Large amplitude conformational change in proteins explored with a plastic network model: adenylate kinase

Dissipative dynamics of enzymes

We explore enzyme conformational dynamics at sub-Å resolution, specifically, temperature effects. The ensemble-averaged mechanical response of the folded enzyme is viscoelastic in the whole temperature range between the warm and cold denaturation transitions. The dissipation parameter γ of the viscoelastic description decreases by a factor of 2 as the temperature is raised from 10 to 45 °C; the elastic parameter K shows a similar decrease. Thus, when probed dynamically, the enzyme softens for increasing temperature. Equilibrium mechanical experiments with the DNA spring (and a different enzyme) also show, qualitatively, a small softening for increasing temperature.

Comparing the intrinsic dynamics of multiple protein structures using elastic network models

BACKGROUND

Elastic network models (ENMs) are based on the simple idea that a protein can be described as a set of particles connected by springs, which can then be used to describe its intrinsic flexibility using, for example, normal mode analysis. Since the introduction of the first ENM by Monique Tirion in 1996, several variants using coarser protein models have been proposed and their reliability for the description of protein intrinsic dynamics has been widely demonstrated. Lately an increasing number of studies have focused on the meaning of slow dynamics for protein function and its potential conservation through evolution. This leads naturally to comparisons of the intrinsic dynamics of multiple protein structures with varying levels of similarity.

SCOPE OF REVIEW

We describe computational strategies for calculating and comparing intrinsic dynamics of multiple proteins using elastic network models, as well as a selection of examples from the recent literature.

MAJOR CONCLUSIONS

The increasing interest for comparing dynamics across protein structures with various levels of similarity, has led to the establishment and validation of reliable computational strategies using ENMs. Comparing dynamics has been shown to be a viable way for gaining greater understanding for the mechanisms employed by proteins for their function. Choices of ENM parameters, structure alignment or similarity measures will likely influence the interpretation of the comparative analysis of protein motion.

GENERAL SIGNIFICANCE

Understanding the relation between protein function and dynamics is relevant to the fundamental understanding of protein structure-dynamics-function relationship. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

A multi-scale-multi-stable model for the rhodopsin photocycle

We report a multi-scale simulation study of the photocycle of the rhodopsins. The quasi-atomistic representation ("united atoms" UA) of retinal is combined with a minimalist coarse grained (CG, one-bead-per amino acid) representation of the protein, in a hybrid UA/CG approach, which is the homolog of QM/MM, but at lower resolution. An accurate multi-stable parameterization of the model allows simulating each state and transition among them, and the combination of different scale representation allows addressing the entire photocycle. We test the model on bacterial rhodopsin, for which more experimental data are available, and then also report results for mammalian rhodopsins. In particular, the analysis of simulations reveals the spontaneous appearance of meta-stable states in quantitative agreement with experimental data.

Energy landscape views for interplays among folding, binding, and allostery of calmodulin domains

Ligand binding modulates the energy landscape of proteins, thus altering their folding and allosteric conformational dynamics. To investigate such interplay, calmodulin has been a model protein. Despite much attention, fully resolved mechanisms of calmodulin folding/binding have not been elucidated. Here, by constructing a computational model that can integrate folding, binding, and allosteric motions, we studied in-depth folding of isolated calmodulin domains coupled with binding of two calcium ions and associated allosteric conformational changes. First, mechanically pulled simulations revealed coexistence of three distinct conformational states: the unfolded, the closed, and the open states, which is in accord with and augments structural understanding of recent single-molecule experiments. Second, near the denaturation temperature, we found the same three conformational states as well as three distinct binding states: zero, one, and two calcium ion bound states, leading to as many as nine states. Third, in terms of the nine-state representation, we found multiroute folding/binding pathways and shifts in their probabilities with the calcium concentration. At a lower calcium concentration, "combined spontaneous folding and induced fit" occurs, whereas at a higher concentration, "binding-induced folding" dominates. Even without calcium binding, we observed that the folding pathway of calmodulin domains can be modulated by the presence of metastable states. Finally, full-length calmodulin also exhibited an intriguing coupling between two domains when applying tension.

Structure-based simulations reveal concerted dynamics of GPCR activation

G protein-coupled receptors (GPCRs) are a vital class of proteins that transduce biological signals across the cell membrane. However, their allosteric activation mechanism is not fully understood; crystal structures of active and inactive receptors have been reported, but the functional pathway between these two states remains elusive. Here, we use structure-based (Gō-like) models to simulate activation of two GPCRs, rhodopsin and the β₂ adrenergic receptor (β₂AR). We used data-derived reaction coordinates that capture the activation mechanism for both proteins, showing that activation proceeds through quantitatively different paths in the two systems. Both reaction coordinates are determined from the dominant concerted motions in the simulations so the technique is broadly applicable. There were two surprising results. First, the main structural changes in the simulations were distributed throughout the transmembrane bundle, and not localized to the obvious areas of interest, such as the intracellular portion of Helix 6. Second, the activation (and deactivation) paths were distinctly nonmonotonic, populating states that were not simply interpolations between the inactive and active structures. These transitions also suggest a functional explanation for β₂AR's basal activity: it can proceed through a more broadly defined path during the observed transitions.

Reaction Path Force Matching: A New Strategy of Fitting Specific Reaction Parameters for Semiempirical Methods in Combined QM/MM Simulations

We present a general strategy of reparametrizing semiempirical (SE) methods against ab initio (AI) methods for combined quantum mechanical and molecular mechanical (QM/MM) simulations of specific chemical reactions in condensed phases. The resulting approach, designated Reaction Path Force Matching (RP-FM), features cycles of sampling configurations along a reaction path on an efficient SE/MM potential energy surface (PES) and adjusting specific reaction parameters (SRPs) in the SE method such that the atomic forces computed at the target AI/MM level are reproduced. Iterative applications of the RP-FM cycle make possible achieving the accuracy of AI/MM simulations without explicitly sampling the computationally expensive AI/MM PES. The bypassed sampling, nevertheless, is implicitly accomplished through the aid of the efficient SE-SRP/MM PES, on which the target-level reaction path is expected to be obtained upon convergence. We demonstrate the effectiveness of the RP-FM procedure for a symmetric proton transfer reaction in the gas phase and in solution. The remarkable agreements between the RP-FM optimized SE-SRP methods and the target AI method on various properties, including energy profiles, potential of mean force free energy profiles, atomic forces, charge populations, and solvation effects, suggest that RP-FM can be used as an efficient and reliable strategy for simulating condensed-phase chemical reactions.

Using multiscale preconditioning to accelerate the convergence of iterative molecular calculations

Iterative procedures for optimizing properties of molecular models often converge slowly owing to the computational cost of accurately representing features of interest. Here, we introduce a preconditioning scheme that allows one to use a less expensive model to guide exploration of the energy landscape of a more expensive model and thus speed the discovery of locally stable states of the latter. We illustrate our approach in the contexts of energy minimization and the string method for finding transition pathways. The relation of the method to other multilevel simulation techniques and possible extensions are discussed.

Discriminating binding mechanisms of an intrinsically disordered protein via a multi-state coarse-grained model

Many proteins undergo a conformational transition upon binding to their cognate binding partner, with intrinsically disordered proteins (IDPs) providing an extreme example in which a folding transition occurs. However, it is often not clear whether this occurs via an "induced fit" or "conformational selection" mechanism, or via some intermediate scenario. In the first case, transient encounters with the binding partner favour transitions to the bound structure before the two proteins dissociate, while in the second the bound structure must be selected from a subset of unbound structures which are in the correct state for binding, because transient encounters of the incorrect conformation with the binding partner are most likely to result in dissociation. A particularly interesting situation involves those intrinsically disordered proteins which can bind to different binding partners in different conformations. We have devised a multi-state coarse-grained simulation model which is able to capture the binding of IDPs in alternate conformations, and by applying it to the binding of nuclear coactivator binding domain (NCBD) to either ACTR or IRF-3 we are able to determine the binding mechanism. By all measures, the binding of NCBD to either binding partner appears to occur via an induced fit mechanism. Nonetheless, we also show how a scenario closer to conformational selection could arise by choosing an alternative non-binding structure for NCBD.

Global transitions of proteins explored by a multiscale hybrid methodology: application to adenylate kinase

Efficient and accurate mapping of transition pathways is a challenging problem in allosteric proteins. We propose here a to our knowledge new methodology called collective molecular dynamics (coMD). coMD takes advantage of the collective modes of motions encoded by the fold, simultaneously evaluating the interactions and energetics via a full-atomic MD simulation protocol. The basic approach is to deform the structure collectively along the modes predicted by the anisotropic network model, upon selecting them via a Monte Carlo/Metropolis algorithm from among the complete pool of all accessible modes. Application to adenylate kinase, an allosteric enzyme composed of three domains, CORE, LID, and NMP, shows that both open-to-closed and closed-to-open transitions are readily sampled by coMD, with large-scale motions of the LID dominating. An energy-barrier crossing occurs during the NMP movements. The energy barrier originates from a switch between the salt bridges K136-D118 at the LID-CORE interface and K57-E170 and D33-R156 at the CORE-NMP and LID-NMP interfaces, respectively. Despite its simplicity and computing efficiency, coMD yields ensembles of transition pathways in close accord with detailed full atomic simulations, lending support to its utility as a multiscale hybrid method for efficiently exploring the allosteric transitions of multidomain or multimeric proteins.

C(α) torsion angles as a flexible criterion to extract secrets from a molecular dynamics simulation

Given the increasing complexity of simulated molecular systems, and the fact that simulation times have now reached milliseconds to seconds, immense amounts of data (in the gigabyte to terabyte range) are produced in current molecular dynamics simulations. Manual analysis of these data is a very time-consuming task, and important events that lead from one intermediate structure to another can become occluded in the noise resulting from random thermal fluctuations. To overcome these problems and facilitate a semi-automated data analysis, we introduce in this work a measure based on C(α) torsion angles: torsion angles formed by four consecutive C(α) atoms. This measure describes changes in the backbones of large systems on a residual length scale (i.e., a small number of residues at a time). Cluster analysis of individual C(α) torsion angles and its fuzzification led to continuous time patches representing (meta)stable conformations and to the identification of events acting as transitions between these conformations. The importance of a change in torsion angle to structural integrity is assessed by comparing this change to the average fluctuations in the same torsion angle over the complete simulation. Using this novel measure in combination with other measures such as the root mean square deviation (RMSD) and time series of distance measures, we performed an in-depth analysis of a simulation of the open form of DNA polymerase I. The times at which major conformational changes occur and the most important parts of the molecule and their interrelations were pinpointed in this analysis. The simultaneous determination of the time points and localizations of major events is a significant advantage of the new bottom-up approach presented here, as compared to many other (top-down) approaches in which only the similarity of the complete structure is analyzed.

A theoretical view of protein dynamics

Proteins are fascinating supramolecular structures, which are able to recognize ligands transforming binding information into chemical signals. They can transfer information across the cell, can catalyse complex chemical reactions, and are able to transform energy into work with much more efficiency than any human engine. The unique abilities of proteins are tightly coupled with their dynamic properties, which are coded in a complex way in the sequence and carefully refined by evolution. Despite its importance, our experimental knowledge of protein dynamics is still rather limited, and mostly derived from theoretical calculations. I will review here, in a systematic way, the current state-of-the-art theoretical approaches to the study of protein dynamics, emphasizing the most recent advances, examples of use and the expected lines of development in the near future.

Analysis of protein conformational transitions using elastic network model

In this chapter, we demonstrate the usage of a coarse-grained elastic network model to analyze protein conformational transitions in the NS3 helicase (NS3hel) of Hepatitis C virus (HCV). This analysis allows us to identify and visualize collective domain motions involved in the conformational transitions and predict the order of structural events during the transitions. It is highly efficient and applicable to many multi-domain protein structures which undergo large conformational changes to fulfill their functions. This method is made available through a Web server ( http://enm.lobos.nih.gov ).

Exploring the conformational transitions of biomolecular systems using a simple two-state anisotropic network model

Biomolecular conformational transitions are essential to biological functions. Most experimental methods report on the long-lived functional states of biomolecules, but information about the transition pathways between these stable states is generally scarce. Such transitions involve short-lived conformational states that are difficult to detect experimentally. For this reason, computational methods are needed to produce plausible hypothetical transition pathways that can then be probed experimentally. Here we propose a simple and computationally efficient method, called ANMPathway, for constructing a physically reasonable pathway between two endpoints of a conformational transition. We adopt a coarse-grained representation of the protein and construct a two-state potential by combining two elastic network models (ENMs) representative of the experimental structures resolved for the endpoints. The two-state potential has a cusp hypersurface in the configuration space where the energies from both the ENMs are equal. We first search for the minimum energy structure on the cusp hypersurface and then treat it as the transition state. The continuous pathway is subsequently constructed by following the steepest descent energy minimization trajectories starting from the transition state on each side of the cusp hypersurface. Application to several systems of broad biological interest such as adenylate kinase, ATP-driven calcium pump SERCA, leucine transporter and glutamate transporter shows that ANMPathway yields results in good agreement with those from other similar methods and with data obtained from all-atom molecular dynamics simulations, in support of the utility of this simple and efficient approach. Notably the method provides experimentally testable predictions, including the formation of non-native contacts during the transition which we were able to detect in two of the systems we studied. An open-access web server has been created to deliver ANMPathway results.

Many biomolecules are like tiny molecular machines that need to change their shapes and visit many states to perform their biological functions. For a complete molecular understanding of a biological process, one needs to have information on the relevant stable states of the system in question, as well as the pathways by which the system travels from one state to another. We report here an efficient computational method that uses the knowledge of experimental structures of a pair of stable states in order to construct an energetically favoravle pathway between them. We adopt a simple representation of the molecular system by replacing the atoms with beads connected by springs and constructing an energy function with two minima around the end-states. We searched for the structure with highest energy that the system is most likely to visit during the transition and created two paths starting from this structure and proceeding toward the end-states. The combined result of these two paths is the minimum energy pathway between the two stable states. We apply this method to study important structural changes in one enzyme and three large proteins that transport small molecules and ions across the cell membrane.

Perspective: Coarse-grained models for biomolecular systems

By focusing on essential features, while averaging over less important details, coarse-grained (CG) models provide significant computational and conceptual advantages with respect to more detailed models. Consequently, despite dramatic advances in computational methodologies and resources, CG models enjoy surging popularity and are becoming increasingly equal partners to atomically detailed models. This perspective surveys the rapidly developing landscape of CG models for biomolecular systems. In particular, this review seeks to provide a balanced, coherent, and unified presentation of several distinct approaches for developing CG models, including top-down, network-based, native-centric, knowledge-based, and bottom-up modeling strategies. The review summarizes their basic philosophies, theoretical foundations, typical applications, and recent developments. Additionally, the review identifies fundamental inter-relationships among the diverse approaches and discusses outstanding challenges in the field. When carefully applied and assessed, current CG models provide highly efficient means for investigating the biological consequences of basic physicochemical principles. Moreover, rigorous bottom-up approaches hold great promise for further improving the accuracy and scope of CG models for biomolecular systems.

Balancing bond, nonbond, and gō-like terms in coarse grain simulations of conformational dynamics

Characterization of the protein conformational landscape remains a challenging problem, whether it concerns elucidating folding mechanisms, predicting native structures or modeling functional transitions. Coarse-grained molecular dynamics simulation methods enable exhaustive sampling of the energetic landscape at resolutions of biological interest. The general utility of structure-based models is reviewed along with their differing levels of approximation. Simple Gō models incorporate attractive native interactions and repulsive nonnative contacts, resulting in an ideal smooth landscape. Non-Gō coarse-grained models reduce the parameter set as needed but do not include bias to any desired native structure. While non-Gō models have achieved limited success in protein coarse-graining, they can be combined with native structured-based potentials to create a balanced and powerful force field. Recent applications of such Gō-like models have yielded insight into complex folding mechanisms and conformational transitions in large macromolecules. The accuracy and usefulness of reduced representations are also revealed to be a function of the mathematical treatment of the intrinsic bonded topology.

Morphing methods to visualize coarse-grained protein dynamics

Morphing was initially developed as a cinematic effect, where one image is seamlessly transformed into another image. The technique was widely adopted by biologists to visualize the transition between protein conformational states, generating an interpolated pathway from an initial to a final protein structure. Geometric morphing seeks to create visually suggestive movies that illustrate structural changes between conformations but do not necessarily represent a biologically relevant pathway, while minimum energy path (MEP) interpolations aim at describing the true transition state between the crystal structure minima in the energy landscape.

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.

Modeling protein conformational transitions by a combination of coarse-grained normal mode analysis and robotics-inspired methods

BACKGROUND

Obtaining atomic-scale information about large-amplitude conformational transitions in proteins is a challenging problem for both experimental and computational methods. Such information is, however, important for understanding the mechanisms of interaction of many proteins.

METHODS

This paper presents a computationally efficient approach, combining methods originating from robotics and computational biophysics, to model protein conformational transitions. The ability of normal mode analysis to predict directions of collective, large-amplitude motions is applied to bias the conformational exploration performed by a motion planning algorithm. To reduce the dimension of the problem, normal modes are computed for a coarse-grained elastic network model built on short fragments of three residues. Nevertheless, the validity of intermediate conformations is checked using the all-atom model, which is accurately reconstructed from the coarse-grained one using closed-form inverse kinematics.

RESULTS

Tests on a set of ten proteins demonstrate the ability of the method to model conformational transitions of proteins within a few hours of computing time on a single processor. These results also show that the computing time scales linearly with the protein size, independently of the protein topology. Further experiments on adenylate kinase show that main features of the transition between the open and closed conformations of this protein are well captured in the computed path.

CONCLUSIONS

The proposed method enables the simulation of large-amplitude conformational transitions in proteins using very few computational resources. The resulting paths are a first approximation that can directly provide important information on the molecular mechanisms involved in the conformational transition. This approximation can be subsequently refined and analyzed using state-of-the-art energy models and molecular modeling methods.

Robust Heterogeneous Anisotropic Elastic Network Model Precisely Reproduces the Experimental B-factors of Biomolecules

A computational method called the progressive fluctuation matching (PFM) is developed for constructing robust heterogeneous anisotropic network models (HANMs) for biomolecular systems. An HANM derived through the PFM approach consists of harmonic springs with realistic positive force constants, and yields the calculated B-factors that are basically identical to the experimental ones. For the four tested protein systems including crambin, trypsin inhibitor, HIV-1 protease, and lysozyme, the root-mean-square deviations between the experimental and the computed B-factors are only 0.060, 0.095, 0.247, and 0.049 Å(2), respectively, and the correlation coefficients are 0.99 for all. By comparing the HANM/ANM normal modes to their counterparts derived from both an atomistic force field and an NMR structure ensemble, it is found that HANM may provide more accurate results on protein dynamics.

Molecular dynamics studies on the conformational transitions of adenylate kinase: a computational evidence for the conformational selection mechanism

Escherichia coli adenylate kinase (ADK) is a monomeric phosphotransferase enzyme that catalyzes reversible transfer of phosphoryl group from ATP to AMP with a large-scale domain motion. The detailed mechanism for this conformational transition remains unknown. In the current study, we performed long time-scale molecular dynamics simulations on both open and closed states of ADK. Based on the structural analyses of the simulation trajectories, we detected over 20 times conformational transitions between the open and closed states of ADK and identified two novel conformations as intermediate states in the catalytic processes. With these findings, we proposed a possible mechanism for the large-scale domain motion of Escherichia coli ADK and its catalytic process: (1) the substrate free ADK adopted an open conformation; (2) ATP bound with LID domain closure; (3) AMP bound with NMP domain closure; (4) phosphoryl transfer occurred with ATP, and AMP converted into two ADPs, and no conformational transition was detected in the enzyme; (5) LID domain opened with one ADP released; (6) another ADP released with NMP domain open. As both open and closed states sampled a wide range of conformation transitions, our simulation strongly supported the conformational selection mechanism for Escherichia coli ADK.

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.

Exploration of conformational transition pathways from coarse-grained simulations

MOTIVATION

A new algorithm to trace conformational transitions in proteins is presented. The method uses discrete molecular dynamics as engine to sample protein conformational space. A multiple minima Go-like potential energy function is used in combination with several enhancing sampling strategies, such as metadynamics, Maxwell Demon molecular dynamics and essential dynamics. The method, which shows an unprecedented computational efficiency, is able to trace a wide range of known experimental transitions. Contrary to simpler methods our strategy does not introduce distortions in the chemical structure of the protein and is able to reproduce well complex non-linear conformational transitions. The method, called GOdMD, can easily introduce additional restraints to the transition (presence of ligand, known intermediate, known maintained contacts, …) and is freely distributed to the community through the Spanish National Bioinformatics Institute (http://mmb.irbbarcelona.org/GOdMD).

AVAILABILITY

Freely available on the web at http://mmb.irbbarcelona.org/GOdMD.

The mechanism of allosteric coupling in choline kinase α1 revealed by the action of a rationally designed inhibitor

Applying a CHOK hold: Combined experimental and computational studies of the binding mode of a rationally designed inhibitor of the dimeric choline kinase α1 (CHOKα1) explain the molecular mechanism of negative cooperativity (see scheme) and how the monomers are connected. The results give insight into how the symmetry of the dimer can be partially conserved despite a lack of conservation in the static crystal structures.

Polymer uncrossing and knotting in protein folding, and their role in minimal folding pathways

We introduce a method for calculating the extent to which chain non-crossing is important in the most efficient, optimal trajectories or pathways for a protein to fold. This involves recording all unphysical crossing events of a ghost chain, and calculating the minimal uncrossing cost that would have been required to avoid such events. A depth-first tree search algorithm is applied to find minimal transformations to fold [Formula: see text], [Formula: see text], [Formula: see text], and knotted proteins. In all cases, the extra uncrossing/non-crossing distance is a small fraction of the total distance travelled by a ghost chain. Different structural classes may be distinguished by the amount of extra uncrossing distance, and the effectiveness of such discrimination is compared with other order parameters. It was seen that non-crossing distance over chain length provided the best discrimination between structural and kinetic classes. The scaling of non-crossing distance with chain length implies an inevitable crossover to entanglement-dominated folding mechanisms for sufficiently long chains. We further quantify the minimal folding pathways by collecting the sequence of uncrossing moves, which generally involve leg, loop, and elbow-like uncrossing moves, and rendering the collection of these moves over the unfolded ensemble as a multiple-transformation "alignment". The consensus minimal pathway is constructed and shown schematically for representative cases of an [Formula: see text], [Formula: see text], and knotted protein. An overlap parameter is defined between pathways; we find that [Formula: see text] proteins have minimal overlap indicating diverse folding pathways, knotted proteins are highly constrained to follow a dominant pathway, and [Formula: see text] proteins are somewhere in between. Thus we have shown how topological chain constraints can induce dominant pathway mechanisms in protein folding.

Molecular machines

Molecular machines (MM) are essential components of living cells. They conduct mechanical work, transport materials into and out of cells, assist in processing enzymatic reactions, and more. Their operations are frequently combined with significant conformational transitions. Computational studies of these conformational transitions and their coupling to molecular functions are discussed. It is argued that coarse descriptions of these molecules which are based on mass density and shape provide useful information on directions of action. It is further argued that MM are likely to have well focused and narrow reaction pathways. The proposal for such pathways is supported by evolutionary analyses of homologous machines. Finally, these observations are used to build atomically detailed models of these systems that are making the link from structure to functions (kinetics and thermodynamics). For that purpose enhanced sampling techniques are required.

Allosteric activation transitions in enzymes and biomolecular motors: insights from atomistic and coarse-grained simulations

The chemical step in enzymes is usually preceded by a kinetically distinct activation step that involves large-scale conformational transitions. In "simple" enzymes this step corresponds to the closure of the active site; in more complex enzymes, such as biomolecular motors, the activation step is more complex and may involve interactions with other biomolecules. These activation transitions are essential to the function of enzymes and perturbations in the scale and/or rate of these transitions are implicated in various serious human diseases; incorporating key flexibilities into engineered enzymes is also considered a major remaining challenge in rational enzyme design. Therefore it is important to understand the underlying mechanism of these transitions. This is a significant challenge to both experimental and computational studies because of the allosteric and multi-scale nature of such transitions. Using our recent studies of two enzyme systems, myosin and adenylate kinase (AK), we discuss how atomistic and coarse-grained simulations can be used to provide insights into the mechanism of activation transitions in realistic systems. Collectively, the results suggest that although many allosteric transitions can be viewed as domain displacements mediated by flexible hinges, there are additional complexities and various deviations. For example, although our studies do not find any evidence for "cracking" in AK, our results do underline the contribution of intra-domain properties (e.g., dihedral flexibility) to the rate of the transition. The study of mechanochemical coupling in myosin highlights that local changes important to chemistry require stabilization from more extensive structural changes; in this sense, more global structural transitions are needed to activate the chemistry in the active site. These discussions further emphasize the importance of better understanding factors that control the degree of co-operativity for allosteric transitions, again hinting at the intimate connection between protein stability and functional flexibility. Finally, a number of topics of considerable future interest are briefly discussed.

Conformational heterogeneity within the LID domain mediates substrate binding to Escherichia coli adenylate kinase: function follows fluctuations

Proteins exist as dynamic ensembles of molecules, implying that protein amino acid sequences evolved to code for both the ground state structure as well as the entire energy landscape of excited states. Accumulating theoretical and experimental evidence suggests that enzymes use such conformational fluctuations to facilitate allosteric processes important for substrate binding and possibly catalysis. This phenomenon can be clearly demonstrated in Escherichia coli adenylate kinase, where experimentally observed local unfolding of the LID subdomain, as opposed to a more commonly postulated rigid-body opening motion, is related to substrate binding. Because "entropy promoting" glycine mutations designed to increase specifically the local unfolding of the LID domain also affect substrate binding, changes in the excited energy landscape effectively tune the function of this enzyme without changing the ground state structure or the catalytic site. Thus, additional thermodynamic information, above and beyond the single folded structure of an enzyme-substrate complex, is likely required for a full and quantitative understanding of how enzymes work.

Exploring the Dynamic Functional Landscape of Adenylate Kinase Modulated by Substrates

Adenylate kinase (ADK) has been explored widely, through both experimental and theoretical studies. However, still less is known about how the functional dynamics of ADK is modulated explicitly by its natural substrates. Here, we report a quantitative study of the dynamic energy landscape for ADK responding to the substrate binding by integrating both experimental investigations and theoretical modeling. We make theoretical predictions which are in remarkable agreement with the single molecule experiments on the substrate-bound complex. With our combined models of ADK in its apo form, in the presence of AMP or ATP, and in complex with both substrates, we specifically address the following key questions: (1) Are there intermediate state(s) during their catalytic cycle and if so how many? (2) How many pathways are there along the open-to-closed transitions and what are their corresponding weights? (3) How do substrates influence the pathway weights and the stability of the intermediates? (4) Which lid's motion is rate-limiting along the turnover cycle, the NMP or the LID domain? Our models predict two major parallel stepwise pathways and two on-pathway intermediates which are denoted as IN (NMP domain open while LID domain closed) and IL (LID domain open and NMP domain closed), respectively. Further investigation of temperature effects suggests that the IN pathway is dominant at room temperature, but the IL pathway is dominant at the optimal temperature. This leads us to propose that the IL pathway is more dominant by entropy and IN pathway by enthalpy. Remarkably, our results show that even with maximum concentrations of natural substrates, ADK still fluctuates between multiple functional states, reflecting an intrinsic capability of large-scale conformational fluctuations which may be essential to its biological function. The results based on the dual-ligands model provide the theoretical validation of random bisubstrate biproducts (Bi-Bi) mechanism for the enzymatic reaction of ADK. Additionally, the pathway flux analysis strongly suggests that the motion of the NMP domain is the rate-determining step for the conformational cycle (opening and closing).

Protein dynamics governed by interfaces of high polarity and low packing density

The folding pathway, three-dimensional structure and intrinsic dynamics of proteins are governed by their amino acid sequences. Internal protein surfaces with physicochemical properties appropriate to modulate conformational fluctuations could play important roles in folding and dynamics. We show here that proteins contain buried interfaces of high polarity and low packing density, coined as LIPs: Light Interfaces of high Polarity, whose physicochemical properties make them unstable. The structures of well-characterized equilibrium and kinetic folding intermediates indicate that the LIPs of the corresponding native proteins fold late and are involved in local unfolding events. Importantly, LIPs can be identified using very fast and uncomplicated computational analysis of protein three-dimensional structures, which provides an easy way to delineate the protein segments involved in dynamics. Since LIPs can be retained while the sequences of the interacting segments diverge significantly, proteins could in principle evolve new functional features reusing pre-existing encoded dynamics. Large-scale identification of LIPS may contribute to understanding evolutionary constraints of proteins and the way protein intrinsic dynamics are encoded.

Coarse-grained and all-atom modeling of structural states and transitions in hemoglobin

Hemoglobin (Hb), an oxygen-binding protein composed of four subunits (α1, α2, β1, and β2), is a well-known example of allosteric proteins that are capable of cooperative ligand binding. Despite decades of studies, the structural basis of its cooperativity remains controversial. In this study, we have integrated coarse-grained (CG) modeling, all-atom simulation, and structural data from X-ray crystallography and wide-angle X-ray scattering (WAXS), aiming to probe dynamic properties of the two structural states of Hb (T and R state) and the transitions between them. First, by analyzing the WAXS data of unliganded and liganded Hb, we have found that the structural ensemble of T or R state is dominated by one crystal structure of Hb with small contributions from other crystal structures of Hb. Second, we have used normal mode analysis to identify two distinct quaternary rotations between the α1β1 and α2β2 dimer, which drive the transitions between T and R state. We have also identified the hot-spot residues whose mutations are predicted to greatly change these quaternary motions. Third, we have generated a CG transition pathway between T and R state, which predicts a clear order of quaternary and tertiary changes involving α and β subunits in Hb. Fourth, we have used the accelerated molecular dynamics to perform an all-atom simulation starting from the T state of Hb, and we have observed a transition toward the R state of Hb. Further analysis of crystal structural data and the all-atom simulation trajectory has corroborated the order of quaternary and tertiary changes predicted by CG modeling.

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.

Multiscale Coarse-Graining via Normal Mode Analysis

A multiscale coarse-graining method called the normal-mode analysis based fluctuation matching (NMA-FM) is developed for constructing coarse-grained models of biomolecular systems. In the framework of normal-mode analysis, an arbitrary fine-grained model can be systematically converted to a more coarse-grained model, while the crucial low-frequency motions of the fine-grained system are able to be reproduced in the coarse-grained modeling. The method relies on the technique of fluctuation matching that has been devised earlier for parametrizing heterogeneous elastic network models based on data from atomistic molecular dynamics simulations. The new approach is quite efficient since it avoids expensive atomistic molecular dynamics simulations and can start from already coarse-grained elastic network models. In the practical aspect, the method is suitable for conformational analyses of large biomacromolecules and calculations of mechanical properties of biomaterials, which is demonstrated by the studied systems including an amyloid dimer, lysozyme and adenylate kinase proteins, and the S2 subdomain of myosin.

Exploring polymorphisms in B-DNA helical conformations

The traditional mesoscopic paradigm represents DNA as a series of base-pair steps whose energy response to equilibrium perturbations is elastic, with harmonic oscillations (defining local stiffness) around a single equilibrium conformation. In addition, base sequence effects are often analysed as a succession of independent XpY base-pair steps (i.e. a nearest-neighbour (NN) model with only 10 unique cases). Unfortunately, recent massive simulations carried out by the ABC consortium suggest that the real picture of DNA flexibility may be much more complex. The paradigm of DNA flexibility therefore needs to be revisited. In this article, we explore in detail one of the most obvious violations of the elastic NN model of flexibility: the bimodal distributions of some helical parameters. We perform here an in-depth statistical analysis of a very large set of MD trajectories and also of experimental structures, which lead to very solid evidence of bimodality. We then suggest ways to improve mesoscopic models to account for this deviation from the elastic regime.

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.