Chapter 8 – Theoretical Biophysics  321

As discussed previously, the r⁻⁶ term models attractive, long-​range forces due to vdW

interactions (also known as dispersion forces) and has a physical origin in modeling the

dipole–​dipole interaction due to electron dispersion (a sixth power term). The r⁻¹² term is

heuristic; it models the electrostatic repulsive force between two unpaired electrons due to

the Pauli exclusion principle, though as such is only used to optimize computational effi­

ciency as the square of the r⁻⁶ term and has no direct physical origin.

For systems containing strong electrostatic forces (e.g., multiple ions), modified versions

of Coulomb’s law can be applied to estimate the potential energy UC–​B called the Coulomb–​

Buckingham potential consisting of the sum of a purely Coulomb potential energy (UC or

electrical potential energy) and the Buckingham potential energy (UB) that models the vdW

interaction:

(8.11)

U

r

q q

r

U

r

A

B r

U

r

U

r

U

r

A

r

C

0

B

B

B

B

C B

B

exp









( ) =

( ) =

−(

) ⁻

( ) =

( ) =

−

1

2

6

4πε ε

B

B

B

0

exp −(

) ⁻

+

B r

U

r

q q

r

r

6

1

2

4πε ε

where

q1 and q2 are the electrical charges on the two interacting ions

εr is the relative dielectric constant of the solvent

ε0 is the electrical permittivity of free space

AB, BB, and CB are constants, with the –​CB/​r⁶ term being the dipole–​dipole interaction

attraction potential energy term as was the case for the Lennard–​Jones potential, but

here the Pauli electrical repulsion potential energy is modeled as the exponential term

AB exp(−BBr).

However, many biological applications have a small or even negligible electrostatic com­

ponent since the Debye length, under physiological conditions, is relatively small compared

to the length scale of atomic separations. The Debye length, denoted by the parameter κ⁻¹, is

a measure of the distance over which electrostatic effects are significant:

(8.12)

κ

ε ε

πλ

−=

=

1

0

2

2

1

8

r

B

A

e

B

A

k T

N q I

N I

where

NA is the Avogadro’s number

qe is the elementary charge on an electron

λ is the ionic strength or ionicity (a measure of the concentration of all ions in the solu­

tion) λB is known as the Bjerrum length of the solution (the distance over which the

electrical potential energy of two elementary charges is comparable to the thermal

energy scale of kBT)

At room temperature, if κ⁻¹ is measured in nm and I in M, this approximates to

(8.13)

κ ⁻≈

1

0 304

.

I

In live cells, I is typically ~0.2–​0.3 M, so κ⁻¹ is ~1 nm.

In classical MD, the effects of bond strengths, angles, and dihedrals are usually empiric­

ally approximated as simple parabolic potential energy functions, meaning that the stiffness