We use Brownian dynamics simulations of a binary mixture of highly charged spherical colloidal particles to test some of the predictions of the random first-order transition (RFOT) theory [Phys. Rev. Lett. 58, 2091 (1987); Phys. Rev. A 40, 1045 (1989)]. In accord with mode-coupling theory and RFOT, we find that as the volume fraction of the colloidal particles ϕ approaches the dynamical transition value ϕ(A), three measures of dynamics show an effective ergodic to nonergodic transition. First, there is a dramatic slowing down of diffusion, with the translational diffusion constant decaying as a power law as ϕ→ϕ(A)(-). Second, the energy metric, a measure of ergodicity breaking in classical many-body systems, shows that the system becomes effectively nonergodic as ϕ(A) is approached. Finally, the time t(*), at which the four-point dynamical susceptibility achieves a maximum, also increases as a power law near ϕ(A). Remarkably, the translational diffusion coefficients, ergodic diffusion coefficient, and (t(*))(-) all vanish as (ϕ(-1)-ϕ(A)(-1))(γ) with both ϕ(A)(≈0.1) and γ being the roughly the same for all three quantities. Above ϕ(A), transport involves crossing free energy barriers. In this regime, the density-density correlation function decays as a stretched exponential [exp-(t/τ(α))(β)] with β≈0.45. The ϕ dependence of the relaxation time τ(α) could be fit using the Vogel-Tamman-Fulcher law with the ideal glass transition at ϕ(K)≈0.47. By using a local entropy measure, we show that the law of large numbers is not obeyed above ϕ(A), and gives rise to subsample to subsample fluctuations in all physical observables. We propose that dynamical heterogeneity is a consequence of violation of law of large numbers.
Chemical physics as a discipline contributes many experimental tools, algorithms, and fundamental theoretical models that can be applied to biological problems. This is especially true now as the molecular level and the systems level descriptions begin to connect, and multi-scale approaches are being developed to solve cutting edge problems in biology. In some cases, the concepts and tools got their start in non-biological fields, and migrated over, such as the idea of glassy landscapes, fluorescence spectroscopy, or master equation approaches. In other cases, the tools were specifically developed with biological physics applications in mind, such as modeling of single molecule trajectories or super-resolution laser techniques. In this introduction to the special topic section on chemical physics of biological systems, we consider a wide range of contributions, all the way from the molecular level, to molecular assemblies, chemical physics of the cell, and finally systems-level approaches, based on the contributions to this special issue. Chemical physicists can look forward to an exciting future where computational tools, analytical models, and new instrumentation will push the boundaries of biological inquiry.
The osmolyte trimethylamine N-oxide (TMAO) stabilizes the tertiary but not the secondary structures of RNA. However, molecular dynamics simulations performed on the PreQ1 riboswitch showed that TMAO destabilizes the tertiary riboswitch structure, leading us to hypothesize that the presence of RNA could result in enhanced population of the protonated form, TMAOP. Constant pH replica exchange simulations showed that a percentage of TMAO is indeed protonated, thus contributing to the stability of the tertiary but not the secondary structure of PreQ1. TMAOP results in an unfavorable dehydration of phosphodiester backbone, which is compensated by electrostatic attraction between TMAOP and the phosphate groups. In addition, TMAOP interacts with specific sites in the tertiary RNA structure, mimicking the behavior of positively charged ions and of the PreQ1 ligand in stabilizing RNA. Finally, we predict that TMAO-induced stabilization of RNA tertiary structures should be strongly pH dependent.
Urea, a polar molecule with a large dipole moment, not only destabilizes folded RNA structures but can also enhance the folding rates of large ribozymes. Unlike the mechanism of urea-induced unfolding of proteins, which is well understood, the action of urea on RNA has barely been explored. We performed extensive all-atom molecular dynamics simulations to determine the molecular underpinnings of urea-induced RNA denaturation. Urea displays its denaturing power in both secondary and tertiary motifs of the riboswitch structure. Our simulations reveal that the denaturation of RNA structures is mainly driven by the hydrogen-bonding and stacking interactions of urea with the bases. Through detailed studies of the simulation trajectories, we found that geminate pairs between urea and bases due to hydrogen bonds and stacks persist only ~0.1-1 ns, which suggests that the urea-base interaction is highly dynamic. Most importantly, the early stage of base-pair disruption is triggered by penetration of water molecules into the hydrophobic domain between the RNA bases. The infiltration of water into the narrow space between base pairs is critical in increasing the accessibility of urea to transiently disrupted bases, thus allowing urea to displace inter-base hydrogen bonds. This mechanism--water-induced disruption of base pairs resulting in the formation of a "wet" destabilized RNA followed by solvation by urea--is the exact opposite of the two-stage denaturation of proteins by urea. In the latter case, initial urea penetration creates a dry globule, which is subsequently solvated by water, leading to global protein unfolding. Our work shows that the ability to interact with both water and polar or nonpolar components of nucleotides makes urea a powerful chemical denaturant for nucleic acids.
A variety of neurodegenerative diseases are associated with amyloid plaques, which begin as soluble protein oligomers but develop into amyloid fibrils. Our incomplete understanding of this process underscores the need to decipher the principles governing protein aggregation. Mechanisms of in vivo amyloid formation involve a number of coconspirators and complex interactions with membranes. Nevertheless, understanding the biophysical basis of simpler in vitro amyloid formation is considered important for discovering ligands that preferentially bind regions harboring amyloidogenic tendencies. The determination of the fibril structure of many peptides has set the stage for probing the dynamics of oligomer formation and amyloid growth through computer simulations. Most experimental and simulation studies, however, have been interpreted largely from the perspective of proteins: the role of solvent has been relatively overlooked in oligomer formation and assembly to protofilaments and amyloid fibrils. In this Account, we provide a perspective on how interactions with water affect folding landscapes of amyloid beta (Aβ) monomers, oligomer formation in the Aβ16-22 fragment, and protofilament formation in a peptide from yeast prion Sup35. Explicit molecular dynamics simulations illustrate how water controls the self-assembly of higher order structures, providing a structural basis for understanding the kinetics of oligomer and fibril growth. Simulations show that monomers of Aβ peptides sample a number of compact conformations. The formation of aggregation-prone structures (N*) with a salt bridge, strikingly similar to the structure in the fibril, requires overcoming a high desolvation barrier. In general, sequences for which N* structures are not significantly populated are unlikely to aggregate. Oligomers and fibrils generally form in two steps. First, water is expelled from the region between peptides rich in hydrophobic residues (for example, Aβ16-22), resulting in disordered oligomers. Then the peptides align along a preferred axis to form ordered structures with anti-parallel β-strand arrangement. The rate-limiting step in the ordered assembly is the rearrangement of the peptides within a confining volume. The mechanism of protofilament formation in a polar peptide fragment from the yeast prion, in which the two sheets are packed against each other and create a dry interface, illustrates that water dramatically slows self-assembly. As the sheets approach each other, two perfectly ordered one-dimensional water wires form. They are stabilized by hydrogen bonds to the amide groups of the polar side chains, resulting in the formation of long-lived metastable structures. Release of trapped water from the pore creates a helically twisted protofilament with a dry interface. Similarly, the driving force for addition of a solvated monomer to a preformed fibril is water release; the entropy gain and favorable interpeptide hydrogen bond formation compensate for entropy loss in the peptides. We conclude by offering evidence that a two-step model, similar to that postulated for protein crystallization, must also hold for higher order amyloid structure formation starting from N*. Distinct water-laden polymorphic structures result from multiple N* structures. Water plays multifarious roles in all of these protein aggregations. In predominantly hydrophobic sequences, water accelerates fibril formation. In contrast, water-stabilized metastable intermediates dramatically slow fibril growth rates in hydrophilic sequences.
We show that the folding rates (k(F)s) of RNA are determined by N, the number of nucleotides. By assuming that the distribution of free-energy barriers separating the folded and the unfolded states is Gaussian, which follows from central limit theorem arguments and polymer physics concepts, we show that k(F)≈k(0)exp(-αN(0.5)). Remarkably, the theory fits experimental rates spanning over 7 orders of magnitude with k(0)~1.0(μs)(-1). Our finding suggests that the speed limit of RNA folding is ~ 1 μs, [corrected] just as it is in the folding of globular proteins.
We use molecular simulations using a coarse-grained model to map the folding landscape of Green Fluorescent Protein (GFP), which is extensively used as a marker in cell biology and biotechnology. Thermal and Guanidinium chloride (GdmCl) induced unfolding of a variant of GFP, without the chromophore, occurs in an apparent two-state manner. The calculated midpoint of the equilibrium folding in GdmCl, taken into account using the Molecular Transfer Model (MTM), is in excellent agreement with the experiments. The melting temperatures decrease linearly as the concentrations of GdmCl and urea are increased. The structural features of rarely populated equilibrium intermediates, visible only in free energy profiles projected along a few order parameters, are remarkably similar to those identified in a number of ensemble experiments in GFP with the chromophore. The excellent agreement between simulations and experiments show that the equilibrium intermediates are stabilized by the chromophore. Folding kinetics, upon temperature quench, show that GFP first collapses and populates an ensemble of compact structures. Despite the seeming simplicity of the equilibrium folding, flux to the native state flows through multiple channels and can be described by the kinetic partitioning mechanism. Detailed analysis of the folding trajectories show that both equilibrium and several kinetic intermediates, including misfolded structures, are sampled during folding. Interestingly, the intermediates characterized in the simulations coincide with those identified in single molecule pulling experiments. Our predictions, amenable to experimental tests, show that MTM is a practical way to simulate the effect of denaturants on the folding of large proteins.
Kinesin walks processively on microtubules in an asymmetric hand-over-hand manner with each step spanning 16 nm. We used molecular simulations to determine the fraction of a single step due to conformational changes in the neck linker, and that due to diffusion of the tethered head. Stepping is determined largely by two energy scales, one favoring neck-linker docking and the other, ε(h)(MT-TH), between the trailing head (TH) and the microtubule. Neck-linker docking and an optimal value of ε(h)(MT-TH) are needed to minimize the probability that the TH takes side steps. There are three major stages in the kinematics of a step. In the first, the neck linker docks, resulting in ∼(5-6) nm movements of the trailing head. The TH moves an additional (6-8) nm in stage II by anisotropic translational diffusion. In the third stage, spanning ∼(3-4) nm, the step is complete with the TH binding to the αβ-tubulin binding site.
Protein conformations change among distinct thermodynamic states as solution conditions (temperature, denaturants, pH) are altered or when they are subjected to mechanical forces. A quantitative description of the changes in the relative stabilities of the various thermodynamic states is needed to interpret and predict experimental outcomes. We provide a framework based on the Molecular Transfer Model (MTM) to account for pH effects on the properties of globular proteins. The MTM utilizes the partition function of a protein calculated from molecular simulations at one set of solution conditions to predict protein properties at another set of solution conditions. To take pH effects into account, we utilized experimentally measured pK(a) values in the native and unfolded states to calculate the free energy of transferring a protein from a reference pH to the pH of interest. We validate our approach by demonstrating that the native-state stability as a function of pH is accurately predicted for chymotrypsin inhibitor 2 (CI2) and protein G. We use the MTM to predict the response of CI2 and protein G subjected to a constant force (f) and varying pH. The phase diagrams of CI2 and protein G as a function of f and pH are dramatically different and reflect the underlying pH-dependent stability changes in the absence of force. The calculated equilibrium free energy profiles as functions of the end-to-end distance of the two proteins show that, at various pH values, CI2 unfolds via an intermediate when subjected to f. The locations of the two transition states move toward the more unstable state as f is changed, which is in accord with the Hammond-Leffler postulate. In sharp contrast, force-induced unfolding of protein G occurs in a single step. Remarkably, the location of the transition state with respect to the folded state is independent of f, which suggests that protein G is mechanically brittle. The MTM provides a natural framework for predicting the outcomes of ensemble and single-molecule experiments for a wide range of solution conditions.
Folding mechanisms in which secondary structures are stabilized through the formation of tertiary interactions are well documented in protein folding but challenge the folding hierarchy normally assumed for RNA. However, it is increasingly clear that RNA could fold by a similar mechanism. P5abc, a small independently folding tertiary domain of the Tetrahymena thermophila group I ribozyme, is known to fold by a secondary structure rearrangement involving helix P5c. However, the extent of this rearrangement and the precise stage of folding that triggers it are unknown. We use experiments and simulations to show that the P5c helix switches to the native secondary structure late in the folding pathway and is directly coupled to the formation of tertiary interactions in the A-rich bulge. P5c mutations show that the switch in P5c is not rate-determining and suggest that non-native interactions in P5c aid folding rather than impede it. Our study illustrates that despite significant differences in the building blocks of proteins and RNA, there may be common ways in which they self-assemble.
We investigate the structural transitions in a polymer induced by spherical and nonspherical crowding particles over a wide range of conditions. The polymer conformations are specified by the radius of gyration and the quality of the solvent in the absence of crowding particles. In the presence of crowding particles, the structures are altered by the volume fraction, size, shape, and polydispersity of the crowders. We show that crowding induces an array of structural changes, ranging from helix, helical hairpin (HH), and multiple helix bundles (HBs), depending on the interplay of multiple length and energy scales including the solvent quality, length of the polymer, temperature, and the characteristics of the crowding agents. In nearly good solvents, the polymer undergoes coil-helix transition in accord with the predictions based on the entropic stabilization mechanism. Higher-order (HH and HB) structures are obtained in poor or moderately poor solvents. In a binary mixture of spherical crowders, the effect of the two components is largely additive with the polymer undergoing greater compaction at higher volume fraction. In contrast to spherical crowders, spherocylinder-like crowders have a dramatically different effect on the diagram of states of the polymer. In the presence of spherocylinders, the polymer prefers to form a nearly ideal helix, especially at low temperatures and high aspect ratios of the crowders, at volume fractions that are not large enough for nematic order. Surprisingly, there is a complete absence of HH and HB in the range of conditions explored here. The dominant formation of spherocylinder-induced helix formation is due to the tendency of the spherocylinders and the polymer to align along the director formed by an increase in nematic order only in the vicinity of the polymer. Our study, which has produced several testable predictions, shows that only by probing the effects of crowding on a polymer (or a protein and RNA) over a wide range of conditions can the diagram of states be quantitatively described.
Force-extension curves (FECs), which quantify the response of a variety of biomolecules subject to mechanical force (f), are often quantitatively fit using worm-like chain (WLC) or freely jointed chain (FJC) models. These models predict that the chain extension, x, normalized by the contour length increases linearly at small f and at high forces scale as x ~ (1 - f(-α)), where α = 0.5 for WLC and unity for FJC. In contrast, experiments on single-stranded DNA (ssDNA) show that over a range of f and ionic concentration, x scales as x ~ ln f, which cannot be explained using WLC or FJC models. Using theory and simulations we show that this unusual behavior in FEC in ssDNA is due to sequence-independent polyelectrolyte effects. We show that the x ~ ln f arises because in the absence of force the tangent correlation function, quantifying chain persistence, decays algebraically on length scales on the order of the Debye length. Our theory, which is most appropriate for monovalent salts, quantitatively fits the experimental data and further predicts that such a regime is not discernible in double-stranded DNA.
A theoretical basis for the molecular transfer model (MTM), which takes into account the effects of denaturants by combining experimental data and molecular models for proteins, is provided. We show that the MTM is a mean field-like model that implicitly takes into account denaturant-induced many body interactions. The MTM in conjunction with the coarse-grained self organized polymer model with side chains (SOP-SC) for polypeptide chains is used to simulate the folding of the src-SH3 domain as a function of temperature (T) and guanidine hydrochloride (GdmCl) concentration [C]. Besides reproducing the thermodynamic aspects of SH3 folding, the SOP-SC also captures the cooperativity of the folding transitions. A number of experimentally testable predictions are also made. First, we predict that the melting temperature T(m)([C]) decreases linearly as [C] increases. Second, we show that the midpoints C(m,i) and melting temperatures T(m,i) at which individual residues acquire 50% of their native contacts differ from the global midpoint (C(m) ≈ 2.5 M) and melting temperature (T(m) = 355 K) at which the folded and unfolded states coexist. Dispersion in C(m,i) is greater than that found for T(m,i). Third, folding kinetics at [C] = 0 M shows that the acquisition of contacts between all the secondary structural elements and global folding occur nearly simultaneously. Finally, from the free energy profiles as a function of the structural overlap function and the radius of gyration of the protein, we find that at a fixed T the transition state moves toward the folded state as [C] increases in accord with the Hammond postulate. In contrast, we predict that along the locus of points T(m)([C]) the location of the transition state does not change. The theory and the models used here are sufficiently general for studying the folding of other single domain proteins.
Riboswitches, structured elements in the untranslated regions of messenger RNAs, regulate gene expression by binding specific metabolites. We introduce a kinetic network model that describes the functions of riboswitches at the systems level. Using experimental data for flavin mononucleotide riboswitch as a guide, we show that efficient function, implying a large dynamic range without compromising the requirement to suppress transcription, is determined by a balance between the transcription speed, the folding and unfolding rates of the aptamer, and the binding rates of the metabolite. We also investigated the effect of negative feedback accounting for binding to metabolites, which are themselves the products of genes that are being regulated. For a range of transcription rates negative feedback suppresses gene expression by nearly 10-fold. Negative feedback speeds the gene expression response time, and suppresses the change of steady-state protein concentration by half relative to that without feedback, when there is a modest spike in DNA concentration. A dynamic phase diagram expressed in terms of transcription speed, folding rates, and metabolite binding rates predicts different scenarios in riboswitch-mediated transcription regulation.
A plausible consequence of the rugged folding energy landscapes inherent to biomolecules is that there may be more than one functionally competent folded state. Indeed, molecule-to-molecule variations in the folding dynamics of enzymes and ribozymes have recently been identified in single-molecule experiments, but without systematic quantification or an understanding of their structural origin. Here, using concepts from glass physics and complementary clustering analysis, we provide a quantitative method to analyse single-molecule fluorescence resonance energy transfer (smFRET) data, thereby probing the isomerization dynamics of Holliday junctions, which display such heterogeneous dynamics over a long observation time (T(obs) ≈ 40 s). We show that the ergodicity of Holliday junction dynamics is effectively broken and that their conformational space is partitioned into a folding network of kinetically disconnected clusters. Theory suggests that the persistent heterogeneity of Holliday junction dynamics is a consequence of internal multiloops with varying sizes and flexibilities frozen by Mg(2+) ions. An annealing experiment using Mg(2+) pulses lends support to this idea by explicitly showing that interconversions between trajectories with different patterns can be induced.
Curvatures in the most probable rupture force (f*) versus log-loading rate (log r(f)) observed in dynamic force spectroscopy (DFS) on biomolecular complexes are interpreted using a one-dimensional free energy profile with multiple barriers or a single barrier with force-dependent transition state. Here, we provide a criterion to select one scenario over another. If the rupture dynamics occurs by crossing a single barrier in a physical free energy profile describing unbinding, the exponent ν, from (1 - f*/f(c))(1/ν) ~ (log r(f)) with f(c) being a critical force in the absence of force, is restricted to 0.5 ≤ ν ≤ 1. For biotin-ligand complexes and leukocyte-associated antigen-1 bound to intercellular adhesion molecules, which display large curvature in the DFS data, fits to experimental data yield ν < 0.5, suggesting that if ligand unbinding is assumed to proceed along one-dimensional pulling coordinate, the dynamics should occur in a energy landscape with multiple-barriers.
Non-coding RNA sequences play a great role in controlling a number of cellular functions, thus raising the need to understand their complex conformational dynamics in quantitative detail. In this perspective, we first show that single molecule pulling when combined with with theory and simulations can be used to quantitatively explore the folding landscape of nucleic acid hairpins, and riboswitches with tertiary interactions. Applications to riboswitches, which are non-coding RNA elements that control gene expression by undergoing dynamical conformational changes in response to binding of metabolites, lead to an organization principle that assembly of RNA is determined by the stability of isolated helices. We also point out the limitations of single molecule pulling experiments, with molecular extension as the only accessible parameter, in extracting key parameters of the folding landscapes of RNA molecules.
The distances over which biological molecules and their complexes can function range from a few nanometres, in the case of folded structures, to millimetres, for example, during chromosome organization. Describing phenomena that cover such diverse length, and also time, scales requires models that capture the underlying physics for the particular length scale of interest. Theoretical ideas, in particular, concepts from polymer physics, have guided the development of coarse-grained models to study folding of DNA, RNA and proteins. More recently, such models and their variants have been applied to the functions of biological nanomachines. Simulations using coarse-grained models are now poised to address a wide range of problems in biology.
Quantitative description of how proteins fold under experimental conditions remains a challenging problem. Experiments often use urea and guanidinium chloride to study folding whereas the natural variable in simulations is temperature. To bridge the gap, we use the molecular transfer model that combines measured denaturant-dependent transfer free energies for the peptide group and amino acid residues, and a coarse-grained C(α)-side chain model for polypeptide chains to simulate the folding of src SH(3) domain. Stability of the native state decreases linearly as [C] (the concentration of guanidinium chloride) increases with the slope, m, that is in excellent agreement with experiments. Remarkably, the calculated folding rate at [C] = 0 is only 16-fold larger than the measured value. Most importantly ln k(obs) (k(obs) is the sum of folding and unfolding rates) as a function of [C] has the characteristic V (chevron) shape. In every folding trajectory, the times for reaching the native state, interactions stabilizing all the substructures, and global collapse coincide. The value of (m(f) is the slope of the folding arm of the chevron plot) is identical to the fraction of buried solvent accessible surface area in the structures of the transition state ensemble. In the dominant transition state, which does not vary significantly at low [C], the core of the protein and certain loops are structured. Besides solving the long-standing problem of computing the chevron plot, our work lays the foundation for incorporating denaturant effects in a physically transparent manner either in all-atom or coarse-grained simulations.
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.