predefined collective variables (CVs). However, in contrast, meta-
dynamics biases in a history-dependent manner by periodically
depositing Gaussians to discourage the simulation from revisiting
the same region of the CV space. Once the CV space is filled, the
free-energy surface can then be recovered by integrating over the
added Gaussians. Normally the simulation is stopped when the
system is diffusing freely through the CV space. The problem of
slow or difficult convergence has been improved through the devel-
opment of well-tempered metadynamics [35](seeNote 9).
In general umbrella sampling is almost certain to converge and
has fewer parameters than metadynamics. However, as mentioned
earlier for both of these approaches the choice of collective variables
is crucial and often difficult. It is generally easier to see problems
with the chosen biasing variables with metadynamics owing to its
diffusivity. Interestingly, other methods like adaptive biasing force
MD offer some of the advantages of both umbrella sampling and
metadynamics [36].
Replica-exchange uses a set of replicas of the same simulation
which have successively improved sampling properties owing to
some variables like temperature. The variables are then exchanged
between replicas such that conformations from the replicas which
sample more rapidly are fed into other replicas with the required
distribution. These exchanges are carried out in a way which pre-
serves the correct distribution. Generally this means adhering to
detailed balance and using a Metropolis test, although more
advanced and efficient arrangements becoming more common
[37–39](seeNote 10).
Recently, easy-to-implement forms of Hamiltonian-exchange
have become popular as they offer enhanced sampling of particular
regions or interactions at much reduced computational expense
compared to temperature replica-exchange [40](seeNote 11).
Even with these highly efficient new methods replica-exchange
approaches alone are not sufficient to sample many of the transi-
tions of interest. Hence, it is now increasingly common to combine
umbrella sampling or metadynamics with replica-exchange in vari-
ous arrangements [41–44]. The combination of biasing collective
variables along with generalized orthogonal sampling from multi-
ple replicas offers well-converged simulations when reasonable vari-
ables can be found (seeNote 12).3.3.2 Celling
Approaches—Markov
State Models (MSM)
The most well-known celling approach is Markov state models
(MSMs) or kinetic transition networks. MSMs are simplified mod-
els defining the regularity of transitions between discretized states
(microstates) derived from dynamics data. Since they are based on
kinetic information these MSMs provide kinetic data such as tran-
sition times and are more readily compared to experimental data of
dynamics [45]. MSMs allow the combination of many shortComputational Study of Protein Conformational Transitions 345