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WIREs Comput Mol Sci
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Modeling molecular kinetics with Milestoning

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Abstract Time scales are of paramount importance in biology. Living systems exploit variations in time scales to aim processes in desired directions. The network of biochemical reactions shapes cellular responses and metabolism. Enzymes speed up the rate of reactions and molecular machines carry on cellular tasks. Significant efforts are invested in studying dynamics of biophysical processes and understanding their mechanisms. Experiments provide important clues, but the data can be sparse. Atomically detailed Molecular Dynamics simulations hold the promise of comprehensive pictures of these events. A challenge for simulations is the wide range of time scales in biology, from femtoseconds to hours. Straightforward Molecular Dynamics simulations of kinetics are typically bound by microseconds and unable to probe slower processes. For example, membrane permeation by a small molecule can take hours, slow events in protein folding, seconds, and enzymatic reactions, hundreds of milliseconds. To address these challenges, we introduce the method of Milestoning. Milestoning is a theory and an algorithm to enhance the sampling of kinetic events using computer simulations. Milestoning exploits short trajectories between interfaces of cells in coarse space. Short trajectories are efficient to compute and provide a sequence of approximations that converge to the exact solution. The theory is discussed, and several examples illustrate the use of Milestoning. We consider an enzymatic reaction, peptide permeation through a phospholipid membrane, and the translocation of the lethal factor through the Anthrax channel. The high versatility of Milestoning suggests that it is a useful tool for investigations of complex biomolecular reactions. This article is categorized under: Structure and Mechanism > Computational Biochemistry and Biophysics Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics
The logarithm of the mean first passage time for the reaction of HIV RT with nucleotide substrate as a function of the iteration number. Note that the value fluctuates near an average suggesting that the Milestoning iterations converge and the remaining source of errors is statistical. Note also that the calculation of a time scale of ~100 milliseconds is conducted with accumulated length of short trajectories lower than a microsecond or about a million time shorter. We conducted this Milestoning calculation at a profound reduction in computational resources compared to straightforward MD simulation. Umbrella sampling and free energy perturbation51 determine free energy profiles efficiently. However, these techniques do not provide time scales. We use approximations (e.g., transition state theory52) that are not always valid to estimate rates and the mean first passage time (MFPT) from the free energy profile. Milestoning provides the MFPT directly
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The free energy profile for the reaction of HIV RT with a nucleotide substrate as a function of the milestone numbers. We show results from multiple Milestoning iterations (Equation (20)). The difference between a single iteration of Milestoning and exact Milestoning is significant but is probably within typical error bars. (Adapted with permission from Reference 37)
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The active site of HIV reverse transcriptase with a nucleotide substrate and two magnesium ions (adapted with permission from Reference 37). r1,…,r10 mark suggestions for coarse variables. We tried several reaction coordinates, however, a simple choice of a one‐dimensional reaction coordinate: r2–r1 works quite well. r1 is the distance of the phosphate to the oxygen with which it forms a new bond, and r2 is the distance along a phosphate oxygen bond to be broken
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The structure of HIV reverse transcriptase bound to DNA and the substrate. Gray ribbons follow the backbone of the protein while the DNA strands are in green and orange sticks. Also shown are two magnesium ions at the active site (green spheres) and the nucleotide that reacts with the DNA (yellow sticks). The protein is a model based on the Protein Data Bank,44 Structure 1RTD.45 The model includes explicit water solvation (not shown). See Reference 46 for more details
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A sketch of a toy Milestoning problem. There are four states presented by blue circles: A reactant (R) is on the left and the product (P) is on the right. There are two intermediate states between the reactant and the product. There are three milestones shown by red lines separating the four states. The probability of transition from the milestone at the center to the milestone on the right is 1 − K. The product is absorbing which means that the transition probability from the product's milestone back to the milestone on the left is zero
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The last milestone a trajectory crosses defines its state. Here the blue trajectory is in state m1 and the green trajectory is in state m2. The blue trajectory is found anywhere in the two cells divided by the blue milestone. Both trajectories can occupy the same point in space (red circle), and they still belong to different states
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An illustration of the derivation of the mean first passage time equation. A trajectory in a two‐dimensional coarse space from reactant to product is split into two segments. In the first segment (black) the trajectory from state i progresses to a nearby milestone j. In the second segment it continues from milestone j to the product (while crossing other milestones along the way)
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The mean first passage time for the N terminal translocation from/to the channel entry (negative values) to/from the ϕ clamp of the anthrax channel (zero position). The top panel is for the forward mean first passage time and the lower panel for the backward reaction. Four colored curves differ in their protonation states. The light blue curve has all the histidine protonated. The dark blue line has all residues (including acids) protonated. The black curve is using a prediction of protonation states according to an empirical pKa algorithm PROPKA60 at each milestone. The red curve has the least positive charge, having all the histidine residues neutral. The dark blue curve is most consistent with experimental finding suggesting nanoseconds of entry time and milliseconds to exit. Reproduced with permission from Reference 1
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The mapping of a trajectory along a coarse variable Q. A trajectory is in state mi if the last milestone crossed is mi. Milestones are indicated in the figure by thin horizontal blue lines. The red line indicates the trajectory's state in a discrete Milestoning space while the black line is the trajectory in continuous space. Between 0 and t1 the trajectory is in the m1 state, and between t6 and t7 it is in the m3 state, and so on. The time of transition between states is known precisely (the time of the crossing). At other times the Milestoning discrete space does not provide the location of the trajectory in continuous space. For example, if the last milestone crossed is m4, the trajectory is in the state m4. However, its Q value can be anywhere in the interval between m3 and m5 (not including the two end points)
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A schematic representation of a coarse Milestoning space in two dimensions. The Milestoning space is partitioned into cells by milestones (thin blue lines), which are boundaries separating cells. The two coordinates Q1 and Q2 represent the coarse space and Y represents the rest of the coordinates. Note that, no partitions are used for the Y coordinates. Two milestones are shown explicitly (mi and mj) as green lines. Also shown is a trajectory in full space (a black curved line). A fragment of the long trajectory between two milestones is sketched in red. The red segment is “touching” the green line of milestone mj
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A schematic drawing of a long trajectory repeatedly switching between States A and B. The states are marked on the vertical axis, and the time is displayed on the horizontal line. Also marked are the entry and exit times into and from the states
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A sketch of the coarse variables that model peptide permeation. The peptide WKW carries two positive charges, one on the lysine and another on the N terminal. We consider the two distances from each of the charges to the nearest phospholipid head on each of the leaflets giving us four coarse variables (adapted with permission from Reference 55)
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The electric field induced by the anthrax channel computed from 4.5 ns atomically detailed simulations. The endosome is on the left, and the cytosol is on the right. The figure is in the same orientation as the atomistic image in the visual abstract
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The free energy profile for the permeation of the peptide WKW through the phospholipid membrane. Note the metastable state D when the peptide crosses the membrane (Figure 12)
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A cartoon and molecular models of a peptide permeating through membrane studied with the Milestoning method. On the right we provide a molecular picture that includes the peptide and aqueous solution. For clarity, the phospholipid molecules were removed from the image. The peptide carries two positive charges that make it challenging to pass the hydrophobic core of the membrane. In the first step (a) the peptide approaches the membrane surface. In (b) it is embedded inside the membrane surface. (c) is a critical step that includes significant membrane distortions. The distrotions allow for deep penetration of one of the charges to the membrane. In step (d) the membrane distortion is reduced. The two charges hang at opposite leaflets. Step (e) and (f) are roughly the mirrors of steps (c) and (b), respectively. The figure is adapted from Reference 55 with permission
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Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics
Molecular and Statistical Mechanics > Molecular Dynamics and Monte-Carlo Methods
Structure and Mechanism > Computational Biochemistry and Biophysics

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