Chapter 8 – Theoretical Biophysics  331

provides a relatively simple and robust way to estimate long-​range electrostatic interactions.

The foundation of the GB approximation is the classical Poisson–​Boltzmann (PB) model of

continuum electrostatics. The PB model uses the Coulomb potential energy with a modified

εr value but also considers the concentration distribution of mobile solvated ions. If Uelec is

the electrostatic potential energy, then the Poisson equation for a solution of ions can be

written as

(8.22)

∇

= −

2

0

U

r

elec

ρ

ε ε

where ρ is the electric charge density. In the case of an SPC, ρ could be modeled as a Dirac

delta function, which results in the standard Coulomb potential formulation. However, for

spatially distributed ions, we can model these as having an electrical charge dq in an incre­

mental volume of dV at a position r of

(8.23)

d

d

q

r

V r

= ( )

( )

ρ

Applying the Coulomb potential to the interaction between these incremental charges in the

whole volume implies

(8.24)

U

r

V

r

r

elec

r

allspace

0

0

0

1

4

( ) ⁼

−

∫

πε ε

ρd

The Poisson–​Boltzmann equation (PBE) then comes from this by modeling the distribution

of ion concentration C as

(8.25)

C

C

zq U

k T

FC

FC

zq U

k T

e

elec

B

e

elec

B

=

−(

)

∴=

=

−(

)

0

0

exp

/

exp

/

ρ

where

z is the valence of the ion

qe is the elementary electron charge

F is the Faraday’s constant

This formulation can be generalized for other types of ions by summing their respective

contributions. Substituting Equations 8.23 and 8.24 into Equation 8.25 results in a second-​

order partial differential equation (PDE), which is the nonlinear Poisson–​Boltzmann

equation (NLPBE). This can be approximated by the linear Poisson–​Boltzmann equation

(LPBE) if qeUelec/​kBT is very small (known as the Debye–​Hückel approximation). Adaptations

to this method can account for the effects of counter ions. The LPBE can be solved exactly.

There are also standard numerical methods for solving the NLPBE, which are obviously com­

putationally slower than solving the LPBE directly, but the NLPBE is a more accurate electro­

static model than the approximate LPBE.

A compromise between the exceptional physical accuracy of explicit solvent modeling

and the computation speed of implicit solvent modeling is to use a hybrid approach of a

multilayered solvent model, which incorporates an additional Onsager reaction field potential

(Figure 8.3). Here, the biomolecule is simulated in the normal way of QM or MM MD (or

both for hybrid QM/​MM), but around each molecule, there is a cavity within which the elec­

trostatic interactions with individual water molecules are treated explicitly. However, outside

this cavity, the solution is assumed to be characterized by a single uniform dielectric constant.

The biomolecule induces electrical polarization in this outer cavity, which in turn creates a

reaction field known as the Onsager reaction field. If a NLPBE is used for the implicit solvent,