The sequence of a protein is not only constrained by its physical and biochemical properties under current selection, but also by features of its past evolutionary history. Understanding the extent and the form that these evolutionary constraints may take is important to interpret the information in protein sequences. To study this problem, we introduce a simple but physical model of protein evolution where selection targets allostery, the functional coupling of distal sites on protein surfaces. This model shows how the geometrical organization of couplings between amino acids within a protein structure can depend crucially on its evolutionary history. In particular, two scenarios are found to generate a spatial concentration of functional constraints: high mutation rates and fluctuating selective pressures. This second scenario offers a plausible explanation for the high tolerance of natural proteins to mutations and for the spatial organization of their least tolerant amino acids, as revealed by sequence analysis and mutagenesis experiments. It also implies a faculty to adapt to new selective pressures that is consistent with observations. The model illustrates how several independent functional modules may emerge within the same protein structure, depending on the nature of past environmental fluctuations. Our model thus relates the evolutionary history of proteins to the geometry of their functional constraints, with implications for decoding and engineering protein sequences.

We study the quantum kicked rotator in the classically fully chaotic regime $K=10$ and for various values of the quantum parameter $k$ using Izrailev's $N$-dimensional model for various $N\le 3000$, which in the limit $N\to \infty$ tends to the exact quantized kicked rotator. By numerically calculating the eigenfunctions in the basis of the angular momentum we find that the localization length $\mathcal{L}$ for fixed parameter values has a certain distribution; in fact, its inverse is Gaussian distributed, in analogy and in connection with the distribution of finite time Lyapunov exponents of Hamilton systems. However, unlike the case of the finite time Lyapunov exponents, this distribution is found to be independent of $N$ and thus survives the limit $N=\infty$. This is different from the tight-binding model of Anderson localization. The reason is that the finite bandwidth approximation of the underlying Hamilton dynamical system in the Shepelyansky picture [Phys. Rev. Lett. 56, 677 (1986)] does not apply rigorously. This observation explains the strong fluctuations in the scaling laws of the kicked rotator, such as the entropy localization measure as a function of the scaling parameter $\Lambda =\mathcal{L}/N$, where $\mathcal{L}$ is the theoretical value of the localization length in the semiclassical approximation. These results call for a more refined theory of the localization length in the quantum kicked rotator and in similar Floquet systems, where we must predict not only the mean value of the inverse of the localization length $\mathcal{L}$ but also its (Gaussian) distribution, in particular the variance. In order to complete our studies we numerically analyze the related behavior of finite time Lyapunov exponents in the standard map and of the $2\times 2$ transfer matrix formalism. This paper extends our recent work [Phys. Rev. E 87, 062905 (2013)].

When applied to strongly nonlinear chaotic dynamics the extended Kalman filter (EKF) is prone to divergence due to the difficulty of correctly forecasting the forecast error probability density function. In operational forecasting applications ensemble Kalman filters circumvent this problem with empirical procedures such as covariance inflation. This paper presents an extension of the EKF that includes nonlinear terms in the evolution of the forecast error estimate. This is achieved starting from a particular square-root implementation of the EKF with assimilation confined in the unstable subspace (EKF-AUS), that is, the span of the Lyapunov vectors with non-negative exponents. When the error evolution is nonlinear, the space where it is confined is no more restricted to the unstable and neutral subspace causing filter divergence. The algorithm presented here, denominated EKF-AUS-NL, includes the nonlinear terms in the error dynamics: These result from the nonlinear interaction among the leading Lyapunov vectors and account for all directions where the error growth may take place. Numerical results show that with the nonlinear terms included, filter divergence can be avoided. We test the algorithm on the Lorenz96 model, showing very promising results.

We consider a nine-partial-differential-equation (1-space and 1-time) model of plane Couette flow in which the degrees of freedom are severely restricted in the streamwise and cross-stream directions to study spanwise localization in detail. Of the many steady Eckhaus (spanwise modulational) instabilities identified of global steady states, none lead to a localized state. Spatially localized, time-periodic solutions were found instead, which arise in saddle node bifurcations in the Reynolds number. These solutions appear global (domain filling) in narrow (small spanwise) domains yet can be smoothly continued out to fully spanwise-localized states in very wide domains. This smooth localization behavior, which has also been seen in fully resolved duct flow (S. Okino, Ph.D. thesis, Kyoto University, Kyoto, 2011), indicates that an apparently global flow structure does not have to suffer a modulational instability to localize in wide domains.

We consider horizontal linear shear flow (shear rate denoted by $\Lambda$) under vertical uniform rotation (ambient rotation rate denoted by ${\Omega }_0$) and vertical stratification (buoyancy frequency denoted by $N$) in unbounded domain. We show that, under a primary vertical velocity perturbation and a radial density perturbation consisting of a one-dimensional standing wave with frequency $N$ and amplitude proportional to $w_{0}sin(\varepsilon Nx/w_0)\approx \varepsilon Nx(\ll 1),$ where $x$ denotes the radial coordinate and $\varepsilon$ a small parameter, a parametric instability can develop in the flow, provided $N^2>8{\Omega }_0(2{\Omega }_0-\Lambda ).$ For astrophysical accretion flows and under the shearing sheet approximation, this implies $N^2>8{\Omega }_0^2\left(2-q\right)$, where $q=\Lambda /{\Omega }_0$ is the local shear gradient. In the case of a stratified constant angular momentum disk, $q=2,$ there is a parametric instability with the maximal growth rate $({\sigma }_m/\varepsilon )=3\sqrt{3}/16$ for any positive value of the buoyancy frequency $N.$ In contrast, for a stratified Keplerian disk, $q=1.5,$ the parametric instability appears only for $N>2{\Omega }_0$ with a maximal growth rate that depends on the ratio ${\Omega }_0/N$ and approaches $(3\sqrt{3}/16)\varepsilon$ for large values of $N.$

We describe the dynamics of vapor nanobubbles in water, on the basis of simulations of a hydrodynamics phase-field model. This situation is relevant to recent experiments, where a water nanobubble is generated around a nanoparticle immersed in water, and heated by an intense laser pulse. We emphasize the importance of nanoscale effects in the dynamics of the nanobubble. We first analyze the evolution of the temperature inside the bubble. We show that the temperature drops by hundredths of kelvins in a few picoseconds, just after nanobubble formation. This is the result of the huge drop of the thermal boundary conductance between the nanoparticle and the fluid accompanying vaporization. Subsequently, the temperature inside the vapor is almost homogeneous and the temperature gradient is concentrated in the liquid, whose thermodynamic state locally follows the saturation line. We discuss also the evolution of the pressure inside the vapor nanobubble. We show that nanobubble generation is accompanied by a pressure wave propagating in the liquid at a velocity close to the liquid speed of sound. The internal pressure inside the vapor just after its formation largely exceeds Laplace pressure and quickly relaxes as a result of the damping generated by the viscous forces. All these considerations shed light on the thermodynamics of the nanobubbles generated experimentally.

Fermi resonance is a phenomenon of quantum mechanical superposition, which most often occurs between normal and overtone modes in molecular systems that are nearly coincident in energy. We find that scarred resonances in deformed dielectric microcavities are the very phenomenon of Fermi resonance, that is, a pair of quasinormal modes interact with each other due to coupling and a pair of resonances are generated through an avoided resonance crossing. Then the quantum number difference of a pair of quasinormal modes, which is a consequence of quantum mechanical superposition, equals periodic orbits, whereby the resonances are localized on the periodic orbits. We derive the relation between the quantum number difference and the periodic orbits and confirm it in an elliptic, a rectangular, and a stadium-shaped dielectric microcavity.

Particles and fields are standard components in numerical calculations like transport simulations in nuclear physics and have well-understood dynamics. Still, a common problem is the interaction between particles and fields due to their different formal description. Particle interactions are discrete, pointlike events while field dynamics is described with continuous partial-differential equations of motion. A workaround is the use of effective theories like the Langevin equation with the drawback of energy conservation violation. We present a method, which allows us to model noncontinuous interactions between particles and scalar fields, allowing us to simulate scattering-like interactions which exchange discrete “quanta” of energy and momentum between fields and particles while obeying energy and momentum conservation and allowing control over interaction strengths and times. In this paper we apply this method to different model systems, starting with a simple harmonic oscillator, which is damped by losing discrete energy quanta. The second and third system consists of an oscillator and a one-dimensional field, which are damped via discrete energy loss and are coupled to a stochastic force, leading to equilibrium states which correspond to statistical Langevin-like systems. The last example is a scalar field in (1 + 3) space-time dimensions, which is coupled to a microcanonical ensemble of particles by incorporating particle production and annihilation processes. Obeying the detailed-balance principle, the system equilibrates to thermal and chemical equilibrium with dynamical fluctuations on the fields, generated dynamically by the discrete interactions.

The progress of realizing colloidal structures mimicking natural forms of organization in condensed matter is inherently limited by the availability of suitable colloidal building blocks. To enable new forms of crystalline and quasicrystalline self-organization of colloids, we develop truncated pyramidal particles that form nematic elastic dipoles with long-range electrostaticlike and geometry-guided low-symmetry short-range interactions. Using a combination of nonlinear optical imaging, laser tweezers, and video microscopy, we characterize colloidal pair interactions and demonstrate unusual forms of self-tiling of these particles into crystalline, quasicrystalline, and other arrays. Our findings are explained using an electrostatics analogy along with liquid crystal elasticity and symmetry breaking considerations, potentially expanding photonic and electro-optic applications of colloids.

A three-stranded DNA with short range base pairings only is known to exhibit a classical analog of the quantum Efimov effect, viz., a three-chain bound state at the two-chain melting point where no two are bound. By using a nonperturbative renormalization-group method for a rigid duplex DNA and a flexible third strand, with base pairings and strand exchange, we show that the Efimov-DNA is associated with a limit cycle type behavior of the flow of an effective three-chain interaction. The analysis also shows that thermally generated bubbles play an essential role in producing the effect. A toy model for the flow equations shows the limit cycle in an extended three-dimensional parameter space of the two-chain coupling and a complex three-chain interaction.

We found that a nontrivial enhancement of network reciprocity for 2 × 2 prisoner's dilemma games can be achieved by coupling two mechanisms. The first mechanism presumes a larger strategy update neighborhood than the conventional first neighborhood on the underlying network. The second is the strategy-shifting rule. At the initial time step, the averaged cooperation extent is assumed to be 0.5. In the case of strategy shifting, an agent adopts a continuous strategy definition during the initial period of a simulation episode (when the global cooperation fraction decreases from its initial value of 0.5; that is, the enduring period). The agent then switches to a discrete strategy definition in the time period afterwards (when the global cooperation fraction begins to increase again; that is, the expanding period). We explored why this particular enhancement comes about, and to summarize, the continuous strategy during the initial period relaxes the conditions for the survival of relatively cooperative clusters, and the large strategy adaptation neighborhood allows those cooperative clusters to expand easily.

Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.

Bridging equilibrium and nonequilibrium statistical physics attracts sustained interest. Hallmarks of nonequilibrium systems include a breakdown of detailed balance, and an absence of a priori potential function corresponding to the Boltzmann-Gibbs distribution, without which classical equilibrium thermodynamical quantities could not be defined. Here, we construct dynamically the potential function through decomposing the system into a dissipative part and a conservative part, and develop a nonequilibrium theory by defining thermodynamical quantities based on the potential function. Concepts for equilibrium can thus be naturally extended to nonequilibrium steady state. We elucidate this procedure explicitly in a class of time-dependent linear diffusive systems without mathematical ambiguity. We further obtain the exact work distribution for an arbitrary control parameter, and work equalities connecting nonequilibrium steady states. Our results provide a direct generalization on Jarzynski equality and Crooks fluctuation theorem to systems without detailed balance.

We consider the thermal undulation, or shape fluctuation, of an almost planar fluid membrane surrounded by the same near-critical binary fluid mixtures on both sides. A weak preferential attraction is assumed between the membrane and one component of the mixture. We use the Gaussian free-energy functional to study the equilibrium average of the undulation amplitude within the linear approximation with respect to the amplitude. According to our result given by a simple analytic formula, the ambient near-criticality tends to suppress the undulation of a membrane, and this suppression effect can overwhelm that of the bending rigidity for small wave numbers. Thus, the ambient near-criticality is suggested to prevent a large membrane from becoming floppy even if the lateral tension vanishes at the equilibrium.

We developed two-photon scanning patterned illumination microscopy (2P-SPIM) for super-resolution two-photon imaging. Our approach used a traditional two-photon microscopy setup with temporally modulated excitation to create patterned illumination fields. Combing nine different illuminations and structured illumination reconstruction, super-resolution imaging was achieved in two-photon microscopy. Using 2P-SPIM we achieved a lateral resolution of 141 nm, which represents an improvement by a factor of 1.9 over the corresponding diffraction limit. We further demonstrated super-resolution cellular imaging by 2P-SPIM to image actin cytoskeleton in mammalian cells and three-dimensional imaging in highly scattering retinal tissue.

Preferential attachment, by which new nodes attach to existing nodes with probability proportional to the existing nodes' degree, has become the standard growth model for scale-free networks, where the asymptotic probability of a node having degree $k$ is proportional to $k^{-\gamma }$. However, the motivation for this model is entirely ad hoc. We use exact likelihood arguments and show that the optimal way to build a scale-free network is to attach most new links to nodes of low degree. Curiously, this leads to a scale-free network with a single dominant hub: a starlike structure we call a superstar network. Asymptotically, the optimal strategy is to attach each new node to one of the nodes of degree $k$ with probability proportional to $\frac{1}{N+\zeta \left(\gamma \right){(k+1)}^{\gamma }}$ (in a $N$ node network): a stronger bias toward high degree nodes than exhibited by standard preferential attachment. Our algorithm generates optimally scale-free networks (the superstar networks) as well as randomly sampling the space of all scale-free networks with a given degree exponent $\gamma$. We generate viable realization with finite $N$ for $1\ll \gamma <2$ as well as $\gamma >2$. We observe an apparently discontinuous transition at $\gamma \approx 2$ between so-called superstar networks and more treelike realizations. Gradually increasing $\gamma$ further leads to reemergence of a superstar hub. To quantify these structural features, we derive a new analytic expression for the expected degree exponent of a pure preferential attachment process and introduce alternative measures of network entropy. Our approach is generic and can also be applied to an arbitrary degree distribution.

We derive explicit, closed-form expressions for the cumulant densities of a multivariate, self-exciting Hawkes point process, generalizing a result of Hawkes in his earlier work on the covariance density and Bartlett spectrum of such processes. To do this, we represent the Hawkes process in terms of a Poisson cluster process and show how the cumulant density formulas can be derived by enumerating all possible “family trees,” representing complex interactions between point events. We also consider the problem of computing the integrated cumulants, characterizing the average measure of correlated activity between events of different types, and derive the relevant equations.

By using the F-expansion method associated with four auxiliary equations, i.e., the Bernoulli equation, the Riccati equation, the Lenard equation, and the hyperbolic equation, we present exact explicit solutions describing the dynamics of matter-wave condensates with time-varying two- and three-body nonlinearities. Condensates are trapped in a harmonic potential and they exchange atoms with the thermal cloud. These solutions include the generalized Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. In addition, we have also found rational function solutions. Solutions constructed here have many free parameters that can be used to manipulate and control some important features of the condensate, such as the position, width, velocity, acceleration, and homogeneous phase. The stability of the solutions is confirmed by their long-time numerical behavior.

We consider a two-dimensional (2D) generalization of a recently proposed model [Gligorić et al., Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual “extended” unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being those considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (also analytically, when possible). Typical scenarios of instability development are exhibited through direct simulations.

In this paper, random-site percolation thresholds for a simple cubic (SC) lattice with site neighborhoods containing next-next-next-nearest neighbors (4NN) are evaluated with Monte Carlo simulations. A recently proposed algorithm with low sampling for percolation thresholds estimation (Bastas et al., arXiv:1411.5834) is implemented for the studies of the top-bottom wrapping probability. The obtained percolation thresholds are $p_C\left(4\mathrm{NN}\right)=0.31160\left(12\right),p_C\left(4\mathrm{NN}+\mathrm{NN}\right)=0.15040\left(12\right),p_C\left(4\mathrm{NN}+2\mathrm{NN}\right)=0.15950\left(12\right),p_C\left(4\mathrm{NN}+3\mathrm{NN}\right)=0.20490\left(12\right),p_C\left(4\mathrm{NN}+2\mathrm{NN}+\mathrm{NN}\right)=0.11440\left(12\right),p_C\left(4\mathrm{NN}+3\mathrm{NN}+\mathrm{NN}\right)=0.11920\left(12\right),p_C\left(4\mathrm{NN}+3\mathrm{NN}+2\mathrm{NN}\right)=0.11330\left(12\right)$, and $p_C\left(4\mathrm{NN}+3\mathrm{NN}+2\mathrm{NN}+\mathrm{NN}\right)=0.10000\left(12\right)$, where 3NN, 2NN, and NN stand for next-next-nearest neighbors, next-nearest neighbors, and nearest neighbors, respectively. As an SC lattice with 4NN neighbors may be mapped onto two independent interpenetrated SC lattices but with a lattice constant that is twice as large, the percolation threshold $p_C$(4NN) is exactly equal to $p_C$(NN). The simplified method of Bastas et al. allows for uncertainty of the percolation threshold value $p_C$ to be reached, similar to that obtained with the classical method but ten times faster.