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.     

Fast chain contraction during protein folding: "foldability" and collapse dynamics

Theory indicates that at least some proteins will undergo a rapid and unimpeded collapse, like a disorganized hydrophobic chain, prior to folding. Yet experiments continue to find signs of an organized, or barrier-limited, collapse in even the fastest (approximately mus) folding proteins. Does the kinetic barrier represent a signature of the equilibrium "foldability" of these molecules? We have measured the rate of chain contraction in two nonfolding analogs of a very fast-collapsing protein. We find that these chains contract on the same time scale (approximately 10(-5)s) as the natural protein, and both pass over an energetic barrier at least as large as that encountered by the protein. The equilibrium foldability of the native structure therefore does not alone determine the dynamics of collapse; even the disordered chains contract approximately 1000x slower than expected for an ideal chain.     

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.     

Simple off-lattice model to study the folding and aggregation of peptides

We present a numerical study of a new protein model. This off-lattice model takes into account both the hydrogen bonds and the amino-acid interactions. It reproduces the folding of a small protein (peptide): morphological analysis of the conformations at low temperature shows two well-known substructures α-helix and β-sheet depending on the chosen sequence. The folding pathway in the scope of this model is studied through a free-energy analysis. We then study the aggregation of proteins. Proteins in the aggregate are mainly bound via hydrogen bonds. Performing a free-energy analysis we show that the addition of a peptide to such an aggregate is not favourable. We qualitatively reproduce the abnormal aggregation of proteins in prion diseases.     

Reconstructing Free Energy Profiles from Nonequilibrium Relaxation Trajectories

Reconstructing free energy profiles is an important problem in bimolecular reactions, protein folding or allosteric conformational changes. Nonequilibrium trajectories are readily measured experimentally, but their statistical significance and relation to equilibrium system properties still call for rigorous methods of assessment and interpretation. Here we introduce methods to compute the equilibrium free energy profile of a given variable from a set of short nonequilibrium trajectories, obtained by externally driving a system out of equilibrium and subsequently observing its relaxation. This protocol is not suitable for the Jarzynski equality since the irreversible work on the system is instantaneous. Assuming that the variable of interest satisfies an overdamped Langevin equation, which is frequently used for modeling biomolecular processes, we show that the trajectories sample a nonequilibrium stationary distribution that can be calculated in closed form. This allows for the estimation of the free energy via an inversion procedure that is analogous to that used in equilibrium and bypasses more complicated path integral methods, which we derive for comparison. We generalize the inversion procedure to systems with a diffusion constant that depends on the reaction coordinate, as is the case in protein folding, as well as to protocols in which the trajectories are initiated at random points. Using only a statistical pool of tens of synthetic trajectories, we demonstrate the versatility of these methods by reconstructing double and multi-well potentials, as well as a proposed profile for the hydrophobic collapse of a protein.     

Kinetics of the Wako-Saitô-Muñoz-Eaton model of protein folding

We consider a simplified model of protein folding, with binary degrees of freedom, whose equilibrium thermodynamics is exactly solvable. Based on this exact solution, the kinetics is studied in the framework of a local equilibrium approach, for which we prove that (i) the free energy decreases with time, (ii) the exact equilibrium is recovered in the infinite time limit, and (iii) the equilibration rate is an upper bound of the exact one. The kinetics is compared to the exact one for a small peptide and to Monte Carlo simulations for a longer protein; then rates are studied for a real protein and a model structure.     

Equilibrium distribution of mutators in the single fitness peak model

This Letter develops an analytically tractable model for determining the equilibrium distribution of mismatch repair deficient strains in unicellular populations. The approach is based on the single fitness peak model, which has been used in Eigen's quasispecies equations in order to understand various aspects of evolutionary dynamics. As with the quasispecies model, our model for mutator-nonmutator equilibrium undergoes a phase transition in the limit of infinite sequence length. This "repair catas-trophe" occurs at a critical repair error probability of epsilon(r)=L(via)/L, where L(via) denotes the length of the genome controlling viability, while L denotes the overall length of the genome. The repair catastrophe therefore occurs when the repair error probability exceeds the fraction of deleterious mutations. Our model also gives a quantitative estimate for the equilibrium fraction of mutators in Escherichia coli.     

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.     

Quantum Approach to Fast Protein-Folding Time

In the traditional random-conformational-search model,various hypotheses with a series of meta-stable intermediate states were proposed to resolve the Levinthal paradox in protein-folding time.Here we introduce a quantum strategy to formulate protein folding as a quantum walk on a definite graph, which provides us a g'eneral framework without making hypotheses.Evaluating it by the mean of first passage time,we find that the folding time via our quantum approach is much shorter than the one obtained via.classical random walks.This idea is expected to evoke more insights for future studies.     

Entropic compression of interacting DNA loops

We solve a model for the equilibrium folding of DNA via the formation of randomly placed (nonspecific) loops. We find that the loop rearrangement entropy drives a significant reduction in loop size as more loops form. The reduction of the most probable loop size occurs over a wide range of forces and is enhanced by interloop cooperativity, indicating that it should be observable in single DNA experiments.     

Equilibrium folding and unfolding dynamics to reveal detailed free energy landscape of src SH3 protein by magnetic tweezers

Src SH3 protein domain is a typical two-state protein which has been confirmed by research of denaturant-induced unfolding dynamics. Force spectroscopy experiments by optical tweezers and atomic force microscopy have measured the force-dependent unfolding rates with different kinds of pulling geometry. However, the equilibrium folding and unfolding dynamics at constant forces has not been reported. Here, using stable magnetic tweezers, we performed equilibrium folding and unfolding dynamic measurement and force-jump measurement of src SH3 domain with tethering points at its N-and C-termini. From the obtained force-dependent transition rates, a detailed two-state free energy landscape of src SH3 protein is constructed with quantitative information of folding free energy, transition state barrier height and position,which exemplifies the capability of magnetic tweezers to study protein folding and unfolding dynamics.     

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.     

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     

Parameterized aeroelastic modeling and flutter analysis for a folding wing

To investigate the flutter characteristics of a folding wing with different configurations, a parameterized aeroelastic model is proposed. First, a parameterized structural model is established based on the substructure synthesis. Afterwards, the parameterized aerodynamic model is derived for each lifting surface using the so-called Doublet Lattice Method (DLM). The correctness of the resulting aeroelastic model is verified via NASTRAN. Finally, some aeroelastic simulations are performed using the proposed aeroelastic model. The results demonstrate that the flutter characteristics of the folding wing are very sensitive to the folding angle. With increasing folding angle, a transition between two unstable modes occurs. Such a transition results in a sudden change of flutter mode shapes and a jump of critical flutter frequency. Besides, there exists a region of folding angle, where the flutter behavior of the folding wing strongly depends on the structural damping.     

Entropy of the Dirac Field in Schwarzschild–de Sitter Space-Time via the Membrane Model

There are two event horizons in Schwarzschild–de Sitter space-time, a blackhole horizon and a cosmological horizon. They have different temperatures. Theradiation between them is of course not in thermal equilibrium. According to themembrane model suggested by us, the two horizons can be thought of as twoindependent thermodynamic systems in equilibrium. Their Dirac field entropiesare calculated via a membrane model. The result shows that the entropy of theDirac field is proportional to the sum of the areas of the two event horizons. Ifwe choose the same cutoff as that of Klein–Gordon field, the entropy of theDirac field is times that of Klein–Gordon field. This agrees with previousresults.     

基于TorchMD的粗粒化分子动力模拟研究

粗粒化模型通过简化原子性质以及原子间的相互作用实现生物大分子长时间尺度的分子动力学模拟. 深度学习通过模拟人类的认知过程实现海量数据的准确分类和回归过程. 本论文将这两种技术进行融合,利用基于深度学习的粗粒化分子动力学模拟技术研究分子在不同状态之间的变化过程,并提出基于TorchMD的分子动力学模拟的分析框架. 在本工作中,MFDP聚类算法被用于在三维的CV变量空间中进行聚类,并确定分子的若干主要状态,在完成聚类的同时,给出各类中的代表分子构象,并给出类之间的分子构象. 这为后续利用String算法分析分子在不同状态间的转换路径打下基础. 通过String算法,迭代搜索得到分子在不同状态之间的变化路径以及对应的势能变化曲线. 通过与已有文献的结果进行对比,验证了基于TorchMD的粗粒化分子动力学模拟的理论框架可以在相对较短的时间尺度里研究分子的变化过程.     

Phase Transition for the Maki–Thompson Rumour Model on a Small-World Network

We consider the Maki–Thompson model for the stochastic propagation of a rumour within a population. In this model the population is made up of “spreaders”, “ignorants” and “stiflers”; any spreader attempts to pass the rumour to the other individuals via pair-wise interactions and in case the other individual is an ignorant, it becomes a spreader, while in the other two cases the initiating spreader turns into a stifler. In a finite population the process will eventually reach an equilibrium situation where individuals are either stiflers or ignorants. We extend the original hypothesis of homogenously mixed population by allowing for a small-world network embedding the model, in such a way that interactions occur only between nearest-neighbours. This structure is realized starting from a k-regular ring and by inserting, in the average, c additional links in such a way that k and c are tuneable parameters for the population architecture. We prove that this system exhibits a transition between regimes of localization (where the final number of stiflers is at most logarithmic in the population size) and propagation (where the final number of stiflers grows algebraically with the population size) at a finite value of the network parameter c. A quantitative estimate for the critical value of c is obtained via extensive numerical simulations.     

Rate determining factors in protein model structures

Previous research has shown a strong correlation of protein folding rates to the native state geometry, yet a complete explanation for this dependence is still lacking. Here we study the rate-geometry relationship with a simple statistical physics model, and focus on two classes of model geometries, representing ideal parallel and antiparallel structures. We find that the logarithm of the rate shows an almost perfect linear correlation with the "absolute contact order", but the slope depends on the particular class considered. We discuss these findings in the light of experimental results.     

Quasi-physical Algorithm for Protein Folding in an Off-Lattice Model

We study a three-dimensional off-lattice protein folding model, which involves two species of residues interacting through Lennard-Jones potentials. By incorporating an extra energy contribution into the original potential function, we replace the original constrained problem with an unconstrained minimization of a mixed potential function. As such an efficient quasi-physical algorithm for solving the protein folding problem is presented. We apply the proposed algorithm to sequences with up to 55 residues and compare the computational results with the putative lowest energy found by several of the most famous algorithms, showing the advantages of our method. The dynamic behavior of the quasi-physlcal algorithm is also discussed.