Chapter 8 – Theoretical Biophysics 339
in insects and even of social interactions in a population, but at the molecular length
scale, there are several good exemplars of this behavior too. For example, bacteria swim
up a concentration gradient of nutrients using a mechanism of a biased random walk
(see later in this chapter), which involves the use of chemoreceptors that are clustered
over the cell membrane. Individual chemoreceptors are two stable conformations, active
(which can transmit the detected signal of a bound nutrient ligand molecule to the inside
of the cell) and inactive (which cannot transmit the bound nutrient ligand signal to the
inside of the cell). Active and inactive states of an isolated chemoreceptor have the same
energy. However, binding of a nutrient ligand molecule lowers the energy of the inactive
state, while chemical adaptation (here in the form of methylation) lowers the energy of
the active state (Figure 8.4b, left panel).
However, in the cell membrane, chemoreceptors are tightly packed, and each in effect
interacts with its four nearest neighbors. Because of steric differences between the active
and inactive states, its energy is lowered by every neighboring chemoreceptor that is in
the same conformational state but raised by every neighboring receptor in the different
state. This interaction can be characterized by an equivalent nearest-neighbor inter
action energy (see Shi and Duke, 1998), which results in the same sort of conformational
spreading as for ferromagnetism, but now with chemoreceptors on the surface of cell
membranes (Figure 8.4b, right panel), and this behavior can be captured in Monte Carlo
simulations (Duke and Bray, 1999) and can be extended to molecular systems with more
challenging geometries, for example, in a 1D ring of proteins as found in the bacterial
flagellar motor (Duke et al., 2001).
This phenomenon of conformational spreading can often be seen in the placement of bio
physics academics at conference dinners, if there is a “free” seating arrangement. I won’t go
into the details of the specific forces that result in increased or decreased energy states, but
you can use your imagination.
Worked Case Example 8.1: Molecular Simulations
A classical MD simulation was performed on a roughly cylindrical protein of diameter
2.4 nm and length 4.2 nm, of molecular weight 28 kDa, in a vacuum that took 5 full days
of computational time on a multicore CPU workstation to simulate 5 ns.
a Estimate with reasoning how long an equivalent simulation would take in minutes if
using a particle mesh Ewald summation method. Alternatively, how long might it take
if truncation was used?
b At best, how long a simulation in picosecond could be achieved using an ab initio
simulation on the same system for a total computational time of 5 days? A conform
ational change involving ~20% of the structure was believed to occur over a time scale
as high as ~1 ns. Is it possible to observe this event using a hybrid QM/MM simulation
with the same total computational time as for part (a)?
c Using classical MD throughout, explicit water was then added with PBCs using a con
fining cuboid with a square base whose minimum distance to the surface of the pro
tein was 2.0 nm in order to be clear of hydration shell effects. What is the molarity of
water under these conditions? Using this information, suggest whether the project
undergraduate student setting up the simulation will be able to witness the final result
before they leave work for the day, assuming that they want to simulate 5 ns under the
same conditions of system temperature and pressure and that there are no internal
hydration cavities in the protein, but they decide to use a high-end GPU instead of the
multicore CPU.
(Assume that the density and molecular weight of water is 1 g/cm−3 and 18 Da, respect
ively, and that Avogadro’s number is ~6.02 × 1023.)
KEY BIOLOGICAL
APPLICATIONS:
MOLECULAR
SIMULATION TOOLS
Simulating multiple molecular
processes including conform
ational changes, topology
transitions, and ligand docking.