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Entropic barriers, transition states, funnels, and exponential protein folding kinetics: A simple model
Abstract
This paper presents an analytically tractable model that captures the most elementary aspect of the protein folding problem, namely that both the energy and the entropy decrease as a protein folds. In this model, the system diffuses within a sphere in the presence of an attractive spherically symmetric potential. The native state is represented by a small sphere in the center, and the remaining space is identified with unfolded states. The folding temperature, the time-dependence of the populations, and the relaxation rate are calculated, and the folding dynamics is analyzed for both golf-course and funnel-like energy landscapes. This simple model allows us to illustrate a surprising number of concepts including entropic barriers, transition states, funnels, and the origin of single exponential relaxation kinetics.
... The golf course landscape provides a useful starting point for the development of a theory of the rate of crystal addition; it is both simple and, with its weak entropy-energy coupling, consistent with our treatments of the crystal-melt transition. Here we consider a solvable version analysed by Bicout and Szabo [42]. ...
... In this case, the inner surface of radius r is treated as an absorbing boundary. If a << 1, Bicout and Szabo [42] found the rate of ordering for the golf course (GC) landscape to be 1 2 2 ( 2) / ...
... The more restricted the extent of the transition state along direction orthogonal to the reaction coordinate relative to the analogous extent in the initial state, the lower the probability of a stochastic trajectory achieving this transition point. This aspect of kinetics has been referred to as an 'entropy' barrier [42,43]. While formally correct, this terminology can obscure the fact that S ∆ can be altered by correlations and/or constraints in the initial state configuration space, features well removed from the transition state itself [44]. ...
It has been established empirically that the rate of addition of molecules to the crystal during crystal growth from the melt is proportional to exp(-|{\Delta}S_fus|/R) where {\Delta}S_fus is the entropy of fusion. Here we show that this entropic slowdown arises directly from the separation of the entropy loss and energy loss processes associated with the freezing of the liquid. We present a theoretical treatment of the kinetics based on a model flat energy landscape and derive an explicit expression for the coupling magnitude in terms of the crystal-melt interfacial free energy. The implications of our work for nucleation kinetics are also discussed.
... Physicists collectively seem to have some similar difficulty in totally discounting Penrose's theory of entanglement by protein-protein interactions (actually, interactions between microtubules) as an explanation for consciousness. The problem for immediate rejection of quantum effects on grounds of large scale movements of a protein chain and solvent is that is actually the action (energy x time) not size or mass per se that matter [Bell, 1980] and the above number 10 11 was to be expected because energies navigated in protein folding are of the order of thermal solvent effects in kinetic theory. The factor (kBT/h) discussed in regard to the Eyring rate theory discussed below is a frequency factor equal to 6 ps −1 at 300 K, which is approximately 10 11 molecules (molecular transformations) per second. ...
... The factor (kBT/h) discussed in regard to the Eyring rate theory discussed below is a frequency factor equal to 6 ps −1 at 300 K, which is approximately 10 11 molecules (molecular transformations) per second. Also, despite the tradition of the Born-Oppenhemer separation being strong and no less strongly upheld in computational protein science, tunneling of atoms certainly occurs in real molecules [Bell, 1980]. Moreover, it is not confined to electrons and hydrogen atoms, but extends to more massive entities. ...
... For example, carbon atoms can tunnel [Carpenter, 1983]. The tunneling effect really relates more to crossing barriers in potential surfaces rather than consideration of individual atoms [Bell, 1980], and hence can apply to large structures. This includes potential surfaces describing conformational transitions, and notably it has been seen in diketopiperazines [Godfrey, 2000] that can be considered as a peptide analog (more precisely, as a ring formed from two peptide groups). ...
... During recent decades, various "folding-funnel" and "landscape" models, [5][6][7][8][9][10][11][12][13][14] as well as computer experiments, [15][16][17][18] have indicated that even a small energy bias against locally unfavor-able configurations can allow the protein chain to avoid sampling all conformations, thus greatly reducing Levinthal's time. However, is this bias the only factor that reduces, for real proteins, the Levinthal's billions of billions…of billions years to a biologically reasonable time? ...
... Various folding-funnel models [6,[8][9][10][11][12][13][14] have become popular to explain and illustrate protein folding. These models stress the energy bias against unfavorable configurations (i.e. ...
... However, the energy funnels, per se, do not provide a solution of the Levinthal's paradox. Strict analysis [27] of the straightforward funnel models [12][13][14] shows that they cannot simultaneously explain both major features observed in protein folding: 1) its non-astronomical time and 2) the co-existence of the native and unfolded protein molecules during the folding process. [26] This explanation (and solution of the paradox) is provided by specific ("capillarity" [10] ) nucleation funnels, which allow for separation of the unfolded and native phases (i.e. the "folding-nucleus" [26] formation) within the folding chain. ...
Complete volume of the protein conformation space is by many orders of magnitude smaller at the level of secondary structure elements than that at the level of amino acid residues: the latter, according to Levinthal's estimate, scales up as ~10^2L with the number L of residues in the chain, while the former, as it is shown in this paper, scales up not faster than ~L^N with the number N of the secondary structure elements, N being about L/15. This drastic decrease in the exponent (L/15 instead of 2L) explains why sampling of the conformation space does not contradict the ability of the protein chain to find its most stable fold.
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... Therefore, methods-based approximations by an overdamped Langevin equations, such as those in Coifman et al. (2005), Coifman et al. (2008), Singer et al. (2009) and Rohrdanz et al. (2011), would be biased, and likely inaccurate. The effective dynamics on M has two high probability regions, separated by regions of large volume where drift is small compared to diffusion ("entropic barriers"), which could make standard approximations of those inaccurate (Bicout and Szabo 2000). ...
... Therefore the total computational cost of obtaining the bursts is at least O( τ D 2 L N /c), where c is the number of parallel cores. Constructing ATLAS requires O(D 2 d N ) calculations to estimate local means, covariances, effective drift, effective diffusion coefficients, landmarks and tangent planes; O(C d DL log L) for constructing and organizing the landmarks using, for example, cover trees (Beygelzimer et al. 2006). A time-step of the ATLAS simulator as in Eq. (3.15), which has time length λτ , has cost O(C d Dd 2 ) by using iterative SVD combining Eq. (3.13,3.14), ...
We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics and high-dimensional, large fast modes. Given only access to a black-box simulator from which short bursts of simulation can be obtained, we design an algorithm that outputs an estimate of the invariant manifold, a process of the effective stochastic dynamics on it, which has averaged out the fast modes, and a simulator thereof. This simulator is efficient in that it exploits of the low dimension of the invariant manifold, and takes time-steps of size dependent on the regularity of the effective process, and therefore typically much larger than that of the original simulator, which had to resolve the fast modes. The algorithm and the estimation can be performed on the fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them.
... Therefore, methods based approximations by an overdamped Langevin equations, such as those in [CLL + 05, CKL + 08, SEKC09, RZMC11], would be biased, and likely inaccurate. The effective dynamics on M has two high probability regions, separated by regions of large volume where drift is small compared to diffusion ("entropic barriers"), which could make standard approximations of those inaccurate [BS00]. ...
... The slow manifold (in the limit → 0) is R(θ) = a 1 + a 2 cos 2 (θ), visualized in fig. 2. The observations z are in Cartesian coordinates, each of which contains a mix of nonlinearly coupled slow and fast components. Note that the drift diverges near the poles, creating a strong repulsion, and is relatively small in other wide regions of the state space, creating entropic barriers [BS00]. ...
We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only access to a black box simulator from which short bursts of simulation can be obtained, we estimate the invariant manifold, a process of the effective (stochastic) dynamics on it, and construct an efficient simulator thereof. These estimation steps can be performed on-the-fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them.
... 9 They also play a role in the protein aggregation into amyloids 23 and in more general protein folding funnel problems. 24 In this paper, we examine a trade-off in the entropic barrier faced by reaching the distant target, which is on the same surface a fixed distance a away, against the reduction in chain confinement. We calculate the mean first time it takes the chain to find the receptor, which is determined by an activation law where the effective potential barrier is purely entropic, −TΔS. ...
... This factor, significantly increasing the time for bridging to a distant target, only arises for the tethered chain. (All polymer work on the related narrow escape problems 19,22,24 has thus far focused on polymers with no attachment to the boundary of the domain, which fundamentally alters the accessibility of the binding site.) One might then naively assume that the binding time will decrease monotonically as the length of the chain increasesthe entropic penalty will become smaller and smaller. ...
The rate of binding of a grafted polymer to the surface is controlled by entropic barriers. Using a mean field approximation of ideal polymer dynamics, we fi rst calculate the characteristic binding time for a tethered ligand reaching for a binding site located on the tethering surface. This time is determined by two separate entropic effects: a barrier for the chain to be stretched sufficiently to reach the distant target, and a restriction on chain conformations near the surface, versus the increase in available phase space for longer chains. The competition between these two constraints determines the optimal (shortest) binding time. The theory is then extended to model bridging between two surfaces, in particular relevant for cell adhesion. Here the tethered ligand reaches for a receptor on a parallel surface, and the binding time depends on the gap between the two constraining surfaces. Again, an optimal binding time is determined for the given tether geometry. The results look similar to those for free particles in the `narrow escape problem', but modi fied by an entropic activation factor introduced by the tether.
... One-dimensional diffusion models based on the Fokker-Planck equation have been used for decades to study the rate of passage of chemical species across an energy barrier separating two adjacent wells [1][2][3][4][5][6][7][8][9][10]. The Fokker-Planck equation [11] is the limit of the Kramers equation [12] in the high friction limit [13,14]. ...
... The protein folding problem is a complex multidisciplinary problem that has been discussed by many workers [3,[23][24][25]. We refer the reader to recent reviews [26,27] that summarize the difficulties for modelling such systems. ...
We consider the one-dimensional bistable Fokker–Planck equation proposed by Polotto et al. (2018), with specific drift and diffusion coefficients so as to model protein folding. In this paper, a pseudospectral method is used to solve the Fokker–Planck equation in terms of the eigenvalues (λ n ) and eigenfunctions (ψ n ) of the Fokker–Planck operator. Nonclassical polynomials, constructed orthogonal with respect to the equilibrium distribution of the Fokker–Planck equation, are used as basis functions. The eigenvalues determined with the pseudospectral method are compared with the Wentzel–Kramers–Brillouin (WKB) and the SUperSYmmetric (SUSY) Wentzel–Kramers–Brillouin (SWKB) approximations. The eigenvalues calculated differ significantly from those reported by Polotto et al. A detailed study of the role of the lowest non-zero eigenvalue, λ 1 , to model the rate coefficient for the transition between the bistable states is provided.
... In particular, various "folding funnel" models [16][17][18][19] have become popular for illustrating and describing fast folding processes. These models, which have their own important advantages and drawbacks, will be considered below in detail as well. ...
... However, it can be shown that the energy funnels per se do not solve the Levinthal's paradox. Strict analysis [37] of the straightforwardly presented funnel models [19,38] shows that close to the midpoint of the folding-unfolding equilibrium they cannot simultaneously explain both the major features observed in protein folding: (1) its nonastronomical time, and (2) the "all-or-none" transition, i.e., coexistence of native and unfolded protein molecules during the folding process. ...
The ability of protein chains to spontaneously form their spatial structures is a long-standing puzzle in molecular biology. This review describes physical theories of rates of overcoming the free-energy barrier separating the natively folded (N) and unfolded (U) states of protein chains in both directions: “U-to-N” and “N-to-U.” In the theory of protein folding rates, a special role is played by the point of thermodynamic (and kinetic) equilibrium between the native and unfolded state of the chain; here, the theory obtains the simplest form. Paradoxically, a theoretical estimate of the folding time is easier to get from consideration of protein unfolding (the “N-to-U” transition) rather than folding, because it is easier to outline a good unfolding pathway of any structure than a good folding pathway that leads to the stable fold, which is yet unknown to the folding protein chain. And since the rates of direct and reverse reactions are equal at the equilibrium point (as follows from the physical ‘detailed balance’ principle), the folding time can be derived from the easier estimated unfolding time: theoretical analysis of the “N-to-U” transition outlines the range of protein folding rates in a good agreement with experiment, although experimentally measured folding times for single-domain globular proteins range from microseconds to hours: the difference (10–11 orders of magnitude) is the same as that between the life span of a mosquito and the age of the Universe. Supplemental theoretical analysis of folding (the “U-to-N” transition), performed at the level of formation and assembly of protein secondary structures, outlines the upper limit of protein folding times (i.e., of the time of search for the most stable fold). Both theories come to essentially the same results; this is not a surprise, because they describe overcoming one and the same free-energy barrier, although the way to the top of this barrier from the side of the unfolded state is very different from the way from the side of the native state; and both theories agree with experiment. In addition, they predict the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control and explain the observed maximal size of the “foldable” protein domains.
... 9 They also play a role in the protein aggregation into amyloids 23 and in more general protein folding funnel problems. 24 In this paper, we examine a trade-off in the entropic barrier faced by reaching the distant target, which is on the same surface a fixed distance a away, against the reduction in chain confinement. We calculate the mean first time it takes the chain to find the receptor, which is determined by an activation law where the effective potential barrier is purely entropic, −TΔS. ...
... This factor, significantly increasing the time for bridging to a distant target, only arises for the tethered chain. (All polymer work on the related narrow escape problems 19,22,24 has thus far focused on polymers with no attachment to the boundary of the domain, which fundamentally alters the accessibility of the binding site.) One might then naively assume that the binding time will decrease monotonically as the length of the chain increasesthe entropic penalty will become smaller and smaller. ...
The rate of binding of a grafted polymer to the surface is controlled by entropic barriers. Using a mean-field approximation of ideal polymer dynamics, we first calculate the characteristic binding time for a tethered ligand reaching for a binding site located on the tethering surface. This time is determined by two separate entropic effects: a barrier for the chain to be stretched sufficiently to reach the distant target and a restriction on chain conformations near the surface, versus the increase in available phase space for longer chains. The competition between these two constraints determines the optimal (shortest) binding time. The theory is then extended to model bridging between two surfaces, in particular relevant for cell adhesion. Here the tethered ligand reaches for a receptor on a parallel surface, and the binding time depends on the gap between the two constraining surfaces. Again, an optimal binding time is determined for the given tether geometry. The results look similar to those for free particles in the “narrow escape problem”, but modified by an entropic activation factor introduced by the tether.
... 18 We validate the rates extracted using the DTC method both by comparison to the corresponding MLPB rates and by analyzing the decay rates of the state autocorrelation functions of the system. [20][21][22] We assess the corresponding REMD efficiency, 23 and we obtain remarkably good agreement with the theoretically predicted errors in estimating the dimer and dissociated populations. ...
... 18 and 21) or from state correlation functions. 20,22 Here, the DTC method 11,12 was applied to dimer formation of NNQQ peptides. We obtained excellent agreement between the rates extracted using the DTC method and our previous, more elaborate maximum likelihood-based method. ...
We show how accurate rates of formation and dissociation of peptide dimers can be calculated using direct transition counting (DTC) from replica-exchange molecular dynamics (REMD) simulations. First, continuous trajectories corresponding to system replicas evolving at different temperatures are used to assign conformational states. Second, we analyze the entire REMD data to calculate the corresponding rates at each temperature directly from the number of transition counts. Finally, we compare the kinetics extracted directly, using the DTC method, with indirect estimations based on trajectory likelihood maximization using short-time propagators and on decay rates of state autocorrelation functions. For systems with relatively low-dimensional intrinsic conformational dynamics, the DTC method is simple to implement and leads to accurate temperature-dependent rates. We apply the DTC rate-extraction method to all-atom REMD simulations of dimerization of amyloid-forming NNQQ tetrapetides in explicit water. In an assessment of the REMD sampling efficiency with respect to standard MD, we find a gain of more than a factor of two at the lowest temperature.
... They show up in polymer translocation through a membrane pore 17,18 , as well as the mean looping time of a polymer chain 5 . They also play a role in the protein aggregation into amyloids 19 , and in more general protein folding funnel problems 20 . ...
... This represents thermal activation over an entropic barrier ∆G = 3 2 k B T a 2 /N b 2 , which is essentually the free energy to stretch the chain ends by a distance a. This factor, significantly increasing the time for bridging to a distant target, only arises for the tethered chain (all polymer work on the related narrow escape problems 15,18,20 has thus far focused on polymers with no attachment to the boundary of the domain, which fundamentally alters the accessibility of the binding site). ...
This paper considers a broadly biologically relevant question of a chain (such as a protein) binding to a sequence of receptors with matching multiple ligands distributed along the chain. This binding is critical in cell adhesion events, and in protein self-assembly. Using a mean field approximation of polymer dynamics, we first calculate the characteristic binding time for a tethered ligand reaching for a specific binding site on the surface. This time is determined by two separate entropic effects: an entropic barrier for the chain to be stretched sufficiently to reach the distant target, and a restriction on chain conformations near the surface. We then derive the characteristic time for a sequence of single binding events, and find that it is determined by the `zipper effect', optimizing the sequence of single and multiple binding steps.
... More recently, various "folding funnel" models [Leopold et al., 1992;Wolynes et al., 1995;Dill, Chan, 1997;Bicout, Szabo, 2000] have become popular for illustrating and describing fast folding processes. ...
... However, it can be shown that the energy funnels per se do not solve the Levinthal's paradox. Strict analysis [Bogatyreva, Finkelstein, 2001] of the straightforwardly presented funnel models [Zwanzig et al, 1992;Bicout, Szabo, 2000] shows that close to the mid-point of the folding-unfolding equilibrium they cannot simultaneously explain the both major features observed in protein folding: (i) its nonastronomical time, and (ii) the "all-or-none" transition, i.e., co-existence of native and unfolded protein molecules during the folding process. ...
The ability of protein chains to spontaneously form their spatial structures is a long-standing puzzle in molecular biology. Experimentally measured folding times of single-domain globular proteins range from microseconds to hours: the difference (10-11 orders of magnitude) is the same as that between the life span of a mosquito and the age of the universe. This review describes physical theories of rates of overcoming the free-energy barrier separating the natively folded (N) and unfolded (U) states of protein chains in both directions: ”U-to-N” and ”N-to-U”. In the theory of protein folding rates a special role is played by the point of thermodynamic (and kinetic) equilibrium between the native and unfolded state of the chain; here, the theory obtains the simplest form. Paradoxically, a theoretical estimate of the folding time is easier to get from consideration of protein unfolding (the ”N-to-U” transition) rather than folding, because it is easier to outline a good unfolding pathway of any structure than a good folding pathway that leads to the stable fold, which is yet unknown to the folding protein chain. And since the rates of direct and reverse reactions are equal at the equilibrium point (as follows from the physical ”detailed balance” principle), the estimated folding time can be derived from the estimated unfolding time. Theoretical analysis of the ”N-to-U” transition outlines the range of protein folding rates in a good agreement with experiment. Theoretical analysis of folding (the ”U-to-N” transition), performed at the level of formation and assembly of protein secondary structures, outlines the upper limit of protein folding times (i.e., of the time of search for the most stable fold). Both theories come to essentially the same results; this is not a surprise, because they describe overcoming one and the same free-energy barrier, although the way to the top of this barrier from the side of the unfolded state is very different from the way from the side of the native state; and both theories agree with experiment. In addition, they predict the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control and explain the observed maximal size of the ”foldable” protein domains.
... Like a golf hole in a flat surface, there are many ways of failing, and only a few of succeeding: this is the entropy barrier. The observed rapid folding is understood through the concept of a guiding funnel leading to the folded state [2][3][4] . Protein folding toy models of a Brownian particle searching outside a golf hole ('unfolded state') for a funnel inside it ('folded state'), also find entropy barriers, at the golf hole edges 3 . ...
... The observed rapid folding is understood through the concept of a guiding funnel leading to the folded state [2][3][4] . Protein folding toy models of a Brownian particle searching outside a golf hole ('unfolded state') for a funnel inside it ('folded state'), also find entropy barriers, at the golf hole edges 3 . ...
Entropy barriers and ageing states appear in martensitic
structural-transition models, slowly re-equilibrating after temperature
quenches, under Monte Carlo dynamics. Concepts from protein folding and ageing
harmonic oscillators turn out to be useful in understanding these
nonequilibrium evolutions. We show how the athermal, non-activated delay time
for seeded parent-phase austenite to convert to product-phase martensite,
arises from an identified entropy barrier in Fourier space. In an ageing state
of low Monte Carlo acceptances, the strain structure factor makes
constant-energy searches for rare pathways, to enter a Brillouin zone `golf
hole' enclosing negative energy states, and to suddenly release entropically
trapped stresses. In this context, a stress-dependent effective temperature can
be defined, that re-equilibrates to the quenched bath temperature.
... The appearance of entropic barriers, reminiscent of the Levinthal paradox in the context of protein folding [45,46], can be illustrated with a simple conceptual model [47]: Let x represents a "reaction coordinate" in the high dimensional configurational space of a system, and assume that in the absence of fluctuations the system follows overdamped dynamics in a smooth and attractive potential ( ) Vx, with its minimum at 0 x = . In the presence of fluctuations, the system's probability to occupy a state, characterized by x is proportional to ...
Recent experimental investigations into Hydra regeneration revealed a remarkable phenomenon: the morphological transformation of a tissue fragment from the incipient spherical configuration to a tube-like structure - the hallmark of a mature Hydra - has the dynamical characteristics of a first-order phase-transition, with calcium field fluctuations within the tissue playing an essential role. This morphological transition was shown to be generated by activation over an energy barrier within an effective potential that underlies morphogenesis. Inspired by this intriguing insight, we propose a novel mechanism where stochastic fluctuations drive the emergence of morphological patterns. Thus, the inherent fluctuations determine the nature of the dynamics and are not incidental noise in the background of the otherwise deterministic dynamics. Instead, they play an important role as a driving force that defines the attributes of the pattern formation dynamics and the nature of the transition itself. Here, we present a simple model that captures the essence of this novel mechanism for morphological pattern formation. Specifically, we consider a one-dimensional tissue arranged as a closed contour embedded in a two-dimensional space, where the local curvature of the contour is coupled to a non-negative scalar field. An effective temperature parameter regulates the strength of the fluctuations in the system. The tissue exhibits fluctuations near a circular shape at sufficiently low coupling strengths, but as the coupling strength exceeds some critical value, the circular state becomes unstable. The nature of the transition to the new state, namely whether it is a first-order-like or a second-order-like transition, depends on the temperature and the effective cutoff on the wavelength of the spatial variations in the system. It is also found that entropic barriers separate the various metastable states of the system.
1. Introduction
... Vendi sampling does not seem to provide a noticeable increase in performance in the early steps of the simulation. Perhaps this is due to the broad entropic basin of Chignolin's unfolded state Bicout and Szabo (2000). Such a basin would allow the Vendi Force to achieve structural diversity without necessarily passing the free energy barrier of interest. ...
Molecular dynamics (MD) is the method of choice for understanding the structure, function, and interactions of molecules. However, MD simulations are limited by the strong metastability of many molecules, which traps them in a single conformation basin for an extended amount of time. Enhanced sampling techniques, such as metadynamics and replica exchange, have been developed to overcome this limitation and accelerate the exploration of complex free energy landscapes. In this paper, we propose Vendi Sampling, a replica-based algorithm for increasing the efficiency and efficacy of the exploration of molecular conformation spaces. In Vendi sampling, replicas are simulated in parallel and coupled via a global statistical measure, the Vendi Score, to enhance diversity. Vendi sampling allows for the recovery of unbiased sampling statistics and dramatically improves sampling efficiency. We demonstrate the effectiveness of Vendi sampling in improving molecular dynamics simulations by showing significant improvements in coverage and mixing between metastable states and convergence of free energy estimates for four common benchmarks, including Alanine Dipeptide and Chignolin.
... To illustrate this concept, Fig. 6a shows a very simple model for the free energy of ligand binding as a function of the distance R of the ligand from the protein. 48 The protein-ligand complex is stabilized by a binding energy V b . Beyond the interaction range of the protein (R 0 ), the free energy decreases due to an entropic contribution, which accounts for the larger space available to the ligand with increasing distance. ...
The dynamics of peptide-protein binding and unbinding of a variant of the RNase S system has been investigated. To initiate the process, a photoswitchable azobenzene moiety has been covalently linked to the S-peptide, thereby switching its binding affinity to the S-protein. Transient fluorescence quenching was measured with the help of a time-resolved fluorometer, which has been specifically designed for these experiments and is based on inexpensive LED's and laser diodes only. One mutant shows on-off behaviour with no specific binding detectable in one of the states of the photoswitch. Unbinding is barrier-less in that case, revealing the intrinsic dynamics of the unbinding event, which occurs on a few 100 microseconds timescale in a strongly stretched-exponential manner.
... This limitation could not be completely overcome by T-RE variants that focus on improving replica exchange or diffusion in the temperature space 35 . The fundamental reason is that the free energy barriers of cooperative conformational transitions are dominated by entropic components 36,37 , which renders tempering (i.e., raising the temperature) ineffective for driving faster transitions. ...
Efficient sampling of the conformational space is essential for quantitative simulations of proteins. The multiscale enhanced sampling (MSES) method accelerates atomistic sampling by coupling it to a coarse‐grained (CG) simulation. Bias from coupling to the CG model is removed using Hamiltonian replica exchange, such that one could benefit simultaneously from the high accuracy of atomistic models and fast dynamics of CG ones. Here, we extend MSES to allow independent control of the effective temperatures of atomistic and CG simulations, by directly scaling the atomistic and CG Hamiltonians. The new algorithm, named MSES with independent tempering (MSES‐IT), supports more sophisticated Hamiltonian and temperature replica exchange protocols to further improve the sampling efficiency. Using a small but nontrivial β‐hairpin, we show that setting the effective temperature of CG model in all conditions to its melting temperature maximizes structural transition rates at the CG level and promotes more efficient replica exchange and diffusion in the condition space. As the result, MSES‐IT drive faster reversible transitions at the atomic level and leads to significant improvement in generating converged conformational ensembles compared to the original MSES scheme.
... 18 the corresponding MLPB rates and by analyzing the decay rates of the state autocorrelation functions of the system. 3,5,37 Interestingly, the MSM approach in conjunction with the DTC method allowed the assess the corresponding REMD efficiency, 38 showing that one can obtain remarkably good agreement with the theoretically predicted errors in estimating the dimer and dissociated populations on the binding dynamics of NNQQ amyloid tetrapetides. 17,18 We highlight here that the calculation of transition probabilities between Markovian conformational states, springing from the use of MSMs in conjunction with enhanced sampling methods such as REMD, has allowed the extension of REMD results beyond accurate calculation of thermodynamic properties alone (e.g., equilibrium populations and corresponding free energies) to accurate kinetic studies and the estimation of transition rates from short segments of REMD trajectories for MD simulations with explicit water molecules ranging from folding of short peptides 26 to unveiling association-dissociation pathways and binding constants. ...
Molecular dynamics (MD) studies of biomolecules require the ability to simulate complex biochemical systems with an increasingly larger number of particles and for longer time scales, a problem that cannot be overcome by computational hardware advances alone. A main problem springs from the intrinsically high-dimensional and complex nature of the underlying free energy landscape of most systems, and from the necessity to sample accurately such landscapes for identifying kinetic and thermodynamic states in the configurations space, and for accurate calculations of both free energy differences and of the corresponding transition rates between states. Here, we review and present applications of two increasingly popular methods that allow long-time MD simulations of biomolecular systems that can open a broad spectrum of new studies. A first approach, Markov State Models (MSMs), relies on identifying a set of configuration states in which the system resides sufficiently long to relax and loose the memory of previous transitions, and on using simulations for mapping the underlying complex energy landscape and for extracting accurate thermodynamic and kinetic information. The Markovian independence of the underlying transition probabilities creates the opportunity to increase the sampling efficiency by using sets of appropriately initialized short simulations rather than typically long MD trajectories, which also enhances sampling. This allows MSM-based studies to unveil bio-molecular mechanisms and to estimate free energy barriers with high accuracy, in a manner that is both systematic and relatively automatic, which accounts for their increasing popularity. The second approach presented, Milestoning, targets accurate studies of the ensemble of pathways connecting specific end-states (e.g., reactants and products) in a similarly systematic, accurate and highly automatic manner. Applications presented range from studies of conformational dynamics and binding of amyloid-forming peptides, cell-penetrating peptides and the DFG-flip dynamics in Abl kinase. As highlighted by the increasing number of studies using both methods, we anticipate that they will open new avenues for the investigation of systematic sampling of reactions pathways and mechanisms occurring on longer time scales than currently accessible by purely computational hardware developments.
... Given the relation F(Q) = f(Q) − TS config (Q) and the funneled character of f(Q), the barrier has a purely entropic origin. 46,47 This is one of the major differences between protein folding and simple chemical reactions of small molecules: 5 in the latter, the barrier appears already in f(Q) as it stems from the high potential energy of a fairly welldefined transition-state structure. ...
The most fundamental aspect of the free energy landscape of proteins is that it is globally funneled such that protein folding is energetically biased. Then, what are the distinctive characteristics of the landscape of intrinsically disordered proteins, apparently lacking such energetic bias, that nevertheless fold upon binding? Here, we address this fundamental issue through the explicit characterization of the free energy landscape of the paradigmatic pKID–KIX system (pKID, phosphorylated kinase-inducible domain; KIX, kinase interacting domain). This is done based on unguided, fully atomistic, explicit-water molecular dynamics simulations with an aggregated simulation time of >30 μs and on the computation of the free energy that defines the landscape. We find that, while the landscape of pKID before binding is considerably shallower than the one for a protein that autonomously folds, it becomes progressively more funneled as the binding of pKID with KIX proceeds. This explains why pKID is disordered in a free state, and the binding of pKID with KIX is a prerequisite for pKID’s folding. In addition, we observe that the key event in completing the pKID–KIX coupled folding and binding is the directed self-assembly where pKID is docked upon the KIX surface to maximize the surface electrostatic complementarity, which, in turn, require pKID to adopt the correct folded structure. This key process shows up as the free energy barrier in the pKID landscape separating the intermediate nonspecific complex state and the specific complex state. The present work not only provides a detailed molecular picture of the coupled folding and binding of pKID but also expands the funneled landscape perspective to intrinsically disordered proteins.
... A prototypical example is regions on the energy landscape with "golf-course" potentials. 5,7,48 . These potentials feature a large region and several small targets within the region. ...
We present Capacity Hopping (CHop), a method for estimating hitting probabilities of small targets in molecular dynamics simulations. Reaching small targets out of a vast number of possible configurations constitutes an entropic barrier. Experimental evidence suggests that entropic barriers are ubiquitous in biomolecular systems, and often characterize the rate-limiting step of biomolecular processes. Hitting probabilities provide valuable information in the presence of entropic barriers. CHop is based on a novel theoretical analysis of hitting probabilities. We show that in the presence of entropic barriers, hitting probabilities are approximately invariant to initial conditions that are modestly far away from the targets. Given this invariance, we show that hitting probabilities must also be invariant to the energy landscape away from the targets, and can be well-approximated in terms of "capacities" of local sets around the targets. Using these theoretical results and a new capacity estimation algorithm, we develop CHop, a method for estimating the approximately constant hitting probabilities of small targets in the presence of entropic barriers. Numerical experiments on a prototypical entropic barrier, a toy model with a golf-course potential, show that CHop is nearly as accurate as direct simulations in estimating hitting probabilities, but 662 times faster.
... The symbols represent histograms for ℘ a (t|x 0 ) deduced from 10 6 Brownian dynamics trajectories, and the lines correspond to the large deviation asymptotic ℘ a (t|x 0 ) w 1 (x 0 )μ 1 e −μ1t deduced from (A6). We conclude that the limit µ 1 λ 1 lead to both, a quite accurate approximationμ 1 µ 1 and to an effectively single exponential decay µ 1 µ 2 of the first passage statistics, which extends previous results [87] (see also [88]). ...
We uncover a duality between relaxation and first passage processes in ergodic reversible Markovian dynamics in both discrete and continuous state-space. The duality exists in the form of a spectral interlacing -- the respective time scales of relaxation and first passage are shown to interlace. Our canonical theory allows for the first time to determine the full first passage time distribution analytically from the simpler relaxation eigenspectrum. The duality is derived and proven rigorously for both discrete state Markov processes in arbitrary dimension and effectively one-dimensional diffusion processes, whereas we also discuss extensions to more complex scenarios. We apply our theory to a simple discrete-state protein folding model and to the Ornstein-Uhlenbeck process, for which we obtain the exact first passage time distribution analytically in terms of a Newton series of determinants of almost triangular matrices. In addition to these stand-alone results the work contains all proofs and extended technical explanations of the results reported in the accompanying letter (Hartich and Godec, submitted).
... Rao and Gosavi (4) now address this question using a structurebased, coarse-grained model in which the only residue pairs with attractive interactions are those formed in a given target structure. Such models are motivated by the energy landscape theory of protein folding, which posits that the folding free-energy landscape is funneled toward a single native state (8)(9)(10)(11). While such a model may at first seem inconsistent with the folding scenario of serpins, it allows direct folding to the alternative latent structure to be studied, without the competing inhibitory fold. By computing separate folding free-energy landscapes for the inhibitory and latent structures, the authors found a free-energy barrier for folding to the latent form that was significantly higher (by ∼4 k B T). ...
... Nevertheless, it is possible to demonstrate that ener gy funnels as such do not resolve Levinthal's paradox. Analysis [52] of strictly formulated funnel models [39,53] shows that at the equilibrium between folded and unfold ed chain forms, these models are unable to explain simul taneously both main features observed during protein folding: (i) non astronomical folding time, and (ii) "all or none" transition, i.e. coexistence of native and unfolded forms of protein molecules during folding. ...
The ability of proteins to spontaneously form their spatial structures is a long-standing puzzle in molecular biology. Experimentally measured rates of spontaneous folding of single-domain globular proteins range from microseconds to hours: the difference–10-11 orders of magnitude–is the same as between the lifespan of a mosquito and the age of the Universe. This review (based on the literature and some personal recollections) describes a winding road to understanding spontaneous folding of protein structure. The main attention is given to the free-energy landscape of conformations of a protein chain–especially to the barrier separating its unfolded (U) and the natively folded (N) states–and to physical the-ories of rates of crossing this barrier in both directions: from U to N, and from N to U. It is shown that theories of both these processes come to essentially the same result and outline the observed range of folding and unfolding rates for single-domain globular proteins. In addition, they predict the maximal size of protein domains that fold under solely thermodynamic (rather than kinetic) control, and explain the observed maximal size of “foldable” protein domains.
... The "energy funnel", centered in the lowest-energy structure, seems to allow protein chains to avoid the "Levinthal's" sampling all conformations. However, it can be shown that the energy funnels per se do not solve the Levinthal's paradox-this, as Eugene Shakhnovich used to say, "Fermat's Last Theorem of protein science"… Strict analysis (Bogatyreva and Finkelstein, 2001) of the straightforwardly presented funnel models (Zwanzig et al., 1992;Bicout and Szabo, 2000) shows that they cannot simultaneously explain both major features observed in protein folding: (1) its non-astronomical time, and (2) co-existence of the native and unfolded protein molecules during the folding process. ...
Two-state folding of small proteins: kinetic analogue of the thermodynamic “all-or-none” transition. Two- and multi-state folding. Theory of transition states. Experimental identification and investigation of unstable transition states in protein folding. Φ-Value analysis. Folding nucleus. Its experimental discovery by protein engineering methods. Nucleation mechanism of protein folding. Native and non-native interactions in the nucleus. Folding nucleus is less specific and less “invariant” than the native protein structure.
... These experimental observations probably originate from a conformational free-energy profile in which the a-helical and random coil ensembles can be characterized as two broad free-energy minima that are separated by a single main free-energy barrier (57)(58)(59). When the free-energy barrier separating these two conformational macrostates is much higher than the barriers between the microstates within each of them, preequilibration within the microstate populations occurs on timescales much faster than that of the population exchange involving the crossing of the main free-energy barrier (64)(65)(66)(67). If the population distribution of all microstates preequilibrates within the time resolution of the T-jump pulse (~5 ns), the relative populations in each distribution will remain constant from this point in time, and single-exponential T-jump relaxation kinetics reflecting the crossing of the main free-energy barrier and the conformational redistribution process between the a-helical and random-coil ensembles is observed (66). ...
Using a combination of ultraviolet circular dichroism, temperature-jump transient-infrared spectroscopy, and molecular dynamics simulations, we investigate the effect of salt bridges between different types of charged amino-acid residue pairs on α-helix folding. We determine the stability and the folding and unfolding rates of 12 alanine-based α-helical peptides, each of which has a nearly identical composition containing three pairs of positively and negatively charged residues (either Glu−/Arg+, Asp−/Arg+, or Glu−/Lys+). Within each set of peptides, the distance and order of the oppositely charged residues in the peptide sequence differ, such that they have different capabilities of forming salt bridges. Our results indicate that stabilizing salt bridges (in which the interacting residues are spaced and ordered such that they favor helix formation) speed up α-helix formation by up to 50% and slow down the unfolding of the α-helix, whereas salt bridges with an unfavorable geometry have the opposite effect. Comparing the peptides with different types of charge pairs, we observe that salt bridges between side chains of Glu− and Arg+ are most favorable for the speed of folding, probably because of the larger conformational space of the salt-bridging Glu−/Arg+ rotamer pairs compared to Asp−/Arg+ and Glu−/Lys+. We speculate that the observed impact of salt bridges on the folding kinetics might explain why some proteins contain salt bridges that do not stabilize the final, folded conformation.
... As one of the major applications, Levinthal's paradox of protein folding 3 has been tackled via folding funnels. [4][5][6][7][8] Similarly, energy landscapes proved fruitful in glass and cluster physics. [9][10][11][12] The case of a diffusivity landscape D(r) has been exploited less intensively, although it has been proven that energy landscapes are connected to diffusivity landscapes, e.g., for the case of protein folding. ...
Diffusion of a particle through an energy and diffusivity landscape is a very general phenomenon in numerous systems of soft and condensed matter. On the one hand, theoretical frameworks such as Langevin and Fokker-Planck equations present valuable accounts to understand these motions in great detail, and numerous studies have exploited these approaches. On the other hand, analytical solutions for correlation functions, as, e.g., desired by experimentalists for data fitting, are only available for special cases. We explore the possibility to use different theoretical methods in the specific picture of time-dependent switching between diffusive states to derive analytical functions that allow to link experimental and simulation results to theoretical calculations. In particular, we present a closed formula for diffusion switching between two states, as well as a general recipe of how to generalize the formula to multiple states.
... The curve denoted by MM and MCB are the results by Marechal and Moreau (1984) and the simulations by , respectively. There are many reactive systems and diffusion processes that are modelled with the Smoluchowski equation (Szabo et al. 1980;Bagchi et al. 1983;Chavanis 2006;Felderhof 2008) including protein folding (Bicout and Szabo 2000), dielectric relaxation (Coffey et al. 2009) and a Smoluchowski equation with a capture term (Spendier et al. 2013) that overlaps in some respects the studies of the nonequilibrium reactive system in Sect. 5.4.4. ...
Spectral and pseudospectral methods in chemistry and physics are based on classical and nonclassical orthogonal polynomials defined in terms of a three term recurrence relation. The coefficients in the three term recurrence relations for the nonclassical polynomials can be calculated with the Gautschi-Stieltjes procedure. The round-off errors that occur with the use of Gram-Schmidt orthogonalization procedure is demonstrated for both classical and nonclassical polynomials. The trapezoidal, Simpson’s and Newton-Cotes integration rules are derived as are the Fejér, Clenshaw-Curtiss, Gauss-Lobatto and Gauss-Radau algorithms. Sinc interpolation based on Fourier sine basis functions is compared with the Lagrange interpolation. Nonclassical Maxwell and Bimodal polynomials orthogonal on the infinite domain with respect to weight functions \(w(x) = x^2\exp (-x^2)\) and \(x^2\exp [-(x^4/4\epsilon -x^2/2\epsilon )]\), respectively, are introduced for kinetic theory problems. The Gaussian quadrature rule based on the nonclassical Rys polynomials orthogonal with respect to the weight function \(w(x) = e^{-cx^2},\; x \in [-1,\ 1]\), used to evaluate integrals in molecular quantum mechanics is presented. For \(c \rightarrow 0\) and \(c \rightarrow \infty \), the Rys polynomials are the Legendre and scaled Hermite polynomials, respectively. Two dimensional quadratures, such as the Lebedev cubature, are used to evaluate two dimensional integrals in density functional theory for electronic structure calculations as well as for the nonlinear Boltzmann equation in kinetic theory. The Stieltjes moment problem is related to the inversion of moment data in chemical physics to reconstruct photoelectron cross sections.
... The curve denoted by MM and MCB are the results by Marechal and Moreau (1984) and the simulations by , respectively. There are many reactive systems and diffusion processes that are modelled with the Smoluchowski equation (Szabo et al. 1980;Bagchi et al. 1983;Chavanis 2006;Felderhof 2008) including protein folding (Bicout and Szabo 2000), dielectric relaxation (Coffey et al. 2009) and a Smoluchowski equation with a capture term (Spendier et al. 2013) that overlaps in some respects the studies of the nonequilibrium reactive system in Sect. 5.4.4. ...
This chapter introduces the basic principles of spectral/pseudospectral methods for the solution of partial differential and/or integral equations that serve to model a large number of physical processes in chemistry and physics. The first part of the chapter defines the spectral space representation of functions and the transformation to the physical space representation. A Hilbert space is defined as well as the definition of self-adjoint operators that occur in quantum mechanics and kinetic theory. The Rayleigh-Ritz variational principle and the method of weighted residuals are discussed. An historical summary of the development of pseudospectral methods in chemistry and physics is presented together with an outline of the book. The science, the mathematical models and the computer algorithms are interrelated.
... The curve denoted by MM and MCB are the results by Marechal and Moreau (1984) and the simulations by , respectively. There are many reactive systems and diffusion processes that are modelled with the Smoluchowski equation (Szabo et al. 1980;Bagchi et al. 1983;Chavanis 2006;Felderhof 2008) including protein folding (Bicout and Szabo 2000), dielectric relaxation (Coffey et al. 2009) and a Smoluchowski equation with a capture term (Spendier et al. 2013) that overlaps in some respects the studies of the nonequilibrium reactive system in Sect. 5.4.4. ...
The orthogonal basis sets most often used in spectral methods are the Chebyshev and Legendre polynomials on a bounded domain, or a Fourier basis set for periodic functions. We discuss in this chapter the expansions of Gaussian and Kappa distributions of kinetic theory in Hermite and Laguerre polynomials on the infinite and semi-infinite intervals, respectively. The spectral convergence properties of these expansions is demonstrated numerically and analytically. The expansions of \(\sin (x)\) in Hermite polynomials, and of the Maxwellian distribution in Chebyshev polynomials are also considered. The basic principles of Fourier series are presented and applied to quantum mechanical wave packets as well as the analysis of free induction decay signals. The resolution of the Gibbs phenomenon with the Gegenbauer reconstruction method is compared with the inverse polynomial reconstruction method. A resolution of the Runge phenonmena is also presented.
... Despite the vast number of degrees of freedom involved, the experimental kinetics of protein folding is often indicative of a very simple process, which can be described by a kinetic scheme containing only two, or a small number of macroscopic states [28]. Such simple kinetics arises from separation in timescales between the re-equilibration of many possible unfolded or partially folded configurations and the main folding transition [29][30][31]. Folding is much slower because the folded state can be directly reached, by an elementary conformational step, from only a small fraction of the chain configurations. These "gateway" states constitute a bottleneck on the folding pathways, which can be represented as a free-energy barrier to folding, provided that a suitable reaction coordinate or coordinates to measure the degree of folding can be defined. ...
... 17−23 The effects of the solvent and the many remaining molecular coordinates not included in this low-dimensional picture is usually approximated in terms of a diffusion process. 20,24 A hierarchical free energy landscape, in particular, represents the dynamics on different time scales by various tiers of the energy landscape, associating, for example, tiers C, B, and A with specific processes on a ps, ns, and μs time scale, respectively. The notion of a hierarchy implies that these processes interact such that the fast processes regulate the slow transitions. ...
Biomolecules exhibit structural dynamics on a number of time scales, including picosecond (ps) motions of a few atoms, nanosecond (ns) local conformational transitions, and microsecond (μs) global conformational rearrangements. Despite this substantial separation of time scales, fast and slow degrees of freedom appear to be coupled in a nonlinear manner; for example, there is theoretical and experimental evidence that fast structural fluctuations are required for slow functional motion to happen. To elucidate a microscopic mechanism of this multiscale behavior, Aib peptide is adopted as a simple model system. Combining extensive molecular dynamics simulations with principal component analysis techniques, a hierarchy of (at least) three tiers of the molecule’s free energy landscape is discovered. They correspond to chiral left- to right-handed transitions of the entire peptide that happen on a μs time scale, conformational transitions of individual residues that take about 1 ns, and the opening and closing of structure-stabilizing hydrogen bonds that occur within tens of ps and are triggered by sub-ps structural fluctuations. Providing a simple mechanism of hierarchical dynamics, fast hydrogen bond dynamics is found to be a prerequisite for the ns local conformational transitions, which in turn are a prerequisite for the slow global conformational rearrangement of the peptide. As a consequence of the hierarchical coupling, the various processes exhibit a similar temperature behavior which may be interpreted as a dynamic transition.
It is estimated that two-thirds of all proteins in higher organisms are composed of multiple domains, many of them containing discontinuous folds. However, to date, most in vitro protein folding studies have focused on small, single-domain proteins. As a model system for a two-domain discontinuous protein, we study the unfolding/refolding of a slow-folding double mutant of the maltose binding protein (DM-MBP) using single-molecule two- and three-color Förster Resonance Energy Transfer experiments. We observe a dynamic folding intermediate population in the N-terminal domain (NTD), C-terminal domain (CTD), and at the domain interface. The dynamic intermediate fluctuates rapidly between unfolded states and compact states, which have a similar FRET efficiency to the folded conformation. Our data reveals that the delayed folding of the NTD in DM-MBP is imposed by an entropic barrier with subsequent folding of the highly dynamic CTD. Notably, accelerated DM-MBP folding is routed through the same dynamic intermediate within the cavity of the GroEL/ES chaperone system, suggesting that the chaperonin limits the conformational space to overcome the entropic folding barrier. Our study highlights the subtle tuning and co-dependency in the folding of a discontinuous multi-domain protein.
It has been established empirically that the rate of addition of molecules to the crystal during crystal growth from the melt is proportional to exp(-|ΔSfus|/R), where ΔSfus is the entropy of fusion. Here we show that this entropic slowdown arises directly from the separation of the entropy loss and energy loss processes associated with the freezing of the liquid. We present a theoretical treatment of the kinetics based on a model flat energy landscape and derive an explicit expression for the coupling magnitude in terms of the crystal-melt interfacial free energy. The implications of our work for nucleation kinetics are also discussed.
We test a post-quench Partial Equilibration Scenario (PES) of Ritort and colleagues, through Monte Carlo quench simulations, of a vector-spin model for a cubic-tetragonal, martensitic structural transition. We confirm the PES signature distribution of an exponential tail in heat releases, scaled in an effective search temperature that controls energy-lowering passages between fixed-energy shells. Our simulations find that this effective temperature vanishes linearly in the deviation of the quench temperatures from a characteristic temperature, where passage bottlenecks in phase space change their topology, and entropy barriers diverge. Equilibration delay times, exponential in the inverse effective temperature, are thus predicted to show singular Vogel-Fulcher behaviour, that is now understood as an arrest of PES heat releases to the bath, and confirmed in experimental data on martensitic alloys.
The primary goal of protein science is to understand how proteins function, which requires understanding the functional dynamics responsible for transitions between different functional structures of a protein. A central concept is the exact reaction coordinates that can determine the value of committor for any protein configuration, which provide the optimal description of functional dynamics. Despite intensive efforts, identifying the exact reaction coordinates (RCs) in complex molecules remains a formidable challenge. Using the recently developed generalized work functional, we report the discovery of the exact RCs for an important functional process-the flap opening of HIV-1 protease. Our results show that this process has six RCs, each one is a linear combination of ~240 backbone dihedrals, providing the precise definition of collectivity and cooperativity in the functional dynamics of a protein. Applying bias potentials along each RC can accelerate flap opening by [Formula: see text] to [Formula: see text] folds. The success in identifying the RCs of a protein with 198 residues represents a significant progress beyond that of the alanine dipeptide, currently the only other complex molecule for which the exact RCs for its conformational changes are known. Our results suggest that the generalized work functional (GWF) might be the fundamental operator of mechanics that controls protein dynamics.
The ability of protein chains to spontaneously form their three-dimensional structures is a long-standing mystery in molecular biology. The most conceptual aspect of this mystery is how the protein chain can find its native, “working” spatial structure (which, for not too big protein chains, corresponds to the global free energy minimum) in a biologically reasonable time, without exhaustive enumeration of all possible conformations, which would take billions of years. This is the so-called “Levinthal’s paradox.” In this review, we discuss the key ideas and discoveries leading to the current understanding of protein folding kinetics, including folding landscapes and funnels, free energy barriers at the folding/unfolding pathways, and the solution of Levinthal’s paradox. A special role here is played by the “all-or-none” phase transition occurring at protein folding and unfolding and by the point of thermodynamic (and kinetic) equilibrium between the “native” and the “unfolded” phases of the protein chain (where the theory obtains the simplest form). The modern theory provides an understanding of key features of protein folding and, in good agreement with experiments, it (i) outlines the chain length-dependent range of protein folding times, (ii) predicts the observed maximal size of “foldable” proteins and domains. Besides, it predicts the maximal size of proteins and domains that fold under solely thermodynamic (rather than kinetic) control. Complementarily, a theoretical analysis of the number of possible protein folding patterns, performed at the level of formation and assembly of secondary structures, correctly outlines the upper limit of protein folding times.
There has been recent success in prediction of the three-dimensional folded native structures of proteins, most famously by the AlphaFold Algorithm running on Google's/Alphabet's DeepMind computer. However, this largely involves machine learning of protein structures and is not a de novo protein structure prediction method for predicting three-dimensional structures from amino acid residue sequences. A de novo approach would be based almost entirely on general principles of energy and entropy that govern protein folding energetics, and importantly do so without the use of the amino acid sequences and structural features of other proteins. Most consider that problem as still unsolved even though it has occupied leading scientists for decades. Many consider that it remains one of the major outstanding issues in modern science. There is crucial continuing help from experimental findings on protein unfolding and refolding in the laboratory, but only to a limited extent because many researchers consider that the speed by which real proteins folds themselves, often from milliseconds to minutes, is itself still not fully understood. This is unfortunate, because a practical solution to the problem would probably have a major effect on personalized medicine, the pharmaceutical industry, biotechnology, and nanotechnology, including for example “smaller” tasks such as better modeling of flexible “unfolded” regions of the SARS-COV-2 spike glycoprotein when interacting with its cell receptor, antibodies, and therapeutic agents. Some important ideas from earlier studies are given before moving on to lessons from periodic and aperiodic crystals, and a possible role for quantum phenomena. The conclusion is that better computation of entropy should be the priority, though that is presented guardedly.
Markov processes are widely used models for investigating kinetic networks. Here, we collate and present a variety of results pertaining to kinetic network models in a unified framework. The aim is to lay out explicit links between several important quantities commonly studied in the field, including mean first passage times (MFPTs), correlation functions, and the Kemeny constant. We provide new insights into (i) a simple physical interpretation of the Kemeny constant, (ii) a relationship to infer equilibrium distributions and rate matrices from measurements of MFPTs, and (iii) a protocol to reduce the dimensionality of kinetic networks based on specific requirements that the MFPTs in the coarse-grained system should satisfy. We prove that this protocol coincides with the one proposed by Hummer and Szabo [J. Phys. Chem. B 119, 9029 (2014)], and it leads to a variational principle for the Kemeny constant. Finally, we introduce a modification of this protocol, which preserves the Kemeny constant. Our work underpinning the theoretical aspects of kinetic networks will be useful in applications including milestoning and path sampling algorithms in molecular simulations.
Small peptides in solution adopt a specific morphology as they function. It is of fundamental interest to examine the structural properties of these small biomolecules in solution and observe how they transition from one conformation to another and form functional structures. In this study, we have examined the structural properties of a simple dipeptide and a five-residue peptide with the application of far-UV circular dichroism (CD) spectroscopy as a function of temperature, fluorescence anisotropy, and all-atom molecular dynamics simulation. Analysis of the temperature dependent CD spectra shows that the simplest dipeptide N-acetyl-tryptophan-amide (NATA) adopts helical, beta sheet, and random coil conformations. At room temperature, NATA is found to have 5% alpha-helical, 37% beta sheet, and 58% random coil conformations. To our knowledge, this type of structural content in a simplest dipeptide has not been observed earlier. The pentapeptide (WK5) is found to have four major secondary structural elements with 8% 310 helix, 14% poly-L-proline II, 8% beta sheet, and 14% turns. A 56% unordered structural population is also present for WK5. The presence of a significant population of 310 helix in a simple pentapeptide is rarely observed. Fluorescence anisotropy decay (FAD) measurements yielded reorientation times of 45 ps for NATA and 120 ps for WK5. The fluorescence anisotropy decay measurements reveal the size differences between the two peptides, NATA and WK5, with possible contributions from differences in shape, interactions with the environment, and conformational dynamics. All-atom molecular dynamics simulations were used to model the structures and motions of these two systems in solution. The predicted structures sampled by both peptides qualitatively agree with the experimental findings. Kinetic modeling with optimal dimensionality reduction suggests that the slowest dynamic processes in the dipeptide involve sidechain transitions occurring on a 1 ns timescale. The kinetics in the pentapeptide monitors the formation of a distorted helical structure from an extended conformation on a timescale of 10 ns. Modeling of the fluorescence anisotropy decay is found to be in good agreement with the measured data and correlates with the main contributions of the measured reorientation times to individual conformers, which we define as dynamic elements. In NATA, the FAD can be well represented as a sum of contributions from representative conformers. This is not the case in WK5, where our analysis suggests the existence of coupling between conformational dynamics and global tumbling. The current study involving detailed experimental measurements and atomically detailed modeling reveals the existence of specific secondary structural elements and novel dynamical features even in the simplest peptide systems.
Alkyl imidazolium chloride ionic liquids (ILs) have been used for numerous biochemical applications. Their hydrophobicity can be tuned by changing the alkyl chain length and the longer-chain ILs can form micelles in aqueous solution. We have investigated the effects of imidazolium chloride ILs on the structure and stability of azurin, which is a very stable Cu2+ redox protein with both alpha-helix and beta-sheet domains. Temperature-dependent infrared (IR) and vibrational circular dichroism (VCD) spectroscopy can provide secondary-structure specific information about how the protein is affected and temperature-jump transient IR measurements can quantify the IL-influenced unfolding dynamics. Using these techniques we can quantify how azurin is destabilized by 1.0 M ILs in aqueous solution. The shorter, less hydrophobic ILs 1-butyl-3-methylimidazolium chloride ([BMIM]Cl) and 1-hexyl-3-methylimidazolium chloride ([HMIM]Cl) likely interact with the alpha-helix domain and decrease protein melting temperature from 82 °C without IL to 55 °C and disturb the overall tertiary structure resulting in a looser, more open shape. Thermodynamic analysis indicates that protein destabilization is due to increased unfolding entropy. 1-octyl-3-methylimidazolium chloride [OMIM]Cl, which forms micelles in solution that may partially solvate the protein, has a more significant destabilizing effect resulting in a melting temperature of 35 °C, larger unfolding entropy, and relaxation kinetics several orders of magnitude faster than with unperturbed azurin. The temperature-independence of the relaxation time constant suggests that in the presence of [OMIM]Cl the protein folding potential energy surface has become very smooth.
Markov state models (MSMs) provide some of the simplest mathematical and physical descriptions of dynamical and thermodynamical properties of complex systems. However, typically, the large dimensionality of biological systems studied makes them prohibitively expensive to work in fully Markovian regimes. In this case, coarse graining can be introduced to capture the key dynamical processes—slow degrees of the system—and reduce the dimension of the problem. Here, we introduce several possible options for such Markovian coarse graining, including previously commonly used choices: the local equilibrium and the Hummer Szabo approaches. We prove that the coarse grained lower dimensional MSM satisfies a variational principle with respect to its slowest relaxation time scale. This provides an excellent framework for optimal coarse graining, as previously demonstrated. Here, we show that such optimal coarse graining to two or three states has a simple physical interpretation in terms of mean first passage times and fluxes between the coarse grained states. The results are verified numerically using both analytic test potentials and data from explicit solvent molecular dynamics simulations of pentalanine. This approach of optimizing and interpreting clustering protocols has broad applicability and can be used in time series analysis of large data.
Folding of proteins to their functional conformation is paramount to life. Though 75% of the proteome consists of multi-domain proteins, our knowledge of folding has been based primarily on studies conducted on single-domain and fast-folding proteins. Nonetheless, the complexity of folding landscapes exhibited by multi-domain proteins has received increased scrutiny in recent years. We study the three-domain protein adenylate kinase from E. coli (AK), which has been shown to fold through a series of pathways involving several intermediate states. We use protein design method to manipulate the folding landscape of AK, and single-molecule FRET spectroscopy to study the effects on the folding process. Mutations introduced in the NMP binding (NMPbind) domain of the protein are found to have unexpected effects on the folding landscape. Thus, while stabilizing mutations in the core of the NMPbind domain retain the main folding pathways of wild-type AK, a destabilizing mutation at the interface between the NMPbind and the CORE domains causes a significant repartition of the flux between the folding pathways. Our results demonstrate the outstanding plasticity of the folding landscape of AK, and reveal how specific mutations in the primary structure are translated into changes in folding dynamics. The combination of methodologies introduced in this work should prove useful for deepening our understanding of the folding process of multi-domain proteins.
We use protein folding energy landscape concepts such as golf course and funnel to study re-equilibration in athermal martensites under systematic temperature quench Monte Carlo simulations. On quenching below a transition temperature, the seeded high-symmetry parent-phase austenite that converts to the low-symmetry product-phase martensite, through autocatalytic twinning or elastic photocopying, has both rapid conversions and incubation delays in the temperature-time-transformation phase diagram. We find the rapid (incubation delays) conversions at low (high) temperatures arises from the presence of large (small) size of golf-course edge that has the funnel inside for negative energy states. In the incubating state, the strain structure factor enters into the Brillouin-zone golf course through searches for finite transitional pathways which close off at the transition temperature with Vogel-Fulcher divergences that are insensitive to Hamiltonian energy scales and log-normal distributions, as signatures of dominant entropy barriers. The crossing of the entropy barrier is identified through energy occupancy distributions, Monte Carlo acceptance fractions, heat emission, and internal work.
Master equations were introduced by Pauli (Sommerfeld at 60 Festschrift, Leipzig, 1928) in the context of quantum correlation functions.
Spectral and pseudospectral methods based on classical and nonclassical polynomial basis sets are used for the solution of the Fokker-Planck and Schrödinger equations. Fokker-Planck equations describe many different processes in chemistry and physics, and their study has attracted considerable attention by researchers in many different fields including astrophysics, finance and biology. Pseudospectral methods of solution of the Fokker-Planck equation are presented for several systems such as the Ornstein-Uhlenbeck model for Brownian motion, electron thermalization in atomic moderators, charged particle relaxation in plasmas and models for chemical reactions based on Kramers’ equation. A Fokker-Planck equation can be transformed to a Schrödinger equation with a potential that belongs to the class of potentials in supersymmetric quantum mechanics and expressed in terms of the superpotential. The quantum harmonic oscillator and the Morse potential belong to this class of Schrödinger equations. The pseudospectral methods developed for the solution of the Fokker-Planck equation based on nonclassical basis sets are also applied to a large number of the Schrödinger equations including the Henon-Heles potential. Fundamental aspects of different pseudospectral methods such as the Discrete Variable Representation, the Quadrature Discretization method, the Lagrange mesh method and Fourier grid methods are discussed.
Solution of “Levinthal’s paradox”: a set of fast folding pathways (a “folding funnel” with phase separation) automatically leads to the most stable structure. It is necessary only to have a sufficient energy gap separating the most stable fold from others. Volume of conformational space at the level of secondary structure formation and assembly. Discussion of very slow folding of stable structure in some proteins: serpins. “Chameleon” proteins. Misfolding. Notes on “energy funnels” and “free-energy landscapes” of the folding protein chains. Consideration of the unfolding and folding sides of the free-energy barrier. The detailed balance law. Protein structures: physics of folding and natural selection of chains capable of folding.
The mean lifetime of a particle in the presence of an absorbing boundary is the mean first passage time to the boundary. For diffusive dynamics on a one dimensional potential, we establish an exact relation between the mean first passage time to x = a, averaged over a (pseudo) equilibrium distribution in the region a ⩽ x<∞, and the time integral of certain equilibrium correlation functions determined by dynamics on the entire (−∞<x<∞) surface in the absence of an absorbing boundary at x = a. As an application, we show that the sum of the forward and reverse rate constants for diffusive barrier crossing obtained using mean first passage times and obtained from the time integral of the normalized number correlation function, are identical for an arbitrary choice of the dividing surface. © 1997 American Institute of Physics.
To understand the kinetics of protein folding, we introduce the concept of a ``transition coordinate'' which is defined to be the coordinate along which the system progresses most slowly. As a practical implementation of this concept, we define the transmission coefficient for any conformation to be the probability for a chain with the given conformation to fold before it unfolds. Since the transmission coefficient can serve as the best possible measure of kinetic distance for a system, we present two methods by which we can determine how closely any parameter of the system approximates the transmission coefficient. As we determine that the transmission coefficient for a short-chain heteropolymer system is dominated by entropic factors, we have chosen to illustrate the methods mentioned by applying them to geometrical properties of the system such as the number of native contacts and the looplength distribution. We find that these coordinates are not good approximations of the transmission coefficient and therefore, cannot adequately describe the kinetics of protein folding.
Protein chains coil into alpha-helices and beta-sheet structures. Knowing the timescales and mechanism of formation of these basic structural elements is essential for understanding how proteins fold. For the past 40 years, alpha-helix formation has been extensively investigated in synthetic and natural peptides, including by nanosecond kinetic studies. In contrast, the mechanism of formation of beta structures has not been studied experimentally. The minimal beta-structure element is the beta-hairpin, which is also the basic component of antiparallel beta-sheets. Here we use a nanosecond laser temperature-jump apparatus to study the kinetics of folding a beta-hairpin consisting of 16 amino-acid residues. Folding of the hairpin occurs in 6 micros at room temperature, which is about 30 times slower than the rate of alpha-helix formation. We have developed a simple statistical mechanical model that provides a structural explanation for this result. Our analysis also shows that folding of a beta-hairpin captures much of the basic physics of protein folding, including stabilization by hydrogen bonding and hydrophobic interactions, two-state behaviour, and a funnel-like, partially rugged energy landscape.
A lattice model of protein folding is developed to distinguish between amino acid sequences that do and do not fold into unique conformations. Although Monte Carlo simulations provide insights into the long-time processes involved in protein folding, these simulations cannot systematically chart the conformational energy surface that enables folding. By assuming that protein folding occurs after chain collapse, a kinetic map of important pathways on this surface is constructed through the use of an analytical theory of probability flow. Convergent kinetic pathways, or "folding funnels," guide folding to a unique, stable, native conformation. Solution of the probability flow equations is facilitated by limiting treatment to diffusion between geometrically similar collapsed conformers. Similarity is measured in terms of a reconfigurational distance. Two specific amino acid sequences are deemed foldable and nonfoldable because one gives rise to a single, large folding funnel leading to a native conformation and the other has multiple pathways leading to several stable conformers. Monte Carlo simulations demonstrate that folding funnel calculations accurately predict the fact of and the pathways involved in folding-specific sequences. The existence of folding funnels for specific sequences suggests that geometrically related families of stable, collapsed conformers fulfill kinetic and thermodynamic requirements of protein folding.
Although the rates of chemical reactions become faster with increasing temperature, the converse may be observed with protein-folding reactions. The rate constant for folding initially increases with temperature, goes through a maximum, and then decreases. The activation enthalpy is thus highly temperature dependent because of a large change in specific heat (delta Cp). Such a delta Cp term is usually presumed to be a consequence of a large decrease in exposure of hydrophobic surfaces to water as the reaction proceeds from the denatured state to the transition state for folding: the hydrophobic side chains are surrounded by "icebergs" of water that melt with increasing temperature, thus making a large contribution to the Cp of the denatured state and a smaller one to the more compact transition state. The rate could also be affected by temperature-induced changes in the conformational population of the ground state: the heat required for the progressive melting of residual structure in the denatured state will contribute to delta Cp. By examining two proteins with different refolding mechanisms, we are able to find both of these two processes; barley chymotrypsin inhibitor 2, which refolds from a highly unfolded state, fits well to a hydrophobic interaction model with a constant delta Cp of activation, whereas barnase, which refolds from a more structured denatured state, deviates from this ideal behavior.
Experimental information on the structure and dynamics of molten globules gives estimates for the energy landscape's characteristics for folding highly helical proteins, when supplemented by a theory of the helix-coil transition in collapsed heteropolymers. A law of corresponding states relating simulations on small lattice models to real proteins possessing many more degrees of freedom results. This correspondence reveals parallels between "minimalist" lattice results and recent experimental results for the degree of native character of the folding transition state and molten globule and also pinpoints the needs of further experiments.
The energy landscape theory of protein folding is a statistical description of a protein's potential surface. It assumes that folding occurs through organizing an ensemble of structures rather than through only a few uniquely defined structural intermediates. It suggests that the most realistic model of a protein is a minimally frustrated heteropolymer with a rugged funnel-like landscape biased toward the native structure. This statistical description has been developed using tools from the statistical mechanics of disordered systems, polymers, and phase transitions of finite systems. We review here its analytical background and contrast the phenomena in homopolymers, random heteropolymers, and protein-like heteropolymers that are kinetically and thermodynamically capable of folding. The connection between these statistical concepts and the results of minimalist models used in computer simulations is discussed. The review concludes with a brief discussion of how the theory helps in the interpretation of results from fast folding experiments and in the practical task of protein structure prediction.
The quantitative description of model protein folding kinetics using a diffusive collective reaction coordinate is examined. Direct folding kinetics, diffusional coefficients and free energy profiles are determined from Monte Carlo simulations of a 27-mer, 3 letter code lattice model, which corresponds roughly to a small helical protein. Analytic folding calculations, using simple diffusive rate theory, agree extremely well with the full simulation results. Folding in this system is best seen as a diffusive, funnel-like process. Comment: LaTeX 12 pages, figures included
The mechanism of protein folding (represented schematically below) is one of the most fascinating problems in the field of chemical reactions. This review presents the progess made recently in understanding key elements of this reaction and describes a solution to the often quoted Levinthal Paradox.
The reversible folding and unfolding of barley chymotrypsin inhibitor 2 (CI2) appears to be a rare example in which both equilibria and kinetics are described by a two-state model. Equilibrium denaturation by guanidinium chloride and heat is completely reversible, and the data can be fitted to a simple two-state model involving only native and denatured forms. The free energy of folding in the absence of denaturant, DELTA-G(H2O) at pH 6.3, is calculated to be 7.03 +/- 0.16 and 7.18 +/- 0.43 kcal mol-1 for guanidinium chloride and thermal denaturation, respectively. Scanning microcalorimetry shows that the ratio of the van't Hoff enthalpy of denaturation to the calorimetric enthalpy of denaturation does not deviate from unity, the value observed for a two-state transition, over the pH range 2.2-3.5. The heat capacity change for denaturation is found to be 0.789 kcal mol-1 K-1. The rate of unfolding of CI2 is first order and increases exponentially with increasing guanidinium chloride concentration. Refolding, however, is complex and involves at least three well-resolved phases. The three phases result from heterogeneity of the unfolded form due to proline isomerization. The fast phase, 77% of the amplitude, corresponds to the refolding of the fraction of the protein that has all its prolines in a native trans conformation. The rate of this major phase decreases exponentially with increasing guanidinium chloride concentration. The unfolding and refolding kinetics can also be fitted to a two-state model. Importantly, DELTA-G(H2O) and m, the constant of proportionality of the free energy of folding with respect to guanidinium chloride concentration, calculated from the kinetic experiments, 7.24 +/- 0.22 kcal mol-1 and 1.86 +/- 0.05 kcal mol-1 M-1, respectively, agree, within experimental error, with the values measured from the equilibrium experiments. This is perhaps the strongest evidence that the unfolding of CI2 follows a simple two-state transition.
Association reactions involving diffusion in one, two, and three‐dimensional finite domains governed by Smoluchowski‐type equations (e.g., interchain reaction of macromolecules, ligand binding to receptors, repressor–operator association of DNA strand) are shown to be often well described by first‐order kinetics and characterized by an average reaction (passage) time τ. An inhomogeneous differential equation is derived which, for problems with high symmetry, yields τ by simple quadrature without taking recourse to detailed cumbersome time‐dependent solutions of the original Smoluchowski equation. The cases of diffusion and nondiffusion controlled processes are included in the treatment. For reaction processes involving free diffusion and intramolecular chain motion, the validity of the passage time approximation is analyzed.
We develop a first passage time description for the kinetics of reactions involving diffusive barrier crossing in a bistable (and also in a more general) potential, a situation realized, for example, in some photoisomerization processes. In case the reactant is in thermal equilibrium, the first passage times account well for the reaction dynamics as shown by comparison with exact numerical calculations. A simple integral expression for the rate constants is presented. For a case involving a reactant initially far off equilibrium, a two relaxation time description for the particle number N(t) is derived and compared with the results of an ’’exact’’ calculation. This description results from a knowledge of N(t = 0), N˙(t = 0), F∞0dt N(t), i.e., the first passage time, and F∞0dt tN(t).
The role of relaxation processes in determining the rates of activated events has long been a point of discussion in chemical physics. In this paper, we re‐examine this issue. We idealize the problem as the classical motion of a particle in a one‐dimensional potential coupled to a heat bath. This situation is described by a kinetic equation with a ’’collision operator’’ glc. An expansion in powers of the damping constant g is developed. This expansion is not limited to the case of high activation barriers. We compare results for various choices of the collision operator and provide a new derivation of Slater’s new rate theory. A Padé approximant approach unifies our low g results with those in the high g, i.e., diffusive, regime.
The dynamics of electron transfer in a non-Debye solvent is described by multidimensional Markovian reaction-diffusion equation. To highlight differences with existing approaches in the simplest possible context, the irreversible outer-sphere reaction in a solvent with a biexponential energy-gap autocorrelation function, Δ(t), is studied in detail. In a Debye solvent, Δ(t) = exp(−t/τL) and the rate can be rigorously expressed as an explicit functional of exp(−t/τL). It has been suggested that the exact rate in a non-Debye solvent can be found by replacing exp(−t/τL) with the appropriate (nonexponential) Δ(t). For a “biexponential” solvent, our approach is based on an anisotropic diffusion equation for motion on a harmonic surface in the presence of a two-dimensional delta function sink. Three approximations, which reduce the solution of this equation to effective one-dimensional ones, are considered and compared with exact Brownian dynamics simulation results. The crudest approximation replaces the non-Debye solvent with an effective Debye one with τeff−1 = (−dΔ/dt)t = 0. The second is obtained by invoking the Wilemski–Fixman-type closure approximation for the equivalent two-dimensional integral equation. This approximation turns out to be identical to the above mentioned “substitution” procedure. When the relaxation times of the two exponentials are sufficiently different, it is shown how the two-dimensional problem can be reduced to a one-dimensional one with a nonlocal sink function. This anisotropic relaxation time approximation is in excellent agreement with simulations when the relaxation times differ by at least a factor of three and the activation energy is greater than kBT. Finally, it is indicated how the influence of intramolecular vibrational modes (i.e., nonlocal sink functions) can be treated within the framework of this formalism. © 1998 American Institute of Physics.
We show, by application to a simple protein folding model, how a stochastic Hamiltonian can be used to study a complex chemical reaction. This model is found to have many metastable states and the properties of these states are investigated. A simple generalization of transition-state theory is developed and used to estimate the folding time for the model. It is found to have a glass phase where the dynamics is very slow. The relevance of our results to protein folding and the general problem of complex chemical reactions is discussed.
A new integral representation of the transition rate holds for any friction and is shown to allow a feasible evaluation in a wide friction range. Analytic approximations include the (high-friction) Kramers result with the leading correction, as well as a low-friction case. The method is complementary to a recent one of Melnikov and Meshkov.
Proteins are involved in virtually every biological process, and their functions range from catalysis of chemical reactions to maintenance of the electrochemical potential across cell membranes. They are synthesized on ribosomes as linear chains of amino acids in a specific order from information encoded within the cellular DNA. To function, it is necessary for these chains to fold into the unique native three-dimensional structure that is characteristic for each protein. This involves a complex molecular recognition phenomenon that depends on the cooperative action of many relatively weak nonbonding interactions. As the number of possible conformations for a polypeptide chain is astronomically large, a systematic search for the native (lowest energy) structure would require an almost infinite length of time. Recently, significant progress has been made towards solving this paradox and understanding the mechanism of folding. This has come about through advances in experimental strategies for following the folding reactions of proteins in the laboratory with biophysical techniques, and through progress in theoretical approaches that simulate the folding process with simplified models. The most recent advances in this area are comparable in significance to those that took place in developing an understanding of reactions of small molecules some thirty years ago. In this article we review the present state of our knowledge of the protein folding reaction and compare the concepts that are emerging with those that are now established for simpler reactions. A major distinction between protein folding and reactions of small molecules is the heterogeneity of the folding process and the resulting complex interplay between entropic and enthalpic contributions to the free energy of the system during the reaction. A unified model for protein folding is outlined based on the effective energy surface of a polypeptide chain and its bias towards the native state. An understanding of folding is important for the analysis of many events involved in cellular regulation, the design of proteins with novel functions, the utilization of sequence information from the various genome projects, and the development of novel therapeutic strategies for treating or preventing debilitating human diseases that are associated with the failure of proteins to fold correctly.
In this article, time correlation function methods are used to discuss classical isomerizationreactions of small nonrigid molecules in liquidsolvents. Molecular expressions are derived for a macroscopic phenomenological rate constant. The form of several of these equations depend upon what ensemble is used when performing averages over initial conditions. All of these formulas, however, reduce to one final physical expression whose value is manifestly independent of ensemble. The validity of the physical expression hinges on a separation of time scales and the plateau value problem. The approximations needed to obtain transition state theory are described and the errors involved are estimated. The coupling of the reaction coordinate to the liquid medium provides the dissipation necessary for the existence of a plateau value for the rate constant, but it also leads to failures of Wigner’s fundamental assumption for transition state theory. We predict that for many isomerizationreactions, the transmission coefficient will differ significantly from unity and that the difference will be a strong function of the thermodynamic state of the liquidsolvent.
The steady flow viscosity of a sample of polyisobutylene of viscosity‐average molecular weight 1.35 million, distributed by the National Bureau of Standards, has been measured from 15° to 100°C. Its logarithm is a linear function of 1/T<sup>2</sup>. Application of the method of reduced variables to dynamic mechanical data from -45° to 100°, previously reported for this polyisobutylene, yields composite curves reduced to 25°C for the real and imaginary parts of the complex compliance and complex shear modulus; the real part of the complex dynamic viscosity; and the mechanical loss tangent. The latter exhibits a broad and peculiarly asymmetric maximum. The reduced time scale extends from 1 to 10<sup>-9</sup> sec. The reduction factors a T obtained in this way are slightly higher than those derived either from the viscosity or from stress relaxation measurements of Tobolsky and associates. The distribution functions of relaxation and retardation times have been calculated by second approximation methods and their detailed shapes are defined in the transition region between rubber‐like and glass‐like behavior. The relaxation distribution function is compared with the idealized distribution of Tobolsky.
In a discussion of the dynamics of protein folding two limiting models
(random-search nucleation and chain propagation., diffusion-collision)
are considered. It is suggested that the latter may have the dominant
role in many proteins.
Levinthal's paradox is that finding the native folded state of a protein by a random search among all possible configurations can take an enormously long time. Yet proteins can fold in seconds or less. Mathematical analysis of a simple model shows that a small and physically reasonable energy bias against locally unfavorable configurations, of the order of a few kT, can reduce Levinthal's time to a biologically significant size.
The refolding of chymotrypsin inhibitor 2 (CI2) is, at least, a triphasic process. The rate constants are 53 s-1 for the major phase (77% of the total amplitude) and 0.43 and 0.024 s-1 for the slower phases (23% of the total amplitude) at 25 degrees C and pH 6.3. The multiphase nature of the refolding reaction results from heterogeneity in the denatured state because of proline isomerization. The fast phase corresponds to the refolding of the fraction of protein that has all its prolines in a native trans conformation in the denatured state. It is not catalyzed by peptidyl-prolyl isomerase. The rate-limiting step of folding for the slower phases, however, is proline isomerization, and they are both catalyzed by peptidyl-prolyl isomerase. The slowest phase has properties consistent with a process involving proline isomerization in a denatured state. In particular, the activation enthalpy is large, 16 kcal mol-1 K-1, and the rate is independent of guanidinium chloride concentration ([GdnHCl]). In comparison, the intermediate phase shows properties consistent with a process involving proline isomerization in a partially structured state. The activation enthalpy is small, 8 kcal mol-1 K-1, and the rate has a strong dependence on [GdnHCl]. Temperature dependences of the rate constants for unfolding and for the fast refolding phase, both in the absence and in the presence of GdnHCl, were used to characterize the thermodynamic nature of the transition state and its relative exposure to solvent. The Eyring plot for unfolding is linear, indicating that there is relatively little change in heat capacity between native state and transition state.(ABSTRACT TRUNCATED AT 250 WORDS)
Diffusion in a spatially rough one-dimensional potential is treated by analysis of the mean first passage time. A general expression is found for the effective diffusion coefficient, which can become very small at low temperatures.
Small, single-module proteins that fold in a single cooperative step may be paradigms for understanding early events in protein-folding pathways generally. Recent experimental studies of the 64-residue chymotrypsin inhibitor 2 (CI2) support a nucleation mechanism for folding, as do some computer stimulations. CI2 has a nucleation site that develops only in the transition state for folding. The nucleus is composed of a set of adjacent residues (an alpha-helix), stabilized by long-range interactions that are formed as the rest of the protein collapses around it. A simple analysis of the optimization of the rate of protein folding predicts that rates are highest when the denatured state has little residual structure under physiological conditions and no intermediates accumulate. This implies that any potential nucleation site that is composed mainly of adjacent residues should be just weakly populated in the denatured state and become structured only in a high-energy intermediate or transition state when it is stabilized by interactions elsewhere in the protein. Hierarchical mechanisms of folding in which stable elements of structure accrete are unfavorable. The nucleation-condensation mechanism of CI2 fulfills the criteria for fast folding. On the other hand, stable intermediates do form in the folding of more complex proteins, and this may be an unavoidable consequence of increasing size and nucleation at more than one site.
A simple model of the kinetics of protein folding is presented. The reaction coordinate is the "correctness" of a configuration compared with the native state. The model has a gap in the energy spectrum, a large configurational entropy, a free energy barrier between folded and partially folded states, and a good thermodynamic folding transition. Folding kinetics is described by a master equation. The folding time is estimated by means of a local thermodynamic equilibrium assumption and then is calculated both numerically and analytically by solving the master equation. The folding time has a maximum near the folding transition temperature and can have a minimum at a lower temperature.
General principles of protein structure, stability, and folding kinetics have recently been explored in computer simulations of simple exact lattice models. These models represent protein chains at a rudimentary level, but they involve few parameters, approximations, or implicit biases, and they allow complete explorations of conformational and sequence spaces. Such simulations have resulted in testable predictions that are sometimes unanticipated: The folding code is mainly binary and delocalized throughout the amino acid sequence. The secondary and tertiary structures of a protein are specified mainly by the sequence of polar and nonpolar monomers. More specific interactions may refine the structure, rather than dominate the folding code. Simple exact models can account for the properties that characterize protein folding: two-state cooperativity, secondary and tertiary structures, and multistage folding kinetics--fast hydrophobic collapse followed by slower annealing. These studies suggest the possibility of creating "foldable" chain molecules other than proteins. The encoding of a unique compact chain conformation may not require amino acids; it may require only the ability to synthesize specific monomer sequences in which at least one monomer type is solvent-averse.
The mechanism of protein folding is being investigated theoretically by the use of both simplified and all-atom models of the polypeptide chain. Lattice heteropolymer simulations of the folding process have led to proposals for the folding mechanism and for the resolution of the Levinthal paradox. Both stability and rapid folding have been shown in model studies to result from the presence of a pronounced global energy minimum corresponding to the native state. Concomitantly, molecular dynamics simulations with detailed atomic models have been used to analyze the initial stages of protein unfolding. Results concerning possible folding intermediates and the role of water in the unfolding process have been obtained. The two types of theoretical approaches are providing information essential for an understanding of the mechanism of protein folding and are useful for the design of experiments to study the mechanism in different proteins.
The understanding, and even the description of protein folding is impeded by the complexity of the process. Much of this complexity can be described and understood by taking a statistical approach to the energetics of protein conformation, that is, to the energy landscape. The statistical energy landscape approach explains when and why unique behaviors, such as specific folding pathways, occur in some proteins and more generally explains the distinction between folding processes common to all sequences and those peculiar to individual sequences. This approach also gives new, quantitative insights into the interpretation of experiments and simulations of protein folding thermodynamics and kinetics. Specifically, the picture provides simple explanations for folding as a two-state first-order phase transition, for the origin of metastable collapsed unfolded states and for the curved Arrhenius plots observed in both laboratory experiments and discrete lattice simulations. The relation of these quantitative ideas to folding pathways, to uniexponential vs. multiexponential behavior in protein folding experiments and to the effect of mutations on folding is also discussed. The success of energy landscape ideas in protein structure prediction is also described. The use of the energy landscape approach for analyzing data is illustrated with a quantitative analysis of some recent simulations, and a qualitative analysis of experiments on the folding of three proteins. The work unifies several previously proposed ideas concerning the mechanism protein folding and delimits the regions of validity of these ideas under different thermodynamic conditions.
In an earlier paper which models the cell-cell (or virus-cell) fusion complex as two partial spherical vesicles joined at a narrow neck (Rubin, R. J., and Yi-der Chen. 1990. Biophys. J. 58:1157-1167), the redistribution by diffusion of lipid-like molecules through the neck between the two fused cell surfaces was studied. In this paper, we extend the study to the calculation of the kinetics of fluorescence increase in a single fusion complex when the lipid-like molecules are fluorescent and self-quenching. The formalism developed in this paper is useful in deducing fusion activation mechanisms from cuvette fluorescence measurements in cell-cell fusion systems. Two different procedures are presented: 1) an exact one which is based on the exact local density functions obtained from diffusion equations in our earlier study; and 2) an approximate one which is based on treating the kinetics of transfer of probes between the two fused cells as a two-state chemical reaction. For typical cell-cell fusion complexes, the fluorescence dequencing curves calculated from the exact and approximate procedures are very similar. Due to its simplicity, the approximate method should be very useful in future applications. The formalism is applied to a typical cell-cell fusion complex to study the sensitivity of dequenching curves to changes in various fusion parameters, such as the radii of the cells, the radius of the pore at the fusion junction, and the number of probes initially loaded to the complex.
Recently, protein-folding models have advanced to the point where folding simulations of protein-like chains of reasonable length (up to 125 amino acids) are feasible, and the major physical features of folding proteins, such as cooperativity in thermodynamics and nucleation mechanisms in kinetics, can be reproduced. This has allowed deep insight into the physical mechanism of folding, including the solution of the so-called 'Levinthal paradox'.
A change in the perception of the protein folding problem has taken place recently. The nature of the change is outlined and the reasons for it are presented. An essential element is the recognition that a bias toward the native state over much of the effective energy surface may govern the folding process. This has replaced the random search paradigm of Levinthal and suggests that there are many ways of reaching the native state in a reasonable time so that a specific pathway does not have to be postulated. The change in perception is due primarily to the application of statistical mechanical models and lattice simulations to protein folding. Examples of lattice model results on protein folding are presented. It is pointed out that the new optimism about the protein folding problem must be complemented by more detailed studies to determine the structural and energetic factors that introduce the biases which make possible the folding of real proteins.
We use two simple models and the energy landscape perspective to study protein folding kinetics. A major challenge has been to use the landscape perspective to interpret experimental data, which requires ensemble averaging over the microscopic trajectories usually observed in such models. Here, because of the simplicity of the model, this can be achieved. The kinetics of protein folding falls into two classes: multiple-exponential and two-state (single-exponential) kinetics. Experiments show that two-state relaxation times have "chevron plot" dependences on denaturant and non-Arrhenius dependences on temperature. We find that HP and HP+ models can account for these behaviors. The HP model often gives bumpy landscapes with many kinetic traps and multiple-exponential behavior, whereas the HP+ model gives more smooth funnels and two-state behavior. Multiple-exponential kinetics often involves fast collapse into kinetic traps and slower barrier climbing out of the traps. Two-state kinetics often involves entropic barriers where conformational searching limits the folding speed. Transition states and activation barriers need not define a single conformation; they can involve a broad ensemble of the conformations searched on the way to the native state. We find that unfolding is not always a direct reversal of the folding process.
Theoretical studies using simplified models of proteins have shed light on the general heteropolymeric aspects of the folding problem. Recent work has emphasized the statistical aspects of folding pathways. In particular, progress has been made in characterizing the ensemble of transition state conformations and elucidating the role of intermediates. These advances suggest a reconciliation between the new ensemble approaches and the classical view of a folding pathway.
Understanding the mechanism of protein secondary structure formation is an essential part of the protein-folding puzzle. Here, we describe a simple statistical mechanical model for the formation of a beta-hairpin, the minimal structural element of the antiparallel beta-pleated sheet. The model accurately describes the thermodynamic and kinetic behavior of a 16-residue, beta-hairpin-forming peptide, successfully explaining its two-state behavior and apparent negative activation energy for folding. The model classifies structures according to their backbone conformation, defined by 15 pairs of dihedral angles, and is further simplified by considering only the 120 structures with contiguous stretches of native pairs of backbone dihedral angles. This single sequence approximation is tested by comparison with a more complete model that includes the 2(15) possible conformations and 15 x 2(15) possible kinetic transitions. Finally, we use the model to predict the equilibrium unfolding curves and kinetics for several variants of the beta-hairpin peptide.
Mossbauer spectroscopy in biological systems
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Viscoelastic relaxation of poly-isobutylene
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Analytical theories of protein folding In: ElberR, ed. Recent developments in theoretical studies of proteins
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Folding dynamics and mechanism of β-hairpin formation
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Viscoelastic relaxation of polyisobutylene
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A statistical mechanical model for b-hairpin kinetics
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Viscoelastic relaxation of polyisobutylene
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