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Molecules provide the ultimate language in terms of which physiology and pathology must be understood. Myriads of proteins participate in elaborate networks of interactions and perform chemical activities coordinating the life of cells. To perform these often amazing tasks, proteins must move and we must think of them as dynamic ensembles of three dimensional structures formed first by folding the polypeptide chains so as to minimize the conflicts between the interactions of their constituent amino acids. It is apparent however that, even when completely folded, not all conflicting interactions have been resolved so the structure remains "locally frustrated". Over the last decades it has become clearer that this local frustration is not just a random accident but plays an essential part of the inner workings of protein molecules. We will review here the physical origins of the frustration concept and review evidence that local frustration is important for protein physiology, protein-protein recognition, catalysis and allostery. Also, we highlight examples showing how alterations in the local frustration patterns can be linked to distinct pathologies. Finally we explore the extensions of the impact of frustration in higher order levels of organization of systems including gene regulatory networks and the neural networks of the brain. more...

During mitosis, near-spherical chromosomes reconfigure into rod-like structures to ensure their accurate segregation to daughter cells. We explore here, the interplay between the nonequilibrium activity of molecular motors in determining the chromosomal organization in mitosis and its characteristic symmetry-breaking events. We present a hybrid motorized chromosome model that highlights the distinct roles of condensin I and II in shaping mitotic chromosomes. Guided by experimental observations, the simulations suggest that condensin II facilitates large-scale scaffold formation, while condensin I is paramount in local helical loop arrangement. Together, these two distinct grappling motors establish the hierarchical helical structure characteristic of mitotic chromosomes, which exhibit striking local and, sometimes global, chirality and contribute to the robust mechanical properties of mitotic chromosomes. Accompanying the emergence of rigidity, the model provides mechanisms of forming defects, including perversions and entanglements, and shows how these may be partially resolved through condensin activity and topoisomerase action. This framework bridges coarse-grained energy landscape models of chromosome dynamics and non-equilibrium molecular dynamics, advancing the understanding of chromosome organization during cell division and beyond., Comment: 45 pages, 6 figures more...

Understanding structural glasses using the Random First Order Transition theory requires a description in both real space and in time, taking into account sample history. A variety of nonlinear objects enter this description, in field theory terms, when there is replica symmetry breaking: instantons, strings, flames and flows. We focus on this real space description and use it to describe the crossover to the energy landscape regime, dynamical heterogeneity, aging and the formation of shear bands in glasses under stress., Comment: This manuscript will appear in the book "Spin Glass Theory and Far Beyond - Replica Symmetry Breaking after 40 Years" more...

We derive an unbiased information theoretic energy landscape for chromosomes at metaphase using a maximum entropy approach that accurately reproduces the details of the experimentally measured pair-wise contact probabilities between genomic loci. Dynamical simulations using this landscape lead to cylindrical, helically twisted structures reflecting liquid crystalline order. These structures are similar to those arising from a generic ideal homogenized chromosome energy landscape. The helical twist can be either right or left handed so chiral symmetry is broken spontaneously. The ideal chromosome landscape when augmented by interactions like those leading to topologically associating domain (TAD) formation in the interphase chromosome reproduces these behaviors. The phase diagram of this landscape shows the helical fiber order and the cylindrical shape persist at temperatures above the onset of chiral symmetry breaking which is limited by the TAD interaction strength. more...

Several recent experiments suggest that rather generally the diffusion of enzymes may be augmented through their activity. We demonstrate that such swimming motility can emerge from the interplay between the enzyme energy landscape and the hydrodynamic coupling of the enzyme to its environment. Swimming thus occurs during the transit time of a transient allosteric change. We estimate the velocity during the transition. The analysis of such a swimming motion suggests the final stroke size is limited by the hydrodynamic size of the enzyme. This limit is quite a bit smaller than the values that can be inferred from the recent experiments. We also show that one proposed explanation of the experiments based on reaction heat effects can be ruled out using an extended hydrodynamic analysis. These results lead us to propose an alternate explanation of the fluorescence correlation measurements. more...

Cytoskeletal networks, which are essentially motor-filament assemblies, play a major role in many developmental processes involving structural remodeling and shape changes. These are achieved by nonequilibrium self-organization processes that generate functional patterns and drive intracellular transport. We construct a minimal physical model that incorporates the coupling between nonlinear elastic responses of individual filaments and force-dependent motor action. By performing stochastic simulations we show that the interplay of motor processes, described as driving anti-correlated motion of the network vertices, and the network connectivity, which determines the percolation character of the structure, can indeed capture the dynamical and structural cooperativity which gives rise to diverse patterns observed experimentally. The buckling instability of individual filaments is found to play a key role in localizing collapse events due to local force imbalance. Motor-driven buckling-induced node aggregation provides a dynamic mechanism that stabilizes the two dimensional patterns below the apparent static percolation limit. Coordinated motor action is also shown to suppress random thermal noise on large time scales, the two dimensional configuration that the system starts with thus remaining planar during the structural development. By carrying out similar simulations on a three dimensional anchored network, we find that the myosin-driven isotropic contraction of a well-connected actin network, when combined with mechanical anchoring that confers directionality to the collective motion, may represent a novel mechanism of intracellular transport, as revealed by chromosome translocation in the starfish oocyte., Comment: 11 pages, 5 figures more...

Biomolecules are the prime information processing elements of living matter. Most of these inanimate systems are polymers that compute their structures and dynamics using as input seemingly random character strings of their sequence, following which they coalesce and perform integrated cellular functions. In large computational systems with a finite interaction-codes, the appearance of conflicting goals is inevitable. Simple conflicting forces can lead to quite complex structures and behaviors, leading to the concept of "frustration" in condensed matter. We present here some basic ideas about frustration in biomolecules and how the frustration concept leads to a better appreciation of many aspects of the architecture of biomolecules, and how structure connects to function. These ideas are simultaneously both seductively simple and perilously subtle to grasp completely. The energy landscape theory of protein folding provides a framework for quantifying frustration in large systems and has been implemented at many levels of description. We first review the notion of frustration from the areas of abstract logic and its uses in simple condensed matter systems. We discuss then how the frustration concept applies specifically to heteropolymers, testing folding landscape theory in computer simulations of protein models and in experimentally accessible systems. Studying the aspects of frustration averaged over many proteins provides ways to infer energy functions useful for reliable structure prediction. We discuss how frustration affects folding, how a large part of the biological functions of proteins are related to subtle local frustration effects and how frustration influences the appearance of metastable states, the nature of binding processes, catalysis and allosteric transitions. We hope to illustrate how Frustration is a fundamental concept in relating function to structural biology., Comment: 97 pages, 30 figures more...

Signatures of glassy dynamics have been identified experimentally for a rich variety of materials in which molecular networks provide rigidity. Here we present a theoretical framework to study the glassy behavior of both passive and active network materials. We construct a general microscopic network model that incorporates nonlinear elasticity of individual filaments and steric constraints due to crowding. Based on constructive analogies between structural glass forming liquids and random field Ising magnets implemented using a heterogeneous self-consistent phonon method, our scheme provides a microscopic approach to determine the mismatch surface tension and the configurational entropy, which compete in determining the barrier for structural rearrangements within the random first order transition theory of escape from a local energy minimum. The influence of crosslinking on the fragility of inorganic network glass formers is recapitulated by the model. For active network materials, the mapping, which correlates the glassy characteristics to the network architecture and properties of nonequilibrium motor processes, is shown to capture several key experimental observations on the cytoskeleton of living cells: Highly connected tense networks behave as strong glass formers; intense motor action promotes reconfiguration. The fact that our model assuming a negative motor susceptibility predicts the latter suggests that on average the motorized processes in living cells do resist the imposed mechanical load. Our calculations also identify a spinodal point where simultaneously the mismatch penalty vanishes and the mechanical stability of amorphous packing disappears., Comment: 23 pages, 12 figures more...

Actomyosin networks are major structural components of the cell. They provide mechanical integrity and allow dynamic remodeling of eukaryotic cells, self-organizing into the diverse patterns essential for development. We provide a theoretical framework to investigate the intricate interplay between local force generation, network connectivity and collective action of molecular motors. This framework is capable of accommodating both regular and heterogeneous pattern formation, arrested coarsening and macroscopic contraction in a unified manner. We model the actomyosin system as a motorized cat's cradle consisting of a crosslinked network of nonlinear elastic filaments subjected to spatially anti-correlated motor kicks acting on motorized (fibril) crosslinks. The phase diagram suggests there can be arrested phase separation which provides a natural explanation for the aggregation and coalescence of actomyosin condensates. Simulation studies confirm the theoretical picture that a nonequilibrium many-body system driven by correlated motor kicks can behave as if it were at an effective equilibrium, but with modified interactions that account for the correlation of the motor driven motions of the actively bonded nodes. Regular aster patterns are observed both in Brownian dynamics simulations at effective equilibrium and in the complete stochastic simulations. The results show that large-scale contraction requires correlated kicking., Comment: 38 pages, 13 figures more...

The interplay between cytoskeletal architecture and the nonlinearity of the interactions due to bucklable filaments plays a key role in modulating the cell's mechanical stability and affecting its structural rearrangements. We study a model of cytoskeletal structure treating it as an amorphous network of hard centers rigidly cross-linked by nonlinear elastic strings, neglecting the effects of motorization. Using simulations along with a self-consistent phonon method, we show that this minimal model exhibits diverse thermodynamically stable mechanical phases that depend on excluded volume, crosslink concentration, filament length and stiffness. Within the framework set by the free energy functional formulation and making use of the random first order transition theory of structural glasses, we further estimate the characteristic densities for a kinetic glass transition to occur in this model system. Network connectivity strongly modulates the transition boundaries between various equilibrium phases, as well as the kinetic glass transition density., Comment: 17 pages, 18 figures more...

Spontaneous directed motion, a hallmark of cell biology, is unusual in classical statistical physics. Here we study, using both numerical and analytical methods, organized motion in models of the cytoskeleton in which constituents are driven by energy-consuming motors. While systems driven by small-step motors are described by an effective temperature and are thus quiescent, at higher order in step size, both homogeneous and inhomogeneous, flowing and oscillating behavior emerges. Motors that respond with a negative susceptibility to imposed forces lead to an apparent negative-temperature system in which beautiful structures form resembling the asters seen in cell division., Comment: 17 pages, 5 figures (2 supplementary figures) more...

A systematic expansion of the many-body master equation for active matter, in which motors power configurational changes as in the cytoskeleton, is shown to yield a description of the steady state and responses in terms of an effective temperature. The effective temperature depends on the susceptibility of the motors and a Peclet number which measures their strength relative to thermal Brownian diffusion. The analytic prediction is shown to agree with previous numerical simulations and experiments. The mapping also establishes a description of aging in active matter that is also kinetically jammed., Comment: 2 figures more...

The folding pathway and rate coefficients of the folding of a knotted protein are calculated for a potential energy function with minimal energetic frustration. A kinetic transition network is constructed using the discrete path sampling approach, and the resulting potential energy surface is visualized by constructing disconnectivity graphs. Owing to topological constraints, the low-lying portion of the landscape consists of three distinct regions, corresponding to the native knotted state and to configurations where either the N- or C-terminus is not yet folded into the knot. The fastest folding pathways from denatured states exhibit early formation of the N-terminus portion of the knot and a rate-determining step where the C-terminus is incorporated. The low-lying minima with the N-terminus knotted and the C-terminus free therefore constitute an off-pathway intermediate for this model. The insertion of both the N- and C-termini into the knot occur late in the folding process, creating large energy barriers that are the rate limiting steps in the folding process. When compared to other protein folding proteins of a similar length, this system folds over six orders of magnitude more slowly., Comment: 19 pages more...

In recent years it has become widely accepted that a dynamical length scale {\xi}_{\alpha} plays an important role in supercooled liquids near the glass transition. We examine the implications of the interplay between the growing {\xi}_{\alpha} and the size of the crystal nucleus, {\xi}_M, which shrinks on cooling. We argue that at low temperatures where {\xi}_{\alpha} > {\xi}_M a new crystallization mechanism emerges enabling rapid development of a large scale web of sparsely connected crystallinity. Though we predict this web percolates the system at too low a temperature to be easily seen in the laboratory, there are noticeable residual effects near the glass transition that can account for several previously observed unexplained phenomena of deeply supercooled liquids including Fischer clusters, and anomalous crystal growth near T_g. more...

Nearly all glass forming liquids display secondary relaxations, dynamical modes seemingly distinct from the primary alpha relaxations. We show that accounting for driving force fluctuations and the diversity of reconfiguring shapes in the random first order transition theory yields a low free energy tail on the activation barrier distribution which shares many of the features ascribed to secondary relaxations. While primary relaxation takes place through activated events involving compact regions, secondary relaxation corresponding to the tail is governed by more ramified, string-like, or percolation-like clusters of particles. These secondary relaxations merge with the primary relaxation peak becoming dominant near the dynamical crossover temperature $T_c$, where they smooth the transition between continuous dynamics described by mode-coupling theory and activated events. more...

As a liquid is progressively supercooled an intriguing weakening of the wavenumber ($q$) dependence of the structural relaxation time $\tau(q)$ in the large q limit is observed both in experiments and simulation studies. Neither the continuous Brownian diffusive dynamics nor the discontinuous activated events can alone explain the anomalous wavenumber dependence. Here we use our recently developed theory that unifies the mode coupling theory (MCT) for continuous dynamics and the random first order transition theory (RFOT) treatment of activated discontinuous motion as a nucleation (instanton) process, to understand the wavenumber dependence of density relaxation. The predicted smooth change in mechanism of relaxation from diffusive to activated, in the crossover regime, is wavevector dependent and is eventually responsible for sub-quadratic, almost linear, $q$ dependence of the relaxation time., Comment: 15 pages and 5 figures, minor changes in the content more...

Complex spatiotemporal structures develop during the process of aging glasses after cooling and of rejuvenating glasses upon heating. The key to understanding these structures is the interplay between the activated reconfiguration events which generate mobility and the transport of mobility. These effects are both accounted for by combining the random first order transition theory of activated events with mode coupling theory in an inhomogeneous setting. The predicted modifications by mobility transport of the time course of the aging regime are modest. In contrast, the rejuvenation process is strongly affected through the propagation of fronts of enhanced mobility originating from the initial reconfiguration events. The structures in a rejuvenating glass resemble flames. An analysis along the lines of combustion theory provides an estimate of the front propagation speed. Heterogeneous rejuvenation naturally should occur for glasses with free surfaces. The analogy with combustion also provides a new way of looking at the uptake of diluents by glasses described by case II and super case II diffusion. more...

Dynamics near the surface of glasses is generally much faster than in the bulk. Neglecting static perturbations of structure at the surface, we use random first order transition theory to show the free energy barrier for activated motion near a free surface should be half that of the bulk at the same temperature. The increased mobility allows the surface layers to descend much further on the energy landscape than the bulk ordinarily does. The simplified RFOT calculation however predicts a limiting value for the configurational entropy a vapor deposited glass may reach as a function of deposition rate. We sketch how mode coupling effects extend the excess free surface mobility into the bulk so that the glass transition temperature is measurably perturbed at depths greater than the naive length scale of dynamic cooperativity., Comment: 5 pages 2 figures more...

In low temperature supercooled liquid, below the ideal mode coupling theory transition temperature, hopping and continuous diffusion are seen to coexist. We present a theory which incorporates interaction between the two processes and shows that hopping can induce continuous diffusion in the otherwise frozen liquid. Several universal features arise from nonlinear interactions between the continuous diffusive dynamics (described here by the mode coupling theory (MCT)) and the activated hopping (described here by the random first order transition theory). We apply the theory to real systems (Salol) to show that the theory correctly predicts the temperature dependence of the non-exponential stretching parameter, $\beta$, and the primary $\alpha$ relaxation timescale, $\tau$. The study explains why, even below the ergodic to non-ergodic transition, the dynamics is well described by MCT. The non-linear coupling between the two dynamical processes modifies the relaxation behavior of the structural relaxation from what would be predicted by a theory with a complete static Gaussian barrier distribution in a manner that may be described as a facilitation effect. Furthermore, the theory explains the observed variation of the stretching exponent $\beta$ with the fragility parameter, $D$., Comment: 8 pages, 4 figures more...

By defining a spatially varying replica overlap parameter for a supercooled liquid referenced to an ensemble of fiducial liquid state configurations we explicitly construct a constrained replica free energy functional that maps directly onto an Ising Hamiltonian with both random fields and random interactions whose statistics depend on liquid structure. Renormalization group results for random magnets when combined with these statistics for the Lennard-Jones glass suggest that discontinuous replica symmetry breaking would occur if a liquid with short range interactions could be equilibrated at a sufficiently low temperature where its mean field configurational entropy would vanish, even though the system strictly retains a finite configurational entropy. more...

Random first order transition theory is used to determine the role of attractive and repulsive interactions in the dynamics of supercooled liquids. Self-consistent phonon theory, an approximate mean field treatment consistent with random first order transition theory, is used to treat individual glassy configurations, while the liquid phase is treated using common liquid state approximations. The transition temperature $T^{*}_{A} $, the temperature where the onset of activated behavior is predicted by mean field theory, the lower crossover temperature $T_{c}^{*}$ where barrierless motions actually occur through fractal or stringy motions, and $T^{*}_{K} $, the Kauzmann temperature, are calculated in addition to $T^{*}_{g} $, the glass transition temperature that corresponds to laboratory cooling rates. Both the isobaric and isochoric behavior in the supercooled regime are studied, providing results for $\Delta C_{V} $ and $\Delta C_{p} $ that can be used to calculate the fragility as a function of density and pressure, respectively. The predicted variations in the $\alpha$-relaxation time with temperature and density conform to the empirical density-temperature scaling relations found by Casalini and Roland. We thereby demonstrate the microscopic origin of their observations. Finally, the relationship first suggested by Sastry between the spinodal temperature and the Kauzmann temperatures, as a function of density, is examined. The present microscopic calculations support the existence of an intersection of these two temperatures at sufficiently low temperatures., Comment: Submitted to J. Phys. Chem. A, June 2007 Replaced with accepted version Sept. 2007 more...

We compare a recent excitation chain argument for the glass transition with the earlier random first order transition theory. The key equation determining the activation barriers and size of cooperatively rearranging regions has the same scaling form in both approaches. The random first order transition theory unambiguously predicts the coefficients in the equation giving results that agree with experiment vis a vis the correlation of activation barriers with thermodynamics and that also agree with experimental determination of correlation lengths following the prescription of Berthier et al. The excitation chain approach, while containing more adjustable parameters accommodates those experimental findings only by using unphysical values for those adjustable parameters., Comment: 3 pages, 1 figure more...

It is believed that, much like a cat's cradle, the cytoskeleton can be thought of as a network of strings under tension. We show that both regular and random bond-disordered networks having bonds that buckle upon compression exhibit a variety of phase transitions as a function of temperature and extension. The results of self-consistent phonon calculations for the regular networks agree very well with computer simulations at finite temperature. The analytic theory also yields a rigidity onset (mechanical percolation) and the fraction of extended bonds for random networks. There is very good agreement with the simulations by Delaney et al. (Europhys. Lett. 2005). The mean field theory reveals a nontranslationally invariant phase with self-generated heterogeneity of tautness, representing ``antiferroelasticity''., Comment: 4 pages, 4 figures more...

The shapes of cooperatively rearranging regions in glassy liquids change from being compact at low temperatures to fractal or ``stringy'' as the dynamical crossover temperature from activated to collisional transport is approached from below. We present a quantitative microscopic treatment of this change of morphology within the framework of the random first order transition theory of glasses. We predict a correlation of the ratio of the dynamical crossover temperature to the laboratory glass transition temperature, and the heat capacity discontinuity at the glass transition, Delta C_p. The predicted correlation agrees with experimental results for the 21 materials compiled by Novikov and Sokolov., Comment: 9 pages, 6 figures more...

Thermodynamics and kinetics are thought to be linked in glass transitions. The quantitative predictions of -relaxation activation barriers provided by the theory of glasses based on random first order transitions are compared with experiment for 44 substances. The agreement found between the predicted activation energies near Tg and experiment is excellent. These predictions depend on the configurational heat capacity change on vitrification and the entropy of melting the crystal which are experimental inputs., Comment: 15 pages, 1 figure, 1 table. submitted to J. Phys. Chem more...

The random first order transition theory of the dynamics of supercooled liquids is extended to treat aging phenomena in nonequilibrium structural glasses. A reformulation of the idea of ``entropic droplets'' in terms of libraries of local energy landscapes is introduced which treats in a uniform way the supercooled liquid (reproducing earlier results) and glassy regimes. The resulting microscopic theory of aging makes contact with the Nayaranaswamy-Moynihan-Tool nonlinear relaxation formalism and the Hodge-Scherer extrapolation of the Adam-Gibbs formula, but deviations from both approaches are predicted and shown to be consistent with experiment. The nonlinearity of glassy relaxation is shown to quantitatively correlate with liquid fragility. The residual nonArrhenius temperature dependence of relaxation observed in quenched glasses is explained. The broadening of relaxation spectra in the nonequilibrium glass with decreasing temperature is quantitatively predicted. The theory leads to the prediction of spatially fluctuating fictive temperatures in the long-aged glassy state, which have non-Gaussian statistics. This can give rise to ``ultra-slow'' relaxations in systems after deep quenches., Comment: Submitted to J. Chem. Phys more...

Many of the large structures of the cell, such as the cytoskeleton, are assembled and maintained far from equilibrium. We study the stabilities of various structures for a simple model of such a far-from-equilibrium organized assembly in which spherical particles move under the influence of attached motors. From the variational solutions of the manybody master equation for Brownian motion with motorized kicking we obtain a closed equation for the order parameter of localization. Thus we obtain the transition criterion for localization and stability limits for the crystalline phase and frozen amorphous structures of motorized particles. The theory also allows an estimate of nonequilibrium effective temperatures characterizing the response and fluctuations of motorized crystals and glasses., Comment: 5 pages, 3 figures more...

According to the Random First Order Transition (RFOT) theory of glasses, the barriers for activated dynamics in supercooled liquids vanish as the temperature of a viscous liquid approaches the dynamical transition temperature from below. This occurs due to a decrease of the surface tension between local meta-stable molecular arrangements much like at a spinodal. The dynamical transition thus represents a crossover from the low $T$ activated bevavior to a collisional transport regime at high $T$. This barrier softening explains the deviation of the relaxation times, as a function of temperature, from the simple $\log \tau \propto 1/s_c$ dependence at the high viscosity to a mode-mode coupling dominated result at lower viscosity. By calculating the barrier softening effects, the RFOT theory provides a {\em unified} microscopic way to interpret structural relaxation data for many distinct classes of structural glass formers over the measured temperature range. The theory also provides an unambiguous procedure to determine the size of dynamically cooperative regions in the presence of barrier renormalization effects using the experimental temperature dependence of the relaxation times and the configurational entropy data. We use the RFOT theory framework to discuss data for tri-naphthyl benzene, salol, propanol and silica as representative systems., Comment: Submitted to J. Chem. Phys more...

We show that the tendency towards microphase separation in microemulsions leads to the formation of a glassy state after sufficiently strong correlations between polar and hydrophobic regions have been established. Glassiness is predicted to occur above a critical volume fraction of surfactant, which is determined by the length of the amphiphilic molecules. Our results are obtained by solving the dynamical equations for the correlation functions of the system and by using a replica approach., Comment: 4 pages, RevTeX, 3 eps figures embedded in the text more...

The motion involved in barrier crossing for protein folding are investigated in terms of the chain dynamics of the polymer backbone, completing the microscopic description of protein folding presented in the previous paper. Local reaction coordinates are identified as collective growth modes of the unstable fluctuations about the saddle-points in the free energy surface. The description of the chain dynamics incorporates internal friction (independent of the solvent viscosity) arising from the elementary isomerizations of the backbone dihedral angles. We find that the folding rate depends linearly on the solvent friction for high viscosity, but saturates at low viscosity because of internal friction. For $\lambda$-repressor, the calculated folding rate prefactor, along with the free energy barrier from the variational theory, gives a folding rate that agrees well with the experimentally determined rate under highly stabilizing conditions, but the theory predicts too large a folding rate at the transition midpoint. This discrepancy obtained using a fairly complete quantitative theory inspires a new set of questions about chain dynamics, specifically detailed motions in individual contact formation., Comment: 18 pages, 8 figures more...

A microscopic theory of the free energy barriers and folding routes for minimally frustrated proteins is presented, greatly expanding on the presentation of the variational approach outlined previously [J. J. Portman, S. Takada, P. G. Wolynes, Phys. Rev. Lett. {\bf 81}, 5237 (1998)]. We choose the $\lambda$-repressor protein as an illustrative example and focus on how the polymer chain statistics influence free energy profiles and partially ordered ensembles of structures. In particular, we investigate the role of chain stiffness on the free energy profile and folding routes. We evaluate the applicability of simpler approximations in which the conformations of the protein molecule along the folding route are restricted to have residues that are either entirely folded or unfolded in contiguous stretches. We find that the folding routes obtained from only one contiguous folded region corresponds to a chain with a much greater persistence length than appropriate for natural protein chains, while the folding route obtained from two contiguous folded regions is able to capture the relatively folded regions calculated within the variational approach. The free energy profiles obtained from the contiguous sequence approximations have larger barriers than the more microscopic variational theory which is understood as a consequence of partial ordering., Comment: 16 pages, 11 figures more...

We present a microscopic variational theory for the free energy surface of a fast folding protein that allows folding kinetics to be resolved to the residue level using Debye-Waller factors as local order parameters. We apply the method to lambda-repressor and compare with site directed mutagenesis experiments. The formation of native structure and the free energy profile along the folding route are shown to be well described by the capillarity approximation but with some fine structure due to local folding topology., Comment: 5 pages, 3 figures, minor corrections, Published Version more...

The non-Markovian nature of polymer motions is accounted for in folding kinetics, using frequency-dependent friction. Folding, like many other problems in the physics of disordered systems, involves barrier crossing on a correlated energy landscape. A variational transition state theory (VTST) that reduces to the usual Bryngelson-Wolynes Kramers approach when the non-Markovian aspects are neglected is used to obtain the rate, without making any assumptions regarding the size of the barrier, or the memory time of the friction. The transformation to collective variables dependent on the dynamics of the system allows the theory to address the controversial issue of what are ``good'' reaction coordinates for folding., Comment: 9 pages RevTeX, 3 eps-figures included, submitted to PRL more...