Statistics of kinetic pathways on biased rough energy landscapes with applications to protein folding

Stochastic paths controlling speed and dissipation

Natural processes occur in a finite amount of time and dissipate energy, entropy, and matter. Near equilibrium, thermodynamic intuition suggests that fast irreversible processes will dissipate more energy and entropy than slow quasistatic processes connecting the same initial and final states. For small systems, recently discovered thermodynamic speed limits suggest that faster processes will dissipate more than slower processes. Here, we test the hypothesis that this relationship between speed and dissipation holds for stochastic paths far from equilibrium. To analyze stochastic paths on finite timescales, we derive an exact expression for the path probabilities of continuous-time Markov chains from the path summation solution to the master equation. We present a minimal model for a driven system in which relative energies of the initial and target states control the speed, and the nonequilibrium currents of a cycle control the dissipation. Although the hypothesis holds near equilibrium, we find that faster processes can dissipate less under far-from-equilibrium conditions because of strong currents. This model serves as a minimal prototype for designing kinetics to sculpt the nonequilibrium path space so that faster paths produce less dissipation.

Microscopic origin of diffusive dynamics in the context of transition path time distributions for protein folding and unfolding

Experimentally measured transition path time distributions are usually analyzed theoretically in terms of a diffusion equation over a free energy barrier. It is though well understood that the free energy profile separating the folded and unfolded states of a protein is characterized as a transition through many stable micro-states which exist between the folded and unfolded states. Why is it then justified to model the transition path dynamics in terms of a diffusion equation, namely the Smoluchowski equation (SE)? In principle, van Kampen has shown that a nearest neighbor Markov chain of thermal jumps between neighboring microstates will lead in a continuum limit to the SE, such that the friction coefficient is proportional to the mean residence time in each micro-state. However, the practical question of how many microstates are needed to justify modeling the transition path dynamics in terms of an SE has not been addressed. This is a central topic of this paper where we compare numerical results for transition paths based on the diffusion equation on the one hand and the nearest neighbor Markov jump model on the other. Comparison of the transition path time distributions shows that one needs at least a few dozen microstates to obtain reasonable agreement between the two approaches. Using the Markov nearest neighbor model one also obtains good agreement with the experimentally measured transition path time distributions for a DNA hairpin and PrP protein. As found previously when using the diffusion equation, the Markov chain model used here also reproduces the experimentally measured long time tail and confirms that the transition path barrier height is ∼3k_BT. This study indicates that in the future, when attempting to model experimentally measured transition path time distributions, one should perhaps prefer a nearest neighbor Markov model which is well defined also for rough energy landscapes. Such studies can also shed light on the minimal number of microstates needed to unravel the experimental data.

Noise-induced quasiperiod and period switching

We employ a typical genetic circuit model to explore how noise can influence dynamic structure. With the increase of a key interactive parameter, the model will deterministically go through two bifurcations and three dynamic structure regions. We find that a quasiperiodic component, which is not allowed by deterministic dynamics, will be generated by noise inducing in the first two regions, and this quasiperiod will be more and more stable along with the increase in noise. In particular, in the second region the quasiperiod will compete with a stable limit cycle and perform a new transient rhythm. Furthermore, we ascertain the entropy production rate and the heat dissipation rate, and discover a minimal value with theoretical elucidation. In the end, we unveil the mechanism of the formation of quasiperiods, and show a practical biological example. We expect this work to be helpful in solving some biological or ecological problems, such as the genetic origin of periodical cicadas and population dynamics with fluctuation.

Physics of biomolecular recognition and conformational dynamics

Biomolecular recognition usually leads to the formation of binding complexes, often accompanied by large-scale conformational changes. This process is fundamental to biological functions at the molecular and cellular levels. Uncovering the physical mechanisms of biomolecular recognition and quantifying the key biomolecular interactions are vital to understand these functions. The recently developed energy landscape theory has been successful in quantifying recognition processes and revealing the underlying mechanisms. Recent studies have shown that in addition to affinity, specificity is also crucial for biomolecular recognition. The proposed physical concept of intrinsic specificity based on the underlying energy landscape theory provides a practical way to quantify the specificity. Optimization of affinity and specificity can be adopted as a principle to guide the evolution and design of molecular recognition. This approach can also be used in practice for drug discovery using multidimensional screening to identify lead compounds. The energy landscape topography of molecular recognition is important for revealing the underlying flexible binding or binding-folding mechanisms. In this review, we first introduce the energy landscape theory for molecular recognition and then address four critical issues related to biomolecular recognition and conformational dynamics: (1) specificity quantification of molecular recognition; (2) evolution and design in molecular recognition; (3) flexible molecular recognition; (4) chromosome structural dynamics. The results described here and the discussions of the insights gained from the energy landscape topography can provide valuable guidance for further computational and experimental investigations of biomolecular recognition and conformational dynamics.

Protein folding as a complex reaction: a two-component potential for the driving force of folding and its variation with folding scenario

The Helmholtz decomposition of the vector field of probability fluxes in a two-dimensional space of collective variables makes it possible to introduce a potential for the driving force of protein folding [Chekmarev, J. Chem. Phys. 139 (2013) 145103]. The potential has two components: one component (Φ) is responsible for the source and sink of the folding flow, which represent, respectively, the unfolded and native state of the protein, and the other (Ψ) accounts for the flow vorticity inherently generated at the periphery of the flow field and provides the canalization of the flow between the source and sink. Both components obey Poisson’s equations with the corresponding source/sink terms. In the present paper, we consider how the shape of the potential changes depending on the scenario of protein folding. To mimic protein folding dynamics projected onto a two-dimensional space of collective variables, the two-dimensional Müller and Brown potential is employed. Three characteristic scenarios are considered: a single pathway from the unfolded to the native state without intermediates, two parallel pathways without intermediates, and a single pathway with an off-pathway intermediate. To determine the probability fluxes, the hydrodynamic description of the folding reaction is used, in which the first-passage folding is viewed as a steady flow of the representative points of the protein from the unfolded to the native state. We show that despite the possible complexity of the folding process, the Φ-component is simple and universal in shape. The Ψ-component is more complex and reveals characteristic features of the process of folding. The present approach is potentially applicable to other complex reactions, for which the transition from the reactant to the product can be described in a space of two (collective) variables.

Modulation of folding energy landscape by charge-charge interactions: linking experiments with computational modeling

The kinetics of folding-unfolding of a structurally diverse set of four proteins optimized for thermodynamic stability by rational redesign of surface charge-charge interactions is characterized experimentally. The folding rates are faster for designed variants compared with their wild-type proteins, whereas the unfolding rates are largely unaffected. A simple structure-based computational model, which incorporates the Debye-Hückel formalism for the electrostatics, was used and found to qualitatively recapitulate the experimental results. Analysis of the energy landscapes of the designed versus wild-type proteins indicates the differences in refolding rates may be correlated with the degree of frustration of their respective energy landscapes. Our simulations indicate that naturally occurring wild-type proteins have frustrated folding landscapes due to the surface electrostatics. Optimization of the surface electrostatics seems to remove some of that frustration, leading to enhanced formation of native-like contacts in the transition-state ensembles (TSE) and providing a less frustrated energy landscape between the unfolded and TS ensembles. Macroscopically, this results in faster folding rates. Furthermore, analyses of pairwise distances and radii of gyration suggest that the less frustrated energy landscapes for optimized variants are a result of more compact unfolded and TS ensembles. These findings from our modeling demonstrates that this simple model may be used to: (i) gain a detailed understanding of charge-charge interactions and their effects on modulating the energy landscape of protein folding and (ii) qualitatively predict the kinetic behavior of protein surface electrostatic interactions.

Non-equilibrium transition state rate theory

Transition state or Kramers' rate theory has been used to quantify the kinetic speed of many chemical, physical and biological equilibrium processes successfully.

How kinetics within the unfolded state affects protein folding: an analysis based on markov state models and an ultra-long MD trajectory

Understanding how kinetics in the unfolded state affects protein folding is a fundamentally important yet less well-understood issue. Here we employ three different models to analyze the unfolded landscape and folding kinetics of the miniprotein Trp-cage. The first is a 208 μs explicit solvent molecular dynamics (MD) simulation from D. E. Shaw Research containing tens of folding events. The second is a Markov state model (MSM-MD) constructed from the same ultralong MD simulation; MSM-MD can be used to generate thousands of folding events. The third is a Markov state model built from temperature replica exchange MD simulations in implicit solvent (MSM-REMD). All the models exhibit multiple folding pathways, and there is a good correspondence between the folding pathways from direct MD and those computed from the MSMs. The unfolded populations interconvert rapidly between extended and collapsed conformations on time scales ≤40 ns, compared with the folding time of ∼5 μs. The folding rates are independent of where the folding is initiated from within the unfolded ensemble. About 90% of the unfolded states are sampled within the first 40 μs of the ultralong MD trajectory, which on average explores ∼27% of the unfolded state ensemble between consecutive folding events. We clustered the folding pathways according to structural similarity into "tubes", and kinetically partitioned the unfolded state into populations that fold along different tubes. From our analysis of the simulations and a simple kinetic model, we find that, when the mixing within the unfolded state is comparable to or faster than folding, the folding waiting times for all the folding tubes are similar and the folding kinetics is essentially single exponential despite the presence of heterogeneous folding paths with nonuniform barriers. When the mixing is much slower than folding, different unfolded populations fold independently, leading to nonexponential kinetics. A kinetic partition of the Trp-cage unfolded state is constructed which reveals that different unfolded populations have almost the same probability to fold along any of the multiple folding paths. We are investigating whether the results for the kinetics in the unfolded state of the 20-residue Trp-cage is representative of larger single domain proteins.

Quantifying the topography of the intrinsic energy landscape of flexible biomolecular recognition

Biomolecular functions are determined by their interactions with other molecules. Biomolecular recognition is often flexible and associated with large conformational changes involving both binding and folding. However, the global and physical understanding for the process is still challenging. Here, we quantified the intrinsic energy landscapes of flexible biomolecular recognition in terms of binding-folding dynamics for 15 homodimers by exploring the underlying density of states, using a structure-based model both with and without considering energetic roughness. By quantifying three individual effective intrinsic energy landscapes (one for interfacial binding, two for monomeric folding), the association mechanisms for flexible recognition of 15 homodimers can be classified into two-state cooperative "coupled binding-folding" and three-state noncooperative "folding prior to binding" scenarios. We found that the association mechanism of flexible biomolecular recognition relies on the interplay between the underlying effective intrinsic binding and folding energy landscapes. By quantifying the whole global intrinsic binding-folding energy landscapes, we found strong correlations between the landscape topography measure Λ (dimensionless ratio of energy gap versus roughness modulated by the configurational entropy) and the ratio of the thermodynamic stable temperature versus trapping temperature, as well as between Λ and binding kinetics. Therefore, the global energy landscape topography determines the binding-folding thermodynamics and kinetics, crucial for the feasibility and efficiency of realizing biomolecular function. We also found "U-shape" temperature-dependent kinetic behavior and a dynamical cross-over temperature for dividing exponential and nonexponential kinetics for two-state homodimers. Our study provides a unique way to bridge the gap between theory and experiments.

Monitoring the folding of Trp-cage peptide by two-dimensional infrared (2DIR) spectroscopy

Protein folding is one of the most fundamental problems in modern molecular biology. Uncovering the detailed folding mechanism requires methods that can monitor the structures at high temporal and spatial resolution. Two-dimensional infrared (2DIR) spectroscopy is a new tool for studying protein structures and dynamics with high time resolution. Using atomistic molecular dynamics simulations, we illustrate the folding process of Trp-cage along the dominant pathway on the free energy landscape by analyzing nonchiral and chiral coherent 2DIR spectra along the pathway. Isotope labeling is used to reveal residue-specific information. We show that the high resolution structural sensitivity of 2DIR can differentiate the ensemble evolution of protein and thus provides a microscopic picture of the folding process.

Bridging steady states with renormalization group analysis

Transitions between different condensed phases, molecular conformations, chemical compositions, or spatiotemporal patterns play important roles in many branches of natural science and at the same time incur serious challenges in their precise characterization. We design an approach for computing connecting orbits bridging steady states based on the renormalization group analysis. The technique is successfully applied to several interesting examples and good analytic results are obtained in a systematic and unified way.

Transition paths, diffusive processes, and preequilibria of protein folding

Fundamental relationships between the thermodynamics and kinetics of protein folding were investigated using chain models of natural proteins with diverse folding rates by extensive comparisons between the distribution of conformations in thermodynamic equilibrium and the distribution of conformations sampled along folding trajectories. Consistent with theory and single-molecule experiment, duration of the folding transition paths exhibits only a weak correlation with overall folding time. Conformational distributions of folding trajectories near the overall thermodynamic folding/unfolding barrier show significant deviations from preequilibrium. These deviations, the distribution of transition path times, and the variation of mean transition path time for different proteins can all be rationalized by a diffusive process that we modeled using simple Monte Carlo algorithms with an effective coordinate-independent diffusion coefficient. Conformations in the initial stages of transition paths tend to form more nonlocal contacts than typical conformations with the same number of native contacts. This statistical bias, which is indicative of preferred folding pathways, should be amenable to future single-molecule measurements. We found that the preexponential factor defined in the transition state theory of folding varies from protein to protein and that this variation can be rationalized by our Monte Carlo diffusion model. Thus, protein folding physics is different in certain fundamental respects from the physics envisioned by a simple transition-state picture. Nonetheless, transition state theory can be a useful approximate predictor of cooperative folding speed, because the height of the overall folding barrier is apparently a proxy for related rate-determining physical properties.

The dominant folding route minimizes backbone distortion in SH3

Energetic frustration in protein folding is minimized by evolution to create a smooth and robust energy landscape. As a result the geometry of the native structure provides key constraints that shape protein folding mechanisms. Chain connectivity in particular has been identified as an essential component for realistic behavior of protein folding models. We study the quantitative balance of energetic and geometrical influences on the folding of SH3 in a structure-based model with minimal energetic frustration. A decomposition of the two-dimensional free energy landscape for the folding reaction into relevant energy and entropy contributions reveals that the entropy of the chain is not responsible for the folding mechanism. Instead the preferred folding route through the transition state arises from a cooperative energetic effect. Off-pathway structures are penalized by excess distortion in local backbone configurations and contact pair distances. This energy cost is a new ingredient in the malleable balance of interactions that controls the choice of routes during protein folding.

Topography of funneled landscapes determines the thermodynamics and kinetics of protein folding

The energy landscape approach has played a fundamental role in advancing our understanding of protein folding. Here, we quantify protein folding energy landscapes by exploring the underlying density of states. We identify three quantities essential for characterizing landscape topography: the stabilizing energy gap between the native and nonnative ensembles δE, the energetic roughness ΔE, and the scale of landscape measured by the entropy S. We show that the dimensionless ratio between the gap, roughness, and entropy of the system Λ=δE/(ΔE√(2S)) accurately predicts the thermodynamics, as well as the kinetics of folding. Large Λ implies that the energy gap (or landscape slope towards the native state) is dominant, leading to more funneled landscapes. We investigate the role of topological and energetic roughness for proteins of different sizes and for proteins of the same size, but with different structural topologies. The landscape topography ratio Λ is shown to be monotonically correlated with the thermodynamic stability against trapping, as characterized by the ratio of folding temperature versus trapping temperature. Furthermore, Λ also monotonically correlates with the folding kinetic rates. These results provide the quantitative bridge between the landscape topography and experimental folding measurements.

An efficient method for modeling kinetic behavior of channel proteins in cardiomyocytes

Characterization of the kinetic and conformational properties of channel proteins is a crucial element in the integrative study of congenital cardiac diseases. The proteins of the ion channels of cardiomyocytes represent an important family of biological components determining the physiology of the heart. Some computational studies aiming to understand the mechanisms of the ion channels of cardiomyocytes have concentrated on Markovian stochastic approaches. Mathematically, these approaches employ Chapman-Kolmogorov equations coupled with partial differential equations. As the scale and complexity of such subcellular and cellular models increases, the balance between efficiency and accuracy of algorithms becomes critical. We have developed a novel two-stage splitting algorithm to address efficiency and accuracy issues arising in such modeling and simulation scenarios. Numerical experiments were performed based on the incorporation of our newly developed conformational kinetic model for the rapid delayed rectifier potassium channel into the dynamic models of human ventricular myocytes. Our results show that the new algorithm significantly outperforms commonly adopted adaptive Runge-Kutta methods. Furthermore, our parallel simulations with coupled algorithms for multicellular cardiac tissue demonstrate a high linearity in the speedup of large-scale cardiac simulations.

Combination of Markov state models and kinetic networks for the analysis of molecular dynamics simulations of peptide folding

Atomistic molecular dynamics simulations of the TZ1 beta-hairpin peptide have been carried out using an implicit model for the solvent. The trajectories have been analyzed using a Markov state model defined on the projections along two significant observables and a kinetic network approach. The Markov state model allowed for an unbiased identification of the metastable states of the system, and provided the basis for commitment probability calculations performed on the kinetic network. The kinetic network analysis served to extract the main transition state for folding of the peptide and to validate the results from the Markov state analysis. The combination of the two techniques allowed for a consistent and concise characterization of the dynamics of the peptide. The slowest relaxation process identified is the exchange between variably folded and denatured species, and the second slowest process is the exchange between two different subsets of the denatured state which could not be otherwise identified by simple inspection of the projected trajectory. The third slowest process is the exchange between a fully native and a partially folded intermediate state characterized by a native turn with a proximal backbone H-bond, and frayed side-chain packing and termini. The transition state for the main folding reaction is similar to the intermediate state, although a more native like side-chain packing is observed.

Kinetic network study of the diversity and temperature dependence of Trp-Cage folding pathways: combining transition path theory with stochastic simulations

We present a new approach to study a multitude of folding pathways and different folding mechanisms for the 20-residue mini-protein Trp-Cage using the combined power of replica exchange molecular dynamics (REMD) simulations for conformational sampling, transition path theory (TPT) for constructing folding pathways, and stochastic simulations for sampling the pathways in a high dimensional structure space. REMD simulations of Trp-Cage with 16 replicas at temperatures between 270 and 566 K are carried out with an all-atom force field (OPLSAA) and an implicit solvent model (AGBNP). The conformations sampled from all temperatures are collected. They form a discretized state space that can be used to model the folding process. The equilibrium population for each state at a target temperature can be calculated using the weighted-histogram-analysis method (WHAM). By connecting states with similar structures and creating edges satisfying detailed balance conditions, we construct a kinetic network that preserves the equilibrium population distribution of the state space. After defining the folded and unfolded macrostates, committor probabilities (P(fold)) are calculated by solving a set of linear equations for each node in the network and pathways are extracted together with their fluxes using the TPT algorithm. By clustering the pathways into folding "tubes", a more physically meaningful picture of the diversity of folding routes emerges. Stochastic simulations are carried out on the network, and a procedure is developed to project sampled trajectories onto the folding tubes. The fluxes through the folding tubes calculated from the stochastic trajectories are in good agreement with the corresponding values obtained from the TPT analysis. The temperature dependence of the ensemble of Trp-Cage folding pathways is investigated. Above the folding temperature, a large number of diverse folding pathways with comparable fluxes flood the energy landscape. At low temperature, however, the folding transition is dominated by only a few localized pathways.

The origin of nonmonotonic complex behavior and the effects of nonnative interactions on the diffusive properties of protein folding

We present a method for calculating the configurational-dependent diffusion coefficient of a globular protein as a function of the global folding process. Using a coarse-grained structure-based model, we determined the diffusion coefficient, in reaction coordinate space, as a function of the fraction of native contacts formed Q for the cold shock protein (TmCSP). We find nonmonotonic behavior for the diffusion coefficient, with high values for the folded and unfolded ensembles and a lower range of values in the transition state ensemble. We also characterized the folding landscape associated with an energetically frustrated variant of the model. We find that a low-level of frustration can actually stabilize the native ensemble and increase the associated diffusion coefficient. These findings can be understood from a mechanistic standpoint, in that the transition state ensemble has a more homogeneous structural content when frustration is present. Additionally, these findings are consistent with earlier calculations based on lattice models of protein folding and more recent single-molecule fluorescence measurements.

Coordinate and time-dependent diffusion dynamics in protein folding

We developed both analytical and simulation methods to explore the diffusion dynamics in protein folding. We found the diffusion as a quantitative measure of escape from local traps along the protein folding funnel with chosen reaction coordinates has two remarkable effects on kinetics. At a fixed coordinate, local escape time depends on the distribution of barriers around it, therefore the diffusion is often time distributed. On the other hand, the environments (local escape barriers) change along the coordinates, therefore diffusion is coordinate dependent. The effects of time-dependent diffusion on folding can lead to non-exponential kinetics and non-Poisson statistics of folding time distribution. The effects of coordinate dependent diffusion on folding can lead to the change of the kinetic barrier height as well as the position of the corresponding transition state and therefore modify the folding kinetic rates as well as the kinetic routes. Our analytical models for folding are based on a generalized Fokker-Planck diffusion equation with diffusion coefficient both dependent on coordinate and time. Our simulation for folding are based on structure-based folding models with a specific fast folding protein CspTm studied experimentally on diffusion and folding with single molecules. The coordinate and time-dependent diffusion are especially important to be considered in fast folding and single molecule studies, when there is a small or no free energy barrier and kinetics is controlled by diffusion while underlying statistics of kinetics become important. Including the coordinate dependence of diffusion will challenge the transition state theory of protein folding. The classical transition state theory will have to be modified to be consistent. The more detailed folding mechanistic studies involving phi value analysis based on the classical transition state theory will also have to be quantitatively modified. Complex kinetics with multiple time scales may allow us not only to explore the folding kinetics but also probe the local landscape and barrier height distribution with single-molecule experiments.

Folding of electrostatically charged beads-on-a-string as an experimental realization of a theoretical model in polymer science

The "beads-on-a-string" model for folding of polymers is a cornerstone of theoretical polymer science. This communication describes a physical model of beads-on-a-string, based on the folding of flexible strings of electrostatically charged beads in two dimensions. The system comprises millimeter-scale Teflon and Nylon-6,6 (spherical or cylindrical) beads (approximately 6 mm in diameter) separated by smaller (approximately 3 mm) poly(methyl methacrylate) (PMMA) spherical beads, threaded on a flexible string. The smaller, uncharged beads define the distances between the larger beads, and control the flexibility of the string. During agitation of the sequence of beads on a planar, horizontal paper surface, tribocharging generates opposite electrostatic charges on the larger Nylon and Teflon beads, but leaves the smaller PMMA beads essentially uncharged; the resulting electrostatic interactions cause the string to fold. Examination and comparison of two models--one physical and one theoretical--may offer a new approach to understanding folding, collapse, and molecular recognition at an abstract level, with particular opportunity to explore the influence of the flexibility of the string and the shape of the beads on the pattern and rate of folding. The physical system is, thus, an analog computer, simulating the theoretical beads-on-a-string model in two dimensions; this system makes it possible to test hypotheses connecting "sequence" to "folding", rapidly and conveniently, while exploring nonlinearities and other complexities omitted from the theoretical model.

Coherent multidimensional vibrational spectroscopy of biomolecules: concepts, simulations, and challenges

The response of complex molecules to sequences of femtosecond infrared pulses provides a unique window into their structure, dynamics, and fluctuating environments. Herein we survey the basic principles of modern two-dimensional infrared (2DIR) spectroscopy, which analogous to those of multidimensional NMR spectroscopy. The perturbative approach for computing the nonlinear optical response of coupled localized chromophores is introduced and applied to the amide backbone transitions of proteins, liquid water, membrane lipids, and amyloid fibrils. The signals are analyzed using classical molecular dynamics simulations combined with an effective fluctuating Hamiltonian for coupled localized anharmonic vibrations whose dependence on the local electrostatic environment is parameterized by an ab initio map. Several simulation methods, (cumulant expansion of Gaussian fluctuation, quasiparticle scattering, the stochastic Liouville equations, direct numerical propagation) are surveyed. Chirality-induced techniques which dramatically enhance the resolution are demonstrated. Signatures of conformational and hydrogen-bonding fluctuations, protein folding, and chemical-exchange processes are discussed.

Configuration-dependent diffusion can shift the kinetic transition state and barrier height of protein folding

We show that diffusion can play an important role in protein-folding kinetics. We explicitly calculate the diffusion coefficient of protein folding in a lattice model. We found that diffusion typically is configuration- or reaction coordinate-dependent. The diffusion coefficient is found to be decreasing with respect to the progression of folding toward the native state, which is caused by the collapse to a compact state constraining the configurational space for exploration. The configuration- or position-dependent diffusion coefficient has a significant contribution to the kinetics in addition to the thermodynamic free-energy barrier. It effectively changes (increases in this case) the kinetic barrier height as well as the position of the corresponding transition state and therefore modifies the folding kinetic rates as well as the kinetic routes. The resulting folding time, by considering both kinetic diffusion and the thermodynamic folding free-energy profile, thus is slower than the estimation from the thermodynamic free-energy barrier with constant diffusion but is consistent with the results from kinetic simulations. The configuration- or coordinate-dependent diffusion is especially important with respect to fast folding, when there is a small or no free-energy barrier and kinetics is controlled by diffusion. Including the configurational dependence will challenge the transition state theory of protein folding. The classical transition state theory will have to be modified to be consistent. The more detailed folding mechanistic studies involving phi value analysis based on the classical transition state theory also will have to be modified quantitatively.

Mosaic energy landscapes of liquids and the control of protein conformational dynamics by glass-forming solvents

Using recent advances in the Random First-Order Transition (RFOT) Theory of glass-forming liquids, we explain how the molecular motions of a glass-forming solvent distort the protein's boundary and slave some of the protein's conformational motions. Both the length and time scales of the solvent imposed constraints are provided by the RFOT theory. Comparison of the protein relaxation rate to that of the solvent provides an explicit lower bound on the size of the conformational space explored by the protein relaxation. Experimental measurements of slaving of myoglobin motions indicate that a major fraction of functionally important motions have significant entropic barriers.

Mathematics of protein pathological misfolding

"Protein folding is defined as a process by which a polypeptide chain performs a search in conformational space with the objective of achieving the so-called native conformation to global free-energy minimum under a given set of physiochemical conditions of the medium." Misfolding then, is the process by which this objective is not achieved. Protein Folding Quality Assessment (PFQA), is characterized by a three-parameter distribution function Phi(T) referred to as the PFQA function. It uses results of protein folding processes to assess the output quality of protein folding. Protein misfolding is implicated in the initial cause of many conformational diseases. Folding of cytosolic protein can be regarded as the performance of the protein after it is produced or manufactured by the synthesis processes. Protein folding through different mechanisms and pathways has been extensively covered in [J.D. Bryngelson, P.G. Wolynes, Spin glass and statistical mechanics of protein folding, Proc. Natl. Acad. Sci. USA 84 (1987) 7524; J. Wang, Statistics, pathways and dynamics of single molecule folding, J. Chem. Phys. 118 (2) (2003) 953; N.D. Socci, J.N. Onuchic, P.G. Wolynes, Diffusive dynamics of the reaction coordinates for protein folding funnels, J. Chem. Phys. 104 (14) (1996); D. Thirumalai, From minimal models to real proteins, time scales for protein folding kinetics, J. Phys. I France 5 (1995) 1457]. The model is based on growth models of Ratkowsky, Richards, etc. [D.A. Ratkowski, T.J. Reeds, Choosing near-linear parameters logistic model for radio-ligand and related assays, Biometrics 42 (1986) 575] for a three-parameters model to handle the quality assessment of the folding process. Thus a complete distribution can be found, thanks to the scale, location and shape parameters.

Structural and Pathway Complexity of β-Strand Reorganization within Aggregates of Human Transthyretin(105−115) Peptide

Interstrand conformational rearrangements of human transthyretin peptide (TTR(105-115)) within dimeric aggregates were simulated by means of molecular dynamics (MD) with implicit solvation model for a total length of 48 micros. The conformations sampled in the MD simulations were clustered to identify free energy minima without any projections of free energy surface. A connected graph was constructed with nodes (=clusters) and edges corresponding to free energy minima and transitions between nodes, respectively. This connected graph which reflects the complexity of the free energy surface was used to extract the transition disconnectivity graph, which reflects the overall free energy barriers between pairs of free energy minima but does not contain information on transition paths. The routes of transitions between important free energy minima were obtained by further processing the original graph and the MD data. We have found that both parallel and antiparallel aggregates are populated. The parallel aggregates with different alignment patterns are separated by nonnegligible free energy barriers. Multiroutes exist in the interstrand conformational reorganization. Most visited routes do not dominant the kinetics, while less visited routes contribute a little each but they are numerous and their total contributions are actually dominant. There are various kinds of reptation motions, including those through a beta-bulge, side-chain aided reptation, and flipping or rotation of a hairpin formed by one strand.

Diabetes mellitus due to misfolding of a beta-cell transcription factor: stereospecific frustration of a Schellman motif in HNF-1alpha

Maturity-onset diabetes of the young (MODY3), a monogenic form of type II diabetes mellitus, results most commonly from mutations in hepatocyte nuclear factor 1alpha (HNF-1alpha). Diabetes-associated mutation G20R perturbs the dimerization domain of HNF-1alpha, an intertwined four-helix bundle. In the wild-type structure G20 participates in a Schellman motif to cap an alpha-helix; its dihedral angles lie in the right side of the Ramachandran plot (alpha(L) region; phi 97 degrees). Substitutions G20R and G20A lead to dimeric molten globules of low stability, suggesting that the impaired function of the diabetes-associated transcription factor is due in large part to a main-chain perturbation rather than to specific features of the Arg side-chain. This hypothesis is supported by the enhanced stability of non-standard analogues containing D-Ala or D-Ser at position 20. The crystal structure of the D-Ala20 analogue, determined to a resolution of 1.4 A, is essentially identical to the wild-type structure in the same crystal form. The mean root-mean-square deviation between equivalent C(alpha) atoms (residues 5-28) is 0.3 A; (phi, psi) angles of D-Ala20 are the same as those of G20 in the wild-type structure. Whereas the side-chain of A20 or R20 would be expected to clash with the preceding carbonyl oxygen (thus accounting for its frustrated energy landscape), the side-chain of D-Ala20 projects into solvent without perturbation of the Schellman motif. Calorimetric studies indicate that the increased stability of the D-Ala20 analogue (DeltaDeltaG(u) 1.5 kcal/mol) is entropic in origin, consistent with a conformational bias toward native-like conformations in the unfolded state. Studies of multiple substitutions at G20 and neighboring positions highlight the essential contributions of a glycine-specific tight turn and adjoining inter-subunit side-chain hydrogen bonds to the stability and architectural specificity of the intertwined dimer. Comparison of L- and D amino acid substitutions thus provides an example of the stereospecific control of an energy landscape by a helix-capping residue.

Absolute rate theories of epigenetic stability

Spontaneous switching events in most characterized genetic switches are rare, resulting in extremely stable epigenetic properties. We show how simple arguments lead to theories of the rate of such events much like the absolute rate theory of chemical reactions corrected by a transmission factor. Both the probability of the rare cellular states that allow epigenetic escape and the transmission factor depend on the rates of DNA binding and unbinding events and on the rates of protein synthesis and degradation. Different mechanisms of escape from the stable attractors occur in the nonadiabatic, weakly adiabatic, and strictly adiabatic regimes, characterized by the relative values of those input rates.

Quantifying kinetic paths of protein folding

We propose a new approach to activated protein folding dynamics via a diffusive path integral framework. The important issues of kinetic paths in this situation can be directly addressed. This leads to the identification of the kinetic paths of the activated folding process, and provides a direct tool and language for the theoretical and experimental community to understand the problem better. The kinetic paths giving the dominant contributions to the long-time folding activation dynamics can be quantitatively determined. These are shown to be the instanton paths. The contributions of these instanton paths to the kinetics lead to the "bell-like" shape folding rate dependence on temperature, which is in good agreement with folding kinetic experiments and simulations. The connections to other approaches as well as the experiments of the protein folding kinetics are discussed.

The complex kinetics of protein folding in wide temperature ranges

The complex protein folding kinetics in wide temperature ranges is studied through diffusive dynamics on the underlying energy landscape. The well-known kinetic chevron rollover behavior is recovered from the mean first passage time, with the U-shape dependence on temperature. The fastest folding temperature T0 is found to be smaller than the folding transition temperature Tf. We found that the fluctuations of the kinetics through the distribution of first passage time show rather universal behavior, from high-temperature exponential Poissonian kinetics to the relatively low-temperature highly non-exponential kinetics. The transition temperature is at Tk and T0 < Tk < Tf. In certain low-temperature regimes, a power law behavior at long time emerges. At very low temperatures (lower than trapping transition temperature T < T0/(4 approximately 6)), the kinetics is an exponential Poissonian process again.

Downhill kinetics of biomolecular interface binding: globally connected scenario

We study the kinetics of the biomolecular binding process at the interface using energy landscape theory. The global kinetic connectivity case is considered for a downhill funneled energy landscape. By solving the kinetic master equation, the kinetic time for binding is obtained and shown to have a U-shape curve-dependence on the temperature. The kinetic minimum of the binding time monotonically decreases when the ratio of the underlying energy gap between native state and average non-native states versus the roughness or the fluctuations of the landscape increases. At intermediate temperatures, fluctuations measured by the higher moments of the binding time lead to non-Poissonian, non-exponential kinetics. At both high and very low temperatures, the kinetics is nearly Poissonian and exponential.

Energy landscape theory, funnels, specificity, and optimal criterion of biomolecular binding

We study the nature of biomolecular binding. We found that in general there exists several thermodynamic phases: a native binding phase, a non-native phase, and a glass or local trapping phase. The quantitative optimal criterion for the binding specificity is found to be the maximization of the ratio of the binding transition temperature versus the trapping transition temperature, or equivalently the ratio of the energy gap of binding between the native state and the average non-native states versus the dispersion or variance of the non-native states. This leads to a funneled binding energy landscape.

Diffusion dynamics, moments, and distribution of first-passage time on the protein-folding energy landscape, with applications to single molecules

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 found that the first passage-time (FPT) distribution undergoes a dynamic transition at a temperature below which the FPT distribution develops a power-law tail, a signature of the intermittent nonexponential kinetic phenomena for the folding dynamics. Possible applications to single-molecule dynamics experiments are discussed.

Kinetic transition in model proteins with a denatured native spinodal

The relaxation kinetics of three-dimensional lattice model proteins with Gō potential is studied. A kinetic transition from an exponential behavior to a nonexponential one with a denatured native spinodal is characterized. The transition temperatures T(k), obtained from simulations and a semiquantitative estimation, are found to be the same. The change in free energy landscape during the transition is discussed microscopically by studying the detailed folding processes of various paths. The connection of T(k) with the foldability is also studied by a Z-score-like quantity T(f)/T(k).

Disorder and funneling effects on exciton migration in treelike dendrimers

The center-bound excitonic diffusion on dendrimers subjected to several types of nonhomogeneous funneling potentials is considered. We first study the mean first passage time (MFPT) for diffusion in a linear potential with different types of correlated and uncorrelated random perturbations. Increasing the funneling force, there is a transition from a phase in which the MFPT grows exponentially with the number of generations g to one in which it does so linearly. Overall the disorder slows down the diffusion, but the effect is much more pronounced in the exponential compared to the linear phase. When the disorder gives rise to uncorrelated random forces there is, in addition, a transition as the temperature T is lowered. This is a transition from a high-T regime in which all paths contribute to the MFPT to a low-T regime in which only a few of them do. We further explore the funneling within a realistic nonlinear potential for extended dendrimers in which the dependence of the lowest excitonic energy level on the segment length was derived using the time-dependent Hatree-Fock approximation. Under this potential the MFPT grows initially linearly with g but crosses over, beyond a molecular-specific and T-dependent optimal size, to an exponential increase. Finally we consider geometrical disorder in the form of a small concentration of long connections as in the small world model. Beyond a critical concentration of connections the MFPT decreases significantly and it changes to a power law or to a logarithmic scaling with g, depending on the strength of the funneling force.

The dimerization domain of HNF-1alpha: structure and plasticity of an intertwined four-helix bundle with application to diabetes mellitus

Maturity-onset diabetes mellitus of the young (MODY) is a human genetic syndrome most commonly due to mutations in hepatocyte nuclear factor-1alpha (HNF-1alpha). Here, we describe the crystal structure of the HNF-1alpha dimerization domain at 1.7 A resolution and assess its structural plasticity. The crystal's low solvent content (23%, v/v) leads to tight packing of peptides in the lattice. Two independent dimers, similar in structure, are formed in the unit cell by a 2-fold crystallographic symmetry axis. The dimers define a novel intertwined four-helix bundle (4HB). Each protomer contains two alpha-helices separated by a sharp non-canonical turn. Dimer-related alpha-helices form anti-parallel coiled-coils, including an N-terminal "mini-zipper" complementary in structure, symmetry and surface characteristics to transcriptional coactivator dimerization cofactor of HNF-1 (DCoH). A confluence of ten leucine side-chains (five per protomer) forms a hydrophobic core. Isotope-assisted NMR studies demonstrate that a similar intertwined dimer exists in solution. Comparison of structures obtained in multiple independent crystal forms indicates that the mini-zipper is a stable structural element, whereas the C-terminal alpha-helix can adopt a broad range of orientations. Segmental alignment of the mini-zipper (mean pairwise root-mean-square difference (rmsd) in C(alpha) coordinates of 0.29 A) is associated with a 2.1 A mean C(alpha) rmsd displacement of the C-terminal coiled-coil. The greatest C-terminal structural variation (4.1 A C(alpha) rmsd displacement) is observed in the DCoH-bound peptide. Diabetes-associated mutations perturb distinct structural features of the HNF-1alpha domain. One mutation (L12H) destabilizes the domain but preserves structural specificity. Adjoining H12 side-chains in a native-like dimer are predicted to alter the functional surface of the mini-zipper involved in DCoH recognition. The other mutation (G20R), by contrast, leads to a dimeric molten globule, as indicated by its 1H-NMR features and fluorescent binding of 1-anilino-8-naphthalene sulfonate. We propose that a glycine-specific turn configuration enables specific interactions between the mini-zipper and the C-terminal coiled-coil.

Supercooled liquids and the glass transition

Glasses are disordered materials that lack the periodicity of crystals but behave mechanically like solids. The most common way of making a glass is by cooling a viscous liquid fast enough to avoid crystallization. Although this route to the vitreous state-supercooling-has been known for millennia, the molecular processes by which liquids acquire amorphous rigidity upon cooling are not fully understood. Here we discuss current theoretical knowledge of the manner in which intermolecular forces give rise to complex behaviour in supercooled liquids and glasses. An intriguing aspect of this behaviour is the apparent connection between dynamics and thermodynamics. The multidimensional potential energy surface as a function of particle coordinates (the energy landscape) offers a convenient viewpoint for the analysis and interpretation of supercooling and glass-formation phenomena. That much of this analysis is at present largely qualitative reflects the fact that precise computations of how viscous liquids sample their landscape have become possible only recently.