Langevin modes analysis of myoglobin

Free Energy Surface and Molecular Mechanism of Slow Structural Transitions in Lipid Bilayers

Lipid membrane remodeling, crucial for many cellular processes, is governed by the coupling of membrane structure and shape fluctuations. Given the importance of the ∼ nm length scale, details of the transition intermediates for conformational change are not fully captured by a continuum-mechanical description. Slow dynamics and the lack of knowledge of reaction coordinates (RCs) for biasing methods pose a challenge for all-atom (AA) simulations. Here, we map system dynamics on Langevin dynamics in a normal mode space determined from an elastic network model representation for the lipid-water Hamiltonian. AA molecular dynamics (MD) simulations are used to determine model parameters, and Langevin dynamics predictions for bilayer structural, mechanical, and dynamic properties are validated against MD simulations and experiments. Transferability to describe the dynamics of a larger lipid bilayer and a heterogeneous membrane-protein system is assessed. A set of generic RCs for pore formation in two tensionless bilayers is obtained by coupling Langevin dynamics to the underlying energy landscape for membrane deformations. Structure evolution is carried out by AA MD, wherein the generic RCs are used in a path metadynamics or an umbrella sampling simulation to determine the thermodynamics of pore formation and its molecular determinants, such as the role of distinct bilayer motions, lipid solvation, and lipid packing.

Coarse-Graining Waters: Unveiling The Effective Hydrophilicity/Hydrophobicity of Individual Protein Atoms and The Roles of Waters' Hydrogens

There have been many coarse-graining methods developed that aim to reduce the sizes of simulated systems and their computational costs. In this work, we develop a new coarse-graining method, called coarse-graining-delta (or δ-CG in short), that reduces the degrees of freedom of the potential energy surface by coarse-graining relative locations of atoms from their unit centers. Our method extends and generalizes the methods used in the coarse-grained normal mode analysis and enables us to study the roles of the individual removed atoms in a system, which have been difficult to study in molecular dynamics simulations. By applying δ-CG to coarse-grain three-point water molecules into single-point solvent particles, we successfully identify the effective hydrophilicity and hydrophobicity of all the individual protein atom types, which collectively correlate well with the known hydrophilic, hydrophobic, and amphipathic characteristics of amino acids. Moreover, our investigation shows that water's hydrogens have two roles in interacting with protein atoms. First, water molecules adjust their poses around different amino acids and their atoms, and the statistical preferences of the hydrogen poses near the atoms determine the effective hydrophilicity and hydrophobicity of the atoms, which have not been successfully addressed before. Second, the collective dynamics of the hydrogens assist the water molecules in escaping from the potential energy wells of the hydrophilic atoms. Our method also shows that coarse-graining a system mathematically leads to breaking antisymmetry of the nonbonded interactions; as a result, two interacting coarse-grained units exert different forces on each other. Our study indicates that the accuracy of coarse-grained force fields, such as the MARTINI force field and the UNRES force field, can be improved in two ways: (i) refining their potential energy functions and coefficients by analyzing the coarse-grained potential energy surface using δ-CG, and (ii) introducing non-antisymmetric interactions.

Impact of Conformational Substates and Energy Landscapes on Understanding Hemoglobin Kinetics and Function

Hans Frauenfelder's discovery of conformational substates in studies of myoglobin carbon monoxide geminate rebinding kinetics at cryogenic temperatures (Austin RH, Beeson KW, Eisenstein L, Frauenfelder H, & Gunsalus IC (1975) Dynamics of Ligand Binding to Myoglobin. Biochemistry 14(24):5355-5373) followed by his introduction of energy landscape theory with Peter Wolynes (Frauenfelder H, Sligar SG, & Wolynes PG (1991) The Energy Landscapes and Motions of Proteins. Science 254(5038):1598-1603) marked the beginning of a new era in the physics and physical chemistry of proteins. Their work played a major role in demonstrating the power and importance of dynamics and of Kramers reaction rate theory for understanding protein function. The biggest impact of energy landscape theory has been in the protein folding field, which is well-known and has been documented in numerous articles and reviews, including a recent one of my own (Eaton WA (2021) Modern Kinetics and Mechanism of Protein Folding: a Retrospective. J. Phys. Chem. B. 125(14):3452-3467). Here I will describe the much less well-known impact of their modern view of proteins on both experimental and theoretical studies of hemoglobin kinetics and function. I will first describe how Frauenfelder's experiments motivated and influenced my own research on myoglobin, which were key ingredients to my work on understanding hemoglobin.

The amounts of thermal vibrations and static disorder in protein X-ray crystallographic B-factors

Crystallographic B-factors provide direct dynamical information on the internal mobility of proteins that is closely linked to function, and are also widely used as a benchmark in assessing elastic network models. A significant question in the field is: what is the exact amount of thermal vibrations in protein crystallographic B-factors? This work sets out to answer this question. First, we carry out a thorough, statistically sound analysis of crystallographic B-factors of over 10 000 structures. Second, by employing a highly accurate all-atom model based on the well-known CHARMM force field, we obtain computationally the magnitudes of thermal vibrations of nearly 1000 structures. Our key findings are: (i) the magnitude of thermal vibrations, surprisingly, is nearly protein-independent, as a corollary to the universality for the vibrational spectra of globular proteins established earlier; (ii) the magnitude of thermal vibrations is small, less than 0.1 Å² at 100 K; (iii) the percentage of thermal vibrations in B-factors is the lowest at low resolution and low temperature (<10%) but increases to as high as 60% for structures determined at high resolution and at room temperature. The significance of this work is that it provides for the first time, using an extremely large dataset, a thorough analysis of B-factors and their thermal and static disorder components. The results clearly demonstrate that structures determined at high resolution and at room temperature have the richest dynamics information. Since such structures are relatively rare in the PDB database, the work naturally calls for more such structures to be determined experimentally.

Structural fluctuation of protein in water around its native state: a new statistical mechanics formulation

A new statistical mechanics formulation of characterizing the structural fluctuation of protein correlated with that of water is presented based on the generalized Langevin equation and the 3D-reference interaction site model (RISM)/RISM theory of molecular liquids. The displacement vector of atom positions, and their conjugated momentum, are chosen for the dynamic variables for protein, while the density fields of atoms and their momentum fields are chosen for water. Projection of other degrees of freedom onto those dynamic variables using the standard projection operator method produces essentially two equations, which describe the time evolution of fluctuation concerning the density field of solvent and the conformation of protein around an equilibrium state, which are coupled with each other. The equation concerning the protein dynamics is formally akin to that of the coupled Langevin oscillators, and is a generalization of the latter, to atomic level. The most intriguing feature of the new equation is that it contains the variance-covariance matrix as the "Hessian" term describing the "force" restoring an equilibrium conformation, which is the second moment of the fluctuation of atom positions. The "Hessian" matrix is naturally identified as the second derivative of the free energy surface around the equilibrium. A method to evaluate the Hessian matrix based on the 3D-RISM/RISM theory is proposed. Proposed also is an application of the present formulation to the molecular recognition, in which the conformational fluctuation of protein around its native state becomes an important factor as exemplified by so called "induced fitting."

Response of rotation-translation blocked proteins using Langevin dynamics on a locally harmonic landscape

Langevin dynamics is used to compute the time evolution of the nonequilibrium motion of the atomic coordinates of a protein in response to ligand dissociation. The protein potential energy surface (PES) is approximated by a harmonic basin about the minimum of the unliganded state. Upon ligand dissociation, the protein undergoes relaxation from the bound to the unbound state. A coarse graining scheme based on rotation translation blocks (RTB) is applied to the relaxation of the two domain iron transport protein, ferric binding protein. This scheme provides a natural and efficient way to freeze out the small amplitude, high frequency motions within each rigid fragment, thereby allowing for the number of dynamical degrees of freedom to be reduced. The results obtained from all flexible atom (constraint free) dynamics are compared to those obtained using RTB-Langevin dynamics. To assess the impact of the assumed rigid fragment clustering on the temporal relaxation dynamics of the protein molecule, three distinct rigid block decompositions were generated and their responses compared. Each of the decompositions was a variant of the one-block-per-residue grouping, with their force and friction matrices being derived from their fully flexible counterpart. Monitoring the time evolution of the distance separating a selected pair of amino acids, the response curves of the blocked decompositions were similar in shape to each other and to the control system in which all atomic degrees of freedom are fully independent. The similar shape of the blocked responses showed that the variations in grouping had only a minor impact on the kinematics. Compared with the all atom responses, however, the blocked responses were faster as a result of the instantaneous transmission of force throughout each rigid block. This occurred because rigid blocking does not permit any intrablock deformation that could store or divert energy. It was found, however, that this accelerated response could be successfully corrected by scaling each eigenvalue in the appropriate propagation matrix by the least-squares fitted slope of the blocked vs nonblocked eigenvalue spectra. The RTB responses for each test system were dominated by small eigenvalue overdamped Langevin modes. The large eigenvalue members of each response dissipated within the first 5 ps, after which the long time response was dominated by a modest set of low energy, overdamped normal modes, that were characterized by highly cooperative, functionally relevant displacements. The response assuming that the system is in the overdamped limit was compared to the full phase space Langevin dynamics results. The responses after the first 5 ps were nearly identical, confirming that the inertial components were significant only in the initial stages of the relaxation. Since the propagator matrix in the overdamped formulation is real-symmetric and does not require the inertial component in the propagator, the computation time and memory footprint was reduced by 1 order of magnitude.

Dissipative electro-elastic network model of protein electrostatics

We propose a dissipative electro-elastic network model to describe the dynamics and statistics of electrostatic fluctuations at active sites of proteins. The model combines the harmonic network of residue beads with overdamped dynamics of the normal modes of the network characterized by two friction coefficients. The electrostatic component is introduced to the model through atomic charges of the protein force field. The overall effect of the electrostatic fluctuations of the network is recorded through the frequency-dependent response functions of the electrostatic potential and electric field at the protein active site. We also consider the dynamics of displacements of individual residues in the network and the dynamics of distances between pairs of residues. The model is tested against loss spectra of residue displacements and the electrostatic potential and electric field at the heme's iron from all-atom molecular dynamics simulations of three hydrated globular proteins.

Coupling of protein and environment fluctuations

We review the concepts of protein dynamics developed over the last 35years and extend applications of the unified model of protein dynamics to heat flow and spatial fluctuations in hydrated myoglobin (Mb) powders. Differential scanning calorimetry (DSC) and incoherent neutron scattering (INS) data on hydration Mb powders are explained by the temperature-dependence of the hydration-shell β(h) process measured by dielectric relaxation spectroscopy (DRS). The unified model explains the temperature dependence of DSC and INS data as a kinetic effect due to a fixed experimental time window and a broad distribution of hydration-shell β(h) fluctuation rates. We review the slaving of large scale protein motions to the bulk solvent α process, and the metastability of Mb molecules in glass forming bulk solvent at low temperatures. This article is part of a Special Issue entitled: "Protein Dynamics: Experimental and Computational Approaches".

Logarithmic decay in single-particle relaxation of hydrated lysozyme powder

We present the self-dynamics of protein amino acids of hydrated lysozyme powder around the physiological temperature by means of molecular dynamics simulations. The self-intermediate scattering functions of the amino acid residue center of mass display a logarithmic decay over 3 decades of time, from 2 ps to 2 ns, followed by an exponential alpha relaxation. This kind of slow dynamics resembles the relaxation scenario within the beta-relaxation time range predicted by mode coupling theory in the vicinity of higher-order singularities. These results suggest a strong analogy between the single-particle dynamics of the protein and the dynamics of colloidal, polymeric, and molecular glass-forming liquids.

A unified model of protein dynamics

Protein functions require conformational motions. We show here that the dominant conformational motions are slaved by the hydration shell and the bulk solvent. The protein contributes the structure necessary for function. We formulate a model that is based on experiments, insights from the physics of glass-forming liquids, and the concepts of a hierarchically organized energy landscape. To explore the effect of external fluctuations on protein dynamics, we measure the fluctuations in the bulk solvent and the hydration shell with broadband dielectric spectroscopy and compare them with internal fluctuations measured with the Mössbauer effect and neutron scattering. The result is clear. Large-scale protein motions are slaved to the fluctuations in the bulk solvent. They are controlled by the solvent viscosity, and are absent in a solid environment. Internal protein motions are slaved to the beta fluctuations of the hydration shell, are controlled by hydration, and are absent in a dehydrated protein. The model quantitatively predicts the rapid increase of the mean-square displacement above approximately 200 K, shows that the external beta fluctuations determine the temperature- and time-dependence of the passage of carbon monoxide through myoglobin, and explains the nonexponential time dependence of the protein relaxation after photodissociation.

Solvent effects in the slow dynamics of proteins

The influence of solvent on the slow internal dynamics of proteins is studied by comparing molecular dynamics simulations of solvated and unsolvated lysozyme. The dynamical trajectories are projected onto the protein's normal modes in order to obtain a separate analysis for each of the associated time scales. The results show that solvent effects are important for the slowest motions (below approximately 1 ps(-1)) but negligible for faster motions. The damping effects seen in the latter show that the principal source of friction in protein dynamics is not the solvent, but the protein itself.

Temperature-dependent protein dynamics: a simulation-based probabilistic diffusion-vibration Langevin description

An enduring challenge in the understanding of internal protein motions is the effective separation and characterization of diffusive and vibrational dynamical components. To address this problem, here nanosecond molecular dynamics trajectories of myoglobin in aqueous solution, performed over a range of temperatures between 120 and 300 K, are subjected to principal component analysis, and the coordinate autocorrelation functions of the resulting principal modes are interpreted using a model combining damped Langevin vibration within potential wells and barrier-crossing diffusion between them. Both the vibrational frequency and the fraction of the mean-square fluctuation arising from vibrational motion undergo transitions with temperature at about 180 K. In contrast, the vibrational friction remains linear with temperature. The diffusional component of the mean-square fluctuation increases dramatically at the dynamical transition. The heights of the energy barriers between the potential wells are estimated, and the associated diffusion constants are calculated using Kramers' rate theory. Model functions of the frequency dependence of the frictional and diffusional quantities are obtained. The dynamic structure factor from the full molecular dynamics trajectory is well reproduced by the model. Overall, the results indicate that a global description of nanosecond temperature-dependent diffusion and vibrational internal protein dynamics can be obtained by applying the results of the present diffusion-vibration model to the vibrational motions obtained from a normal-mode analysis.

Langevin model of the temperature and hydration dependence of protein vibrational dynamics

The modification of internal vibrational modes in a protein due to intraprotein anharmonicity and solvation effects is determined by performing molecular dynamics (MD) simulations of myoglobin, analyzing them using a Langevin model of the vibrational dynamics and comparing the Langevin results to a harmonic, normal mode model of the protein in vacuum. The diagonal and off-diagonal Langevin friction matrix elements, which model the roughness of the vibrational potential energy surfaces, are determined together with the vibrational potentials of mean force from the MD trajectories at 120 K and 300 K in vacuum and in solution. The frictional properties are found to be describable using simple phenomenological functions of the mode frequency, the accessible surface area, and the intraprotein interaction (the displacement vector overlap of any given mode with the other modes in the protein). The frictional damping of a vibrational mode in vacuum is found to be directly proportional to the intraprotein interaction of the mode, whereas in solution, the friction is proportional to the accessible surface area of the mode. In vacuum, the MD frequencies are lower than those of the normal modes, indicating intramolecular anharmonic broadening of the associated potential energy surfaces. Solvation has the opposite effect, increasing the large-amplitude vibrational frequencies relative to in vacuum and thus vibrationally confining the protein atoms. Frictional damping of the low-frequency modes is highly frequency dependent. In contrast to the damping effect of the solvent, the vibrational frequency increase due to solvation is relatively temperature independent, indicating that it is primarily a structural effect. The MD-derived vibrational dynamic structure factor and density of states are well reproduced by a model in which the Langevin friction and potential of mean force parameters are applied to the harmonic normal modes.

Distance versus energy fluctuations and electron transfer in single protein molecules

Stochastic nature due to distance and energy fluctuations of single protein molecules involved in electron-transfer (ET) reactions is studied. Distance fluctuations have been assumed previously for causing the slow fluctuations in the ET rates between a donor-acceptor pair constrained to a native protein. Although the observed t(-1/2) power law can be derived using Langevin dynamics with a simple chain model, some discrepancies exist. The friction coefficient and the Rouse segment time constant deduced from experimental data are several orders of magnitude too large, even though the extracted force constant is reasonable. Therefore, questions are raised about the distance-fluctuation mechanism and the activationless ET hypothesis. As an alternative mechanism, we considered fluctuations in activation energy and analyzed the data from two different single protein experiments to determine spectral distribution of energy fluctuations.

Internal friction controls the speed of protein folding from a compact configuration

Several studies have found millisecond protein folding reactions to be controlled by the viscosity of the solvent: Reducing the viscosity allows folding to accelerate. In the limit of very low solvent viscosity, however, one expects a different behavior. Internal interactions, occurring within the solvent-excluded interior of a compact molecule, should impose a solvent-independent upper limit to folding speed once the bulk diffusional motions become sufficiently rapid. Why has this not been observed? We have studied the effect of solvent viscosity on the folding of cytochrome c from a highly compact, late-stage intermediate configuration. Although the folding rate accelerates as the viscosity declines, it tends toward a finite limiting value approximately 10(5) s(-1) as the viscosity tends toward zero. This limiting rate is independent of the cosolutes used to adjust solvent friction. Therefore, interactions within the interior of a compact denatured polypeptide can limit the folding rate, but the limiting time scale is very fast. It is only observable when the solvent-controlled stages of folding are exceedingly rapid or else absent. Interestingly, we find a very strong temperature dependence in these "internal friction"-controlled dynamics, indicating a large energy scale for the interactions that govern reconfiguration within compact, near-native states of a protein.

Configurational diffusion down a folding funnel describes the dynamics of DNA hairpins

Elucidating the mechanism of folding of polynucleotides depends on accurate estimates of free energy surfaces and a quantitative description of the kinetics of structure formation. Here, the kinetics of hairpin formation in single-stranded DNA are measured after a laser temperature jump. The kinetics are modeled as configurational diffusion on a free energy surface obtained from a statistical mechanical description of equilibrium melting profiles. The effective diffusion coefficient is found to be strongly temperature-dependent in the nucleation step as a result of formation of misfolded loops that do not lead to subsequent zipping. This simple system exhibits many of the features predicted from theoretical studies of protein folding, including a funnel-like energy surface with many folding pathways, trapping in misfolded conformations, and non-Arrhenius folding rates.

Kinetics of diffusion-assisted reactions in microheterogeneous systems

This review is focused on the basic theory of diffusion-assisted reactions in microheterogeneous systems, from porous solids to self-organized colloids and biomolecules. Rich kinetic behaviors observed experimentally are explained in a unified fashion using simple concepts of competing distance and time scales of the reaction and the embedding structure. We mainly consider pseudo-first-order reactions, such as luminescence quenching, described by the Smoluchowski type of equation for the reactant pair distribution function with a sink term defined by the reaction mechanism. Microheterogeneity can affect the microscopic rate constant. It also enters the evolution equation through various spatial constraints leading to complicated boundary conditions and, possibly, to the reduction of dimensionality of the diffusion space. The reaction coordinate and diffusive motion along this coordinate are understood in a general way, depending on the problem at hand. Thus, the evolution operator can describe translational and rotational diffusion of molecules in a usual sense, it can be a discrete random walk operator when dealing with hopping of adsorbates in solids, or it can correspond to conformational fluctuations in proteins. Mathematical formulation is universal but physical consequences can be different. Understanding the principal features of reaction kinetics in microheterogeneous systems enables one to extract important structural and dynamical information about the host environments by analyzing suitably designed experiments, it helps building effective strategies for computer simulations, and ultimately opens possibilities for designing systems with controllable reactivity properties.