Quantum aspects of molecular correlations in biological catalysis

Hydrogen bonds are of central importance in many natural phenomena in physical, chemical, and biological systems. In particular, several critical cellular processes require structural and electronic complementarity between molecules mimicking a lock and a key, and the degree of this complementarity is mainly achieved through the number and strength of intermolecular hydrogen bonds. From an information-theoretic point of view, the biomolecular complementarity described in this way is nothing but a type of correlation generated through hydrogen bonds. There are various experimental and theoretical findings suggesting the occurrence of proton tunneling through the hydrogen bonds formed during some of the critical cellular processes, including nucleic acid base pairing and enzyme catalysis. This raises the question for a relation between the quantum correlations generated by proton tunneling events and the biological functions of the hydrogen-bonded biomolecules performed inside the cell. In this respect, the object of this thesis is not only to discover how an intermolecular interaction can generate quantum correlations via proton tunnelings through hydrogen bonds, but also to investigate how an enzyme can use them to catalyze a biochemical reaction. In order to test the hypothesis that the tunneling of protons through hydrogen bonds generates intermolecular quantum correlations, we use the following plan. First, we construct a fermionic Hamiltonian for the motion of a single proton in a hydrogen-bonded system and then convert it into a two-qubit pseudo-spin Hamiltonian using the Jordan-Wigner transformation. This pseudo-spin formalism enables us to quantify the amount of correlations generated by the proton mobility using the tools of quantum information theory. It also enables us to develop a Markovian master equation approach for the open system dynamics of the proton under the influence of the stretch vibrations of single covalent bonds involving the hydrogen when memory effects are neglected. Second, we extend our pseudo-spin formalism to a twelve-qubit model of a single hexameric H2O ring. After constructing a map from the actual 3-d structures of water ice to the present 2-d model of a single hexamer based on the single-hexamer proton relocation events occurring in them, we find that the violation of the ice rules in the model corresponds to the proton-disordering phase transition in hexagonal water ice. We then consider the steady-state solution of the master equation which reveals the role of vibrations in the emergence of ice rules and depends only on two parameters. We fix these parameters using i) the unit probability of pseudo-spins to be found inside the ice-rule preserving subspace as an indicator of proton-ordered phase and ii) the entropy of pseudo-spins as a measure of proton-disorder. Using these model parameters, we show that l1 form of coherence of the pseudo-spin steady state captures the behaviour of coherent proton mobility observed in low-temperature experiments that suspect a concerted tunneling of six protons. We investigate the nature and the amount of pairwise pseudo-spin correlations by means of the entanglement of formation, the concurrence, the quantum discord, the geometric discord, and the quantum mutual information of the reduced states. We conclude that individual quantum tunneling events in the bonds of water ice are classically rather than quantum correlated at low temperatures, a point that can't be addressed by the previous studies. We also show that although the quantum correlations in hydrogen bonds seem to be insignificant when compared to their classical counterparts, temperature dependence of quantum coherent proton mobility still originates from them. Thirdly, we move on to our main hypothesis in this thesis which states that an enzyme, at least in principle, can derive its catalytic potency from the quantum correlations generated through intermolecular hydrogen bonds. To test this hypothesis, we take into account a prototypical enzyme-catalyzed nucleotide tautomerization reaction starting by the formation of two weak hydrogen bonds between an enzyme and its substrate. We describe the substrate recognition process by an induced-fit mechanism in which the initial hydrogen bonds are assumed to trigger a slow and continuous conformational change in the binding site of the enzyme. The two biomolecules are assumed to decouple from each other at the end of the binding stage after which the enzyme rapidly undergoes a subsequent conformational change to return back to its initial state. In this respect, we use our pseudo-spin formalism to construct a time-independent three-qubit Hamiltonian for the tautomeric transitions of the nucleotide. We extend our master equation approach to include the effects of conformational dynamics of the nucleotide on the intramolecular proton dynamics. We describe the enzyme-nucleotide hydrogen bonds by the two-qubit pseudo-spin model assuming that the protons stay in the ground state of the time-dependent total pseudo-spin interaction Hamiltonian during the slow conformational change of the enzyme which is induced upon substrate binding. On the other hand, the effect of the subsequent rapid conformational change of the enzyme on the proton dynamics is modeled by a post-selection measurement on the pseudo-spins of the enzyme. The quantum correlations generated through the intermolecular hydrogen bonds at the end of binding state are found to be transferred to the nucleotide when the enzyme suddenly returns back to its initial state in this way. The resulting intramolecular correlations inside the nucleotide constitute a transition state which actually corresponds to a quantum coherent superposition of the substrate and product forms. We analyze the open system dynamics of the nucleotide using our master equation approach starting after this transition state. The stationary solution is found to depend on only two parameters, which are estimated from the literature. In particular, we show that the nucleotide reaches an equilibrium such that both probabilities corresponding to substrate and product forms are equal to 1/2 when the overall entanglement generated through hydrogen bonds is chosen to be maximal at the end of binding stage. The transition state has a free energy that equals to the average of the free energies of the substrate and the product forms, a value much lower than expected in biochemical reactions. According to our results, the equilibrium probabilities are found to be unaffected when the quantum correlations generated through hydrogen bonds at the end of binding stage rapidly decay to classical correlations before the subsequent conformational change. The calculations show that the average energy is also independent from such a decoherence. On the other hand, it does make a difference to the free energies. Although this is not a significant difference at all, the direction of the change is opposite to that which one expects. The free energy of the transition state drops further as a result of the increase in the entropy that is produced by the decoherence. The rate enhancement the enzyme brings about in this way is found to be in the range of 22 to 27 orders of magnitudes for different amounts of intermolecular entanglement generation at the end of the binding stage. We compare this enzyme-catalyzed reaction with the water-mediated tautomerization of the nucleotide in which intermolecular quantum correlations are also generated through water-nucleotide hydrogen bonds. A maximal entanglement generation through intermolecular hydrogen bonds produces the product with an equilibrium probability of 1/4 in this case. Moreover, it produces cationic and ionic transition states with the same equilibrium probability. Besides this, the water-mediated tautomerization requires the right orientations of the molecule which is very unlikely inside the cellular environment. Hence we conclude that the cyclic and classical conformational change of the enzyme increases the likelihood of the formation of quantum hydrogen bonds with the nucleotide and makes them a more efficient resource for the tautomerization process.