Electrostatics

Basic electrostatics

• involve stationary or slowly moving charges, sizes vary
• Interactions between charges are guided by:
  • coulomb’s law, gauss’s law, electrostatic potential, electric field
  • Default is vacuum, extra terms (dielectric const) are needed for materials. TLDR dielectric dampens the effect of the forces
• This physics matters because it describes the majority of AA interactions in solution
• Assumptions made for protein chem:
  • Charged AA are assumed to always be charged no WRT location
  • neglect explicit solvent molecules, do vacuum with dielectric approximations
  • Short time scale

Introduction to Poisson-Boltzman

• Poisson-boltzman is commonly written as ${\nabla }^2\varphi =-\frac{\rho }{{\in }_0}$
• You have to define the problem in 3D space as the nabla is a gradient
  • There is a boundary between the bulk solve and the protein surface where the dielectric constant transitions and must be made with a smooth function
  • Ions are distrubted in energy with a boltzman distribution, which determines their relative location and frequency within the boundary layer

PB equation and solutions

• Solving
  • Can only be solved analytically for very simple systems
  • Otherwise need numerical methods like finite difference
    • Discretize space into a grid, solve using laplace operator
    • At the edges of the grid use boundary condition potential = 0. Allows you to solve for adjacent cells, then fill in the gid

Protein applications

• Hydration energies
  • in a vacuum and water
• Binding energies between proteins, antibodies, small molecules, etc.
  • Largely contributed by electrostatic factors (in addition to steric) $\Delta G_{binding}=\Delta G_{complex}-(\Delta G_{receptor}-\Delta G_{ligand})$ $"thereisnogodbutgod"$
• Molecular dynamics
  • Forces between discrete objects can be incorporated into traditional MD force fields
  • Better for calculating strongly charged objects like salt bridges
• Brownian Dynamics
  • Simulation of solution diffusion, matters in quiescent contexts
• pKa and Titration
  • pH is a critical factor in stability
  • residues charges are often determined by subtle interactions between protonated/deprotonated residues

Calculating Protonation states

Conclusions