Three applications of path integrals: equilibrium and kinetic isotope effects, and the temperature dependence of the rate constant of the [1,5] sigmatropic hydrogen shift in (Z)-1,3-pentadiene

Recent experiments have confirmed the importance of nuclear quantum effects even in large biomolecules at physiological temperature. Here we describe how the path integral formalism can be used to describe rigorously the nuclear quantum effects on equilibrium and kinetic properties of molecules. Specifically, we explain how path integrals can be employed to evaluate the equilibrium (EIE) and kinetic (KIE) isotope effects, and the temperature dependence of the rate constant. The methodology is applied to the [1,5] sigmatropic hydrogen shift in pentadiene. Both the KIE and the temperature dependence of the rate constant confirm the importance of tunneling and other nuclear quantum effects as well as of the anharmonicity of the potential energy surface. Moreover, previous results on the KIE were improved by using a combination of a high level electronic structure calculation within the harmonic approximation with a path integral anharmonicity correction using a lower level method.Comment: 9 pages, 4 figure

Unraveling the role of protein dynamics in dihydrofolate reductase catalysis

Protein dynamics have controversially been proposed to be at the heart of enzyme catalysis, but identification and analysis of dynamical effects in enzyme-catalyzed reactions have proved very challenging. Here, we tackle this question by comparing an enzyme with its heavy (15N, 13C, 2H substituted) counterpart, providing a subtle probe of dynamics. The crucial hydride transfer step of the reaction (the chemical step) occurs more slowly in the heavy enzyme. A combination of experimental results, quantum mechanics/molecular mechanics simulations, and theoretical analyses identify the origins of the observed differences in reactivity. The generally slightly slower reaction in the heavy enzyme reflects differences in environmental coupling to the hydride transfer step. Importantly, the barrier and contribution of quantum tunneling are not affected, indicating no significant role for “promoting motions” in driving tunneling or modulating the barrier. The chemical step is slower in the heavy enzyme because protein motions coupled to the reaction coordinate are slower. The fact that the heavy enzyme is only slightly less active than its light counterpart shows that protein dynamics have a small, but measurable, effect on the chemical reaction rate

On the curvature in logarithmic plots of rate coefficients for chemical reactions

In terms of the reduced potential energy barrier ζ = ΔuTS/kT, the rate coefficients for chemical reactions are usually expressed as proportional to e-ζ. The coupling between vibrational modes of the medium to the reaction coordinate leads to a proportionality of the regularized gamma function of Euler Q(a,ζ) = Γ(a,ζ)/Γ(a), with a being the number of modes coupled to the reaction coordinate. In this work, the experimental rate coefficients at various temperatures for several chemical reactions were fitted to the theoretical expression in terms of Q(a,ζ) to determine the extent of its validity and generality. The new expression affords lower deviations from the experimental points in 29 cases out of 38 and it accounts for the curvature in the logarithmic plots of rate coefficients versus inverse temperature. In the absence of tunneling, conventional theories predict the curvature of these plots to be identically zero

Activity-Dependent Exocytosis of Lysosomes Regulates the Structural Plasticity of Dendritic Spines

Lysosomes have traditionally been viewed as degradative organelles, though a growing body of evidence suggests that they can function as Ca2+ stores. Here, we examined the function of these stores in hippocampal pyramidal neurons. We found that back-propagating action potentials (bpAPs) could elicit Ca2+ release from lysosomes in the dendrites. This Ca2+ release triggered the fusion of lysosomes with the plasma membrane, resulting in the release of Cathepsin B. Cathepsin B increased the activity of matrix metalloproteinase 9 (MMP-9), an enzyme involved in extracellular matrix (ECM) remodelling and synaptic plasticity. Inhibition of either lysosomal Ca2+ signalling or Cathepsin B release prevented the maintenance of dendritic spine growth induced by Hebbian activity. This impairment could be rescued by exogenous application of active MMP-9. Our findings suggest that activity-dependent exocytosis of Cathepsin B from lysosomes regulates the long-term structural plasticity of dendritic spines by triggering MMP-9 activation and ECM remodelling

The evolution of multiple active site configurations in a designed enzyme

Developments in computational chemistry, bioinformatics, and laboratory evolution have facilitated the de novo design and catalytic optimization of enzymes. Besides creating useful catalysts, the generation and iterative improvement of designed enzymes can provide valuable insight into the interplay between the many phenomena that have been suggested to contribute to catalysis. In this work, we follow changes in conformational sampling, electrostatic preorganization, and quantum tunneling along the evolutionary trajectory of a designed Kemp eliminase. We observe that in the Kemp Eliminase KE07, instability of the designed active site leads to the emergence of two additional active site configurations. Evolutionary conformational selection then gradually stabilizes the most efficient configuration, leading to an improved enzyme. This work exemplifies the link between conformational plasticity and evolvability and demonstrates that residues remote from the active sites of enzymes play crucial roles in controlling and shaping the active site for efficient catalysis

Effects of Macromolecular Crowding on Protein Conformational Changes

Many protein functions can be directly linked to conformational changes. Inside cells, the equilibria and transition rates between different conformations may be affected by macromolecular crowding. We have recently developed a new approach for modeling crowding effects, which enables an atomistic representation of “test” proteins. Here this approach is applied to study how crowding affects the equilibria and transition rates between open and closed conformations of seven proteins: yeast protein disulfide isomerase (yPDI), adenylate kinase (AdK), orotidine phosphate decarboxylase (ODCase), Trp repressor (TrpR), hemoglobin, DNA β-glucosyltransferase, and Ap4A hydrolase. For each protein, molecular dynamics simulations of the open and closed states are separately run. Representative open and closed conformations are then used to calculate the crowding-induced changes in chemical potential for the two states. The difference in chemical-potential change between the two states finally predicts the effects of crowding on the population ratio of the two states. Crowding is found to reduce the open population to various extents. In the presence of crowders with a 15 Å radius and occupying 35% of volume, the open-to-closed population ratios of yPDI, AdK, ODCase and TrpR are reduced by 79%, 78%, 62% and 55%, respectively. The reductions for the remaining three proteins are 20–44%. As expected, the four proteins experiencing the stronger crowding effects are those with larger conformational changes between open and closed states (e.g., as measured by the change in radius of gyration). Larger proteins also tend to experience stronger crowding effects than smaller ones [e.g., comparing yPDI (480 residues) and TrpR (98 residues)]. The potentials of mean force along the open-closed reaction coordinate of apo and ligand-bound ODCase are altered by crowding, suggesting that transition rates are also affected. These quantitative results and qualitative trends will serve as valuable guides for expected crowding effects on protein conformation changes inside cells

Computational Treatment of Metalloproteins

Metalloproteins present a considerable challenge for modeling, especially when the starting point is far from thermodynamic equilibrium. Examples include formidable problems such as metalloprotein folding and structure prediction upon metal addition, removal, or even just replacement; metalloenzyme design, where stabilization of a transition state of the catalyzed reaction in the specific binding pocket around the metal needs to be achieved; docking to metal-containing sites and design of metalloenzyme inhibitors. Even more conservative computations, such as elucidations of the mechanisms and energetics of the reaction catalyzed by natural metalloenzymes, are often nontrivial. The reason is the vast span of time and length scales over which these proteins operate, and thus the resultant difficulties in estimating their energies and free energies. It is required to perform extensive sampling, properly treat the electronic structure of the bound metal or metals, and seamlessly merge the required techniques to assess energies and entropies, or their changes, for the entire system. Additionally, the machinery needs to be computationally affordable. Although a great advancement has been made over the years, including some of the seminal works resulting in the 2013 Nobel Prize in chemistry, many aforementioned exciting applications remain far from reach. We review the methodology on the forefront of the field, including several promising methods developed in our lab that bring us closer to the desired modern goals. We further highlight their performance by a few examples of applications