Publications

1993
Straub, J. E. ; Thirumalai, D. Exploring the energy landscape in proteins. Proc Natl Acad Sci U S A 90, 809-13.Abstract
We present two methods to probe the energy landscape and motions of proteins in the context of molecular dynamics simulations of the helix-forming S-peptide of RNase A and the RNase A-3'-UMP enzyme-product complex. The first method uses the generalized ergodic measure to compute the rate of conformational space sampling. Using the dynamics of nonbonded forces as a means of probing the time scale for ergodicity to be obtained, we argue that even in a relatively short time (< 10 psec) several different conformational substrates are sampled. At longer times, barriers on the order of a few kcal/mol (1 cal = 4.184 J) are involved in the large-scale motion of proteins. We also present an approximate method for evaluating the distribution of barrier heights g(EB) using the instantaneous normal-mode spectra of a protein. For the S-peptide, we show that g(EB) is adequately represented by a Poisson distribution. By comparing with previous work on other systems, we suggest that the statistical characteristics of the energy landscape may be a "universal" feature of all proteins.
exploring-the-energy-landscape-in-proteins.pdf
Camacho, C. J. ; Thirumalai, D. Kinetics and thermodynamics of folding in model proteins. Proc Natl Acad Sci U S A 90, 6369-72.Abstract
Monte Carlo simulations on a class of lattice models are used to probe the thermodynamics and kinetics of protein folding. We find two transition temperatures: one at T theta, when chains collapse from a coil to a compact phase, and the other at Tf (< T theta), when chains adopt a conformation corresponding to their native state. The kinetics are probed by several correlation functions and are interpreted in terms of the underlying energy landscape. The transition from the coil to the native state occurs in three distinct stages. The initial stage corresponds to a random collapse of the protein chain. At intermediate times tau c, during which much of the native structure is acquired, there are multiple pathways. For longer times tau r (> tau c) the decay is exponential, suggestive of a late transition state. The folding time scale (approximately tau r) varies greatly depending on the model. Implications of our results for in vitro folding of proteins are discussed.
kinetics-and-thermodynamics-of-folding-in-model-proteins.pdf
Camacho, C. J. ; Thirumalai, D. Minimum energy compact structures of random sequences of heteropolymers. Phys Rev Lett 71, 2505-2508.
Straub, J. E. ; Thirumalai, D. Theoretical probes of conformational fluctuations in S-peptide and RNase A/3'-UMP enzyme product complex. Proteins 15, 360-73.Abstract
The dynamic properties of the RNase A/3'-UMP enzyme/product complex and the S-peptide of RNase A have been investigated by molecular dynamics simulations using suitable generalization of ideas introduced to probe the energy landscape in structural glasses. We introduce two measures, namely, the kinetic energy fluctuation metric and the force metric, both of which are used to calculate the time needed for sampling the conformation space of the molecules. The calculation of the fluctuation metric requires a single trajectory whereas the force metric is computed using two independent trajectories. The vacuum MD simulations show that for both systems the time required for kinetic energy equipartitioning is surprisingly long even at high temperatures. We show that the force metric is a powerful means of probing the nature and relative importance of conformational substates which determine the dynamics at low temperatures. In particular the time dependence of the non-bonded force metric is used to demonstrate that at low temperatures the system is predominantly localized in a single cluster of conformational substates. The force metric is used to show that relaxation of long range (in sequence space) interactions must be mediated by a sequence of local dihedral angle transitions. We also argue that the time needed for compact structure formation is intimately related to the time needed for the relaxation of the dihedral angle degrees of freedom. The time for non-bonded interactions, which drive protein molecules to fold under appropriate conditions, to relax becomes extremely long as the temperature is lowered suggesting that the formation of maximally compact structure in proteins must be a very slow process.
theoretical-probes-of-conformational-fluctuations-in-s-peptide-and-rnase-a3-ump-enzyme-product-complex.pdf
1992
Ha, B. Y. ; Thirumalai, D. Conformations of a polyelectrolyte chain. Phys Rev A 46, R3012-R3015. conformations-of-polyelectrolyte-chain.pdf
Mountain, R. D. ; Thirumalai, D. Ergodicity and activated dynamics in supercooled liquids. Phys Rev A 45, R3380-R3383.
Honeycutt, J. D. ; Thirumalai, D. The nature of folded states of globular proteins. Biopolymers 32, 695-709.Abstract
We suggest, using dynamical simulations of a simple heteropolymer modelling the alpha-carbon sequence in a protein, that generically the folded states of globular proteins correspond to statistically well-defined metastable states. This hypothesis, called the metastability hypothesis, states that there are several free energy minima separated by barriers of various heights such that the folded conformations of a polypeptide chain in each of the minima have similar structural characteristics but have different energies from one another. The calculated structural characteristics, such as bond angle and dihedral angle distribution functions, are assumed to arise from only those configurations belonging to a given minimum. The validity of this hypothesis is illustrated by simulations of a continuum model of a heteropolymer whose low temperature state is a well-defined beta-barrel structure. The simulations were done using a molecular dynamics algorithm (referred to as the "noisy" molecular dynamics method) containing both friction and noise terms. It is shown that for this model there are several distinct metastable minima in which the structural features are similar. Several new methods of analyzing fluctuations in structures belonging to two distinct minima are introduced. The most notable one is a dynamic measure of compactness that can in principle provide the time required for maximal compactness to be achieved. The analysis shows that for a given metastable state in which the protein has a well-defined folded structure the transition to a state of higher compactness occurs very slowly, lending credence to the notion that the system encounters a late barrier in the process of folding to the most compact structure. The examination of the fluctuations in the structures near the unfolding----folding transition temperature indicates that the transition state for the unfolding to folding process occurs closer to the folded state.
1991
Bhattacharjee, J. K. ; Thirumalai, D. Drag reduction in turbulent flows by polymers. Phys Rev Lett 67, 196-199.
Shaw, M. R. ; Thirumalai, D. Free polymer in a colloidal solution. Phys Rev A 44, R4797-R4800. free-polymer-in-a-colloidal-solution.pdf
1990
Thirumalai, D. ; Mountain, R. D. Ergodic convergence properties of supercooled liquids and glasses. Phys Rev A 42, 4574-4587.
Honeycutt, J. D. ; Thirumalai, D. Metastability of the folded states of globular proteins. Proc Natl Acad Sci U S A 87, 3526-9.Abstract
The possibility that several metastable minima exist in which the folded forms of a polypeptide chain have similar structural characteristics but different energies is suggested. The validity of this hypothesis is illustrated with the aid of simulation methods on a model protein that folds into a beta-barrel structure. Some implications of this hypothesis such as the existence of multiple pathways with intermediates for protein folding are discussed.
metastability-of-the-folded-states-of-globular-proteins.pdf
1989
Thirumalai, D. ; Mountain, R. D. ; Kirkpatrick, T. R. Ergodic behavior in supercooled liquids and in glasses. Phys Rev A Gen Phys 39, 3563-3574.
Kirkpatrick, T. R. ; Thirumalai, D. ; Wolynes, P. G. Scaling concepts for the dynamics of viscous liquids near an ideal glassy state. Phys Rev A Gen Phys 40, 1045-1054. scaling-concepts-for-the-dynamics-of-viscous-liquids-near-an-ideal-glassy-state.pdf
1988
Thirumalai, D. Isolated polymer molecule in a random environment. Phys Rev A Gen Phys 37, 269-276.
Kirkpatrick, T. R. ; Thirumalai, D. Comparison between dynamical theories and metastable states in regular and glassy mean-field spin models with underlying first-order-like phase transitions. Phys Rev A Gen Phys 37, 4439-4448.
Bagchi, B. ; Thirumalai, D. Freezing of a colloidal liquid subject to shear flow. Phys Rev A Gen Phys 37, 2530-2538.
Thirumalai, D. ; Kirkpatrick, T. R. Mean-field Potts glass model: Initial-condition effects on dynamics and properties of metastable states. Phys Rev B Condens Matter 38, 4881-4892.
Kirkpatrick, T. R. ; Thirumalai, D. Mean-field soft-spin Potts glass model: Statics and dynamics. Phys Rev B Condens Matter 37, 5342-5350.
1987
Kirkpatrick, T. R. ; Thirumalai, D. Dynamics of the structural glass transition and the p-spin-interaction spin-glass model. Phys Rev Lett 58, 2091-2094.
Mountain, R. D. ; Thirumalai, D. Molecular-dynamics study of glassy and supercooled states of a binary mixture of soft spheres. Phys Rev A Gen Phys 36, 3300-3311.

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