Structure optimization by heuristic algorithm in a coarse-grained off-lattice model

A heuristic algorithm is presented for a three-dimensional off-lattice AB model consisting of hydrophobic (A) and hydrophilic (B) residues in Fibonacci sequences. By incorporating extra energy contributions into the original potential function, we convert the constrained optimization problem of AB model into an unconstrained optimization problem which can be solved by the gradient method. After the gradient minimization leads to the basins of the local energy minima, the heuristic off-trap strategy and subsequent neighborhood search mechanism are then proposed to get out of local minima and search for the lower-energy configurations. Furthermore, in order to improve the efficiency of the proposed algorithm, we apply the improved version called the new PERM with importance sampling (nPERMis) of the chain-growth algorithm, pruned-enriched-Rosenbluth method (PERM), to face-centered-cubic (FCC)-lattice to produce the initial configurations. The numerical results show that the proposed methods are very promising for finding the ground states of proteins. In several cases, we found the ground state energies are lower than the best values reported in the present literature.     

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     

蛋白质折叠的非格子模型

用粒子群算法对蛋白质的简化模型进行研究,体系中各原子间的相互作用由一物理势表示.通过全局优化体系的势函数得到蛋白质的结构数据,与实验测得的结构数据之间的均方根偏差RMSD为6.12(A).记录了能量最小值及回转半径随程序运行步骤的变化以及RMSD值与最低能量值的分布关系,这些结果反映了蛋白质折叠的一些基本特征.     

蛋白质表示简化的物理研究

文中概要地描述了“蛋白质折叠”的问题及其统计研究方法 ,并针对根据氨基酸之间相互作用特性进行的氨基酸和蛋白质简化研究进行了较为细致的论述和讨论 ;并指出通过这样的分析 ,我们不仅可以对蛋白质具体的简化字母表示有一定的了解 ,而且对于我们进一步了解蛋白质序列 结构关系有很大的帮助。     

蛋白质表示简化的物理研究

文中概要地描述了“蛋白质折叠”的问题及其统计研究方法 ,并针对根据氨基酸之间相互作用特性进行的氨基酸和蛋白质简化研究进行了较为细致的论述和讨论 ;并指出通过这样的分析 ,我们不仅可以对蛋白质具体的简化字母表示有一定的了解 ,而且对于我们进一步了解蛋白质序列 结构关系有很大的帮助。     

Challenges in protein folding simulations: Timescale, representation, and analysis

Experimental studies of protein folding processes are frequently hampered by the fact that only low resolution structural data can be obtained with sufficient temporal resolution. Molecular dynamics simulations offer a complementary approach, providing extremely high resolution spatial and temporal data on folding processes. The effectiveness of such simulations is currently hampered by continuing questions regarding the ability of molecular dynamics force fields to reproduce the true potential energy surfaces of proteins, and ongoing difficulties with obtaining sufficient sampling to meaningfully comment on folding mechanisms. We review recent progress in the simulation of three common model systems for protein folding, and discuss how recent advances in technology and theory are allowing protein folding simulations to address their current shortcomings.     

The Rubik’s cube problem revisited: a statistical thermodynamic approach

Inspired by the protein folding problem, we propose a Rubik’s cube model and study its thermodynamic and kinetic behavior. We find that the energy landscape contains a tiny funnel-like region, as the dynamics towards the native state is mostly diffusive. In particular, from Monte Carlo simulations we observe exponential kinetics in the first-passage-time distribution towards the native state at all temperatures considered, while the complexity of the energy landscape is exhibited through a stretched-exponential relaxation of the energy autocorrelation function. The rollover feature in the mean first passage time, as observed in many protein-folding dynamics studies, is captured again in our model and discussed under the statistical energy landscape approach.     

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.     

Secondary structure prediction of beta-hairpin peptide tryptophan zipper-I

We have investigated the folding properties of tryptophan zipper-I molecule which folds into a stable β-hairpin motif in aqueous solution as suggested by nuclear magnetic resonance (NMR) experiments. An all-atom presentation, including hydrogen, was used with an implicit solvent. As a simulation technique, simulated tempering algorithm was used to obtain equilibrium conformations of the molecule at ten distinct temperatures. Our minimum energy configuration obtained from simulated tempering algorithm is a β-hairpin motif with 1.30 Å backbone root-mean-square deviation from the reference PDB structure (1le0.pdb). Several quantities and exhaustive folding free energy landscapes were determined and discussed in order to clarify the folding behavior.     

葡萄球菌核酸酶蛋白的时间分辨荧光与热力学特性

葡萄球菌核酸酶(SNase)是一种小型球状蛋白,其变体常用来研究蛋白质的折叠过程。不同于之前报道的研究方法和技术手段,采用时间相关单光子计数(TCSPC)及飞秒荧光上转换技术,结合紫外吸收谱和稳态荧光光谱,研究了SNase蛋白变体Δ+PHS和Δ+PHS+I92A的荧光动力学,以及不同温度下蛋白结构与热稳定性的关系,证明蛋白质内色氨酸残基可作为一种内源性探针对蛋白变体的结构折叠和热稳定性进行印证和研究。衰减相关光谱(DAS)表明了两种变体随温度变化的不同趋势,在此基础上进一步分析了这两种变体的结构折叠及热稳定性的差异。皮秒时间分辨发射光谱(TRES)显示色氨酸残基存在0.5 ns的连续光谱弛豫过程,而光谱移动量可作为SNase变体蛋白结构紧密程度的判断依据。飞秒上转换数据分析结果中,0.5 ps的DAS在光谱蓝端为正、红端为负,表明了色氨酸残基受到弛豫效应的影响。200 ps的寿命则说明色氨酸残基与周围猝灭基团之间存在电子转移过程。时间分辨荧光各向异性(anisotropy)的分析结果则说明了色氨酸残基在蛋白质体系内具有独立的局部运动,且其强弱与变体的热稳定性和热运动的整体效果有关。测量和分析色氨酸残基的时间分辨荧光性质为深入研究SNase蛋白的结构和功能提供了新的思路。     

A simple coarse-grained model for interacting protein complex

ABSTRACT

We present a simple coarse-grained model in which each amino acid residue is represented by one coarse-grained particle for interacting protein complex. In order to determine the coarse-grained potential function of the interaction between amino acid residues, free energy profile as a function of the distance between amino acid side chains is investigated by using all-atom molecular dynamics simulations with thermodynamic integration method. The Langevin dynamics simulation with Gō-like model and our coarse-grained model reproduces homotetramer complex structure of GCN4-pLI and shows that interaction between hydrophobic amino acid residues promote the association of GCN4-pLI monomers.     

Stochastic annealing

We show how to simulate a system in thermal equilibrium when the energy cannot be evaluated exactly: the error distribution needs to be symmetric, but it does not need to be known. We also solve the Ceperley-Dewing version of this problem, where the error distribution is taken to be fully known. These underlying ideas give an effective optimization strategy for problems where the evaluation of each design can be sampled only statistically, including an application to protein folding.     

Folding/unfolding kinetics of lattice proteins studied using a simple statistical mechanical model for protein folding, I: Dependence on native structures and amino acid sequences

The folding/unfolding kinetics of a three-dimensional lattice protein was studied using a simple statistical mechanical model for protein folding that we developed earlier. We calculated a characteristic relaxation rate for the free energy profile starting from a completely unfolded structure (or native structure) that is assumed to be associated with a folding rate (or an unfolding rate). The chevron plot of these rates as a function of the inverse temperature was obtained for four lattice proteins, namely, proteins a1, a2, b1, and b2, in order to investigate the dependency of the folding and unfolding rates on their native structures and amino acid sequences. Proteins a1 and a2 fold to the same native conformation, but their amino acid sequences differ. The same is the case for proteins b1 and b2, but their native conformation is different from that of proteins a1 and a2. However, the chevron plots of proteins a1 and a2 are very similar to each other, and those of proteins b1 and b2 differ considerably. Since the contact orders of proteins b1 and b2 are identical, the differences in their kinetics should be attributed to the amino acid sequences and consequently to the interactions between the amino acid residues. A detailed analysis revealed that long-range interactions play an important role in causing the difference in the folding rates. The chevron plots for the four proteins exhibit a chevron rollover under both strongly folding and strongly unfolding conditions. The slower relaxation time on the broad and flat free energy surfaces of the unfolding conformations is considered to be the main origin of the chevron rollover, although the free energy surfaces have features that are rather complicated to be described in detail here. Finally, in order to concretely examine the relationship between changes in the free energy profiles and the chevron plots, we illustrate some examples of single amino acid substitutions that increase the folding rate.     

Curvature of the energy landscape and folding of model proteins

We study the geometric properties of the energy landscape of coarse-grained, off-lattice models of polymers by endowing the configuration space with a suitable metric, depending on the potential energy function, such that the dynamical trajectories are the geodesics of the metric. Using numerical simulations, we show that the fluctuations of the curvature clearly mark the folding transition, and that this quantity allows to distinguish between polymers having a proteinlike behavior (i.e., that fold to a unique configuration) and polymers which undergo a hydrophobic collapse but do not have a folding transition. These geometrical properties are defined by the potential energy without requiring any prior knowledge of the native configuration.     

A torsional potential for graphene derived from fitting to DFT results

We present a simple torsional potential for graphene to accurately describe its out-of-plane deformations. The parameters of the potential are derived through appropriate fitting with suitable DFT calculations regarding the deformation energy of graphene sheets folded around two different folding axes, along an armchair or along a zig-zag direction. Removing the energetic contribution of bending angles, using a previously introduced angle bending potential, we isolate the purely torsional deformation energy, which is then fitted to simple torsional force fields. The presented out-of-plane torsional potential can accurately fit the deformation energy for relatively large torsional angles up to 0.5 rad. To test our proposed potential, we apply it to the problem of the vertical displacement of a single carbon atom out of the graphene plane and compare the obtained deformation energy with corresponding DFT calculations. The dependence of the deformation energy on the vertical displacement of the pulled carbon atom is indistinguishable in these two cases, for displacements up to about 0.5 Å. The presented potential is applicable to other sp² carbon structures.     

Protein structural codes and nucleation sites for protein folding

One of the long-standing controversial arguments in protein folding is Levinthal's paradox. We have recently proposed a new nucleation hypothesis and shown that the nucleation residues are the most conserved sequences in protein. To avoid the complicated effect of tertiary interactions, we limit our search for structural codes to the nucleation residues. Starting with the hypotheses of secondary structure nucleation and conservation of residues important for folding, we have analysed 762 folds classified as unique by SCOP. Segments of 17 residues around the top 20% conserved amino acids are analysed, resulting in approximately 100 clusters each for the main secondary structure classes of helix, sheet and coil. Helical clusters have the longest correlation range, coils the shortest (four residues). Strong specific sequence-structure correlation is observed for coil but not for helix and sheet, suggesting a mapping relationship between the sequence and the structure for coil. We propose that the central sequences in these clusters form `structural codes', a useful basis set for identifying nucleation sites, protein fragments stable in isolation, and secondary structural patterns in proteins (particularly turns and loops).     

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.     

Generating ensemble averages for small proteins from extended conformations by Monte Carlo simulations

Since it is not feasible to determine the structure of every protein by experiment, algorithms delivering the folded conformation of a protein solely from its amino acid sequence are desirable. Here the diffusion-process controlled-Monte Carlo approach has been applied to generating ensemble averages for three small proteins with 31, 36, and 46 residues. Starting from extended conformations and using an energy model that was developed on other protein models, the simulations find nativelike structures deviating by 3 A rms from the experimental structures for the main chain atoms. The balance between long-range and short-range interactions is discussed briefly in the context of stability and folding.