Elasticity Dominated Surface Segregation of Small Molecules in Polymer Mixtures.

We study the phenomenon of migration of the small molecular weight component of a binary polymer mixture to the free surface using mean field and self-consistent field theories. By proposing a free energy functional that incorporates polymer-matrix elasticity explicitly, we compute the migrant volume fraction and show that it decreases significantly as the sample rigidity is increased. A wetting transition, observed for high values of the miscibility parameter can be prevented by increasing the matrix rigidity. Estimated values of the bulk modulus suggest that the effect should be observable experimentally for rubberlike materials. This provides a simple way of controlling surface migration in polymer mixtures and can play an important role in industrial formulations, where surface migration often leads to decreased product functionality.

Delayed self-regulation and time-dependent chemical drive leads to novel states in epigenetic landscapes.

The epigenetic pathway of a cell as it differentiates from a stem cell state to a mature lineage-committed one has been historically understood in terms of Waddington's landscape, consisting of hills and valleys. The smooth top and valley-strewn bottom of the hill represent their undifferentiated and differentiated states, respectively. Although mathematical ideas rooted in nonlinear dynamics and bifurcation theory have been used to quantify this picture, the importance of time delays arising from multistep chemical reactions or cellular shape transformations have been ignored so far. We argue that this feature is crucial in understanding cell differentiation and explore the role of time delay in a model of a single-gene regulatory circuit. We show that the interplay of time-dependent drive and delay introduces a new regime where the system shows sustained oscillations between the two admissible steady states. We interpret these results in the light of recent perplexing experiments on inducing the pluripotent state in mouse somatic cells. We also comment on how such an oscillatory state can provide a framework for understanding more general feedback circuits in cell development.

Allostery without conformation change: modelling protein dynamics at multiple scales.

The original ideas of Cooper and Dryden, that allosteric signalling can be induced between distant binding sites on proteins without any change in mean structural conformation, has proved to be a remarkably prescient insight into the rich structure of protein dynamics. It represents an alternative to the celebrated Monod-Wyman-Changeux mechanism and proposes that modulation of the amplitude of thermal fluctuations around a mean structure, rather than shifts in the structure itself, give rise to allostery in ligand binding. In a complementary approach to experiments on real proteins, here we take a theoretical route to identify the necessary structural components of this mechanism. By reviewing and extending an approach that moves from very coarse-grained to more detailed models, we show that, a fundamental requirement for a body supporting fluctuation-induced allostery is a strongly inhomogeneous elastic modulus. This requirement is reflected in many real proteins, where a good approximation of the elastic structure maps strongly coherent domains onto rigid blocks connected by more flexible interface regions.

Micelle shape transitions in block copolymer/homopolymer blends: comparison of self-consistent field theory with experiment.

Diblock copolymers blended with homopolymer may self-assemble into spherical, cylindrical, or lamellar aggregates. Transitions between these structures may be driven by varying the homopolymer diblock molecular weight or composition. Using self-consistent field theory (SCFT), we reproduce these effects. Our results are compared to x-ray scattering and transmission electron microscopy measurements by Kinning et al. and good agreement is found, although the tendency to form cylindrical and lamellar structures is sometimes overestimated due to our neglect of edge effects due to the finite size of these aggregates. Our results demonstrate that SCFT can provide detailed information on the self-assembly of isolated block copolymer aggregates.

Molecular physics of a polymer engineering instability: experiments and computation.

Entangled polymer melts exhibit a variety of flow instabilities that limit production rates in industrial applications. We present both experimental and computational findings, using flow of monodisperse linear polystyrenes in a contraction-expansion geometry, which illustrate the formation and development of one such flow instability. This viscoelastic disturbance is observed at the slit outlet and subsequently produces large-scale fluid motions upstream. A numerical linear stability study using the molecular structure based Rolie-Poly model confirms the instability and identifies important parameters within the model, which gives physical insight into the underlying mechanism. Chain stretch was found to play a critical role in the instability mechanism, which partially explains the effectiveness of introducing a low-molecular weight tail into a polymer blend to increase its processability.

Demixing instability in coil-rod blends undergoing polycondensation reactions.

The authors extend their earlier work on the stability of a reacting binary polymer blend with respect to demixing [D. J. Read, Macromolecules 31, 899 (1998); P. I. C. Teixeira et al., Macromolecules 33, 387 (2000)] to the case where one of the polymers is rod-like and may order nematically. As before, the authors combine the random phase approximation for the free energy with a Markov chain model for the chemistry to obtain the spinodal as a function of the relevant degrees of reaction. These are then calculated by assuming a simple second-order chemical kinetics. Results are presented, for linear systems, which illustrate the effects of varying the proportion of coils and rods, their relative sizes, and the strength of the nematic interaction between the rods.

Diffusive searches in high-dimensional spaces and apparent 'two-state' behaviour in protein folding.

We extend a simple model for protein folding as a high-dimensional diffusive search. By solving a steady-state diffusion equation on a hypersphere centred on an absorbing 'native state' we find the general property that the kinetics of such a search will always be nearly single exponential. This explains the common observation of such simple 'two-state' folding kinetics in models that contain considerable intermediate structure. It also suggests that the experimental signature of single-exponential folding kinetics does not imply a simple two-state structure to the folding space.

A theory for heterogeneous states of polymer melts produced by single chain crystal melting.

We consider the polymer physics of a possible molecular mechanism for disentangled melt states observed recently. We find that the route from molten, but only marginally-overlapping single crystals to the equilibrium state must pass over a free-energy barrier that becomes large at high molecular weights. The penalty arises from elastic distortions of the entanglement network as the chains diffuse. A critical molecular weight arises naturally from the competition of elastic distortion and the free-energy of confinement. We calculate the barrier and critical molecular weight at both a simple single-chain level, and at a "one-loop" level of co-operative motion, finding that many-chain effects alter the physics quantitatively, but not qualitatively. Several new experiments are suggested.

Small angle neutron scattering observation of chain retraction after a large step deformation.

The process of retraction in entangled linear chains after a fast nonlinear stretch was detected from time-resolved but quenched small angle neutron scattering (SANS) experiments on long, well-entangled polyisoprene chains. The statically obtained SANS data cover the relevant time regime for retraction, and they provide a direct, microscopic verification of this nonlinear process as predicted by the tube model. Clear, quantitative agreement is found with recent theories of contour length fluctuations and convective constraint release, using parameters obtained mainly from linear rheology. The theory captures the full range of scattering vectors once the crossover to fluctuations on length scales below the tube diameter is accounted for.

Protein folding in high-dimensional spaces: hypergutters and the role of nonnative interactions.

We explore the consequences of very high dimensionality in the dynamical landscape of protein folding. Consideration of both typical range of stabilizing interactions, and folding rates themselves, leads to a model of the energy hypersurface that is characterized 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 nonnative interactions may be important in stabilizing these low-dimensional diffusive searches on the folding pathway, as well as native interactions; 3), sequential barriers will commonly be found, even in "two-state" proteins; 4), very early times will show characteristic departures from single-exponential kinetics; and 5), contributions of nonnative interactions to Phi-values and "Chevron plots" 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 different models of protein folding. It suggests that kinetic strategies for fast folding may be encoded rather generally in nonnative as well as in native interactions. The predictions are related to very recent findings in experiment and simulation.

Neutron-mapping polymer flow: scattering, flow visualization, and molecular theory.

Flows of complex fluids need to be understood at both macroscopic and molecular scales, because it is the macroscopic response that controls the fluid behavior, but the molecular scale that ultimately gives rise to rheological and solid-state properties. Here the flow field of an entangled polymer melt through an extended contraction, typical of many polymer processes, is imaged optically and by small-angle neutron scattering. The dual-probe technique samples both the macroscopic stress field in the flow and the microscopic configuration of the polymer molecules at selected points. The results are compared with a recent "tube model" molecular theory of entangled melt flow that is able to calculate both the stress and the single-chain structure factor from first principles. The combined action of the three fundamental entangled processes of reptation, contour length fluctuation, and convective constraint release is essential to account quantitatively for the rich rheological behavior. The multiscale approach unearths a new feature: Orientation at the length scale of the entire chain decays considerably more slowly than at the smaller entanglement length.

Molecular observation of contour-length fluctuations limiting topological confinement in polymer melts.

In order to study the mechanisms limiting the topological chain confinement in polymer melts, we have performed neutron-spin-echo investigations of the single-chain dynamic-structure factor from polyethylene melts over a large range of chain lengths. While at high molecular weight the reptation model is corroborated, a systematic loosening of the confinement with decreasing chain length is found. The dynamic-structure factors are quantitatively described by the effect of contour-length fluctuations on the confining tube, establishing this mechanism on a molecular level in space and time.

Theoretical and finite-element investigation of the mechanical response of spinodal structures.

In recent years there have been major advances in our understanding of the mechanisms of phase separation in polymer and copolymer blends, to the extent that good control of phase-separated morphology is a real possibility. Many groups are studying the computational simulation of polymer phase separation. In the light of this, we are exploring methods which will give insight into the mechanical response of multiphase polymers. We present preliminary results from a process which allows the production of a two-dimensional finite-element mesh from the contouring of simulated composition data. We examine the stretching of two-phase structures obtained from a simulation of linear Cahn-Hilliard spinodal phase separation. In the simulations, we assume one phase to be hard, and the other soft, such that the shear modulus ratio G is large (>or= 10(3)). We indicate the effect of varying composition on the material modulus and on the distribution of strains through the stretched material. We also examine in some detail the symmetric structures obtained at 50% composition, in which both phases are at a percolation threshold. Inspired by simulation results for the deformation of these structures, we construct a "scaling" theory, which reproduces the main features of the deformation. Of particular interest is the emergence of a lengthscale, below which the deformation is non-affine. This length is proportional to G(1/4), and hence is still quite small for all reasonable values of this ratio. The same theory predicts that the effective composite modulus scales also as G(1/4), , which is supported by the simulations.

Structure and dynamics of self-assembling beta-sheet peptide tapes by dynamic light scattering.

Oligomeric peptides can be designed which undergo one-dimensional self-assembly in solution to form beta-sheet tapes a single molecule in thickness and micrometers in length.(1) In this paper, we present the first systematic investigation of the size, shape, dynamics, and interactions of beta-sheet tapes formed by a self-assembling 24-residue peptide, K24, in 2-chloroethanol, over a wide range of peptide concentrations c (c: 10(-7)-1 mM), using photon correlation spectroscopy. The tapes behave like semiflexible chains with persistence lengths of several hundred nanometers and much longer contour lengths, even at c approximately 0.1 nM. The polarized q-dependent light-scattering intensity I fits a model of a prolate object with major and minor axes alpha approximately 630 nm and beta approximately 40 nm. This is an unexpected result in view of the previous theoretical predictions that tapelike polymers could form oblate coinlike structures in solution.(2) This experimentally observed behavior is attributed to the pronounced twist and bend of the beta-tapes, which do not allow them to form the coinlike structures, but instead they favor the formation of elongated polymers. At c approximately 10(-2) mM, the tapes are seen to start overlapping and forming networks with unusually large mesh sizes (e.g., ca. 400 nm at 15 mciroM), much larger than those of conventional polymers. With increasing peptide concentration the mesh size decreases and the network becomes a physical gel at c approximately 0.4 mM. These semidilute solutions are characterized by one main relaxation mode associated with the cooperative diffusion of the entangled tape network, and a weaker slower mode, associated with gel cluster formation. The concentration dependence of xi (xi(c) approximately c(-0.34)) is much weaker compared to the expected scaling for Gaussian or swollen chains (xi(c) approximately c(-1) for Gaussian chains, or xi(c) approximately c(-3/4) for swollen ones), but is not inconsistent with the expected scaling for rigid rods. On the basis of the concentration dependencies of the light-scattering intensity I and of the cooperative diffusion coefficient D, the cooperative friction coefficient f(c) is found to display a stronger concentration dependence (f(c) approximately c(1.34)) than in the case of semidilute flexible and semiflexible polymer solutions (f(c) approximately c(0.5)).(3) Thus, we may conclude that the network of entangled tapes approximates in its behavior that of semirigid polymers.

Hierarchical self-assembly of chiral rod-like molecules as a model for peptide beta -sheet tapes, ribbons, fibrils, and fibers.

A generic statistical mechanical model is presented for the self-assembly of chiral rod-like units, such as beta-sheet-forming peptides, into helical tapes, which with increasing concentration associate into twisted ribbons (double tapes), fibrils (twisted stacks of ribbons), and fibers (entwined fibrils). The finite fibril width and helicity is shown to stem from a competition between the free energy gain from attraction between ribbons and the penalty because of elastic distortion of the intrinsically twisted ribbons on incorporation into a growing fibril. Fibers are stabilized similarly. The behavior of two rationally designed 11-aa residue peptides, P(11)-I and P(11)-II, is illustrative of the proposed scheme. P(11)-I and P(11)-II are designed to adopt the beta-strand conformation and to self-assemble in one dimension to form antiparallel beta-sheet tapes, ribbons, fibrils, and fibers in well-defined solution conditions. The energetic parameters governing self-assembly have been estimated from the experimental data using the model. The 8-nm-wide fibrils consist of eight tapes, are extremely robust (scission energy approximately 200 k(B)T), and sufficiently rigid (persistence length l(fibril) approximately 20-70 microm) to form nematic solutions at peptide concentration c approximately 0.9 mM (volume fraction approximately 0.0009 vol/vol), which convert to self-supporting nematic gels at c > 4 mM. More generally, these observations provide a new insight into the generic self-assembling properties of beta-sheet-forming peptides and shed new light on the factors governing the structures and stability of pathological amyloid fibrils in vivo. The model also provides a prescription of routes to novel macromolecules based on a variety of self-assembling chiral units, and protocols for extraction of the associated energy changes.

Responsive gels formed by the spontaneous self-assembly of peptides into polymeric beta-sheet tapes.

Molecular self-assembly is becoming an increasingly popular route to new supramolecular structures and molecular materials. The inspiration for such structures is commonly derived from self-assembling systems in biology. Here we show that a biological motif, the peptide beta-sheet, can be exploited in designed oligopeptides that self-assemble into polymeric tapes and with potentially useful mechanical properties. We describe the construction of oligopeptides, rationally designed or based on segments of native proteins, that aggregate in suitable solvents into long, semi-flexible beta-sheet tapes. These become entangled even at low volume fractions to form gels whose viscoelastic properties can be controlled by chemical (pH) or physical (shear) influences. We suggest that it should be possible to engineer a wide range of properties in these gels by appropriate choice of the peptide primary structure.