温控分子动力学研究微管蛋白活性肽链的折叠机制

运用温控和常温分子动力学方法, 研究了微管蛋白活性部位Pep1-28肽链的折叠机制, 总模拟时间为380.0 ns. 对于温控分子动力学, 逐渐降温可以清晰显示Pep1-28肽链的折叠途径, 发生明显折叠的温度约为550 K, 其折叠和去折叠可逆机制为U(>1200 K)←→I1(1200-1000 K)←→I2(800 K)←→I3(600 K)←→I4(450 K)←→F1(400 K)←→F2(300 K), 其中U为去折叠态构象, I1、I2、I3和I4是折叠过程中的四个重要的中间态构象, F1和F2是两个结构相近的折叠态构象. 对于常温(300 K)分子动力学, 其构象转变和折叠过程相当迅速, 很难观察到有效、稳定的中间态构象. 尤其引人注意的是, 其折叠态结构陷入了能量局域极小点, 与温控(300 K)的有较大差异, 两者能量差高达297.53 kJ·mol-1. 可见, 温控分子动力学方法不仅清晰地显示多肽和蛋白质折叠过程的重要中间态构象, 为折叠和去折叠机制提供直接、可靠的依据, 而且还有助于跨越较高的构象能垒, 促使多肽和蛋白质折叠以形成全局能量最低的稳定结构.     

An algorithm for the uniform sampling of iso-energy surfaces and for the calculation of microcanonical averages

In this article an algorithm is proposed to efficiently perform the uniform sampling of an iso-energy surface corresponding to a fixed potential energy U of a molecular system, and for calculating averages of certain quantities over microstates having this energy (microcanonical averages). The developed sampling technique is based upon the combination of a recently proposed method for performing constant potential energy molecular dynamics simulations [Rapallo, A. J Chem Phys 2004, 121, 4033] with well-established thermostatting techniques used in the framework of standard molecular dynamics simulations, such as the Andersen thermostat, and the Nose-Hoover chain thermostat. The proposed strategy leads to very accurate and drift-free potential energy conservation during the whole sampling process, and, very important, specially when dealing with high-dimensional or complicated potential functions, it does not require the calculation of the potential energy function hessian. The technique proved to be very reliable for sampling both low- and high-dimensional surfaces.     

Two-dimensional replica exchange approach for peptide-peptide interactions

The replica exchange molecular dynamics (REMD) method has emerged as a standard approach for simulating proteins and peptides with rugged underlying free energy landscapes. We describe an extension to the original methodology--here termed umbrella-sampling REMD (UREMD)--that offers specific advantages in simulating peptide-peptide interactions. This method is based on the use of two dimensions in the replica cascade, one in temperature as in conventional REMD, and one in an umbrella sampling coordinate between the center of mass of the two peptides that aids explicit exploration of the complete association-dissociation reaction coordinate. To mitigate the increased number of replicas required, we pursue an approach in which the temperature and umbrella dimensions are linked at only fully associated and dissociated states. Coupled with the reweighting equations, the UREMD method aids accurate calculations of normalized free energy profiles and structural or energetic measures as a function of interpeptide separation distance. We test the approach on two families of peptides: a series of designed tetrapeptides that serve as minimal models for amyloid fibril formation, and a fragment of a classic leucine zipper peptide and its mutant. The results for these systems are compared to those from conventional REMD simulations, and demonstrate good convergence properties, low statistical errors, and, for the leucine zippers, an ability to sample near-native structures.     

Inherent structure analysis of protein folding

An analysis in terms of the inherent structures (IS, local minima) of the multidimensional potential energy landscape is applied to proteins. Detailed calculations are performed for the 46 bead BLN model, which folds into a four-stranded beta-barrel. Enhanced sampling has allowed determination of 239 199 IS states, believed to encompass nearly all the compact, low-energy states, and of well-averaged thermodynamic quantities at low temperature. The density of states shows distinct lobes for compact and extended states, and entropic barriers for the collapse and local ordering transitions. A two-dimensional scatterplot or density of states clearly shows the multifunnel structure of the energy landscape. The anharmonic vibrational free energy is found to play a crucial role in protein folding. The problem of determining the folding transition in a multifunnel system is discussed, and novel indicators of folding are introduced. A particularly clear picture is obtained through the occupation probabilities, pi, of individual low-lying IS, which become finite below the collapse temperature; it is suggested that poor foldability corresponds to a large "misfolding interval" where the excited state pi>0 exceeds that of the native state p0.     

Free energy landscapes of electron transfer system in dipolar environment below and above the rotational freezing temperature

Electron transfer reaction in a polar solvent is modeled by a solute dipole surrounded by dipolar molecules with simple rotational dynamics posted on the three-dimensional distorted lattice sites. The interaction energy between the solute and solvent dipoles as a reaction coordinate is adopted and free energy landscapes are calculated by generating all possible states for a 26 dipolar system and by employing Wang-Landau sampling algorithm for a 92 dipolar system. For temperatures higher than the energy scale of dipole-dipole interactions, the free energy landscapes for the small reaction coordinate region have quadratic shape as predicted by Marcus [Rev. Mod. Phys. 65, 599 (1993)] whereas for the large reaction coordinate region, the landscapes exhibit a nonquadratic shape. When the temperature drops, small notched structures appear on the free energy profiles because of the frustrated interactions among dipoles. The formation of notched structure is analyzed with statistical approach and it is shown that the amplitude of notched structure depend upon the segment size of the reaction coordinate and is characterized by the interaction energy among the dipoles. Using simulated free energy landscapes, the authors calculate the reaction rates as a function of the energy gap for various temperatures. At high temperature, the reactions rates follow a bell shaped (inverted parabolic) energy gap law in the small energy gap regions, while it becomes steeper than the parabolic shape in a large energy gap regions due to the nonquadratic shape of the free energy landscape. The peak position of parabola also changes as the function of temperature. At low temperature, the profile of the reaction rates is no longer smooth because of the many local minima of the free energy landscape.     

Foldamer dynamics expressed via Markov state models. I. Explicit solvent molecular-dynamics simulations in acetonitrile, chloroform, methanol, and water

In this article, we analyze the folding dynamics of an all-atom model of a polyphenylacetylene (pPA) 12-mer in explicit solvent for four common organic and aqueous solvents: acetonitrile, chloroform, methanol, and water. The solvent quality has a dramatic effect on the time scales in which pPA 12-mers fold. Acetonitrile was found to manifest ideal folding conditions as suggested by optimal folding times on the order of approximately 100-200 ns, depending on temperature. In contrast, chloroform and water were observed to hinder the folding of the pPA 12-mer due to extreme solvation conditions relative to acetonitrile; chloroform denatures the oligomer, whereas water promotes aggregation and traps. The pPA 12-mer in a pure methanol solution folded in approximately 400 ns at 300 K, compared relative to the experimental 12-mer folding time of approximately 160 ns measured in a 1:1 v/v THF/methanol solution. Requisite in drawing the aforementioned conclusions, analysis techniques based on Markov state models are applied to multiple short independent trajectories to extrapolate the long-time scale dynamics of the 12-mer in each respective solvent. We review the theory of Markov chains and derive a method to impose detailed balance on a transition-probability matrix computed from simulation data.     

Molecular dynamics coupled with a virtual system for effective conformational sampling

An enhanced conformational sampling method is proposed: virtual‐system coupled canonical molecular dynamics (VcMD). Although VcMD enhances sampling along a reaction coordinate, this method is free from estimation of a canonical distribution function along the reaction coordinate. This method introduces a virtual system that does not necessarily obey a physical law. To enhance sampling the virtual system couples with a molecular system to be studied. Resultant snapshots produce a canonical ensemble. This method was applied to a system consisting of two short peptides in an explicit solvent. Conventional molecular dynamics simulation, which is ten times longer than VcMD, was performed along with adaptive umbrella sampling. Free‐energy landscapes computed from the three simulations mutually converged well. The VcMD provided quicker association/dissociation motions of peptides than the conventional molecular dynamics did. The VcMD method is applicable to various complicated systems because of its methodological simplicity. © 2018 Wiley Periodicals, Inc.     

Nonempirical statistical theory for molecular evaporation from nonrigid clusters

We propose a nonempirical statistical theory to give the reaction rate and the kinetic energy distribution of fragments for molecular evaporation from highly nonrigid atomic and van der Waals clusters. To quantify the theory, an efficient and accurate method to evaluate the absolute value of classical density of states (the Thomas-Fermi density in phase space) and the flux at the so-called dividing surface is critically important, and we have devised such an efficient method. The theory and associated methods are verified by numerical comparison with the corresponding molecular dynamics simulation through the study of Ar(2) evaporation from Ar(8) cluster, in which evaporation is strongly coupled with structural isomerization dynamics. It turns out that the nonempirical statistical theory gives quite an accurate reaction rate. We also study the kinetic energy release (KER) arising from these evaporations and its Boltzmann-like distribution both for atomic and diatomic evaporations. This provides a general relation between the KER and temperature of the fragments.     

Calibration of force-field dependency in free energy landscapes of peptide conformations by quantum chemical calculations

The free energy landscapes of peptide conformations were calibrated by ab initio quantum chemical calculations, after the enhanced conformational diversity search using the multicanonical molecular dynamics simulations. Three different potentials of mean force for an isolated dipeptide were individually obtained by the multicanonical molecular dynamics simulations using the conventional force fields, AMBER parm94, AMBER parm96, and CHARMm22. Each potential of mean force was then calibrated based upon the umbrella sampling algorithm from the adiabatic energy map that was calculated separately by the ab initio molecular orbital method, and all of the calibrated potentials of mean force coincided well. The calibration method was also applied to the simulations of a peptide dimer in explicit water models, and it was shown that the calibrated free energy landscapes did not depend on the force field used in the classical simulations, as far as the conformational space was sampled well. The current calibration method fuses the classical free energy calculation with the quantum chemical calculation, and it should generally make simulations for biomolecular systems much more reliable when combining with enhanced conformational sampling.     

Length dependent folding kinetics of phenylacetylene oligomers: structural characterization of a kinetic trap

Using simulation to study the folding kinetics of 20-mer poly-phenylacetylene (pPA) oligomers, we find a long time scale trapped kinetic phase in the cumulative folding time distribution. This is demonstrated using molecular dynamics to simulate an ensemble of over 100 folding trajectories. The simulation data are fit to a four-state kinetic model which includes the typical folded and unfolded states, along with an intermediate state, and most surprisingly, a kinetically trapped state. Topologically diverse conformations reminiscent of alpha helices, beta turns, and sheets in proteins are observed, along with unique structures in the form of knots. The nonhelical conformations are implicated, on the basis of structural correlations to kinetic parameters, to contribute to the trapped kinetic behavior. The strong solvophobic forces which mediate the folding process and produce a stable helical folded state also serve to overstabilize the nonhelical conformations, ultimately trapping them. From our simulations, the folding time is predicted to be on the order of 2.5-12.5 mus in the presence of the trapped kinetic phase. The folding mechanism for these 20-mer chains is compared with the previously reported folding mechanism for the pPA 12-mer chains. A linear scaling relationship between the chain length and the mean first passage time is predicted in the absence of the trapped kinetic phase. We discuss the major implications of this discovery in the design of self-assembling nanostructures.     

Diffusive dynamics on multidimensional rough free energy surfaces

The dynamics of processes relevant to chemistry and biophysics on rough free energy landscapes is investigated using a recently developed algorithm to solve the Smoluchowski equation. Two different processes are considered: ligand rebinding in MbCO and protein folding. For the rebinding dynamics of carbon monoxide (CO) to native myoglobin (Mb) from locations around the active site, the two-dimensional free energy surface (FES) is constructed using extensive molecular dynamics simulations. The surface describes the minima in the A state (bound MbCO), CO in the distal pocket and in the Xe4 pocket, and the transitions between these states and allows to study the diffusion of CO in detail. For the folding dynamics of protein G, a previously determined two-dimensional FES was available. To follow the diffusive dynamics on these rough free energy surfaces, the Smoluchowski equation is solved using the recently developed hierarchical discrete approximation method. From the relaxation of the initial nonequilibrium distribution, experimentally accessible quantities such as the rebinding time for CO or the folding time for protein G can be calculated. It is found that the free energy barrier for CO in the Xe4 pocket and in the distal pocket (B state) closer to the heme iron is approximately 6 kcal/mol which is considerably larger than the inner barrier which separates the bound state and the B state. For the folding of protein G, a barrier of approximately 10 kcal/mol between the unfolded and the folded state is consistent with folding times of the order of milliseconds.     

Application of torsion angle molecular dynamics for efficient sampling of protein conformations

We investigate the application of torsion angle molecular dynamics (TAMD) to augment conformational sampling of peptides and proteins. Interesting conformational changes in proteins mainly involve torsional degrees of freedom. Carrying out molecular dynamics in torsion space does not only explicitly sample the most relevant degrees of freedom, but also allows larger integration time steps with elimination of the bond and angle degrees of freedom. However, the covalent geometry needs to be fixed during internal coordinate dynamics, which can introduce severe distortions to the underlying potential surface in the extensively parameterized modern Cartesian-based protein force fields. A "projection" approach (Katritch et al. J Comput Chem 2003, 24, 254-265) is extended to construct an accurate internal coordinate force field (ICFF) from a source Cartesian force field. Torsion crossterm corrections constructed from local molecular fragments, together with softened van der Waals and electrostatic interactions, are used to recover the potential surface and incorporate implicit bond and angle flexibility. MD simulations of dipeptide models demonstrate that full flexibility in both the backbone phi/psi and side chain chi1 angles are virtually restored. The efficacy of TAMD in enhancing conformational sampling is then further examined by folding simulations of small peptides and refinement experiments of protein NMR structures. The results show that an increase of several fold in conformational sampling efficiency can be reliably achieved. The current study also reveals some complicated intrinsic properties of internal coordinate dynamics, beyond energy conservation, that can limit the maximum size of the integration time step and thus the achievable gain in sampling efficiency.     

Advances in Enhanced Sampling Molecular Dynamics Simulations for Biomolecules

Molecular dynamics simulation has emerged as a powerful computational tool for studying biomolecules as it can provide atomic insights into the conformational transitions involved in biological functions. However, when applied to complex biological macromolecules, the conformational sampling ability of conventional molecular dynamics is limited by the rugged free energy landscapes, leading to inherent timescale gaps between molecular dynamics simulations and real biological processes. To address this issue, several advanced enhanced sampling methods have been proposed to improve the sampling efficiency in molecular dynamics. In this review, the theoretical basis, practical applications, and recent improvements of both constraint and unconstrained enhanced sampling methods are summarized. Furthermore, the combined utilizations of different enhanced sampling methods that take advantage of both approaches are also briefly discussed.     

Helix formation in a pentapeptide: experiment and force-field dependent dynamics

We used a combined approach of experiment and simulation to determine the helical population and folding pathway of a small helix forming blocked pentapeptide, Ac-(Ala)(5)-NH(2). Experimental structural characterization of this blocked peptide was carried out with far UV circular dichroism spectroscopy, FTIR, and NMR measurements. These measurements confirm the presence of the α-helical state in a buffer solution. Direct molecular dynamics and replica-exchange simulations of the pentapeptide were performed using several popular force fields with explicit solvent. The simulations yielded statistically reliable estimates of helix populations, melting curves, folding, and nucleation times. The distributions of conformer populations are used to measure folding cooperativity. Finally, a statistical analysis of the sample of helix-coil transition paths was performed. The details of the calculated helix populations, folding kinetics and pathways vary with the employed force field. Interestingly, the helix populations, folding, and unfolding times obtained from most of the studied force fields are in qualitative agreement with each other and with available experimental data, with the deviations corresponding to several kcal/mol in energy at 300 K. Most of the force fields also predict qualitatively similar transition paths, with unfolding initiated at the C-terminus. Accuracy of potential energy parameters, rather than conformational sampling may be the limiting factor in current molecular simulations.     

Convergence and sampling efficiency in replica exchange simulations of peptide folding in explicit solvent

Replica exchange methods (REMs) are increasingly used to improve sampling in molecular dynamics (MD) simulations of biomolecular systems. However, despite having been shown to be very effective on model systems, the application of REM in complex systems such as for the simulation of protein and peptide folding in explicit solvent has not been objectively tested in detail. Here we present a comparison of conventional MD and temperature replica exchange MD (T-REMD) simulations of a beta-heptapeptide in explicit solvent. This system has previously been shown to undergo reversible folding on the time scales accessible to MD simulation and thus allows a direct one-to-one comparison of efficiency. The primary properties compared are the free energy of folding and the relative populations of different conformers as a function of temperature. It is found that to achieve a similar degree of precision T-REMD simulations starting from a random set of initial configurations were approximately an order of magnitude more computationally efficient than a single 800 ns conventional MD simulation for this system at the lowest temperature investigated (275 K). However, whereas it was found that T-REMD simulations are more than four times more efficient than multiple independent MD simulations at one temperature (300 K) the actual increase in conformation sampling was only twofold. The overall gain in efficiency using REMD resulted primarily from the ordering of different conformational states over temperature, as opposed to a large increase of conformational sampling. It is also shown that in this system exchanges are accepted primarily based on (random) fluctuations within the solvent and are not strongly correlated with the instantaneous peptide conformation raising questions in regard to the efficiency of T-REMD in larger systems.     

An improved replica-exchange sampling method: temperature intervals with global energy reassignment

In a molecular dynamics (MD) simulation, representative sampling over the entire phase space is desired to obtain an accurate canonical distribution at a given temperature. For large molecules, such as proteins, this is problematic because systems tend to become trapped in local energy minima. The extensively used replica-exchange molecular dynamics (REMD) simulation technique overcomes this kinetic-trapping problem by allowing Boltzmann-weighted configuration exchange processes to occur between numerous thermally adjacent and compositionally identical simulations that are thermostated at sequentially higher temperatures. While the REMD method provides much better sampling than conventional MD, there are two substantial difficulties that are inherent in its application: (1) the large number of replicas that must be used to span a designated temperature range and (2) the subsequent long time required for configurations sampled at high temperatures to exchange down for potential inclusion within the low-temperature ensemble of interest. In this work, a new method based on temperature intervals with global energy reassignment (TIGER) is presented that overcomes both of these problems. A TIGER simulation is conducted as a series of short heating-sampling-quenching cycles. At the end of each cycle, the potential energies of all replicas are simultaneously compared at the same temperature using a Metropolis sampling method and then globally reassigned to the designated temperature levels. TIGER is compared with regular MD and REMD methods for the alanine dipeptide in water. The results indicate that TIGER increases sampling efficiency while substantially reducing the number of central processing units required for a comparable conventional REMD simulation.     

Relaxation of caloric curves on complex potential energy surfaces

Time-dependent caloric curves of model systems with complex energy landscapes are calculated by solving master equation kinetics in stepwise heating or cooling protocols. By considering in detail a simple two-state harmonic model, we show that both the transition temperature and the associated latent heat vary significantly if the sampling time is not long enough. Microcanonical characteristics, including possible S-bends in the caloric curve, are also qualitatively affected by insufficient sampling. The geometry of S-bends as a function of the observation time agrees quantitatively with the predictions of catastrophe theory. For two Lennard-Jones clusters with 13 and 31 atoms the relations between the transition temperatures and the sampling time are shown to follow scaling laws, in agreement with the results of molecular dynamics simulations [J. Chem. Phys. 113, 1315 (2000)].