Dynamics and dissipation in enzyme catalysis

Semiclassical instanton theory for reaction rates at any temperature: How a rigorous real-time derivation solves the crossover temperature problem

Instanton theory relates the rate constant for tunneling through a barrier to the periodic classical trajectory on the upturned potential energy surface, whose period is τ = ℏ/(kBT). Unfortunately, the standard theory is only applicable below the "crossover temperature," where the periodic orbit first appears. This paper presents a rigorous semiclassical (ℏ → 0) theory for the rate that is valid at any temperature. The theory is derived by combining Bleistein's method for generating uniform asymptotic expansions with a real-time modification of Richardson's flux-correlation function derivation of instanton theory. The resulting theory smoothly connects the instanton result at low temperature to the parabolic correction to Eyring transition state theory at high-temperature. Although the derivation involves real time, the final theory only involves imaginary-time (thermal) properties, consistent with the standard version of instanton theory. Therefore, it is no more difficult to compute than the standard theory. The theory is illustrated with application to model systems, where it is shown to give excellent numerical results. Finally, the first-principles approach taken here results in a number of advantages over previous attempts to extend the imaginary free-energy formulation of instanton theory. In addition to producing a theory that is a smooth (continuously differentiable) function of temperature, the derivation also naturally incorporates hyperasymptotic (i.e., multi-orbit) terms and provides a framework for further extensions of the theory.

Coherence in Chemistry: Foundations and Frontiers

Coherence refers to correlations in waves. Because matter has a wave-particle nature, it is unsurprising that coherence has deep connections with the most contemporary issues in chemistry research (e.g., energy harvesting, femtosecond spectroscopy, molecular qubits and more). But what does the word "coherence" really mean in the context of molecules and other quantum systems? We provide a review of key concepts, definitions, and methodologies, surrounding coherence phenomena in chemistry, and we describe how the terms "coherence" and "quantum coherence" refer to many different phenomena in chemistry. Moreover, we show how these notions are related to the concept of an interference pattern. Coherence phenomena are indeed complex, and ambiguous definitions may spawn confusion. By describing the many definitions and contexts for coherence in the molecular sciences, we aim to enhance understanding and communication in this broad and active area of chemistry.

Quantum rates in dissipative systems with spatially varying friction

We investigate whether making the friction spatially dependent on the reaction coordinate introduces quantum effects into the thermal reaction rates for dissipative reactions. Quantum rates are calculated using the numerically exact multi-configuration time-dependent Hartree method, as well as the approximate ring-polymer molecular dynamics (RPMD), ring-polymer instanton methods, and classical molecular dynamics. By conducting simulations across a wide range of temperatures and friction strengths, we can identify the various regimes that govern the reactive dynamics. At high temperatures, in addition to the spatial-diffusion and energy-diffusion regimes predicted by Kramer's rate theory, a (coherent) tunneling-dominated regime is identified at low friction. At low temperatures, incoherent tunneling dominates most of Kramer's curve, except at very low friction, when coherent tunneling becomes dominant. Unlike in classical mechanics, the bath's influence changes the equilibrium time-independent properties of the system, leading to a complex interplay between spatially dependent friction and nuclear quantum effects even at high temperatures. More specifically, a realistic friction profile can lead to an increase (or decrease) of the quantum (classical) rates with friction within the spatial-diffusion regime, showing that classical and quantum rates display qualitatively different behaviors. Except at very low frictions, we find that RPMD captures most of the quantum effects in the thermal reaction rates.

Anisotropic dynamics of an interfacial enzyme active site observed using tethered substrate analogs and ultrafast 2D IR spectroscopy

Enzymes accelerate the rates of biomolecular reactions by many orders of magnitude compared to bulk solution, and it is widely understood that this catalytic effect arises from a combination of polar pre-organization and electrostatic transition state stabilization. A number of recent reports have also implicated ultrafast (femtosecond-picosecond) timescale motions in enzymatic activity. However, complications arising from spatially-distributed disorder, the occurrence of multiple substrate binding modes, and the influence of hydration dynamics on solvent-exposed active sites still confound many experimental studies. Here we use ultrafast two-dimensional infrared (2D IR) spectroscopy and covalently-tethered substrate analogs to examine dynamical properties of the promiscuous Pyrococcus horikoshii ene-reductase (PhENR) active site in two binding configurations mimicking proposed "inactive" and "reactive" Michaelis complexes. Spectral diffusion measurements of aryl-nitrile substrate analogs reveal an end-to-end tradeoff between fast (sub-ps) and slow (>5 ps) motions. Fermi resonant aryl-azide analogs that sense interactions of coupled oscillators are described. Lineshape and quantum beat analyses of these probes reveal characteristics that correlate with aryl-nitrile frequency fluctuation correlation functions parameters, demonstrating that this anisotropy is an intrinsic property of the water-exposed active site, where countervailing gradients of fast dynamics and disorder in the reactant ground state are maintained near the hydration interface. Our results suggest several plausible factors leading to state-selective rate enhancement and promiscuity in PhENR. This study also highlights a strategy to detect perturbations to vibrational modes outside the transparent window of the mid-IR spectrum, which may be extended to other macromolecular systems.

A multi-layered computational structural genomics approach enhances domain-specific interpretation of Kleefstra syndrome variants in EHMT1

This study investigates the functional significance of assorted variants of uncertain significance (VUS) in euchromatic histone lysine methyltransferase 1 (EHMT1), which is critical for early development and normal physiology. EHMT1 mutations cause Kleefstra syndrome and are linked to various human cancers. However, accurate functional interpretations of these variants are yet to be made, limiting diagnoses and future research. To overcome this, we integrate conventional tools for variant calling with computational biophysics and biochemistry to conduct multi-layered mechanistic analyses of the SET catalytic domain of EHMT1, which is critical for this protein function. We use molecular mechanics and molecular dynamics (MD)-based metrics to analyze the SET domain structure and functional motions resulting from 97 Kleefstra syndrome missense variants within the domain. Our approach allows us to classify the variants in a mechanistic manner into SV (Structural Variant), DV (Dynamic Variant), SDV (Structural and Dynamic Variant), and VUS (Variant of Uncertain Significance). Our findings reveal that the damaging variants are mostly mapped around the active site, substrate binding site, and pre-SET regions. Overall, we report an improvement for this method over conventional tools for variant interpretation and simultaneously provide a molecular mechanism for variant dysfunction.

Alternate Strategies to Induce Dynamically Modulated Transient Transcription Machineries

Emulating native transient transcription machineries modulating temporal gene expression by synthetic circuits is a major challenge in the area of systems chemistry. Three different methods to operate transient transcription machineries and to modulate the gated transcription processes of target RNAs are introduced. One method involves the design of a reaction module consisting of transcription templates being triggered by promoter fuel strands transcribing target RNAs and in parallel generating functional DNAzymes in the transcription templates, modulating the dissipative depletion of the active templates and the transient operation of transcription circuits. The second approach involves the application of a reaction module consisting of two transcription templates being activated by a common fuel promoter strand. While one transcription template triggers the transcription of the target RNA, the second transcription template transcribes the anti-fuel strand, displacing the promoter strand associated with the transcription templates, leading to the depletion of the transcription templates and to the dynamic transient modulation of the transcription process. The third strategy involves the assembly of a reaction module consisting of a reaction template triggered by a fuel promoter strand transcribing the target RNA. The concomitant nickase-stimulated depletion of the promoter strand guides the transient modulation of the transcription process. Via integration of two parallel fuel-triggered transcription templates in the three transcription reaction modules and application of template-specific blocker units, the parallel and gated transiently modulated transcription of two different RNA aptamers is demonstrated. The nickase-stimulated transiently modulated transcription reaction module is applied as a functional circuit guiding the dynamic expression of gated, transiently operating, catalytic DNAzymes.

Equilibrium-nonequilibrium ring-polymer molecular dynamics for nonlinear spectroscopy

Two-dimensional Raman and hybrid terahertz-Raman spectroscopic techniques provide invaluable insight into molecular structures and dynamics of condensed-phase systems. However, corroborating experimental results with theory is difficult due to the high computational cost of incorporating quantum-mechanical effects in the simulations. Here, we present the equilibrium-nonequilibrium ring-polymer molecular dynamics (RPMD), a practical computational method that can account for nuclear quantum effects on the two-time response function of nonlinear optical spectroscopy. Unlike a recently developed approach based on the double Kubo transformed (DKT) correlation function, our method is exact in the classical limit, where it reduces to the established equilibrium-nonequilibrium classical molecular dynamics method. Using benchmark model calculations, we demonstrate the advantages of the equilibrium-nonequilibrium RPMD over classical and DKT-based approaches. Importantly, its derivation, which is based on the nonequilibrium RPMD, obviates the need for identifying an appropriate Kubo transformed correlation function and paves the way for applying real-time path-integral techniques to multidimensional spectroscopy.

Structural dynamics support electrostatic interactions in the active site of Adenylate Kinase

Electrostatic preorganization as well as structural and dynamic heterogeneity are often used to rationalize the remarkable catalytic efficiency of enzymes. However, they are often presented as incompatible because the generation of permanent electrostatic effects implies that the protein structure remains rigid. Here, we use a metric, electric fields, that can treat electrostatic contributions and dynamics effects on equal footing, for a unique perspective on enzymatic catalysis. We find that the residues that contribute the most to electrostatic interactions with the substrate in the active site of Adenylate Kinase (our working example) are also the most flexible residues. Further, entropy-tuning mutations raise flexibility at the picosecond timescale where more conformations can be visited on short time periods, thereby softening the sharp heterogeneity normally visible at the microsecond timescale.

Path Integrals for Nonadiabatic Dynamics: Multistate Ring Polymer Molecular Dynamics

This review focuses on a recent class of path-integral-based methods that simulate nonadiabatic dynamics in the condensed phase using only classical molecular dynamics trajectories in an extended phase space. Specifically, a semiclassical mapping protocol is used to derive an exact, continuous, Cartesian variable path-integral representation for the canonical partition function of a system in which multiple electronic states are coupled to nuclear degrees of freedom. Building on this exact statistical foundation, multistate ring polymer molecular dynamics methods are developed for the approximate calculation of real-time thermal correlation functions. The remarkable promise of these multistate ring polymer methods, their successful applications, and their limitations are discussed in detail.Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Energy Bilocalization Effect and the Emergence of Molecular Functions in Proteins

Proteins are among the most complex molecular structures, which have evolved to develop broad functions, such as energy conversion and transport, information storage and processing, communication, and regulation of chemical reactions. However, the mechanisms by which these dynamical entities coordinate themselves to perform biological tasks remain hotly debated. Here, a physical theory is presented to explain how functional dynamical behavior possibly emerge in complex/macro molecules, thanks to the effect that we term bilocalization of thermal vibrations. More specifically, our approach allows us to understand how structural irregularities lead to a partitioning of the energy of the vibrations into two distinct sets of molecular domains, corresponding to slow and fast motions. This shape-encoded spectral allocation, associated to the genetic sequence, provides a close access to a wide reservoir of dynamical patterns, and eventually allows the emergence of biological functions by natural selection. To illustrate our approach, the SPIKE protein structure of SARS-COV2 is considered.

On the "Matsubara heating" of overtone intensities and Fermi splittings

Classical molecular dynamics (MD) and imaginary-time path-integral dynamics methods underestimate the infrared absorption intensities of overtone and combination bands by typically an order of magnitude. Plé et al. [J. Chem. Phys. 155, 2863 (2021)] have shown that this is because such methods fail to describe the coupling of the centroid to the Matsubara dynamics of the fluctuation modes; classical first-order perturbation theory (PT) applied to the Matsubara dynamics is sufficient to recover most of the lost intensity in simple models and gives identical results to quantum (Rayleigh-Schrödinger) PT. Here, we show numerically that the results of this analysis can be used as post-processing correction factors, which can be applied to realistic (classical MD or path-integral dynamics) simulations of infrared spectra. We find that the correction factors recover most of the lost intensity in the overtone and combination bands of gas-phase water and ammonia and much of it for liquid water. We then re-derive and confirm the earlier PT analysis by applying canonical PT to Matsubara dynamics, which has the advantage of avoiding secular terms and gives a simple picture of the perturbed Matsubara dynamics in terms of action-angle variables. Collectively, these variables "Matsubara heat" the amplitudes of the overtone and combination vibrations of the centroid to what they would be in a classical system with the oscillators (of frequency Ω_i) held at their quantum effective temperatures [of ℏΩ_i coth(βℏΩ_i/2)/2k_B]. Numerical calculations show that a similar neglect of "Matsubara heating" causes path-integral methods to underestimate Fermi resonance splittings.

Isotopic separation of helium through graphyne membranes: a ring polymer molecular dynamics study

Microscopic-level understanding of the separation mechanism for two-dimensional (2D) membranes is an active area of research due to potential implications of this class of membranes for various technological processes. Helium (He) purification from the natural resources is of particular interest due to the shortfall in its production. In this work, we applied the ring polymer molecular dynamics (RPMD) method to graphdiyne (Gr2) and graphtriyne (Gr3) 2D membranes having variable pore sizes for the separation of He isotopes, and compare for the first time with rigorous quantum calculations. We found that the transmission rate through Gr3 is many orders of magnitude greater than Gr2. The selectivity of either isotope at low temperatures is a consequence of a delicate balance between the zero-point energy effect and tunneling of ⁴He and ³He. In particular, a remarkable tunneling effect is reported on the Gr2 membrane at 10 K, leading to a much larger permeation of the lighter species as compared to the heavier isotope. RPMD provides an efficient approach for studying the separation of He isotopes, taking into account quantum effects of light nuclei motions at low temperatures, which classical methods fail to capture.

Comparative studies of IR spectra of deprotonated serine with classical and thermostated ring polymer molecular dynamics simulations

Here we report the vibrational spectra of deprotonated serine calculated from the classical molecular dynamics (MD) simulations and thermostated ring-polymer molecular dynamics (TRPMD) simulation with third-order density-functional tight-binding. In our earlier study [Inakollu and Yu, "A systematic benchmarking of computational vibrational spectroscopy with DFTB3: Normal mode analysis and fast Fourier transform dipole autocorrelation function," J. Comput. Chem. 39, 2067 (2018)] of deprotonated serine, we observed a significant difference in the vibrational spectra with the classical MD simulations compared to the infrared multiple photon dissociation spectra. It was postulated that this is due to neglecting the nuclear quantum effects (NQEs). In this work, NQEs are considered in spectral calculation using the TRPMD simulations. With the help of potential of mean force calculations, the conformational space of deprotonated serine is analyzed and used to understand the difference in the spectra of classical MD and TRPMD simulations at 298.15 and 100 K. The high-frequency vibrational bands in the spectra are characterized using Fourier transform localized vibrational mode (FT-νN AC) and interatomic distance histograms. At room temperature, the quantum effects are less significant, and the free energy profiles in the classical MD and the TRPMD simulations are very similar. However, the hydrogen bond between the hydroxyl-carboxyl bond is slightly stronger in TRPMD simulations. At 100 K, the quantum effects are more prominent, especially in the 2600-3600 cm-1, and the free energy profile slightly differs between the classical MD and TRPMD simulations. Using the FT-νN AC and the interatomic distance histograms, the high-frequency vibrational bands are discussed in detail.

Investigating Tunneling-Controlled Chemical Reactions through Ab Initio Ring Polymer Molecular Dynamics

We use the ab initio ring polymer molecular dynamics (RPMD) approach to investigate tunneling-controlled reactions in methylhydroxycarbene. Nuclear tunneling effects enable molecules to overcome the barriers which cannot be overcome classically. Under low-temperature conditions, intrinsic quantum tunneling effects can facilitate the chemical reaction in a pathway that is favored neither thermodynamically nor kinetically. This behavior is referred to as the tunneling-controlled chemical reaction and is regarded as the third paradigm of chemical reaction controls. In this work, we use the ab initio RPMD approach to incorporate the tunneling effects in our quantum dynamics simulations and investigate the reaction kinetics of two competitive reaction pathways at various temperatures. The reaction rate constants obtained here agree extremely well with the experimentally measured rates. We demonstrate the feasibility of using ab initio RPMD rate calculations in a realistic molecular system and provide an interesting and important example for future investigations of reaction mechanisms dominated by quantum tunneling effects.

Non-adiabatic Matsubara dynamics and non-adiabatic ring-polymer molecular dynamics

We present the non-adiabatic Matsubara dynamics, a general framework for computing the time-correlation function (TCF) of electronically non-adiabatic systems. This new formalism is derived based on the generalized Kubo-transformed TCF using the Wigner representation for both the nuclear degrees of freedom and the electronic mapping variables. By dropping the non-Matsubara nuclear normal modes in the quantum Liouvillian and explicitly integrating these modes out from the expression of the TCF, we derived the non-adiabatic Matsubara dynamics approach. Further making the approximation to drop the imaginary part of the Matsubara Liouvillian and enforce the nuclear momentum integral to be real, we arrived at the non-adiabatic ring-polymer molecular dynamics (NRPMD) approach. We have further justified the capability of NRPMD for simulating the non-equilibrium TCF. This work provides the rigorous theoretical foundation for several recently proposed state-dependent RPMD approaches and offers a general framework for developing new non-adiabatic quantum dynamics methods in the future.

Recent progress in approximate quantum dynamics methods for the study of proton-coupled electron transfer reactions

Proton-coupled electron transfer (PCET) reactions are ubiquitous natural processes at the heart of energy conversion reactions in photosynthesis and respiration, DNA repair, and diverse enzymatic reactions. Theoretical formulation and computational method developments have eyed modeling of thermal and photoinduced PCET for the last three decades. The accumulation of these studies, collected in dozens of reviews, accounts, and perspectives, has firmly established the influence of quantum effects, including non-adiabatic electronic transitions, vibrational relaxation, zero-point energy, and proton tunneling, on the rate and mechanism of PCET reactions. Here, we focus on some recently-developed methods, spanning the last eight years, that can quantitatively capture these effects in the PCET context and provide efficient means for their qualitative description in complex systems. The theoretical background of each method and their accuracy with respect to exact results are discussed and the results of relevant PCET simulations based on each method are presented.

Ring polymer quantization of the photon field in polariton chemistry

We use the ring polymer (RP) representation to quantize the radiation field inside an optical cavity to investigate polariton quantum dynamics. Using a charge transfer model coupled to an optical cavity, we demonstrate that the RP quantization of the photon field provides accurate rate constants of the polariton mediated electron transfer reaction compared to Fermi's golden rule. Because RP quantization uses extended phase space to describe the photon field, it significantly reduces the computational costs compared to the commonly used Fock state description of the radiation field. Compared to the other quasi-classical descriptions of the photon field, such as the classical Wigner based mean-field Ehrenfest model, the RP representation provides a much more accurate description of the polaritonic quantum dynamics because it alleviates the potential quantum distribution leakage problem associated with the photonic degrees of freedom (DOF). This work demonstrates the possibility of using the ring polymer description to treat the quantized radiation field in polariton chemistry, offering an accurate and efficient approach for future investigations in cavity quantum electrodynamics.

Catalytic Principles from Natural Enzymes and Translational Design Strategies for Synthetic Catalysts

As biocatalysts, enzymes are characterized by their high catalytic efficiency and strong specificity but are relatively fragile by requiring narrow and specific reactive conditions for activity. Synthetic catalysts offer an opportunity for more chemical versatility operating over a wider range of conditions but currently do not reach the remarkable performance of natural enzymes. Here we consider some new design strategies based on the contributions of nonlocal electric fields and thermodynamic fluctuations to both improve the catalytic step and turnover for rate acceleration in arbitrary synthetic catalysts through bioinspired studies of natural enzymes. With a focus on the enzyme as a whole catalytic construct, we illustrate the translational impact of natural enzyme principles to synthetic enzymes, supramolecular capsules, and electrocatalytic surfaces.

An improved path-integral method for golden-rule rates

We present a simple method for the calculation of reaction rates in the Fermi golden-rule limit, which accurately captures the effects of tunneling and zero-point energy. The method is based on a modification of the recently proposed golden-rule quantum transition state theory (GR-QTST) of Thapa, Fang, and Richardson [J. Chem. Phys. 150, 104107 (2019)]. While GR-QTST is not size consistent, leading to the possibility of unbounded errors in the rate, our modified method has no such issue and so can be reliably applied to condensed phase systems. Both methods involve path-integral sampling in a constrained ensemble; the two methods differ, however, in the choice of constraint functional. We demonstrate numerically that our modified method is as accurate as GR-QTST for the one-dimensional model considered by Thapa and co-workers. We then study a multidimensional spin-boson model, for which our method accurately predicts the true quantum rate, while GR-QTST breaks down with an increasing number of boson modes in the discretization of the spectral density. Our method is able to accurately predict reaction rates in the Marcus inverted regime without the need for the analytic continuation required by Wolynes theory.

The physical origin of rate promoting vibrations in enzymes revealed by structural rigidity

Enzymes are the most efficient catalysts known to date. However, decades of research have failed to fully explain the catalytic power of enzymes, and most of the current attempts to uncloak the details of atomic motions at active sites remain incomplete. Here, a straightforward manner for understanding the interplay between the complex or irregular enzyme topology and dynamical effects at catalytic sites is introduced, by revealing how fast localized vibrations form spontaneously in the stiffest parts of the scaffold. While shedding light on a physical mechanism that allowed the selection of the picosecond (ps) timescale to increase the catalytic proficiency, this approach exposes the functional importance of localized motions as a by-product of the stability-function tradeoff in enzyme evolution. From this framework of analysis—directly accessible from available diffraction data—experimental strategies for engineering the catalytic rate in enzymatic proteins are proposed.

Revisiting nuclear tunnelling in the aqueous ferrous–ferric electron transfer

We find that golden-rule quantum transition-state theory predicts nearly an order of magnitude less tunnelling than some of the previous estimates. This may indicate that the spin-boson model of electron transfer is not valid in the quantum regime.

Path integral methods for reaction rates in complex systems

We shall use this introduction to the Faraday Discussion on quantum effects in complex systems to review the recent progress that has been made in using imaginary time path integral methods to calculate chemical reaction rates.

The low temperature D+ + H2→ HD + H+ reaction rate coefficient: a ring polymer molecular dynamics and quasi-classical trajectory study

The reaction between D⁺ and H₂ plays an important role in astrochemistry at low temperatures and also serves as a prototype for a simple ion-molecule reaction. Its ground X[combining tilde]¹A' state has a very small thermodynamic barrier (up to 1.8 × 10^-2 eV) and the reaction proceeds through the formation of an intermediate complex lying within the potential well with a depth of at least 0.2 eV, thus representing a challenge for dynamical studies. In the present work, we analyze the title reaction within the temperature range of 20-100 K by means of ring polymer molecular dynamics (RPMD) and quasi-classical trajectory (QCT) methods over the full-dimensional global potential energy surface developed by Aguado et al. [A. Aguado, O. Roncero, C. Tablero, C. Sanz and M. Paniagua, J. Chem. Phys., 2000, 112, 1240]. The computed thermal RPMD and QCT rate coefficients are found to be almost independent of temperature and fall within the range of 1.34-2.01 × 10^-9 cm³ s^-1. They are also in very good agreement with previous time-independent quantum mechanical and statistical quantum method calculations. Furthermore, we observe that the choice of asymptotic separation distance between the reactants can markedly alter the rate coefficient in the low temperature regime (20-50 K). Therefore it is of utmost importance to correctly assign the value of this parameter for dynamical studies, particularly at very low temperatures of astrochemical importance. We finally conclude that the experimental rate measurements for the title reaction are highly desirable in future.

Effect of Asp122 Mutation on the Hydride Transfer in E. coli DHFR Demonstrates the Goldilocks of Enzyme Flexibility

Dihydrofolate reductase (DHFR) catalyzes the reduction of dihydrofolate (DHF) to tetrahydrofolate (THF) in the presence of NADPH. The key hydride transfer step in the reaction is facilitated by a combination of enzyme active site preorganization and correlated protein motions in the Michaelis-Menten (E:NADPH:DHF) complex. The present theoretical study employs mutagenesis to examine the relation between structural and functional properties of the enzyme. We mutate Asp122 in Escherichia coli DHFR, which is a conserved amino acid in the DHFR family. The consequent effect of the mutation on enzyme catalysis is examined from an energetic, structural and short-time dynamic perspective. Our investigations suggest that the structural and short-time dynamic perturbations caused by Asp122X mutations (X = Asn, Ser, Ala) are along the reaction coordinate and lower the rate of hydride transfer. Importantly, analysis of the correlated and principle component motions in the enzyme suggest that the mutation alters the coupled motions that are present in the wild-type enzyme. In the case of D122N and D122S, the mutations inhibit coupled motion, whereas in the case of D122A, the mutation enhances coupled motion, although all mutations result in similar rate reduction. These results emphasize a Goldilocks principle of enzyme flexibility, that is, enzymes should neither be too rigid nor too flexible.

Approximating Matsubara dynamics using the planetary model: Tests on liquid water and ice

Matsubara dynamics is the quantum-Boltzmann-conserving classical dynamics which remains when real-time coherences are taken out of the exact quantum Liouvillian [T. J. H. Hele et al., J. Chem. Phys. 142, 134103 (2015)]; because of a phase-term, it cannot be used as a practical method without further approximation. Recently, Smith et al. [J. Chem. Phys. 142, 244112 (2015)] developed a "planetary" model dynamics which conserves the Feynman-Kleinert (FK) approximation to the quantum-Boltzmann distribution. Here, we show that for moderately anharmonic potentials, the planetary dynamics gives a good approximation to Matsubara trajectories on the FK potential surface by decoupling the centroid trajectory from the locally harmonic Matsubara fluctuations, which reduce to a single phase-less fluctuation particle (the "planet"). We also show that the FK effective frequency can be approximated by a direct integral over these fluctuations, obviating the need to solve iterative equations. This modification, together with use of thermostatted ring-polymer molecular dynamics, allows us to test the planetary model on water (gas-phase, liquid, and ice) using the q-TIP4P/F potential surface. The "planetary" fluctuations give a poor approximation to the rotational/librational bands in the infrared spectrum, but a good approximation to the bend and stretch bands, where the fluctuation lineshape is found to be motionally narrowed by the vibrations of the centroid.

Harmonic-phase path-integral approximation of thermal quantum correlation functions

We present an approximation to the thermal symmetric form of the quantum time-correlation function in the standard position path-integral representation. By transforming to a sum-and-difference position representation and then Taylor-expanding the potential energy surface of the system to second order, the resulting expression provides a harmonic weighting function that approximately recovers the contribution of the phase to the time-correlation function. This method is readily implemented in a Monte Carlo sampling scheme and provides exact results for harmonic potentials (for both linear and non-linear operators) and near-quantitative results for anharmonic systems for low temperatures and times that are likely to be relevant to condensed phase experiments. This article focuses on one-dimensional examples to provide insights into convergence and sampling properties, and we also discuss how this approximation method may be extended to many-dimensional systems.

Quantifying the limits of transition state theory in enzymatic catalysis

While being one of the most popular reaction rate theories, the applicability of transition state theory to the study of enzymatic reactions has been often challenged. The complex dynamic nature of the protein environment raised the question about the validity of the nonrecrossing hypothesis, a cornerstone in this theory. We present a computational strategy to quantify the error associated to transition state theory from the number of recrossings observed at the equicommittor, which is the best possible dividing surface. Application of a direct multidimensional transition state optimization to the hydride transfer step in human dihydrofolate reductase shows that both the participation of the protein degrees of freedom in the reaction coordinate and the error associated to the nonrecrossing hypothesis are small. Thus, the use of transition state theory, even with simplified reaction coordinates, provides a good theoretical framework for the study of enzymatic catalysis.

The role of the Met20 loop in the hydride transfer in Escherichia coli dihydrofolate reductase

A key question concerning the catalytic cycle of Escherichia coli dihydrofolate reductase (ecDHFR) is whether the Met²⁰ loop is dynamically coupled to the chemical step during catalysis. A more basic, yet unanswered question is whether the Met²⁰ loop adopts a closed conformation during the chemical hydride transfer step. To examine the most likely conformation of the Met²⁰ loop during the chemical step, we studied the hydride transfer in wild type (WT) ecDHFR using hybrid quantum mechanics-molecular mechanics free energy simulations with the Met²⁰ loop in a closed and disordered conformation. Additionally, we investigated three mutant forms (I14X; X = Val, Ala, Gly) of the enzyme that have increased active site flexibility and donor-acceptor distance dynamics in closed and disordered Met²⁰ loop states. We found that the conformation of the Met²⁰ loop has a dramatic effect on the ordering of active site hydration, although the Met²⁰ loop conformation only has a moderate effect on the hydride transfer rate and donor-acceptor distance dynamics. Finally, we evaluated the pK_a of the substrate N5 position in closed and disordered Met²⁰ loop states and found a strong correlation between N5 basicity and the conformation of the Met²⁰ loop.

Molecular Basis for K63-Linked Ubiquitination Processes in Double-Strand DNA Break Repair: A Focus on Kinetics and Dynamics

Cells are exposed to thousands of DNA damage events on a daily basis. This damage must be repaired to preserve genetic information and prevent development of disease. The most deleterious damage is a double-strand break (DSB), which is detected and repaired by mechanisms known as non-homologous end-joining (NHEJ) and homologous recombination (HR), which are components of the DNA damage response system. NHEJ is an error-prone first line of defense, whereas HR invokes error-free repair and is the focus of this review. The functions of the protein components of HR-driven DNA repair are regulated by the coordinated action of post-translational modifications including lysine acetylation, phosphorylation, ubiquitination, and SUMOylation. The latter two mechanisms are fundamental for recognition of DSBs and reorganizing chromatin to facilitate repair. We focus on the structures and molecular mechanisms for the protein components underlying synthesis, recognition, and cleavage of K63-linked ubiquitin chains, which are abundant at damage sites and obligatory for DSB repair. The forward flux of the K63-linked ubiquitination cascade is driven by the combined activity of E1 enzyme, the heterodimeric E2 Mms2-Ubc13, and its cognate E3 ligases RNF8 and RNF168, which is balanced through the binding and cleavage of chains by the deubiquitinase BRCC36, and the proteasome, and through the binding of chains by recognition modules on repair proteins such as RAP80. We highlight a number of aspects regarding our current understanding for the role of kinetics and dynamics in determining the function of the enzymes and chain recognition modules that drive K63 ubiquitination.

Reduction of Folate by Dihydrofolate Reductase from Thermotoga maritima

Mammalian dihydrofolate reductases (DHFRs) catalyze the reduction of folate more efficiently than the equivalent bacterial enzymes do, despite typically having similar efficiencies for the reduction of their natural substrate, dihydrofolate. In contrast, we show here that DHFR from the hyperthermophilic bacterium Thermotoga maritima can catalyze reduction of folate to tetrahydrofolate with an efficiency similar to that of reduction of dihydrofolate under saturating conditions. Nuclear magnetic resonance and mass spectrometry experiments showed no evidence of the production of free dihydrofolate during either the EcDHFR- or TmDHFR-catalyzed reductions of folate, suggesting that both enzymes perform the two reduction steps without release of the partially reduced substrate. Our results imply that the reaction proceeds more efficiently in TmDHFR than in EcDHFR because the more open active site of TmDHFR facilitates protonation of folate. Because T. maritima lives under extreme conditions where tetrahydrofolate is particularly prone to oxidation, this ability to salvage folate may impart an advantage to the bacterium by minimizing the squandering of a valuable cofactor.