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Broadly, one convenient way to characterize enhanced sampling methods is to
categorize them as either tempering or external biasing methods. In tempering
methods, temperature is exploited to enhance the sampling of the system. The canonical form of these methods is simulated annealing [33], a generalized probabilistic
method used to quickly reach a global optimum. The probability of accepting an
exchange relies on a temperature variable which either promotes or restricts exploration of new states. This process rapidly becomes computationally intensive, which
introduced a need for more advanced methods with respect to molecular simulations. Replica exchange is the most generalized sampling approach used in MD,
where several replicas of a molecular configuration are simulated at the same time.
A swap of two replicas can occur under an acceptance probability that preserves
detailed balance. Replicas are weakly coupled and only interact under an exchange
attempt. This promotes a highly parallelizable and enhanced method for sampling a
system. While it is not a requirement, the most common variant of replica exchange
involves simulating replicas at different temperatures; this is commonly known as
parallel tempering (PT) [34, 35]. Other sampling subsets of replica exchange include
Hamiltonian replica exchange [36] and solute tempering [37]. These methods and
many more such as adiabatic free energy dynamics [38] and temperature accelerated
molecular dynamics (TAMD) [39] are described in much greater detail in a number
of prominent reviews [5, 40–42].
In external biasing methods, sampling of the system is enhanced by applying an
external bias on the slow modes of the system. These slow modes of the system
can be described by defining a few choice collective variables (CVs). A CV is a
differentiable function that provides a low-dimensional projection of all atomic coordinates in the system. This function can be as simple as the distance between two
atoms or be extended to more complex ones like Pickett angles and Cremer-Pople
coordinates [43], which can quantify the conformation of N-member rings. A good
CV (or set of CVs) should capture the slow dynamic evolution of the system (e.g.,
breaking/forming bonds, adsorption/desorption on a surface, protein unfolding, etc.)
and distinguish between states of interest. CVs can provide meaningful insight about
the energy differences between two stable states, but should not be confused with a
reaction coordinate, which carries a special connotation in the literature. Reaction
coordinates are capable of uniquely quantifying the dynamics of a system as it transitions between two stable states. Discussion on the identification and assessment of
reaction coordinates has been thoroughly accomplished elsewhere [44]. Choosing
a CV is a non-trivial task, as it can be difficult to gain chemical intuition about a
system prior to launching a simulation.
External biasing methods greatly benefit from reducing the free energy landscape
of a system in terms of a few CVs. One early example of an external biasing method
is umbrella sampling (US), developed by Torrie and Valleau [45]. In US, a static
restraint (or bias) in the form of a harmonic potential is applied along multiple intervals along a CV. In the simplest terms, a bias is an additional term added to the
potential energy of the system. In turn, this allows for greater sampling of highenergy regions. Sampling many overlapping “umbrellas” in phase space allows for
the reconstruction of the underlying free energy surface. This can be done by using
S. Alamdari et al.
Broadly, one convenient way to characterize enhanced sampling methods is to
categorize them as either tempering or external biasing methods. In tempering
methods, temperature is exploited to enhance the sampling of the system. The canonical form of these methods is simulated annealing [33], a generalized probabilistic
method used to quickly reach a global optimum. The probability of accepting an
exchange relies on a temperature variable which either promotes or restricts exploration of new states. This process rapidly becomes computationally intensive, which
introduced a need for more advanced methods with respect to molecular simulations. Replica exchange is the most generalized sampling approach used in MD,
where several replicas of a molecular configuration are simulated at the same time.
A swap of two replicas can occur under an acceptance probability that preserves
detailed balance. Replicas are weakly coupled and only interact under an exchange
attempt. This promotes a highly parallelizable and enhanced method for sampling a
system. While it is not a requirement, the most common variant of replica exchange
involves simulating replicas at different temperatures; this is commonly known as
parallel tempering (PT) [34, 35]. Other sampling subsets of replica exchange include
Hamiltonian replica exchange [36] and solute tempering [37]. These methods and
many more such as adiabatic free energy dynamics [38] and temperature accelerated
molecular dynamics (TAMD) [39] are described in much greater detail in a number
of prominent reviews [5, 40–42].
In external biasing methods, sampling of the system is enhanced by applying an
external bias on the slow modes of the system. These slow modes of the system
can be described by defining a few choice collective variables (CVs). A CV is a
differentiable function that provides a low-dimensional projection of all atomic coordinates in the system. This function can be as simple as the distance between two
atoms or be extended to more complex ones like Pickett angles and Cremer-Pople
coordinates [43], which can quantify the conformation of N-member rings. A good
CV (or set of CVs) should capture the slow dynamic evolution of the system (e.g.,
breaking/forming bonds, adsorption/desorption on a surface, protein unfolding, etc.)
and distinguish between states of interest. CVs can provide meaningful insight about
the energy differences between two stable states, but should not be confused with a
reaction coordinate, which carries a special connotation in the literature. Reaction
coordinates are capable of uniquely quantifying the dynamics of a system as it transitions between two stable states. Discussion on the identification and assessment of
reaction coordinates has been thoroughly accomplished elsewhere [44]. Choosing
a CV is a non-trivial task, as it can be difficult to gain chemical intuition about a
system prior to launching a simulation.
External biasing methods greatly benefit from reducing the free energy landscape
of a system in terms of a few CVs. One early example of an external biasing method
is umbrella sampling (US), developed by Torrie and Valleau [45]. In US, a static
restraint (or bias) in the form of a harmonic potential is applied along multiple intervals along a CV. In the simplest terms, a bias is an additional term added to the
potential energy of the system. In turn, this allows for greater sampling of highenergy regions. Sampling many overlapping “umbrellas” in phase space allows for
the reconstruction of the underlying free energy surface. This can be done by using