Diffusive model of protein folding dynamics with Kramers turnover in rate

We study the folding kinetics of a three-helix bundle protein using a coarse polymer model. The folding dynamics can be accurately represented by one-dimensional diffusion along a reaction coordinate selected to capture the transition state. By varying the solvent friction, we show that position-dependent diffusion coefficients are determined by microscopic transitions on a rough energy landscape. A maximum in the folding rate at intermediate friction is explained by "Kramers turnover" in these microscopic dynamics that modulates the rate via the diffusion coefficient; overall folding remains diffusive even close to zero friction. For water friction, we find that the "attempt frequency" (or "speed limit") in a Kramers model of folding is about 2 micros-1, with an activation barrier of about 2kBT, and a folding transition path duration of approximately equal to 100 ns, 2 orders of magnitude less than the folding time of approximately equal to 10 micros.     

There and back again: Two views on the protein folding puzzle

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.     

Folding of lattice protein chains with modified Gō potential

We propose a modified Gō model in which the pairwise interaction energies vary as local environment changes. The stability difference between the surface and the core is also well considered in this model. Thermodynamic and kinetic studies suggest that this model has improved folding cooperativity and foldability in contrast with the Gō model. The free energy landscape of this model has broad barriers and narrow denatured states, which is consistent with that of the two-state folding proteins and is lacked for the Gō model. The role of non-native interactions in protein folding is also studied. We find that appropriate consideration of the contribution of the non-native interactions may increase the folding rate around the transition temperature. Our results show that conformation-dependent interaction between the residues is a realistic representation of potential functions in protein folding. Received 10 April 2002 / Received in final form 20 August 2002 Published online 19 December 2002 RID="a" ID="a"e-mail: wangwei@nju.edu.cn     

Two state behavior in a solvable model of beta-hairpin folding

Understanding the mechanism of protein secondary structure formation is an essential part of the protein-folding puzzle. Here we describe a simple model for the formation of a beta hairpin, motivated by the fact that folding of a beta hairpin captures much of the basic physics of protein folding. The modeled hairpin is composed of two interacting Gaussian chains with one pairwise (two-body) and two many-body interactions. We show that these many-body interactions, arising from side chain packing effects, are responsible for producing an "all-or-none" folding transition. We also estimate the (single exponential) folding/unfolding rate via calculating the thermodynamic weight of the "critical" droplet/bubble.     

Time resolved collapse of a folding protein observed with small angle x-ray scattering

High-intensity, "pink" beam from an undulator was used in conjunction with microfabricated rapid-fluid mixing devices to monitor the early events in protein folding with time resolved small angle x-ray scattering. This Letter describes recent work on the protein bovine beta-lactoglobulin where collapse from an expanded to a compact set of states was directly observed on the millisecond time scale. The role of chain collapse, one of the initial stages of protein folding, is not currently understood. The characterization of transient, compact states is vital in assessing the validity of theories and models of the folding process.     

Reconstructing the free-energy landscape of a mechanically unfolded model protein

The equilibrium free-energy landscape of an off-lattice model protein as a function of an internal (reaction) coordinate is reconstructed from out-of-equilibrium mechanical unfolding manipulations. This task is accomplished via two independent methods: by employing an extended version of the Jarzynski equality (EJE) and the protein inherent structures (ISs). In a range of temperatures around the "folding transition" we find a good quantitative agreement between the free energies obtained via EJE and IS approaches. This indicates that the two methodologies are consistent and able to reproduce equilibrium properties of the examined system. Moreover, for the studied model the structural transitions induced by pulling can be related to thermodynamical aspects of folding.     

Polymer collapse in the presence of hydrodynamic interactions

We investigate numerically the dynamical behaviour of a polymer chain collapsing in a dilute solution. The rate of collapse is measured with and without the presence of hydrodynamic interactions. We find that hydrodynamic interactions both accelerate polymer collapse and alter the folding pathway.     

Physics of protein folding

Protein physics is grounded on three fundamental experimental facts: protein, this long heteropolymer, has a well defined compact three-dimensional structure; this structure can spontaneously arise from the unfolded protein chain in appropriate environment; and this structure is separated from the unfolded state of the chain by the “all-or-none” phase transition, which ensures robustness of protein structure and therefore of its action. The aim of this review is to consider modern understanding of physical principles of self-organization of protein structures and to overview such important features of this process, as finding out the unique protein structure among zillions alternatives, nucleation of the folding process and metastable folding intermediates. Towards this end we will consider the main experimental facts and simple, mostly phenomenological theoretical models. We will concentrate on relatively small (single-domain) water-soluble globular proteins (whose structure and especially folding are much better studied and understood than those of large or membrane and fibrous proteins) and consider kinetic and structural aspects of transition of initially unfolded protein chains into their final solid (“native”) 3D structures.     

Specific and nonspecific collapse in protein folding funnels

Experiments with fast folding proteins are beginning to address the relationship between collapse and folding. We investigate how different scenarios for folding can arise depending on whether the folding and collapse transitions are concurrent or whether a nonspecific collapse precedes folding. Many earlier studies have focused on the limit in which collapse is fast compared to the folding time; in this work we focus on the opposite limit where, at the folding temperature, collapse and folding occur simultaneously. Real proteins exist in both of these limits. The folding mechanism varies substantially in these two regimes. In the regime of concurrent folding and collapse, nonspecific collapse now occurs at a temperature below the folding temperature (but slightly above the glass transition temperature).     

Structural determinant of protein designability

Here we present an approximate analytical theory for the relationship between a protein structure's contact matrix and the shape of its energy spectrum in amino acid sequence space. We demonstrate a dependence of the number of sequences of low energy in a structure on the eigenvalues of the structure's contact matrix, and then use a Monte Carlo simulation to test the applicability of this analytical result to cubic lattice proteins. We find that the lattice structures with the most low-energy sequences are the same as those predicted by the theory. We argue that, given sufficiently strict requirements for foldability, these structures are the most designable, and we propose a simple means to test whether the results in this paper hold true for real proteins.     

Folding of lattice protein model chains with fixed ends

Using Monte Carlo simulations, we have studied the folding dynamics and thermodynamics of geometricall yconstrained lattice protein model chains. The constraints are realized by fixing one or both terminals of the chains. By comparing the results with that of the free-end chains, we find that the folding behaviours of the end-constralned chains are not completely similar to that of the free-end chains. Both kinds of constraints on the chain ends affect the folding dynamics of the chains: i.e., the folding rate, but not the thermodynamics. The thermodynamic behaviour of the one-end-fixed chains shows less difference from that of the free-end chains, while the thermodynamic behaviour of the two-end-fixed chains has obvious difference from that of the free-end chains. The origin of these differences comes from the differences of the ergodicitv of the chains in the conformational space.     

Glassy dynamics of side-chain ordering in a simple model of protein folding

We introduce a modified version of protein lattice models in which monomers have several spin states, representing side-chain rotamers. Completion of folding corresponds to reaching the native backbone configuration with complete ordering of side chains. We find that as temperature is lowered, side-chain ordering becomes much slower than backbone folding. The presence of side chains leads to nonexponential kinetics and a broad distribution of relaxation times.     

Single chain force spectroscopy - Reading the sequence of HP protein models

We study the elastic properties of single A/B random copolymer chains, with a specific sequence and use them as theoretical model for so called HP proteins. HP proteins carry hydrophilic (P) and hydrophobic (H) monomers. We predict a rich structure in the force-extension relations which can be attributed to the information in the sequence. The variational method is used to probe local minima on the path of stretching and releasing for the chain molecules. At a given force, we find multiple configurations which are separated by energy barriers. A collapsed globular configuration consists of several domains which unravel cooperatively. Upon stretching, the unfolding path shows a stepwise pattern corresponding to the unfolding of each domain. While releasing, several cores can be created simultaneously in the middle of the chain, resulting in a different path of collapse. The long-range interactions and stiffness of the chain simplify the potential landscape given by the disorder in sequence. Received 5 March 2002 / Received in final form 16 May 2002 Published online 13 August 2002     

Onset of tethered chain overcrowding

We proposed an approach to precisely control the density of tethered chains on solid substrates using PEO-b-PS and PLLA-b-PS. As the crystallization temperature Tx increased, the PEO or PLLA lamellar crystal thickness d(L) increased as well as the reduced tethering density sigma; of the PS chains. The onset of tethered PS chains overcrowding in solution occurs at sigma(*) approximately 3.7-3.8 as evidenced by an abrupt change in the slope between (d(L))(-1) and Tx. This results from the extra surface free energy created by the tethered chain that starts to affect the growth barrier of the crystalline blocks.     

Kinetics of protein adsorption/desorption mediated by pH-responsive polymer layer

We propose a new way of regulating protein adsorption by using a p H-responsive polymer. According to the theoretical results obtained from the molecular theory and kinetic approaches, both thermodynamics and kinetics of protein adsorption are verified to be well controlled by the solution p H. The kinetics and the amount of adsorbed proteins at equilibrium are greatly increased when the solution environment changes from acid to neutral. The reason is that the increased p H promotes the dissociation of the weak polyelectrolyte, resulting in more charged monomers and more stretched chains.Thus the steric repulsion within the polymer layer is weakened, which effectively lowers the barrier felt by the protein during the process of adsorption. Interestingly, we also find that the kinetics of protein desorption is almost unchanged with the variation of p H. It is because although the barrier formed by the polymer layer changes along with the change of p H,the potential at contact with the surface varies equally. Our results may provide useful insights into controllable protein adsorption/desorption in practical applications.     

A study of Barstar folding events using boundary value simulations

From extensive biophysical studies of protein folding, two competing mechanisms emerged: hydrophobic collapse and the framework model. Our protein of choice is Barstar—a barnase inhibitor. The approximation algorithm we used to study Barstar folding trajectories is called SDEL—stochastic difference equation in length. Using the native structure as the final boundary value and a collection of unfolded structures as the varying initial boundary value, SDEL calculates an ensemble of least action pathways between these boundaries. The results are atomically detailed folding pathways, with as many intermediate structures as you request in the input. We generated 12 pathways, starting from a structurally wide selection of unfolded conformations. Using the protein's radius of gyration as our primary reaction coordinate, we tracked H-bonds, dihedral angles, native and non-native contacts, and energy along the folding pathways. This paper will follow our findings, with special emphasis on pinpointing hydrophobic collapse as a more appropriate mechanism for Barstar. Comparison with pathway predictions for Barstar using experimental techniques will also be discussed.     

Step Density Profiles in Localized Chains

We consider two types of strongly disordered one-dimensional Hamiltonian systems coupled to baths (energy or particle reservoirs) at the boundaries: strongly disordered quantum spin chains and disordered classical harmonic oscillators. These systems are believed to exhibit localization, implying in particular that the conductivity decays exponentially in the chain length L. We ask however for the profile of the (very slowly) transported quantity in the steady state. We find that this profile is a step-function, jumping in the middle of the chain from the value set by the left bath to the value set by the right bath. This is confirmed by numerics on a disordered quantum spin chain of 9 spins and on much longer chains of harmonic oscillators. From theoretical arguments, we find that the width of the step grows not faster than \(\sqrt{L}\), and we confirm this numerically for harmonic oscillators. In this case, we also observe a drastic breakdown of local equilibrium at the step, resulting in a heavily oscillating temperature profile.     

Modeling diffusion of adsorbed polymer with explicit solvent

Computer simulations of a polymer chain of length N strongly adsorbed at the solid-liquid interface in the presence of explicit solvent are used to delineate the factors affecting the N dependence of the polymer lateral diffusion coefficient, D(||). We find that surface roughness has a large influence, and D(||) scales as D(||) approximately N(-x), with x approximately 3/4 and x approximately 1 for ideal smooth and corrugated surfaces, respectively. The first result is consistent with the hydrodynamics of a "particle" of radius of gyration R(G) approximately N(nu) (nu=0.75) translating parallel to a planar interface, while the second implies that the friction of the adsorbed chains dominates. These results are discussed in the context of recent measurements.     

Fractal globule as a molecular machine

A fractal (crumpled) polymer globule, which is an unusual equilibrium state of a condensed unknotted macromolecule that is experimentally found in the DNA folding in human chromosomes, has been formed through the hierarchical collapse of a polymer chain. The relaxation dynamics of the elastic network constructed through the contact matrix of the fractal globule has been studied. It has been found that the fractal globule in its dynamic properties is similar to a molecular machine.     

Conformations of proteins in equilibrium

We introduce a simple theoretical approach for an equilibrium study of proteins with known native-state structures. We test our approach with results on well-studied globular proteins, chymotrypsin inhibitor (2ci2), barnase, and the alpha spectrin SH3 domain, and present evidence for a hierarchical onset of order on lowering the temperature with significant organization at the local level even at high temperatures. A further application to the folding process of HIV-1 protease shows that the model can be reliably used to identify key folding sites that are responsible for the development of drug resistance.