Chapter 8 – Theoretical Biophysics  337

Another important issue to bear in mind is that for supercomputing resources you usually

have a queue to wait in for your job to start on a cluster. If you care about the absolute time

required to get an output from a molecular simulation, the real door-​to-​door time if you like,

it may be worth waiting longer for a calculation to run on a smaller machine in order to save

the queue time on a larger one.

KEY POINT 8.5

Driven by the video-​gaming industry, GPUs have an improvement in speed of multicore

processors that can result in reductions in computational time of two orders of magni­

tude compared to a multicore CPU.

8.2.10  ISING MODELS

The Ising model was originally developed from quantum statistical mechanics to explain

the emergent properties of ferromagnetism (see Chapter 5) by using simple rules of local

interactions between spin-​up and spin-​down states. These localized cooperative effects result

in emergent behavior at the level of the whole sample. For example, the existence of either

ferromagnetic (ordered) or paramagnetic (disordered) bulk emergent features in the sample,

and phase transition behavior between the two bulk states, which is temperature dependent

up until a critical point (in this example known as the Curie temperature). Many of the key

features of cooperativity between smaller length scale subunits, leading to a different larger

length scale emergent behavior are also present in the Monod–​Wyman–​Changeux (MWC)

model, also known as the symmetry model, which describes the mechanism of coopera­

tive ligand binding in some molecular complexes, for example, the binding of oxygen to the

tetramer protein complex hemoglobin (see Chapter 2), as occurs inside red blood cells for

oxygenating tissues.

In the MWC model, the phenomenon is framed in terms of allostery; that is, the binding

of one subunit affects the ability of it and/​or others in the complex to bind a ligand molecule,

in which biochemists often characterize by the more heuristic Hill equation for the fraction θ

of bound ligand at concentration C that has a dissociation rate constant of Kd (see Chapter 7):

(8.29)

θ =

+

C

K

C

n

d

n

where n is the empirical Hill coefficient such that n =​ 1 indicates no cooperativity, n < 1 is

negatively cooperative, and n > 1 is positively cooperative.

Hemoglobin has a highly positive cooperativity with n =​ 2.8 indicating that 2–​3 of the 4

subunits in the tetramer interact cooperatively during oxygen binding, which results in a typ­

ically sigmoidal shaped all-​or-​nothing type of response if θ is plotted as a function of oxygen

concentration. However, the Ising model puts this interaction on a more generic footing by

evoking the concept of an interaction energy that characterizes cooperativity and thus can be

generalized into several systems, biological and otherwise.

In the original Ising model, the magnetic potential energy of a sample in which the ith

atom has a magnetic moment (or spin) σi is given by the Hamiltonian equation:

(8.30)

H

K

B

ij

i

j

i

i

=

−

∑

∑

σ σ

σ

where K is a coupling constant with the first term summation over adjacent atoms with mag­

netic moments of spin +​1 or −1 depending on whether they are up or down and B is an