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Explicit Solvation

Abstract

Rules for explicit solvation

1. It takes a LOT of solvent molecules to look like a solution

  • too few and it looks like a cluster, which is not a solution
  • This makes it incredibly expensive
  • We have tools to make this more simple - PBC, Ewald sums, QM/MM, split basis?

Benefits:

  • The solvent IS actually there, so we can performa a solvent density analysis (molecules per unit\(^2\))
  • Density differences mean that you can look at the specific interactions with regions of the molecule (favourable/unfavourable solvation conditions)

E.g. Solvation of tRNA

In this example, we can see that while there is typically only favourable solvation along the \(\ce{PO4}\) spine in RNA (2AA:IsoC and G:C), in this case (G:U), we have hydration in the typically hydrophobic minor groove.

This extra solvation also correlates with enzymatic methylation of the base pair in this particular species.

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2. Equilibrium properties (e.g. free energies) require averaging over phase space (MM/MD)

  • There are far too many minima if we do this though optimisation, making a QM-only solvation process near impossible to do rigorously

  • Solvation really should be done through MC or MD, until ergodic behaviour is identified.

    • To make this more rigorous, different starting trajectories should be chosen and the system should converge to the same average geometry
    • This also allows for averages of flexible molecules with dynamic structures (particularly biopolymers)
    • You could also choose a small portion of MD steps to then run QM calculations on, reducing the amount of possible configurations
    • You can obtain structural details that are associated specifically with the solvation shell. Specific solvent-solute molecular interactions.

Radial distribution function

Tells us the distribution of distances between atoms \(A\) and \(B\). (The probability of finding an atom at distance \(r\))

Where:

  • \(V=\) Volume that all the atoms take up
  • \(g_{AB}=\) Radial distribution of atoms \(A\) to \(B\)
  • \(r=\) The distance between the atoms
  • \(\frac{1}{N_A\cdot N_B}=\) A normalisation value for the numbers of atoms \(A\) (\(N_A\)) and \(B\) (\(N_B\))
  • \(\sum\limits_{i=1}^{N_A}\sum\limits_{j=1}^{N_B}=\) Summing over all the occurrences of atoms \(A\), from \(i=1\) to \(N_A\) and \(B\), from \(j=1\) to \(N_B\)
  • \(\delta\big[r-r_{A_iB_j}\big]=\) Kronecker delta function - Checking whether the distance between the atoms is equal to the value of \(r\) we’ve specified (spits out a True/False statement)
\[ \frac{1}{V}g_{AB}(r)=\frac{1}{N_A\cdot N_B}\bigg\langle\sum\limits_{i=1}^{N_A}\sum\limits_{j=1}^{N_B}\delta\big[r-r_{A_iB_j}\big]\bigg\rangle \]

In the plot below, the distances (\(r\)) of the shells of hydration are

image