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We study the forced rupture of adhesive contacts between monomers that are not covalently linked in a Rouse chain. When the applied force ($f$) to the chain end is less than the critical force for rupture ($f_c$), the {\it reversible} rupture process is coupled to the internal Rouse modes. If $f/f_{c}$$>$1 the rupture is {\it irreversible}. In both limits, the non-exponential distribution of contact lifetimes, which depends sensitively on the location of the contact, follows the double-exponential (Gumbel) distribution. When two contacts are well separated along the chain, the rate limiting step in the {\it sequential} rupture kinetics is the disruption of the contact that is in the chain interior. If the two contacts are close to each other, they cooperate to sustain the stress, which results in an ``all-or-none'' transition., Comment: 9 pages, 3 figures, accepted to Phys. Rev. Lett

E. Coli. dihydrofolate reductase (DHFR) undergoes conformational transitions between the closed (CS) and occluded (OS) states which, respectively, describe whether the active site is closed or occluded by the Met20 loop. A sequence-based approach is used to identify a network of residues that represents the allostery wiring diagram. We also use a self-organized polymer model to monitor the kinetics of the CS->OS and the reverse transitions. a sliding motion of Met20 loop is observed. The residues that facilitate the Met20 loop motion are part of the network of residues that transmit allosteric signals during the CS->OS transition., Comment: 43 pages, 12 figures

We investigate the kinetics of loop formation in flexible ideal polymer chains (Rouse model), and polymers in good and poor solvents. We show for the Rouse model, using a modification of the theory of Szabo, Schulten, and Schulten, that the time scale for cyclization is $\tau_c\sim \tau_0 N^2$ (where $\tau_0$ is a microscopic time scale and $N$ is the number of monomers), provided the coupling between the relaxation dynamics of the end-to-end vector and the looping dynamics is taken into account. The resulting analytic expression fits the simulation results accurately when $a$, the capture radius for contact formation, exceeds $b$, the average distance between two connected beads. Simulations also show that, when $a < b$, $\tau_c\sim N^{\alpha_\tau}$, where $1.5<{\alpha_\tau}\le 2$ in the range $7

Force induced stretching of polymers is important in a variety of contexts. We have used theory and simulations to describe the response of homopolymers, with $N$ monomers, to force ($f$) in good and poor solvents. In good solvents and for {{sufficiently large}} $N$ we show, in accord with scaling predictions, that the mean extension along the $f$ axis $\sim f$ for small $f$, and $\sim f^{{2/3}}$ (the Pincus regime) for intermediate values of $f$. The theoretical predictions for $\la Z\ra$ as a function of $f$ are in excellent agreement with simulations for N=100 and 1600. However, even with N=1600, the expected Pincus regime is not observed due to the the breakdown of the assumptions in the blob picture for finite $N$. {{We predict the Pincus scaling in a good solvent will be observed for $N\gtrsim 10^5$}}. The force-dependent structure factors for a polymer in a poor solvent show that there are a hierarchy of structures, depending on the nature of the solvent. For a weakly hydrophobic polymer, various structures (ideal conformations, self-avoiding chains, globules, and rods) emerge on distinct length scales as $f$ is varied. A strongly hydrophobic polymer remains globular as long as $f$ is less than a critical value $f_c$. Above $f_c$, an abrupt first order transition to a rod-like structure occurs. Our predictions can be tested using single molecule experiments., Comment: 24 pages, 7 figures

We analyze the dependence of cooperativity of the thermal denaturation transition and folding rates of globular proteins on the number of amino acid residues, $N$, using lattice models with side chains,off-lattice Go models and the available experimental data. A dimensionless measure of cooperativity, $\Omega_c$ ($0 < \Omega_c < \infty$), scales as $\Omega_c \sim N^{\zeta}$. The results of simulations and the analysis of experimental data further confirm the earlier prediction that $\zeta$ is universal with $\zeta = 1 +\gamma$, where exponent $\gamma$ characterizes the susceptibility of a self-avoiding walk. This finding suggests that the structural characteristics in the denaturated state are manifested in the folding cooperativity at the transition temperature. The folding rates $k_F$ for the Go models and a dataset of 69 proteins can be fit using $k_F = k_F^0 \exp(-cN^\beta)$. Both $\beta = 1/2$ and 2/3 provide a good fit of the data. We find that $k_F = k_F^0 \exp(-cN^{{1/2}})$, with the average (over the dataset of proteins) $k_F^0 \approx (0.2\mu s)^{-1}$ and $c \approx 1.1$, can be used to estimate folding rates to within an order of magnitude in most cases. The minimal models give identical $N$ dependence with $c \approx 1$. The prefactor for off-lattice Go models is nearly four orders of magnitude larger than the experimental value., Comment: 17 pages, 3 figures, 2 tables

Single molecule mechanical unfolding experiments are beginning to provide profiles of the complex energy landscape of biomolecules. In order to obtain reliable estimates of the energy landscape characteristics it is necessary to combine the experimental measurements with sound theoretical models and simulations. Here, we show how by using temperature as a variable in mechanical unfolding of biomolecules in laser optical tweezer or AFM experiments the roughness of the energy landscape can be measured without making any assumptions about the underlying reaction oordinate. The efficacy of the formalism is illustrated by reviewing experimental results that have directly measured roughness in a protein-protein complex. The roughness model can also be used to interpret experiments on forced-unfolding of proteins in which temperature is varied. Estimates of other aspects of the energy landscape such as free energy barriers or the transition state (TS) locations could depend on the precise model used to analyze the experimental data. We illustrate the inherent difficulties in obtaining the transition state location from loading rate or force-dependent unfolding rates. Because the transition state moves as the force or the loading rate is varied it is in general difficult to invert the experimental data unless the curvature at the top of the one dimensional free energy profile is large, i.e the barrier is sharp. The independence of the TS location on force holds good only for brittle or hard biomolecules whereas the TS location changes considerably if the molecule is soft or plastic. We also comment on the usefulness of extension of the molecule as a surrogate reaction coordinate especially in the context of force-quench refolding of proteins and RNA., Comment: 44 pages, 7 figures

The chaperonin GroEL-GroES, a machine which helps some proteins to fold, cycles through a number of allosteric states, the $T$ state, with high affinity for substrate proteins (SPs), the ATP-bound $R$ state, and the $R^{\prime\prime}$ ($GroEL-ADP-GroES$) complex. Structures are known for each of these states. Here, we use a self-organized polymer (SOP) model for the GroEL allosteric states and a general structure-based technique to simulate the dynamics of allosteric transitions in two subunits of GroEL and the heptamer. The $T \to R$ transition, in which the apical domains undergo counter-clockwise motion, is mediated by a multiple salt-bridge switch mechanism, in which a series of salt-bridges break and form. The initial event in the $R \to R^{\prime\prime}$ transition, during which GroEL rotates clockwise, involves a spectacular outside-in movement of helices K and L that results in K80-D359 salt-bridge formation. In both the transitions there is considerable heterogeneity in the transition pathways. The transition state ensembles (TSEs) connecting the $T$, $R$, and $R^{\prime\prime}$ states are broad with the the TSE for the $T \to R$ transition being more plastic than the $R\to R^{\prime\prime}$ TSE. The results suggest that GroEL functions as a force-transmitting device in which forces of about (5-30) pN may act on the SP during the reaction cycle., Comment: 32 pages, 10 figures (Longer version than the one published)

Using self-organized polymer models, we predict mechanical unfolding and refolding pathways of ribo-zymes, and the green fluorescent protein. In agreement with experiments, there are between six and eight unfolding transitions in the Tetrahymena ribozyme. Depending on the loading rate, the number of rips in the force-ramp unfolding of the Azoarcus ribozymes is between two and four. Force-quench refolding of the P4-P6 subdomain of the Tetrahymena ribozyme occurs through a compact intermediate. Subsequent formation of tertiary contacts between helices P5b-P6a and P5a/P5c-P4 leads to the native state. The force-quench refolding pathways agree with ensemble experiments. In the dominant unfolding route, the N-terminal a helix of GFP unravels first, followed by disruption of the N terminus b strand. There is a third intermediate that involves disruption of three other strands. In accord with experiments, the force-quench refolding pathway of GFP is hierarchic, with the rate-limiting step being the closure of the barrel., Comment: 33 pages 7 figures

Determination of sizes and flexibilities of RNA molecules is important in understanding the nature of packing in folded structures and in elucidating interactions between RNA and DNA or proteins. Using the coordinates of the structures of RNA in the Protein Data Bank we find that the size of the folded RNA structures, measured using the radius of gyration, $R_G$, follows the Flory scaling law, namely, $R_G =5.5 N^{1/3}$ \AA where N is the number of nucleotides. The shape of RNA molecules is characterized by the asphericity $\Delta$ and the shape $S$ parameters that are computed using the eigenvalues of the moment of inertia tensor. From the distribution of $\Delta$, we find that a large fraction of folded RNA structures are aspherical and the distribution of $S$ values shows that RNA molecules are prolate ($S>0$). The flexibility of folded structures is characterized by the persistence length $l_p$. By fitting the distance distribution function $P(r)$ to the worm-like chain model we extracted the persistence length $l_p$. We find that $l_p\approx 1.5 N^{0.33}$ \AA. The dependence of $l_p$ on $N$ implies the average length of helices should increases as the size of RNA grows. We also analyze packing in the structures of ribosomes (30S, 50S, and 70S) in terms of $R_G$, $\Delta$, $S$, and $l_p$. The 70S and the 50S subunits are more spherical compared to most RNA molecules. The globularity in 50S is due to the presence of an unusually large number (compared to 30S subunit) of small helices that are stitched together by bulges and loops. Comparison of the shapes of the intact 70S ribosome and the constituent particles suggests that folding of the individual molecules might occur prior to assembly., Comment: 28 pages, 8 figures, J. Chem. Phys. in press

Mechanical unfolding of RNA structures, ranging from hairpins to ribozymes, using laser optical tweezer (LOT) experiments have begun to reveal the features of the energy landscape that cannot be easily explored using conventional experiments. Upon application of constant force ($f$), RNA hairpins undergo cooperative transitions from folded to unfolded states whereas subdomains of ribozymes unravel one at a time. Here, we use a self-organized polymer (SOP) model and Brownian dynamics simulations to probe mechanical unfolding at constant force and constant-loading rate of four RNA structures of varying complexity. Our work shows (i) the response of RNA to force is largely determined by the native structure; (ii) only by probing mechanical unfolding over a wide range of forces can the underlying energy landscape be fully explored., Comment: 26 pages, 6 figures, Biophys. J. (in press)

We investigate the solvation of methane and methane decorated with charges in spherically confined water droplets. Free energy profiles for a single methane molecule in droplets, ranging in diameter D, from 1 to 4 nm, show that the droplet surfaces are strongly favorable as compared to the interior. From the temperature dependence of the free energy in D=3 nm, we show that this effect is entropically driven. The potentials of mean force (PMFs) between two methane molecules show that the solvent separated minimum in the bulk is completely absent in confined water, independent of the droplet size since the solute particles are primarily associated with the droplet surface. The tendency of methanes with charges (Mq+ and Mq- with q+ = q- = 0.4e, where e is the electronic charge) to be pinned at the surface depends dramatically on the size of the water droplet. When D=4 nm, the ions prefer the interior whereas for D<4 nm the ions are localized at the surface, but with much less tendency than for methanes. Increasing the ion charge to e makes the surface strongly unfavorable. Reflecting the charge asymmetry of the water molecule, negative ions have a stronger preference for the surface compared to positive ions of the same charge magnitude. With increasing droplet size, the PMFs between Mq+ and Mq- show decreasing influence of the boundary due to the reduced tendency for surface solvation. We also show that as the solute charge density decreases the surface becomes less unfavorable. The implications of our results for the folding of proteins in confined spaces are outlined., Comment: 27 pages, 9 figures plus supporting material (6 pages, 5 figures) To be published in The Journal of the American Chemical Society

The need to understand the assembly kinetics of fibril formation has become urgent because of the realization that soluble oligomers of amyloidogenic peptides may be even more neurotoxic than the end product, namely, the amyloid fibrils. In order to fully understand the routes to fibril formation one has to characterize the major species in the assembly pathways. The characterization of the energetics and dynamics of oligomers (dimers, trimers etc) is difficult using experiments alone because they undergo large conformational fluctuations. In this context, carefully planned molecular dynamics simulation studies, computations using coarse-grained models, and bioinformatic analysis have given considerable insights into the early events in the route to fibril formation. Here, we describe progress along this direction using examples taken largely from our own work. In this chapter, we focus on aspects of protein aggregation using Abeta-peptides and prion proteins as examples., Comment: 27 pages, 7 figures, book chapter

Using coarse-grained model we have explored forced-unfolding of RNA hairpin as a function of $f_S$ and the loading rate ($r_f$). The simulations and theoretical analysis have been done without and with the handles that are explicitly modeled by semiflexible polymer chains. The mechanisms and time scales for denaturation by temperature jump and mechanical unfolding are vastly different. The directed perturbation of the native state by $f_S$ results in a sequential unfolding of the hairpin starting from their ends whereas thermal denaturation occurs stochastically. From the dependence of the unfolding rates on $r_f$ and $f_S$ we show that the position of the unfolding transition state (TS) is not a constant but moves dramatically as either $r_f$ or $f_S$ is changed. The TS movements are interpreted by adopting the Hammond postulate for forced-unfolding. Forced-unfolding simulations of RNA, with handles attached to the two ends, show that the value of the unfolding force increases (especially at high pulling speeds) as the length of the handles increases. The pathways for refolding of RNA from stretched initial conformation, upon quenching $f_S$ to the quench force $f_Q$, are highly heterogeneous. The refolding times, upon force quench, are at least an order of magnitude greater than those obtained by temperature quench. The long $f_Q$-dependent refolding times starting from fully stretched states are analyzed using a model that accounts for the microscopic steps in the rate limiting step which involves the trans to gauche transitions of the dihedral angles in the GAAA tetraloop. The simulations with explicit molecular model for the handles show that the dynamics of force-quench refolding is strongly dependent on the interplay of their contour length and the persistence length, and the RNA persistence length., Comment: 42 pages, 15 figures, Biophys. J. (in press)

Loop formation between monomers in the interior of semiflexible chains describes elementary events in biomolecular folding and DNA bending. We calculate analytically the interior distance distribution function for semiflexible chains using a mean-field approach. Using the potential of mean force derived from the distance distribution function we present a simple expression for the kinetics of interior looping by adopting Kramers theory. For the parameters, that are appropriate for DNA, the theoretical predictions in comparison to the case are in excellent agreement with explicit Brownian dynamics simulations of worm-like chain (WLC) model. The interior looping times ($\tau_{IC}$) can be greatly altered in cases when the stiffness of the loop differs from that of the dangling ends. If the dangling end is stiffer than the loop then $\tau_{IC}$ increases for the case of the WLC with uniform persistence length. In contrast, attachment of flexible dangling ends enhances rate of interior loop formation. The theory also shows that if the monomers are charged and interact via screened Coulomb potential then both the cyclization ($\tau_c$) and interior looping ($\tau_{IC}$) times greatly increase at low ionic concentration. Because both $\tau_c$ and $\tau_{IC}$ are determined essentially by the effective persistence length ($l_p^{(R)}$) we computed $l_p^{(R)}$ by varying the range of the repulsive interaction between the monomers. For short range interactions $l_p^{(R)}$ nearly coincides with the bare persistence length which is determined largely by the backbone chain connectivity. This finding rationalizes the efficacy of describing a number of experimental observations (response of biopolymers to force and cyclization kinetics) in biomolecules using WLC model with an effective persistence length., Comment: 38 pages, 8 Figures, J. Chem. Phys. (in press)

In the current AFM experiments the distribution of unfolding times, P(t), is measured by applying a constant stretching force f_s from which the apparent unfolding rate is obtained. To describe the complexity of the underlying energy landscape requires additional probes that can incorporate the dynamics of tension propagation and relaxation of the polypeptide chain upon force quench. We introduce a theory of force correlation spectroscopy (FCS) to map the parameters of the energy landscape of proteins. In the FCS the joint distribution, P(T,t) of folding and unfolding times is constructed by repeated application of cycles of stretching at constant fs, separated by release periods T during which the force is quenched to f_q

We determine the persistence length, $l_p$, for a bacterial group I ribozyme as a function of concentration of monovalent and divalent cations by fitting the distance distribution functions $P(r)$ obtained from small angle X-ray scattering intensity data to the asymptotic form of the calculated $P_{WLC}(r)$ for a worm-like chain (WLC). The $l_p$ values change dramatically over a narrow range of \Mg concentration from $\sim$21 \AA in the unfolded state (\textbf{U}) to $\sim$10 \AA in the compact ($\mathrm{I_C}$) and native states. Variations in $l_p$ with increasing \Na concentration are more gradual. In accord with the predictions of polyelectrolyte theory we find $l_p \propto 1/ \kappa^2$ where $\kappa$ is the inverse Debye-screening length., Comment: 4 page. Phys. Rev. Lett. in press

We investigate the folding and forced-unbinding transitions of adsorbed semiflexible polymer chains using theory and simulations. These processes describe biologically relevant phenomena that include adhesive interactions between proteins and tethering of receptors to cell walls. The binding interface is modeled as a solid surface and the worm-like chain is used for the semiflexible chain. Using Langevin simulations we examine the ordering kinetics of racquet-like and toroidal structures in the presence of attractive interaction between the surface and the polymer chain. For a range of interactions, temperature, and the persistence length l_p we obtained the monomer density distribution n(x) for all the relevant morphologies. The simulated results for n(x) are in good agreement with theory. The formation of toroids on the surface appears to be a first order transition. Chain-surface interaction is probed by subjecting the surface structures to a pulling force f. The average extension x as a function of f exhibit sigmoidal profile with sharp all-or-none transition at the unfolding f_c which increases for more structured states. Simulated x compare well with the theoretical predictions. The critical force f_c is a function of l_s/l_c for a fixed temperature, where l_c, l_s are the length scales that express the strength of the intramolecular and chain-surface attraction, respectively. For a fixed l_s, f_c increases as l_p decreases., Comment: 26 pages, 6 figures, accepted to J. Phys. Chem. B

We develop a formalism for single molecule dynamic force spectroscopy to map the energy landscape of protein-protein complex ($P_1$$P_2$). The joint distribution $P(\tau_1,\tau_2)$ of unbinding lifetimes $\tau_1$ and $\tau_2$ measurable in a compression-tension cycle, which accounts for the internal relaxation dynamics of the proteins under tension, shows that the histogram of $\tau_1$ is not Poissonian. The theory is applied to the forced unbinding of protein $P_1$, modeled as a wormlike chain, from $P_1$$P_2$. We propose a new class of experiments which can resolve the effect of internal protein dynamics on the unbinding lifetimes., Comment: 12 pages, 3 figures, accepted to Phys. Rev. Lett

We calculate the mean end-to-end distance ($R$) of a self-avoiding polymer encapsulated in an infinitely long cylinder with radius $D$. A self-consistent perturbation theory is used to calculate $R$ as a function of $D$ for impenetrable hard walls and soft walls. In both cases, $R$ obeys the predicted scaling behavior in the limit of large and small $D$. The crossover from the three dimensional behavior ($D\to\infty$) to the fully stretched one dimensional case ($D\to 0$) is non-monotonic. The minimum value of $R$ is found at $D\sim 0.46 R_F$, where $R_F$ is the Flory radius of $R$ at $D \to \infty$. The results for soft walls map onto the hard wall case with a larger cylinder radius., Comment: 8 pages, 2 figures, to be published in the Journal of Chemical Physics

Mechanical unfolding trajectories, generated by applying constant force in optical tweezer experiments, show that RNA hairpins and the P5abc subdomain of the group I intron unfold reversibly. We use coarse-grained Go-like models for RNA hairpins to explore forced-unfolding over a broad range of temperatures. A number of predictions that are amenable to experimental tests are made. At the critical force the hairpin jumps between folded and unfolded conformations without populating any discernible intermediates. The phase diagram in the force-temperature (f,T) plane shows that the hairpin unfolds by an all-or-none process. The cooperativity of the unfolding transition increases dramatically at low temperatures. Free energy of stability, obtained from time averages of mechanical unfolding trajectories, coincide with ensemble averages which establishes ergodicity. The hopping time between the the native basin of attraction (NBA) and the unfolded basin increases dramatically along the phase boundary. Thermal unfolding is stochastic whereas mechanical unfolding occurs in "quantized steps" with great variations in the step lengths. Refolding times, upon force quench, from stretched states to the NBA is "at least an order of magnitude" greater than folding times by temperature quench. Upon force quench from stretched states the NBA is reached in at least three stages. In the initial stages the mean end-to-end distance decreases nearly continuously and only in the last stage there is a sudden transition to the NBA. Because of the generality of the results we propose that similar behavior should be observed in force quench refolding of proteins., Comment: 23 pages, 6 Figures. in press (Proc. Natl. Acad. Sci.)

Finite size effects on the calorimetric cooperatity of the folding-unfolding transition in two-state proteins are considered using the Go lattice models with and without side chains. We show that for models without side chains a dimensionless measure of calorimetric cooperativity kappa2 defined as the ratio of the van't Hoff to calorimetric enthalpy does not depend on the number of amino acids N. The average value of kappa2 is about 3/4 which is lower than the experimental value kappa2=1. For models with side chains kappa2 approaches unity as kappa2 \sim N^mu, where exponent mu=0.17. Above the critical chain length Nc =135 these models can mimic the truly all-or-non folding-unfolding transition., Comment: 3 eps figures. To appear in the special issue of Physica A

Finite size effects on the cooperative thermal denaturation of proteins are considered. A dimensionless measure of cooperativity, Omega, scales as N^zeta, where N is the number of amino acids. Surprisingly, we find that zeta is universal with zeta = 1 + gamma, where the exponent gamma characterizes the divergence of the susceptibility for a self-avoiding walk. Our lattice model simulations and experimental data are consistent with the theory. Our finding rationalizes the marginal stability of proteins and substantiates the earlier predictions that the efficient folding of two-state proteins requires the folding transition temperature to be close to the collapse temperature., Comment: 3 figures. Physical Review Letters (in press)

Function of proteins or a network of interacting proteins often involves communication between residues that are well separated in sequence. The classic example is the participation of distant residues in allosteric regulation. Bioinformatic and structural analysis methods have been introduced to infer residues that are correlated. Recently, increasing attention has been paid to obtain the sequence properties that determine the tendency of disease related proteins (Abeta peptides, prion proteins, transthyretin etc.) to aggregate and form fibrils. Motivated in part by the need to identify sequence characteristics that indicate a tendency to aggregate, we introduce a general method that probes covariations in charged residues along the sequence in a given protein family. The method, which involves computing the Sequence Correlation Entropy (SCE) using the quenched probability Psk(i,j) of finding a residue pair at a given sequence separation sk, allows us to classify protein families in terms of their SCE. Our general approach may be a useful way in obtaining evolutionary covariations of amino acid residues on a genome wide level., Comment: 23 pages, 4 figures

The asymmetry in the shapes of folded and unfolded states are probed using two parameters, one being a measure of the sphericity and the other that describes the shape. For the folded states, whose interiors are densely packed, the radii of gyration (Rg) and these two parameters are calculated using the coordinates of the experimentally determined structures. Although Rg scales as expected for maximally compact structures, the distributions of the shape parameters show that there is considerable asymmetry in the shapes of folded structures. The degree of asymmetry is greater for proteins that form oligomers. Analysis of the two- and three-body contacts in the native structures shows that the presence of near equal number of contacts between backbone and side-chains and between side-chains gives rise to dense packing. We suggest that proteins with relatively large values of shape parameters can tolerate volume mutations without greatly affecting the network of contacts or their stability. To probe shape characteristics of denatured states we have developed a model of a WW-like domain. The shape parameters, which are calculated using Langevin simulations, change dramatically in the course of coil to globule transition. Comparison of the values of shape parameters between the globular state and the folded state of WW domain shows that both energetic (especially dispersion in the hydrophobic interactions) and steric effects are important in determining packing in proteins., Comment: 22 pages, 7 figures

We analyze the dependence of thermal denaturation transition and folding rates of globular proteins on the number of amino acid residues, N. Using lattice Go models we show that DeltaT/T_F ~ N^-1, where T_F is the folding transition temperature and DeltaT is the folding transition width. This finding is consistent with finite size effects expected for the systems undergoing a phase transition from a disordered to an ordered phase. The dependence of the folding rates k_F on N for lattice models and the dataset of 57 proteins and peptides shows that k_F = k_F^0 exp(-CN^beta) provides a good fit, if 0 < beta <= 2/3 and C is a constant. We find that k_F = k_F^0 exp(-1.1N^0.5) with k_F^0 =(0.4x10^-6 s)^-1 can estimate optimal protein folding rates to within an order of magnitude in most cases. By using this fit for a set of proteins with beta-sheet topology we find that k_F^0 is approximately equal to k_U^0, the prefactor for unfolding rates. The maximum ratio of k_U^0/k_F^0 is 10 for this class of proteins., Comment: 11 pages, 3 figures

We present a new method to extract distance and orientation dependent potentials between amino acid side chains using a database of protein structures and the standard Boltzmann device. The importance of orientation dependent interactions is first established by computing orientational order parameters for proteins with alpha-helical and beta-sheet architecture. Extraction of the anisotropic interactions requires defining local reference frames for each amino acid that uniquely determine the coordinates of the neighboring residues. Using the local reference frames and histograms of the radial and angular correlation functions for a standard set of non-homologue protein structures, we construct the anisotropic pair potentials. The performance of the orientation dependent potentials was studied using a large database of decoy proteins. The results demonstrate that the new distance and orientation dependent residue-residue potentials present a significantly improved ability to recognize native folds from a set of native and decoy protein structures., Comment: Submitted to "The Journal of Chemical Physics"

Using three-dimensional Go lattice models with side chains for proteins, we investigate the dependence of folding times on protein length. In agreement with previous theoretical predictions, we find that the folding time grows as a power law with the chain length N with exponent $\lambda \approx 3.6$ for the Go model, in which all native interactions (i.e., between all side chains and backbone atoms) are uniform. If the interactions between side chains are given by pairwise statistical potentials, which introduce heterogeneity in the contact energies, then the power law fits yield large $\lambda$ values that typically signifies a crossover to an underlying activated process. Accordingly, the dependence of folding time is best described by the stretched exponential \exp(\sqrt{N}). The study also shows that the incorporation of side chains considerably slows down folding by introducing energetic and topological frustration., Comment: 6 pages, 5 eps figures

The folding kinetics of three-dimensional lattice Go models with side chains is studied using two different Monte Carlo move sets. A flexible move set based on single, double and triple backbone moves is found to be far superior compared to the standard Monte Carlo dynamics. It was found that the folding time grows as a power law with the chain length and the corresponding exponent $\lambda \approx 3.6$ for Go models. The study shows that the incorporation of side chains dramatically slows down folding rates., Comment: 4 pages, CCP2001 proceeding paper

We study the bending rigidity of highly charged stiff polyelectrolytes, for both a single chain and many chains forming a bundle. A theory is developed to account for the interplay between competitive binding of counterions and charge correlations in softening the polyelectrolyte (PE) chains. The presence of even a small concentration of multivalent counterions leads to a dramatic reduction in the bending rigidity of the chains that are nominally stiffened by the repulsion between their backbone charges. The variation of the bending rigidity as a function of $f_{0}$, the fraction of charged monomers on the chain, does not exhibits simple scaling behavior; it grows with increasing $f_{0}$ below a critical value of $f_{0}$. Beyond the critical value, however, the chain becomes softer as $f_{0}$ increases. The bending rigidity also exhibits intriguing dependence on the concentration of multivalent counterion $n_{2}$; for highly charged PEs, the bending rigidity decreases as $n_2$ increases from zero, while it increases with increasing $n_{2}$ beyond a certain value of $n_{2}$. When polyelectrolyte chains form a $N$-loop condensate (e.g., a toroidal bundle formed by $N$ turns (winds) of the chain), the inter-loop coupling further softens the condensate, resulting in the bending free energy of the condensate that scales as $N$ for large $N$., Comment: 11 pages, 2 figures

We discuss recent theoretical developments in the study of simple lattice models of proteins. Such models are designed to understand general features of protein structures and mechanism of folding. Among the topics covered are (i) the use of lattice models to understand the selection of the limited set of viable protein folds; (ii) the relationship between structure and sequence spaces; (iii) the application of lattice models for studying folding mechanisms (topological frustration, kinetic partitioning mechanism). Classification of folding scenarios based on the intrinsic thermodynamic properties of a sequence (namely, the collapse and folding transition temperatures) is outlined. A brief discussion of random heteropolymer model is also presented., Comment: latex, 32 pages, 10 Postscript figures, to be published in Encyclopedia of Chemical Physics and Protein Chemistry. IoP Publishing, Bristol, UK

We provide a theory for the dynamics of collapse of strongly charged polyelectrolytes (PEs) and flexible polyampholytes (PAs) using Langevin equation. After the initial stage, in which counterions condense onto PE, the mechanism of approach to the globular state is similar for PE and PA. In both instances, metastable pearl-necklace structures form in characteristic time scale that is proportional to N^{4/5} where N is the number of monomers. The late stage of collapse occurs by merger of clusters with the largest one growing at the expense of smaller ones (Lifshitz- Slyozov mechanism). The time scale for this process T_{COLL} N. Simulations are used to support the proposed collapse mechanism for PA and PE., Comment: 14 pages, 2 figures

The number of protein structures is far less than the number of sequences. By imposing simple generic features of proteins (low energy and compaction) on all possible sequences we show that the structure space is sparse compared to the sequence space. Even though the sequence space grows exponentially with N (the number of amino acids) we conjecture that the number of low energy compact structures only scales as ln N. This implies that many sequences must map onto countable number of basins in the structure space. The number of sequences for which a given fold emerges as a native structure is further reduced by the dual requirements of stability and kinetic accessibility. The factor that determines the dual requirement is related to the sequence dependent temperatures, T_\theta (collapse transition temperature) and T_F (folding transition temperature). Sequences, for which \sigma =(T_\theta-T_F)/T_\theta is small, typically fold fast by generically collapsing to the native-like structures and then rapidly assembling to the native state. Such sequences satisfy the dual requirements over a wide temperature range. We also suggest that the functional requirement may further reduce the number of sequences that are biologically competent. The scheme developed here for thinning of the sequence space that leads to foldable structures arises naturally using simple physical characteristics of proteins. The reduction in sequence space leading to the emergence of foldable structures is demonstrated using lattice models of proteins., Comment: latex, 18 pages, 8 figures, to be published in the conference proceedings "Stochastic Dynamics and Pattern Formation in Biological Systems"

The impact of two landmark papers one by Edwards and the other by de Gennes on the field of polymer physics is highlighted., Comment: Latex, no figures. to appear in Theor. Chem. Acc

The collapse kinetics of strongly charged polyelectrolytes in poor solvents is investigated by Langevin simulations and scaling arguments. The rate of collapse increases sharply as the valence of counterions, z, increases from one to four. The combined system of the collapsed chain and the condensed counterions forms a Wigner crystal when the solvent quality is not too poor provided z >= 2. For very poor solvents the morphology of the collapsed structure resembles a Wigner glass. For a fixed z and quality of the solvent the efficiency of collapse decreases dramatically as the size of the counterion increases. A valence dependent diagram of states in poor solvents is derived., Comment: 4pages, 4 figures

Single molecule force spectroscopy reveals unfolding of domains in titin upon stretching. We provide a theoretical framework for these experiments by computing the phase diagrams for force-induced unfolding of single domain proteins using lattice models. The results show that two-state folders (at zero force) unravel cooperatively whereas stretching of non-two-state folders occurs through intermediates. The stretching rates of individual molecules show great variations reflecting the heterogeneity of force-induced unfolding pathways. The approach to the stretched state occurs in a step-wise "quantized" manner. Unfolding dynamics depends sensitively on topology. The unfolding rates increase exponentially with force f till an optimum value which is determined by the barrier to unfolding when f=0. A mapping of these results to proteins shows qualitative agreement with force-induced unfolding of Ig-like domains in titin. We show that single molecule force spectroscopy can be used to map the folding free energy landscape of proteins in the absence of denaturants., Comment: 12 pages, Latex, 6 ps figures