Reaction Coordinate, Free Energy, and Rate of Intramolecular Proton Transfer in Human Carbonic Anhydrase II

pH-Dependent Structure and Dynamics of the Catalytic Domains of Human Carbonic Anhydrase II and IX

Extensive computer simulation studies have been carried out to probe the pH-dependent structure and dynamics of the two most efficient isoenzymes II and IX of human carbonic anhydrase (HCA) that control the pH in the human body. The equilibrium structure and hydration of their catalytic domains are found to be largely unaffected by the variation of pH in the range studied, in close agreement with the known experimental results. In contrast, a significant effect of the change in pH is observed for the first time on the local electrostatic potential of the active site walls and the dynamics of active site water molecules. We also report for the first time the free energy and kinetics of coupled fluctuations of orientation and protonation states of the well-known His-mediated proton shuttle (His-64) in both isozymes at pH 7 and 8. The transitions between different tautomers of in or out conformations of His-64 side chain range between 109 and 106 s-1 depending on pH. Possible implications of these results on conformation-dependent pKa of His-64 side chain and its role in driving the catalysis toward hydration of CO2 or dehydration of HCO3- with varying pH are discussed.

Diffusion in a two-dimensional energy landscape in the presence of dynamical correlations and validity of random walk model

Diffusion in a multidimensional energy surface with minima and barriers is a problem of importance in statistical mechanics and it also has wide applications, such as protein folding. To understand it in such a system, we carry out theory and simulations of a tagged particle moving on a two-dimensional periodic potential energy surface, both in the presence and absence of noise. Langevin dynamics simulations at multiple temperatures are carried out to obtain the diffusion coefficient of a solute particle. Friction is varied from zero to large values. Diffusive motion emerges in the limit of a long time, even in the absence of noise. Noise destroys the correlations and increases diffusion at small friction. Diffusion thus exhibits a nonmonotonic friction dependence at the intermediate value of the damping, ultimately converging to our theoretically predicted value. The latter is obtained using the well-established relationship between diffusion and random walk. An excellent agreement is obtained between theory and simulations in the high-friction limit but not so in the intermediate regime. We explain the deviation in the low- to intermediate-friction regime using the modified random walk theory. The rate of escape from one cell to another is obtained from the multidimensional rate theory of Langer. We find that enhanced dimensionality plays an important role. To quantify the effects of noise on the potential-imposed coherence on the trajectories, we calculate the Lyapunov exponent. At small friction values, the Lyapunov exponent mimics the friction dependence of the rate.

Perspective: Path Sampling Methods Applied to Enzymatic Catalysis

This Perspective reviews the use of Transition Path Sampling methods to study enzymatically catalyzed chemical reactions. First applied by our group to an enzymatic reaction over 15 years ago, the method has uncovered basic principles in enzymatic catalysis such as the protein promoting vibration, and it has also helped harmonize such ideas as electrostatic preorganization with dynamic views of enzyme function. It is now being used to help uncover principles of protein design necessary to artificial enzyme creation.

Structure and Dynamics of the Isozymes II and IX of Human Carbonic Anhydrase

Human carbonic anhydrases (HCAs) are responsible for the pH control and sensing in our body and constitute key components in the central pH paradigm connected to cancer therapeutics. However, little or no molecular level studies are available on the pH-dependent stability and functional dynamics of the known isozymes of HCA. The main objective of this Article is to report the first bench-marking study on the structure and dynamics of the two most efficient isozymes, HCA II and IX, at neutral pH using classical molecular dynamics (MD) and constant pH MD (CpHMD) simulations combined with umbrella sampling, transition path sampling, and Markov state models. Starting from the known crystal structures of HCA II and the monomeric catalytic domain of HCA IX (labeled as HCA IX-c), we have generated classical MD and CpHMD trajectories (of length 1 μs each). In all cases, the overall stability, RMSD, and secondary structure segments of the two isozymes are found to be quite similar. Functionally important dynamics of these two enzymes have been probed in terms of active site hydration, coordination of the Zn(II) ion to a transient excess water, and the formation of putative proton transfer paths. The most important difference between the two isozymes is observed for the side-chain fluctuations of His-64 that is expected to shuttle an excess proton out of the active site as a part of the rate-determining intramolecular proton transfer reaction. The relative stability of the stable inward and outward conformations of the His-64 side-chain and the underlying free energy surfaces are found to depend strongly on the isozyme. In each case, a lower free energy barrier is detected between predominantly inward conformations from predominantly outward ones when simulated under constant pH conditions. The kinetic rate constants of interconversion between different free energy basins are found to span 10⁷-10⁸ s^-1 with faster conformational transitions predicted at constant pH condition. The estimated rate constants and free energies are expected to validate if the fluctuation of the His-64 side-chain in HCA IX may have a significance similar to that known in the multistep catalytic cycle of HCA II.

Non-Markovian rate theory on a multidimensional reaction surface: Complex interplay between enhanced configuration space and memory

A theory of barrier crossing rate on a multidimensional reaction energy surface is presented. The theory is a generalization of the earlier theoretical schemes to higher dimensions, with the inclusion of non-Markovian friction along both the reactive and the nonreactive coordinates. The theory additionally includes the bilinear coupling between the reactive and the nonreactive modes at the Hamiltonian level. Under suitable conditions, we recover the rate expressions of Langer and Hynes and establish a connection with the rate treatment of Pollak. Within the phenomenology of generalized Langevin equation description, our formulation provides an improvement over the existing ones because we explicitly include both the non-Markovian effects along the reaction coordinate and the bilinear coupling at the Hamiltonian level. At intermediate-to-large friction, an increase in dimensionality by itself tends to reduce the rate, while the inclusion of the memory effects increases the rate. The theory predicts an increase in rate when off-diagonal friction terms are included. We present a model calculation to study isomerization of a stilbene-like molecule using the prescription of Hochstrasser and co-workers on a two-dimensional reaction energy surface, employing Zwanzig–Bixon hydrodynamic theory of frequency-dependent friction. The calculated rate shows a departure from the predictions of Langer’s theory and also from the two-dimensional transition state theory.

Dynamics of Excitons in Conjugated Molecules and Organic Semiconductor Systems

The exciton, an excited electron-hole pair bound by Coulomb attraction, plays a key role in photophysics of organic molecules and drives practically important phenomena such as photoinduced mechanical motions of a molecule, photochemical conversions, energy transfer, generation of free charge carriers, etc. Its behavior in extended π-conjugated molecules and disordered organic films is very different and very rich compared with exciton behavior in inorganic semiconductor crystals. Due to the high degree of variability of organic systems themselves, the exciton not only exerts changes on molecules that carry it but undergoes its own changes during all phases of its lifetime, that is, birth, conversion and transport, and decay. The goal of this review is to give a systematic and comprehensive view on exciton behavior in π-conjugated molecules and molecular assemblies at all phases of exciton evolution with emphasis on rates typical for this dynamic picture and various consequences of the above dynamics. To uncover the rich variety of exciton behavior, details of exciton formation, exciton transport, exciton energy conversion, direct and reverse intersystem crossing, and radiative and nonradiative decay are considered in different systems, where these processes lead to or are influenced by static and dynamic disorder, charge distribution symmetry breaking, photoinduced reactions, electron and proton transfer, structural rearrangements, exciton coupling with vibrations and intermediate particles, and exciton dissociation and annihilation as well.

In the footsteps of an inhibitor unbinding from the active site of human carbonic anhydrase II

The crystal structure of human carbonic anhydrase (HCA) II bound to an inhibitor molecule, 6-hydroxy-2-thioxocoumarin (FC5), shows FC5 to be located in a hydrophobic pocket at the active site. The present work employs classical molecular dynamics (MD) simulation to follow the FC5 molecule for 1 μs as it unbinds from its binding location, adopts the path of substrate/product diffusion (path 1) to leave the active site at around 75 ns. It is then found to undergo repeated binding and unbinding at different locations on the surface of the enzyme in water. Several transient excursions through different regions of the enzyme are also observed prior to its exit from the active site. These transient paths are combined with functionally relevant cavities/channels to enlist five additional pathways (path 2-6). Pathways 1-6 are subsequently explored using steered MD and umbrella sampling simulations. A free energy barrier of 0.969 kcal mol^-1 is encountered along path 1, while barriers in the range of 0.57-2.84 kcal mol^-1 are obtained along paths 2, 4 and 5. We also analyze in detail the interaction between FC5 and the enzyme along each path as the former leaves the active site of HCA II. Our results indicate path 1 to be the major exit pathway for FC5, although competing contributions may also come from the paths 2, 4 and 5.Communicated by Ramaswamy H. Sarma.

Nonlinear Reaction Coordinate of an Enzyme Catalyzed Proton Transfer Reaction

We present an in-depth study on the theoretical calculation of an optimum reaction coordinate as a linear or nonlinear combination of important collective variables (CVs) sampled from an ensemble of reactive transition paths for an intramolecular proton transfer reaction catalyzed by the enzyme human carbonic anhydrase (HCA) II. The linear models are optimized by likelihood maximization for a given number of CVs. The nonlinear models are based on an artificial neural network with the same number of CVs and optimized by minimizing the root-mean-square error in comparison to a training set of committor estimators generated for the given transition. The nonlinear reaction coordinate thus obtained yields the free energy of activation and rate constant as 9.46 kcal mol^-1 and 1.25 × 10⁶ s^-1, respectively. These estimates are found to be in quantitative agreement with the known experimental results. We have also used an extended autoencoder model to show that a similar analysis can be carried out using a single CV only. The resultant free energies and kinetics of the reaction slightly overestimate the experimental data. The implications of these results are discussed using a detailed microkinetic scheme of the proton transfer reaction catalyzed by HCA II.

Conformational Change of H64 and Substrate Transportation: Insight Into a Full Picture of Enzymatic Hydration of CO2 by Carbonic Anhydrase

The enzymatic hydration of CO₂ into HCO₃ ^- by carbonic anhydrase (CA) is highly efficient and environment-friendly measure for CO₂ sequestration. Here extensive MM MD and QM/MM MD simulations were used to explore the whole enzymatic process, and a full picture of the enzymatic hydration of CO₂ by CA was achieved. Prior to CO₂ hydration, the proton transfer from the water molecule (WT1) to H64 is the rate-limiting step with the free energy barrier of 10.4 kcal/mol, which leads to the ready state with the Zn-bound OH^-. The nucleophilic attack of OH^- on CO₂ produces HCO₃ ^- with the free energy barrier of 4.4 kcal/mol and the free energy release of about 8.0 kcal/mol. Q92 as the key residue manipulates both CO₂ transportation to the active site and release of HCO₃ ^-. The unprotonated H64 in CA prefers in an inward orientation, while the outward conformation is favorable energetically for its protonated counterpart. The conformational transition of H64 between inward and outward correlates with its protonation state, which is mediated by the proton transfer and the product release. The whole enzymatic cycle has the free energy span of 10.4 kcal/mol for the initial proton transfer step and the free energy change of -6.5 kcal/mol. The mechanistic details provide a comprehensive understanding of the entire reversible conversion of CO₂ into bicarbonate and roles of key residues in chemical and nonchemical steps for the enzymatic hydration of CO₂.

Reaction Coordinate and Thermodynamics of Base Flipping in RNA

Base flipping is a key biophysical event involved in recognition of various ligands by ribonucleic acid (RNA) molecules. However, the mechanism of base flipping in RNA remains poorly understood, in part due to the lack of atomistic details on complex rearrangements in neighboring bases. In this work, we applied transition path sampling (TPS) methods to study base flipping in a double-stranded RNA (dsRNA) molecule that is known to interact with RNA-editing enzymes through this mechanism. We obtained an ensemble of 1000 transition trajectories to describe the base-flipping process. We used the likelihood maximization method to determine the refined reaction coordinate (RC) consisting of two collective variables (CVs), a distance and a dihedral angle between nucleotides that form stacking interactions with the flipping base. The free energy profile projected along the refined RC revealed three minima, two corresponding to the initial and final states and one for a metastable state. We suggest that the metastable state likely represents a wobbled conformation of nucleobases observed in NMR studies that is often characterized as the flipped state. The analyses of reactive trajectories further revealed that the base flipping is coupled to a global conformational change in a stem-loop of dsRNA.

Kohn-Sham Density Functional Calculations Reveal Proton Wires in the Enolization and Carboxylase Reactions Catalyzed by Rubisco

Ribulose 1,5-bisphosphate (RuBP) carboxylase-oxygenase (Rubisco) plays a fundamental role in the carbon cycle by fixing the atmospheric CO₂ used in photosynthesis. Rubisco is all the more remarkable because it must catalyze some difficult multistep reaction chemistry involving proton transfers within the one active site. In the present study, we have used Kohn-Sham density functional theory at the B3LYP/6-31G* level with basis set superposition error and dispersion corrections (B3LYP-gCP-D3) to examine the possibility that the proton transfers can take place through molecular wires (including active-site water molecules) via the classical Grotthuss proton-shuttle mechanism. The results support an essential role for water molecules found in the crystal structures of Rubisco complexes as facilitators of proton transport in all the rate-limiting (catalytic) reaction steps through a network of short proton wires within the Rubisco active site. We suggest that completion of the initial product turnover (cycle) requires two excess protons produced in the initial carbamylation that is required for Rubisco activation. By use of proton wires, a large number of reaction steps may be accommodated within a single active site without necessitating the input of excessive conformational strain energy arising from the movement of residue side chains into positions where direct protonation of substrates can occur. The involvement of the identified types of proton wires in the kinetic mechanism is capable of providing a unique explanation for various experimental observations, including deuterium isotope effects and the results of site-directed mutagenesis experiments, and may thus provide a realistic solution to the problem of Rubisco's challenging chemistry.

Orthogonal order parameters to model the reaction coordinate of an enzyme catalyzed reaction

The choice of suitable collective variables in formulating an optimal reaction coordinate is a challenging task for activated transitions between a pair of stable states especially when dealing with biochemical changes such as enzyme catalyzed reactions. A detailed benchmarking study is carried out on the choice of collective variables that can distinguish between the stable states unambiguously. We specifically address the issue if these variables may be directly used to model the optimal reaction coordinate, or if it would be better to use their orthogonalized counterparts. The proposed computational scheme is applied to the rate determining intramolecular proton transfer step in the enzyme human carbonic anhydrase II. The optimum reaction coordinate is determined with and without orthogonalization of the collective variables pertinent to a key conformational fluctuation and the actual proton transfer step at the active site of the enzyme. Suitability of the predicted reaction coordinates in different processes is examined in terms of the free energy profile projected along the reaction coordinate, the rate constant of transition and the underlying molecular mechanism of barrier crossing. Our results indicate that a better agreement with earlier simulation and experimental data is obtained when the orthogonalized collective variables are used to model the reaction coordinate.

Unraveling Anhydrous Proton Conduction in Hydroxygraphane

We predict that graphane functionalized with hydroxyl groups, hydroxygraphane, can conduct protons in the complete absence of water, as shown from density functional theory calculations. Hydroxygraphane's anhydrous intrinsic proton conductivity results from the self-assembling two-dimensional network of hydrogen bonds on its surface. We show that the proton conduction occurs through a Grotthuss-like mechanism, as protons hop between neighboring hydroxyl groups, aided by their rotation. Our calculations predict that hydroxygraphane has a direct bandgap of 3.43 eV, a phonon dispersion spectrum with no instabilities, and a 2-D Young's modulus and Poisson's ratio stiffer than those for graphane-the parent material for hydroxygraphane. Hence, hydroxygraphane has the desired electronic and mechanical properties to make it a viable candidate for a proton exchange membrane material capable of operating under anhydrous or low-humidity conditions.