Genetic circuits with feedback such as the toggle switch often exhibit bistability, namely, two stable states with rare spontaneous transitions between them. These systems can be characterized by the average time between such transitions (referred to as the switching time). However, commonly used deterministic models, based on rate equations, do not account for these fluctuation-induced transitions. Stochastic methods, such as the direct integration of the master equation, do account for the transitions. However, they cannot be used to evaluate the switching time. In order to obtain the switching time, one needs to use Monte Carlo simulations. These methods require the accumulation of statistical data, which limits their accuracy. They may become infeasible when the switching time is long. Here we present an accurate and efficient method for the calculation of the switching time. The method consists of coupled recursion equations for the transition times between microscopic states of the system. Using a suitable definition of the two macroscopic bistable states (in terms of the microscopic states) and the probabilities obtained from the master equation, the method provides the switching time between the two states of the system. The method is demonstrated for the genetic toggle switch. It can be used to evaluate the switching times in a broad range of bistable and multistable systems. We also show that it is suitable for the evaluation of the oscillation periods in oscillatory systems such as the repressilator.

It is evident from a wide range of experimental findings that ion channel gating is inherently stochastic. The issue of “memory effects” (diffusional retardation due to local changes in water viscosity) in ionic flow has been recently addressed using Brownian dynamics simulations. The results presented indicate such memory effects are negligible, unless the diffusional barrier is much higher than that of free solute. In this paper using differential stochastic methods we conclude that the Markovian property of exponential dwell times gives rise to a high barrier, resulting in diffusional memory effects that cannot be ignored in determining ionic flow through channels. We have addressed this question using a generalized Langevin equation that contains a combination of Markovian and non-Markovian processes with different time scales. This approach afforded the development of an algorithm that describes an oscillatory ionic diffusional sequence. The resulting oscillatory function behavior, with exponential decay, was obtained at the weak non-Markovian limit with two distinct time scales corresponding to the processes of ionic diffusion and drift. This will be analyzed further in future studies using molecular dynamics simulations. We propose that the rise of time scales and memory effects is related to differences of shear viscosity in the cytoplasm and extracellular matrix.

A clustering algorithm based on the Hausdorff distance is analyzed and compared to the single, complete, and average linkage algorithms. The four clustering procedures are applied to a toy example and to the time series of financial data. The dendrograms are scrutinized and their features compared. The Hausdorff linkage relies on firm mathematical grounds and turns out to be very effective when one has to discriminate among complex structures.

Bipartite graphs have received some attention in the study of social networks and of biological mutualistic systems. A generalization of a previous model is presented, that evolves the topology of the graph in order to optimally account for a given contact preference rule between the two guilds of the network. As a result, social and biological graphs are classified as belonging to two clearly different classes. Projected graphs, linking the agents of only one guild, are obtained from the original bipartite graph. The corresponding evolution of its statistical properties is also studied. An example of a biological mutualistic network is analyzed in detail, and it is found that the model provides a very good fitting of all the main statistical features. The model also provides a proper qualitative description of the same features observed in social webs, suggesting the possible reasons underlying the difference in the organization of these two kinds of bipartite networks.

In an effort to develop a chemically reactive interaction potential suitable for application to the study of conventional, organic explosives, we have modified the diatomic $AB$ potential of Brenner et al. [Phys. Rev. Lett. 70, 2174 (1993); 76, 2202(E) (1996)] such that it exhibits improved detonation characteristics. In particular, equilibrium molecular dynamics (MD) calculations of the modified potential demonstrate that the detonation products have an essentially diatomic, rather than polymeric, composition and that the detonation Hugoniot has the classic, concave-upward form. Nonequilibrium MD calculations reveal the separation of scales between chemical and hydrodynamic effects essential to the Zel’dovitch, von Neumann, and Döring theory.

After making modifications to the reactive empirical bond-order potential for molecular dynamics (MD) of Brenner et al. [Phys. Rev. Lett.70, 2174 (1993); 76, 2202(E) (1996)] in order to make the model behave in a more conventional manner, we discover that the updated model exhibits detonation instability, a first for MD. The instability is analyzed in terms of the accepted theory.

A lattice Boltzmann model is developed to simulate finite-rate catalytic surface chemistry. Diffusive wall boundary conditions are established to account for catalytic reactions in multicomponent mixtures. Implementation of wall boundary conditions with chemical reactions is based on a general second-order accurate interpolation scheme. Results of lattice Boltzmann simulations for a four-component mixture with a global catalytic methane oxidation reaction in a straight channel are in excellent agreement with a finite volume Navier-Stokes solver in terms of both the flow field and species concentrations.

The extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment. We investigate this effect using, as an example, a stochastic branching-annihilation process with a time-dependent branching rate. The population extinction is treated in eikonal approximation, where it is described as an instanton trajectory of a proper reaction Hamiltonian. The modulation of the environment perturbs this trajectory and synchronizes it with the modulation phase. We calculate the corresponding change in the action along the instanton using perturbation techniques supported by numerical calculations. The techniques include a first-order theory with respect to the modulation amplitude, a second-order theory in the spirit of the Kapitsa pendulum effect, and adiabatic theory valid for low modulation frequencies.

We study shock wave formation and propagation in an experimental vertically driven quasi-two-dimensional granular gas. We measure the moments of the single particle velocity distribution as a function of space and time. The space-time fields of the velocity moments show acoustic waves with a serrated substructure on the scale of a particle diameter. We show that this substructure is the result of collisional transport in which sequential collisions each transport momentum and energy by one particle diameter.

Insights into the melting of two-dimensional simple atomic systems are presented from investigation of the effect of the range of the interatomic potentials on the existence of hexatic phase, using molecular-dynamics simulations under isobaric-isothermal as well as isochoric-isothermal conditions. We find that longer-ranged interatomic potentials are important for the formation of stable hexatic phases. A schematic plot of the phase diagrams with a hexatic regime is presented capturing the overall shape of the phase boundaries and the behavior of the system. As the range of the potential is varied, the pressure-temperature phase diagram exhibits distinct topologies. For soft longer-ranged Morse potentials, the hexatic phase can coexist with the gaseous phase. On the other hand, as the range gets shorter, the onset point of the hexatic phase, i.e., the lower bound pressure of the hexatic region in the phase diagram, gets shifted upward, and the hexatic phase region no longer touches the gaseous phase.

The rates of chain aggregation of poly(methyl methacrylate) (PMMA) in acetonitrile (AcN) and in the mixed solvent of $\mathrm{Ac}\mathrm{N}+\text{water}$ $(10\mathrm{vol}\%)$ were determined by static light scattering and compared with the rates of chain collapse [Maki et al., J. Chem. Phys. 126, 134901 (2007)]. Dilute solutions of PMMA with the molecular weight $m_w=6.4\times {10}^6$ and in the concentration range of $(0.8–5)\times {10}^{-4}\mathrm{g}∕{\mathrm{cm}}^3$ were quenched below the cloud point, and the weight-average molecular weight $M_w$ and $z$-average square radius of gyration ${⟨S^2⟩}_z$ for clusters of PMMA chains were measured as a function of the time $t$ after the quench and the concentration $c$. The measurement of chain aggregation was carried out up to the cluster size of $M_w∕m_w\sim 30$, which required time periods of hours to several days depending on the concentration and solvent. The chain aggregation in $\mathrm{Ac}\mathrm{N}+\text{water}$ occurred much faster than that in AcN. The growth of clusters in both the solvents was represented by the exponential function as $M_w\sim e^{gct}$ and ${⟨S^2⟩}_z\sim e^{hct}$, where $g$ and $h$ represent the intrinsic rate of chain aggregation. The ratio $\sigma$ of the intrinsic rate in $\mathrm{Ac}\mathrm{N}+\text{water}$ to that in AcN was estimated to be 9 by taking a rough average of the ratios 9.4 obtained from $g$ and 8.8 from $h$. This value is comparable to the ratio 11 of the rate of chain collapse of PMMA in $\mathrm{Ac}\mathrm{N}+\text{water}$ $(10\mathrm{vol}\%)$ to that in AcN. This close value of the ratios indicates that the nature of solvent would affect the rates of chain collapse and chain aggregation through a similar mechanism.

DNA synthesis in Xenopus frog embryos initiates stochastically in time at many sites (origins) along the chromosome. Stochastic initiation implies fluctuations in the time to complete and may lead to cell death if replication takes longer than the cell cycle time $(\approx 25\min )$. Surprisingly, although the typical replication time is about $20\min$, in vivo experiments show that replication fails to complete only about 1 in 300 times. How is replication timing accurately controlled despite the stochasticity? Biologists have proposed two solutions to this “random-completion problem.” The first solution uses randomly located origins but increases their rate of initiation as $S$ phase proceeds, while the second uses regularly spaced origins. In this paper, we investigate the random-completion problem using a type of model first developed to describe the kinetics of first-order phase transitions. Using methods from the field of extreme-value statistics, we derive the distribution of replication-completion times for a finite genome. We then argue that the biologists’ first solution to the problem is not only consistent with experiment but also nearly optimizes the use of replicative proteins. We also show that spatial regularity in origin placement does not alter significantly the distribution of replication times and, thus, is not needed for the control of replication timing.

We propose an approach for inferring strength of coupling between two systems from their transient dynamics. This is of vital importance in cases where most information is carried by the transients, for instance, in evoked potentials measured commonly in electrophysiology. We show viability of our approach using nonlinear and linear measures of synchronization on a population model of thalamocortical loop and on a system of two coupled Rössler-type oscillators in nonchaotic regime.

Existing models of network growth typically have one or two parameters or strategies which are fixed for all times. We introduce a general framework where feedback on the current state of a network is used to dynamically alter the values of such parameters. A specific model is analyzed where limited resources are shared among arriving nodes, all vying to connect close to the root. We show that tunable feedback leads to growth of larger, more efficient networks. Exact results show that linear scaling of resources with system size yields crossover to a trivial condensed state, which can be considerably delayed with sublinear scaling.

We address the fundamental issue of network scalability in terms of dynamics and topology. In particular, we consider different network topologies and investigate, for every given topology, the dependence of certain dynamical properties on the network size. By focusing on network synchronizability, we find both analytically and numerically that globally coupled networks and random networks are scalable, but locally coupled regular networks are not. Scale-free networks are scalable for certain types of node dynamics. We expect our findings to provide insights into the ubiquity and workings of networks arising in nature and to be potentially useful for designing technological networks as well.

We have performed a series of large-scale atomistic simulations of the Rayleigh-Taylor instability including up to $5.7\times {10}^9$ particles and spanning time and length scales of up to $170\mathrm{ns}$ and $45\mu \mathrm{m}$, respectively. The results suggest that atomistic fluid dynamics simulations exhibit the same scaling as solutions of the continuum Navier-Stokes equations. Furthermore, a comparison with macroscopic Rayleigh-Taylor experiments suggests that the results of such atomistic simulations can, in fact, be scaled up to macroscopic dimensions, even for complex, nonstationary flows.

We describe the consequences of time-reversal invariance of the Stokes equations for the hydrodynamic scattering of two low-Reynolds-number swimmers. For swimmers that are related to each other by a time-reversal transformation, this leads to the striking result that the angle between the two swimmers is preserved by the scattering. The result is illustrated for the particular case of a linked-sphere model swimmer. For more general pairs of swimmers, not related to each other by time reversal, we find that hydrodynamic scattering can alter the angle between their trajectories by several tens of degrees. For two identical contractile swimmers, this can lead to the formation of a bound state.

The plasmonic venue to realize invisibility and cloaking [A. Alù and N. Engheta, Phys. Rev. E 72, 016623 (2005)] is analyzed here in terms of its limitations and its frequency dispersion relative to the cloak size. Intrinsic limits due to causality and comparison with transformation-based cloaking techniques are discussed and analyzed. An interestingly simple low-dispersion cloak is also suggested for background materials with larger refractive index. These results may shed light on this scattering cancellation phenomenon, suggesting potential applications in scattering reduction and noninvasive probing.

We study numerically the effect of a scale-free topology on the signal-to-noise ratio of networked two-state systems and find a double resonance phenomenon, i.e., a resonance on coupling strength and a stochastic resonance on noise strength. This finding suggests an alternative approach of self-tuning, i.e., tuning from the scale-free topology, instead of the self-tuning of potential. A heuristic theory through a starlike network is presented to explain the double resonance.

We propose an algorithm of coupling design for a long-term anticipating synchronization of chaos and demonstrate its efficacy for typical chaotic systems: namely, the Rössler system, the double-scroll Chua circuit, and the Lorenz system. The maximum prediction time attained with our algorithm is several times larger than with the diagonal coupling usually used in the literature.