Protein dynamics by the combination of high-speed AFM and computational modeling

High-speed atomic force microscopy (HS-AFM) allows direct observation of biological molecules in dynamic action. However, HS-AFM has no atomic resolution. This article reviews recent progress of computational methods to infer high-resolution information, including the construction of 3D atomistic structures, from experimentally acquired resolution-limited HS-AFM images.

Time-correlation functions for odd Langevin systems

We investigate the statistical properties of fluctuations in active systems that are governed by non-symmetric responses. Both an underdamped Langevin system with an odd resistance tensor and an overdamped Langevin system with an odd elastic tensor are studied. For a system in thermal equilibrium, the time-correlation functions should satisfy time-reversal symmetry and the anti-symmetric parts of the correlation functions should vanish. For the odd Langevin systems, however, we find that the anti-symmetric parts of the time-correlation functions can exist and that they are proportional to either the odd resistance coefficient or the odd elastic constant. This means that the time-reversal invariance of the correlation functions is broken due to the presence of odd responses in active systems. Using the short-time asymptotic expressions of the time-correlation functions, one can estimate an odd elastic constant of an active material such as an enzyme or a motor protein.

Mechanochemical enzymes and protein machines as hydrodynamic force dipoles: the active dimer model

Mechanochemically active enzymes change their shapes within every turnover cycle. Therefore, they induce circulating flows in the solvent around them and behave as oscillating hydrodynamic force dipoles. Because of non-equilibrium fluctuating flows collectively generated by the enzymes, mixing in the solution and diffusion of passive particles within it are expected to get enhanced. Here, we investigate the intensity and statistical properties of such force dipoles in the minimal active dimer model of a mechanochemical enzyme. In the framework of this model, novel estimates for hydrodynamic collective effects in solution and in lipid bilayers under rapid rotational diffusion are derived, and available experimental and computational data is examined.

Shear viscosity of two-state enzyme solutions

We discuss the shear viscosity of a Newtonian solution of catalytic enzymes and substrate molecules. The enzyme is modeled as a two-state dimer consisting of two spherical domains connected with an elastic spring. The enzymatic conformational dynamics is induced by the substrate binding and such a process is represented by an additional elastic spring. Employing the Boltzmann distribution weighted by the waiting times of enzymatic species in each catalytic cycle, we obtain the shear viscosity of dilute enzyme solutions as a function of substrate concentration and its physical properties. The substrate affinity distinguishes between fast and slow enzymes, and the corresponding viscosity expressions are obtained. Furthermore, we connect the obtained viscosity with the diffusion coefficient of a tracer particle in enzyme solutions.

Enhanced Diffusion of Catalytically Active Enzymes

The past decade has seen an increasing number of investigations into enhanced diffusion of catalytically active enzymes. These studies suggested that enzymes are actively propelled as they catalyze reactions or bind with ligands (e.g., substrates or inhibitors). In this Outlook, we chronologically summarize and discuss the experimental observations and theoretical interpretations and emphasize the potential contradictions in these efforts. We point out that the existing multimeric forms of enzymes or isozymes may cause artifacts in measurements and that the conformational changes upon substrate binding are usually not sufficient to give rise to a diffusion enhancement greater than 30%. Therefore, more rigorous experiments and a more comprehensive theory are urgently needed to quantitatively validate and describe the enhanced enzyme diffusion.

Simple mechanics of protein machines

While belonging to the nanoscale, protein machines are so complex that tracing even a small fraction of their cycle requires weeks of calculations on supercomputers. Surprisingly, many aspects of their operation can be however already reproduced by using very simple mechanical models of elastic networks. The analysis suggests that, similar to other self-organized complex systems, functional collective dynamics in such proteins is effectively reduced to a low-dimensional attractive manifold.

Absolute diffusion measurements of active enzyme solutions by NMR

The diffusion of enzymes is of fundamental importance for many biochemical processes. Enhanced or directed enzyme diffusion can alter the accessibility of substrates and the organization of enzymes within cells. Several studies based on fluorescence correlation spectroscopy report enhanced diffusion of enzymes upon interaction with their substrate or inhibitor. In this context, major importance is given to the enzyme fructose-bisphosphate aldolase, for which enhanced diffusion has been reported even though the catalysed reaction is endothermic. Additionally, enhanced diffusion of tracer particles surrounding the active aldolase enzymes has been reported. These studies suggest that active enzymes can act as chemical motors that self-propel and give rise to enhanced diffusion. However, fluorescence studies of enzymes can, despite several advantages, suffer from artefacts. Here, we show that the absolute diffusion coefficients of active enzyme solutions can be determined with Pulsed Field Gradient Nuclear Magnetic Resonance (PFG-NMR). The advantage of PFG-NMR is that the motion of the molecule of interest is directly observed in its native state without the need for any labelling. Furthermore, PFG-NMR is model-free and thus yields absolute diffusion constants. Our PFG-NMR experiments of solutions containing active fructose-bisphosphate aldolase from rabbit muscle do not show any diffusion enhancement for the active enzymes, nor the surrounding molecules. Additionally, we do not observe any diffusion enhancement of aldolase in the presence of its inhibitor pyrophosphate.

Coarse-Grained Protein Dynamics Studies Using Elastic Network Models

Elastic networks have been used as simple models of proteins to study their slow structural dynamics. They consist of point-like particles connected by linear Hookean springs and hence are convenient for linear normal mode analysis around a given reference structure. Furthermore, dynamic simulations using these models can provide new insights. As the computational cost associated with these models is considerably lower compared to that of all-atom models, they are also convenient for comparative studies between multiple protein structures. In this review, we introduce examples of coarse-grained molecular dynamics studies using elastic network models and their derivatives, focusing on the nonlinear phenomena, and discuss their applicability to large-scale macromolecular assemblies.

Diffusion Measurements of Swimming Enzymes with Fluorescence Correlation Spectroscopy

Self-propelled chemical motors are chemically powered micro- or nanosized swimmers. The energy required for these motors' active motion derives from catalytic chemical reactions and the transformation of a fuel dissolved in the solution. While self-propulsion is now well established for larger particles, it is still unclear if enzymes, nature's nanometer-sized catalysts, are potentially also self-powered nanomotors. Because of its small size, any increase in an enzyme's diffusion due to active self-propulsion must be observed on top of the enzyme's passive Brownian motion, which dominates at this scale. Fluorescence correlation spectroscopy (FCS) is a sensitive method to quantify the diffusion properties of single fluorescently labeled molecules in solution. FCS experiments have shown a general increase in the diffusion constant of a number of enzymes when the enzyme is catalytically active. Diffusion enhancements after addition of the enzyme's substrate (and sometimes its inhibitor) of up to 80% have been reported, which is at least 1 order of magnitude higher than what theory would predict. However, many factors contribute to the FCS signal and in particular the shape of the autocorrelation function, which underlies diffusion measurements by fluorescence correlation spectroscopy. These effects need to be considered to establish if and by how much the catalytic activity changes an enzyme's diffusion. We carefully review phenomena that can play a role in FCS experiments and the determination of enzyme diffusion, including the dissociation of enzyme oligomers upon interaction with the substrate, surface binding of the enzyme to glass during the experiment, conformational changes upon binding, and quenching of the fluorophore. We show that these effects can cause changes in the FCS signal that behave similar to an increase in diffusion. However, in the case of the enzymes F₁-ATPase and alkaline phosphatase, we demonstrate that there is no measurable increase in enzyme diffusion. Rather, dissociation and conformational changes account for the changes in the FCS signal in the former and fluorophore quenching in the latter. Within the experimental accuracy of our FCS measurements, we do not observe any change in diffusion due to activity for the enzymes we have investigated. We suggest useful control experiments and additional tests for future FCS experiments that should help establish if the observed diffusion enhancement is real or if it is due to an experimental or data analysis artifact. We show that fluorescence lifetime and mean intensity measurements are essential in order to identify the nature of the observed changes in the autocorrelation function. While it is clear from theory that chemically active enzymes should also act as self-propelled nanomotors, our FCS measurements show that the associated increase in diffusion is much smaller than previously reported. Further experiments are needed to quantify the contribution of the enzymes' catalytic activity to their self-propulsion. We hope that our findings help to establish a useful protocol for future FCS studies in this field and help establish by how much the diffusion of an enzyme is enhanced through catalytic activity.

Exothermicity Is Not a Necessary Condition for Enhanced Diffusion of Enzymes

Recent experiments have revealed that the diffusivity of exothermic and fast enzymes is enhanced when they are catalytically active, and different physical mechanisms have been explored and quantified to account for this observation. We perform measurements on the endothermic and relatively slow enzyme aldolase, which also shows substrate-induced enhanced diffusion. We propose a new physical paradigm, which reveals that the diffusion coefficient of a model enzyme hydrodynamically coupled to its environment increases significantly when undergoing changes in conformational fluctuations in a substrate concentration dependent manner, and is independent of the overall turnover rate of the underlying enzymatic reaction. Our results show that substrate-induced enhanced diffusion of enzyme molecules can be explained within an equilibrium picture and that the exothermicity of the catalyzed reaction is not a necessary condition for the observation of this phenomenon.

Diffusion in systems crowded by active force-dipole molecules

Experimental studies of systems containing active proteins that undergo conformational changes driven by catalytic chemical reactions have shown that the diffusion coefficients of passive tracer particles and active molecules are larger than the corresponding values when chemical activity is absent. Various mechanisms have been proposed for such behavior, including, among others, force dipole interactions of molecular motors moving on filaments and collective hydrodynamic effects arising from active proteins. Simulations of a multi-component system containing active dumbbell molecules that cycle between open and closed states, a passive tracer particle and solvent molecules are carried out. Consistent with experiments, it is shown that the diffusion coefficients of both passive particles and the dumbbells themselves are enhanced when the dumbbells are active. The dependence of the diffusion enhancement on the volume fraction of dumbbells is determined, and the effects of crowding by active dumbbell molecules are shown to differ from those due to inactive molecules.

Passive and active colloidal chemotaxis in a microfluidic channel: mesoscopic and stochastic models

Chemotaxis is the response of a particle to a gradient in the chemical composition of the environment. While it was originally observed for biological organisms, it is of great interest in the context of synthetic active particles such as nanomotors. Experimental demonstration of chemotaxis for chemically-powered colloidal nanomotors was reported in the literature in the context of chemo-attraction in a still fluid or in a microfluidic channel where the gradient is sustained by a specific inlet geometry. In this work, we use mesoscopic particle-based simulations of the colloid and solvent to demonstrate chemotaxis in a microfluidic channel. On the basis of this particle-based model, we evaluate the chemical concentration profiles in the presence of passive or chemically active colloids, compute the chemotactic force acting upon them and propose a stochastic model that rationalises our findings on colloidal chemotaxis. Our model is also able to explain the results of an earlier simulation work that uses a simpler geometry and to extend its interpretation.

Diffusional correlations among multiple active sites in a single enzyme

Simulations of the enzymatic dynamics of a model enzyme containing multiple substrate binding sites indicate the existence of diffusional correlations in the chemical reactivity of the active sites. A coarse-grain, particle-based, mesoscopic description of the system, comprising the enzyme, the substrate, the product and solvent, is constructed to study these effects. The reactive and non-reactive dynamics is followed using a hybrid scheme that combines molecular dynamics for the enzyme, substrate and product molecules with multiparticle collision dynamics for the solvent. It is found that the reactivity of an individual active site in the multiple-active-site enzyme is reduced substantially, and this effect is analyzed and attributed to diffusive competition for the substrate among the different active sites in the enzyme.

Coarse-grain simulations of active molecular machines in lipid bilayers

A coarse-grain method for simulations of the dynamics of active protein inclusions in lipid bilayers is described. It combines the previously proposed hybrid simulations of bilayers [M.-J. Huang, R. Kapral, A. S. Mikhailov, and H.-Y. Chen, J. Chem. Phys. 137, 055101 (2012)], based on molecular dynamics for the lipids and multi-particle collision dynamics for the solvent, with an elastic-network description of active proteins. The method is implemented for a model molecular machine which performs active conformational motions induced by ligand binding and its release after reaction. The situation characteristic for peripheral membrane proteins is considered. Statistical investigations of the effects of single active or passive inclusions on the shape of the membrane are carried out. The results show that the peripheral machine produces asymmetric perturbations of the thickness of two leaflets of the membrane. It also produces a local saddle in the midplane height of the bilayer. Analysis of the power spectrum of the fluctuations of the membrane midplane shows that the conformational motion of the machine perturbs these membrane fluctuations. The hydrodynamic lipid flows induced by cyclic conformational changes in the machine are analyzed. It is shown that such flows are long-ranged and should provide an additional important mechanism for interactions between active inclusions in biological membranes.

Perspective: nanomotors without moving parts that propel themselves in solution

Self-propelled nanomotors use chemical energy to produce directed motion. Like many molecular motors they suffer strong perturbations from the environment in which they move as a result of thermal fluctuations and do not rely on inertia for their propulsion. Such tiny motors are the subject of considerable research because of their potential applications, and a variety of synthetic motors have been made and are being studied for this purpose. Chemically powered self-propelled nanomotors without moving parts that rely on asymmetric chemical reactions to effect directed motion are the focus of this article. The mechanisms they use for propulsion, how size and fuel sources influence their motion, how they cope with strong molecular fluctuations, and how they behave collectively are described. The practical applications of such nanomotors are largely unrealized and the subject of speculation. Since molecular motors are ubiquitous in biology and perform a myriad of complex tasks, the hope is that synthetic motors might be able to perform analogous tasks. They may have the potential to change our perspective on how chemical dynamics takes place in complex systems.

Entropic mechanism of allosteric communication in conformational transitions of dihydrofolate reductase

The mechanism of allosteric conformational transitions of Escherichia coli dihydrofolate reductase (DHFR) is investigated theoretically by applying a newly developed coarse-grained model. Functional forms of interaction potentials in the model depend on the local structural environments around those interactions to represent the many-residue effects due to atomic packing in each local region, and hence, this model is called "the chameleon model". The chameleon model consistently describes the free-energy landscape of two conformational transitions in the catalytic cycle of DHFR, which we call conformational transition 1 (CT1) and conformational transition 2 (CT2); CT1 is accompanied by the hydride transfer reaction, and CT2 is accompanied by the product ligand release. The transition state of CT1 is entropically stabilized by the disordering of loops at the peripheral regions of the protein, which enhances the positively correlated fluctuations at the center part of the protein, showing that the allosteric communication between distant regions through the central region is intrinsically associated with the entropic stabilization of the transition state. The transition state of CT2 is entropically stabilized through the mechanism that enhances the breathing motion of two domains, showing that the difference in the distribution of interactions brings about the difference in the transition mechanism between CT1 and CT2. The chameleon model opens a way to consistently describe the dynamical energy landscape of enzymatic reactions.

Complex intramolecular mechanics of G-actin--an elastic network study

Systematic numerical investigations of conformational motions in single actin molecules were performed by employing a simple elastic-network (EN) model of this protein. Similar to previous investigations for myosin, we found that G-actin essentially behaves as a strain sensor, responding by well-defined domain motions to mechanical perturbations. Several sensitive residues within the nucleotide-binding pocket (NBP) could be identified, such that the perturbation of any of them can induce characteristic flattening of actin molecules and closing of the cleft between their two mobile domains. Extending the EN model by introduction of a set of breakable links which become effective only when two domains approach one another, it was observed that G-actin can possess a metastable state corresponding to a closed conformation and that a transition to this state can be induced by appropriate perturbations in the NBP region. The ligands were roughly modeled as a single particle (ADP) or a dimer (ATP), which were placed inside the NBP and connected by elastic links to the neighbors. Our approximate analysis suggests that, when ATP is present, it stabilizes the closed conformation of actin. This may play an important role in the explanation why, in the presence of ATP, the polymerization process is highly accelerated.

Structure and flexibility of the C-ring in the electromotor of rotary F(0)F(1)-ATPase of pea chloroplasts

A ring of 8-15 identical c-subunits is essential for ion-translocation by the rotary electromotor of the ubiquitous F(O)F(1)-ATPase. Here we present the crystal structure at 3.4Å resolution of the c-ring from chloroplasts of a higher plant (Pisum sativum), determined using a native preparation. The crystal structure was found to resemble that of an (ancestral) cyanobacterium. Using elastic network modeling to investigate the ring's eigen-modes, we found five dominant modes of motion that fell into three classes. They revealed the following deformations of the ring: (I) ellipsoidal, (II) opposite twisting of the luminal circular surface of the ring against the stromal surface, and (III) kinking of the hairpin-shaped monomers in the middle, resulting in bending/stretching of the ring. Extension of the elastic network analysis to rings of different c(n)-symmetry revealed the same classes of dominant modes as in P. sativum (c(14)). We suggest the following functional roles for these classes: The first and third classes of modes affect the interaction of the c-ring with its counterparts in F(O), namely subunits a and bb'. These modes are likely to be involved in ion-translocation and torque generation. The second class of deformation, along with deformations of subunits γ and ε might serve to elastically buffer the torque transmission between F(O) and F(1).

Modeling of solvent flow effects in enzyme catalysis under physiological conditions

A stochastic model for the dynamics of enzymatic catalysis in explicit, effective solvents under physiological conditions is presented. Analytically-computed first passage time densities of a diffusing particle in a spherical shell with absorbing boundaries are combined with densities obtained from explicit simulation to obtain the overall probability density for the total reaction cycle time of the enzymatic system. The method is used to investigate the catalytic transfer of a phosphoryl group in a phosphoglycerate kinase-ADP-bis phosphoglycerate system, one of the steps of glycolysis. The direct simulation of the enzyme-substrate binding and reaction is carried out using an elastic network model for the protein, and the solvent motions are described by multiparticle collision dynamics which incorporates hydrodynamic flow effects. Systems where solvent-enzyme coupling occurs through explicit intermolecular interactions, as well as systems where this coupling is taken into account by including the protein and substrate in the multiparticle collision step, are investigated and compared with simulations where hydrodynamic coupling is absent. It is demonstrated that the flow of solvent particles around the enzyme facilitates the large-scale hinge motion of the enzyme with bound substrates, and has a significant impact on the shape of the probability densities and average time scales of substrate binding for substrates near the enzyme, the closure of the enzyme after binding, and the overall time of completion of the cycle.

Coarse-grained molecular simulations of large biomolecules

Recently, we saw a dramatic increase in the number of researches that rely on coarse-grained (CG) simulations for large biomolecules. Here, first, we briefly describe recently developed and used CG models for proteins and nucleic acids. Balance between structure-based and physico-chemical terms is a key issue. We also discuss the multiscale algorithms used to derive CG parameters. Next, we comment on the dynamics used in CG simulations with an emphasis on the importance of hydrodynamic interactions. We then discuss the pros and cons of CG simulations. Finally, we overview recent exciting applications of CG simulations. Publicly available tools and software for CG simulations are also summarized.