Chapter 8 – Theoretical Biophysics  323

KEY POINT 8.1

In P³M, short-​range forces are solved in real space; long-​range forces are solved in

reciprocal space.

P³M has a similar computation scaling improvement to PM but improves the accuracy of the

force field estimates by introducing a cutoff radius threshold on interactions. Normal pair­

wise calculations using the original point positions are performed to determine the short-​

range forces (i.e., below the preset cutoff radius). But above the cutoff radius, the PM method

is used to calculate long-​range forces. To avoid aliasing effects, the cutoff radius must be less

than half the mesh width, and a typical value would be ~1 nm. P³M is obviously computation­

ally slower than PM since it scales as ~O(wn log n +​ (1 –​ w)n²) where 0 < w < 1 and w is the

proportion of pairwise interactions that are deemed long-​range. However, if w in practice is

close to 1, then the computation scaling factor is not significantly different from that of PM.

Other more complex many-​body empirical potential energy models exist, for example, the

Tersoff potential energy (UT). This is normally written as the half potential for a directional

interaction between atom number i and j in the system. For simple pairwise potential energy

models such as the Lennard–​Jones potential, the pairwise interactions are assumed to be

symmetrical, that is, the half potential in the i → j direction is the same as that in the j → i dir­

ection, and so the sum of the two half potentials simply equate to one single pairwise poten­

tial energy for that atomic pair. However, the Tersoff potential energy model approximates

the half potential as being that due to an appropriate symmetrical half potential due US, for

example, a UL–​J or UB–​C model as appropriate to the system, but then takes into account the

bond order of the system with a term that models the weakening of the interaction between

i and j atoms due to the interaction between the i and k atoms with a potential energy UA,

essentially through sharing of electron density between the i and k number atoms:

(8.15)

U

r

U

r

B U

r

ij

ij

ij

ij

ij

ij

T

s

A







( ) ⁼

( )⁺

( )

∑

∑

1

2

1

2

where the term Bij is the bond order term, but is not a constant but rather a function of a

coordination term Gij of each bond, that is, Bij =​ B(Gij), such that

(8.16)

G

f

r

g

f r

r

ij

k

ik

jik

ij

ik

=

( ) (

)

−

(

)

∑c

θ

where

fc and f are functions dependent on just the relative displacements between the ith and

jth and kth atoms

g is a less critical function that depends on the relative bond angle between these three

atoms centered on the ith atom

8.2.3  MONTE CARLO METHODS

Monte Carlo methods are in essence very straightforward but enormously valuable for

modeling a range of biological processes, not exclusively for molecular simulations, but, for

example, they can be successfully applied to simulate a range of ensemble thermodynamic

properties. Monte Carlo methods can be used to simulate the effects of complex biological

systems using often relatively simple underlying rules but which can enable emergent prop­

erties of that system to evolve stochastically that are too difficult to predict deterministically.

If there is one single theoretical biophysics technique that a student would be well advised to

come to grips with then it is the Monte Carlo method.