Reconstructing the free-energy landscape of a mechanically unfolded model protein

Thermal versus mechanical unfolding in a model protein

Force spectroscopy techniques are often used to learn about the free energy landscape of single biomolecules, typically by recovering free energy quantities that, extrapolated to zero force, are compared to those measured in bulk experiments. However, it is not always clear how the information obtained from a mechanically perturbed system can be related to the information obtained using other denaturants since tensioned molecules unfold and refold along a reaction coordinate imposed by the force, which is not likely to be meaningful in its absence. Here, we explore this dichotomy by investigating the unfolding landscape of a model protein, which is unfolded first mechanically through typical force spectroscopy-like protocols and next thermally. When unfolded by nonequilibrium force extension and constant force protocols, we recover a simple two-barrier landscape as the protein reaches the extended conformation through a metastable intermediate. Interestingly, folding-unfolding equilibrium simulations at low forces suggested a totally different scenario, where this metastable state plays little role in the unfolding mechanism, and the protein unfolds through two competing pathways [R. Tapia-Rojo et al., J. Chem. Phys. 141, 135102 (2014)]. Finally, we use Markov state models to describe the configurational space of the unperturbed protein close to the critical temperature. The thermal dynamics is well understood by a one-dimensional landscape along an appropriate reaction coordinate, however it is very different from the mechanical picture. In this sense, the results of our protein model for the mechanical and thermal descriptions provide incompatible views of the folding/unfolding landscape of the system, and the estimated quantities to zero force result are hard to interpret.

Thermal stabilization of dihydrofolate reductase using monte carlo unfolding simulations and its functional consequences

Design of proteins with desired thermal properties is important for scientific and biotechnological applications. Here we developed a theoretical approach to predict the effect of mutations on protein stability from non-equilibrium unfolding simulations. We establish a relative measure based on apparent simulated melting temperatures that is independent of simulation length and, under certain assumptions, proportional to equilibrium stability, and we justify this theoretical development with extensive simulations and experimental data. Using our new method based on all-atom Monte-Carlo unfolding simulations, we carried out a saturating mutagenesis of Dihydrofolate Reductase (DHFR), a key target of antibiotics and chemotherapeutic drugs. The method predicted more than 500 stabilizing mutations, several of which were selected for detailed computational and experimental analysis. We find a highly significant correlation of r = 0.65–0.68 between predicted and experimentally determined melting temperatures and unfolding denaturant concentrations for WT DHFR and 42 mutants. The correlation between energy of the native state and experimental denaturation temperature was much weaker, indicating the important role of entropy in protein stability. The most stabilizing point mutation was D27F, which is located in the active site of the protein, rendering it inactive. However for the rest of mutations outside of the active site we observed a weak yet statistically significant positive correlation between thermal stability and catalytic activity indicating the lack of a stability-activity tradeoff for DHFR. By combining stabilizing mutations predicted by our method, we created a highly stable catalytically active E. coli DHFR mutant with measured denaturation temperature 7.2°C higher than WT. Prediction results for DHFR and several other proteins indicate that computational approaches based on unfolding simulations are useful as a general technique to discover stabilizing mutations.

All-atom molecular simulations have provided valuable insight into the workings of molecular machines and the folding and unfolding of proteins. However, commonly employed molecular dynamics simulations suffer from a limitation in accessible time scale, making it difficult to model large-scale unfolding events in a realistic amount of simulation time without employing unrealistically high temperatures. Here, we describe a rapid all-atom Monte Carlo simulation approach to simulate unfolding of the essential bacterial enzyme Dihydrofolate Reductase (DHFR) and all possible single point-mutants. We use these simulations to predict which mutants will be more thermodynamically stable (i.e., reside more often in the native folded state vs. the unfolded state) than the wild-type protein, and we confirm our predictions experimentally, creating several highly stable and catalytically active mutants. Thermally stable active engineered proteins can be used as a starting point in directed evolution experiments to evolve new functions on the background of this additional “reservoir of stability.” The stabilized enzyme may be able to accumulate a greater number of destabilizing yet functionally important mutations before unfolding, protease digestion, and aggregation abolish its activity.

Mechanical unfolding of a simple model protein goes beyond the reach of one-dimensional descriptions

We study the mechanical unfolding of a simple model protein. The Langevin dynamics results are analyzed using Markov-model methods which allow to describe completely the configurational space of the system. Using transition-path theory we also provide a quantitative description of the unfolding pathways followed by the system. Our study shows a complex dynamical scenario. In particular, we see that the usual one-dimensional picture: free-energy vs end-to-end distance representation, gives a misleading description of the process. Unfolding can occur following different pathways and configurations which seem to play a central role in one-dimensional pictures are not the intermediate states of the unfolding dynamics.

Stochastic thermodynamics, fluctuation theorems and molecular machines

Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.

Evolution of the potential energy landscape with static pulling force for two model proteins

The energy landscape is analyzed for off-lattice bead models of protein L and protein G as a function of a static pulling force. Two different pairs of attachment points (pulling directions) are compared in each case, namely, residues 1/56 and 10/32. For the terminal residue pulling direction 1/56, the distinct global minimum structures are all extended, aside from the compact geometry that correlates with zero force. The helical turns finally disappear at the highest pulling forces considered. For the 10/32 pulling direction, the changes are more complicated, with a variety of competing arrangements for beads outside the region where the force is directly applied. These alternatives produce frustrated energy landscapes, with low-lying minima separated by high barriers. The calculated folding pathways in the absence of force are in good agreement with previous work. The N-terminal hairpin folds first for protein L and the C-terminal hairpin for protein G, which exhibits an intermediate. However, for a relatively low static force, where the global minimum retains its structure, the folding mechanisms change, sometimes dramatically, depending on the protein and the attachment points. The scaling relations predicted by catastrophe theory are found to hold in the limit of short path lengths.

Direction-dependent mechanical unfolding and green fluorescent protein as a force sensor

An Ising-like model of proteins is used to investigate the mechanical unfolding of the green fluorescent protein along different directions. When the protein is pulled from its ends, we recover the major and minor unfolding pathways observed in experiments. Upon varying the pulling direction, we find the correct order of magnitude and ranking of the unfolding forces. Exploiting the direction dependence of the unfolding force at equilibrium, we propose a force sensor whose luminescence depends on the applied force.

Discrete breathers in a realistic coarse-grained model of proteins

We report the results of molecular dynamics simulations of an off-lattice protein model featuring a physical force-field and amino-acid sequence. We show that localized modes of nonlinear origin, discrete breathers (DBs), emerge naturally as continuations of a subset of high-frequency normal modes residing at specific sites dictated by the native fold. DBs are time-periodic, space-localized vibrational modes that exist generically in nonlinear discrete systems and are known for their resilience and ability to concentrate energy for long times. In the case of the small β-barrel structure that we consider, DB-mediated localization occurs on the turns connecting the strands. At high energies, DBs stabilize the structure by concentrating energy on a few sites, while their collapse marks the onset of large-amplitude fluctuations of the protein. Furthermore, we show how breathers develop as energy-accumulating centres following perturbations even at distant locations, thus mediating efficient and irreversible energy transfers. Remarkably, due to the presence of angular potentials, the breather induces a local static distortion of the native fold. Altogether, the combination of these two nonlinear effects may provide a ready means for remotely controlling local conformational changes in proteins.

Stochastic thermodynamics of single enzymes and molecular motors

For a single enzyme or molecular motor operating in an aqueous solution of non-equilibrated solute concentrations, a thermodynamic description is developed on the level of an individual trajectory of transitions between states. The concept of internal energy, intrinsic entropy and free energy for states follows from a microscopic description using one assumption on time scale separation. A first-law energy balance then allows the unique identification of the heat dissipated in one transition. Consistency with the second law on the ensemble level enforces both stochastic entropy as third contribution to the entropy change involved in one transition and the local detailed balance condition for the ratio between forward and backward rates for any transition. These results follow without assuming weak coupling between the enzyme and the solutes, ideal solution behavior or mass action law kinetics. The present approach highlights both the crucial role of the intrinsic entropy of each state and the physically questionable role of chemiostats for deriving the first law for molecular motors subject to an external force under realistic conditions.

Exact nonequilibrium work generating function for a small classical system

We obtain the exact nonequilibrium work generating function (NEWGF) for a small system consisting of a massive Brownian particle connected to internal and external springs. The external work is provided to the system for a finite-time interval. The Jarzynski equality, obtained in this case directly from the NEWGF, is shown to be valid for the present model, in an exact way regardless of the rate of external work.

Unfolding times for proteins in a force clamp

The escape process from the native valley for proteins subjected to a constant stretching force is examined using a model for a beta barrel. For a wide range of forces, the unfolding dynamics can be treated as one-dimensional diffusion, parametrized in terms of the end-to-end distance. In particular, the escape times can be evaluated as first passage times for a Brownian particle moving on the protein free-energy landscape, using the Smoluchowski equation. At strong forces, the unfolding process can be viewed as a diffusive drift away from the native state, while at weak forces thermal activation is the relevant mechanism. An escape-time analysis within this approach reveals a crossover from an exponential to an inverse Gaussian escape-time distribution upon passing from weak to strong forces. Moreover, a single expression valid at weak and strong forces can be devised both for the average unfolding time as well as for the corresponding variance. The analysis offers a possible explanation of recent experimental findings for the proteins ddFLN4 and ubiquitin.

Critical examination of the inherent-structure-landscape analysis of two-state folding proteins

Recent studies attracted the attention on the inherent-structure-landscape (ISL) approach as a reduced description of proteins allowing to map their full thermodynamic properties. However, the analysis has been so far limited to a single topology of a two-state folding protein, and the simplifying assumptions of the method have not been examined. In this work, we construct the thermodynamics of four two-state folding proteins of different sizes and secondary structure by molecular dynamics (MD) simulations using the ISL method and critically examine possible limitations of the method. Our results show that the ISL approach correctly describes the thermodynamics function, such as the specific heat, on a qualitative level. Using both analytical and numerical methods, we show that some quantitative limitations cannot be overcome with enhanced sampling or the inclusion of harmonic corrections.

Temperature and chemical denaturant dependence of forced unfolding of titin I27

Single-molecule force measurement opens a new door for investigating detailed biomolecular interactions and their thermodynamic properties by pulling molecules apart while monitoring the force exerted on them. Recent advances in the nonequilibrium work theorem allows one to determine the free-energy landscapes of these events. Such information is valuable for understanding processes such as protein and RNA folding and receptor-ligand binding. Here, we used force as a physical parameter under the traditional chemical and temperature denaturing environment to alter the protein folding energy landscape and compared the change in the unfolding free-energy barrier of the I27 domain of human cardiac titin. We found that the trends in protein unfolding free-energy barriers are consistent for single-molecule force measurements and bulk chemical and temperature studies. The results suggest that the information from single-molecule pulling experiments are meaningful and useful for understanding the mechanism of folding of titin I27.

Velocity convergence of free energy surfaces from single-molecule measurements using Jarzynski's equality

We studied the velocity dependence of mechanical unfolding of single protein molecules with the atomic force microscope. We showed that with enough realizations, the free energy surfaces reconstructed from Jarzynski's equality converge with respect to pulling velocity, in good agreement with theory. Using the I27 domain of titin as an example, we estimated the required number of realizations for a given pulling velocity, and suggested the optimal range of velocities for single-molecule experiments. The results demonstrate that Jarzynski's equality is a powerful and practical tool for reconstructing free energy landscapes.

Changing the mechanical unfolding pathway of FnIII10 by tuning the pulling strength

We investigate the mechanical unfolding of the tenth type III domain from fibronectin (FnIII(10)) both at constant force and at constant pulling velocity, by all-atom Monte Carlo simulations. We observe both apparent two-state unfolding and several unfolding pathways involving one of three major, mutually exclusive intermediate states. All three major intermediates lack two of seven native beta-strands, and share a quite similar extension. The unfolding behavior is found to depend strongly on the pulling conditions. In particular, we observe large variations in the relative frequencies of occurrence for the intermediates. At low constant force or low constant velocity, all three major intermediates occur with a significant frequency. At high constant force or high constant velocity, one of them, with the N- and C-terminal beta-strands detached, dominates over the other two. Using the extended Jarzynski equality, we also estimate the equilibrium free-energy landscape, calculated as a function of chain extension. The application of a constant pulling force leads to a free-energy profile with three major local minima. Two of these correspond to the native and fully unfolded states, respectively, whereas the third one can be associated with the major unfolding intermediates.

Free-energy landscape of mechanically unfolded model proteins: extended Jarzinsky versus inherent structure reconstruction

The equilibrium free-energy landscape of off-lattice model heteropolymers as a function of an internal coordinate, namely the end-to-end distance, is reconstructed from out-of-equilibrium steered molecular dynamics data. This task is accomplished via two independent methods: By employing an extended version of the Jarzynski equality and the inherent structure formalism. A comparison of the free energies estimated with these two schemes with equilibrium results obtained via the umbrella sampling technique reveals a good quantitative agreement among all the approaches in a range of temperatures around the "folding transition" for the two examined sequences. In particular, for the sequence with good foldability properties, the mechanically induced structural transitions can be related to thermodynamical aspects of folding. Moreover, for the same sequence the knowledge of the landscape profile allows for a good estimation of the lifetimes of the native configuration for temperatures ranging from the folding to the collapse temperature. For the random sequence, mechanical and thermal unfolding appear to follow different paths along the landscape.

Geometry of the energy landscape and folding transition in a simple model of a protein

A geometric analysis of the global properties of the energy landscape of a minimalistic model of a polypeptide is presented, which is based on the relation between dynamical trajectories and geodesics of a suitable manifold, whose metric is completely determined by the potential energy. We consider different sequences, some with a definite proteinlike behavior, a unique native state and a folding transition, and others undergoing a hydrophobic collapse with no tendency to a unique native state. The global geometry of the energy landscape appears to contain relevant information on the behavior of the various sequences: in particular, the fluctuations of the curvature of the energy landscape, measured by means of numerical simulations, clearly mark the folding transition and allow the proteinlike sequences to be distinguished from the others.

Application range of Jarzynski's equation for boundary-switching processes

Jarzynski's equation (JE) has been known to relate free energy change of a system to statistical distribution of work done on the system for an arbitrary process. In the present work, we first establish the validity condition of JE for boundary switching processes. The validity condition of JE is examined for an example of spontaneous irreversible processes, for which, obviously, JE does not hold. We find that the free energy difference between two configurational states with different phase-space volume cannot be correctly estimated by JE for any adiabatic boundary switching process.