Electrostatic models

[jarvist]

The fastest (but not necessarily least accurate!) method to treat the interaction of a peptide with the cell membrane is via electrostatics. One of the simplest models here is to consider the materials you are interested in being where you are considering the materials as being a dielectric continuum.

One central paper I've been reading with great interest is:

Cahill, K., 2012. Models of membrane electrostatics. Phys. Rev. E 85, 051921. https://doi.org/10.1103/PhysRevE.85.051921

Cahill is very much a physicist. Considering a charge, solving Gauss' law and the static form of Faraday's law, with the boundary conditions imposed by the cell membrane of transitioning between different dielectric continuum (i.e. water ε=80, phospholipid head groups ε=195, lipids ε=2), you get a set of recursive relationships stating the potentials. Essentially each transition between dielectric medium generates a series of image charges, screened by different dielectric constants. Cahill is at pains to point out that these solutions go beyond the mean-field Poisson-Boltzmann solution, which requires both cations and anions to feel the same potential, and therefore you can't have a membrane which simultaneously repels both charges. Cahill's theory does exactly this. Cahill also runs a Monte-Carlo code based on these expressions to calculate the equlibrium densities of ions (numeric results for a 150 mM K+ / 10 mM Na-,

2012 Cahill PRE, tasks:

• [x] Implement codes to Figure 1
• [x] Email Cahill, re: his MC code. (Missing from website - a statement at the top says that the disk crashed.) (Emeritus, so may not respond.)
• [ ] Re-implement Monte-Carlo method for ions and check against Figure 3

The dielectric constants came from modelling. In the particular situation of AMPs, we want to have good models for both bacteria and mammal cell membranes, so that we can calculate the difference in attraction / electrostatics. So we'd need a way of calculating (probably also by simulation) the dielectric constants in cell membranes we are interested in.

Stern, H.A., Feller, S.E., 2003. Calculation of the dielectric permittivity profile for a nonuniform system: Application to a lipid bilayer simulation. The Journal of Chemical Physics 118, 3401–3412. https://doi.org/10.1063/1.1537244

2003 Stern and Feller JCP: Sampling dielectric constants:

• [ ] Re-simulate their molecular dynamics
  • [ ] Figure out whether Eqn. 83 (integrate over partial charges) is sufficient, or one needs to do the whole fitting dipole moments (by SVD of their constructive definitiin) and associating them with the covalent bonds, and then summing over the dipole moments, Eqn 81.
• [ ] Write code to reproduce their figures

[jarvist]

Copy of figure 1 from the Pluto Notebook: https://github.com/Frost-group/Nornour/blob/main/1000-KevinCahill-MembraneElectrostatics/figure1.jl