show a decrease in the energy of the system due to dipolar interactions at different critical fields confirming the lowering of the system energy through lateral coalescence. The observed zippering phenomenon is perfectly reversible.

The mathematical model of rotating electrohydrodynamic flows in a thin suspended liquid film is proposed and studied. The flows are driven by the given difference of potentials in one direction and constant external electric field ${\mathit{E}}_{\text{out}}$ in another direction in the plane of a film. To derive the model, we employ the spatial averaging over the normal coordinate to a film that leads to the average Reynolds stress that is proportional to ${\left|{\mathit{E}}_{\text{out}}\right|}^3$. This stress generates tangential velocity in the vicinity of the edges of a film that, in turn, causes the rotational motion of a liquid. The proposed model is used to explain the experimental observations of the liquid film motor.

The vertical size segregation of granular materials, a process commonly associated with the Brazil-nut effect, has generally been thought to proceed faster the greater the size difference of the particles. We experimentally investigate sheared dense bidisperse granular materials as a function of the size ratio of the two species and find that the mixing rate at low confining pressure behaves as expected from percolation-based arguments. However, we also observe an anomalous effect for the resegregation rates, wherein the segregation rate is a nonmonotonic function of the particle size ratio with a maximum for intermediate particle size ratio. Combined with the fact that increasing the confining pressure significantly suppresses both mixing and segregation rates of particles of sufficiently dissimilar size, we propose that the anomalous behavior may be attributed to a species-dependent distribution of forces within the system.

In the present work we examine in detail the formation of a depletion zone in the trapping reaction in networks, with a single perfect trap. We monitor the particle density $\rho \left(r\right)$ with respect to the distance $r$ from the trap. We show using Monte Carlo simulations that the depletion zone is absent in regular, Erdos-Renyi (ER), and scale-free (SF) networks. The density profiles show significant differences for these cases. The particles are homogeneously distributed in regular and ER networks with the depletion effect appearing in very sparse ER networks. In SF networks we reveal the important role of the hubs, which due to their high random walk centrality are critical in the trapping reaction. In addition, the degree distribution plays a significant role in the distribution of the particles recovering the depletion zone formation for high $\gamma$ values. The mean connectivity of the network is found to play a significant role in both ER and SF networks.