Reaction coordinates are vital tools for qualitative and quantitative analysis of molecular processes. They provide a simple picture of reaction progress and essential input for calculations of free energies and rates. Iso-committor surfaces are considered the optimal reaction coordinate. We present an algorithm to compute efficiently a sequence of isocommittor surfaces. These surfaces are considered an optimal reaction coordinate. The algorithm analyzes Milestoning results to determine the committor function. It requires only the transition probabilities between the milestones, and not transition times. We discuss the following numerical examples: (i) a transition in the Mueller potential; (ii) a conformational change of a solvated peptide; and (iii) cholesterol aggregation in membranes.

Determining a reaction coordinate (RC) or an order parameter as a starting point for computational chemistry calculations is a common approach to investigate kinetics. It offers a one-dimensional progress parameter for complex molecular processes in high dimensions. A RC can lead to more efficient computations of rates and simpler interpretations of the simulation results. How to choose an optimal reaction coordinate is a topic of considerable debate. There are two principal uses of a RC that motivate the search for an optimal reaction coordinate.

A RC offers efficient ways to study equilibrium and kinetic observables in complex systems by following the general direction of the coordinate. Methods like umbrella sampling [

A RC allows the projection of the dynamics onto a one-dimensional curve and therefore suggests a simple picture of the mechanism. Examples of methods that provides a one-dimensional view of the reaction progress include the Locally Updated Planes (LUP [

Both types of reaction coordinates are computed in continuous space, or on a network [

There are two main realizations of a reaction coordinate that connects the states of reactants and products. These views offer theoretical perspective but do not necessarily coincide with the uses mentioned above. The first view is that the reaction coordinate is a curve (a line) that links the reactant and product states. We denote this representation by RL (Reaction Line). The Steepest Descent Path (SDP), which is the line with a minimum energy barrier between the reactant and product states, is an example. Approaches such as LUP [

The second view of an RC is a progression of non-crossing surfaces from the reactant to the product basins. We denote this second representation by RS (Reaction Surfaces). If the system is of dimension

To avoid the problem of hypersurfaces that cross and therefore do not provide a unique reaction coordinate, the use of Voronoi tessellation was proposed [

In the next section, we define the committor and the iso-committor surfaces. These functions are computationally expensive and in practice approximations are used to estimate their value. We follow the discussion about the committor by a brief introduction of the method of Milestoning. We then show that the Milestoning algorithm allows for efficient calculations of iso-committor surfaces. The iso-committor surfaces we compute, go beyond the plane assumption [

Consider a system of two metastable states

For overdamped Langevin dynamics, a partial differential equation (PDE) determines the iso-committor surfaces [

Given the staggering costs of computing the exact committor surfaces, it is not a surprise that approximate approaches to determine this function are used. Among these approaches are the use of planes in a specific orientation with respect to the SDP curve [

In the present manuscript, we suggest an alternative calculation of the committor function using Milestoning with several coarse variables [

We consider a system with two metastable states, a reactant state

A critical operator of the Milestoning theory is the matrix

The kinetic of the complete process is determined by the distribution of the first passage times from reactants to products. The complete distribution is hard to determine computationally, and typically only the first moment or the mean first passage time is reported. Milestoning, however, makes it possible to determine all the moments of the distribution from local information. The local information is the moments of the milestones’ life times, or the time moments of

To use Milestoning the following two conditions must be met. First, the milestones must be able to differentiate between reactants and products, i.e., there is a subset of milestones that bounds the reactants and another subset of milestones that bound the products and the two sets have zero overlap. Second, there is a sequence of transitions between milestones connecting the reactant and product with non-zero probability. In other words we require that the Mean First Passage Time (MFPT) between the reactant and the product is finite.

A fundamental Milestoning equation determines the stationary flux

We use bold-faced symbols to denote vectors and matrices. The length of the vector

If we consider the limit of long time

According to Equation (1) the stationary flux is the left eigenvector of the kernel with an eigenvalue one. The matrix

In the exact variant of Milestoning [

Equation (3) is then plugged in Equation (1) to provide an equation for the milestone weights—

The lower formula in Equation (4) is easy to solve since the number of milestones and not the distribution within the milestone determines the dimensionality of the matrix. The complication using Equation (4) is that, in general,

The superscript denotes the iteration number. For example,

In the present manuscript, we use Milestoning with a single iteration. However, the procedure described here is directly applicable also for exact Milestoning [

Instead of running many short trajectories between milestones it is also possible to use Milestoning as an analysis tool and to extract the necessary transitions from a long trajectory. A long trajectory, with many reactive events at equilibrium is an exact representation of the dynamics and it can be used to study the committor function and the time scales of the reaction. Hence, the distribution

When the milestones are iso-committors or so-called optimal milestones [

The theory and algorithm that we propose below follow closely the mathematical approach described in reference [

The current formulation uses only a single function that is readily available in Milestoning simulations: the kernel. The kernel,

In the Milestoning context, we define the committor function as “Given that we start a trajectory at milestone

The committor

First, we consider the calculation of the flux by power iteration ^{−3}) to

The committor function is obtained by similar power iterations of the matrix

Interestingly the matrix converged after only 4 iterations (with errors less than 10^{−3}) to:

Equation (7) is also an iterative adjustment of a vector, consider:

Equation (8) defines iterations of matrix-vector multiplications that converge to the same desired result as Equation (7) in the limit of

We can use Equation (8) as a base for an iterative algorithm to determine

The dimensionality of the matrix is the number of milestones,

The second algorithm to determine the committors from a Milestoning kernel is based on solving linear equations. The advantage of the linear formulation is that many algorithms are available to address this type of linear problems. Consider trajectories that start at milestone

If we insert the boundary conditions

We can write more compactly Equation (10) as

Equation (11) is a straightforward linear equation for the committor coefficients

The solution of Equation (11) is the same as the solution of Equation (8). We show that by substituting in (11) the solution from Equation (8)

The last equality is a result of the convergence of the power iterations, which proves that the solution of Equation (8) solves Equation (11). Since the solution is unique we demonstrate the equivalence of the two approaches.

For an illustration, we are using a long time trajectory to determine the kernel in the examples provided in this manuscript. The long path is decomposed to trajectory fragments between milestones to obtain transition statistics. Instead of a long trajectory, it is possible to use an ensemble of short trajectories with initial conditions sampled at the milestones [

We provide three examples of calculations of iso-committor “surfaces”. In the discrete case the iso-committor is a set of points, not necessarily a surface. The first example is a “toy” problem, a transition between two metastable states on the Mueller potential [

A milestone is a surface in the reduced space of coarse variables. The choice of the surfaces is flexible. For example, it can be a boundary between Voronoi cells, as proposed by Vanden Eijnden [

In the three examples, we used a long time trajectory and Milestoning analysis to estimate the transition kernel. Once the transition kernel was determined, we proceeded with the calculations of the committor as described in previous sections. To determine events of milestone crossing we use centers of Voronoi cells [

At most one event of crossing should be detected between sequential time steps. Crossing corners of cells can be interpreted as events of multiple transitions and are a concern. There is a need to use a small time step and frequently save the values of the coarse variables to avoid more than one crossing between two sequential configurations. In the first example, which is a Milestoning calculation on a fine mesh, transitions through corners are detected. We therefore add milestones to account for “corner” transitions.

In

The iso committor surfaces in

Here we consider a transition between metastable states of solvated alanine tri-peptide (

The trajectory length was 101.86 nanoseconds, and it samples configurations from the NVE ensemble. The box size was ^{−6}.

A natural choice for coarse variables to describe backbone transitions in peptide chains are the

For each dihedral angle, we define 12 milestones covering the range between −180 to 180 degrees. The program Miles analyzed the trajectory [

The graph makes it possible to visualize a preferred pathway, which is going from the reactants to the product (green line in

The prediction of the committor values at the milestone is approximate, since the committor is in principle a function of the complete phase space vector and not only of the coarse variables. To test out choice of coarse variables we launch trajectories from milestones with commitment predicted close to 0.5. We sample 100 trajectories for each milestone from the canonical distribution and integrate them in time to termination either at the reactant or the product. The result are summarized in

Biological membranes frequently include cholesterols in addition to the phospholipid components. The fraction of these molecules embedded in DMPC membrane may exceed 50 percent by weight. It is therefore of considerable interest to model thermodynamic and kinetic of the heterogeneous membrane-cholesterol environment. Here we consider the transition of a cholesterol molecule between two clusters. An example of this phenomenon is shown in

The simulation was conducted in the DMPC membrane with 40 percent cholesterol at 300 K using the MOIL program [

To examine the accuracy of the calculated committor, we considered 9 milestones that have committor values close to 0.5 (0.47 <

We derived and illustrated a novel algorithm to determine committor functions. The set of iso-committor surfaces with values

The limitation of the algorithm is the computational costs, which is proportional to the number of active milestones. If the number of coarse variables is large, the number of milestones is likely to be large as well. It is growing exponentially with the number of coarse variables in the worst-case scenario.

This research was supported in part by grants from NIH GM059796 and GM111364 and from the Welch foundation F-1896. It was motivated by extensive discussions on the committor function during a weeklong conference in Leiden on “Reaction Coordinates from Molecular Trajectories”.

Ron Elber developed the basic theory and formulated the power iteration algorithm. Juan M. Bello-Rivas refined the algorithm and formulated the linear equation for the committor. Piao Ma computed the committor function for the Mueller potential, Juan M. Bello-Rivas conducted the simulation and analysis of the peptide. Alfredo E. Cardenas and Arman Fathizadeh performed the simulations and analysis of the membrane system.

The authors declare no conflict of interest.

A schematic drawing of a Milestoning trajectory. We initiate trajectories at milestone

The iso-committor surfaces for the Mueller potential computed for overdamped Langevin dynamics. See text for more details.

A schematic drawing of a blocked trialanine. We also show the three dihedral angles that are considered for the coarse variables:

The probability density of the dihedral angle

The committor function, colored-coded according to the bar on the right, is determined on a square grid and shown on the canonical probability density of two coarse variables

(

The committor function for exchanging a molecule between two adjacent clusters of cholesterol molecules embedded in a DMPC membrane. The iso-committors are shown as a function of the distance between the centers of masses of the molecules and their relative tilting angle. The gray lines are equipotential curves of the free energy landscape. The committor function (color coded) was computed on the grid by solving the linear equation (second algorithm, Equation (11)). Note that the committor function depends primarily on the distance and less on the orientation. The iso-committor surfaces are determined in the space of coarse variables and are therefore approximate.

The committor values estimated from 20 trajectories starting from each of nine different milestones with C ~ 0.5 with different initial velocities.

The committor values of 15 milestones are reported. The second column was obtained by running 100 trajectories from initial points sampled according to the canonical distribution until termination at milestones leading to the reactant or product. The values reported in the third column are the values of the committor obtained from Milestoning. The Milestoning index in the first column is given for ease of reproducibility of the data.

Milestone Index | Committor Value by Trajectories | Committor Value by Milestoning |
---|---|---|

1741 | 0.49 | 0.45 |

2611 | 0.63 | 0.60 |

2767 | 0.46 | 0.44 |

3481 | 0.65 | 0.56 |

3492 | 0.51 | 0.40 |

4496 | 0.48 | 0.41 |

4507 | 0.58 | 0.50 |

5232 | 0.70 | 0.42 |

6236 | 0.69 | 0.45 |

6247 | 0.71 | 0.52 |

6972 | 0.59 | 0.43 |

7976 | 0.63 | 0.45 |

7987 | 0.76 | 0.52 |

8132 | 0.55 | 0.40 |

8711 | 0.56 | 0.40 |