The application of all-atom force fields (and explicit or implicit solvent models) to protein homology-modeling tasks such as side-chain and loop prediction remains challenging both because of the expense of the individual energy calculations and because of the difficulty of sampling the rugged all-atom energy surface. Here we address this challenge for the problem of loop prediction through the development of numerous new algorithms, with an emphasis on multiscale and hierarchical techniques. As a first step in evaluating the performance of our loop prediction algorithm, we have applied it to the problem of reconstructing loops in native structures; we also explicitly include crystal packing to provide a fair comparison with crystal structures. In brief, large numbers of loops are generated by using a dihedral angle-based buildup procedure followed by iterative cycles of clustering, side-chain optimization, and complete energy minimization of selected loop structures. We evaluate this method by using the largest test set yet used for validation of a loop prediction method, with a total of 833 loops ranging from 4 to 12 residues in length. Average/median backbone rootmean-square deviations (RMSDs) to the native structures (superimposing the body of the protein, not the loop itself) are 0.42/0.24 Å for 5 residue loops, 1.00/0.44 Å for 8 residue loops, and 2.47/1.83 Å for 11 residue loops. Median RMSDs are substantially lower than the averages because of a small number of outliers; the causes of these failures are examined in some detail, and many can be attributed to errors in assignment of protonation states of titratable residues, omission of ligands from the simulation, and, in a few cases, probable errors in the experimentally determined structures. When these obvious problems in the data sets are filtered out, average RMSDs to the native structures improve to 0.43 Å for 5 residue loops, 0.84 Å for 8 residue loops, and 1.63 Å for 11 residue loops. In the vast majority of cases, the method locates energy minima that are lower than or equal to that of the minimized native loop, thus indicating that sampling rarely limits prediction accuracy. The overall results are, to our knowledge, the best reported to date, and we attribute this success to the combination of an accurate all-atom energy function, efficient methods for loop buildup and side-chain optimization, and, especially for the longer loops, the hierarchical refinement protocol.

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We have developed an improved sampling algorithm and energy model for protein loop prediction, the combination of which has yielded the first methodology capable of achieving good results for the prediction of loop backbone conformations of 11 residue length or greater. Applied to our newly constructed test suite of 104 loops ranging from 11 to 13 residues, our method obtains average/median global backbone root-mean-square deviations (RMSDs) to the native structure (superimposing the body of the protein, not the loop itself) of 1.00/0.62 A for 11 residue loops, 1.15/0.60 A for 12 residue loops, and 1.25/0.76 A for 13 residue loops. Sampling errors are virtually eliminated, while energy errors leading to large backbone RMSDs are very infrequent compared to any previously reported efforts, including our own previous study. We attribute this success to both an improved sampling algorithm and, more critically, the inclusion of a hydrophobic term, which appears to approximately fix a major flaw in SGB solvation model that we have been employing. A discussion of these results in the context of the general question of the accuracy of continuum solvation models is presented.

Understanding how RNA molecules navigate their rugged folding landscapes holds the key to describing their roles in a variety of cellular functions. To dissect RNA folding at the molecular level, we performed simulations of three pseudoknots (MMTV and SRV-1 from viral genomes and the hTR pseudoknot from human telomerase) using coarse-grained models. The melting temperatures from the specific heat profiles are in good agreement with the available experimental data for MMTV and hTR. The equilibrium free energy profiles, which predict the structural transitions that occur at each melting temperature, are used to propose that the relative stabilities of the isolated helices control their folding mechanisms. Kinetic simulations, which corroborate the inferences drawn from the free energy profiles, show that MMTV folds by a hierarchical mechanism with parallel paths, i.e., formation of one of the helices nucleates the assembly of the rest of the structure. The SRV-1 pseudoknot, which folds in a highly cooperative manner, assembles in a single step in which the preformed helices coalesce nearly simultaneously to form the tertiary structure. Folding occurs by multiple pathways in the hTR pseudoknot, the isolated structural elements of which have similar stabilities. In one of the paths, tertiary interactions are established before the formation of the secondary structures. Our work shows that there are significant sequence-dependent variations in the folding landscapes of RNA molecules with similar fold. We also establish that assembly mechanisms can be predicted using the stabilities of the isolated secondary structures.kinetic partitioning mechanism ͉ parallel pathways ͉ ribosomal frameshifting ͉ RNA folding T he RNA folding problem has taken center stage in molecular biology because these molecules play a vital role in a variety of cellular functions. The percentage of the transcribed noncoding sequences in mice and human genomes exceed 90% (1), and the functional importance of the rest of the noncoding RNA is now only beginning to be understood (2). The noncoding roles of ribosomal RNA (rRNA) and transfer RNA (tRNA) are well known, but the discovery of ribozymes (RNA enzymes) has sparked an intense search for the functional roles of the rest of the noncoding RNA molecules (2, 3), which include catalysis, replication, transcriptional and translational regulation, and ligand binding (4), just to name a few. Consequently, it is important to determine the mechanisms by which RNA molecules fold so that a deeper understanding of the structure-function relationship can emerge.Although great progress has been made in understanding of how RNA molecules fold (5Ϫ10) global principles that determine the sequence dependent folding mechanisms have not fully emerged. It is argued that RNA folding mechanisms must be inherently simple because of the apparent separation in the energy scales that describe the different levels of structural organization (11). Furthermore, RNA molecules are constructed from only four nucleotides (nts), ...

We study the effect of the osmolyte, Trimethylamine N-Oxide (TMAO), which accumulates in cells in response to osmotic stress, on the stability of RNA hairpins. All atom molecular dynamics (MD) simulations of a nucleotide and the 22-nucleotide RNA hairpin P5GA in an aqueous TMAO solution show that TMAO preferentially interacts with the base through the formation of a single hydrogen bond. To circumvent the difficulties of adequately sampling the conformational space of polynucleotides, we used coarse-grained models (including one that is inspired by the results of all-atom MD simulations of a single nucleotide) to probe the effects of osmoyltes on the stability of P5GA. If, as revealed by our MD simulations, the cosolute specifically interacts with only one base at a time, then we find practically no change in hairpin stability as measured by Delta T m = T m(Phi) - T m, where T m(Phi) and T m are the melting temperatures at volume fraction Phi of the osmolyte and Phi = 0, respectively. This finding is in qualitative agreement with recent experiments. If the interactions between the RNA and osmolytes are repulsive, which is appropriate for mimicking the effects of crowding, Delta T m can vary from 5 to 15 K depending on the size of the osmolyte and the nature of RNA-osmolyte interactions. Cosolutes that interact favorably with multiple bases simultaneously can stabilize the hairpin more than a crowding agent of the same size. The implications of our predictions for experiments are briefly outlined.

Single-molecule force spectroscopy methods can be used to generate folding trajectories of biopolymers from arbitrary regions of the folding landscape. We illustrate the complexity of the folding kinetics and generic aspects of the collapse of RNA and proteins upon force quench by using simulations of an RNA hairpin and theory based on the de Gennes model for homopolymer collapse. The folding time, τ F , depends asymmetrically on δf S = f S − f m and δf Q = f m − f Q where f S ( f Q ) is the stretch (quench) force and f m is the transition midforce of the RNA hairpin. In accord with experiments, the relaxation kinetics of the molecular extension, R(t), occurs in three stages: A rapid initial decrease in the extension is followed by a plateau and finally, an abrupt reduction in R(t) occurs as the native state is approached. The duration of the plateau increases as λ = τ Q /τ F decreases (where τ Q is the time in which the force is reduced from f S to f Q ). Variations in the mechanisms of force-quench relaxation as λ is altered are reflected in the experimentally measurable time-dependent entropy, which is computed directly from the folding trajectories. An analytical solution of the de Gennes model under tension reproduces the multistage stage kinetics in R(t). The prediction that the initial stages of collapse should also be a generic feature of polymers is validated by simulation of the kinetics of toroid (globule) formation in semiflexible (flexible) homopolymers in poor solvents upon quenching the force from a fully stretched state. Our findings give a unified explanation for multiple disparate experimental observations of protein folding.T he folding of RNA molecules (1) and proteins (2-5) should be thought of as dynamic changes in the distribution of conformations that result in collapse, the formation of intermediates, and barrier crossings to reach the folded structures. Such a statistical, mechanical perspective of the folding process is finding support in single-molecule measurements that manipulate the structural ensembles by using an external mechanical force ( f ) (6-8). The development of the force-clamp methods (6,8), which allow the application of a constant force to specific locations on a biomolecule, have made it possible to explore in the most straightforward manner the folding of proteins and RNA initiated from regions of the folding landscape that are inaccessible in conventional ensemble experiments. The use of an initial stretching force, f S , which fully unfolds the biomolecule, followed by a subsequent quench to a sufficiently low force, f Q , which populates the native basin of attraction (NBA), holds the promise of unearthing all aspects of the folding reaction, including the dynamics of the collapse process and the extent to which the folding pathways are heterogeneous.Although there is great heterogeneity in the folding trajectories, Fernandez and Li (6) noted that upon the quench f S → f Q , ubiquitin (Ub) folded in three stages, as reflected in the timedependent changes in the ex...

We present a panoramic view of the utility of coarse-grained (CG) models to study folding and functions of proteins and RNA. Drawing largely on the methods developed in our group over the last twenty years, we describe a number of key applications ranging from folding of proteins with disulfide bonds to functions of molecular machines. After presenting the theoretical basis that justifies the use of CG models, we explore the biophysical basis for the emergence of a finite number of folds from lattice models. The lattice model simulations of approach to the folded state show that non-native interactions are relevant only early in the folding process -a finding that rationalizes the success of structure-based models that emphasize native interactions. Applications of off-lattice C α and models that explicitly consider side chains (C α -SCM) to folding of β-hairpin and effects of macromolecular crowding are briefly discussed. Successful applications of a new class of off-lattice models, referred to as the Self- We also present two distinct models for RNA, namely, the Three Site Interaction (TIS) model and the SOP model, that probe forced unfolding and force quench refolding of a simple hairpin and Azoarcus ribozyme. The unfolding pathways of Azoarcus ribozyme depend on the loading rate, while constant force and constant loading rate simulations of the hairpin show that both forced-unfolding and force-quench refolding pathways are heterogeneous. The location of the transition state moves as force is varied. The predictions based on the SOP model show that force-induced unfolding pathways of the ribozyme can be dramatically changed by varying the loading rate. We conclude with a discussion of future prospects for the use of coarse-grained models in addressing problems of outstanding interest in biology.

Helicases, involved in a number of cellular functions, are motors that translocate along single-stranded nucleic acid and couple the motion to unwinding double-strands of a duplex nucleic acid. The junction between double- and single-strands creates a barrier to the movement of the helicase, which can be manipulated in vitro by applying mechanical forces directly on the nucleic acid strands. Single-molecule experiments have demonstrated that the unwinding velocities of some helicases increase dramatically with increase in the external force, while others show little response. In contrast, the unwinding processivity always increases when the force increases. The differing responses of the unwinding velocity and processivity to force have lacked explanation. By generalizing a previous model of processive unwinding by helicases, we provide a unified framework for understanding the dependence of velocity and processivity on force and the nucleic acid sequence. We predict that the sensitivity of unwinding processivity to external force is a universal feature that should be observed in all helicases. Our prediction is illustrated using T7 and NS3 helicases as case studies. Interestingly, the increase in unwinding processivity with force depends on whether the helicase forces basepair opening by direct interaction or if such a disruption occurs spontaneously due to thermal fluctuations. Based on the theoretical results, we propose that proteins like single-strand binding proteins associated with helicases in the replisome may have coevolved with helicases to increase the unwinding processivity even if the velocity remains unaffected.