Understanding how monomeric proteins fold under in vitro conditions is crucial to describing their functions in the cellular context. Significant advances both in theory and experiments have resulted in a conceptual framework for describing the folding mechanisms of globular proteins. The experimental data and theoretical methods have revealed the multifaceted character of proteins. Proteins exhibit universal features that can be determined using only the number of amino acid residues (N) and polymer concepts. The sizes of proteins in the denatured and folded states, cooperativity of the folding transition, dispersions in the melting temperatures at the residue level, and time scales of folding are to a large extent determined by N. The consequences of finite N especially on how individual residues order upon folding depends on the topology of the folded states. Such intricate details can be predicted using the Molecular Transfer Model that combines simulations with measured transfer free energies of protein building blocks from water to the desired concentration of the denaturant. By watching one molecule fold at a time, using single molecule methods, the validity of the theoretically anticipated heterogeneity in the folding routes, and the N-dependent time scales for the three stages in the approach to the native state have been established. Despite the successes of theory, of which only a few examples are documented here, we conclude that much remains to be done to solve the "protein folding problem" in the broadest sense.
Within the frame of an effective, coarse-grained hydrophobic-polar protein model, we employ multicanonical Monte Carlo simulations to investigate free-energy landscapes and folding channels of exemplified heteropolymer sequences, which are permutations of each other. Despite the simplicity of the model, the knowledge of the free-energy landscape in dependence of a suitable system order parameter enables us to reveal complex folding characteristics known from real bioproteins and synthetic peptides, such as two-state folding, folding through weakly stable intermediates, and glassy metastability…
ZSPA-1 is an engineered protein that binds to its parent, the three-helix-bundle Z domain of staphylococcal protein A. Uncomplexed ZSPA-1 shows a reduced helix content and a melting behavior that is less cooperative, compared with the wild-type Z domain. Here we show that the difference in folding behavior between these two sequences can be partly understood in terms of an off-lattice model with 5-6 atoms per amino acid and a minimalistic potential, in which folding is driven by backbone hydrogen bonding and effective hydrophobic attraction.
In the past years, the folding kinetics of many small single-domain proteins has been characterized by mutational Phi-value analysis. In this article, a simple, essentially parameter-free model is introduced which derives folding routes from native structures by minimizing the entropic loop-closure cost during folding. The model predicts characteristic folding sequences of structural elements such as helices and beta-strand pairings. Based on few simple rules, the kinetic impact of these structural elements is estimated from the routes and compared to average experimental Phi-values for the he…
Proteins with nontrivial topology, containing knots and slipknots, have the ability to fold to their native states without any additional external forces invoked. A mechanism is suggested for folding of these proteins, such as YibK and YbeA, which involves an intermediate configuration with a slipknot. It elucidates the role of topological barriers and backtracking during the folding event. It also illustrates that native contacts are sufficient to guarantee folding in around 1-2% of the simulations, and how slipknot intermediates are needed to reduce the topological bottlenecks. As expected, …
Phi-values are experimental measures of how the kinetics of protein folding is changed by single-site mutations. Phi-values measure energetic quantities, but are often interpreted in terms of the structures of the transition state ensemble. Here we describe a simple analytical model of the folding kinetics in terms of the formation of protein substructures. The model shows that Phi-values have both structural and energetic components. In addition, it provides a natural and general interpretation of "nonclassical" Phi-values (i.e., less than zero, or greater than one). The model reproduces the …
In this study, we explore nucleation and the transition state ensemble of the ribosomal protein S6 using a Monte Carlo Go model in conjunction with restraints from experiment. The results are analyzed in the context of extensive experimental and evolutionary data. The roles of individual residues in the folding nucleus are identified and the order of events in the S6 folding mechanism is explored in detail. Interpretation of our results agrees with, and extends the utility of, experiments that shift f-values by modulating denaturant concentration and presents strong evidence for the realism of…
Reaching a ground state of a spin system is analogous to a protein evolving into its native state. We study the ``folding'' times for various random Ising spin systems and determine characteristic temperatures that relate to the ``folding''. Under optimal kinetic conditions, the ``folding'' times scale with the system size as a power law with a non-universal exponent. This is similar to what happens in model proteins. On the other hand, the scaling behavior of the characteristic temperatures is different than in model proteins. Both in the spin systems and in proteins, the folding properties d…
The extent of coupling between the folding of a protein and its binding to a substrate varies from protein to protein. Some proteins have highly structured native states in solution, while others are natively disordered and only fold fully upon binding. In this Letter, we use Monte Carlo simulations to investigate how disordered polymer chains grafted around a binding site affect the folding and binding of three model proteins. The protein that approaches the substrate fully folded is more hindered during the binding process than those whose folding and binding are cooperative. The polymer cha…
Scaling of folding properties of proteins is studied in a toy system -- the lattice Go model with various two- and three- dimensional geometries of the maximally compact native states. Characteristic folding times grow as power laws with the system size. The corresponding exponents are not universal. Scaling of the thermodynamic stability also indicates size-related deterioration of the folding properties.
We introduce a powerful iterative algorithm to compute protein folding pathways, with realistic all-atom force fields. Using the path integral formalism, we explicitly derive a modified Langevin equation which samples directly the ensemble of reactive pathways, exponentially reducing the cost of simulating thermally activated transitions. The algorithm also yields a rigorous stochastic estimate of the reaction coordinate. After illustrating this approach on a simple toy model, we successfully validate it against the results of ultra-long plain MD protein folding simulations for a fast folding …
We investigate the interplay between post-translational folding and escape of two small single-domain proteins at the ribosomal exit tunnel by using Langevin dynamics with coarse-grained models. It is shown that at temperatures lower or near the temperature of the fastest folding, folding proceeds concomitantly with the escape process, resulting in vectorial folding and enhancement of foldability of nascent proteins. The concomitance between the two processes, however, deteriorates as temperature increases. Our folding simulations as well as free energy calculation by using umbrella sampling s…
The assumption that similar structures have similar folding probabilities ($p_{fold}$) leads naturally to a procedure to evaluate $p_{fold}$ for every snapshot saved along an equilibrium folding-unfolding trajectory of a structured peptide or protein. The procedure utilizes a structurally homogeneous clustering and does not require any additional simulation. It can be used to detect multiple folding pathways as shown for a three-stranded antiparallel $β$-sheet peptide investigated by implicit solvent molecular dynamics simulations.
Two proteins, one belonging to the mainly alpha class and the other belonging to the alpha/beta class, are selected to test a kinetic mechanism for protein folding. Targeted molecular dynamics is applied to generate folding pathways for those two proteins, starting from two well defined initial conformations: a fully extended and a alpha-helical conformation. The results show that for both proteins the alpha-helical initial conformation provides overall lower energy pathways to the native state. For the alpha/beta protein, 30 % (40%) of the pathways from an initial alpha-helix (fully extended)…
We propose a general theory to describe the distribution of protein-folding transition paths. We show that transition paths follow a predictable sequence of high-free-energy transient states that are separated by free-energy barriers. Each transient state corresponds to the assembly of one or more discrete, cooperative units, which are determined directly from the native structure. We show that the transition state on a folding pathway is reached when a small number of critical contacts are formed between a specific set of substructures, after which folding proceeds downhill in free energy. Th…
Novel numerical techniques, validated by an analysis of barnase and chymotrypsin inhibitor, are used to elucidate the paramount role played by the geometry of the protein backbone in steering the folding to the correct native state. It is found that, irrespective of the sequence, the native state of a protein has exceedingly large number of conformations with a given amount of structural overlap compared to other compact artificial backbones; moreover the conformational entropies of unrelated proteins of the same length are nearly equal at any given stage of folding. These results are suggesti…
We study the dynamics of protein folding via statistical energy-landscape theory. In particular, we concentrate on the local-connectivity case with the folding progress described by the fraction of native conformations. We obtain information for the first passage-time (FPT) distribution and its moments. The results show a dynamic transition temperature below which the FPT distribution develops a power-law tail, a signature of the intermittency phenomena of the folding dynamics. We also discuss the possible application of the results to single-molecule dynamics experiments.
We introduce a method for calculating the extent to which chain non-crossing is important in the most efficient, optimal trajectories or pathways for a protein to fold. This involves recording all unphysical crossing events of a ghost chain, and calculating the minimal uncrossing cost that would have been required to avoid such events. A depth-first tree search algorithm is applied to find minimal transformations to fold $α$, $β$, $α/β$, and knotted proteins. In all cases, the extra uncrossing/non-crossing distance is a small fraction of the total distance travelled by a ghost chain. Different…
The folding vs. adsorption behaviour of a coarse-grained off-lattice protein model near an attractive surface is presented within the frame of a Multicanonical Monte Carlo simulations. In the polymer-surface model, the Lennard-Jones potential is assumed as an interaction potential between the effective monomers and the attractive surface. Thermodynamic properties and some structural parameters for the minimum energy conformations are calculated for comparison of the folding and adsorption cases.
Here we study the effects of many-body interactions on rate and mechanism in protein folding, using the results of molecular dynamics simulations on numerous coarse-grained C-alpha-model single-domain proteins. After adding three-body interactions explicitly as a perturbation to a Go-like Hamiltonian with native pair-wise interactions only, we have found 1) a significantly increased correlation with experimental phi-values and folding rates, 2) a stronger correlation of folding rate with contact order, matching the experimental range in rates when the fraction of three-body energy in the nativ…
We have developed a simple, phenomenological, Monte-Carlo code that predicts the three-dimensional structure of globular proteins from the DNA sequences that define them. We have applied this code to two small proteins, the villin headpiece (1VII) and cole1 rop (1ROP). Our code folds both proteins to within 5 A rms of their native structures.
The process of protein folding from an unfolded state to a biologically active, folded conformation is governed by many parameters e.g the sequence of amino acids, intermolecular interactions, the solvent, temperature and chaperon molecules. Our study, based on random matrix modeling of the interactions, shows however that the evolution of the statistical measures e.g Gibbs free energy, heat capacity, entropy is single parametric. The information can explain the selection of specific folding pathways from an infinite number of possible ways as well as other folding characteristics observed in …
A novel combination of discontinuous molecular dynamics and the Langevin equation, together with an intermediate-resolution model, are used to carry out long (several $μ$s) simulation and study folding transition and transport of proteins in slit nanopores. Both attractive ($U^+$) and repulsive ($U^-$) interaction potentials between the proteins and the pore walls are considered. Near the folding temperature $T_f$ and in the presence of $U^+$ the proteins undergo a repeating sequence of folding/partially-folding/ unfolding transitions, while $T_f$ decreases with decreasing pore sizes. The oppo…
We model protein folding as a physical stochastic process as follows. The unfolded protein chain is treated as a random coil described by SAW (self-avoiding walk). Folding is induced by hydrophobic forces and other interactions, such as hydrogen bonding, which can be taken into account by imposing conditions on SAW. The resulting model is termed CSAW (conditioned self-avoiding walk. Conceptually, the mathematical basis is a generalized Langevin equation. In practice, the model is implemented on a computer by combining SAW and Monte Carlo. To illustrate the flexibility and capabilities of the m…
Different aspects of protein folding are illustrated by simplified polymer models. Stressing the diversity of side chains (residues) leads one to view folding as the freezing transition of an heteropolymer. Technically, the most common approach to diversity is randomness, which is usually implemented in two body interactions (charges, polar character,..). On the other hand, the (almost) universal character of the protein backbone suggests that folding may also be viewed as the crystallization transition of an homopolymeric chain, the main ingredients of which are the peptide bond and chirality…
Folding channels and free-energy landscapes of hydrophobic-polar heteropolymers are discussed on the basis of a minimalistic off-lattice coarse-grained model. We investigate how rearrangements of hydrophobic and polar monomers in a heteropolymer sequence lead to completely different folding behaviors. Studying three exemplified sequences with the same content of hydrophobic and polar residues, we can reproduce within this simple model two-state folding, folding through intermediates, as well as metastability.
We propose a universal elastic energy for proteins, which depends only on the radius of gyration $R_{g}$ and the residue number $N$. It is constructed using physical arguments based on the hydrophobic effect and hydrogen bonding. Adjustable parameters are fitted to data from the computer simulation of the folding of a set of proteins using the CSAW (conditioned self-avoiding walk) model. The elastic energy gives rise to scaling relations of the form $R_{g}\sim N^ν$ in different regions. It shows three folding stages characterized by the progression with exponents $ν= 3/5, 3/7, 2/5$, which we i…
Monte Carlo simulations of protein folding show the emergence of a strong correlation between the relative contact order parameter, CO, and the folding time, t, of two-state folding proteins for longer chains with number of amino acids, N>=54, and higher contact order, CO > 0.17. The correlation is particularly strong for N=80 corresponding to slow and more complex folding kinetics. These results are qualitatively compatible with experimental data where a general trend towards increasing t with CO is indeed observed in a set of proteins with chain length ranging from 41 to 154 amino acids.
We investigate the effect of macromolecular crowding on protein folding, using purely repulsive crowding particles and a self-organizing polymer model of protein folding. We find that the thermodynamics of folding for typical alpha-, beta- and alpha/beta-proteins are well described by an adaptation of the scaled particle theory (SPT). In this approach, the native state, transition state, and the unfolded protein are treated as effective hard spheres with radii approximately independent of the size and concentration of the crowders. The same model predicts the effect of crowding on the folding …
In this paper, we introduce an approach to the protein folding problem from the point of view of statistical physics. Protein folding is a stochastic process by which a polypeptide folds into its characteristic and functional 3D structure from random coil. The process involves an intricate interplay between global geometry and local structure, and each protein seems to present special problems. We introduce CSAW (conditioned self-avoiding walk), a model of protein folding that combines the features of self-avoiding walk (SAW) and the Monte Carlo method. In this model, the unfolded protein chai…
We study folding dynamics of protein-like sequences on square lattice using physical move set that exhausts all possible conformational changes. By analytically solving the master equation, we follow the time-dependent probabilities of occupancy of all 802,075 conformations of 16-mers over 7-orders of time span. We find that (i) folding rates of these protein-like sequences of same length can differ by 4-orders of magnitude, (ii) folding rates of sequences of the same conformation can differ by a factor of 190, and (iii) parameters of the native structures, designability, and thermodynamic pro…
We offer simple solutions to three kinematic problems that occur in the folding of proteins. We show how to construct suitably local elementary Monte Carlo moves, how to close a loop, and how to fold a loop without breaking the bond that closes it.
We develop a multi-scale approach to simulate hydrated nanobio systems under realistic condi- tions (e.g., nanoparticles and protein solutions at physiological conditions over time-scales up to hours). We combine atomistic simulations of water at bio-interfaces (e.g., proteins or membranes) and nano-interfaces (e.g., nanoparticles or graphene sheets) and coarse-grain models of hydration water for protein folding and protein design. We study protein self-assembly and crystallization, in bulk or under confinement, and the kinetics of protein adsorption onto nanoparticles, verify- ing our predict…
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 proper…
Chevron rollovers of some proteins imply that their logarithmic folding rates are nonlinear in native stability. This is predicted by lattice and continuum Gō models to arise from diminished accessibilities of the ground state from transiently populated compact conformations under strongly native conditions. Despite these models' native-centric interactions, the slowdown is due partly to kinetic trapping caused by some of the folding intermediates' nonnative topologies. Notably, simple two-state folding kinetics of small single-domain proteins are not reproduced by common Gō-like schemes.
We consider equilibrium folding transitions in lattice protein models with and without side chains. A dimensionless measure, $Omega_{c}$, is introduced to quantitatively assess the degree of cooperativity in lattice models and in real proteins. We show that larger values of $Ω_{c}$ resembling those seen in proteins are obtained in lattice models with side chains (LMSC). The enhanced cooperativity in LMSC is due to the possibility of denser packing of side chains in the interior of the model protein. We also establish that $Ω_{c}$ correlates extremely well with (σ= (T_θ -T_{f} )/T_θ), where (T_…
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.
We study the thermodynamic and kinetic consequences of the competition between single-protein folding and protein-protein aggregation using a phenomenological model, in which the proteins can be in the unfolded (U), misfolded (M) or folded (F) states. The phase diagram shows the coexistence between a phase with aggregates of misfolded proteins and a phase of isolated proteins (U or F) in solution. The spinodal at low protein concentrations shows non-monotonic behavior with temperature, with implications for the stability of solutions of folded proteins at low temperatures. We follow the dynami…
We study the protein folding problem on the base of the quantum approach we proposed recently by considering the model of protein chain with nine amino-acid residues. We introduced the concept of distance space and its projections on a $XY$-plane, and two characteristic quantities, one is called compactness of protein structure and another is called probability ratio involving shortest path. Our results not only confirmed the fast quantum folding time but also unveiled the existence of quantum intelligence hidden behind in choosing protein folding pathways.
The interplay between structure-search of the native structure and desolvation in protein folding has been explored using a minimalist model. These results support a folding mechanism where most of the structural formation of the protein is achieved before water is expelled from the hydrophobic core. This view integrates water expulsion effects into the funnel energy landscape theory of protein folding. Comparisons to experimental results are shown for the SH3 protein. After the folding transition, a near-native intermediate with partially solvated hydrophobic core is found. This transition is…
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 $λ$-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 applica…
We introduce a novel generalization of the discrete nonlinear Schrödinger equation. It supports solitons that describe how proteins fold. As an example we scrutinize the villin headpiece HP35, an archetypal protein for testing both experimental and theoretical approaches to protein folding. Using explicit soliton profiles we construct its carbon backbone with an unprecedented accuracy.
A coarse-grained off-lattice model that is not biased in any way to the native state is proposed to fold proteins. To predict the native structure in a reasonable time, the model has included the essential effects of water in an effective potential. Two new ingredients, the dipole-dipole interaction and the local hydrophobic interaction, are introduced and are shown to be as crucial as the hydrogen bonding. The model allows successful folding of the wild-type sequence of protein G and may have provided important hints to the study of protein folding.
We test a simplified, local version of the helix model on two synthetic and two natural proteins, to study its efficiency in predicting the native secondary structure. The results we obtain are very good for the synthetic sequences, poorer for the two natural ones. This suggests that non-local terms play a fundamental role in determining the secondary structure, even if in some cases local terms alone may be sufficient.
Extensive Monte Carlo folding simulations for four proteins of various structural classes are carried out, using a single atomistic potential. In all cases, collapse occurs at a very early stage, and proteins fold into their native-like conformations at appropriate temperatures. The results demonstrate that the folding mechanism is controlled not only by thermodynamic factors but also by kinetic factors: The way a protein folds into its native structure, is also determined by the convergence point of early folding trajectories, which cannot be obtained by the free energy surface.
When described by a low-dimensional reaction coordinate, the rates of protein folding are determined by a subtle interplay between free-energy barriers and friction. While it is commonplace to extract free-energy profiles from molecular trajectories, a direct evaluation of friction is far more elusive, and one typically evaluates it indirectly via memoryless reaction rate theories. Here, using memory-kernel extraction methods founded on a generalised Langevin equation (GLE) formalism, we directly calculate the memory-dependent friction for eight fast-folding proteins, taken from a published se…
The dynamics of folding of proteins is studied by means of a phenomenological master equation. The energy distribution is taken as a truncated exponential for the misfolded states plus a native state sitting below the continuum. The influence of the gap on the folding dynamics is studied, for various models of the transition probabilities between the different states of the protein. We show that for certain models, the relaxation to the native state is accelerated by increasing the gap, whereas for others it is slowed down .
We employ simulations of model proteins to study folding on rugged energy landscapes. We construct ``first-passage'' networks as the system transitions from unfolded to native states. The nodes and bonds in these networks correspond to basins and transitions between them in the energy landscape. We find power-laws between the folding time and number of nodes and bonds. We show that these scalings are determined by the fractal properties of first-passage networks. Reliable folding is possible in systems with rugged energy landscapes because first passage networks have small fractal dimension.
A protein model with the pairwise interaction energies varying as local environment changes, i.e., including some kinds of collective effect between the contacts, is proposed. Lattice Monte Carlo simulations on the thermodynamical characteristics and free energy profile show a well-defined two-state behavior and cooperativity of folding for such a model. As a comparison, related simulations for the usual Gō model, where the interaction energies are independent of the local conformations, are also made. Our results indicate that the evolution of interactions during the folding process plays an …
We present a statistical mechanics approach to the protein folding problem. We first review some of the basic properties of proteins, and introduce some physical models to describe their thermodynamics. These models rely on a random heteropolymeric description of these non random biomolecules. Various kinds of randomness are investigated, and the connection with disordered systems is discussed. We conclude by a brief study of the dynamics of proteins.
Applying multicanonical simulations we investigated folding properties of off-lattice heteropolymers employing a mesoscopic hydrophobic-polar model. We study for various sequences folding channels in the free-energy landscape by comparing the equilibrium conformations with the folded state in terms of an angular overlap parameter. Although all investigated heteropolymer sequences contain the same content of hydrophobic and polar monomers, our analysis of the folding channels reveals a variety of characteristic folding behaviors known from realistic peptides.
A general theoretical framework is developed using free energy functional methods to understand the effects of heterogeneity in the folding of a well-designed protein. Native energetic heterogeneity arising from non-uniformity in native stability, as well as entropic heterogeneity intrinsic to the topology of the native structure are both investigated as to their impact on the folding free energy landscape and resulting folding mechanism. Given a minimally frustrated protein, both structural and energetic heterogeneity lower the thermodynamic barrier to folding, and designing in sufficient het…
The dynamics of contact pair formation between various hydrophobic residues during folding of a model protein Hp-36 is investigated by Brownian dynamics simulation. Hydropathy scale and non-local helix propensity of amino acids are used to model the complex interaction potential. The resulting structure of the model protein mimics the native state of the real protein with a $RMSD$ of 4.5 Å. A contact pair distance time correlation function (CPCF), $C_{P}^{ij}(t)$, is introduced which shows multistage decay, including a {\it slow late stage dynamics} for a few specific pairs. {\it These pairs d…
A phenomenological model hamiltonian to describe the folding of a protein with any given sequence is proposed. The protein is thought of as a collection of pieces of helices; as a consequence its configuration space increases with the number of secondary structure elements rather than with the number of residues. The hamiltonian presents both local (i.e. single helix, accounting for the stiffness of the chain) and non local (interactions between hydrophobically-charged helices) terms, and is expected to provide a first tool for studying the folding of real proteins. The partition function for …
Focusing on a small set of proteins that i) fold in a concerted, all-or-none fashion and ii) do not contain knots or slipknots, we show that the Gauss linking integral, the torsion and the number of sequence-distant contacts provide information regarding the folding rate. Our results suggest that the global topology/geometry of the proteins shifts from right-handed to left-handed with decreasing folding rate, and that this topological change is associated with an increase in the number of more sequence-distant contacts.
A new lattice protein model with a four-helix bundle ground state is analyzed by a parameter-space Monte Carlo histogram technique to evaluate the effects of an extensive variety of model potentials on folding thermodynamics. Cooperative helical formation and contact energies based on a 5-letter alphabet are found to be insufficient to satisfy calorimetric and other experimental criteria for two-state folding. Such proteinlike behaviors are predicted, however, by models with polypeptide-like local conformational restrictions and environment-dependent hydrogen bonding-like interactions.
The results of minimal model calculations suggest that the stability and the kinetic accessibility of the native state of small globular proteins are controlled by few "hot" sites. By mean of molecular dynamics simulations around the native conformation, which simulate the protein and the surrounding solvent at full--atom level, we generate an energetic map of the equilibrium state of the protein and simplify it with an Eigenvalue decomposition. The components of the Eigenvector associated with the lowest Eigenvalue indicate which are the "hot" sites responsible for the stability and for the f…
We argue that the first order folding transitions of proteins observed at physiological chemical conditions end in a critical point for a given temperature and chemical potential of the surrounding water. We investigate this critical point using a hierarchical Hamiltonian and determine its universality class. This class differs qualitatively from those of other known models.
A transfer-matrix formalism is introduced to evaluate exactly the partition function of the Munoz-Eaton model, relating the folding kinetics of proteins of known structure to their thermodynamics and topology. This technique can be used for a generic protein, for any choice of the energy and entropy parameters, and in principle allows the model to be used as a first tool to characterize the dynamics of a protein of known native state and equilibrium population. Applications to a $β$-hairpin and to protein CI-2, with comparisons to previous results, are also shown.
