Heterogeneity in biological molecules, resulting in molecule-to-molecule variations in their dynamics and function, is an emerging theme. To elucidate the consequences of heterogeneous behavior at the single molecule level, we propose an exactly solvable model in which the unfolding rate due to mechanical force depends parametrically on an auxiliary variable representing an entropy barrier arising from fluctuations in internal dynamics. When the rate of fluctuations--a measure of dynamical disorder--is comparable to or smaller than the rate of force-induced unbinding, we show that there are two experimentally observable consequences: nonexponential survival probability at constant force, and a heavy-tailed rupture force distribution at constant loading rate. By fitting our analytical expressions to data from single molecule pulling experiments on proteins and DNA, we quantify the extent of disorder. We show that only by analyzing data over a wide range of forces and loading rates can the role of disorder due to internal dynamics be quantitatively assessed.
Mechanical forces acting on cell adhesion receptor proteins regulate a range of cellular functions by formation and rupture of noncovalent interactions with ligands. Typically, force decreases the lifetimes of intact complexes ("slip bonds"), making the discovery that these lifetimes can also be prolonged ("catch bonds") a surprise. We created a microscopic analytic theory by incorporating the structures of selectin and integrin receptors into a conceptual framework based on the theory of stochastic equations, which quantitatively explains a wide range of experimental data (including catch bonds at low forces and slip bonds at high forces). Catch bonds arise due to force-induced remodeling of hydrogen bond networks, a finding that also accounts for unbinding in structurally unrelated integrin-fibronectin and actomyosin complexes. For the selectin family, remodeling of hydrogen bond networks drives an allosteric transition resulting in the formation of the maximum number of hydrogen bonds determined only by the structure of the receptor and independent of the ligand. A similar transition allows us to predict the increase in the number of hydrogen bonds in a particular allosteric state of α5β1 integrin-fibronectin complex, a conformation which is yet to be crystallized. We also make a testable prediction that a single point mutation (Tyr51Phe) in the ligand associated with selectin should dramatically alter the nature of the catch bond compared with the wild type. Our work suggests that nature uses a ductile network of hydrogen bonds to engineer function over a broad range of forces.
Protein aggregation, linked to many of diseases, is initiated when monomers access rogue conformations that are poised to form amyloid fibrils. We show, using simulations of src SH3 domain, that mechanical force enhances the population of the aggregation-prone (N(⁎)) states, which are rarely populated under force free native conditions but are encoded in the spectrum of native fluctuations. The folding phase diagrams of SH3 as a function of denaturant concentration ([C]), mechanical force (f), and temperature exhibit an apparent two-state behavior, without revealing the presence of the elusive N(⁎) states. Interestingly, the phase boundaries separating the folded and unfolded states at all [C] and f fall on a master curve, which can be quantitatively described using an analogy to superconductors in a magnetic field. The free energy profiles as a function of the molecular extension (R), which are accessible in pulling experiments, (R), reveal the presence of a native-like N(⁎) with a disordered solvent-exposed amino-terminal β-strand. The structure of the N(⁎) state is identical with that found in Fyn SH3 by NMR dispersion experiments. We show that the timescale for fibril formation can be estimated from the population of the N(⁎) state, determined by the free energy gap separating the native structure and the N(⁎) state, a finding that can be used to assess fibril forming tendencies of proteins. The structures of the N(⁎) state are used to show that oligomer formation and likely route to fibrils occur by a domain-swap mechanism in SH3 domain.
Intermediate filaments (IFs) are key to the mechanical strength of metazoan cells. Their basic building blocks are dimeric coiled coils mediating hierarchical assembly of the full-length filaments. Here we use single-molecule force spectroscopy by optical tweezers to assess the folding and stability of coil 2B of the model IF protein vimentin. The coiled coil was unzipped from its N and C termini. When pulling from the C terminus, we observed that the coiled coil was resistant to force owing to the high stability of the C-terminal region. Pulling from the N terminus revealed that the N-terminal half is considerably less stable. The mechanical pulling assay is a unique tool to study and control seed formation and structure propagation of the coiled coil. We then used rigorous theory-based deconvolution for a model-free extraction of the energy landscape and local stability profiles. The data obtained from the two distinct pulling directions complement each other and reveal a tripartite stability of the coiled coil: a labile N-terminal half, followed by a medium stability section and a highly stable region at the far C-terminal end. The different stability regions provide important insight into the mechanics of IF assembly.
The 99 amino acid C-terminal fragment of amyloid precursor protein (C99), consisting of a single transmembrane (TM) helix, is known to form homodimers. Homodimers can be processed by γ-secretase to produce amyloid-β (Aβ) protein, which is implicated in Alzheimer's disease (AD). While knowledge of the structure of C99 homodimers is of great importance, experimental NMR studies and simulations have produced varying structural models, including right-handed and left-handed coiled-coils. In order to investigate the structure of this critical protein complex, simulations of the C99(15-55) homodimer in POPC membrane bilayer and DPC surfactant micelle environments were performed using a multiscale approach that blends atomistic and coarse-grained models. The C99(15-55) homodimer adopts a dominant right-handed coiled-coil topology consisting of three characteristic structural states in a bilayer, only one of which is dominant in the micelle. Our structural study, which provides a self-consistent framework for understanding a number of experiments, shows that the energy landscape of the C99 homodimer supports a variety of slowly interconverting structural states. The relative importance of any given state can be modulated through environmental selection realized by altering the membrane or micelle characteristics.
Concepts rooted in physics are becoming increasingly important in biology as we transition to an era in which quantitative descriptions of all processes from molecular to cellular level are needed. In this perspective I discuss two unexpected findings of universal behavior, uncommon in biology, in the self-assembly of proteins and RNA. These findings, which are surprising, reveal that physics ideas applied to biological problems, ranging from folding to gene expression to cellular movement and communication between cells, might lead to discovery of universal principles operating in adoptable living systems.
Cellular signaling involves the transmission of environmental information through cascades of stochastic biochemical reactions, inevitably introducing noise that compromises signal fidelity. Each stage of the cascade often takes the form of a kinase-phosphatase push-pull network, a basic unit of signaling pathways whose malfunction is linked with a host of cancers. We show that this ubiquitous enzymatic network motif effectively behaves as a Wiener-Kolmogorov optimal noise filter. Using concepts from umbral calculus, we generalize the linear Wiener-Kolmogorov theory, originally introduced in the context of communication and control engineering, to take nonlinear signal transduction and discrete molecule populations into account. This allows us to derive rigorous constraints for efficient noise reduction in this biochemical system. Our mathematical formalism yields bounds on filter performance in cases important to cellular function—such as ultrasensitive response to stimuli. We highlight features of the system relevant for optimizing filter efficiency, encoded in a single, measurable, dimensionless parameter. Our theory, which describes noise control in a large class of signal transduction networks, is also useful both for the design of synthetic biochemical signaling pathways and the manipulation of pathways through experimental probes such as oscillatory input.
We present a thermodynamically robust coarse-grained model to simulate folding of RNA in monovalent salt solutions. The model includes stacking, hydrogen bond, and electrostatic interactions as fundamental components in describing the stability of RNA structures. The stacking interactions are parametrized using a set of nucleotide-specific parameters, which were calibrated against the thermodynamic measurements for single-base stacks and base-pair stacks. All hydrogen bonds are assumed to have the same strength, regardless of their context in the RNA structure. The ionic buffer is modeled implicitly, using the concept of counterion condensation and the Debye-Hückel theory. The three adjustable parameters in the model were determined by fitting the experimental data for two RNA hairpins and a pseudoknot. A single set of parameters provides good agreement with thermodynamic data for the three RNA molecules over a wide range of temperatures and salt concentrations. In the process of calibrating the model, we establish the extent of counterion condensation onto the single-stranded RNA backbone. The reduced backbone charge is independent of the ionic strength and is 60% of the RNA bare charge at 37 °C. Our model can be used to predict the folding thermodynamics for any RNA molecule in the presence of monovalent ions.
We solve a two-dimensional model for polymer chain folding in the presence of mechanical pulling force (f) exactly using equilibrium statistical mechanics. Using analytically derived expression for the partition function we determine the phase diagram for the model in the f-temperature (T) plane. A square root singularity in the susceptibility indicates a second order phase transition from a folded to an unfolded state at a critical force (fc) in the thermodynamic limit of infinitely long polymer chain. The temperature dependence of fc shows a reentrant phase transition, which is reflected in an increase in fc as T increases below a threshold value. As a result, for a range of f values, the unfolded state is stable at both low and high temperatures. The high temperature unfolded state is stabilized by entropy whereas the low temperature unfolded state is dominated by favorable energy. The exact calculation could serve as a benchmark for testing approximate theories that are used in analyzing single molecule pulling experiments.
Using theoretical arguments and extensive Monte Carlo (MC) simulations of a coarse-grained three-dimensional off-lattice model of a β-hairpin, we demonstrate that the equilibrium critical force, Fc, needed to unfold the biopolymer increases nonlinearly with increasing volume fraction occupied by the spherical macromolecular crowding agent. Both scaling arguments and MC simulations show that the critical force increases as Fc ≈ φc(α). The exponent α is linked to the Flory exponent relating the size of the unfolded state of the biopolymer and the number of amino acids. The predicted power law dependence is confirmed in simulations of the dependence of the isothermal extensibility and the fraction of native contacts on φc. We also show using MC simulations that Fc is linearly dependent on the average osmotic pressure (P) exerted by the crowding agents on the β-hairpin. The highly significant linear correlation coefficient of 0.99657 between Fc and P makes it straightforward to predict the dependence of the critical force on the density of crowders. Our predictions are amenable to experimental verification using laser optical tweezers.
In single-molecule laser optical tweezer (LOT) pulling experiments, a protein or RNA is juxtaposed between DNA handles that are attached to beads in optical traps. The LOT generates folding trajectories under force in terms of time-dependent changes in the distance between the beads. How to construct the full intrinsic folding landscape (without the handles and beads) from the measured time series is a major unsolved problem. By using rigorous theoretical methods--which account for fluctuations of the DNA handles, rotation of the optical beads, variations in applied tension due to finite trap stiffness, as well as environmental noise and limited bandwidth of the apparatus--we provide a tractable method to derive intrinsic free-energy profiles. We validate the method by showing that the exactly calculable intrinsic free-energy profile for a generalized Rouse model, which mimics the two-state behavior in nucleic acid hairpins, can be accurately extracted from simulated time series in a LOT setup regardless of the stiffness of the handles. We next apply the approach to trajectories from coarse-grained LOT molecular simulations of a coiled-coil protein based on the GCN4 leucine zipper and obtain a free-energy landscape that is in quantitative agreement with simulations performed without the beads and handles. Finally, we extract the intrinsic free-energy landscape from experimental LOT measurements for the leucine zipper.
It is established that prion protein is the sole causative agent in a number of diseases in humans and animals. However, the nature of conformational changes that the normal cellular form, PrP(C), undergoes in its conversion to a self-replicating state is still not fully understood. The ordered C-terminus of PrP(C) proteins has three helices (H1-H3). Here, we use statistical coupling analysis (SCA) to infer covariations at various locations using a family of evolutionarily related sequences and the response of mouse and human PrP(C)s to mechanical force to decipher the initiation sites for the transition from PrP(C) to an aggregation-prone PrP* state. Sequence-based SCA predicts that the clustered residues in nonmammals are localized in the stable core (near H1) of PrP(C), whereas in mammalian PrP(C), they are localized in frustrated helices H2 and H3 where most of the pathogenic mutations are found. Force-extension curves and free energy profiles as a function of extension of mouse and human PrP(C) in the absence of a disulfide (SS) bond between residues Cys179 and Cys214, generated by applying mechanical force to the ends of the molecule, show a sequence of unfolding events starting first with rupture of H2 and H3. This is followed by disruption of structure in two strands. Helix H1, stabilized by three salt bridges, resists substantial force before unfolding. Force extension profiles and the dynamics of rupture of tertiary contacts also show that even in the presence of an SS bond the instabilities in most of H3 and parts of H2 still determine the propensity to form the PrP* state. In mouse PrP(C) with an SS bond, there are ∼10 residues that retain their order even at high forces. Both SCA and single-molecule force simulations show that in the conversion from PrP(C) to PrP(SC) major conformational changes occur (at least initially) in H2 and H3, which because of their sequence compositions are frustrated in the helical state. Implications of our findings for the structural model for the scrapie form of PrP(C) are discussed.
A quantitative theory of protein folding should make testable predictions using theoretical models and simulations performed under conditions that closely mimic those used in experiments. Typically, in laboratory experiments folding or unfolding is initiated using denaturants or external mechanical force, whereas theories and simulations use temperature as the control parameter, thus making it difficult to make direct comparisons with experiments. The molecular transfer model (MTM), which incorporates environmental changes using measured quantities in molecular simulations, overcomes these difficulties. Predictions of the folding thermodynamics and kinetics of a number of proteins using MTM simulations are in remarkable agreement with experiments. The MTM and all atom simulations demonstrating the presence of dry globules represent major advances in the proteins folding field.
The molecular motor myosin V (MyoV) exhibits a wide repertoire of pathways during the stepping process, which is intimately connected to its biological function. The best understood of these is the hand-over-hand stepping by a swinging lever arm movement toward the plus end of actin filaments. Single-molecule experiments have also shown that the motor "foot stomps," with one hand detaching and rebinding to the same site, and back-steps under sufficient load. The complete taxonomy of MyoV's load-dependent stepping pathways, and the extent to which these are constrained by motor structure and mechanochemistry, are not understood. Using a polymer model, we develop an analytical theory to describe the minimal physical properties that govern motor dynamics. We solve the first-passage problem of the head reaching the target-binding site, investigating the competing effects of backward load, strain in the leading head biasing the diffusion in the direction of the target, and the possibility of preferential binding to the forward site due to the recovery stroke. The theory reproduces a variety of experimental data, including the power stroke and slow diffusive search regimes in the mean trajectory of the detached head, and the force dependence of the forward-to-backward step ratio, run length, and velocity. We derive a stall force formula, determined by lever arm compliance and chemical cycle rates. By exploring the MyoV design space, we predict that it is a robust motor whose dynamical behavior is not compromised by reasonable perturbations to the reaction cycle and changes in the architecture of the lever arm.
As a consequence of the rugged landscape of RNA molecules their folding is described by the kinetic partitioning mechanism according to which only a small fraction (φF) reaches the folded state while the remaining fraction of molecules is kinetically trapped in misfolded intermediates. The transition from the misfolded states to the native state can far exceed biologically relevant time. Thus, RNA folding in vivo is often aided by protein cofactors, called RNA chaperones, that can rescue RNAs from a multitude of misfolded structures. We consider two models, based on chemical kinetics and chemical master equation, for describing assisted folding. In the passive model, applicable for class I substrates, transient interactions of misfolded structures with RNA chaperones alone are sufficient to destabilize the misfolded structures, thus entropically lowering the barrier to folding. For this mechanism to be efficient the intermediate ribonucleoprotein complex between collapsed RNA and protein cofactor should have optimal stability. We also introduce an active model (suitable for stringent substrates with small φF), which accounts for the recent experimental findings on the action of CYT-19 on the group I intron ribozyme, showing that RNA chaperones do not discriminate between the misfolded and the native states. In the active model, the RNA chaperone system utilizes chemical energy of adenosine triphosphate hydrolysis to repeatedly bind and release misfolded and folded RNAs, resulting in substantial increase of yield of the native state. The theory outlined here shows, in accord with experiments, that in the steady state the native state does not form with unit probability.
Riboswitches are RNA elements that allosterically regulate gene expression by binding cellular metabolites. The SAM-III riboswitch, one of several classes that binds S-adenosylmethionine (SAM), represses translation upon binding SAM (OFF state) by encrypting the ribosome binding sequence. We have carried out simulations of the RNA by applying mechanical force (f) to the ends of SAM-III, with and without SAM, to get quantitative insights into the f-dependent structural changes. Force-extension (z) curves (FECs) for the apo (ON) state, obtained in simulations in which f is increased at a constant loading rate, show three intermediates, with the first one being the rupture of SAM binding region, which is greatly stabilized in the OFF state. Force-dependent free energy profiles, G(z,f), as a function of z, obtained in equilibrium constant force simulations, reveal the intermediates observed in FECs. The predicted stability difference between the ON and OFF states using G(z,f) is in excellent agreement with experiments. Remarkably, using G(z,f)s and estimate of an effective diffusion constant at a single value of f allows us to predict the f-dependent transition rates using theory of first passage times for both the apo and holo states. To resolve the kinetics of assembly of SAM-III riboswitch in structural terms, we use force stretch-quench pulse sequences in which the force on RNA is maintained at a low (fq) value starting from a high value for a time period tq. Variation of tq over a wide range results in resolution of elusive states involved in the SAM binding pocket and leads to accurate determination of folding times down to fq = 0. Quantitative measure of the folding kinetics, obtained from the folding landscape, allows us to propose that, in contrast to riboswitches regulating transcription, SAM-III functions under thermodynamic control provided the basal concentration of SAM exceeds a small critical value. All of the predictions are amenable to tests in single molecule pulling experiments.