Naively, one way to have an MD system visit a hard-to-reach point z in CV space is simply to create a realization of the configuration x at that point (i.e., such that θ(x) = z). This is an inverse problem, since the number of degrees of freedom in x is usually much larger than in z. One way to perform this inversion is by introducing external forces that guide the configuration to the desired point from some easy-to-create initial state; both targeted MD [9] and steered MD [10] are ways to do this. Of course, one would like MD to explore CV space in the vicinity of z, so after creating the configuration x, one would just let it run. Unfortunately, this would likely result in the system drifting away from z rather quickly, and there would be no way from such calculations to estimate the likelihood of observing an unbiased long MD simulation visit z. However, there is information in the fact that the system drifts away; if one knows on average which direction and how strongly the system would like to move if initialized at z, this would be a measure of negative gradient of the free energy, −(∂F/∂z), or the “mean force”. We have then a glimpse of a three-step method to compute F (i.e., the statistics of CVs) over a meaningfully broad extent of CV space: (1)

visit a select number of local points in that space, and at each one,

The discussion so far leaves open the correct way to compute the local free-energy gradients. A gradient is a local quantity, so a natural choice is to compute it from an MD simulation localized at a point in CV space by a constraint. Consider a long MD simulation with a holonomic constraint fixing the system at the point z. Uniform samples from this constrained trajectory x(t) then represent an ensemble at fixed z over which the averaging needed to convert gradients in potential energy to gradients in free energy could be done. However, this constrained ensemble has the undesired property that the velocities θ̇(x) are zero. This is a bit problematic because virtually none of the samples plucked from a long unconstrained MD simulation (as is implied by Equation (1)), would have θ̇ = 0, and θ̇ = 0 acts as a set of M unphysical constraints on the system velocities ẋ, since θ ˙ j = ∑ i ( ∂ θ j / ∂ x i ) x ˙ i. Probably the best-known example of a method to correct for this bias is the so-called “blue-moon” sampling method [12–15] or the constrained ensemble method [16,17]. The essence of the method is a decomposition of free energy gradients into components along the CV gradients and thermal components orthogonal to them: (4) ∂ F ∂ z j = 〈 b j ( x ) ⋅ ∇ V ( x ) − k B T ∇ ⋅ b j ( x ) 〉 θ ( x ) = zwhere 〈·〉_θ(x)=z denotes averaging across samples drawn uniformly from the MD simulation constrained at θ̇(x) = z, and the b_j(x) is the vector field orthogonal to the gradients of every component k of θ for k ≠ j: (5) b j ( x ) ⋅ ∇ θ k ( x ) = δ j kwhere δ_jk is the Kroenecker delta. (For brevity, we have omitted the consideration of holonomic constraints other than that on the CV; the reader is referred to the paper by Ciccotti et al. for details [15].) The vector fields b_j for each θ_j can be constructed by orthogonalization. The first term in the angle brackets in Equation (4) implements the chain rule one needs to account for how energy V changes with z through all the ways z can change with x. The second term corrects for the thermal bias imposed by the constraint.

Although nowhere near exhaustive, below is a listing of common types of problems to which blue-moon sampling has been applied with some representative examples: (1)

sampling conformations of small flexible molecules and peptides [18–20];

The blue-moon approach requires multiple independent constrained MD simulations to cover the region of CV space in which one wants internal statistics. The care taken in choosing these quadrature points can often dictate the accuracy of the resulting free energy reconstruction. It is therefore sometimes advantageous to consider ways to avoid having to choose such points ahead of time, and adaptive methods attempt to address this problem. One example is the adaptive-biasing force (ABF) algorithm of Darve et al. [6,35] The essence of ABF is two-fold: (1) recognition that external bias forces of the form ∇_xθ_j (∂F/∂z_j) for j = 1, …, M exactly oppose mean forces and should lead to more uniform sampling of CV space; and (2) that these bias forces can be converged upon adaptively during a single unconstrained MD simulation.

The first of those two ideas is motivated by the fact that “forces” that keep normal MD simulations effectively confined to free energy minima are mean forces on the collective variables projected onto the atomic coordinates, and balancing those forces against their exact opposite should allow for thermal motion to take the system out of those minima. The second idea is a bit more subtle; after all, in a running MD simulation with no CV constraints, the constrained ensemble expression for the mean force (Equation (4)) does not directly apply, because a constrained ensemble is not what is being sampled. However, Darve et al. showed how to relate these ensembles so that the samples generated in the MD simulation could be used to build mean forces [35]. Further, they showed using a clever choice of the fields of Equation (4) an equivalence between (i) the spatial gradients needed to computed forces, and (ii) time-derivatives of the CVs [6]: (6) ∂ F ∂ z i = − k B T 〈 d d t ( M θ d θ i d t ) 〉 θ = zwhere M_θ is the transformed mass matrix given by (7) M θ − 1 = J θ M − 1 J θwhere J_θ is the M × 3N matrix with elements ∂θ_i/∂x_j (i = 1 … M, j = 1 … 3N), and M is the diagonal matrix of atomic masses. Equation (7) is the result of a particular choice for the fields b_j(x). This reformulation of the instantaneous mean forces computed on-the-fly makes ABF exceptionally easy to implement in most modern MD packages. Darve et al. present a clear demonstration of the ABF algorithm in a pseudocode [6] that attests to this fact.

ABF has found rather wide application in CV-based free energy calculations in recent years. Below is a representative sample of some types of problems subjected to ABF calculations in the recent literature: (1)

Peptide backbone angle sampling [36,37];

Both blue-moon sampling and ABF are based on statistics in the constrained ensemble. However, estimation of mean forces need not only use this ensemble. One can instead relax the constraint and work with a “mollified” version of the free energy: (8) F κ ( z ) = − k B T   ln   〈 δ κ [ θ ( x ) − z ] 〉where δ_κ refers to the Gaussian (or “mollified delta function”): (9) δ κ = β κ 2 π exp   [ − 1 2 β κ | θ ( x ) − z | 2 ]where β is just shorthand for 1/k_BT. Since lim_βκ→∞δ_κ = δ, we know that lim_βκ→∞ F_κ = F. One way to view this Gaussian is that it “smoothes out” the true free energy to a tunable degree; the factor 1 / β κ is a length-scale in CV space below which details are smeared.

Because the Gaussian has continuous gradients, it can be used directly in an MD simulation. Suppose we have a CV space θ(x), and we extend our MD system to include variables z such that the combined set (x, z) obeys the following extended potential: (10) U ( x , z ) = V ( x ) + ∑ j = 1 M 1 2 κ | θ j ( x ) − z j | 2where V(x) is the interatomic potential, and κ is a constant. Clearly, if we fix z, then the resulting free energy is to within an additive constant the mollified free energy of Equation (8). (The additive constant is related to the prefactor of the mollified delta function and has nothing to do with the number of CVs.) Further, we can directly express the gradient of this mollified free energy with respect to z: [54] (11) ∇ z F κ = − 〈 κ [ θ ( x ) − z ] 〉This suggests that, instead of using constrained ensemble MD to accumulate mean forces, we could work in the restrained ensemble and get very good approximations to the mean force. By “restrained”, we refer to the fact that the term giving rise to the mollified delta function in the configurational integral is essentially a harmonic restraining potential with a “spring constant” κ. In this restrained-ensemble approach, no velocities are held fixed, and the larger we choose κ the more closely we can approximate the true free energy. Notice however that large values of κ could lead to numerical instabilities in integrating equations of motion, and a balance should be found. (In practice, we have found that for CVs with dimensions of length, values of κ less than about 1,000 kcal/mol/Ų can be stably handled, and values of around 100 kcal/mol/Ų are typically adequate.)

Temperature-accelerated MD (TAMD) [7] takes advantage of the restrained-ensemble approach to directly evolve the variables z in such a way to accelerate the sampling of CV space. First, consider how the atomic variables x evolve under the extended potential (assuming Langevin dynamics): (12) m i x ¨ i = − ∂ V ( x ) ∂ x i − κ ∑ j = 1 m [ θ j ( x ) − z j ] ∂ θ j ( x ) ∂ x i − γ m i x ˙ i + η i ( t ; β )Here, m_i is the mass of x_i, γ is the friction coefficient for the Langevin thermostat, and η is the thermostat white noise satisfying the fluctuation-dissipation theorem at physical temperature β⁻¹: (13) 〈 η i ( t ; β ) η j ( t ′ ; β ) 〉 = β − 1 γ m i δ i j δ ( t − t ′ )Key to TAMD is that the z are treated as slow variables that evolve according to their own equations of motion, which here we take as diffusive (though other choices are possible [7]): (14) γ ¯ m ¯ j z ˙ j = κ [ θ j ( x ) − z j ] + ξ j ( t ; β ¯ )Here, γ̄ is a fictitious friction, m̄_j is a mass, and the first term on the right-hand side represents the instantaneous force on variable z_j, and the second term represents thermal noise at the fictitious thermal energy β̄⁻¹ ≠ β⁻¹.

The advantage of TAMD is that if (1) γ̄ is chosen sufficiently large so as to guarantee that the slow variables indeed evolve slowly relative to the fundamental variables; and (2) κ is sufficiently large such that θ(x(t)) ≈ z(t) at any given time, then the force acting on z is approximately equal to minus the gradient of the free energy (Equation (11)) [7]. This is because the MD integration repeatedly samples κ[θ(x) − z] for an essentially fixed (but actually very slowly moving) z, so z evolution effectively feels these samples as a mean force. In other words, the dynamics of z(t) is effectively (15) γ ¯ m ¯ j z ˙ j = − ∂ F ( z ) ∂ z j + ξ j ( t ; β ¯ )This shows that the z-dynamics describes an equilibrium constant-temperature ensemble at fictitious temperature β̄⁻¹ acted on by the “potential” F (z), which is the free energy evaluated at the physical temperature β⁻¹. That is, under TAMD, z conforms to a probability distribution of the form exp [−β̄F(z; β)], whereas under normal MD it would conform to exp [−βF (z; β)]. The all-atom MD simulation (at β) simply serves to approximate the local gradients of F (z). Sampling is enhanced by taking β̄⁻¹>β⁻¹, which has the effect of attenuating the ruggedness of F. TAMD therefore can accelerate a trajectory z(t) through CV space by increasing the likelihood of visiting points with relatively low physical Boltzmann factors. This borrows directly from the main idea of adiabatic free-energy dynamics [55] (AFED), in that one deliberately makes some variables hot (to overcome barriers) but slow (to keep them adiabatically separated from all other variables). In TAMD, however, the use of the mollified free energy means no cumbersome variable transformations are required. (The authors of AFED refer to TAMD as “driven”-AFED, or d-AFED [56].) It is also worth mentioning in this review that TAMD borrows heavily from an early version of metadynamics [57], which was formulated as a way to evolve the auxiliary variables z on a mollified free energy. However, unlike metadynamics (which we discuss below in Section 2.3.3.), there is no history-dependent bias in TAMD.

Unlike TI, ABF, and the methods of umbrella sampling and metadynamics discussed in the next section, TAMD is not a method for direct calculation of the free energy. Rather, it is a way to overcome free energy barriers in a chosen CV space quickly without visiting irrelevant regions of CV space. (However, we discuss briefly a method in Section 4.2.2. in which TAMD gradients are used in a spirit similar to ABF to reconstruct a free energy.) That is, we consider TAMD a way to efficiently explore relevant regions CV space that are practically inaccessible to standard MD simulation. It is also worth pointing out that, unlike ABF, TAMD does not operate by opposing the natural gradients in free energy, but rather by using them to guide accelerated sampling. ABF can only use forces in locations in CV space the trajectory has visited, which means nothing opposes the trajectory going to regions of very high free energy. However, under TAMD, an acceleration of β̄⁻¹ = 6 kcal/mol on the CVs will greatly accelerate transitions over barriers of 6-12 kcal/mol, but will still not (in theory) accelerate excursions to regions requiring climbs of hundreds of kcal/mol. TAMD and ABF have in common the ability to handle rather high-dimensional CVs.

Although it was presented theoretically in 2006 [7], TAMD was not applied directly to large-scale MD until much later [58]. Since then, there has been growing interest in using TAMD in a variety of applications requiring enhanced sampling: (1)

TAMD-enhanced flexible fitting of all-atom protein and RNA models into low-resolution electron microscopy density maps [59,60];

Finally, we mention briefly that TAMD can be used as a quick way to generate trajectories from which samples can be drawn for subsequent mean-force estimation for later reconstruction of a multidimensional free energy; this is the essence of the single-sweep method [68], which is an efficient means of computing multidimensional free energies. Rather than using straight numerical TI, single sweep posits the free energy as a basis function expansion and uses standard optimization methods to find the expansion coefficients that best reproduce the measured mean forces. Single-sweep has been used to map diffusion pathways of CO and H₂O in myoglobin [64,65].

In the previous section, we considered methods that achieve enhanced sampling by using mean forces: in TI, these are integrated to reconstruct a free energy; in ABF, these are built on-the-fly to drive uniform CV sampling; and in TAMD, these are used on-the-fly to guide accelerated evolution of CVs. In this section, we consider methods that achieve enhanced sampling by means of controlled bias potentials. As a class, we refer to these as non-Boltzmann sampling methods.

Non-Boltzmann sampling is generally a way to derive statistics on a system whose energetics differ from the energetics used to perform the sampling. Imagine we have an MD system with bare interatomic potential V(x), and we add a bias ΔV(x) to arrive at a biased total potential: (16) V b ( x ) = V ( x ) + Δ V ( x )The statistics of the CVs on this biased potential are then given as (17) P b ( z ) = ∫ d x   e − β V 0 ( x ) e − β Δ V ( x ) δ [ θ ( x ) − z ] ∫ d x   e − β V 0 ( x ) e − β Δ V ( x ) = ∫ d x   e − β V 0 ( x ) e − β Δ V δ [ θ ( x ) − z ] ∫ d x   e − β V 0 ( x ) ∫ d x   e − β V 0 ( x ) ∫ d x   e − β V 0 ( x ) e − β Δ V ( x ) = 〈 e − β Δ V ( x ) δ [ θ ( x ) − z ] 〉 〈 e − β Δ V ( x ) 〉where 〈·〉 denotes ensemble averaging on the unbiased potential V (x). Further, if we take the bias potential ΔV to be explicitly a function only of the CVs θ, then it becomes invariant in the averaging of the numerator thanks to the delta function, and we have (18) P b ( x ) = e − β Δ V ( x ) 〈 δ [ θ ( x ) − z ] 〉 〈 e − β Δ V [ θ ( x ) ] 〉Finally, since the unbiased statistics are P (z) = 〈δ[θ(x) − z]〉, we arrive at (19) P ( z ) = P b ( z ) e β Δ V ( z ) 〈     e − β Δ V [ θ ( x ) ] 〉Taking samples from an ergodic MD simulation on the biased potential V_b, Equation (19) provides the recipe for reconstructing the statistics the CVs would present were they generated using the unbiased potential V. However, the probability P (z) is implicit in this equation, because (20) 〈 e − β Δ V 〉 = ∫ d z P ( z ) e − β Δ V [ θ ( x ) ]This is not really a problem, since we can treat 〈e^−βΔV〉 as a constant we can get from normalizing P_b(z)e^βΔV(z).

How does one choose ΔV so as to enhance the sampling of CV space? Evidently, from the standpoint of non-Boltzmann sampling, the closer the bias potential is to the negative free energy −F (z), the more uniform the sampling of CV space will be. To wit: if ΔV [θ(x)] = −F [θ (x)], then e^βΔV(z) = e^−βF(z) = P (z), and Equation (19) can be inverted for P_b to yield (21) P b ( z ) = 1 〈 e β F ( z ) 〉 = 1 ∫ d z P ( z ) e β F ( z ) = 1 ∫ d z e − β F e β F = 1 ∫ d zSo we see that taking the bias potential to be the negative free energy makes all states z in CV space equiprobable. This is indeed the limit to which ABF strives by applying negative mean forces, for example [6].

We usually do not know the free energy ahead of time; if we did, we would already know the statistics of CV space and no enhanced sampling would be necessary. Moreover, perfectly uniform sampling of the entire CV space is usually far from necessary, since most CV spaces have many irrelevant regions that should be ignored. And in reference to the mean-force methods of the last section, uniform sampling is likely not necessary to achieve accurate mean force values; how good an estimate of ∇F is at some point z₀ should not depend on how well we sampled at some other point z₁. Yet achieving uniform sampling is an idealization since, if we do, this means we know the free energy. We now consider two other biasing methods that aim for this ideal, either in relatively small regions of CV space using fixed biases, or over broader extents using adaptive biases.

Umbrella sampling is the standard way of using non-Boltzmann sampling to overcome free energy barriers. In its debut [69], umbrella sampling used a function w(x) that weights hard-to-sample configurations, equivalent to adding a bias potential of the form (22) Δ V ( x ) = − k B T   ln   w ( x )w is found by trial-and-error such that configurations that are easy to sample on the unbiased potential are still easy to sample; that is, w acts like an “umbrella” covering both the easy- and hard-to-sample regions of configuration space. Nearly always, w is an explicit function of the CVs, w(x) = W [θ(x)].

Coming up with the umbrella potential that would enable exploration of CV space with a single umbrella sampling simulation that takes the system far from its initial point is not straightforward. Akin to TI, it is therefore advantageous to combine results from several independent trajectories, each with its own umbrella potential that localizes it to a small volume of CV space that overlaps with nearby volumes. The most popular way to combine the statistics of such a set of independent umbrella sampling runs is the weighted-histogram analysis method (WHAM) [70].

To compute statistics of CV space using WHAM, one first chooses the points in CV space that define the little local neighborhoods, or “windows” to be sampled and chooses the bias potential used to localize the sampling. Not knowing how the free energy changes in CV space makes the first task somewhat challenging, since more densely packed windows are preferred in regions where the free energy changes rapidly; however, since the calculations are independent, more can be added later if needed. A convenient choice for the bias potential is a simple harmonic spring that tethers the trajectory to a reference point z_i in CV space: (23) Δ V i ( x ) = 1 2 κ | θ ( x ) − z i | 2which means the dynamics of the atomic variables x are identical to Equation (12) at fixed z = z_i. The points {z_i} and the value of κ (which may be point-dependent) must be chosen such that θ[x(t)] from any one window’s trajectory makes excursions into the window of each of its nearest neighbors in CV space.

Each window-restrained trajectory is directly histogrammed to yield apparent (i.e., biased) statistics on θ; let us call the biased probability in the ith window P_b,i(z). Equation (19) again gives the recipe to reconstruct the unbiased statistics P_i(z) for z in the window of z_i: (24) P i ( z ) = P b , i ( z ) e 1 2 β κ | z − z i | 2 〈 e − β 1 2 κ | θ ( x ) − z i | 2 〉We could use Equation (24) directly assuming the biased MD trajectory is ergodic, but we know that regions far from the reference point will be explored very rarely and thus their free energy would be estimated with large uncertainty. This means that, although we can use sampling to compute P_b,i knowing it effectively vanishes outside the neighborhood of z_i, we cannot use sampling to compute 〈 e − β 1 2 κ | θ ( x ) − z i | 2 〉.

WHAM solves this problem by renormalizing the probabilities in each window into a single composite probability. Where there is overlap among windows, WHAM renormalizes such that the statistical variance of the probability is minimal. That is, it treats the factor 〈 e − β 1 2 κ | θ ( x ) − z i | 2 〉 as an undetermined constant C_i for each window, and solves for specific values such that the composite unbiased probability P (z) is continuous across all overlap regions with minimal statistical error. An alternative to WHAM, termed “umbrella integration”, solves the problem of renormalization across windows by constructing the composite mean force [71,72].

The literature on umbrella sampling is vast (by simulation standards), so we present here a very condensed listing of some of its more recent application areas with representative citations: (1)

Small molecule conformational sampling [73–76];

As already mentioned, one of the difficulties of the umbrella sampling method is the choice and construction of the bias potential. As we already saw with the relationship among TI, ABF, and TAMD, an adaptive method for building a bias potential in a running MD simulation may be advantageous. Metadynamics [8,143] represents just such a method.

Metadynamics is rooted in the original idea of “local elevation” [144], in which a supplemental bias potential is progressively grown in the dihedral space of a molecule to prevent it from remaining in one region of configuration space. However, at variance with metadynamics, local elevation does not provide any means to reconstruct the unbiased free-energy landscape and as such it is mostly aimed at fast generation of plausible conformers.

In metadynamics, configurational variables x evolve in response to a biased total potential: (25) V ( x ) = V 0 ( x ) + Δ V ( x , t )where V₀ is the bare interatomic potential and ΔV (x, t) is a time-dependent bias potential. The key element of metadynamics is that the bias is built as a sum of Gaussian functions centered on the points in CV space already visited: (26) Δ V [ θ ( x ) , t ] = w ∑ t ′ = τ G , 2 τ G , …                 t ′ < t exp ( − | θ [ x ( t ) ] − θ [ x ( t ′ ) ] | 2 2 δ θ 2 )Here, w is the height of each Gaussian, τ_G is the size of the time interval between successive Gaussian depositions, and δθ is the Gaussian width. It has been first empirically [145] then analytically [146] demonstrated that in the limit in which the CVs evolve according to a Langevin dynamics, the bias indeed converges to the negative of the free energy, thus providing an optimal bias to enhance transition events. Multiple simulations can also be used to allow for a quicker filling of the free-energy landscape [147].

The difference between the metadynamics estimate of the free energy and the true free energy can be shown to be related to the diffusion coefficient of the collective variables and to the rate at which the bias is grown. A possible way to decrease this error as a simulation progresses is to decrease the growth rate of the bias. Well-tempered metadynamics [148] used an optimized schedule to decrease the deposition rate of bias by modulating the Gaussian height: (27) w = ω 0 τ G e − Δ V ( θ , t ) k B Δ THere, ω₀ is the initial “deposition rate”, measured Gaussian height per unit time, and ΔT is a parameter that controls the degree to which the biased trajectory makes excursions away from free-energy minima. It is possible to show that using well-tempered metadynamics the bias does not converge to the negative of the free-energy but to a fraction of it, thus resulting in sampling the CVs at an effectively higher temperature T + ΔT, where normal metadynamics is recovered for ΔT → ∞. We notice that other deposition schedules can be used aimed, e.g., at maximizing the number of round-trips in the CV space [149]. Importantly, it is possible to recover equilibrium Boltzmann statistics of unbiased collective variables from samples drawn throughout a well-tempered metadynamics trajectory [150]; it does not seem clear that one can do this from an ABF trajectory. Finally, it is possible to tune the shape of the Gaussians on the fly using schemes based on the geometric compression of the phase space or on the variance of the CVs [151].

In the well-tempered ensemble, the parameter ΔT can be used to tune the size of the explored region, in a fashion similar to the fictitious temperature in TAMD. So both TAMD and well-tempered metadynamics can be used to explore relevant regions of CV space while surmounting relevant free energy barriers. However, there are important distictions between the two methods. First, the main source of error in TAMD rests with how well mean-forces are approximated, and adiabatic separation, realizable only when the auxiliary variables z never move, is the only way to guarantee they are perfectly accurate. In practical application, TAMD never achieves perfect adiabatic separation. In contrast, because the deposition rate of decreases as a well-tempered trajectory progresses, errors related to poor adiabatic separation are progressively damped. Second, as already mentioned, TAMD alone cannot report the free energy, but it also is therefore not practically limited by the dimensionality of CV space; multicomponent gradients are just as accurately calculated in TAMD as are single-component gradients. Metadynamics, as a histogram-filling method, must exhaustively sample a finite region around any point to know the free energy and its gradients are correct, which can sometimes limit its utility.

Metadynamics is a powerful method whose popularity continues to grow. In either its original formulation or in more recent variants, metadynamics has been employed successfully in several fields, some of which we point out below with some representative examples: (1)

Chemical reactions [57,152];

Given a potential V (x), any multidimensional CV θ(x) has a mathematically determined free energy F (z), and in principle the free-energy methods we describe here (and others) can use and/or compute it. However, this does not guarantee that F is meaningful, and a poor choice for θ(x) can render the results of even the most sophisticated free-energy methods useless for understanding the nature of actual metastable states and the transitions among them. This puts two major requirements on any CV space: (1)

Metastable states and transition states must be unambiguously identified as energetically separate regions in CV space.

The first of these may seem obvious: CVs are chosen to provide a low-dimensional description of some important process, say a conformational change or a chemical reaction or a binding event, and one can not describe a process without being able to discriminate states. However, it is not always easy to find CVs that do this. Even given representative configurations of two distinct metastable states, standard MD from these two different initial configurations may sample partially overlapping regions of CV space, making ambiguous the assignation of an arbitrary configuration to a state. It may be in this case that the two representative configurations actually belong to the same state, or that if there are two states, that no matter what CV space is overlaid, the barrier separating them is so small that, on MD timescales, they can be considered rapidly exchanging substates of some larger state.

However, a third possibility exists: the two MD simulations mentioned above may in fact represent very different states. The overlap might just be an artifact of neglecting to include one or more CVs that are truly necessary to distinguish those states. If there is a significant free energy barrier along this neglected variable, an MD simulation will not cross it, yet may still sample regions in CV space also sampled by an MD simulation launched from the other side of this hidden barrier. And it is even worse: if TI or umbrella sampling is used along a pathway in CV space that neglects an important variable, the free-energy barriers along that pathway might be totally meaningless.

Hidden barriers can be a significant problem in CV-based free-energy calculations. Generally speaking, one only learns of a hidden barrier after postulating its existence and testing it with a new calculation. Detecting them is not straightforward and often involves a good deal of CV space exploration. Methods such as TAMD and well-tempered metadynamics offer this capability, but much more work could be done in the automated detection of hidden barriers and the “right” CVs (e.g., [169–171]).

An obvious way of reducing the likelihood of hidden barriers is to use increase the dimensionality of CV space. TAMD is well-suited to this because it is a gradient method, but standard metadynamics, because it is a histogram-filling method, is not. A recent variant of metadynamics termed “reconnaissance metadynamics” [172] does have the capability of handling high-dimensional CV spaces. In reconnaissance metadynamics, bias potential kernels are deposited at the CV space points identified as centers of clusters detected and measured by an on-the-fly clusterization scheme. These kernels are hyperspherically symmetric but grow as cluster sizes grow and are able to push a system out of a CV space basin to discover other basins. As such, reconnaissance metadynamics is an automated way of identifying free-energy minima in high-dimensional CV spaces. It has been applied the identification of configurations of small clusters of molecules [173] and identification of protein-ligand binding poses [162].

There are very few “best practices” codified for choosing CVs for any given system. Most CVs are developed ad hoc based on the processes that investigators would like to study, for instance, center-of-mass distance between two molecules for studying binding/unbinding, or torsion angles for studying conformational changes, or number of contacts for studying order-disorder transitions. Cartesian coordinates of centers of mass of groups of atoms are also often used as CVs, as they are functions of these coordinates.

The potential energy V (x) is also an example of a 1-D CV, and there have been several examples of using it in CV-based enhanced sampling methods, such as umbrella sampling [174], metadynamics [175] well-tempered metadynamics [176]. In a recent work based on steered MD, it has been shown that also relevant reductions of the potential energy (e.g., the electrostatic interaction free-energy) can be used as effective CVs [177]. The basic rationale for enhanced sampling of V is that states with higher potential energy often correspond to transition states, and one need make no assumptions about precise physical mechanisms. Key to its successful use as a CV, as it is for any CV, is a proper accounting for its entropy;i.e., the classical density-of-states.

Coarse-graining of particle positions onto Eulerian fields was used early on in enhanced sampling [178]; here, the value of the field at any Cartesian point is a CV, and the entire field represents a very high-dimensional CV. This idea has been put to use recently in the “indirect umbrella sampling” method of Patel et al. [179] for computing free energies of solvation, and string method (Section 4.2.1.) calculations of lipid bilayer fusion [180]. In a similar vein, there have been recent attempts at variables designed to count the recurrency of groups of atoms positioned according to given templates, such as α-helices paired β-strands in proteins [181].

We finally mention the possibility of building collective variables based on set of frames which might be available from experimental data or generated by means of previous MD simulations. Some of these variables are based on the idea of computing the distances between the present configuration and a set of precomputed snapshots. These distances, here indicated with d_i, where i is the index of the snapshot, are then combined to obtain a coarse representation of the present configuration, which is then used as a CV. As an example, one might combine the distances as (28) s = ∑ i e − λ d i i ∑ i e − λ d iIf the parameter λ is properly chosen, this function returns a continuous interpolation between the indexes of the snapshots which are closer to the present conformation. If the snapshots are disposed along a putative path connecting two experimental structures, this CV can be used as a path CV to monitor and bias the progression along the path [182]. A nice feature of path CVs is that it is straighforward to also monitor the distance from the putative path. The standard way to do it is by looking at the distance from the closest reference snapshot, which can be approximately computed with the following continuous function: (29) z = − λ − 1 log ∑ i e − λ d iThis approach, modified to use internal coordinates, was used recently by Zinovjev et al. to study the aqueous phase reaction of pyruvate to salycilate, and in the CO bond-breaking/proton transfer in PchB [183].

A generalization to multidimensional paths (i.e., sheets) can be obtained by assigning a generic vector v_i to each of the precomputed snapshots and computing its average [184]: (30) s = ∑ i e − λ d i v i ∑ i e − λ d i

The string method is generally an approach to find pathways of minimal energy connecting two points in phase space [228]. When working in CVs, the string method is used to find minimal free-energy paths (MFEP’s) [229]. String method calculations involve multiple replicas, each representing a point z_s in CV space at position s along a discretized string connecting two points of interest (reactant and product states, say). The forces on each replica’s z_s are computed and their z_s’s updated, as in TAMD, with the addition of forces that act to keep the z’s equidistant along the string (so-called reparameterization forces): (34) γ ¯ z ˙ j ( s , t ) = ∑ k [ M ˜ j k ( x ( s , t ) ) κ [ θ k ( x ( s , t ) ) − z k ( s , t ) ] ] + η z ( t ) + λ ( s , t ) ∂ z j ∂ sHere, M̃_jk is the metric tensor mapping distances on the manifold of atomic coordinates to the manifold of CV space, η is thermal noise and λ ( s , t ) ∂ z j ∂ s represents the reparameterization force tangent to the string that is sufficient to maintain equidistant images along the string. String method has been used to study activation of the insulin-receptor kinase [63], docking of insulin to its receptor [230], and myosin [231]. In these examples, the update of the string coordinates is done at a lower frequency than the atomic variables in each image.

In contrast, in the on-the-fly variant of string method in CVs, the friction on the z_s’s is set high enough to make the effective averaging of the forces approach the true mean forces, and the z updates occur in lockstep with the x updates of the MD system [232]. Just as in TAMD, the atomic variables obey an equation of motion like Equation (12) tethering them to the z_s. Stober and Abrams recently demonstrated an implementation of on-the-fly string method to study the thermodynamics of the normal-to-amyloidogenic transition of β2-microglobulin [233]. Unique in this approach was the construction of a single composite MD system containing 27 individual β2 molecules restrained to points on 3 × 3 × 3 grid inside a single large solvent box. Zinovjev et al. used a combination of the on-the-fly string method and of path-collective variables (see Equations (28) and (29)) in a quantum-mechanics/molecular-mechanics approach to study a methyltransferase reaction [234].

Because TAMD provides mean-force estimates as it is exploring CV space, it stands to reason that those mean forces could be used to compute a free energy. In contrast, in the single-sweep method [68], the TAMD forces are only used in the CV space exploration phase, not the free-energy calculation itself. Recently, Abrams and Vanden-Eijnden proposed a method for using TAMD directly to parameterize a free energy; that is, to determine the best set of some parameters λ on which a free energy of known functional form depends [235]: (35) F ( z ) = F ( z ; λ * )The approach, termed “on-the-fly free energy parameterization”, uses forces from a running TAMD simulation to progressively optimize λ using a time-averaged gradient error: (36) E ( λ ) = 1 2 t ∫ 0 t | ∇ z F [ z ( s ) , λ ( t ) ] + κ [ θ ( x ( s ) ) − z ( s ) ] | 2 d sIf constructed so that F is linear in λ = (λ₁, λ₂, …, λ_M), minimization of E can be expressed as a simple linear algebra problem (37) ∑ j A i j λ j = b i ,           i = 1 , … , Mand the running TAMD simulation provides progressively better estimates of A and b until the λ converge. In the cited work, it was shown that this method is an efficient way to derive potentials of mean force between particles in coarse-grained molecular simulations as basis-function expansions. It is currently being investigated as a means to parameterize free energies associated with conformational changes of proteins.

Chen, Cuendet, and Tuckermann developed a very similar approach that in addition to parameterizing a free energy using d-AFED-computed gradients uses a metadynamics-like bias on the potential [236]. These authors demonstrated efficient reconstruction of the four-dimensional free-energy of vacuum alanine dipeptide with this approach.