Biophysics Tools and Techniques for the Physics of Life (Mark C. Leake) (1)

328 8.2 Molecular Simulation Methods

KEY POINT 8.3

Classical (MM) MD methods are fast but suffer inaccuracies due to approximations

of the underlying potential energy functions. Also, they cannot simulate chemical

reactions involving the making/breaking of covalent bonds. QM MD methods generate

very accurate spatial information, but they are computationally enormously expen

sive. Hybrid QM/MM offers a compromise in having a comparable speed to MM for

rendering very accurate spatial detail to a restricted part of a simulated structure.

8.2.5 STEERED MD

Steered molecular dynamics (SMD) simulations, or force probe simulations, use the same

core simulation algorithms of either MM or QM simulation methods but in addition apply

external mechanical forces to a molecule (most commonly, but not exclusively, a protein)

in order to manipulate its structure. A pulling force causes a change in molecular conform

ation, resulting in a new potential energy at each point on the pulling pathway, which can be

calculated at each step of the simulation. For example, this can be used to probe the force

dependence on protein folding and unfolding processes, and of the binding of a ligand to

receptor, or of the strength of the molecular adhesion interactions between two touching

cells. These are examples of thermodynamically nonequilibrium states and are maintained

by the input of external mechanical energy into the system by the action of pulling on the

molecule.

As discussed previously (see Chapter 2), all living matter is in a state of thermodynamic

nonequilibrium, and this presents more challenges in theoretical analysis for molecular

simulations. Energy-dissipating processes are essential to biology though they are frequently

left out of mathematical/computational models, primarily for three reasons. First, historical

approaches inevitably derive from equilibrium formulations, as they are mathematically more

tractable. Second, and perhaps most importantly, in many cases, equilibrium approximations

seem to account for experimentally derived data very well. Third, the theoretical framework

for tackling nonequilibrium processes is far less intuitive than that for equilibrium processes.

This is not to say we should not try to model these features, but perhaps should restrict this

modeling only to processes that are poorly described by equilibrium models.

Applied force changes can affect the molecular conformation both by changing the rela

tive positions of covalently bonded atoms and by breaking and making bonds. Thus, SMD

can often involve elements of both classical and QM MD, in addition to Monte Carlo

methods, for example, to poll for the likelihood of a bond breaking event in a given small

time interval. The mathematical formulation for these types of bond state calculations relate

to continuum approaches of the Kramers theory and are described under reaction–diffusion

analysis discussed later in this chapter.

SMD simulations mirror the protocols of single-molecule pulling experiments, such as those

described in Chapter 6 using optical and/or magnetic tweezers and AFM. These can be broadly

divided into molecular stretches using a constant force (i.e., a force clamp) that, in the experiment,

result in stochastic changes to the end-to-end length of the molecule being stretched, and con

stant velocity experiments, in which the rate of change of probe head displacement relative to the

attached molecule with respect to time t is constant (e.g., the AFM tip is being ramped up and

down in height from the sample surface using a triangular waveform). If the ramp speed is v and

the effective stiffness of the force transducer used (e.g., an AFM tip, optical tweezers) is k, then we

can model the external force Fext due to potential energy Uext as

(8.21)

k v t

r t

ext

( ) =

−

(

) ⁻

( ) −( )

(

)

(

)

( ) = −∇

( )



.