Non-equilibrium pulling

When the distance between two groups is changed continuously, work is applied to the system, which means that the system is no longer in equilibrium. Although in the limit of very slow pulling the system is again in equilibrium, for many systems this limit is not reachable within reasonable computational time. However, one can use the Jarzynski relation 135 to obtain the equilibrium free-energy difference \(\Delta G\) between two distances from many non-equilibrium simulations:

(330)\[\Delta G_{AB} = -k_BT \log \left\langle e^{-\beta W_{AB}} \right\rangle_A\]

where \(W_{AB}\) is the work performed to force the system along one path from state A to B, the angular bracket denotes averaging over a canonical ensemble of the initial state A and \(\beta=1/k_B T\).

Collective variables: the pull code

The pull code applies forces or constraints on collective variables (sometimes referred to as reaction coordinates). The basic collective pull coordinates are a distance, angle and dihedral angle between centers of mass of groups atoms, the so-called “pull groups”. More complex collective variables can be defined using The transformation pull coordinate. A pull group can be part of one or more pull coordinates. Furthermore, a coordinate can also operate on a single group and an absolute reference position in space. The distance between a pair of groups can be determined in 1, 2 or 3 dimensions, or can be along a user-defined vector. The reference distance can be constant or can change linearly with time. Normally all atoms are weighted by their mass, but an additional weighting factor can also be used.

Fig. 38 Schematic picture of pulling a lipid out of a lipid bilayer with umbrella pulling. \(V_{rup}\) is the velocity at which the spring is retracted, \(Z_{link}\) is the atom to which the spring is attached and \(Z_{spring}\) is the location of the spring.

Several different pull types, i.e. ways to apply the pull force, are supported, and in all cases the reference distance can be constant or linearly changing with time.

1. Umbrella pulling A harmonic potential is applied between the centers of mass of two groups. Thus, the force is proportional to the displacement.

2. Constraint pulling The distance between the centers of mass of two groups is constrained. The constraint force can be written to a file. This method uses the SHAKE algorithm but only needs 1 iteration to be exact if only two groups are constrained.

3. Constant force pulling A constant force is applied between the centers of mass of two groups. Thus, the potential is linear. In this case there is no reference distance of pull rate.

4. Flat bottom pulling Like umbrella pulling, but the potential and force are zero for coordinate values below (pull-coord?-type = flat-bottom) or above (pull-coord?-type = flat-bottom-high) a reference value. This is useful for restraining e.g. the distance between two molecules to a certain region.

5. External potential This takes the potential acting on the reaction coordinate from another module. Current only the Accelerated Weight Histogram method (see sec. Adaptive biasing with AWH) is supported, which provides adaptive biasing of pull coordinates.

In addition, there are different types of reaction coordinates, so-called pull geometries. These are set with the mdp option pull-coord?-geometry.

Definition of the center of mass

In GROMACS, there are three ways to define the center of mass of a group. The standard way is a “plain” center of mass, possibly with additional weighting factors. With periodic boundary conditions it is no longer possible to uniquely define the center of mass of a group of atoms. Therefore, a reference atom is used. For determining the center of mass, for all other atoms in the group, the closest periodic image to the reference atom is used. This uniquely defines the center of mass. By default, the middle (determined by the order in the topology) atom is used as a reference atom, but the user can also select any other atom if it would be closer to center of the group.

When there are large pull groups, such as a lipid bilayer, pull-pbc-ref-prev-step-com can be used to avoid potential large movements of the center of mass in case that atoms in the pull group move so much that the reference atom is too far from the intended center of mass. With this option enabled the center of mass from the previous step is used, instead of the position of the reference atom, to determine the reference position. The position of the reference atom is still used for the first step. For large pull groups it is important to select a reference atom that is close to the intended center of mass, i.e. do not use pull-group?-pbcatom = 0.

For a layered system, for instance a lipid bilayer, it may be of interest to calculate the PMF of a lipid as function of its distance from the whole bilayer. The whole bilayer can be taken as reference group in that case, but it might also be of interest to define the reaction coordinate for the PMF more locally. The mdp option pull-coord?-geometry = cylinder does not use all the atoms of the reference group, but instead dynamically only those within a cylinder with radius pull-cylinder-r around the pull vector going through the pull group. This only works for distances defined in one dimension, and the cylinder is oriented with its long axis along this one dimension. To avoid jumps in the pull force, contributions of atoms are weighted as a function of distance (in addition to the mass weighting), for atom \(i\):

(331)\[\begin{split}\begin{aligned} w_i(r_i < r_\mathrm{cyl}) & = & 1-2 \left(\frac{r_i}{r_\mathrm{cyl}}\right)^2 + \left(\frac{r_i}{r_\mathrm{cyl}}\right)^4 \\ w_i(r_i \geq r_\mathrm{cyl}) & = & 0\end{aligned}\end{split}\]

Note that the radial dependence on the weight causes a radial force on both cylinder group and the other pull group:

(332)\[\begin{split}\begin{aligned} F^\mathrm{radial}_i(r_i < r_\mathrm{cyl}) & = & F^\mathrm{pull} a_i \frac{1}{\sum_i w_i}\frac{4}{r_\mathrm{cyl}^4} r_i (r_i^2 - r_\mathrm{cyl}^2) \\ F^\mathrm{radial}_i(r_i \geq r_\mathrm{cyl}) & = & 0\end{aligned}\end{split}\]

where \(F^\mathrm{pull}\) is the pull force working between the groups and \(a_i\) is the axial distance of atom \(i\) to the center of mass of the cylinder group. This is an undesirable, but unavoidable effect. To minimize this effect, the cylinder radius should be chosen sufficiently large. The effective mass is 0.47 times that of a cylinder with uniform weights and equal to the mass of uniform cylinder of 0.79 times the radius.

Fig. 39 Comparison of a plain center of mass reference group versus a cylinder reference group applied to interface systems. C is the reference group. The circles represent the center of mass of two groups plus the reference group, \(d_c\) is the reference distance.

For a group of molecules in a periodic system, a plain reference group might not be well-defined. An example is a water slab that is connected periodically in \(x\) and \(y\), but has two liquid-vapor interfaces along \(z\). In such a setup, water molecules can evaporate from the liquid and they will move through the vapor, through the periodic boundary, to the other interface. Such a system is inherently periodic and there is no proper way of defining a “plain” center of mass along \(z\). A proper solution is to using a cosine shaped weighting profile for all atoms in the reference group. The profile is a cosine with a single period in the unit cell. Its phase is optimized to give the maximum sum of weights, including mass weighting. This provides a unique and continuous reference position that is nearly identical to the plain center of mass position in case all atoms are all within a half of the unit-cell length. See ref 136 for details.

When relative weights \(w_i\) are used during the calculations, either by supplying weights in the input or due to cylinder geometry or due to cosine weighting, the weights need to be scaled to conserve momentum:

(333)\[w'_i = w_i \left. \sum_{j=1}^N w_j \, m_j \right/ \sum_{j=1}^N w_j^2 \, m_j\]

where \(m_j\) is the mass of atom \(j\) of the group. The mass of the group, required for calculating the constraint force, is:

(334)\[M = \sum_{i=1}^N w'_i \, m_i\]

The definition of the weighted center of mass is:

(335)\[\mathbf{r}_{com} = \left. \sum_{i=1}^N w'_i \, m_i \, \mathbf{r}_i \right/ M\]

From the centers of mass the AFM, constraint, or umbrella force \(\mathbf{F}_{\!com}\) on each group can be calculated. The force on the center of mass of a group is redistributed to the atoms as follows:

(336)\[\mathbf{F}_{\!i} = \frac{w'_i \, m_i}{M} \, \mathbf{F}_{\!com}\]

Definition of the pull direction

The most common setup is to pull along the direction of the vector containing the two pull groups, this is selected with pull-coord?-geometry = distance. You might want to pull along a certain vector instead, which is selected with pull-coord?-geometry = direction. But this can cause unwanted torque forces in the system, unless you pull against a reference group with (nearly) fixed orientation, e.g. a membrane protein embedded in a membrane along x/y while pulling along z. If your reference group does not have a fixed orientation, you should probably use pull-coord?-geometry = direction-relative, see Fig. 40. Since the potential now depends on the coordinates of two additional groups defining the orientation, the torque forces will work on these two groups.

Fig. 40 The pull setup for geometry direction-relative. The “normal” pull groups are 1 and 2. Groups 3 and 4 define the pull direction and thus the direction of the normal pull forces (red). This leads to reaction forces (blue) on groups 3 and 4, which are perpendicular to the pull direction. Their magnitude is given by the “normal” pull force times the ratio of \(d_p\) and the distance between groups 3 and 4.

Definition of the angle and dihedral pull geometries

Four pull groups are required for pull-coord?-geometry = angle. In the same way as for geometries with two groups, each consecutive pair of groups \(i\) and \(i+1\) define a vector connecting the COMs of groups \(i\) and \(i+1\). The angle is defined as the angle between the two resulting vectors. E.g., the mdp option pull-coord?-groups = 1 2 2 4 defines the angle between the vector from the COM of group 1 to the COM of group 2 and the vector from the COM of group 2 to the COM of group 4. The angle takes values in the closed interval [0, 180] deg. For pull-coord?-geometry = angle-axis the angle is defined with respect to a reference axis given by pull-coord?-vec and only two groups need to be given. The dihedral geometry requires six pull groups. These pair up in the same way as described above and so define three vectors. The dihedral angle is defined as the angle between the two planes spanned by the two first and the two last vectors. Equivalently, the dihedral angle can be seen as the angle between the first and the third vector when these vectors are projected onto a plane normal to the second vector (the axis vector). As an example, consider a dihedral angle involving four groups: 1, 5, 8 and 9. Here, the mdp option pull-coord?-groups = 8 1 1 5 5 9 specifies the three vectors that define the dihedral angle: the first vector is the COM distance vector from group 8 to 1, the second vector is the COM distance vector from group 1 to 5, and the third vector is the COM distance vector from group 5 to 9. The dihedral angle takes values in the interval (-180, 180] deg and has periodic boundaries.

The transformation pull coordinate

The transformation pull coordinate is a “meta” pull coordinate that can be used to define more complex collective variables. It can transform one or more other pull coordinates using an arbitrary mathematical expression. This is a powerful tool for generating complex collective variables. A simple example is a contact coordinate using a non-linear transformation of a distance. More complex examples are a (non-)linear combination of two or more pull coordinates or a sum of contacts.

Typically, the force constant for pull coordinate(s) the transformation coordinates acts on should be zero. This avoids unintended addition of direct forces on the pull coordinate(s) to the indirect forces from the transition pull coordinate. This is not a requirement, but having both a direct and indirect, from the tranformation coordinate, force working on them is almost never desirable. If the transformation is a linear combination of multiple distances, it is useful to normalize the coefficients such that the transformation coordinate also has units of nanometer. That makes both the choice of the force constant and the interpretation easier.

Here are two examples of pull sections of the mdp input that use a tranformation coordinate setups. The first is a contact reaction coordinate that is 1 at contact and 0 at larger distances:

pull                     = yes
pull-ngroups             = 2
pull-ncoords             = 2

pull-group1-name         = groupA
pull-group2-name         = groupB

pull-coord1-type         = umbrella
pull-coord1-geometry     = distance
pull-coord1-groups       = 1 2
pull-coord1-dim          = Y Y Y
pull-coord1-k            = 0      ; avoid forces working directly on this distance

pull-coord2-type         = umbrella
pull-coord2-geometry     = transformation
pull-coord2-expression   = 1/(1 + exp(50*(x1 - 1.8*0.3)))  ; x1 refers to the value of coord1
pull-coord2-init         = 1      ; this restrains the distance to having the contact
pull-coord2-k            = 100

The second example is an average of two distances:

pull                     = yes
pull-ngroups             = 4
pull-ncoords             = 3

pull-group1-name         = groupA
pull-group2-name         = groupB
pull-group3-name         = groupC
pull-group4-name         = groupD

pull-coord1-type         = umbrella
pull-coord1-geometry     = distance
pull-coord1-groups       = 1 2
pull-coord1-dim          = Y Y Y
pull-coord1-k            = 0      ; avoid forces working directly on this distance

pull-coord2-type         = umbrella
pull-coord2-geometry     = distance
pull-coord2-groups       = 3 4
pull-coord2-dim          = Y Y Y
pull-coord2-k            = 0      ; avoid forces working directly on this distance

pull-coord3-type         = umbrella
pull-coord3-geometry     = transformation
pull-coord3-expression   = 0.5*(x1 + x2)  ; x1 and x2 refer to the value of coord1 and coord2
pull-coord3-init         = 0.8    ; restrains the average distance to 0.8 nm
pull-coord3-k            = 1000

Limitations

There is one theoretical limitation: strictly speaking, constraint forces can only be calculated between groups that are not connected by constraints to the rest of the system. If a group contains part of a molecule of which the bond lengths are constrained, the pull constraint and LINCS or SHAKE bond constraint algorithms should be iterated simultaneously. This is not done in GROMACS. This means that for simulations with constraints = all-bonds in the mdp file pulling is, strictly speaking, limited to whole molecules or groups of molecules. In some cases this limitation can be avoided by using the free energy code, see sec. Calculating a PMF using the free-energy code. In practice, the errors caused by not iterating the two constraint algorithms can be negligible when the pull group consists of a large amount of atoms and/or the pull force is small. In such cases, the constraint correction displacement of the pull group is small compared to the bond lengths.