‘‘Lozenge’’ contour plots in scattering from polymer networks - reply

A Reply to the Comment by S. Westermann, et al

Reptation and contour-length fluctuations in melts of linear polymers

We present an analytical theory of stress relaxation in monodisperse linear polymer melts that contains contributions from both reptation and contour-length fluctuations, modeled as in our previous work on arm retraction in star polymers. Our approach resolves two long-standing problems with reptation theory: it predicts a zero-shear viscosity eta scaling as eta similar to N-3.4 over a broad range in chain length N before reaching an asymptotic N3 dependence, and a power law omega(-alpha) in the dynamic loss modulus G "(omega) with 0 < alpha < 1/4 depending on chain length, in agreement with experiment

Are there ergodic limits to evolution? Ergodic exploration of genome space and convergence

We examine the analogy between evolutionary dynamics and statistical mechanics to include the fundamental question of ergodicity—the representative exploration of the space of possible states (in the case of evolution this is genome space). Several properties of evolutionary dynamics are identified that allow a generalization of the ergodic dynamics, familiar in dynamical systems theory, to evolution. Two classes of evolved biological structure then arise, differentiated by the qualitative duration of their evolutionary time scales. The first class has an ergodicity time scale (the time required for representative genome exploration) longer than available evolutionary time, and has incompletely explored the genotypic and phenotypic space of its possibilities. This case generates no expectation of convergence to an optimal phenotype or possibility of its prediction. The second, more interesting, class exhibits an evolutionary form of ergodicity—essentially all of the structural space within the constraints of slower evolutionary variables have been sampled; the ergodicity time scale for the system evolution is less than the evolutionary time. In this case, some convergence towards similar optima may be expected for equivalent systems in different species where both possess ergodic evolutionary dynamics. When the fitness maximum is set by physical, rather than co-evolved, constraints, it is additionally possible to make predictions of some properties of the evolved structures and systems. We propose four structures that emerge from evolution within genotypes whose fitness is induced from their phenotypes. Together, these result in an exponential speeding up of evolution, when compared with complete exploration of genomic space. We illustrate a possible case of application and a prediction of convergence together with attaining a physical fitness optimum in the case of invertebrate compound eye resolution

A viscoelastic theory of turbulent fluid permeated with fibril magnetic fields

The solar convection zone is a turbulent plasma interacting with a magnetic field. Its magnetic field is often described as fibrillar since it consists of slender flux tubes occupying a small fraction of the total volume. It is well known that plasma flow will exert a force on these magnetic fibrils, but few models have accounted for the back-reaction of the fibrils on the flow. We present a model in which the back-reaction of the fibrils on the flow is manifest as viscoelastic properties. On short timescales the fibrils react elastically with a shear modulus proportional to their overall magnetic energy density. On longer timescales they produce an effective viscosity resulting from collective aerodynamic drag. The viscosity due to flux tubes in the solar convection zone can be comparable to that attributed to turbulence there. These forces might have observable effects on the convection zone flows

The Role of High-Dimensional Diffusive Search, Stabilization, and Frustration in Protein Folding

Proteins are polymeric molecules with many degrees of conformational freedom whose internal energetic interactions are typically screened to small distances. Therefore, in the high-dimensional conformation space of a protein, the energy landscape is locally relatively flat, in contrast to low-dimensional representations, where, because of the induced entropic contribution to the full free energy, it appears funnel-like. Proteins explore the conformation space by searching these flat subspaces to find a narrow energetic alley that we call a hypergutter and then explore the next, lower-dimensional, subspace. Such a framework provides an effective representation of the energy landscape and folding kinetics that does justice to the essential characteristic of high-dimensionality of the search-space. It also illuminates the important role of nonnative interactions in defining folding pathways. This principle is here illustrated using a coarse-grained model of a family of three-helix bundle proteins whose conformations, once secondary structure has formed, can be defined by six rotational degrees of freedom. Two folding mechanisms are possible, one of which involves an intermediate. The stabilization of intermediate subspaces (or states in low-dimensional projection) in protein folding can either speed up or slow down the folding rate depending on the amount of native and nonnative contacts made in those subspaces. The folding rate increases due to reduced-dimension pathways arising from the mere presence of intermediate states, but decreases if the contacts in the intermediate are very stable and introduce sizeable topological or energetic frustration that needs to be overcome. Remarkably, the hypergutter framework, although depending on just a few physically meaningful parameters, can reproduce all the types of experimentally observed curvature in chevron plots for realizations of this fold

A medieval multiverse?: Mathematical modelling of the thirteenth century universe of Robert Grosseteste

In his treatise on light, written about 1225, Robert Grosseteste describes a cosmological model in which the universe is created in a big-bang-like explosion and subsequent condensation. He postulates that the fundamental coupling of light and matter gives rises to the material body of the entire cosmos. Expansion is arrested when matter reaches a minimum density and subsequent emission of light from the outer region leads to compression and rarefaction of the inner bodily mass so as to create nine celestial spheres, with an imperfect residual core. In this paper, we reformulate the Latin description in terms of a modern mathematical model, teasing out consequences implicit in the text, but which the author would not have had the tools to explore. The equations which describe the coupling of light and matter are solved numerically, subjected to initial conditions and critical criteria consistent with the text. Formation of a universe with a non-infinite number of perfected spheres is extremely sensitive to the initial conditions, the intensity of the light and the transparency of these spheres. In this ‘medieval multiverse’, only a small range of opacity and initial density profiles leads to a stable universe with nine perfected spheres. As in current cosmological thinking, the existence of Grosseteste’s universe relies on a very special combination of fundamental parameters

Cross-slot extensional rheometry and the steady-state extensional response of long chain branched polymer melts

Stress-optical measurements at a flow stagnation point in confined geometries such as the cross-slot provide an elegant way to perform extensional testing for polymer melts. This technique is especially useful for samples which have a steady-state that cannot be reached (easily) in standard elongational rheometry, for example, highly branched polymers which show a non-homogeneous deformation that occurs in stretching experiments for Hencky strains above 4. In contrast to filament stretching, the cross-slot provides one point at which steady-state extensional flow may be sustained indefinitely. In this study, a Cambridge multi-pass rheometer [ Coventry, K. D., and M. R. Mackley, J. Rheol. 52, 401–415 (2008) ] is used to generate planar elongational flow in a cross-slot geometry for different polyethylene melts. The experimental results are compared to finite element flow simulations using the multi-mode Pompom constitutive equations. The steady-state elongational viscosity at the stagnation point is computed from the flow-induced stress birefringence and the strain-rate determined from numerical calculations of the flow field. We apply this technique to a range of different branched high- and low-density polyethylene melts. This demonstrates both the effectiveness of this technique and shows how the stress distribution in a complex flow depends on molecular structure. Cross slot extensional rheometry therefore provides a very promising technique for parameterizing molecular constitutive equations for LCB melts

Transient overshoot extensional rheology of long-chain branched polyethylenes: Experimental and numerical comparisons between filament stretching and cross-slot flow

This work analyses the high-strain extensional behavior of long-chain branched polyethylenes, employing two novel extensional rheometer devices, the filament stretching rheometer and the cross-slot extensional rheometer. The filament stretching rheometer uses an active feedback loop to control the imposed strain rate on a filament, allowing Hencky strains of around 7 to be reached. The cross-slot extensional rheometer uses optical birefringence patterns to determine the steady-state extensional viscosity from planar stagnation point flow. The two methods probe different strain-rate regimes and in this paper we demonstrate the agreement when the operating regimes overlap and explore the steady-state extensional viscosity in the full strain-rate regime that these two complimentary techniques offer. For long-chain branched materials, the cross-slot birefringence images show a double cusp pattern around the outflow centre line (named W-cusps). Using constitutive modeling of the observed transient overshoot in extension seen in the filament stretching rheometer and using finite element simulations we show that the overshoot explains the W-cusps seen in the cross-slot extensional rheometer, further confirming the agreement between the two experimental techniques. © 2013 The Society of Rheology

Dumbbell transport and deflection in a spatially periodic potential

We present theoretical results on the deterministic and stochastic motion of a dumbbell carried by a uniform flow through a three-dimensional spatially periodic potential. Depending on parameters like the flow velocity, there are two different kinds of movement: transport along a potential valley and stair-like motion oblique to the potential trenches. The crossover between these two regimes, as well as the deflection angle, depends on the size of the dumbbell. Moreover, thermal fluctuations cause a resonance-like variation in the deflection angle as a function of the dumbbell extension.Comment: 5 pages, 8 figure

Protein folding in high-dimensional spaces:hypergutters and the role of non-native interactions

We explore the consequences of very high dimensionality in the dynamical landscape of protein folding. Consideration of both typical range of stabilising interactions, and folding rates themselves, leads to a model of the energy hypersurface that is characterised by the structure of diffusive "hypergutters" as well as the familiar "funnels". Several general predictions result: (1) intermediate subspaces of configurations will always be visited; (2) specific but non-native interactions are important in stabilising these low-dimensional diffusive searches on the folding pathway; (3) sequential barriers will commonly be found, even in "two-state"proteins; (4) very early times will show charactreristic departures from single-exponential kinetics; (5) contributions of non-native interactions to phi-values are calculable, and may be significant. The example of a three-helix bundle is treated in more detail as an illustration. The model also shows that high-dimensional structures provide conceptual relations between the "folding funnel", "diffusion-collision", "nucleation-condensation" and "topomer search" models of protein folding. It suggests that kinetic strategies for fast folding may be encoded rather generally in non-native, rather than native interactions. The predictions are related to very recent findings in experiment and simulation.Comment: Submitted to Biophys.