338 8.2  Molecular Simulation Methods

applied external magnetic field. The second term is an external field interaction energy, which

increases the probability for spins to line up with the external field to minimize this energy.

One simple prediction of this model is that at low B the nearest-​neighbor interaction energy

will bias nearest neighbors toward aligning with the external field, and this results in clustering

of aligned spins, or conformational spreading in the sample of spin states (Figure 8.4a).

This same general argument can be applied to any system that involves a nearest-​

neighbor interaction between components that exist in two states of different energy,

whose state can be biased probabilistically by an external field of some kind. At the long

biological length scale, this approach can characterize flocking in birds and swarming

FIGURE 8.4  Ising modeling in molecular simulations. (a) Spin-​up (black) and spin-​down

(gray) of magnetic domains at low density in a population are in thermal equilibrium, such that

the relative proportion of each state in the population will change as expected by the impos­

ition of a strong external magnetic B-​field, which is either parallel (lower energy) or antiparallel

(higher energy) to the spin state of the magnetic domain (left panel). However, the Ising model

of statistical quantum mechanics predicts an emergent behavior of conformational spreading

of the state due to an interaction energy between neighboring domains, which results in spatial

clustering of one state in a densely packed population as occurs in a real ferromagnetic sample,

even in a weak external magnetic field (dashed circle right panel). (b) A similar effect can be seen

for chemoreceptor expressed typically in the polar regions of cell membranes in certain rodlike

bacteria such as Escherichia coli. Chemoreceptors are either in an active conformation (black),

which results in transmitting the ligand detection signal into the body of the cell or an inactive

conformation (gray) that will not transmit this ligand signal into the cell. These two states are in

dynamic equilibrium, but the relative proportion of each can be biased by either the binding

of the ligand (raises the energy state) or methylation of the chemoreceptor (lowers the energy

state) (left panel). The Ising model correctly predicts an emergent behavior of conformational

spreading through an interaction energy between neighboring chemoreceptors, resulting in

clustering of the state on the surface of the cell membrane (dashed circle, right panel).