Using self-organized polymer models, we predict mechanical unfolding and refolding pathways of ribozymes, and the green fluorescent protein. In agreement with experiments, there are between six and eight unfolding transitions in the Tetrahymena ribozyme. Depending on the loading rate, the number of rips in the force-ramp unfolding of the Azoarcus ribozymes is between two and four. Force-quench refolding of the P4-P6 subdomain of the Tetrahymena ribozyme occurs through a compact intermediate. Subsequent formation of tertiary contacts between helices P5b-P6a and P5a/P5c-P4 leads to the native state. The force-quench refolding pathways agree with ensemble experiments. In the dominant unfolding route, the N-terminal alpha helix of GFP unravels first, followed by disruption of the N terminus beta strand. There is a third intermediate that involves disruption of three other strands. In accord with experiments, the force-quench refolding pathway of GFP is hierarchic, with the rate-limiting step being the closure of the barrel.

Mechanical unfolding trajectories, generated by applying constant force in optical-tweezer experiments, show that RNA hairpins and the P5abc subdomain of the group I intron unfold reversibly. We use coarse-grained Go-like models for RNA hairpins to explore forced unfolding over a broad range of temperatures. A number of predictions that are amenable to experimental tests are made. At the critical force, the hairpin jumps between folded and unfolded conformations without populating any discernible intermediates. The phase diagram in the force-temperature (f, T) plane shows that the hairpin unfolds by an all-or-none process. The cooperativity of the unfolding transition increases dramatically at low temperatures. Free energy of stability, obtained from time averages of mechanical unfolding trajectories, coincides with ensemble averages, which establishes ergodicity. The hopping time between the native basin of attraction (NBA) and the unfolded basin increases dramatically along the phase boundary. Thermal unfolding is stochastic, whereas mechanical unfolding occurs in ''quantized steps'' with great variations in the step lengths. Refolding times, upon force quench, from stretched states to the NBA are at least an order of magnitude greater than folding times by temperature quench. Upon force quench from stretched states, the NBA is reached in at least three stages. In the initial stages, the mean end-to-end distance decreases nearly continuously, and there is a sudden transition to the NBA only in the last stage. Because of the generality of the results, we propose that similar behavior should be observed in force quench refolding of proteins.U nraveling the complexity of the energy landscape of RNA molecules requires exploration of their assembly and unfolding over a wide range of external conditions. In the last decade, a combination of experiments, theoretical arguments, and simulations have been used to decipher the folding mechanisms of RNA molecules (1-3). These studies have shown that RNA folding depends critically on a number of factors, including valence and shape of counterions (4) as well as temperature. Somewhat more surprising, recent experiments have shown that the folding mechanisms depend sensitively on the initial folding conditions (5). In conventional experiments, the difficult-tocharacterize unfolded conformations are typically generated by elevated temperature or by lowering the counterion concentration. In contrast, well defined and vastly different initial conditions can be realized by applying force. Indeed, in remarkable experiments, Bustamante and coworkers (6, 7) have generated mechanical unfolding trajectories for RNA hairpins and Tetrahymena thermophila ribozyme. These experiments, which used constant external force to denature folded RNA, show that unfolding involves multiple routes in which a number of kinetic intermediates are sampled in the transition from the folded state to a stretched conformation (6, 7). The lifetimes of the intermediates vary considerably, which is indicative ...

Visualizing the navigation of an ensemble of unfolded molecules through the bumpy energy landscape in search of the native state gives a pictorial view of biomolecular folding. This picture, when combined with concepts in polymer theory, provides a unified theory of RNA and protein folding. Just as for proteins, the major folding free energy barrier for RNA scales sublinearly with the number of nucleotides, which allows us to extract the elusive prefactor for RNA folding. Several folding scenarios can be anticipated by considering variations in the energy landscape that depend on sequence, native topology, and external conditions. RNA and protein folding mechanism can be described by the kinetic partitioning mechanism (KPM) according to which a fraction (Phi) of molecules reaches the native state directly, whereas the remaining fraction gets kinetically trapped in metastable conformations. For two-state folders Phi approximately 1. Molecular chaperones are recruited to assist protein folding whenever Phi is small. We show that the iterative annealing mechanism, introduced to describe chaperonin-mediated folding, can be generalized to understand protein-assisted RNA folding. The major differences between the folding of proteins and RNA arise in the early stages of folding. For RNA, folding can only begin after the polyelectrolyte problem is solved, whereas protein collapse requires burial of hydrophobic residues. Cross-fertilization of ideas between the two fields should lead to an understanding of how RNA and proteins solve their folding problems.

The chaperonin GroEL-GroES, a machine that helps proteins to fold, cycles through a number of allosteric states, the T state, with high affinity for substrate proteins, the ATP-bound R state, and the R؆ (GroEL-ADP-GroES) complex. Here, we use a self-organized polymer model for the GroEL allosteric states and a general structurebased technique to simulate the dynamics of allosteric transitions in two subunits of GroEL and the heptamer. The T 3 R transition, in which the apical domains undergo counterclockwise motion, is mediated by a multiple salt-bridge switch mechanism, in which a series of salt-bridges break and form. The initial event in the R 3 R؆ transition, during which GroEL rotates clockwise, involves a spectacular outside-in movement of helices K and L that results in K80-D359 salt-bridge formation. In both the transitions there is considerable heterogeneity in the transition pathways. The transition state ensembles (TSEs) connecting the T, R, and R؆ states are broad with the TSE for the T 3 R transition being more plastic than the R 3 R؆ TSE.allostery ͉ self-organized polymer model T he hallmark of allostery in biomolecules is the conformational changes at distances far from the sites at which ligands bind (1-3). The potential link between large scale allosteric transitions and function is most vividly illustrated in biological nanomachines (4, 5). Sequence (6-8) or structure-based (9, 10) methods have been proposed to predict the allosteric wiring diagram. However, to fully understand the role of allostery it is important to dynamically monitor the structural changes that occur in the transition from one state to another (11)(12)(13)(14)(15). Here, we propose a method for determining allosteric mechanisms in biological systems with applications to dynamics of such processes in the chaperonin GroEL, an ATP-fueled nanomachine, which facilitates folding of proteins [substrate proteins (SPs)] that are otherwise destined to aggregate (16,17).GroEL has two heptameric rings, stacked back-to-back. SPs are captured by GroEL in the T state ( Fig. 1) while ATP-binding triggers a transition to the R state. Binding of the co-chaperonin GroES requires dramatic movements in the A domains which double the volume of the central cavity. Comparison of the structures of the T, R, and the RЉ [GroEL-(ADP) 7 -GroES] indicates that the equatorial (E) domain, which serves as an anchor (16), undergoes comparatively fewer structural changes. Although structural and mutational studies (18-20) have identified many residues that affect GroEL function, only few studies have explored the dynamics of allosteric transitions between the various states (21-23).Here, we use the self-organized polymer model of GroEL and a novel technique (see Methods) to monitor the order of events in the T 3 R, R 3 RЉ, and T 3 RЉ transitions. By simulating the dynamics of ligand-induced conformational changes in the heptamer and also in two subunits, we have obtained an unprecedented view of the key interactions that drive the various allosteric transiti...

the distances over which biological molecules and their complexes can function range from a few nanometres, in the case of folded structures, to millimetres, for example, during chromosome organization. Describing phenomena that cover such diverse length, and also time, scales requires models that capture the underlying physics for the particular length scale of interest. theoretical ideas, in particular, concepts from polymer physics, have guided the development of coarse-grained models to study folding of Dna, rna and proteins. more recently, such models and their variants have been applied to the functions of biological nanomachines. simulations using coarse-grained models are now poised to address a wide range of problems in biology. Minimal models that capture the essence of complex phenomena has a rich history in the natural sciences. In condensed matter physics, insights into many phenomena have emerged from analytic theories of models, which use effective many-body Hamiltonians that succinctly capture the essence of the problems 1 . Examples include phase transitions, superfluidity and superconductivity. However, complex problems, such as spin glasses 2 , structural glasses 3,4 and a host of problems in biology like protein and RNA folding and functions of macromolecules, have resisted solutions using purely theoretical methods. These and other problems in material science, in which a wide range of time, energy, and length scales are intertwined, require well-designed computer simulations, which capture the essential features of the systems. Although the temptation to use detailed atomic simulations in protein folding and more complicated problems is hard to resist, such an approach has given us only limited insights. In contrast, since the first classical molecular dynamics simulation that reported phase transition in hard-sphere systems 5 , it has been clear that coarse-grained (CG) models are often the only way to describe phenomena that involve an interplay of multiple energy and time scales. Nowhere is the need for CG models greater than in biology in which self-assembly of macromolecules and their functions, which involve multiple partners, occur on time and length scales that cover many orders of magnitude. In the context of protein and RNA folding, simulations using CG models, guided by theoretical concepts [6][7][8][9][10] , have unearthed the principles of self-assembly. More recently, models, which were introduced to describe folding of isolated proteins [11][12][13][14][15] , have also been adopted and extended in novel ways to predict functions of large complexes such as ribosomes , protein insertion into membranes 20 and a number of motors [21][22][23][24][25][26][27][28] . These developments have resulted in a quiet revolution, which has provided molecular insights into a variety of biological processes.In the last two decades, fundamental breakthroughs in structural organization and dynamics of proteins, RNA and DNA have been achieved using theoretical concepts from polymer physics 29 and...

RNA molecules are exquisitely sensitive to the properties of counterions. The folding equilibrium of the Tetrahymena ribozyme is measured by non-denaturing gel electrophoresis in the presence of divalent group IIA metal cations. The stability of the folded ribozyme increases with the charge density (ζ) of the cation. Similar scaling is found when the free energy of the RNA folded in small and large metal cations is measured by urea denaturation. Brownian dynamics simulations of a polyelectrolyte show that the experimental observations can be explained by non-specific ion-RNA interactions in the absence of site-specific metal chelation. The experimental and simulation results establish that RNA stability is largely determined by a combination of counterion charge and the packing efficiency of condensed cations that depends on the excluded volume of the cations.

Nanomanipulation of biomolecules by using single-molecule methods and computer simulations has made it possible to visualize the energy landscape of biomolecules and the structures that are sampled during the folding process. We use simulations and single-molecule force spectroscopy to map the complex energy landscape of GFP that is used as a marker in cell biology and biotechnology. By engineering internal disulfide bonds at selected positions in the GFP structure, mechanical unfolding routes are precisely controlled, thus allowing us to infer features of the energy landscape of the wild-type GFP. To elucidate the structures of the unfolding pathways and reveal the multiple unfolding routes, the experimental results are complemented with simulations of a self-organized polymer (SOP) model of GFP. The SOP representation of proteins, which is a coarse-grained description of biomolecules, allows us to perform forced-induced simulations at loading rates and time scales that closely match those used in atomic force microscopy experiments. By using the combined approach, we show that forced unfolding of GFP involves a bifurcation in the pathways to the stretched state. After detachment of an N-terminal ␣-helix, unfolding proceeds along two distinct pathways. In the dominant pathway, unfolding starts from the detachment of the primary N-terminal ␤-strand, while in the minor pathway rupture of the last, C-terminal ␤-strand initiates the unfolding process. The combined approach has allowed us to map the features of the complex energy landscape of GFP including a characterization of the structures, albeit at a coarse-grained level, of the three metastable intermediates.AFM experiments ͉ coarse-grained simulations ͉ cross-link mutants ͉ pathway bifurcation ͉ plasticity of energy landscape P rotein structures, which are astounding examples of selforganization in living systems, reach their folded states by navigating through a rugged energy landscape. Considerable progress has been made in understanding the folding mechanisms of small, single-domain proteins by using a combination of theory and experiments (1-5). Folding of many of these proteins can be approximately described as being two-state-like, that is, their energy landscape does not exhibit pronounced local minima corresponding to partially folded or misfolded structures. However, the folding energy landscapes of larger proteins, with complex topology, can be difficult to characterize because of the presence of multiple metastable intermediate structures (6,7). A detailed characterization of the structures of the intermediate states and the associated energetics is a challenge for the experimentalist. In part, the difficulty arises because complex proteins generally fold slowly and tend to aggregate in bulk experiments (8), a problem that can be avoided in mechanical unfolding of single proteins. Indeed, single-molecule mechanical methods have recently provided new possibilities for directly probing the energy landscapes of proteins and RNA because they can monito...

Understanding how monomeric proteins fold under in vitro conditions is crucial to describing their functions in the cellular context. Significant advances in theory and experiments have resulted in a conceptual framework for describing the folding mechanisms of globular proteins. The sizes of proteins in the denatured and folded states, cooperativity of the folding transition, dispersions in the melting temperatures at the residue level, and timescales of folding are, to a large extent, determined by N, the number of residues. The intricate details of folding as a function of denaturant concentration can be predicted by using a novel coarse-grained molecular transfer model. By watching one molecule fold at a time, using single-molecule methods, investigators have established the validity of the theoretically anticipated heterogeneity in the folding routes and the N-dependent timescales for the three stages in the approach to the native state. Despite the successes of theory, of which only a few examples are documented here, we conclude that much remains to be done to solve the protein folding problem in the broadest sense.