Publications

1998
Klimov, D. K. ; Betancourt, M. R. ; Thirumalai, D. Virtual atom representation of hydrogen bonds in minimal off-lattice models of alpha helices: effect on stability, cooperativity and kinetics. Fold Des 3 481-96.Abstract
BACKGROUND: The most conspicuous feature of a right-handed alpha helix is the presence of hydrogen bonds between the backbone carbonyl oxygen and NH groups along the chain. A simple off-lattice model that includes hydrogen bond interactions using virtual atoms is used to examine the stability, cooperativity and kinetics of the helix-coil transition. RESULTS: We have studied the thermodynamics (using multiple histogram method) and kinetics (by Brownian dynamics simulations) of 16-mer minimal off-lattice models of four-turn alpha-helix sequences. The carbonyl and NH groups are represented as virtual moieties located between two alpha-carbon atoms along the polypeptide chain. The characteristics of the native conformations of the model helices, such as the helical pitch and angular correlations, coincide with those found in real proteins. The transition from coil to helix is quite broad, which is typical of these finite-sized systems. The cooperativity, as measured by a dimensionless parameter, omegac, that takes into account the width and the slope of the transition curves, is enhanced when hydrogen bonds are taken into account. The value of omegac for our model is consistent with that inferred from experiment for an alanine-based helix-forming peptide. The folding time tauF ranges from 6 to 1000 ns in the temperature range 0.7-1.9 T(F), where T(F) is the helix-coil transition temperature. These values are in excellent agreement with the results from recent fast folding experiments. The temperature dependence of tauF exhibits a nearly Arrhenius behavior. Thermally induced unfolding occurs on a time scale that is less than 40-170 ps depending on the final temperature. Our calculations also predict that, although tauF can be altered by changes in the sequence, the dynamic range over which such changes take place is not as large as that predicted for beta-turn formation. CONCLUSIONS: Hydrogen bonds not only affect the stability of alpha-helix formation but also have profound influence on the kinetics. The excellent agreement between our calculations and experiments suggests that these models can be used to investigate the effects of sequence, temperature and viscosity on the helix-coil transition.
virtual-atom-representation-of-hydrogen-bonds-in-minimal-off-lattice-models-of-a-helices-effect-on-stability-cooperativity-and-kinetics.pdf
Thirumalai, D. ; Klimov, D. K. Fishing for folding nuclei in lattice models and proteins. Fold Des 3 R112-8; discussion R107.
1997
Pan, J. ; Thirumalai, D. ; Woodson, S. A. Folding of RNA involves parallel pathways. J Mol Biol 273, 7-13.Abstract
Folding kinetics of large RNAs are just beginning to be investigated. We show that the Tetrahymena self-splicing RNA partitions into a population that rapidly reaches the native state, and a slowly folding population that is trapped in metastable misfolded structures. Transitions from the misfolded structures to the native state involve partial unfolding. The total yield of native RNA is increased by iterative annealing of the inactive population, and mildly denaturing conditions increase the rate of folding at physiological temperatures. These results provide the first evidence that an RNA can fold by multiple parallel paths.
Mohanty, D. ; Elber, R. ; Thirumalai, D. ; Beglov, D. ; Roux, B. Kinetics of peptide folding: computer simulations of SYPFDV and peptide variants in water. J Mol Biol 272, 423-42.Abstract
The folding of Ser-Tyr-Pro-Phe-Asp-Val (SYPFDV), and sequence variants of this peptide (SYPYD and SYPFD) are studied computationally in an explicit water environment. An atomically detailed model of the peptide is embedded in a sphere of TIP3P water molecules and its optimal structure is computed by simulated annealing. At distances from the peptide that are beyond a few solvation shells, a continuum solvent model is employed. The simulations are performed using a mean field approach that enhances the efficiency of sampling peptide conformations. The computations predict a small number of conformations as plausible folded structures. All have a type VI turn conformation for the peptide backbone, similar to that found using NMR. However, some of the structures differ from the experimentally proposed ones in the packing of the proline ring with the aromatic residues. The second most populated structure has, in addition to a correctly folded backbone, the same hydrophobic packing as the conformation measured by NMR. Our simulations suggest a kinetic mechanism that consists of three separate stages. The time-scales associated with these stages are distinct and depend differently on temperature. Electrostatic interactions play an initial role in guiding the peptide chain to a roughly correct structure as measured by the end-to-end distance. At the same time or later the backbone torsions rearrange due to local tendency of the proline ring to form a turn: this step depends on solvation forces and is helped by loose hydrophobic interactions. In the final step, hydrophobic residues pack against each other. We also show the existence of an off the pathway intermediate, suggesting that even in the folding of a small peptide "misfolded" structures can form. The simulations clearly show that parallel folding paths are involved. Our findings suggest that the process of peptide folding shares many of the features expected for the significantly larger protein molecules.
Guo, Z. ; Thirumalai, D. The nucleation-collapse mechanism in protein folding: evidence for the non-uniqueness of the folding nucleus. Fold Des 2 377-91.Abstract
BACKGROUND: Recent experimental and theoretical studies have shown that several small proteins reach the native state by a nucleation-collapse mechanism. Studies based on lattice models have been used to suggest that the critical nucleus is specific, leading to the notion that the transition state may be unique. On the other hand, results of studies using off-lattice models show that the critical nuclei should be viewed as fluctuating mobile structures, thus implying non-unique transition states. RESULTS: The microscopic underpinnings of the nucleation-collapse mechanism in protein folding are probed using minimal off-lattice models and Langevin dynamics. We consider a 46-mer continuum model which has a native beta-barrel-like structure. The fast-folding trajectories reach the native state by a nucleation-collapse process. An algorithm based on the self-organized neural nets is used to identify the critical nuclei for a large number of rapidly folding trajectories. This method, which reduces the determination of the critical nucleus to one of 'pattern recognition', unambiguously shows that the folding nucleus is not unique. The only common characteristics of the mobile critical nuclei are that they are small (containing on average 15-22 residues) and are largely composed of residues near the loop regions of the molecule. The structures of the transition states, corresponding to the critical nuclei, show the existence of spatially localized ordered regions that are largely made up of residues that are close to each other. These structures are stabilized by a few long-range contacts. The structures in the ensemble of transition states exhibit a rather diverse degree of similarity to the native conformation. CONCLUSIONS: The multiplicity of delocalized nucleation regions can explain the two-state folding by a nucleation-collapse mechanism for small single-domain proteins (such as chymotrypsin inhibitor 2) and their mutants. Because there are many distinct critical nuclei, we predict that the folding kinetics of fast-folding proteins will not be drastically changed even if some of the residues in a 'typical' nucleus are altered.
the-nucleation-collapse-mechanism-in-protein-folding-evidence-for-the-non-uniqueness-of-the-folding-nucleus.pdf
Veitshans, T. ; Klimov, D. ; Thirumalai, D. Protein folding kinetics: timescales, pathways and energy landscapes in terms of sequence-dependent properties. Fold Des 2 1-22.Abstract
BACKGROUND: Recent experimental and theoretical studies have revealed that protein folding kinetics can be quite complex and diverse depending on various factors such as size of the protein sequence and external conditions. For example, some proteins fold apparently in a kinetically two-state manner, whereas others follow complex routes to the native state. We have set out to provide a theoretical basis for understanding the diverse behavior seen in the refolding kinetics of proteins in terms of properties that are intrinsic to the sequence. RESULTS: The folding kinetics of a number of sequences for off-lattice continuum models of proteins is studied using Langevin simulations at two different values of the friction coefficient. We show for these models that there is a remarkable correlation between folding time, tau F, and sigma = (T theta - TF)/T theta, where T theta and TF are the equilibrium collapse and folding transition temperatures, respectively. The microscopic dynamics reveals that several scenarios for the kinetics of refolding arise depending on the range of values of sigma. For relatively small sigma, the chain reaches the native conformation by a direct native conformation nucleation collapse (NCNC) mechanism without being trapped in any detectable intermediates. For moderate and large values of sigma, the kinetics is described by the kinetic partitioning mechanism, according to which a fraction of molecules phi (kinetic partition factor) reach the native conformation via the NCNC mechanism. The remaining fraction attains the native state by off-pathway processes that involve trapping in several misfolded structures. The rate-determining step in the off-pathway processes is the transition from the misfolded structures to the native state. The partition factor phi is also determined by sigma: the smaller the value of sigma, the larger is phi. The qualitative aspects of our results are found to be independent of the friction coefficient. The simulation results and theoretical arguments are used to obtain estimates for timescales for folding via the NCNC mechanism in small proteins, those with less than about 70 amino acid residues. CONCLUSIONS: We have shown that the various scenarios for folding of proteins, and possibly other biomolecules, can be classified solely in terms of sigma. Proteins with small values of sigma reach the native conformation via a nucleation collapse mechanism and their energy landscape is characterized by having one dominant native basin of attraction (NBA). On the other hand, proteins with large sigma get trapped in competing basins of attraction (CBAs) in which they adopt misfolded structures. Only a small fraction of molecules access the native state rapidly when sigma is large. For these sequences, the majority of the molecules approach the native state by a three-stage multipathway mechanism in which the rate-determining step involves a transition from one of the CBAs to the NBA.
protein-folding-kinetics-time_scales-pathways-and-energy-landscapes-in-terms-of-sequence-dependent-properties.pdf
1996
Bryngelson, J. D. ; Thirumalai, D. Bryngelson and Thirumalai Reply. Phys Rev Lett 77, 4277.
Todd, M. J. ; Lorimer, G. H. ; Thirumalai, D. Chaperonin-facilitated protein folding: optimization of rate and yield by an iterative annealing mechanism. Proc Natl Acad Sci U S A 93, 4030-5.Abstract
We develop a heuristic model for chaperonin-facilitated protein folding, the iterative annealing mechanism, based on theoretical descriptions of "rugged" conformational free energy landscapes for protein folding, and on experimental evidence that (i) folding proceeds by a nucleation mechanism whereby correct and incorrect nucleation lead to fast and slow folding kinetics, respectively, and (ii) chaperonins optimize the rate and yield of protein folding by an active ATP-dependent process. The chaperonins GroEL and GroES catalyze the folding of ribulose bisphosphate carboxylase at a rate proportional to the GroEL concentration. Kinetically trapped folding-incompetent conformers of ribulose bisphosphate carboxylase are converted to the native state in a reaction involving multiple rounds of quantized ATP hydrolysis by GroEL. We propose that chaperonins optimize protein folding by an iterative annealing mechanism; they repeatedly bind kinetically trapped conformers, randomly disrupt their structure, and release them in less folded states, allowing substrate proteins multiple opportunities to find pathways leading to the most thermodynamically stable state. By this mechanism, chaperonins greatly expand the range of environmental conditions in which folding to the native state is possible. We suggest that the development of this device for optimizing protein folding was an early and significant evolutionary event.
chaperonin-facilitated-protein-folding-optimization-of-rate-and-yield-by-an-iterative-annealing-mechanism.pdf
Klimov, D. K. ; Thirumalai, D. Criterion that determines the foldability of proteins. Phys Rev Lett 76, 4070-4073. criterion-that-determines-the-foldability-of-proteins.pdf
Camacho, C. J. ; Thirumalai, D. Denaturants can accelerate folding rates in a class of globular proteins. Protein Sci 5 1826-32.Abstract
We present a lattice Monte Carlo study to examine the effect of denaturants on the folding rates of simplified models of proteins. The two-dimensional model is made from a three-letter code mimicking the presence of hydrophobic, hydrophilic, and cysteine residues. We show that the rate of folding is maximum when the effective hydrophobic interaction epsilon H is approximately equal to the free energy gain epsilon S upon forming disulfide bonds. In the range 1 < or = epsilon H/ epsilon S < or = 3, multiple paths that connect several intermediates to the native state lead to fast folding. It is shown that at a fixed temperature and epsilon S the folding rate increases as epsilon H decreases. An approximate model is used to show that epsilon H should decrease as a function of the concentration of denaturants such as urea or guanidine hydrochloride. Our simulation results, in conjunction with this model, are used to show that increasing the concentration of denaturants can lead to an increase in folding rates. This occurs because denaturants can destabilize the intermediates without significantly altering the energy of the native conformation. Our findings are compared with experiments on the effects of denaturants on the refolding of bovine pancreatic trypsin inhibitor and ribonuclease T1. We also argue that the phenomenon of denaturant-enhanced folding of proteins should be general.
denaturants-can-accelerate-folding-rates-in-a-class-of-globular-proteins.pdf
Klimov, D. K. ; Thirumalai, D. Factors governing the foldability of proteins. Proteins 26, 411-41.Abstract
We use a three-dimensional lattice model of proteins to investigate systematically the global properties of the polypeptide chains that determine the folding to the native conformation starting from an ensemble of denatured conformations. In the coarse-grained description, the polypeptide chain is modeled as a heteropolymer consisting of N beads confined to the vertices of a simple cubic lattice. The interactions between the beads are taken from a random gaussian distribution of energies, with a mean value B0 < 0 that corresponds to the overall average hydrophobic interaction energy. We studied 56 sequences all with a unique ground state (native conformation) covering two values of N (15 and 27) and two values of B0. The smaller value of magnitude of B0 was chosen so that the average fraction of hydrophobic residues corresponds to that found in natural proteins. The higher value of magnitude of B0 was selected with the expectation that only the fully compact conformations would contribute to the thermodynamic behavior. For N = 15 the entire conformation space (compact as well as noncompact structures) can be exhaustively enumerated so that the thermodynamic properties can be exactly computed at all temperatures. The thermodynamic properties for the 27-mer chain were calculated using the slow cooling technique together with standard Monte Carlo simulations. The kinetics of approach to the native state for all the sequences was obtained using Monte Carlo simulations. For all sequences we find that there are two intrinsic characteristic temperatures, namely, T theta and Tf. At the temperature T theta the polypeptide chain makes a transition to a collapsed structure, while at Tf the chain undergoes a transition to the native conformation. We show that foldability of sequences can be characterized entirely in terms of these two temperatures. It is shown that fast folding sequences have small values of sigma = (T theta - Tf)/T theta whereas slow folders have larger values of sigma (the range of sigma is 0 < sigma < 1). The calculated values of the folding times correlate extremely well with sigma. An increase in sigma from 0.1 to 0.7 can result in an increase of 5-6 orders of magnitudes in folding times. In contrast, we demonstrate that there is no useful correlation between folding times and the energy gap between the native conformation and the first excited state at any N for any value of B0. In particular, in the parameter space of the model, many sequences with varying energy gaps, all with roughly the same folding time, can be easily engineered. Folding sequences in this model, therefore, can be classified based solely on the value of sigma. Fast folders have small values of sigma (typically less than about 0.1), moderate folders have values of sigma in the approximate range between 0.1 and 0.6, while for slow folders sigma exceeds 0.6. The precise boundary between these categories depends crucially on N and on the model. The range of sigma for fast folders decreases with the length of the chain. At temperatures close to Tf fast folders reach the native conformation via a native conformation nucleation collapse mechanism without forming any detectable intermediates, whereas only a fraction of molecule phi (T) reaches the native conformation by this process for moderate folders. The remaining fraction reaches the native state via three-stage multipathway process. For slow folders phi (T) is close to zero at all temperatures. The simultaneous requirement of native state stability and kinetic accessibility can be achieved at high enough temperatures for those sequences with small values of sigma. The utility of these results for de novo design of proteins is briefly discussed.
factors-governing-the-foldability-of-proteins.pdf
Bryngelson, J. D. ; Thirumalai, D. Internal constraints induce localization in an isolated polymer molecule. Phys Rev Lett 76, 542-545.
Guo, Z. ; Thirumalai, D. Kinetics and thermodynamics of folding of a de novo designed four-helix bundle protein. J Mol Biol 263, 323-43.Abstract
A simple continuum model of a de novo designed model of a four-helix bundle is presented. The thermodynamics and kinetics of the model are studied using Langevin simulations. We use a three-letter minimal off-lattice representation of a de novo designed four-helix bundle protein. The native state of the model, which can be thought of as an alpha-carbon representation of the peptide chain, is a caricature of the sequence designed by Ho and Degrado and shows several characteristics found in the naturally occurring four-helix bundles. These include the structural aspects and the relative stability of the native conformation. The model four-helix bundle shows two characteristic temperatures T theta and Tf. The former is the temperature above which the structure resembles that of the random coil. Below the first-order folding transition temperature Tf the chain adopts the native conformation corresponding to the four-helix bundle. It is shown that in order to obtain a unique native structure a proper free energy balance between secondary and tertiary interactions is needed. The thermal denaturation starting from the unique native conformation indicates that at least a three-state analysis is required. The intermediates in the equilibrium thermal denaturation are all found to be native-like. The kinetics of refolding starting from an ensemble of denatured states shows that the acquisition of the native conformation takes place via a kinetic partitioning mechanism. A fraction of molecules, phi, reaches the native state by a topology inducing nucleation collapse mechanism, while the remainder (1-phi) follows a complex three-stage multipathway process. We suggest, in accord with our earlier studies, that phi is essentially determined by the intrinsic temperature scales T theta and Tf. Our studies indicate that better design of proteins can be achieved by making T theta as close to Tf as possible. Experimental implications for de novo design of proteins are briefly discussed.
Thirumalai, D. ; Bhattacharjee, J. K. Polymer-induced drag reduction in turbulent flows. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 53, 546-551.
Thirumalai, D. ; Ashwin, V. ; Bhattacharjee, J. K. Dynamics of Random Hydrophobic-Hydrophilic Copolymers with Implications for Protein Folding. Phys Rev Lett 77, 5385-5388. dynamics-of-random-hydrophobic-hydrophilic-copolymers-with-implications-for-protein-folding.pdf
1995
Wolynes, P. G. ; Onuchic, J. N. ; Thirumalai, D. Navigating the folding routes. Science 267, 1619-20. navigating-the-folding-routes.pdf
Camacho, C. J. ; Thirumalai, D. Theoretical predictions of folding pathways by using the proximity rule, with applications to bovine pancreatic trypsin inhibitor. Proc Natl Acad Sci U S A 92, 1277-81.Abstract
We propose a phenomenological theory that accounts for entropic effects due to loop formation to predict pathways in the kinetics of protein folding. The theory, the basis of which lies in multiple folding pathways and a three-stage kinetics, qualitatively reproduces most of the kinetic measurements in the refolding of bovine pancreatic trypsin inhibitor. The resulting pathways show that nonnative kinetic transients are involved in the productive routes leading to the formation of native intermediates. Our theory emphasizes the importance of the random origin of chain folding initiation structures in directing protein folding.
theoretical-predictions-of-folding-pathways-by-using-the-proximity-rule-with-applications-to-bovine-pancreatic-trypsin-inhibitor.pdf
Camacho, C. J. ; Thirumalai, D. Modeling the role of disulfide bonds in protein folding: entropic barriers and pathways. Proteins 22, 27-40.Abstract
The role of disulfide bonds in directing protein folding is studied using lattice models. We find that the stability and the specificity of the disulfide bond interactions play quite different roles in the folding process: Under some conditions, the stability decreases the overall rate of folding; the specificity, however, by yielding a simpler connectivity of intermediates, always increases the rate of folding. This conclusion is intimately related to the selection mechanism entailed by entropic driving forces, such as the loop formation probability, and entropic barriers separating the native and the many native-like metastable states. The folding time is found to be a minimum for a certain range of the effective disulfide bond interaction. Examination of a model, which allows for the formation of disulfide bonded intermediates, suggests that folding proceeds via a three-stage multiple pathways kinetics. We show that there are pathways to the native state involving only native-like intermediates, as well as those that are mediated by nonnative intermediates. These findings are interpreted in terms of the appropriate energy landscape describing the barriers connecting low energy conformations. The consistency of our conclusions with several experimental studies is also discussed.
Wolynes, P. G. ; Onuchic, J. N. ; Thirumalai, D. Response. Science 268, 960-1.
1993
Thirumalai, D. ; Mountain, R. D. Activated dynamics, loss of ergodicity, and transport in supercooled liquids. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 47, 479-489.

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