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.     

Foldamer simulations: novel computational methods and applications to poly-phenylacetylene oligomers

We apply several methods to probe the ensemble kinetic and structural properties of a model system of poly-phenylacetylene (pPA) oligomer folding trajectories. The kinetic methods employed included a brute force accounting of conformations, a Markovian state matrix method, and a nonlinear least squares fit to a minimalist kinetic model used to extract the folding time. Each method gave similar measures for the folding time of the 12-mer chain, calculated to be on the order of 7 ns for the complete folding of the chain from an extended conformation. Utilizing both a linear and a nonlinear scaling relationship between the viscosity and the folding time to correct for a low simulation viscosity, we obtain an upper and a lower bound for the approximate folding time within the range 70 ns相似文献   

Foldamer dynamics expressed via Markov state models. II. State space decomposition

The structural landscape of poly-phenylacetylene (pPA), otherwise known as m-phenylene ethynylene oligomers, has been shown to consist of a very diverse set of conformations, including helices, turns, and knots. Defining a state space decomposition to classify these conformations into easily identifiable states is an important step in understanding the dynamics in relation to Markov state models. We define the state decomposition of pPA oligomers in terms of the sequence of discretized dihedral angles between adjacent phenyl rings along the oligomer backbone. Furthermore, we derive in mathematical detail an approach to further reduce the number of states by grouping symmetrically equivalent states into a single parent state. A more challenging problem requires a formal definition for knotted states in the structural landscape. Assuming that the oligomer chain can only cross the ideal helix path once, we propose a technique to define a knotted state derived from a helical state determined by the position along the helical nucleus where the chain crosses the ideal helix path. Several examples of helical states and knotted states from the pPA 12-mer illustrate the principles outlined in this article.     

Two-state folding observed in individual protein molecules

The folding dynamics of small proteins are often described in terms of a simple two-state kinetic model. Within this notion, the behavior of individual molecules is expected to be stochastic, with a protein molecule residing in either the unfolded or the folded state for extended periods of time, with intermittent rapid jumps across the free energy barrier. However, a direct observation of this bistable behavior has not been made to date. Rather, previous reports of folding trajectories of individual proteins have shown an unexpected degree of complexity. This raises the question whether the simple kinetic properties derived from classical experiments on large ensembles of molecules are reflected in the folding paths taken by individual proteins. Here we report single-molecule folding/unfolding trajectories observed by fluorescence resonance energy transfer for a protein that meets all criteria of a two state-system. The trajectories, measured on molecules immobilized in lipid vesicles, demonstrate the anticipated bistable behavior, with steplike transitions between folded and unfolded conformations. They further allow us to put an upper bound on the barrier crossing time.     

Simulation of chaperonin effect on protein folding: a shift from nucleation-condensation to framework mechanism

The iterative annealing mechanism (IAM) of chaperonin-assisted protein folding is explored in a framework of a well-established coarse-grained protein modeling tool, which enables the study of protein dynamics in a time-scale well beyond classical all-atom molecular mechanics. The chaperonin mechanism of action is simulated for two paradigm systems of protein folding, B domain of protein A (BdpA) and B1 domain of protein G (GB1), and compared to chaperonin-free simulations presented here for BdpA and recently published for GB1. The prediction of the BdpA transition state ensemble (TSE) is in perfect agreement with experimental findings. It is shown that periodic distortion of the polypeptide chains by hydrophobic chaperonin interactions can promote rapid folding and leads to a decrease in folding temperature. It is also demonstrated how chaperonin action prevents kinetically trapped conformations and modulates the observed folding mechanisms from nucleation-condensation to a more framework-like.     

Helical Folding of Conjugated Oligo(phenyleneethynylene): Chain‐Length Dependence,Solvent Effects,and Intermolecular Assembly

As a representative folding system that features a conjugated backbone, a series of monodispersed (o‐phenyleneethynylene)‐alt‐(p‐phenyleneethynylene) (PE) oligomers of varied chain length and different side chains were studied. Molecules with the same backbone but different side‐chain structures were shown to exhibit similar helical conformations in respectively suitable solvents. Specifically, oligomers with dodecyloxy side chains folded into the helical structure in apolar aliphatic solvents, whereas an analogous oligomer with tri(ethylene glycol) (Tg) side chains adopted the same conformation in polar solvents. The fact that the oligomers with the same backbone manifested a similar folded conformation independent of side chains and the nature of the solvent confirmed the concept that the driving force for folding was the intramolecular aromatic stacking and solvophobic interactions. Although all were capable of inducing folding, different solvents were shown to bestow slightly varied folding stability. The chain‐length dependence study revealed a nonlinear correlation between the folding stability with backbone chain length. A critical size of approximately 10 PE units was identified for the system, beyond which folding occurred. This observation corroborated the helical nature of the folded structure. Remarkably, based on the absorption and emission spectra, the effective conjugation length of the system extended more effectively under the folded state than under random conformations. Moreover, as evidenced by the optical spectra and dynamic light‐scattering studies, intermolecular association took place among the helical oligomers with Tg side chains in aqueous solution. The demonstrated ability of such a conjugated foldamer in self‐assembling into hierarchical supramolecular structures promises application potential for the system.     

Molecular simulation of polymer assisted protein refolding

Protein refolding in vitro, the formation of the tertiary structure that enables the protein to display its biological function, can be significantly enhanced by adding a polymer of an appropriate hydrophobicity and concentration into the refolding buffer. A molecular simulation of the refolding of a two-dimensional simple lattice protein was presented. A protein folding map recording the occurrence frequency of specified conformations was derived, from which the refolding thermodynamics and kinetics were interpreted. It is shown that, in the absence of polymer, the protein falls into the "energy trapped" conformations characterized by a high intramolecular hydrophobic interaction, denoted as HH contact, and a high magnitude of the structure overlap function, chi. This makes it difficult for the protein to fold to the native state. The polymer with a suitable chain length, concentration, and hydrophobicity has formed complex with partially folded protein and created diversified intermediates with low chi. This gives more pathways for the protein to fold to the native state. At a given hydrophobicity, the short chain polymer has a broader concentration range where it assists protein folding than those of long chains. The above simulation agrees well with the experimental results reported elsewhere [Cleland et al., J. Biol. Chem. 267, 13327 (1992); ibid., Bio/Technology 10, 1013 (1992); Chen et al., Enzyme Microb. Technol. 32, 120 (2003); Lu et al., Biochem. Eng. J. 24, 55 (2005); ibid., J. Chem. Phys. 122, 134902 (2005); ibid., Biochem. Eng. J. (to be published)] and is of fundamental importance for the design and application of polymers for protein refolding.     

Folding dynamics of Trp-cage in the presence of chemical interference and macromolecular crowding. I

Proteins fold and function in the crowded environment of the cell's interior. In the recent years it has been well established that the so-called "macromolecular crowding" effect enhances the folding stability of proteins by destabilizing their unfolded states for selected proteins. On the other hand, chemical and thermal denaturation is often used in experiments as a tool to destabilize a protein by populating the unfolded states when probing its folding landscape and thermodynamic properties. However, little is known about the complicated effects of these synergistic perturbations acting on the kinetic properties of proteins, particularly when large structural fluctuations, such as protein folding, have been involved. In this study, we have first investigated the folding mechanism of Trp-cage dependent on urea concentration by coarse-grained molecular simulations where the impact of urea is implemented into an energy function of the side chain and/or backbone interactions derived from the all-atomistic molecular dynamics simulations with urea through a Boltzmann inversion method. In urea solution, the folding rates of a model miniprotein Trp-cage decrease and the folded state slightly swells due to a lack of contact formation between side chains at the terminal regions. In addition, the equilibrium m-values of Trp-cage from the computer simulations are in agreement with experimental measurements. We have further investigated the combined effects of urea denaturation and macromolecular crowding on Trp-cage's folding mechanism where crowding agents are modeled as hard-spheres. The enhancement of folding rates of Trp-cage is most pronounced by macromolecular crowding effect when the extended conformations of Trp-cast dominate at high urea concentration. Our study makes quantitatively testable predictions on protein folding dynamics in a complex environment involving both chemical denaturation and macromolecular crowding effects.     

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.     

Exploring protein structure and dynamics under denaturing conditions by single-molecule FRET analysis

Proteins are highly complex biopolymers, exhibiting a substantial degree of structural variability in their properly folded, native state. In the presence of denaturants, this heterogeneity is greatly enhanced, and fluctuations take place among vast numbers of folded and unfolded conformations via many different pathways. To better understand protein folding it is necessary to explore the structural and energetic properties of the folded and unfolded polypeptide chain, as well as the trajectories along which the chain navigates through its multi-dimensional conformational energy landscape. In recent years, single-molecule fluorescence spectroscopy has been established as a powerful tool in this research area, as it allows one to monitor the structure and dynamics of individual polypeptide chains in real time with atomic scale resolution using F?rster resonance energy transfer (FRET). Consequently, time trajectories of folding transitions can be directly observed, including transient intermediates that may exist along these pathways. Here we illustrate the power of single-molecule fluorescence with our recent work on the structure and dynamics of the small enzyme RNase H in the presence of the chemical denaturant guanidinium chloride (GdmCl). For FRET analysis, a pair of fluorescent dyes was attached to the enzyme at specific locations. In order to observe conformational changes of individual protein molecules for up to several hundred seconds, the proteins were immobilized on nanostructured, polymer coated glass surfaces specially developed to have negligible interactions with folded and unfolded proteins. The single-molecule FRET analysis gave insight into structural changes of the unfolded polypeptide chain in response to varying the denaturant concentration, and the time traces revealed stepwise transitions in the FRET levels, reflecting conformational dynamics. Barriers in the free energy landscape of RNase H were estimated from the kinetics of the transitions.     

Helical Folding Competing with Unfolded Aggregation in Phenylene Ethynylene Foldamers

The folding and aggregation behavior of a pair of oligo(phenylene ethynylene) (OPE) foldamers are investigated by means of UV/Vis absorption and circular dichroism spectroscopy. With identical OPE backbones, two foldamers, 1 with alkyl side groups and 2 with triethylene glycol side chains, manifest similar helical conformations in solutions in n‐hexane and methanol, respectively. However, disparate and competing folding and aggregation processes are observed in alternative solvents. In cyclohexane, oligomer 1 initially adopts the helical conformation, but the self‐aggregation of unfolded chains, as a minor component, gradually drives the folding–unfolding transition eventually to the unfolded aggregate state completely. In contrast, in aqueous solution (CH₃OH/H₂O) both folded and unfolded oligomer 2 appear to undergo self‐association; aggregates of the folded chains are thermodynamically more stable. In solutions with a high H₂O content, self‐aggregation among unfolded oligomers is kinetically favored; these oligomers very slowly transform into aggregates of helical structures with greater thermodynamic stability. The folded–unfolded conformational switch thus takes place with the free (nonaggregated) molecules, and the very slow folding transition is due to the low concentration of molecularly dispersed oligomers.     

Topology-based models and NMR structures in protein folding simulations

Topology-based interaction potentials are simplified models that use the native contacts in the folded structure of a protein to define an energetically unfrustrated folding funnel. They have been widely used to analyze the folding transition and pathways of different proteins through computer simulations. Obviously, they need a reliable, experimentally determined folded structure to define the model interactions. In structures elucidated through NMR spectroscopy, a complex treatment of the raw experimental data usually provides a series of models, a set of different conformations compatible with the available experimental data. Here, we use an efficient coarse-grained simulation technique to independently consider the contact maps from every different NMR model in a protein whose structure has been resolved by the use of NMR spectroscopy. For lambda-Cro repressor, a homodimeric protein, we have analyzed its folding characteristics with a topology-based model. We have focused on the competition between the folding of the individual chains and their binding to form the final quaternary structure. From 20 different NMR models, we find a predominant three-state folding behavior, in agreement with experimental data on the folding pathway for this protein. Individual NMR models, however, show distinct characteristics, which are analyzed both at the level of the interplay between tertiary/quaternary structure formation and also regarding the thermal stability of the tertiary structure of every individual chain.     

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.     

Viscosity scaling for the glassy phase of protein folding

Although commendable progress has been made in the understanding of the physics of protein folding, a key unresolved issue is whether Kramers' diffusion model of chemical reactions is generally applicable to activated barrier crossing events during folding. To examine the solvent viscosity effect on the folding transition of native-like trapped intermediates, laser flash photolysis has been used to measure the microsecond folding kinetics of a natively folded state of CO-liganded ferrocytochrome c (M-state) in the 1-250 cP range of glycerol viscosity at pH 7.0, 20 degrees C. The single rate coefficient for the folding of the M-state to the native state of the protein (i.e., the M --> N folding process) decreases initially when the solvent viscosity is low (<10 cP), but saturates at higher viscosity, indicating that Kramers model is not general enough for scaling the viscosity dependence of post-transition folding involving glassy dynamics. Analysis based on the Grote-Hynes idea of time dependent friction in conjunction with defect diffusion dynamics can account for the observed non-Kramers scaling.     

Kinetics of cooperative protein folding involving two separate conformational families

The master equation that describes the kinetics of protein folding is solved numerically for a portion of Staphylococcal Protein A by a Laplace transformation. The calculations are carried out with 50 local-minimum conformations belonging to two conformational families. The master equation allows for transitions among all the 50 conformations in the evolution toward the final folded equilibrium distribution of conformations. It is concluded that the native protein folds in a fast cooperative process. The global energy minimum of a native protein can be reached after a sufficiently long folding time regardless of the initial state and the existence of a large number of local energy minima. Conformations representing non-native states of the protein can transform to the native state even if they do not belong to the native conformational family. Given a starting conformation, the protein molecule can fold to its final conformation through different paths. Finally, when the folding reaches the equilibrium distribution, the protein molecule adopts a set of conformations in which the global minimum has the largest average probability.     

Molecular dynamics simulation of folding of a short helical peptide with many charged residues

A molecular dynamics simulation of the folding of conantokin-T (con-T), a short helical peptide with 5 helical turns of 21 amino acids with 10 charged residues, was carried out to examine folding pathways for this peptide and to predict the folding rate. In the 18 trajectories run at 300 K, 16 trajectories folded, with an averaged folding time of approximately 50 ns. Two trajectories did not fold in up to 200 ns simulation. The folded structure in folded trajectories is in good agreement with experimental structure. An analysis of the trajectories showed that, at the beginning of a few nanoseconds, helix formation started from residues 5-9 with assistance of a hydrophobic clustering involving Tyr5, Met8, and Leu9. The peptide formed a U-shape mainly due to charge-charge interactions between charged residues at the N- and C-terminus segments. In the next approximately 10 ns, several nonnative charge-charge interactions were broken and nonnative Gla10-Lys18 (this denotes a salt bridge between Gal10 and Lys18) and/or Gla10-Lys19 interactions appeared more frequently in this folding step and the peptide became a fishhook J-shape. From this structure, the peptide folded to the folded state in 7 of all 16 folded trajectories in approximately 15 ns. Alternatively, in approximately 30 ns, the con-T went to a conformation in an L-shape with 4 helical turns and a kink at the Arg13 and Gla14 segment in the other 9 trajectories. Con-T in the L-shape then required another approximately 15 ns to fold into the folded state. In addition, in overall folding times, the former 7 trajectories folded faster with the total folding times all shorter than 45 ns, while the latter 9 trajectories folded at a time longer than 45 ns, resulting in an average folding time of approximately 50 ns. Two major folding intermediates found in 2 nonfolded trajectories are stabilized by charge clusters of 5 and 6 charged residues, respectively. With inclusion of friction and solvent-solvent interactions, which were ignored in the present GB/SA solvation model, the folding time obtained above should be multiplied by a factor of 1.25-1.7 according to a previous, similar simulation study. This results in a folding time of 65-105 ns, slightly shorter than the folding time of 127 ns for an alanine-based peptide of the same length. This suggests that the energy barrier of folding for this type of peptides with many charged residues is slightly lower than alanine-based helical peptides by less than 1 kcal/mol.     

Light-fueled dynamic covalent crosslinking of single polymer chains in non-equilibrium states

While polymer synthesis proceeds predominantly towards the thermodynamic minimum, living systems operate on the reverse principle – consuming fuel to maintain a non-equilibrium state. Herein, we report the controlled formation of 3D macromolecular architectures based on light-fueled covalent non-equilibrium chemistry. In the presence of green light (525 nm) and a bivalent triazolinedione (TAD) crosslinker, naphthalene-containing polymers can be folded into single chain nanoparticles (SCNPs). At ambient temperature, the cycloaddition product of TAD with naphthalene reverts and the SCNP unfolds into its linear parent polymer. The reported SCNP is the first example of a reversible light triggered folding of single polymer chains and can readily be repeated for several cycles. The folded state of the SCNP can either be preserved through a constant supply of light fuel, kinetic trapping or through a chemical modification that makes the folded state thermodynamically favored. Whereas small molecule bivalent TAD/naphthalene cycloaddition products largely degraded after 3 days in solution, even in the presence of fuel, the SCNP entities were found to remain intact, thereby indicating the light-fueled stabilization of the SCNP to be an inherent feature of the confined macromolecular environment.

Synthetic polymers consume green light as fuel for intramolecular crosslinking, yielding non-equilibrium single chain nanoparticles that can be light-stabilised, kinetically and chemically trapped, or else unfold in the absence of light fuel.     

Measuring pK(a) values in protein folding transition state ensembles by NMR spectroscopy

Protein folding kinetic data have been obtained for the marginally stable N-terminal SH3 domain of the Drosophila protein drk as a function of pH in order to investigate the electrostatic properties of Asp8 in the folding transition state ensemble. The slow exchange between folded and unfolded forms of the protein gives rise to separate NMR resonances for both folded and unfolded states at equilibrium. As a result, kinetic data can be derived from magnetization transfer between these two states without the need for denaturants. Using the fact that ionization of Asp8 dominates the electrostatic behavior of the protein between pH 2 and 3, along with pKa values for titrating groups in both folded and unfolded states that have been determined in a previous study, values of 2.9 +/- 0.1 and 3.3 +/- 0.2 are obtained for the pKa of Asp8 in the transition state for the wild-type protein and for a His7Ala mutant, respectively. The data are consistent with the partial formation in the transition state ensemble of an Asp8 side chain carboxylate-a Lys21 backbone amide interaction that represents a highly conserved contact in folded SH3 domains.     

Energetic basis of human telomeric DNA folding into G-quadruplex structures

Recent theoretical studies performed on the folding/unfolding mechanism of the model telomeric human DNA, 5'-AGGGTTAGGGTTAGGGTTAGGG-3' (Tel22), have indicated that in the presence of K(+) ions Tel22 folds into two hybrid G-quadruplex structures characterized by one double and two reversal TTA loops arranged in a different way. They predicted a new unfolding pathway from the initial mixture of hybrid G-quadruplexes via the corresponding intermediate triplex structures into the final, fully unfolded state. Significantly, no experimental evidence supporting the suggested pathway has been reported. In the current work, we performed a comprehensive global thermodynamic analysis of calorimetric (DSC, ITC) and spectroscopic (CD) data obtained on monitoring the folding/unfolding of Tel22 induced by changes of temperature and K(+) concentration. We show that unfolding of Tel22 may be described as a monomolecular equilibrium three-state process that involves thermodynamically distinguishable folded (F), intermediate (I), and unfolded (U) state. Considering that calorimetric methods cannot distinguish between energetically similar G-quadruplex or triplex conformations predicted by the theoretical model one can conclude that our results represent the first experimental support of the suggested unfolding/folding mechanism of Tel22. This conclusion is confirmed by the fact that the estimated number of K(+) ions released upon each unfolding step in our thermodynamic model agrees well with the corresponding values predicted by the theoretical model and that the observed changes in enthalpy, entropy, and heat capacity accompanying the F → I and I → U transitions can be reasonably explained only if the intermediate state I is considered to be a triplex structural conformation.