# How can enzyme/substrate reactions that adhere (largely) to quantum theory also require 'Newtonian' consideration of gravity?

I'd just like to ask for a little clarification here due to confusion from interdisciplinary studies.

I'm currently reading the 1976 paper related to the recent 2013 Nobel Prize for Chemistry, by 2 of the laureates Warshel and Levitt — Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and Steric Stabilisation of the Carbonium Ion in the Reaction of Lysosyme — and have recently been reading into quantum chemistry, as well as studying biochemistry and have had further lectures in Physics to boot.

I'm wondering if the points made here are comparable to points I've heard in mechanics lectures, about a cannonball not being modelled as a quantum object, since every subatomic particle would be too computationally intensive, so a coarse-grain model is used (i.e. Newton's "classical" laws).

I wouldn't be so quick to draw parallels between mechanics lectures and enzyme biochemistry but it seems to be in the language they are using here (see highlights below).

From what I understand (in mechanics) only 3 of the 4 'fundamental forces' (strong, weak, and EM) only strictly apply at the quantum/microscopic scale, and scientists are as it stands struggling to work gravity into the setup. Is this worth considering in this paper - if an enzyme is considered all of a sudden due to its size as a quantum object does this entail ignoring gravity QM now in favour of 'Newtonian', inertial frame-of-reference type mechanics, or considering it in some other 'Newtonian' manner?

The introductory section is as follows (full paper here):

The conformation of the enzyme-substrate complex can be adequately studied using empirical energy functions based on the “classical” contributions of bond stretching, bond angle bending, bond twisting, non-bonded interactions, etc. The mechanism and energetics of the enzymic reaction can only be studied using a quantum mechanical approach. Previous quantum mechanical calculations on enzymic reactions have been limited in several respects. In the first place, they deal with an over-simplified model system consisting of only small fractions of the atoms involved in the real enzyme-substrate interaction. Second, all available quantum mechanical methods treat the reaction as if in a vacuum and are, therefore, unable to include the dielectric, which has a very important effect on the energy and charge distribution of the system. This limitation does not depend on the actual quantum mechanical scheme, being equally valid for the simplest Hiickel treatment and most extensive ab initio calculation.

Because the treatment of the whole enzyme-substrate complex quantum mechanically is computationally impossible, it is necessary to simplify the problem and use a hybrid classical/quantum mechanical approach. The method described here considers the whole enzyme-substrate complex: the energy and charge distribution of those atoms that are directly involved in the reaction are evaluated quantum mechanically, while the potential energy surfac e of the rest of the system, including the urrounding solvent, is evaluated classically. An important feature of the method is the treatment of the “dielectric” effect due to both the protein atoms and the surrounding water molecules. Our dielectric model is based on a direct calculation of the electrostatic field due to the dipoles induced by polarizing the protein atoms, and the dipoles induced by orienting the surrounding water molecules. With the inclusion of the microscopic dielectric effect we feel that our model for enzymic reactions includes all the important factors that may contribute to the potential surface of the enzymic reaction. Because we use analytical expressions to evaluate the energy and first derivative with respect to the atomic Cartesian co-ordinates, it is possible to allow the system to relax using a convergent energy minimization method. Here we use the method to study the mechanism of the lysozyme reaction, with particular emphasis on factors contributing to the stabilization of the carbonium ion intermediate. We first outline our theoretical scheme, paying attention to the partition of the potential surface into classical and quantum mechanical parts and to details of the microscopic dielectric model. Then we use the method to investigate factors that contribute to the stabilization of the carbonium ion intermediate and deal with the relative importance of each factor. The effect of steric strain is considered for both the ground and the carbonium states and found to be of minor importance. Electrostatic stabilization is treated at some length with special emphasis on the balance between stabilization by induced dipoles (polarizability) and stabilization by charge- charge interactions. The results demonstrate the importance of electrostatic stabilization for this reaction.

• Atomistic calculations comfortably neglect 3 out of 4 fundamental interactions regardless of whether they are QM or classical - the strong and weak forces attenuate too rapidly to meaningfully factor into chemical interatomic interactions, whilst gravity is so much weaker than electromagnetism as to be irrelevant. – Richard Terrett Oct 22 '13 at 3:22
• Nuclei are heavier and more classical than electron. Just like quantum mechanics contains Newtonian mechanics as a classical limit. Although here the Newtonian mechanics does not correspond to gravity, it used other potential (force field). This we aldready saw in the potential energy surface defined by Born-Oppenheimer approximation. – user26143 Dec 4 '13 at 11:53
• Newtonian gravity is a red herring here as gravity is far too weak to be involved in enzyme chemistry. The Newtonian or classical mechanics involved is a simplification of the molecular structure of most of the enzyme that models is, effectively, as a bunch of atoms connected by springs ( a little like a macroscopic atomic model built from balls with bonds represented by springs). The model uses simple classical approximations of how springs behave (with additions to cope with rotation as well as stretching). This is easier to compute and a good approximation. – matt_black Dec 31 '13 at 13:20

Unfortunately there's been something of a nomenclature mismatch here. While classical mechanics refers to Newtonian SUVAT-style things, "classical" molecular dynamics or geometry optimisation refers to, as the paper puts it: "empirical energy functions based on the 'classical' contributions of bond stretching, bond angle bending, bond twisting, non-bonded interactions, etc."

In pure "classical force fields", these functions use parameters produced from analysis of various known geometries, and are collectively optimised using linear algebra optimisation techniques to obtain an approximation of a molecular or ionic system. Simple point-wise electrostatics may also be included.

This paper refers to the use of a combination of quantum mechanical techniques (modelling the complex reaction centre) with classical force field techniques (modelling the rest of the protein) and a dielectric model, modelling the interaction between the two partitions and the effect of the bulk solvent on the protein.

If you'd like to look into this in more detail, I'd suggest looking for mention of "QM/MM" (Quantum Mechanics/Molecular Mechanics) or "ONIOM" techniques, as such models are now known. The use of simulated dielectrics for solvents is now largely covered under "PCM" (polarisable continuum model) and "COSMO" (Conductor-like Screening Model) techniques.

If you're curious about "classical" molecular force-fields, I'd suggest looking up one of the packages that perform such calculations, such as DL POLY or GROMACS.

• Thank you, yes I thought it was too simple to be true! Sounds abstract when you're new to it but reading 'parameter sets' as bond parameters helps. I've been looking into the terminology like CFFs, dielectrics etc. as it seems odd not to understand the fundamentals here. I don't even know how I'd begin to get into 'linear algebra optimisation', my maths education only went so far as single variable calculus unfortunately, this seems advanced. DL_POLY looks v familiar from Dalton Trans. figures... V v interesting, thanks – Louis Maddox Oct 21 '13 at 22:52
• You're basically just trying to find a combination of geometric options that gives you what looks like the minimum energy, but it turns out you can represent that as the approximate solution to a equation multiplying a vector by a matrix. Linear algebra features heavily in computational chemistry (both classical and quantum mechanical), so if you're getting into the field I'd recommend getting familiar with the terminology, even if you don't understand the guts of it. – Aesin Oct 21 '13 at 23:12

What they mean by

Because the treatment of the whole enzyme-substrate complex quantum mechanically is computationally impossible, [...]

is basically a consequence of the results from the inner workings of the quantum chemical formulation of the theory. If one goes through the equations (it takes a while), one arrives at an expression that tells you that the required number of calculations scales with $N^4$, where $N$ is the number of basis functions. Imagine the wavefunction being a (hyperdimensional) vector in a coordinate system. Say you had a 3-dimensional vector but only two basis vectors to express it, then your representation of the vector would be very inaccurate, so you add another basis vector to your set of basis vectors and then you can accurately describe the vector by $\vec{v} = a\vec{e}_x + b\vec{e}_y + c\vec{e}_z$.

In enzymes, the huge amount of atoms would lead to gigantic numbers of required calculations which make it largely impossible to calculate completely quantum mechanically. So it's an impracticality, not a theoretical problem.