Frozen states of a disordered globular heteropolymer

Sequence Design of Random Heteropolymers as Protein Mimics

Random heteropolymers (RHPs) have been computationally designed and experimentally shown to recapitulate protein-like phase behavior and function. However, unlike proteins, RHP sequences are only statistically defined and cannot be sequenced. Recent developments in reversible-deactivation radical polymerization allowed simulated polymer sequences based on the well-established Mayo-Lewis equation to more accurately reflect ground-truth sequences that are experimentally synthesized. This led to opportunities to perform bioinformatics-inspired analysis on simulated sequences to guide the design, synthesis, and interpretation of RHPs. We compared batches on the order of 10000 simulated RHP sequences that vary by synthetically controllable and measurable RHP characteristics such as chemical heterogeneity and average degree of polymerization. Our analysis spans across 3 levels: segments along a single chain, sequences within a batch, and batch-averaged statistics. We discuss simulator fidelity and highlight the importance of robust segment definition. Examples are presented that demonstrate the use of simulated sequence analysis for in-silico iterative design to mimic protein hydrophobic/hydrophilic segment distributions in RHPs and compare RHP and protein sequence segments to explain experimental results of RHPs that mimic protein function. To facilitate the community use of this workflow, the simulator and analysis modules have been made available through an open source toolkit, the RHPapp.

Fingerprint of semi-crystalline structure memory in the thermal and ionic conduction properties of amorphous ureasil-polyether hybrid solid electrolytes

Correlations among the structure, thermal properties, and ionic conductivity of solid polymer electrolytes (SPEs) were studied using a ureasil-polyethylene oxide (U-PEO) organic-inorganic hybrid prepared according to a simple sol-gel route, employing a low molecular weight PEO macromer (M w = 1900 g mol-1). The behavior of an amorphous sample loaded with lithium triflate (LiTFSI) at an optimum ratio between ether oxygen and lithium (EO/Li+ = 15) was compared with that of a semicrystalline sample prepared without salt loading. The temperature range investigated by differential scanning calorimetry (DSC), Raman spectroscopy, small angle X-ray scattering (SAXS), and complex impedance spectroscopy covered both the glass transition and the melting temperature of the U-PEO. The gauche to trans conformational transformation of the (O-C-C-O)Li+ sequence showed similarity between the temperature evolution of the semi-crystalline U-PEO and amorphous U-PEO:Li+ samples, providing an indication of the local structural memory of crystalline state in the amorphous SPE. The linear thermal expansion of the average correlation distance between the siloxane crosslink nodes and the long-distance period of the lamellar semi-crystalline edifice were determined by SAXS. Comparison of the expansion curves suggested that although the siloxane nodes were excluded from the PEO crystalline edifice, the sharp expansion of the amorphous region between the lamellae during melting permitted modulation of the free volume of the hybrid network. In addition, the temperature-induced Li+-EO decomplexation observed by Raman spectroscopy explained the change of the average activation energy of the conduction process revealed by the different Arrhenius regimes. These results evidence the key role of the ionic conductivity decoupling from the segmental motion of chain pair channels on the improvement of ion mobility through the free volume between chains. This concept may inspire materials chemistry researchers to design optimized structures of polymer electrolytes with minimized structural memory of crystaline building blocks and improved ionic conductivity.

Percolation-like phase transitions in network models of protein dynamics

In broad terms, percolation theory describes the conditions under which clusters of nodes are fully connected in a random network. A percolation phase transition occurs when, as edges are added to a network, its largest connected cluster abruptly jumps from insignificance to complete dominance. In this article, we apply percolation theory to meticulously constructed networks of protein folding dynamics called Markov state models. As rare fluctuations are systematically repressed (or reintroduced), we observe percolation-like phase transitions in protein folding networks: whole sets of conformational states switch from nearly complete isolation to complete connectivity in a rapid fashion. We analyze the general and critical properties of these phase transitions in seven protein systems and discuss how closely dynamics on protein folding landscapes relate to percolation on random lattices.

A statistical model of intra-chromosome contact maps

A statistical model describing a fine structure of the intra-chromosome maps obtained by a genome-wide chromosome conformation capture method (Hi-C) is proposed. The model combines hierarchical chain folding with a quenched heteropolymer structure of primary chromatin sequences. It is conjectured that the observed Hi-C maps are statistical averages over many different ways of hierarchical genome folding. It is shown that the existence of a quenched primary structure coupled with hierarchical folding induces a full range of features observed in experimental Hi-C maps: hierarchical elements, chess-board intermittency and large-scale compartmentalization.

Equilibrium properties of realistic random heteropolymers and their relevance for globular and naturally unfolded proteins

Random heteropolymers do not display the typical equilibrium properties of globular proteins, but are the starting point to understand the physics of proteins and, in particular, to describe their non-native states. So far, they have been studied with mean-field models in the thermodynamic limit, or with computer simulations of very small chains on lattice. After describing a self-adjusting parallel-tempering technique to sample efficiently the low-energy states of frustrated systems without the need of tuning the system-dependent parameters of the algorithm, we apply it to random heteropolymers moving in continuous space. We show that if the mean interaction between monomers is negative, the usual description through the random-energy model is nearly correct, provided that it is extended to account for noncompact conformations. If the mean interaction is positive, such a simple description breaks out and the system behaves in a way more similar to Ising spin glasses. The former case is a model for the denatured state of globular proteins, the latter of naturally unfolded proteins, whose equilibrium properties thus result as qualitatively different.

A probabilistic approach to the effect of water hydrogen bonds on the kinetics of protein folding and protein denaturation

Previously, we presented a review of our kinetic models for the nucleation mechanism of protein folding and for the protein thermal denaturation in a barrierless way. A protein was treated as a random heteropolymer consisting of hydrophobic, hydrophilic, and neutral beads. As a crucial idea of the model, an overall potential around the cluster of native residues wherein a residue performs a chaotic motion was considered as the combination of the average dihedral, effective pairwise, and confining potentials. The overall potential as a function of the distance from the cluster has a double well shape. This allowed one to develop kinetic models for the nucleation mechanism of protein folding (NMPF) and barrierless protein denaturation (BPD) by using the mean first passage time analysis. In the original models, however, hydrogen bonding effects were taken into account only indirectly which affected the accuracy of the models because hydrogen bonding does play a crucial role in the folding, stability, and denaturation of proteins. To improve the NMPF and BPD models and explicitly take into account the hydrogen bonding "water-water" and "water-protein residue", we have developed a probabilistic hydrogen bond (PHB) model for the effect of hydrogen bond networks of water molecules around two solute particles (immersed in water) on their interaction, and have then combined the PHB model with the NMPF and BPD models. In this paper, that can be regarded as sequel of our previous review, we analyze the modified NMPF and BPD models that explicitly take into account the effect of water-water hydrogen bonding on these processes. As expected, the application of the modified models to the folding/unfolding of two model proteins (one short, consisting of 124 residues and the other large, consisting of 2500 residues) demonstrate that the hydrogen bond networks play a very important role in the protein folding/unfolding phenomena.

Effect of hydrogen bond networks on the nucleation mechanism of protein folding

We have recently developed a kinetic model for the nucleation mechanism of protein folding (NMPF) in terms of ternary nucleation by using the first passage time analysis. A protein was considered as a random heteropolymer consisting of hydrophobic, hydrophilic (some of which are negatively or positively ionizable), and neutral beads. The main idea of the NMPF model consisted of averaging the dihedral potential in which a selected residue is involved over all possible configurations of all neighboring residues along the protein chain. The combination of the average dihedral, effective pairwise (due to Lennard-Jones-type and electrostatic interactions), and confining (due to the polymer connectivity constraint) potentials gives rise to an overall potential around the cluster that, as a function of the distance from the cluster center, has a double-well shape. This allows one to evaluate the protein folding time. In the original NMPF model hydrogen bonding was not taken into account explicitly. To improve the NMPF model and make it more realistic, in this paper we modify our (previously developed) probabilistic hydrogen bond model and combine it with the former. Thus, a contribution due to the disruption of hydrogen bond networks around the interacting particles (cluster of native residues and residue in the protein unfolded part) appears in the overall potential field around a cluster. The modified model is applied to the folding of the same model proteins that were examined in the original model: a short protein consisting of 124 residues (roughly mimicking bovine pancreatic ribonuclease) and a long one consisting of 2500 residues (as a representative of large proteins with superlong polypeptide chains), at pH=8.3 , 7.3, and 6.3. The hydrogen bond contribution now plays a dominant role in the total potential field around the cluster (except for very short distances thereto where the repulsive energy tends to infinity). It is by an order of magnitude stronger for hydrophobic residues than for hydrophilic ones. The range of "residue-cluster" distances, at which the hydrogen bond effect exists, is twice as long for hydrophobic residues as for hydrophilic ones.

Thermal denaturation of a native protein via spinodal decomposition in the framework of first-passage-time analysis

In this paper we present a kinetic model for the thermal unfolding of a native protein. Due to a sufficiently large temperature increase or decrease, the rate with which a cluster of native residues within a protein emits residues becomes larger than the rate of absorption of residues from the unfolded part of the protein in the whole range of cluster sizes up to the size of the whole protein. This leads to the unfolding of the protein in a barrierless way, i.e., as spinodal decomposition. Using the formalism of the first passage time analysis [previously applied also to the problem of protein folding via nucleation by the authors, J. Phys. Chem. B 111, 886 (2007); J. Chem. Phys. 126, 175103 (2007)], one can determine the temperature dependence of the rates of emission and absorption of residues by the cluster. Knowing these rates as functions of temperature and cluster size, one can find the threshold temperatures of cold and hot barrierless denaturation as well as the unfolding times at temperatures lower and higher, respectively, than those threshold values. For a numerical illustration, the method is applied to the thermal unfolding of a model protein consisting of 2500 residues.

An analytical study of the interplay between geometrical and energetic effects in protein folding

Analytical studies have several advantages for an understanding of the mechanisms of protein folding such as the interplay between geometrical and energetic effects. In this paper, we introduce a Gaussian filament with a C(alpha) structure-based (Go) potential as a new theoretical scheme based on a Hamiltonian approach. This model takes into account geometrical information in a realistic fashion without the need of phenomenological descriptions. In order to make this model more appropriate for comparison with protein folding simulations and experiments, we introduce a many-body interaction into the potential term to enhance cooperativity. We apply our new analytical model to a beta-hairpin-type peptide and compare our results with a molecular dynamics simulation of a structure-based model.

A ternary nucleation model for the nucleation pathway of protein folding

Recently [Y. S. Djikaev and E. Ruckenstein, J. Phys. Chem. B 111, 886 (2007)], the authors proposed a kinetic model for the nucleation mechanism of protein folding where a protein was modeled as a heteropolymer consisting of hydrophobic and hydrophilic beads and the composition of the growing cluster of protein residues was assumed to be constant and equal to the overall protein composition. Here, they further develop the model by considering a protein as a three-component heteropolymer and by allowing the composition of the growing cluster of protein residues to vary independently of the overall one. All the bonds in the heteropolymer (now consisting of hydrophobic, hydrophilic, and neutral beads) have the same constant length, and all the bond angles are equal and fixed. As a crucial idea of the model, an overall potential around the cluster wherein a residue performs a chaotic motion is considered to be a combination of the average dihedral and average pairwise potentials assigned to the bead. The overall potential as a function of the distance from the cluster center has a double well shape which allows one to determine its emission and absorption rates by using a first passage time analysis. Knowing these rates as functions of three independent variables of a ternary cluster, one can develop a self-consistent kinetic theory for the nucleation mechanism of folding of a protein using a ternary nucleation formalism and evaluate the size and composition of the nucleus and the protein folding time. As an illustration, the model is applied to the folding of bovine pancreatic ribonuclease consisting of 124 amino acids whereof 40 are hydrophobic, 81 hydrophilic, and 3 neutral. With a reasonable choice of diffusion coefficients of the residues in the native state and potential parameters, the model predicts folding times in the range of 1-100 s.

Model for the Nucleation Mechanism of Protein Folding

A nucleation-like pathway of protein folding involves the formation of a cluster containing native residues that grows by including residues from the unfolded part of the protein. This pathway is examined by using a heteropolymer as a protein model. The model heteropolymer consists of hydrophobic and hydrophilic beads with fixed bond lengths and bond angles. The total energy of the heteropolymer is determined by the pairwise repulsive/attractive interactions between nonlinked beads and by the contribution from the dihedral angles involved. The parameters of these interactions can be rigorously defined, unlike the ill-defined surface tension of a cluster of protein residues that constitutes the basis of a previous nucleation model. The main idea underlying the new model consists of averaging the dihedral potential of a selected residue over all possible configurations of all neighboring residues along the protein chain. The resulting average dihedral potential depends on the distance between the selected residue and the cluster center. Its combination with the average pairwise potential of the selected residue and with a confining potential caused by the bonds between the residues leads to an overall potential around the cluster that has a double-well shape. Residues in the inner (closer to the cluster) well are considered as belonging to the folded cluster, whereas those in the outer well are treated as belonging to the unfolded part of the protein. Transitions of residues from the inner well into the outer one and vice versa are considered as elementary emission and absorption events, respectively. The double-well character of the potential well around the cluster allows one to determine the rates of both emission and absorption of residues by the cluster using a first passage time analysis. Once these rates are found as functions of the cluster size, one can develop a self-consistent kinetic theory for the nucleation mechanism of folding of a protein. The model allows one to evaluate the size of the nucleus and the protein folding time. The latter is evaluated as the sum of the times necessary for the first nucleation event to occur and for the nucleus to grow to the maximum size (of the folded protein). Depending on the diffusion coefficients of the native residues in the range from 10(-6) to 10(-8) cm2/s, numerical calculations for a protein of 2500 residues suggest that the folding time ranges from several seconds to several hundreds of seconds.

Large-time dynamics and aging of a polymer chain in a random potential

We study the out-of-equilibrium large-time dynamics of a Gaussian polymer chain in a quenched random potential. The dynamics studied is a simple Langevin dynamics commonly referred to as the Rouse model. The equations for the two-time correlation and response functions are derived within the Gaussian variational approximation. In order to implement this approximation faithfully, we employ the supersymmetric representation of the Martin-Siggia-Rose dynamical action. For a short-ranged correlated random potential the equations are solved analytically in the limit of large times using certain assumptions concerning the asymptotic behavior. Two possible dynamical behaviors are identified depending upon the time separation: a stationary regime and an aging regime. In the stationary regime time translation invariance holds and so does the fluctuation dissipation theorem. The aging regime which occurs for large time separations of the two-time correlation functions is characterized by a history dependence and the breakdown of certain equilibrium relations. The large-time limit of the equations yields equations among the order parameters that are similar to the equations obtained in statics using replicas. In particular the aging solution corresponds to the broken replica solution. But there is a difference in one equation that leads to important consequences for the solution. The stationary regime corresponds to the motion of the polymer inside a local minimum of the random potential, whereas in the aging regime the polymer hops between different minima. As a by-product we also solve exactly the dynamics of a chain in a random potential with quadratic correlations.

Embedding a native state into a random heteropolymer model: the dynamic approach

We study a random heteropolymer model with Langevin dynamics, in the supersymmetric formulation. Employing a procedure similar to one that has been used in static calculations, we construct an ensemble in which the affinity of the system for a native state is controlled by a "selection temperature" T0. In the limit of high T0, the model reduces to a random heteropolymer, while for T0-->0 the system is forced into the native state. Within the Gaussian variational approach that we employed previously for the random heteropolymer, we explore the phases of the system for high and low T0. For high T0, the system exhibits a (dynamical) spin-glass phase, like that found for the random heteropolymer, below a temperature T(g). For low T0, we find an ordered phase, characterized by a nonzero overlap with the native state, below a temperature T(n) proportional to 1/T(0)>T(g). However, the random-globule phase remains locally stable below T(n), down to the dynamical glass transition at T(g). Thus, in this model, folding is rapid for temperatures between T(g) and T(n), but below T(g) the system can get trapped in conformations uncorrelated with the native state. At a lower temperature, the ordered phase can also undergo a dynamical glass transition, splitting into substates separated by large barriers.

Random heteropolymer dynamics

We study the Langevin dynamics of the standard random heteropolymer model by mapping the problem to a supersymmetric field theory using the Martin-Siggia-Rose formalism. The resulting model is solved nonperturbatively employing a Gaussian variational approach. In constructing the solution, we assume that the chain is very long and impose the translational invariance which is expected to be present in the bulk of the globule by averaging over the center of mass coordinate. In this way we derive equations of motion for the correlation and response functions C(t,t') and R(t,t'). The order parameters are extracted from the asymptotic behavior of these functions. We find a dynamical phase diagram with frozen (glassy) and melted (ergodic) phases. In the glassy phase the system fails to reach equilibrium and exhibits aging of the type found in p-spin glasses. Within the approximations used in this study, the random heteropolymer model can be mapped to the problem of a manifold in a random potential with power law correlations.

Localization of a polymer in random media: relation to the localization of a quantum particle

In this paper we consider in detail the connection between the problem of a polymer in a random medium and that of a quantum particle in a random potential. We are interested in a system of finite volume where the polymer is known to be localized inside a low minimum of the potential. We show how the end-to-end distance of a polymer that is free to move can be obtained from the density of states of the quantum particle using extreme value statistics. We give a physical interpretation to the recently discovered one-step replica-symmetry-breaking solution for the polymer [Phys. Rev. E 61, 1729 (2000)] in terms of the statistics of localized tail states. Numerical solutions of the variational equations for chains of different length are performed and compared with quenched averages computed directly by using the eigenfunctions and eigenenergies of the Schrödinger equation for a particle in a one-dimensional random potential. The quantities investigated are the radius of gyration of a free Gaussian chain, its mean square distance from the origin and the end-to-end distance of a tethered chain. The probability distribution for the position of the chain is also investigated. The glassiness of the system is explained and is estimated from the variance of the measured quantities.

Mode-coupling theory for heteropolymers

We study the Langevin dynamics of a heteropolymer by means of a mode-coupling approximation scheme, giving rise to a set of coupled integro-differential equations relating the response and correlation functions. The analysis shows that there is a regime at low temperature characterized by out-of-equilibrium dynamics, with violation of time-translational invariance and of the fluctuation-dissipation theorem. The onset of aging dynamics at low temperatures gives insight into the nature of the slow dynamics of a disordered polymer. We also introduce a renormalization-group treatment of our mode-coupling equations, which supports our analysis, and might be applicable to other systems.

Frustrations in polymer conformation in gels and their minimization through molecular imprinting

We report an experimental realization of a gel system in which frustrations exist and can be minimized, thus meeting two crucial criteria predicted to enable memory of conformations in polymers. The gels consist of a thermosensitive major monomer component and two minor components. One minor component is positively charged and will form complexes around negatively charged target molecules placed in solution. The complexes can be imprinted into the gel by then cross-linking the second minor component, which will form cross-links additional to those in the major polymer matrix. The complexes are destroyed and reformed upon swelling and reshrinking of the gels, showing that memorization has been achieved.

Proteins with selected sequences: a heteropolymeric study

Protein sequences are expected not to be random but selected in order to form a stable native structure that is kinetically accessible. Therefore our model contains a selective temperature in sequence space (see [S. Ramanathan and E. Shakhnovich, Phys. Rev. E 50, 1303 (1994)] ) to optimize the sequence for the target conformation statistically. Replica calculations, which go beyond quadratic approximations in the field-theoretical Hamiltonian, are presented. A phase diagram indicating the temperatures and selective temperatures at which transitions to a frozen globule, i.e., the native state, occur is obtained. It is shown that going beyond the quadratic approximation in the field Hamiltonian is very important, since it results in a significant change of the phase diagram. Moreover, we suggest that a one-step replica permutation symmetry scheme is sufficient to solve the model. In addition to this we present a result for the sequence correlation function along the chain in the case of a short-ranged potential between the monomers. A correlation function between monomers that form a contact in the native state is given depending on the temperature and the interaction parameter.

Dynamics of structural models with a long-range interaction: glassy versus nonglassy behavior

By making use of the Langevin dynamics and its generating functional (GF) formulation, the influence of the long-range nature of the interaction on the tendency of the glass formation is systematically investigated. In doing so, two types of models are considered: (i) the nondisordered model with a pure repulsive type of interaction, and (ii) the model with a randomly distributed strength of interaction (a quenched disordered model). The long-ranged potential of interaction is scaled with a number of particles N in such a way as to enable for the GF the saddle-point treatment as well as the systematic 1/N expansion around it. We show that the nondisordered model has no glass transition, which is in line with the mean-field limit of the mode-coupling theory (MCT) predictions. On the other hand, the model with a long-range interaction that has a quenched disorder leads to MC equations which are generic for the p-spin glass model and polymeric manifold in a random media.

Unexpected scenario of glass transition in polymer globules: an exactly enumerable model

We introduce a lattice model of glass transition in polymer globules. This model exhibits ergodicity breaking in which the disjoint regions of phase space do not arise uniformly, but as small chambers whose number increases exponentially with polymer density. Chamber sizes obey power law distribution, making phase space similar to a fractal foam. This clearly demonstrates the importance of the phase space geometry and topology in describing any glass-forming system, such as semicompact polymers during protein folding.