Random walker models for durotaxis.

Motile biological cells in tissue often display the phenomenon of durotaxis, i.e. they tend to move towards stiffer parts of substrate tissue. The mechanism for this behavior is not completely understood. We consider simplified models for durotaxis based on the classic persistent random walker scheme. We show that even a one-dimensional model of this type sheds interesting light on the classes of behavior cells might exhibit. Our results strongly indicate that cells must be able to sense the gradient of stiffness in order to show the effects observed in experiment. This is in contrast to the claims in recent publications that it is sufficient for cells to be more persistent in their motion on stiff substrates to show durotaxis: i.e. it would be enough to sense the value of the stiffness. We show that these cases give rise to extremely inefficient transport towards stiff regions. Gradient sensing is almost certainly the selected behavior.

Optimal Wall-to-Wall Transport by Incompressible Flows.

We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields u. Given an enstrophy budget ⟨|∇u|^{2}⟩≤Pe^{2} we construct steady two-dimensional flows that transport at rates Nu(u)≳Pe^{2/3}/(logPe)^{4/3} in the large enstrophy limit. Combined with the known upper bound Nu(u)≲Pe^{2/3} for any such enstrophy-constrained flow, we conclude that maximally transporting flows satisfy Nu∼Pe^{2/3} up to possible logarithmic corrections. Combined with known transport bounds in the context of Rayleigh-Bénard convection, this establishes that while suitable flows approaching the "ultimate" heat transport scaling Nu∼Ra^{1/2} exist, they are not always realizable as buoyancy-driven flows. The result is obtained by exploiting a connection between the wall-to-wall optimal transport problem and a closely related class of singularly perturbed variational problems arising in the study of energy-driven pattern formation in materials science.

Resonant activation of population extinctions.

Understanding the mechanisms governing population extinctions is of key importance to many problems in ecology and evolution. Stochastic factors are known to play a central role in extinction, but the interactions between a population's demographic stochasticity and environmental noise remain poorly understood. Here we model environmental forcing as a stochastic fluctuation between two states, one with a higher death rate than the other. We find that, in general, there exists a rate of fluctuations that minimizes the mean time to extinction, a phenomenon previously dubbed "resonant activation." We develop a heuristic description of the phenomenon, together with a criterion for the existence of resonant activation. Specifically, the minimum extinction time arises as a result of the system approaching a scenario wherein the severity of rare events is balanced by the time interval between them. We discuss our findings within the context of more general forms of environmental noise and suggest potential applications to evolutionary models.

Gene expression dynamics with stochastic bursts: Construction and exact results for a coarse-grained model.

We present a theoretical framework to analyze the dynamics of gene expression with stochastic bursts. Beginning with an individual-based model which fully accounts for the messenger RNA (mRNA) and protein populations, we propose an expansion of the master equation for the joint process. The resulting coarse-grained model reduces the dimensionality of the system, describing only the protein population while fully accounting for the effects of discrete and fluctuating mRNA population. Closed form expressions for the stationary distribution of the protein population and mean first-passage times of the coarse-grained model are derived and large-scale Monte Carlo simulations show that the analysis accurately describes the individual-based process accounting for mRNA population, in contrast to the failure of commonly proposed diffusion-type models.

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection.

An alternative computational procedure for numerically solving a class of variational problems arising from rigorous upper-bound analysis of forced-dissipative infinite-dimensional nonlinear dynamical systems, including the Navier-Stokes and Oberbeck-Boussinesq equations, is analyzed and applied to Rayleigh-Bénard convection. A proof that the only steady state to which this numerical algorithm can converge is the required global optimal of the relevant variational problem is given for three canonical flow configurations. In contrast with most other numerical schemes for computing the optimal bounds on transported quantities (e.g., heat or momentum) within the "background field" variational framework, which employ variants of Newton's method and hence require very accurate initial iterates, the new computational method is easy to implement and, crucially, does not require numerical continuation. The algorithm is used to determine the optimal background-method bound on the heat transport enhancement factor, i.e., the Nusselt number (Nu), as a function of the Rayleigh number (Ra), Prandtl number (Pr), and domain aspect ratio L in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries (Rayleigh's original 1916 model of convection). The result of the computation is significant because analyses, laboratory experiments, and numerical simulations have suggested a range of exponents α and ß in the presumed Nu∼Pr(α)Ra(ß) scaling relation. The computations clearly show that for Ra≤10(10) at fixed L=2√[2],Nu≤0.106Pr(0)Ra(5/12), which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra regime.

Demographic stochasticity and evolution of dispersion II: spatially inhomogeneous environments.

Demographic stochasticity, the random fluctuations arising from the intrinsic discreteness of populations and the uncertainty of individual birth and death events, is an essential feature of population dynamics. Nevertheless theoretical investigations often neglect this naturally occurring noise due to the mathematical complexity of stochastic models. This paper reports the results of analytical and computational investigations of models of competitive population dynamics, specifically the competition between species in heterogeneous environments with different phenotypes of dispersal, fully accounting for demographic stochasticity. A novel asymptotic approximation is introduced and applied to derive remarkably simple analytical forms for key statistical quantities describing the populations' dynamical evolution. These formulas characterize the selection processes that determine which (if either) competitor has an evolutionary advantage. The theory is verified by large-scale numerical simulations. We discover that the fluctuations can (1) break dynamical degeneracies, (2) support polymorphism that does not exist in deterministic models, (3) reverse the direction of the weak selection and cause shifts in selection regimes, and (4) allow for the emergence of evolutionarily stable dispersal rates. Dynamical mechanisms and time scales of the fluctuation-induced phenomena are identified within the theoretical approach.

Demographic stochasticity and evolution of dispersion I. Spatially homogeneous environments.

The selection of dispersion is a classical problem in ecology and evolutionary biology. Deterministic dynamical models of two competing species differing only in their passive dispersal rates suggest that the lower mobility species has a competitive advantage in inhomogeneous environments, and that dispersion is a neutral characteristic in homogeneous environments. Here we consider models including local population fluctuations due to both individual movements and random birth and death events to investigate the effect of demographic stochasticity on the competition between species with different dispersal rates. In this paper, the first of two, we focus on homogeneous environments where deterministic models predict degenerate dynamics in the sense that there are many (marginally) stable equilibria with the species' coexistence ratio depending only on initial data. When demographic stochasticity is included the situation changes. A novel large carrying capacity ([Formula: see text]) asymptotic analysis, confirmed by direct numerical simulations, shows that a preference for faster dispersers emerges on a precisely defined [Formula: see text] time scale. We conclude that while there is no evolutionarily stable rate for competitors to choose in these models, the selection mechanism quantified here is the essential counterbalance in spatially inhomogeneous models including demographic fluctuations which do display an evolutionarily stable dispersal rate.

Ultimate state of two-dimensional Rayleigh-Bénard convection between free-slip fixed-temperature boundaries.

Rigorous upper limits on the vertical heat transport in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number Nu is bounded in terms of the Rayleigh number Ra according to Nu≤0.2891Ra(5/12) uniformly in the Prandtl number Pr. This scaling challenges some theoretical arguments regarding asymptotic high Rayleigh number heat transport by turbulent convection.

Bounds on heat transport in Bénard-Marangoni convection.

For Pearson's model of Bénard-Marangoni convection, the Nusselt number Nu is proven to be bounded as a function Marangoni number Ma according to Nu

Demographic stochasticity versus spatial variation in the competition between fast and slow dispersers.

Dispersal is an important strategy that allows organisms to locate and exploit favorable habitats. The question arises: given competition in a spatially heterogeneous landscape, what is the optimal rate of dispersal? Continuous population models predict that a species with a lower dispersal rate always drives a competing species to extinction in the presence of spatial variation of resources. However, the introduction of intrinsic demographic stochasticity can reverse this conclusion. We present a simple model in which competition between the exploitation of resources and stochastic fluctuations leads to victory by either the faster or slower of two species depending on the environmental parameters. A simplified limiting case of the model, analyzed by closing the moment and correlation hierarchy, quantitatively predicts which species will win in the complete model under given parameters of spatial variation and average carrying capacity.

Comparison of turbulent thermal convection between conditions of constant temperature and constant flux.

We report the results of high-resolution direct numerical simulations of two-dimensional Rayleigh-Bénard convection for Rayleigh numbers up to Ra=10;{10} in order to study the influence of temperature boundary conditions on turbulent heat transport. Specifically, we considered the extreme cases of fixed heat flux (where the top and bottom boundaries are poor thermal conductors) and fixed temperature (perfectly conducting boundaries). Both cases display identical heat transport at high Rayleigh numbers fitting a power law Nu approximately 0.138xRa;{0.285} with a scaling exponent indistinguishable from 2/7=0.2857... above Ra=10;{7}. The overall flow dynamics for both scenarios, in particular, the time averaged temperature profiles, are also indistinguishable at the highest Rayleigh numbers.

Multiscale mixing efficiencies for steady sources.

Multiscale mixing efficiencies for passive scalar advection are defined in terms of the suppression of variance weighted at various length scales. We consider scalars maintained by temporally steady but spatially inhomogeneous sources, stirred by statistically homogeneous and isotropic incompressible flows including fully developed turbulence. The mixing efficiencies are rigorously bounded in terms of the Péclet number and specific quantitative features of the source. Scaling exponents for the bounds at high Péclet number depend on the spectrum of length scales in the source, indicating that molecular diffusion plays a more important quantitative role than that implied by classical eddy diffusion theories.

Smooth and rough boundaries in turbulent Taylor-Couette flow.

We examine the torque required to drive the smooth or rough cylinders in turbulent Taylor-Couette flow. With rough inner and outer walls the scaling of the dimensionless torque G is found to be consistent with pure Kolmogorov scaling G approximately Re2. The results are interpreted within the Grossmann-Lohse theory for the relative role of the energy dissipation rates in the boundary layers and in the bulk; as the boundary layers are destroyed through the wall roughness, the torque scaling is due only to the bulk contribution. For the case of one rough and one smooth wall, we find that the smooth cylinder dominates the dissipation rate scaling, i.e., there are corrections to Kolmogorov scaling. A simple model based on an analogy to electrical circuits is advanced as a phenomenological organization of the observed relative drag functional forms. This model leads to a qualitative prediction for the mean velocity profile within the bulk of the flow.

Constructive role of noise: Fast fluctuation asymptotics of transport in stochastic ratchets.

The constructive role of random fluctuations is studied in the context of transport in stochastic ratchets. We discuss the interplay of independent white (thermal) and discrete (external) noises and their generation of transport in anisotropic potentials. The constructive cooperation of such fluctuations is most apparent in the asymptotic limit of fast discrete-valued noise, a limit which presents some interesting mathematical features. We describe the asymptotic analysis of the current in the limit of fast external noise, pointing out the strong qualitative dependence of the current on the interplay of the independent noise sources and its surprising sensitivity to the regularity of the underlying anisotropic ratchet potential. (c) 1998 American Institute of Physics.