Chapter 8 – Theoretical Biophysics 329
where
t0 is some initial time
r is the position of an atom
u is the unitary vector for the pulling direction of the force probe
Uext can then be summed into the appropriate internal potential energy formulation used to
give a revised total potential energy function relevant to each individual atom.
The principal issue with SMD is a mismatch of time scales and force scales between simu
lation outputs compared to actual experimental data. For example, single-molecule pulling
experiments involve generating forces between zero and up to ~10–100 pN over a time
scale of typically a second to observe a molecular unfolding event in a protein. However, the
equivalent time scale in SMD is more like ~10−9 s, extending as high as ~10−6 s in exceptional
cases. To stretch a molecule, a reasonable distance compared to its own length scale, for
example, ~10 nm, after ~1 ns of simulation implies a probe ramp speed equivalent to ~10 m
s−1 (or ~0.1 Å ps−1 in the units often used in SMD). However, in an experiment, ramp rates
are limited by viscous drag effects between the probe and the sample, so speeds equivalent
to ~10−6 m s−1 are more typical, seven or more orders of magnitude slower than in the SMD
simulations. As discussed in the section on reaction–diffusion analysis later in this chapter,
this results in a significantly lower probability of molecular unfolding for the simulations,
making it nontrivial to interpret the simulated unfolding kinetics. However, they do still
provide valuable insight into the key mechanistic events of importance for force dependent
molecular processes and enable estimates to be made for the free energy differences between
different folded intermediate states of proteins as a function of pulling force, which provides
valuable biological insight.
A key feature of SMD is the importance of water-solvent molecules. Bond rupture, for
example, with hydrogen bonds, in particular, is often mediated through the activities of water
molecules. The ways in which the presence of water molecules are simulated in general MD
are discussed as follows.
8.2.6 SIMULATING THE EFFECTS OF WATER MOLECULES AND SOLVATED IONS
The primary challenge of simulating the effects of water molecules on a biomolecule
structure is computational. Molecular simulations that include an explicit solvent take
into account the interactions of all individual water molecules with the biomolecule. In
other words, the atoms of each individual water molecule are included in the MD simu
lation at a realistic density, which can similarly be applied to any solvated ions in the
solution. This generates the most accuracy from any simulation, but the computational
expense can be significant (the equivalent molarity of water is higher than you might
imagine; see Worked Case Example 8.1).
The potential energy used is typically Lennard–Jones (which normally is only applied to
the oxygen atom of the water molecule interacting with the biomolecule) with the addition
of the Coulomb potential. Broadly, there are two explicit solvent models: the fixed charge
explicit solvent model and the polarizable explicit solvent model. The latter characterizes
the ability of water molecules to become electrically polarized by the nearby presence of the
biomolecule and is the most accurate physical description but computationally most costly.
An added complication is that there are >40 different water models used by different research
groups that can account for the electrostatic properties of water (e.g., see Guillot, 2002),
which include different bond angles and lengths between the oxygen and hydrogen atoms,
different dielectric permittivity values, enthalpic and electric polarizability properties, and
different assumptions about the number of interaction sites between the biomolecule and the
water’s oxygen atom through its lone pair electrons. However, the most common methods
include single point charge (SPC), TIP3P, and TIP4P models that account for most biophys
ical properties of water reasonably well.