Toan, N. M. ; Morrison, G. ; Hyeon, C. ; Thirumalai, D. Kinetics of loop formation in polymer chains. J Phys Chem B 112, 6094-106.Abstract
We investigate the kinetics of loop formation in ideal flexible polymer chains (the Rouse model), and polymers in good and poor solvents. We show for the Rouse model, using a modification of the theory of Szabo, Schulten, and Schulten, that the time scale for cyclization is tau(c) approximately tau(0)N(2) (where tau(0) is a microscopic time scale and N is the number of monomers), provided the coupling between the relaxation dynamics of the end-to-end vector and the looping dynamics is taken into account. The resulting analytic expression fits the simulation results accurately when a, the capture radius for contact formation, exceeds b, the average distance between two connected beads. Simulations also show that when a < b, tau(c) approximately N(alpha)(tau), where 1.5 < alpha(tau) < or = 2 in the range 7 < N < 200 used in the simulations. By using a diffusion coefficient that is dependent on the length scales a and b (with a < b), which captures the two-stage mechanism by which looping occurs when a < b, we obtain an analytic expression for tauc that fits the simulation results well. The kinetics of contact formation between the ends of the chain are profoundly effected when interactions between monomers are taken into account. Remarkably, for N < 100, the values of tau(c) decrease by more than 2 orders of magnitude when the solvent quality changes from good to poor. Fits of the simulation data for tau(c) to a power law in N (tau(c) approximately N(alpha)(tau)) show that alpha(tau) varies from about 2.4 in a good solvent to about 1.0 in poor solvents. The effective exponent alpha(tau) decreases as the strength of the attractive monomer-monomer interactions increases. Loop formation in poor solvents, in which the polymer adopts dense, compact globular conformations, occurs by a reptation-like mechanism of the ends of the chain. The time for contact formation between beads that are interior to the chain in good solvents changes nonmonotonically as the loop length varies. In contrast, the variation in interior loop closure time is monotonic in poor solvents. The implications of our results for contact formation in polypeptide chains, RNA, and single-stranded DNA are briefly outlined.
Hyeon, C. ; Thirumalai, D. Multiple probes are required to explore and control the rugged energy landscape of RNA hairpins. J Am Chem Soc 130, 1538-9. multiple-probes-are-required-to-explore-and-control-the-rugged-energy-landscape-of-rna-hairpins.pdf
Tarus, B. ; Straub, J. E. ; Thirumalai, D. Structures and free-energy landscapes of the wild type and mutants of the Abeta(21-30) peptide are determined by an interplay between intrapeptide electrostatic and hydrophobic interactions. J Mol Biol 379, 815-29.Abstract
The initial events in protein aggregation involve fluctuations that populate monomer conformations, which lead to oligomerization and fibril assembly. The highly populated structures, driven by a balance between hydrophobic and electrostatic interactions in the protease-resistant wild-type Abeta(21-30) peptide and mutants E22Q (Dutch), D23N (Iowa), and K28N, are analyzed using molecular dynamics simulations. Intrapeptide electrostatic interactions were connected to calculated pK(a) values that compare well with the experimental estimates. The pK(a) values of the titratable residues show that E22 and D23 side chains form salt bridges only infrequently with the K28 side chain. Contacts between E22-K28 are more probable in "dried" salt bridges, whereas D23-K28 contacts are more probable in solvated salt bridges. The strength of the intrapeptide hydrophobic interactions increases as D23N
Jun, S. ; Thirumalai, D. ; Ha, B. - Y. Compression and stretching of a self-avoiding chain in cylindrical nanopores. Phys Rev Lett 101, 138101.Abstract
Force-induced deformations of a self-avoiding chain confined inside a cylindrical cavity, with diameter D, are probed using molecular dynamics simulations, scaling analysis, and analytical calculations. We obtain and confirm a simple scaling relation -fD approximately R(-9/4) in the strong-compression regime, while for weak deformations, we find fD = -A(R/R0) + B(R/R0)(-2), where A and B are constants, f the external force, and R the chain extension (with R0 its unperturbed value). For a strong stretch, we present a universal, analytical force-extension relation. Our results can be used to analyze the behavior of biomolecules in confinement.
O'Brien, E. P. ; Ziv, G. ; Haran, G. ; Brooks, B. R. ; Thirumalai, D. Effects of denaturants and osmolytes on proteins are accurately predicted by the molecular transfer model. Proc Natl Acad Sci U S A 105, 13403-8.Abstract
Interactions between denaturants and proteins are commonly used to probe the structures of the denatured state ensemble and their stabilities. Osmolytes, a class of small intracellular organic molecules found in all taxa, also profoundly affect the equilibrium properties of proteins. We introduce the molecular transfer model, which combines simulations in the absence of denaturants or osmolytes, and Tanford's transfer model to predict the dependence of equilibrium properties of proteins at finite concentration of osmolytes. The calculated changes in the thermodynamic quantities (probability of being in the native basin of attraction, m values, FRET efficiency, and structures of the denatured state ensemble) with GdmCl concentration [C] for the protein L and cold shock protein CspTm compare well with experiments. The radii of gyration of the subpopulation of unfolded molecules for both proteins decrease (i.e., they undergo a collapse transition) as [C] decreases. Although global folding is cooperative, residual secondary structures persist at high denaturant concentrations. The temperature dependence of the specific heat shows that the folding temperature (T(F)) changes linearly as urea and trimethylamine N-oxide (TMAO) concentrations increase. The increase in T(F) in TMAO can be as large as 20 degrees C, whereas urea decreases T(F) by as much as 35 degrees C. The stabilities of protein L and CspTm also increase linearly with the concentration of osmolytes (proline, sorbitol, sucrose, TMAO, and sarcosine).
O'Brien, E. P. ; Stan, G. ; Thirumalai, D. ; Brooks, B. R. Factors governing helix formation in peptides confined to carbon nanotubes. Nano Lett 8 3702-8.Abstract
The effect of confinement on the stability and dynamics of peptides and proteins is relevant in the context of a number of problems in biology and biotechnology. We have examined the stability of different helix-forming sequences upon confinement to a carbon nanotube using Langevin dynamics simulations of a coarse-grained representation of the polypeptide chain. We show that the interplay of several factors that include sequence, solvent conditions, strength (lambda) of nanotube-peptide interactions, and the nanotube diameter (D) determines confinement-induced stability of helicies. In agreement with predictions based on polymer theory, the helical state is entropically stabilized for all sequences when the interaction between the peptide and the nanotube is weakly hydrophobic and D is small. However, there is a strong sequence dependence as the strength of the lambda increases. For an amphiphilic sequence, the helical stability increases with lambda, whereas for polyalanine the diagram of states is a complex function of lambda and D. In addition, decreasing the size of the "hydrophobic patch" lining the nanotube, which mimics the chemical heterogeneity of the ribosome tunnel, increases the helical stability of the polyalanine sequence. Our results provide a framework for interpreting a number of experiments involving the structure formation of peptides in the ribosome tunnel as well as transport of biopolymers through nanotubes.
Hyeon, C. ; Morrison, G. ; Thirumalai, D. Force-dependent hopping rates of RNA hairpins can be estimated from accurate measurement of the folding landscapes. Proc Natl Acad Sci U S A 105, 9604-9.Abstract
The sequence-dependent folding landscapes of nucleic acid hairpins reflect much of the complexity of biomolecular folding. Folding trajectories, generated by using single-molecule force-clamp experiments by attaching semiflexible polymers to the ends of hairpins, have been used to infer their folding landscapes. Using simulations and theory, we study the effect of the dynamics of the attached handles on the handle-free RNA free-energy profile F(o)(eq)(z(m)), where z(m) is the molecular extension of the hairpin. Accurate measurements of F(o)(eq)(z(m)) requires stiff polymers with small L/l(p), where L is the contour length of the handle, and l(p) is the persistence length. Paradoxically, reliable estimates of the hopping rates can only be made by using flexible handles. Nevertheless, we show that the equilibrium free-energy profile F(o)(eq)(z(m)) at an external tension f(m), the force (f) at which the folded and unfolded states are equally populated, in conjunction with Kramers' theory, can provide accurate estimates of the force-dependent hopping rates in the absence of handles at arbitrary values of f. Our theoretical framework shows that z(m) is a good reaction coordinate for nucleic acid hairpins under tension.
Vaitheeswaran, S. ; Thirumalai, D. Interactions between amino acid side chains in cylindrical hydrophobic nanopores with applications to peptide stability. Proc Natl Acad Sci U S A 105, 17636-41.Abstract
Confinement effects on protein stability are relevant in a number of biological applications ranging from encapsulation in the cylindrical cavity of a chaperonin, translocation through pores, and structure formation in the exit tunnel of the ribosome. Consequently, free energies of interaction between amino acid side chains in restricted spaces can provide insights into factors that control protein stability in nanopores. Using all-atom molecular dynamics simulations, we show that 3 pair interactions between side chains--hydrophobic (Ala-Phe), polar (Ser-Asn) and charged (Lys-Glu)--are substantially altered in hydrophobic, water-filled nanopores, relative to bulk water. When the pore holds water at bulk density, the hydrophobic pair is strongly destabilized and is driven to large separations corresponding to the width and the length of the cylindrical pore. As the water density is reduced, the preference of Ala and Phe to be at the boundary decreases, and the contact pair is preferred. A model that accounts for the volume accessible to Phe and Ala in the solvent-depleted region near the pore boundary explains the simulation results. In the pore, the hydrogen-bonded interactions between Ser and Asn have an enhanced dependence on their relative orientations, as compared with bulk water. When the side chains of Lys and Glu are restrained to be side by side, parallel to each other, then salt bridge formation is promoted in the nanopore. Based on these results, we argue and demonstrate that for a generic amphiphilic sequence, cylindrical confinement is likely to enhance thermodynamic stability relative to the bulk.
Li, M. S. ; Klimov, D. K. ; Straub, J. E. ; Thirumalai, D. Probing the mechanisms of fibril formation using lattice models. J Chem Phys 129, 175101.Abstract
Using exhaustive Monte Carlo simulations we study the kinetics and mechanism of fibril formation using lattice models as a function of temperature (T) and the number of chains (M). While these models are, at best, caricatures of peptides, we show that a number of generic features thought to govern fibril assembly are captured by the toy model. The monomer, which contains eight beads made from three letters (hydrophobic, polar, and charged), adopts a compact conformation in the native state. In both the single-layered protofilament (seen for M10) structures, the monomers are arranged in an antiparallel fashion with the "strandlike" conformation that is perpendicular to the fibril axis. Partial unfolding of the folded monomer that populates an aggregation prone conformation (N(*)) is required for ordered assembly. The contacts in the N(*) conformation, which is one of the four structures in the first "excited" state of the monomer, are also present in the native conformation. The time scale for fibril formation is a minimum in the T-range when the conformation N(*) is substantially populated. The kinetics of fibril assembly occurs in three distinct stages. In each stage there is a cascade of events that transforms the monomers and oligomers to ordered structures. In the first "burst" stage, highly mobile oligomers of varying sizes form. The conversion to the N(*) conformation occurs within the oligomers during the second stage in which a vast number of interchain contacts are established. As time progresses, a dominant cluster emerges that contains a majority of the chains. In the final stage, the aggregation of N(*) particles serve as a template onto which smaller oligomers or monomers can dock and undergo conversion to fibril structures. The overall time for growth in the latter stages is well described by the Lifshitz-Slyazov growth kinetics for crystallization from supersaturated solutions. The detailed analysis shows that elements of the three popular models, namely, nucleation and growth, templated assembly, and nucleated conformational conversion are present at various stages of fibril assembly.
Lin, J. - C. ; Thirumalai, D. Relative stability of helices determines the folding landscape of adenine riboswitch aptamers. J Am Chem Soc 130, 14080-1.Abstract
Riboswitches, whose folding is controlled by binding of metabolites to the aptamer domain, regulate downstream gene expression. Folding properties of the aptamer strongly influence the conformation of the downstream expression platform, which controls transcription termination or translation initiation. We have characterized the energy landscape of the add riboswitch aptamer quantitatively by unfolding and refolding the molecule with mechanical force using the coarse-grained self-organized polymer model and Brownian dynamics simulation. Multiple folding states have been found during the folding process of the aptamer, both with and without adenine, consistent with single molecule studies of purine riboswitches. Adenine binding stabilizes the folded structure and significantly decreases the unfolding rate of the aptamer, the folding of which is in competition with the formation of the downstream stem-loop structure in the complete riboswitch. These results provide insights into the mechanism of gene regulation by the RNA switches.
Barsegov, V. ; Morrison, G. ; Thirumalai, D. Role of internal chain dynamics on the rupture kinetic of adhesive contacts. Phys Rev Lett 100, 248102.Abstract
We study the forced rupture of adhesive contacts between monomers that are not covalently linked in a Rouse chain. When the applied force (f) to the chain end is less than the critical force for rupture (f{c}), the reversible rupture process is coupled to the internal Rouse modes. If f/f{c}>1 the rupture is irreversible. In both limits, the nonexponential distribution of contact lifetimes, which depends sensitively on the location of the contact, follows the double-exponential (Gumbel) distribution. When two contacts are well separated along the chain, the rate limiting step in the sequential rupture kinetics is the disruption of the contact that is in the chain interior. If the two contacts are close to each other, they cooperate to sustain the stress, which results in an "all-or-none" transition.
Hua, L. ; Zhou, R. ; Thirumalai, D. ; Berne, B. J. Urea denaturation by stronger dispersion interactions with proteins than water implies a 2-stage unfolding. Proc Natl Acad Sci U S A 105, 16928-33.Abstract
The mechanism of denaturation of proteins by urea is explored by using all-atom microseconds molecular dynamics simulations of hen lysozyme generated on BlueGene/L. Accumulation of urea around lysozyme shows that water molecules are expelled from the first hydration shell of the protein. We observe a 2-stage penetration of the protein, with urea penetrating the hydrophobic core before water, forming a "dry globule." The direct dispersion interaction between urea and the protein backbone and side chains is stronger than for water, which gives rise to the intrusion of urea into the protein interior and to urea's preferential binding to all regions of the protein. This is augmented by preferential hydrogen bond formation between the urea carbonyl and the backbone amides that contributes to the breaking of intrabackbone hydrogen bonds. Our study supports the "direct interaction mechanism" whereby urea has a stronger dispersion interaction with protein than water.
Pincus, D. L. ; Cho, S. S. ; Hyeon, C. ; Thirumalai, D. Minimal models for proteins and RNA from folding to function. Prog Mol Biol Transl Sci 84, 203-50. minimal-models-for-protein-and-rna-from-folding-to-function.pdf
Chen, J. ; Bryngelson, J. D. ; Thirumalai, D. Estimations of the size of nucleation regions in globular proteins. J Phys Chem B 112, 16115-20.Abstract
Folding of many single-domain proteins has been described using the nucleation-collapse (NC) mechanism. According to NC, folding (formation of secondary structures and tertiary interactions) and chain collapse occur synchronously upon formation of native-like structures involving a critical number of residues. Using simple nucleation theory together with structure-based thermodynamic data, the average size of the most probable nucleus N(R)*, for single-domain proteins, is estimated to be between 15 and 30 residues. We argue that finite-sized fluctuations in this estimate can be large so that nearly half of the residues of a 100 residue protein can be part of the folding nucleus. Inclusion of surface area changes in the folded and unfolded states are important in the determination of N(R)*.
Thirumalai, D. ; Klimov, D. K. Intermediates and transition states in protein folding. Methods Mol Biol 350, 277-303.Abstract
The complex role played by intermediates is dissected using experimental data on apomyoglobin (apoMb), simple theoretical concepts, and simulations of kinetics of simple minimal off-lattice models. The folding of moderate-to-large-sized proteins often occurs through passage of an ensemble of intermediates. In the case of apoMb there is dominant kinetic intermediate I that also occurs at equilibrium. The cooperativity of transition of U<-->I (U represents the ensemble of unfolded states) in apoMb at pH 4.0 is determined not only by the sequence but also by the anion concentration. Point mutations can substantially alter the cooperativity of formation of I. Another class of intermediates arise owing to bottlenecks in the rugged energy landscape that arises from topological frustration. As a result of the rough energy landscape, folding is predicted to follow the kinetic partitioning mechanism (KPM). According to KPM a fraction of molecules reaches the native state rapidly, while the remaining fraction is kinetically trapped in intermediates. The folding of lysozyme at pH 5.5 follows KPM. Our perspective also shows that the fraction of fast folding trajectories can be altered by changing pH, for example. These observations are clearly illustrated in simple off-lattice models of proteins. The simulations show that equilibrium intermediates occur "on-pathway" and have substantial probability to be revisited after the native state is reached, while kinetic intermediates are almost never sampled after native state is reached. In addition, kinetic intermediates are higher in free energy than equilibrium intermediates. We also discuss the consequences of multiple routes and intermediates on the transition state ensemble (TSE) in folding. Whenever multiple routes to the native state dominate, Phi-values can be larger than unity or less than zero. There appears to be a relationship between the diversity of structures in the denatured state ensemble and the extent to which the TSE is plastic. Simulations of beta-hairpins are used to illustrate these ideas.
Hyeon, C. ; Thirumalai, D. Mechanical unfolding of RNA: from hairpins to structures with internal multiloops. Biophys J 92, 731-43.Abstract
Mechanical unfolding of RNA structures, ranging from hairpins to ribozymes, using laser optical tweezer experiments have begun to reveal the features of the energy landscape that cannot be easily explored using conventional experiments. Upon application of constant force (f), RNA hairpins undergo cooperative transitions from folded to unfolded states whereas subdomains of ribozymes unravel one at a time. Here, we use a self-organized polymer model and Brownian dynamics simulations to probe mechanical unfolding at constant force and constant-loading rate of four RNA structures of varying complexity. For simple hairpins, such as P5GA, application of constant force or constant loading rate results in bistable cooperative transitions between folded and unfolded states without populating any intermediates. The transition state location (DeltaxFTS) changes dramatically as the loading rate is varied. At loading rates comparable to those used in laser optical tweezer experiments, the hairpin is plastic, with DeltaxFTS being midway between folded and unfolded states; whereas at high loading rates, DeltaxFTS moves close to the folded state, i.e., RNA is brittle. For the 29-nucleotide TAR RNA with the three-nucleotide bulge, unfolding occurs in a nearly two-state manner with an occasional pause in a high free energy metastable state. Forced unfolding of the 55 nucleotides of the Hepatitis IRES domain IIa, which has a distorted L-shaped structure, results in well-populated stable intermediates. The most stable force-stabilized intermediate represents straightening of the L-shaped structure. For these structures, the unfolding pathways can be predicted using the contact map of the native structures. Unfolding of a RNA motif with internal multiloop, namely, the 109-nucleotide prohead RNA that is part of the 29 DNA packaging motor, at constant value of rf occurs with three distinct rips that represent unraveling of the paired helices. The rips represent kinetic barriers to unfolding. Our work shows 1), the response of RNA to force is largely determined by the native structure; and 2), only by probing mechanical unfolding over a wide range of forces can the underlying energy landscape be fully explored.
Koculi, E. ; Hyeon, C. ; Thirumalai, D. ; Woodson, S. A. Charge density of divalent metal cations determines RNA stability. J Am Chem Soc 129, 2676-82.Abstract
RNA molecules are exquisitely sensitive to the properties of counterions. The folding equilibrium of the Tetrahymena ribozyme is measured by nondenaturing gel electrophoresis in the presence of divalent group IIA metal cations. The stability of the folded ribozyme increases with the charge density (zeta) 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 nonspecific 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.
Stan, G. ; Lorimer, G. H. ; Thirumalai, D. ; Brooks, B. R. Coupling between allosteric transitions in GroEL and assisted folding of a substrate protein. Proc Natl Acad Sci U S A 104, 8803-8.Abstract
Escherichia coli chaperonin, GroEL, helps proteins fold under nonpermissive conditions. During the reaction cycle, GroEL undergoes allosteric transitions in response to binding of a substrate protein (SP), ATP, and the cochaperonin GroES. Using coarse-grained representations of the GroEL and GroES structures, we explore the link between allosteric transitions and the folding of a model SP, a de novo-designed four-helix bundle protein, with low spontaneous yield. The ensemble of GroEL-bound SP is less structured than the bulk misfolded structures. Upon binding, which kinetically occurs in two stages, the SP loses not only native tertiary contacts but also experiences a decrease in helical content. During multivalent binding and the subsequent ATP-driven transition of GroEL the SP undergoes force-induced stretching. Upon encapsulation, which occurs upon GroES binding, the SP finds itself in a "hydrophilic" cavity in which it can reach the folded conformation. Surprisingly, we find that the yield of the native state in the expanded GroEL cavity is relatively small even after it remains in it for twice the spontaneous folding time. Thus, in accord with the iterative annealing mechanism, multiple rounds of binding, partial unfolding, and release of the SP are required to enhance the yield of the folded SP.
O'Brien, E. P. ; Dima, R. I. ; Brooks, B. ; Thirumalai, D. Interactions between hydrophobic and ionic solutes in aqueous guanidinium chloride and urea solutions: lessons for protein denaturation mechanism. J Am Chem Soc 129, 7346-53.Abstract
In order to clarify the mechanism of denaturant-induced unfolding of proteins we have calculated the interactions between hydrophobic and ionic species in aqueous guanidinium chloride and urea solutions using molecular dynamics simulations. Hydrophobic association is not significantly changed in urea or guanidinium chloride solutions. The strength of interaction between ion pairs is greatly diminished by the guanidinium ion. Although the changes in electrostatic interactions in urea are small, examination of structures, using appropriate pair functions, of urea and water around the solutes show strong hydrogen bonding between urea's carbonyl oxygen and the positively charged solute. Our results strongly suggest protein denaturation occurs by the direct interaction model according to which the most commonly used denaturants unfold proteins by altering electrostatic interactions either by solvating the charged residues or by engaging in hydrogen bonds with the protein backbone. To further validate the direct interaction model we show that, in urea and guanidinium chloride solutions, unfolding of an unusually stable helix (H1) from mouse PrPC (residues 144-153) occurs by hydrogen bonding of denaturants to charged side chains and backbone carbonyl groups.
Nguyen, P. H. ; Li, M. S. ; Stock, G. ; Straub, J. E. ; Thirumalai, D. Monomer adds to preformed structured oligomers of Abeta-peptides by a two-stage dock-lock mechanism. Proc Natl Acad Sci U S A 104, 111-6.Abstract
Nonfibrillar soluble oligomers, which are intermediates in the transition from monomers to amyloid fibrils, may be the toxic species in Alzheimer's disease. To monitor the early events that direct assembly of amyloidogenic peptides we probe the dynamics of formation of (Abeta(16-22))(n) by adding a monomer to a preformed (Abeta(16-22))(n-1) (n = 4-6) oligomer in which the peptides are arranged in an antiparallel beta-sheet conformation. All atom molecular dynamics simulations in water and multiple long trajectories, for a cumulative time of 6.9 mus, show that the oligomer grows by a two-stage dock-lock mechanism. The largest conformational change in the added disordered monomer occurs during the rapid ( approximately 50 ns) first dock stage in which the beta-strand content of the monomer increases substantially from a low initial value. In the second slow-lock phase, the monomer rearranges to form in register antiparallel structures. Surprisingly, the mobile structured oligomers undergo large conformational changes in order to accommodate the added monomer. The time needed to incorporate the monomer into the fluid-like oligomer grows even when n = 6, which suggests that the critical nucleus size must exceed six. Stable antiparallel structure formation exceeds hundreds of nanoseconds even though frequent interpeptide collisions occur at elevated monomer concentrations used in the simulations. The dock-lock mechanism should be a generic mechanism for growth of oligomers of amyloidogenic peptides.