# Calculating Iso-Committor Surfaces as Optimal Reaction Coordinates with Milestoning

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Committor

#### 2.2. Milestoning

#### 2.3. The Committor Function and Milestoning

^{−3}) to ${q}^{(10)}=\left(0,0,0,0.333\right)$. The elements of the stationary vector are all zero excluding the product entry.

^{−3}) to:

## 3. Results

#### 3.1. Example 1: A Transition between Metastable States on the Mueller Model Potential

**K**. We used power iterations of the transition matrix to determine the committor. After 3000 iterations we verified convergence by requiring that the sum of row elements that should be exactly zero, is smaller than 0.05, i.e., we require

#### 3.2. Example 2: A Conformational Transition in Alanine Tri-Peptide

^{−6}.

#### 3.3. Example 3: Exchanging Cholesterol Molecules between Aggregates Embedded in Dimyristoylphosphatidylcholine (DMPC) Membrane

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**A schematic drawing of a Milestoning trajectory. We initiate trajectories at milestone $i$ according to a known flux distribution function at the interface: ${q}_{i}$. We terminate the trajectories when they hit for the first time a different milestone $j$ ($j\ne i$). An example of a single trajectory is displayed. The pathway starts from the red milestone and terminates at a divider $j$. From multiple trajectories we estimate the matrix ${K}_{ij}$, which is the probability that a path initiated at milestone $i$ will terminate at milestone $j$, ${K}_{ij}\cong {n}_{ij}/{n}_{i}$ where ${n}_{i}$ is the number of trajectories initiated at milestone $i$ and ${n}_{ij}$ the number of trajectories initiated at $i$ that terminated at $j$. Note that ${n}_{i}={\displaystyle \sum _{j}{n}_{ij}}$.

**Figure 2.**The iso-committor surfaces for the Mueller potential computed for overdamped Langevin dynamics. See text for more details.

**Figure 3.**A schematic drawing of a blocked trialanine. We also show the three dihedral angles that are considered for the coarse variables: ${\psi}_{1},{\psi}_{2}$ and ${\psi}_{3}$. They are shown as rotations around ${C}_{\alpha}-CO$ bonds. In the simulations it turns out that ${\psi}_{2}$ has a restricted distribution (see also Figure 4) and therefore was not considered as a coarse variable in the calculation described below.

**Figure 4.**The probability density of the dihedral angle ${\psi}_{2}$ of alanine tripeptide. The values of the torsion were extracted from a trajectory and binned. The distribution consists of a single maximum near 160 degrees, and another metastable state near 50 degrees, which is barely populated.

**Figure 5.**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 $\left({\psi}_{1},{\psi}_{3}\right)$ of tri-alanine. The green arrowed line is a schematic reaction coordinate to guide the eye. It starts at the reactant and follows a steepest descent direction in committor space to the product. The iso-committor surfaces are determined in the space of coarse variables and are therefore approximate. See text for more details.

**Figure 6.**(

**Left**): A top view of the membrane box. Cholesterol molecules are illustrated with colored spheres and the DMPC molecules are shown with gray lines. The two cholesterol clusters of interest are shown in red and yellow with an orange cholesterol molecule transitions between the two clusters. (

**Right**): A schematic representation of the coarse variables that determine the relative position of two cholesterol molecules 1 and 2. The dashed line shows the distance between their centers of mass and the green solid arrows represent the tilt vector for each cholesterol molecule. The tilt vectors are unit vectors along the line connecting atoms C3 and C17 of the cholesterol molecule. The tilt angle is defined as the angle between the tilt vectors. The plot was prepared with the VMD program [43].

**Figure 7.**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.

**Figure 8.**The committor values estimated from 20 trajectories starting from each of nine different milestones with C ~ 0.5 with different initial velocities.

**Table 1.**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 |

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**MDPI and ACS Style**

Elber, R.; Bello-Rivas, J.M.; Ma, P.; Cardenas, A.E.; Fathizadeh, A. Calculating Iso-Committor Surfaces as Optimal Reaction Coordinates with Milestoning. *Entropy* **2017**, *19*, 219.
https://doi.org/10.3390/e19050219

**AMA Style**

Elber R, Bello-Rivas JM, Ma P, Cardenas AE, Fathizadeh A. Calculating Iso-Committor Surfaces as Optimal Reaction Coordinates with Milestoning. *Entropy*. 2017; 19(5):219.
https://doi.org/10.3390/e19050219

**Chicago/Turabian Style**

Elber, Ron, Juan M. Bello-Rivas, Piao Ma, Alfredo E. Cardenas, and Arman Fathizadeh. 2017. "Calculating Iso-Committor Surfaces as Optimal Reaction Coordinates with Milestoning" *Entropy* 19, no. 5: 219.
https://doi.org/10.3390/e19050219