Statistical temperature molecular dynamics: application to coarse-grained beta-barrel-forming protein models

Recently the authors proposed a novel sampling algorithm, "statistical temperature molecular dynamics" (STMD) [J. Kim et al., Phys. Rev. Lett. 97, 050601 (2006)], which combines ingredients of multicanonical molecular dynamics and Wang-Landau sampling. Exploiting the relation between the statistical temperature and the density of states, STMD generates a flat energy distribution and efficient sampling with a dynamic update of the statistical temperature, transforming an initial constant estimate to the true statistical temperature T(U), with U being the potential energy. Here, the performance of STMD is examined in the Lennard-Jones fluid with diverse simulation conditions, and in the coarse-grained, off-lattice BLN 46-mer and 69-mer protein models, exhibiting rugged potential energy landscapes with a high degree of frustration. STMD simulations combined with inherent structure (IS) analysis allow an accurate determination of protein thermodynamics down to very low temperatures, overcoming quasiergodicity, and illuminate the transitions occurring in folding in terms of the energy landscape. It is found that a thermodynamic signature of folding is significantly suppressed by accurate sampling, due to an incoherent contribution from low-lying non-native IS in multifunneled landscapes. It is also shown that preferred accessibility to such IS during the collapse transition is intimately related to misfolding or poor foldability.     

Folding of human telomerase RNA pseudoknot using ion-jump and temperature-quench simulations

Globally RNA folding occurs in multiple stages involving chain compaction and subsequent rearrangement by a number of parallel routes to the folded state. However, the sequence-dependent details of the folding pathways and the link between collapse and folding are poorly understood. To obtain a comprehensive picture of the thermodynamics and folding kinetics we used molecular simulations of coarse-grained model of a pseudoknot found in the conserved core domain of the human telomerase (hTR) by varying both temperature (T) and ion concentration (C). The phase diagram in the [T,C] plane shows that the boundary separating the folded and unfolded state for the finite 47-nucleotide system is relatively sharp, implying that from a thermodynamic perspective hTR behaves as an apparent two-state system. However, the folding kinetics following single C-jump or T-quench is complicated, involving multiple channels to the native state. Although globally folding kinetics triggered by T-quench and C-jump are similar, the kinetics of chain compaction are vastly different, which reflects the role of initial conditions in directing folding and collapse. Remarkably, even after substantial reduction in the overall size of hTR, the ensemble of compact conformations are far from being nativelike, suggesting that the search for the folded state occurs among the ensemble of low-energy fluidlike globules. The rate of unfolding, which occurs in a single step, is faster upon C-decrease compared to a jump in temperature. To identify "hidden" states that are visited during the folding process we performed simulations by periodically interrupting the approach to the folded state by lowering C. These simulations show that hTR reaches the folded state through a small number of connected clusters that are repeatedly visited during the pulse sequence in which the folding or unfolding is interrupted. The results from interrupted folding simulations, which are in accord with non-equilibrium single-molecule folding of a large ribozyme, show that multiple probes are needed to reveal the invisible states that are sampled by RNA as it folds. Although we have illustrated the complexity of RNA folding using hTR as a case study, general arguments and qualitative comparisons to time-resolved scattering experiments on Azoarcus group I ribozyme and single-molecule non-equilibrium periodic ion-jump experiments establish the generality of our findings.     

Sampling of states for estimating the folding funnel entropy and energy landscape of a model alpha-helical hairpin peptide

Protein folding times are many orders of magnitude shorter than would occur if the peptide chain randomly sampled possible configurations, which implies that protein folding is a directed process. The detailed shape of protein's energy landscape determines the rate and reliability of folding to the native state, but the large number of structural degrees of freedom generates an energy landscape that is hard to visualize because of its high dimensionality. A commonly used picture is that of an energy funnel leading from high energy random coil state down to the low energy native state. As lattice computer models of protein dynamics become more realistic, the number of possible configurations becomes too large to count directly. Statistical mechanic and thermodynamic approaches allow us to count states in an approximate manner to quantify the entropy and energy of the energy landscape within a folding funnel for an alpha-helical protein. We also discuss the problems that arise in attempting to count the huge number of individual states of the random coil at the top of the funnel.     

Amyloid Polymorphism in the Protein Folding and Aggregation Energy Landscape

Protein folding involves a large number of steps and conformations in which the folding protein samples different thermodynamic states characterized by local minima. Kinetically trapped on‐ or off‐pathway intermediates are metastable folding intermediates towards the lowest absolute energy minima, which have been postulated to be the natively folded state where intramolecular interactions dominate, and the amyloid state where intermolecular interactions dominate. However, this view largely neglects the rich polymorphism found within amyloid species. We review the protein folding energy landscape in view of recent findings identifying specific transition routes among different amyloid polymorphs. Observed transitions such as twisted ribbon→crystal or helical ribbon→nanotube, and forbidden transitions such helical ribbon?crystal, are discussed and positioned within the protein folding and aggregation energy landscape. Finally, amyloid crystals are identified as the ground state of the protein folding and aggregation energy landscape.     

Temperature dependent free energy surface of polymer folding from equilibrium and quench studies

Langevin dynamics simulation studies have been employed to calculate the temperature dependent free energy surface and folding characteristics of a 500 monomer long linear alkane (polyethylene) chain with a realistic interaction potential. Both equilibrium and temperature quench simulation studies have been carried out. Using the shape anisotropy parameter (S) of the folded molecule as the order parameter, we find a weakly first order phase transition between the high-temperature molten globule and low-temperature rodlike crystalline states separated by a small barrier of the order of k(B)T. Near the melting temperature (580 K), we observe an intriguing intermittent fluctuation with pronounced "1/f noise characteristics" between these two states with large difference in shape and structure. We have also studied the possibilities of different pathways of folding to states much below the melting point. At 300 K starting from the all-trans linear configuration, the chain folds stepwise into a very regular fourfold crystallite with very high shape anisotropy. Whereas, when quenched from a high temperature (900 K) random coil regime, we identify a two step transition from the random coiled state to a molten globulelike state and, further, to a anisotropic rodlike state. The trajectory reveals an interesting coupling between the two order parameters, namely, radius of gyration (R(g)) and the shape anisotropy parameter (S). The rodlike final state of the quench trajectory is characterized by lower shape anisotropy parameter and significantly larger number of gauche defects as compared to the final state obtained through equilibrium simulation starting from all-trans linear chain. The quench study shows indication of a nucleationlike pathway from the molten globule to the rodlike state involving an underlying rugged energy landscape.     

Canonical and micro-canonical analysis of folding of trpzip2: An all-atom replica exchange Monte Carlo simulation study

The density of states of trpzip2, a β-hairpin peptide, has been explored at all-atom level. Replica exchange Monte Carlo method was used for sufficient sampling over a wide range of temperature. Micro-canonical analysis was performed to confirm that the phase transition behavior of this two-state folder is first-order-like. Canonical analysis of heat capacity suggests that hydrogen bonding interaction exerts a considerable positive influence on folding cooperativity, in contrast, hydrophobic interaction is insufficient for high degree of folding cooperativity. Furthermore, we explain physical nature of the folding process from free energy landscape perspective and extensively analyse hydrogen bonding and stacking energy.     

A study of density of states and ground states in hydrophobic-hydrophilic protein folding models by equi-energy sampling

We propose an equi-energy (EE) sampling approach to study protein folding in the two-dimensional hydrophobic-hydrophilic (HP) lattice model. This approach enables efficient exploration of the global energy landscape and provides accurate estimates of the density of states, which then allows us to conduct a detailed study of the thermodynamics of HP protein folding, in particular, on the temperature dependence of the transition from folding to unfolding and on how sequence composition affects this phenomenon. With no extra cost, this approach also provides estimates on global energy minima and ground states. Without using any prior structural information of the protein the EE sampler is able to find the ground states that match the best known results in most benchmark cases. The numerical results demonstrate it as a powerful method to study lattice protein folding models.     

Charge-state dependent compaction and dissociation of protein complexes: insights from ion mobility and molecular dynamics

Collapse to compact states in the gas phase, with smaller collision cross sections than calculated for their native-like structure, has been reported previously for some protein complexes although not rationalized. Here we combine experimental and theoretical studies to investigate the gas-phase structures of four multimeric protein complexes during collisional activation. Importantly, using ion mobility-mass spectrometry (IM-MS), we find that all four macromolecular complexes retain their native-like topologies at low energy. Upon increasing the collision energy, two of the four complexes adopt a more compact state. This collapse was most noticeable for pentameric serum amyloid P (SAP) which contains a large central cavity. The extent of collapse was found to be highly correlated with charge state, with the surprising observation that the lowest charge states were those which experience the greatest degree of compaction. We compared these experimental results with in vacuo molecular dynamics (MD) simulations of SAP, during which the temperature was increased. Simulations showed that low charge states of SAP exhibited compact states, corresponding to collapse of the ring, while intermediate and high charge states unfolded to more extended structures, maintaining their ring-like topology, as observed experimentally. To simulate the collision-induced dissociation (CID) of different charge states of SAP, we used MS to measure the charge state of the ejected monomer and assigned this charge to one subunit, distributing the residual charges evenly among the remaining four subunits. Under these conditions, MD simulations captured the unfolding and ejection of a single subunit for intermediate charge states of SAP. The highest charge states recapitulated the ejection of compact monomers and dimers, which we observed in CID experiments of high charge states of SAP, accessed by supercharging. This strong correlation between theory and experiment has implications for further studies as well as for understanding the process of CID and for applications to gas-phase structural biology more generally.     

二维HP格点模型中的紧密高分子链构象及其热力学性质的研究

采用二维HP模型用精确计数法和MonteCarlo方法研究了链长为N(≤ 2 2 )的紧密高分子链的构象和热力学性质 .发现不同HP序列的紧密高分子链的平均自由能和平均配分函数与链长N存在关系 :〈F〉=aN+b ,　ln〈Z〉=a′N +b′ .同时发现对于可折叠成基态且简并度为 1的紧密高分子链 ,其平均自由能和平均配分函数与链长N也存在相似的关系 .在HP模型中对于链长为N的紧密高分子链 ,存在着 2 N + 1 个不同的HP序列 .我们发现可以折叠成基态且简并度为 1的蛋白质分子的HP序列数目NS 为NS =a× 2 N+ 1 　 (a =0 0 2 5 ) ,对应的HP序列中 ,疏水基团 (H)数目的含量为 4 0 %～ 6 0 %的序列出现的几率最大 .同时在这些紧密高分子链中有些具有相同的结构 ,发现结构的‘简并度’为 3 3～ 4 0 (10≤N≤ 16 ) .在紧密高分子链折叠过程中 ,折叠的初期能量下降比较快 ,折叠的中期能量下降比较缓慢 ,折叠的后期能量下降也是比较快     

Electronic structure, X-ray absorption, and optical spectroscopy of LaCoO(3) in the ground-state and excited-states

We present the magnetic and optical properties of various combinations of ordered spin state configurations between low-spin (LS) state, intermediate-spin (IS) state, and high-spin (HS) state of LaCoO(3) . In this study, we use the state-of-the-art first principles calculations based on generalized gradient (GGA) + Hubbard U approach. The excited-state properties of different spin configurations of LaCoO(3) such as the X-ray absorption spectra, optical conductivity, reflectivity, and electron energy loss are calculated. We have demonstrated that the optical spectra results can be used for analyzing the spin state of Co(3+) ion. The first specie is the local excitation of IS cobalt ions in the LS ground state. The second excitation leads to the stabilization of the mixed IS/HS Co(3+) metallic state. At low temperature, the comparison between O 2p and Co 3d projected density of states with the experimental valence band spectra indicates significant IS Co(3+) ions and this is in sharp contrast to the HS state which is negligible. The line shape of O 2s and Co 3d core level spectra are well reproduced in this study. The present results are in excellent agreement with the available experimental data. The variation in the spectra of different configurations of LaCoO(3) suggests a changing in the spin state as the temperature is enhanced from 90 to 500 K.     

Ubiquitin: a small protein folding paradigm

For the past twenty years, the small, 76-residue protein ubiquitin has been used as a model system to study protein structure, stability, folding and dynamics. In this time, ubiquitin has become a paradigm for both the experimental and computational folding communities. The folding energy landscape is now uniquely characterised with a plethora of information available on not only the native and denatured states, but partially structured states, alternatively folded states and locally unfolded states, in addition to the transition state ensemble. This Perspective focuses on the experimental characterisation of ubiquitin using a comprehensive range of biophysical techniques.     

聚乙烯单链松驰过程的分子模拟

利用分子模拟方法研究了不同链长聚乙烯单链折叠过程和相关的松弛和坍塌机理．发现在链长短于１０００ＣＨ２单元时,聚乙烯的链段主要通过整体塌缩机理进行折叠和取向;而链长超过１０００ＣＨ２单元时,可以明显地观察到局部取向团簇的形成,聚乙烯单链通过局部塌缩机理进行折叠和取向．通过对各阶段团簇数目,体系取向链段长度的表征,说明体系在模拟时间范围内表现了很强的松驰特性．     

高分子熔体结晶的两维计算机模拟

高分子熔体结晶的两维计算机模拟胡文兵，于同隐，卜海山（复旦大学高分子科学系，复旦大学材料科学系，上海，２００４３３）关键词计算机模拟，有序相变，高分子１９５６年Ｆｌｏｒｙ［１］从平均场近似的格子模型证明：高分子链的非柔顺性会导致长链的完全有序排列．此...     

A lattice protein with an amyloidogenic latent state: stability and folding kinetics

We have designed a model lattice protein that has two stable folded states, the lower free energy native state and a latent state of somewhat higher energy. The two states have a sizable part of their structures in common (two "alpha-helices") and differ in the content of "alpha-helices" and "beta-strands" in the rest of their structures; i.e. for the native state, this part is alpha-helical, and for the latent state it is composed of beta-strands. Thus, the lattice protein free energy surface mimics that of amyloidogenic proteins that form well organized fibrils under appropriate conditions. A Go-like potential was used and the folding process was simulated with a Monte Carlo method. To gain insight into the equilibrium free energy surface and the folding kinetics, we have combined standard approaches (reduced free energy surfaces, contact maps, time-dependent populations of the characteristic states, and folding time distributions) with a new approach. The latter is based on a principal coordinate analysis of the entire set of contacts, which makes possible the introduction of unbiased reaction coordinates and the construction of a kinetic network for the folding process. The system is found to have four characteristic basins, namely a semicompact globule, an on-pathway intermediate (the bifurcation basin), and the native and latent states. The bifurcation basin is shallow and consists of the structure common to the native and latent states, with the rest disorganized. On the basis of the simulation results, a simple kinetic model describing the transitions between the characteristic states was developed, and the rate constants for the essential transitions were estimated. During the folding process the system dwells in the bifurcation basin for a relatively short time before it proceeds to the native or latent state. We suggest that such a bifurcation may occur generally for proteins in which native and latent states have a sizable part of their structures in common. Moreover, there is the possibility of introducing changes in the system (e.g., mutations), which guide the system toward the native or misfolded state.     

Gaussian excitations model for glass-former dynamics and thermodynamics

We describe a model for the thermodynamics and dynamics of glass-forming liquids in terms of excitations from an ideal glass state to a Gaussian manifold of configurationally excited states. The quantitative fit of this three parameter model to the experimental data on excess entropy and heat capacity shows that "fragile" behavior, indicated by a sharply rising excess heat capacity as the glass transition is approached from above, occurs in anticipation of a first-order transition--usually hidden below the glass transition--to a "strong" liquid state of low excess entropy. The distinction between fragile and strong behavior of glass formers is traced back to an order of magnitude difference in the Gaussian width of their excitation energies. Simple relations connect the excess heat capacity to the Gaussian width parameter, and the liquid-liquid transition temperature, and strong, testable, predictions concerning the distinct properties of energy landscape for fragile liquids are made. The dynamic model relates relaxation to a hierarchical sequence of excitation events each involving the probability of accumulating sufficient kinetic energy on a separate excitable unit. Super-Arrhenius behavior of the relaxation rates, and the known correlation of kinetic with thermodynamic fragility, both follow from the way the rugged landscape induces fluctuations in the partitioning of energy between vibrational and configurational manifolds. A relation is derived in which the configurational heat capacity, rather than the configurational entropy of the Adam-Gibbs equation, controls the temperature dependence of the relaxation times, and this gives a comparable account of the experimental observations without postulating a divergent length scale. The familiar coincidence of zero mobility and Kauzmann temperatures is obtained as an approximate extrapolation of the theoretical equations. The comparison of the fits to excess thermodynamic properties of laboratory glass formers, and to configurational thermodynamics from simulations, reveals that the major portion of the excitation entropy responsible for fragile behavior resides in the low-frequency vibrational density of states. The thermodynamic transition predicted for fragile liquids emerges from beneath the glass transition in case of laboratory water and the unusual heat capacity behavior observed for this much studied liquid can be closely reproduced by the model.     

The protein folding network indicates that the ultrafast folding mutant of villin headpiece subdomain has a deeper folding funnel

Protein folding is a dynamic process with continuous transitions among different conformations. In this work, the dynamics in the protein folding network of villin headpiece subdomain (HP35) has been investigated based on multiple reversible folding trajectories of HP35 and its ultrafast folding mutant where sub-angstrom folding was achieved. The four folding states were clearly separated on the network, validating the classification of the states. Examination of the eight conformers with different formation of the individual helices revealed high plasticity of the three helices in all the four states. A consistent feature between the wild type and mutant protein is the dominant conformer 111 (all three helices formed) in the folded state and conformers 111 and 011 (helices II and III formed) in the major intermediate state, indicating the critical role of helices II and III in the folding mechanism. When compared to the wild type, the folding landscape of the ultrafast folding mutant exhibited a deeper folding funnel towards the folded state. The very beginning of the folding (0-10 ns) was very similar for both protein variants but it soon diverged and displayed different folding pathways. Although going through the major intermediate state is the dominant pathway for both, it was also observed that some folding went through the minor intermediate state for the mutant. The intriguing difference resulting from the mutation at two residues in helix III has been carefully analyzed and discussed in details.     

Enhanced Wang Landau sampling of adsorbed protein conformations

Using computer simulations to model the folding of proteins into their native states is computationally expensive due to the extraordinarily low degeneracy of the ground state. In this paper, we develop an efficient way to sample these folded conformations using Wang Landau sampling coupled with the configurational bias method (which uses an unphysical "temperature" that lies between the collapse and folding transition temperatures of the protein). This method speeds up the folding process by roughly an order of magnitude over existing algorithms for the sequences studied. We apply this method to study the adsorption of intrinsically disordered hydrophobic polar protein fragments on a hydrophobic surface. We find that these fragments, which are unstructured in the bulk, acquire secondary structure upon adsorption onto a strong hydrophobic surface. Apparently, the presence of a hydrophobic surface allows these random coil fragments to fold by providing hydrophobic contacts that were lost in protein fragmentation.     

Do Charge State Signatures Guarantee Protein Conformations?

The extent to which proteins in the gas phase retain their condensed-phase structure is a hotly debated issue. Closely related to this is the degree to which the observed charge state reflects protein conformation. Evidence from electron capture dissociation, hydrogen/deuterium exchange, ion mobility, and molecular dynamics shows clearly that there is often a strong correlation between the degree of folding and charge state, with the most compact conformations observed for the lowest charge states. In this article, we address recent controversies surrounding the relationship between charge states and folding, focussing also on the manipulation of charge in solution and its effect on conformation. 'Supercharging' reagents that have been used to effect change in charge state can promote unfolding in the electrospray droplet. However for several protein complexes, supercharging does not appear to perturb the structure in that unfolding is not detected. Consequently, a higher charge state does not necessarily imply unfolding. Whilst the effect of charge manipulation on conformation remains controversial, there is strong evidence that a folded, compact state of a protein can survive in the gas phase, at least on a millisecond timescale. The exact nature of the side-chain packing and secondary structural elements in these compact states, however, remains elusive and prompts further research.     

Exploring the energy landscape of a small RNA hairpin

The energy landscape of a small RNA tetraloop hairpin is explored by temperature jump kinetics and base-substitution. The folding kinetics are single-exponential near the folding transition midpoint T(m). An additional fast phase appears below the midpoint, and an additional slow phase appears above the midpoint. Stem mutation affects the high-temperature phase, while loop mutation affects the low-temperature phase. An adjusted 2-D lattice model reproduces the temperature-dependent phases, although it oversimplifies the structural interpretation. A four-state free energy landscape model is generated based on the lattice model. This model explains the thermodynamics and multiphase kinetics over the full temperature range of the experiments. An analysis of three variants shows that one of the intermediate RNA structures is a stacking-related trap affected by stem but not loop modification, while the other is an early intermediate that forms some stem and loop structure. Even a very fast-folding 8-mer RNA with an ideal tetraloop sequence has a rugged energy landscape, ideal for testing analytical and computational models.