Dmitrii E. Makarov

Van der waals energies in density functional theory

In principle, density functional theory yields the correct ground-state densities and energies of electronic systems under the action of a static external potential. However, traditional approximations fail to include Van der Waals energies between separated systems. This paper proposes a practical procedure for remedying this difficulty. Our method allows seamless calculations between small and large inter-system distances. The asymptotic H-He and He--He interactions are calculated as a first illustration, with very accurate results.

Quantifying internal friction in unfolded and intrinsically disordered proteins with single-molecule spectroscopy

Internal friction, which reflects the “roughness” of the energy landscape, plays an important role for proteins by modulating the dynamics of their folding and other conformational changes. However, the experimental quantification of internal friction and its contribution to folding dynamics has remained challenging. Here we use the combination of single-molecule Förster resonance energy transfer, nanosecond fluorescence correlation spectroscopy, and microfluidic mixing to determine the reconfiguration times of unfolded proteins and investigate the mechanisms of internal friction contributing to their dynamics. Using concepts from polymer dynamics, we determine internal friction with three complementary, largely independent, and consistent approaches as an additive contribution to the reconfiguration time of the unfolded state. We find that the magnitude of internal friction correlates with the compactness of the unfolded protein: its contribution dominates the reconfiguration time of approximately 100 ns of the compact unfolded state of a small cold shock protein under native conditions, but decreases for more expanded chains, and approaches zero both at high denaturant concentrations and in intrinsically disordered proteins that are expanded due to intramolecular charge repulsion. Our results suggest that internal friction in the unfolded state will be particularly relevant for the kinetics of proteins that fold in the microsecond range or faster. The low internal friction in expanded intrinsically disordered proteins may have implications for the dynamics of their interactions with cellular binding partners.

The topomer search model: A simple, quantitative theory of two‐state protein folding kinetics

Most small, single‐domain proteins fold with the uncomplicated, single‐exponential kinetics expected for diffusion on a smooth energy landscape. Despite this energetic smoothness, the folding rates of these two‐state proteins span a remarkable million‐fold range. Here, we review the evidence in favor of a simple, mechanistic description, the topomer search model, which quantitatively accounts for the broad scope of observed two‐state folding rates. The model, which stipulates that the search for those unfolded conformations with a grossly correct topology is the rate‐limiting step in folding, fits observed rates with a correlation coefficient of ∼0.9 using just two free parameters. The fitted values of these parameters, the pre‐exponential attempt frequency and a measure of the difficulty of ordering an unfolded chain, are consistent with previously reported experimental constraints. These results suggest that the topomer search process may dominate the relative barrier heights of two‐state protein‐folding reactions.

How the folding rate constant of simple, single-domain proteins depends on the number of native contacts

Experiments have shown that the folding rate constants of two dozen structurally unrelated, small, single-domain proteins can be expressed in terms of one quantity (the contact order) that depends exclusively on the topology of the folded state. Such dependence is unique in chemical kinetics. Here we investigate its physical origin and derive the approximate formula ln(k) = ln(N) + a + bN, were N is the number of contacts in the folded state, and a and b are constants whose physical meaning is understood. This formula fits well the experimentally determined folding rate constants of the 24 proteins, with single values for a and b.

Theoretical studies of the mechanical unfolding of the muscle protein titin: Bridging the time-scale gap between simulation and experiment

Brute-force, fully atomistic simulations of single molecule mechanical unfolding experiments are not feasible because current simulation time scales are about six orders of magnitude shorter than the time scales explored by experiments. To circumvent this difficulty, we have constructed a model, in which the unfolding dynamics of the I27 domain of the muscle protein titin is described as diffusive motion along a single unfolding coordinate R (equal to the domain extension) in the presence of an external driving potential and the potential of mean force G(R). The effect of the remaining degrees of freedom is described in terms of a viscous force with a friction coefficient η. The potential of mean force G(R) is computed from a series of equilibrium molecular dynamics trajectories performed with constrained values of R and η is extracted from a series of steered molecular dynamics simulations, in which R is increased at a constant rate and the mechanical response of the molecule is monitored as a function o...

Molecular catch bonds and the anti-Hammond effect in polymer mechanochemistry.

While the field of polymer mechanochemistry has traditionally focused on the use of mechanical forces to accelerate chemical processes, theoretical considerations predict an underexplored alternative: the suppression of reactivity through mechanical perturbation. Here, we use electronic structure calculations to analyze the mechanical reactivity of six mechanophores, or chemical functionalities that respond to mechanical stress in a controlled manner. Our computational results indicate that appropriately directed tensile forces could attenuate (as opposed to facilitate) mechanochemical phenomena. Accompanying experimental studies supported the theoretical predictions and demonstrated that relatively simple computational models may be used to design new classes of mechanically responsive materials. In addition, our computational studies and theoretical considerations revealed the prevalence of the anti-Hammond (as opposed to Hammond) effect (i.e., the increased structural dissimilarity between the reactant and transition state upon lowering of the reaction barrier) in the mechanical activation of polyatomic molecules.

Kinetic Monte Carlo simulation of titin unfolding

Recently, it has become possible to unfold a single protein molecule titin, by pulling it with an atomic-force-microscope tip. In this paper, we propose and study a stochastic kinetic model of this unfolding process. Our model assumes that each immunoglobulin domain of titin is held together by six hydrogen bonds. The external force pulls on these bonds and lowers the energy barrier that prevents the hydrogen bond from breaking; this increases the rate of bond breaking and decreases the rate of bond healing. When all six bonds are broken, the domain unfolds. Since the experiment controls the pulling rate, not the force, the latter is calculated from a wormlike chain model for the protein. In the limit of high pulling rate, this kinetic model is solved by a novel simulation method. In the limit of low pulling rate, we develop a quasiequilibrium rate theory, which is tested by simulations. The results are in agreement with the experiments: the distribution of the unfolding force and the dependence of the me...

Chemical reactions modulated by mechanical stress: Extended Bell theory

A number of recent studies have shown that mechanical stress can significantly lower or raise the activation barrier of a chemical reaction. Within a common approximation due to Bell [Science 200, 618 (1978)], this barrier is linearly dependent on the applied force. A simple extension of Bells theory that includes higher order corrections in the force predicts that the force-induced change in the activation energy will be given by -FΔR - ΔχF(2)∕2. Here, ΔR is the change of the distance between the atoms, at which the force F is applied, from the reactant to the transition state, and Δχ is the corresponding change in the mechanical compliance of the molecule. Application of this formula to the electrocyclic ring-opening of cis and trans 1,2-dimethylbenzocyclobutene shows that this extension of Bells theory essentially recovers the force dependence of the barrier, while the original Bell formula exhibits significant errors. Because the extended Bell theory avoids explicit inclusion of the mechanical stress or strain in electronic structure calculations, it allows a computationally efficient characterization of the effect of mechanical forces on chemical processes. That is, the mechanical susceptibility of any reaction pathway is described in terms of two parameters, ΔR and Δχ, both readily computable at zero force.

Computer Simulations and Theory of Protein Translocation

The translocation of proteins through pores is central to many biological phenomena, such as mitochondrial protein import, protein degradation, and delivery of protein toxins to their cytosolic targets. Because proteins typically have to pass through constrictions that are too narrow to accommodate folded structures, translocation must be coupled to protein unfolding. The simplest model that accounts for such co-translocational unfolding assumes that both translocation and unfolding are accomplished by pulling on the end of the polypeptide chain mechanically. In this Account, we describe theoretical studies and computer simulations of this model and discuss how the time scales of translocation depend on the pulling force and on the protein structure. Computationally, this is a difficult problem because biologically or experimentally relevant time scales of translocation are typically orders of magnitude slower than those accessible by fully atomistic simulations. For this reason, we explore one-dimensional free energy landscapes along suitably defined translocation coordinates and discuss various approaches to their computation. We argue that the free energy landscape of translocation is often bumpy because confinement partitions the proteins configuration space into distinct basins of attraction separated by large entropic barriers. Favorable protein-pore interactions and nonnative interactions within the protein further contribute to the complexity. Computer simulations and simple scaling estimates show that forces of just 2-6 pN are often sufficient to ensure transport of unstructured polypeptides, whereas much higher forces are typically needed to translocate folded protein domains. The unfolding mechanisms found from simulations of translocation are different from those observed in the much better understood case of atomic force microscopy (AFM) pulling studies, in which proteins are unraveled by stretching them between their N- and C-termini. In contrast to AFM experiments, single-molecule experimental studies of protein translocation have just started to emerge. We describe one example of a collaborative study, in which dwell times of beta-hairpin-forming peptides inside the alpha-hemolysin pore were both measured experimentally and estimated using computer simulations. Analysis of the simulated trajectories has explained the experimental finding that more stable hairpins take, on the average, longer to traverse the pore. Despite the insight we have gained, the general relationship between the structure of proteins and their resistance to mechanically driven co-translocational unfolding remains poorly understood. Future theoretical progress likely will be made in conjunction with single-molecule experiments and will require realistic models to account for specific protein-pore interactions and for solvent effects.

Translocation of a knotted polypeptide through a pore

We use Langevin dynamics simulations to study how the presence of a deep knot affects the time it takes to thread a polypeptide chain through a narrow pore by pulling mechanically at its end. The polypeptide was designed to contain a knotted unstructured segment inserted between two beta-hairpins, which prevented the knot from slipping off the chain ends. In the range of forces studied (40-200 pN), the mean translocation time increased with the knot complexity. The type 5(2) knot, which was recently discovered in the structure of human ubiquitin hydrolase and is the most complex knot found in the protein databank, slows down translocation by about two orders of magnitude, as compared to the unknotted chain. In contrast to the unknotted chain case, the translocation mechanism of knotted chains involves multiple slippage events suggesting that the corresponding free energy landscape is rugged and involves multiple metastable minima.