Compaction and tensile forces determine the accuracy of folding landscape parameters from single molecule pulling experiments

We establish a framework for assessing whether the transition state location of a biopolymer, which can be inferred from single molecule pulling experiments, corresponds to the ensemble of structures that have equal probability of reaching either the folded or unfolded states (P(fold)=0.5). Using results for the forced unfolding of a RNA hairpin, an exactly soluble model, and an analytic theory, we show that P(fold) is solely determined by s, an experimentally measurable molecular tensegrity parameter, which is a ratio of the tensile force and a compaction force that stabilizes the folded state. Applications to folding landscapes of DNA hairpins and a leucine zipper with two barriers provide a structural interpretation of single molecule experimental data. Our theory can be used to assess whether molecular extension is a good reaction coordinate using measured free energy profiles.

Myosin VI: how do charged tails exert control?

Molecular dynamics simulations and single molecule experiments are used to suggest that charged helices in the medial tail domain participate in myosin VI dimerization (Kim et al., 2010), which reinforces the mechanism that unfolding of the three helix bundle in the proximal tail serves as a lever arm extension.

Factors governing fibrillogenesis of polypeptide chains revealed by lattice models

Using lattice models we explore the factors that determine the tendencies of polypeptide chains to aggregate by exhaustively sampling the sequence and conformational space. The morphologies of the fibril-like structures and the time scales (τ(fib)) for their formation depend on a balance between hydrophobic and Coulomb interactions. The extent of population of an ensemble of N* structures, which are fibril-prone structures in the spectrum of conformations of an isolated protein, is the major determinant of τ(fib). This observation is used to determine the aggregating sequences by exhaustively exploring the sequence space, thus providing a basis for genome wide search of fragments that are aggregation prone.

Rigor to post-rigor transition in myosin V: link between the dynamics and the supporting architecture

The detachment kinetics from actin upon ATP binding is a key step in the reaction cycle of myosin V. We show that a network of residues, constituting the allostery wiring diagram (AWD), that trigger the rigor (R) to post-rigor (PR) transition, span key structural elements from the ATP and actin-binding regions. Several of the residues are in the 33 residue helix (H18), P loop, and switch I. Brownian dynamics simulations show that a hierarchy of kinetically controlled local structural changes leads to the opening of the "cleft" region, resulting in the detachment of the motor domain from actin. Movements in switch I and P loop facilitate changes in the rest of the motor domain, in particular the rotation of H18, whose stiffness within the motor domain is crucial in the R --> PR transition. The finding that residues in the AWD also drive the kinetics of the R --> PR transition shows how the myosin architecture regulates the allosteric movements during the reaction cycle.

Theoretical perspectives on protein folding

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.

Theory of Biopolymer Stretching at High Forces

We provide a unified theory for the high force entropic elasticity of biopolymers solely in terms of the persistence length, ξ_p , and the monomer spacing, a. When the force f>ℱ _h ~ k_BTξ_p /a² the biopolymers behave as freely jointed chains (FJCs) while in the range ℱ _l ~ k_BT/ξ_p <f<ℱ _h the worm-like chain (WLC) is a better model. We show that ξ_p can be estimated from the force extension curve (FEC) at the extension x ≈ ¹/₂ (normalized by the contour length of the biopolymer). After validating the theory using simulations, we provide a quantitative analysis of the FECs for a diverse set of biopolymers (dsDNA, ssRNA, ssDNA, polysaccharides, and unstructured PEVK domain of titin) for x ≥ ¹/₂. The success of a specific polymer model (FJC or WLC) to describe the FEC of a given biopolymer is naturally explained by the theory. Only by probing the response of biopolymers over a wide range of forces can the f-dependent elasticity be fully described.

Structures of beta-amyloid peptide 1-40, 1-42, and 1-55-the 672-726 fragment of APP-in a membrane environment with implications for interactions with gamma-secretase

Aggregation of Amyloid beta (Abeta) peptide has been linked to the neurodegenerative Alzheimer's Disease and implicated in other amyloid diseases including cerebral amyloid angiopathy. Abeta peptide is generated by cleavage of the amyloid precursor protein (APP) by transmembrane proteases. It is crucial to determine the structures of beta-amyloid peptides in a membrane to provide a molecular basis for the cleavage mechanism. We report the structures of amyloid beta peptide (Abeta(1-40) and Abeta(1-42)) as well as the 672-726 fragment of APP (referred to as Abeta(1-55)) in a membrane environment determined by replica-exchange molecular dynamics simulation. Abeta(1-40) is found to have two helical domains A (13-22) and B(30-35) and a type I beta-turn at 23-27. The peptide is localized at the interface between membrane and solvent. Substantial fluctuations in domain A are observed. The dominant simulated tertiary structure of Abeta(1-40) is observed to be similar to the simulated Abeta(1-42) structure. However, there are differences observed in the overall conformational ensemble, as characterized by the two-dimensional free energy surfaces. The fragment of APP (Abeta(1-55)) is observed to have a long transmembrane helix. The position of the transmembrane region and ensemble of membrane structures are elucidated. The conformational transition between the transmembrane Abeta(1-55) structure, prior to cleavage, and the Abeta(1-40) structure, following cleavage, is proposed.

Thermodynamic perspective on the dock-lock growth mechanism of amyloid fibrils

The mechanism of addition of a soluble unstructured monomer to a preformed ordered amyloid fibril is a complex process. On the basis of the kinetics of monomer disassociation of Abeta(1-40) from the amyloid fibril, it has been suggested that deposition is a multistep process involving a rapid reversible association of the unstructured monomer to the fibril surface (docking) followed by a slower conformational rearrangement leading to the incorporation onto the underlying fibril lattice (locking). By exploiting the vast time scale separation between the dock and lock processes and using molecular dynamics simulation of deposition of the disordered peptide fragment (35)MVGGVV(40) from the Abeta peptide onto the fibril with known crystal structure, we provide a thermodynamic basis for the dock-lock mechanism of fibril growth. Free energy profiles, computed using implicit solvent model and enhanced sampling methods with the distance (delta(C)) between the center of mass of the peptide and the fibril surface as the order parameter, show three distinct basins of attraction. When delta(C) is large, the monomer is compact and unstructured and the favorable interactions with the fibril results in stretching of the peptide at delta(C) approximately 13 A. As delta(C) is further decreased, the peptide docks onto the fibril surface with a structure that is determined by a balance between intrapeptide and peptide fibril interactions. At delta(C) approximately 4 A, a value that is commensurate with the spacing between beta-strands in the fibril, the monomer expands and locks onto the fibril. Using simulations with implicit solvent model and all atom molecular dynamics in explicit water, we show that the locked monomer, which interacts with the underlying fibril, undergoes substantial conformational fluctuations and is not stable. The cosolutes urea and TMAO destabilize the unbound phase and stabilize the docked phase. Interestingly, small crowding particles enhance the stability of the fibril-bound monomer only marginally. We predict that the experimentally measurable critical monomer concentration, C(R), at which the soluble unbound monomer is in equilibrium with the ordered fibril, increases sharply as temperature is increased under all solution conditions.

Urea destabilizes RNA by forming stacking interactions and multiple hydrogen bonds with nucleic acid bases

Urea titration of RNA by urea is an effective approach to investigate the forces stabilizing this biologically important molecule. We used all atom molecular dynamics simulations using two urea force fields and two RNA constructs to elucidate in atomic detail the destabilization mechanism of folded RNA in aqueous urea solutions. Urea denatures RNA by forming multiple hydrogen bonds with the RNA bases and has little influence on the phosphodiester backbone. Most significantly we discovered that urea engages in stacking interactions with the bases. We also estimate, for the first time, the m-value for RNA, which is a measure of the strength of urea-RNA interactions. Our work provides a conceptual understanding of the mechanism by which urea enhances RNA folding rates.

Metal ion dependence of cooperative collapse transitions in RNA

Positively charged counterions drive RNA molecules into compact configurations that lead to their biologically active structures. To understand how the valence and size of the cations influences the collapse transition in RNA, small-angle X-ray scattering was used to follow the decrease in the radius of gyration (R(g)) of the Azoarcus and Tetrahymena ribozymes in different cations. Small, multivalent cations induced the collapse of both ribozymes more efficiently than did monovalent ions. Thus, the cooperativity of the collapse transition depends on the counterion charge density. Singular value decomposition of the scattering curves showed that folding of the smaller and more thermostable Azoarcus ribozyme is well described by two components, whereas collapse of the larger Tetrahymena ribozyme involves at least one intermediate. The ion-dependent persistence length, extracted from the distance distribution of the scattering vectors, shows that the Azoarcus ribozyme is less flexible at the midpoint of transition in low-charge-density ions than in high-charge-density ions. We conclude that the formation of sequence-specific tertiary interactions in the Azoarcus ribozyme overlaps with neutralization of the phosphate charge, while tertiary folding of the Tetrahymena ribozyme requires additional counterions. Thus, the stability of the RNA structure determines its sensitivity to the valence and size of the counterions.

How accurate are polymer models in the analysis of Förster resonance energy transfer experiments on proteins?

Single molecule Förster resonance energy transfer (FRET) experiments are used to infer the properties of the denatured state ensemble (DSE) of proteins. From the measured average FRET efficiency, <E>, the distance distribution P(R) is inferred by assuming that the DSE can be described as a polymer. The single parameter in the appropriate polymer model (Gaussian chain, wormlike chain, or self-avoiding walk) for P(R) is determined by equating the calculated and measured <E>. In order to assess the accuracy of this "standard procedure," we consider the generalized Rouse model (GRM), whose properties [<E> and P(R)] can be analytically computed, and the Molecular Transfer Model for protein L for which accurate simulations can be carried out as a function of guanadinium hydrochloride (GdmCl) concentration. Using the precisely computed <E> for the GRM and protein L, we infer P(R) using the standard procedure. We find that the mean end-to-end distance can be accurately inferred (less than 10% relative error) using <E> and polymer models for P(R). However, the value extracted for the radius of gyration (R(g)) and the persistence length (l(p)) are less accurate. For protein L, the errors in the inferred properties increase as the GdmCl concentration increases for all polymer models. The relative error in the inferred R(g) and l(p), with respect to the exact values, can be as large as 25% at the highest GdmCl concentration. We propose a self-consistency test, requiring measurements of <E> by attaching dyes to different residues in the protein, to assess the validity of describing DSE using the Gaussian model. Application of the self-consistency test to the GRM shows that even for this simple model, which exhibits an order-->disorder transition, the Gaussian P(R) is inadequate. Analysis of experimental data of FRET efficiencies with dyes at several locations for the cold shock protein, and simulations results for protein L, for which accurate FRET efficiencies between various locations were computed, shows that at high GdmCl concentrations there are significant deviations in the DSE P(R) from the Gaussian model.

Dynamics of tRNA at different levels of hydration

The influence of hydration on the nanosecond timescale dynamics of tRNA is investigated using neutron scattering spectroscopy. Unlike protein dynamics, the dynamics of tRNA is not affected by methyl group rotation. This allows for a simpler analysis of the influence of hydration on the conformational motions in RNA. We find that hydration affects the dynamics of tRNA significantly more than that of lysozyme. Both the characteristic length scale and the timescale of the conformational motions in tRNA depend strongly on hydration. Even the characteristic temperature of the so-called "dynamical transition" appears to be hydration-dependent in tRNA. The amplitude of the conformational motions in fully hydrated tRNA is almost twice as large as in hydrated lysozyme. We ascribe these differences to a more open and flexible structure of hydrated RNA, and to a larger fraction and different nature of hydrophilic sites. The latter leads to a higher density of water that makes the biomolecule more flexible. All-atom molecular-dynamics simulations are used to show that the extent of hydration is greater in tRNA than in lysozyme. We propose that water acts as a "lubricant" in facilitating enhanced motion in solvated RNA molecules.

Molecular origin of constant m-values, denatured state collapse, and residue-dependent transition midpoints in globular proteins

Experiments show that for many two-state folders the free energy of the native state, DeltaG(ND)([C]), changes linearly as the denaturant concentration, [C], is varied. The slope {m = [dDeltaG(ND)([C])]/(d[C])}, is nearly constant. According to the transfer model, the m-value is associated with the difference in the surface area between the native (N) and denatured (D) state, which should be a function of DeltaR(g)(2), the difference in the square of the radius of gyration between the D and N states. Single-molecule experiments show that the R(g) of the structurally heterogeneous denatured state undergoes an equilibrium collapse transition as [C] decreases, which implies m also should be [C]-dependent. We resolve the conundrum between constant m-values and [C]-dependent changes in R(g) using molecular simulations of a coarse-grained representation of protein L, and the molecular transfer model, for which the equilibrium folding can be accurately calculated as a function of denaturant (urea) concentration. In agreement with experiment, we find that over a large range of denaturant concentration (>3 M) the m-value is a constant, whereas under strongly renaturing conditions (<3 M), it depends on [C]. The m-value is a constant above [C] > 3 M because the [C]-dependent changes in the surface area of the backbone groups, which make the largest contribution to m, are relatively narrow in the denatured state. The burial of the backbone and hydrophobic side chains gives rise to substantial surface area changes below [C] < 3 M, leading to collapse in the denatured state of protein L. Dissection of the contribution of various amino acids to the total surface area change with [C] shows that both the sequence context and residual structure are important. There are [C]-dependent variations in the surface area for chemically identical groups such as the backbone or Ala. Consequently, the midpoints of transition of individual residues vary significantly (which we call the Holtzer effect) even though global folding can be described as an all-or-none transition. The collapse is specific in nature, resulting in the formation of compact structures with appreciable populations of nativelike secondary structural elements. The collapse transition is driven by the loss of favorable residue-solvent interactions and a concomitant increase in the strength of intrapeptide interactions with a decreasing [C]. The strength of these interactions is nonuniformly distributed throughout the structure of protein L. Certain secondary structure elements have stronger [C]-dependent interactions than others in the denatured state.

Transmembrane structures of amyloid precursor protein dimer predicted by replica-exchange molecular dynamics simulations

Abeta peptide is an essential protein in the pathogenesis of Alzheimer's disease and is derived from amyloid precursor protein (APP) in the membrane by beta- and gamma-secretase cleavage. An experimental study has shown that a pairwise replacement of Gly with Leu in APP enhances homodimerization but leads to a drastic reduction of Abeta secretion. To resolve this apparent discrepancy, we predicted the wild-type (WT) and mutant APP dimer conformations by replica-exchange molecular dynamics simulations using the implicit membrane model IMM1. The simulations illustrate large conformational differences between the WT and mutant APP fragments in the membrane. Dimerization of the WT is due to two C(alpha)-H...O hydrogen bonds between two APP fragments, whereas dimerization of the mutant is due to hydrophobic interactions. In the mutant, each APP fragment is more tilted, and the gamma-cleavage site is shifted toward the center of the membrane. This position produces a mismatch between the active site of gamma-secretase and the gamma-cleavage site of APP that might prohibit Abeta production.

Crowding effects on the structural transitions in a flexible helical homopolymer

We elucidate the structural transitions in a helical off-lattice homopolymer induced by crowding agents, as a function of the number of monomers (N) and volume fraction (varphi c) of crowding particles. At varphic=0, the homopolymer undergoes transitions from a random coil to a helix, helical hairpin HH, and helix bundle HB structures depending on N, and temperature. Crowding induces chain compaction that can promote HH or HB formation depending on varphic. Typically, the helical content decreases which is reflected in the decrease in the transition temperatures that depend on varphic, N, and the size of the crowding particles.

Water-mediated interactions between hydrophobic and ionic species in cylindrical nanopores

We use Metropolis Monte Carlo and umbrella sampling to calculate the free energies of interaction of two methane molecules and their charged derivatives in cylindrical water-filled pores. Confinement strongly alters the interactions between the nonpolar solutes and completely eliminates the solvent separated minimum (SSM) that is seen in bulk water. The free energy profiles show that the methane molecules are either in contact or at separations corresponding to the diameter and the length of the cylindrical pore. Analytic calculations that estimate the entropy of the solutes, which are solvated at the pore surface, qualitatively explain the shape of the free energy profiles. Adding charges of opposite sign and magnitude 0.4e or e (where e is the electronic charge) to the methane molecules decreases their tendency for surface solvation and restores the SSM. We show that confinement induced ion-pair formation occurs whenever l(B)/D approximately O(1), where l(B) is the Bjerrum length and D is the pore diameter. The extent of stabilization of the SSM increases with ion charge density as long as l(B)/D<1. In pores with D<or=1.2 nm, in which the water is strongly layered, increasing the charge magnitude from 0.4e to e reduces the stability of the SSM. As a result, ion-pair formation that occurs with negligible probability in the bulk is promoted. In larger diameter pores that can accommodate a complete hydration layer around the solutes, the stability of the SSM is enhanced.

Crowding effects on the mechanical stability and unfolding pathways of ubiquitin

The interiors of cells are crowded, thus making it important to assess the effects of macromolecules on the folding of proteins. Using the self-organized polymer (SOP) model, which is a coarse-grained representation of polypeptide chains, we probe the mechanical stability of ubiquitin (Ub) monomers and trimers ((Ub)(3)) in the presence of monodisperse spherical crowding agents. Crowding increases the volume fraction (Phi(c))-dependent average force (f(u)(Phi(c))), relative to the value at Phi(c) = 0, needed to unfold Ub and the polyprotein. For a given Phi(c), the values of f(u)(Phi(c)) increase as the diameter (sigma(c)) of the crowding particles decreases. The average unfolding force f(u)(Phi(c)) depends on the ratio D/R(g), where D approximately sigma(c)(pi/6Phi(c))(1/3), with R(g) being the radius of gyration of Ub (or (Ub)(3)) in the unfolded state. Examination of the unfolding pathways shows that, relative to Phi(c) = 0, crowding promotes reassociation of ruptured secondary structural elements. Both the nature of the unfolding pathways and f(u)(Phi(c)) for (Ub)(3) are altered in the presence of crowding particles, with the effect being most dramatic for the subunit that unfolds last. We predict, based on SOP simulations and theoretical arguments, that f(u)(Phi(c)) approximately Phi(c)(1/3nu), where nu is the Flory exponent that describes the unfolded (random coil) state of the protein.

Collapse transition in proteins

The coil-globule transition, a tenet of the physics of polymers, has been identified in recent years as an important unresolved aspect of the initial stages of the folding of proteins. We describe the basics of the collapse transition, starting with homopolymers and continuing with proteins. Studies of denatured-state collapse under equilibrium are then presented. An emphasis is placed on single-molecule fluorescence experiments, which are particularly useful for measuring properties of the denatured state even under conditions of coexistence with the folded state. Attempts to understand the dynamics of collapse, both theoretically and experimentally, are then described. Only an upper limit for the rate of collapse has been obtained so far. Improvements in experimental and theoretical methodology are likely to continue to push our understanding of the importance of the denatured-state thermodynamics and dynamics for protein folding in the coming years.

Semiflexible chains in confined spaces

We develop an analytical method for studying the properties of a noninteracting wormlike chain (WLC) in confined geometries. The mean-field-like theory replaces the rigid constraints of confinement with average constraints, thus allowing us to develop a tractable method for treating a WLC wrapped on the surface of a sphere, and fully encapsulated within it. The efficacy of the theory is established by reproducing the exact correlation functions for a WLC confined to the surface of a sphere. In addition, the coefficients in the free energy are exactly calculated. We also describe the behavior of a surface-confined chain under external tension that is relevant for single molecule experiments on histone-DNA complexes. The force-extension curves display spatial oscillations, and the extension of the chain, whose maximum value is bounded by the sphere diameter, scales as f(-1) at large forces, in contrast to the unconfined chain that approaches the contour length as f(-1/2). A WLC encapsulated in a sphere, that is relevant for the study of the viral encapsulation of DNA, can also be treated using the mean-field approach. The predictions of the theory for various correlation functions are in excellent agreement with Langevin simulations. We find that strongly confined chains are highly structured by examining the correlations using a local winding axis. The predicted pressure of the system is in excellent agreement with simulations but, as is known, is significantly lower than the pressures seen for DNA packaged in viral capsids.

Probing the mechanisms of fibril formation using lattice models

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.

Factors governing helix formation in peptides confined to carbon nanotubes

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.

Relative stability of helices determines the folding landscape of adenine riboswitch aptamers

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.

Compression and stretching of a self-avoiding chain in cylindrical nanopores

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.

Force-dependent hopping rates of RNA hairpins can be estimated from accurate measurement of the folding landscapes

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.

Role of internal chain dynamics on the rupture kinetic of adhesive contacts

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.

Effects of trimethylamine N-oxide (TMAO) and crowding agents on the stability of RNA hairpins

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.

Kinetics of loop formation in polymer chains

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.

Kinetic model for the coupling between allosteric transitions in GroEL and substrate protein folding and aggregation

The bacterial chaperonin GroEL and the co-chaperonin GroES assist in the folding of a number of structurally unrelated substrate proteins (SPs). In the absence of chaperonins, SP folds by the kinetic partitioning mechanism (KPM), according to which a fraction of unfolded molecules reaches the native state directly, while the remaining fraction gets trapped in a potentially aggregation-prone misfolded state. During the catalytic reaction cycle, GroEL undergoes a series of allosteric transitions (T<-->R-->R"-->T) triggered by SP capture, ATP binding and hydrolysis, and GroES binding. We developed a general kinetic model that takes into account the coupling between the rates of the allosteric transitions and the folding and aggregation of the SP. Our model, in which the GroEL allosteric rates and SP-dependent folding and aggregation rates are independently varied without prior assumption, quantitatively fits the GroEL concentration-dependent data on the yield of native ribulose bisphosphate carboxylase/oxygenase (Rubisco) as a function of time. The extracted kinetic parameters for the GroEL reaction cycle are consistent with the available values from independent experiments. In addition, we also obtained physically reasonable parameters for the kinetic steps in the reaction cycle that are difficult to measure. If experimental values for GroEL allosteric rates are used, the time-dependent changes in native-state yield at eight GroEL concentrations can be quantitatively fit using only three SP-dependent parameters. The model predicts that the differences in the efficiencies (as measured by yields of the native state) of GroEL, single-ring mutant (SR1), and variants of SR1, in the rescue of mitochondrial malate dehydrogenase, citrate synthase, and Rubisco, are related to the large variations in the allosteric transition rates. We also show that GroEL/S mutants that efficiently fold one SP at the expense of all others are due to a decrease in the rate of a key step in the reaction cycle, which implies that wild-type GroEL has evolved as a compromise between generality and specificity. We predict that, under maximum loading conditions and saturating ATP concentration, the efficiency of GroEL (using parameters for Rubisco) depends predominantly on the rate of R-->R" transition, while the equilibrium constant of the T<-->R has a small effect only. Both under sub- and superstoichiometric GroEL concentrations, enhanced efficiency is achieved by rapid turnover of the reaction cycle, which is in accord with the predictions of the iterative annealing mechanism. The effects are most dramatic at substoichiometric conditions (most relevant for in vivo situations) when SP aggregation can outcompete capture of SP by chaperonins.

Dissecting contact potentials for proteins: relative contributions of individual amino acids

Knowledge-based contact potentials are routinely used in fold recognition, binding of peptides to proteins, structure prediction, and coarse-grained models to probe protein folding kinetics. The dominant physical forces embodied in the contact potentials are revealed by eigenvalue analysis of the matrices, whose elements describe the strengths of interaction between amino acid side chains. We propose a general method to rank quantitatively the importance of various inter-residue interactions represented in the currently popular pair contact potentials. Eigenvalue analysis and correlation diagrams are used to rank the inter-residue pair interactions with respect to the magnitude of their relative contributions to the contact potentials. The amino acid ranking is shown to be consistent with a mean field approximation that is used to reconstruct the original contact potentials from the most relevant amino acids for several contact potentials. By providing a general, relative ranking score for amino acids, this method permits a detailed, quantitative comparison of various contact interaction schemes. For most contact potentials, between 7 and 9 amino acids of varying chemical character are needed to accurately reconstruct the full matrix. By correlating the identified important amino acid residues in contact potentials and analysis of about 7800 structural domains in the CATH database we predict that it is important to model accurately interactions between small hydrophobic residues. In addition, only potentials that take interactions involving the protein backbone into account can predict dense packing in protein structures.

Kinetics of interior loop formation in semiflexible chains

Loop formation between monomers in the interior of semiflexible chains describes elementary events in biomolecular folding and DNA bending. We calculate analytically the interior distance distribution function for semiflexible chains using a mean field approach. Using the potential of mean force derived from the distance distribution function we present a simple expression for the kinetics of interior looping by adopting Kramers theory. For the parameters, that are appropriate for DNA, the theoretical predictions in comparison with the case are in excellent agreement with explicit Brownian dynamics simulations of wormlike chain (WLC) model. The interior looping times (tauIC) can be greatly altered in the cases when the stiffness of the loop differs from that of the dangling ends. If the dangling end is stiffer than the loop then tauIC increases for the case of the WLC with uniform persistence length. In contrast, attachment of flexible dangling ends enhances rate of interior loop formation. The theory also shows that if the monomers are charged and interact via screened Coulomb potential then both the cyclization (tauc) and interior looping (tauIC) times greatly increase at low ionic concentration. Because both tauc and tauIC are determined essentially by the effective persistence length [lp(R)] we computed lp(R) by varying the range of the repulsive interaction between the monomers. For short range interactions lp(R) nearly coincides with the bare persistence length which is determined largely by the backbone chain connectivity. This finding rationalizes the efficacy of describing a number of experimental observations (response of biopolymers to force and cyclization kinetics) in biomolecules using WLC model with an effective persistence length.

Effect of finite size on cooperativity and rates of protein folding

We analyze the dependence of cooperativity of the thermal denaturation transition and folding rates of globular proteins on the number of amino acid residues, N, using lattice models with side chains, off-lattice Go models, and the available experimental data. A dimensionless measure of cooperativity, Omega(c) (0 < Omega(c) < infinity), scales as Omega(c) approximately N(zeta). The results of simulations and the analysis of experimental data further confirm the earlier prediction that zeta is universal with zeta = 1 + gamma, where exponent gamma characterizes the susceptibility of a self-avoiding walk. This finding suggests that the structural characteristics in the denaturated state are manifested in the folding cooperativity at the transition temperature. The folding rates k(F) for the Go models and a dataset of 69 proteins can be fit using k(F) = k(F)0 exp(-cN(beta)). Both beta = 1/2 and 2/3 provide a good fit of the data. We find that k(F) = k(F)0 exp(-cN(1/2)), with the average (over the dataset of proteins) k(F)0 approximately (0.2 micros)(-1) and c approximately 1.1, can be used to estimate folding rates to within an order of magnitude in most cases. The minimal models give identical N dependence with c approximately 1. The prefactor for off-lattice Go models is nearly 4 orders of magnitude larger than the experimental value.

Effects of Crowding and Confinement on the Structures of the Transition State Ensemble in Proteins

Characterization of the structures of the transition state ensemble is a key step in describing the folding reaction. Using two variants of a coarse-grained model of the three-stranded beta-sheet WW domain and a fully automated progress variable clustering (PVC) algorithm, we have dissected the effect of macromolecular crowding and confinement on the changes in the transition state structures in comparison to bulk. Each amino acid is represented using a Calpha atom and a side chain. The distance between the Calpha atom and center of mass of the side chain is taken to be its effective van der Waals radius. For the bulk case, we predict using the PVC algorithm, which does not assume knowledge of the underlying folding reaction coordinate, that there are two classes of structures in the transition state ensemble (TSE). The structures in both of the classes are compact. The dominant cluster is more structured than the structures in the less populated class. In accord with bulk experiments, the residues in strands beta2 and beta3 and the interactions at the beta2-beta3 interface are structured. When only excluded volume interactions between the crowding particles and the WW domain are taken into account or when the protein is confined to an inert spherical pore, the overall structure of the TSE does not change dramatically. However, in this entropy dominated regime, the width of the TSE decreases and the structures become more oblate and less spherical as the volume fraction of crowding particle increases or when the pore radius decreases. It suggests that the shape changes, which are computed using the moment of inertia tensor, may represent the slow degrees of freedom during the folding process. When non-native interactions between side chains and interactions with the cavity of the pores are taken into account, the TSE becomes considerably broader. Although the topology in the transition has a fold similar to the native state, the structures are far more plastic than in the bulk. The TSE is sensitive to the size of the pore as well as interactions between the pore and the protein. The differences between the two cases (confinement in an inert pore and when pore-protein interactions are considered) arise due to the increased importance of enthalpic interactions in the transition state as the strength of the protein-pore interaction increases.

Allosteric transitions in the chaperonin GroEL are captured by a dominant normal mode that is most robust to sequence variations

The Escherichia coli chaperonin GroEL, which helps proteins to fold, consists of two heptameric rings stacked back-to-back. During the reaction cycle GroEL undergoes a series of allosteric transitions triggered by ligand (substrate protein, ATP, and the cochaperonin GroES) binding. Based on an elastic network model of the bullet-shaped double-ring chaperonin GroEL-(ADP)(7)-GroES structure (R''T state), we perform a normal mode analysis to explore the energetically favorable collective motions encoded in the R''T structure. By comparing each normal mode with the observed conformational changes in the R''T --> TR'' transition, a single dominant normal mode provides a simple description of this highly intricate allosteric transition. A detailed analysis of this relatively high-frequency mode describes the structural and dynamic changes that underlie the positive intra-ring and negative inter-ring cooperativity. The dynamics embedded in the dominant mode entails highly concerted structural motions with approximate preservation of sevenfold symmetry within each ring and negatively correlated ones between the two rings. The dominant normal mode (in comparison with the other modes) is robust to parametric perturbations caused by sequence variations, which validates its functional importance. Response of the dominant mode to local changes that mimic mutations using the structural perturbation method technique leads to a wiring diagram that identifies a network of key residues that regulate the allosteric transitions. Many of these residues are located in intersubunit interfaces, and may therefore play a critical role in transmitting allosteric signals between subunits.

Interactions between Hydrophobic and Ionic Solutes in Aqueous Guanidinium Chloride and Urea Solutions: Lessons for Protein Denaturation Mechanism

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.

Coupling between allosteric transitions in GroEL and assisted folding of a substrate protein

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.

Charge density of divalent metal cations determines RNA stability

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.

Size, shape, and flexibility of RNA structures

Determination of sizes and flexibilities of RNA molecules is important in understanding the nature of packing in folded structures and in elucidating interactions between RNA and DNA or proteins. Using the coordinates of the structures of RNA in the Protein Data Bank we find that the size of the folded RNA structures, measured using the radius of gyration R(G), follows the Flory scaling law, namely, R(G)=5.5N(1/3) A, where N is the number of nucleotides. The shape of RNA molecules is characterized by the asphericity Delta and the shape S parameters that are computed using the eigenvalues of the moment of inertia tensor. From the distribution of Delta, we find that a large fraction of folded RNA structures are aspherical and the distribution of S values shows that RNA molecules are prolate (S>0). The flexibility of folded structures is characterized by the persistence length l(p). By fitting the distance distribution function P(r), that is computed using the coordinates of the folded RNA, to the wormlike chain model we extracted the persistence length l(p). We find that l(p) approximately 1.5N(0.33) A which might reflect the large separation between the free energies that stabilize secondary and tertiary structures. The dependence of l(p) on N implies that the average length of helices should increase as the size of RNA grows. We also analyze packing in the structures of ribosomes (30S, 50S, and 70S) in terms of R(G), Delta, S, and l(p). The 70S and the 50S subunits are more spherical compared to most RNA molecules. The globularity in 50S is due to the presence of an unusually large number (compared to 30S subunit) of small helices that are stitched together by bulges and loops. Comparison of the shapes of the intact 70S ribosome and the constituent particles suggests that folding of the individual molecules might occur prior to assembly.

Pathways and kinetic barriers in mechanical unfolding and refolding of RNA and proteins

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.

Monomer adds to preformed structured oligomers of Abeta-peptides by a two-stage dock-lock mechanism

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.

Dynamic Competition between Catch and Slip Bonds in Selectins Bound to Ligands

Atomic force measurements of unbinding rates (or off-rates) of ligands bound to a class of cell adhesion molecules from the selectin family show a transition from catch to slip bonds as the value of external force (f) is increased. At low forces (<10 pN), the unbinding rates decrease (catch regime), while, at high forces, the rates increase in accord with the Bell model (slip regime). The energy landscape underlying the catch-slip transition can be captured by a two-state model that considers the possibility of redistribution of population from the force-free bound state to the force-stabilized bound state. The excellent agreement between theory and experiments is used to extract the parameters characterizing the energy landscape of the complex by fitting the calculated curves to lifetime data (obtained at constant f) for the monomeric form of PSGL-1 (sPSGL-1). We used the constant force parameters to predict the distributions of unbinding times and unbinding forces as a function of the loading rate. The general two-state model, which also correctly predicts the absence of catch bonds in the binding of antibodies to selectins, is used to resolve the energy landscape parameters characterizing adhesive interactions of P- and L-selectins with physiological ligands such as sPSGL-1 and endoglycan and antibodies such as G1 and DREG56. Despite high sequence similarity, the underlying shapes of the energy landscape of P-selectin and L-selectin interacting with sPSGL-1 are markedly different. The underlying energy landscape of the selectin cell adhesion complex is sensitive to the nature of the ligand. The unified description of selectins bound to physiological ligands and antibodies in conjunction with experimental data can be used to extract the key parameters that describe the dynamics of cell adhesion complexes.