Figure 4.2: Flowchart for swarm MD calculation. A large munber of processors independently perform molecular dynamics (MD) calculations.
Furthermore, the crucial step to this process is that database of states is shared by all processors.
4.4 State Constrained MD and Programmed State Constrained MD
Adaptive sampling methods177,204that seek rare-configurations so that new MD tra- jectories can be seeded from such configurations, are generally used for calculating thermodynamic properties. Relevant kinetic information can be gained by efficiently extending the MSM validity time. If state S is poorly sampled in a dynamical tra- jectory even though it is relevant, one will require a longer trajectory with the hope that at some point enough transitions from the state will be sampled. Such situa- tions can be tackled usingstate-constrained MD (SC-MD) calculations. In SC-MD calculations, one performs MD in state S while checking for a transition at regular intervals. Once a transition is detected, the MD calculation is stopped, the waiting
4.4 State Constrained MD and Programmed State Constrained MD
Figure 4.3: Flowchart for a state-constrained calculation. The overall steps are similar to a swarm calculation (see fig. 4.2), except that the system is returned to a chosen state S each time an escape is found.
time and final state are noted. A fresh independent MD calculation is seeded fromS such that the system is in thermal equilibrium. The flowchart of SC-MD calculation is shown in Fig. 4.3. More transitions from S are sought. This prevents the system from freely diffusing over the potential energy landscape and confines it to a partic- ular state for the purpose of detecting kinetic pathways from that state, calculating the rates, and lowering leakage flux of state S. Core- and full-network models (Eqs.
3.13 and 3.12) can be constructed efficiently with programmed state constrained MD (PSC-MD) by automatically targeting states with the largest leakage flux and performing SC-MD calculations in those states. The flowchart of PSC-MD method is given in Fig. 4.4.
PSC-MD method provides a way to increase the validity time of MSM by lowering the largest contribution to the leakage ratekleak systematically. In this approach, the first step is to calculate the validity time (Eq. 3.18) of an existing MSM constructed from swarm MD calculations starting from initial state S by solving core-(and full-)
4.4 State Constrained MD and Programmed State Constrained MD
Figure 4.4: Flow chart for programmed state constrained MD (PSC-MD). Note that here MD can be replaced with another dynamical method.
network models (as in Eq. 3.12). Next, the leakage flux (FS in Eq. 3.11) from all the core states to the missing states and the periphery states at validity time (τV) are calculated separately, and the missing pathway or the periphery state associated with the largest leakage flux is identified. If a periphery state S0 is found to be accountable for maximum leakage flux from core states then, a state-constrained simulation is performed in periphery state S0 until sufficient number of transitions is obtained from state S0 and state S0 will be considered as a new core state in core-network model. In the other case, if the largest leakage flux is due to a miss- ing pathway from a core state S then, a state constrained is carried out in state S to get desirable number of transitions from state S. Thereafter, next cycle of the process begins with validity time calculation of new core-(and-full) network model.
As a consequence, one-by-one the periphery states are included within core-network model. Thus, the validity time is extended systematically by reducing the leakage flux to the periphery/missing states from the core-states.
4.4 State Constrained MD and Programmed State Constrained MD Let us consider the case where the largest leakage is due to a kinetic pathway from a core state S to a peripheral state S0. The MD time spent in state S0 on average in a trajectory of duration τM D, which is given by,
tS0 =πavg
S0 τM D (4.4)
can be made significantly large to ensure that the validity time is not lowered by inclusion of S0 as core state. Here, πavg
S0 = τM D−1
tM D
R
0
πS0(t)dt; the occupancy of the system in state S0 on average. Rarely-visited states with small values ofπS0 present a challenge as they require long MD trajectories. Suppose we require that the fastest rate kf from stateS0 is selected at leastm times in the MD calculation, we conclude that on average kfπavg
S0 τM D = m, that is, extending the validity time is limited by the rate kf which determines the duration of MD as,
τM D = m kfπavg
S0
. (4.5)
where mdenotes the number of transitions needed for estimating the rate with rea- sonable statistical accuracy. This issue can be tackled using SC-MD calculations where an MD calculation is started in the state S0 and it is returned to the same state once an escape is detected. The duration of MD required is τM D = m/kf. The computational speed-up over regular MD given by 1/πavg
S0 can be phenomenal in most biomolecular systems due to the typically small values of πavg
S0 .
The PSC-MD scheme usually introduces many periphery states in the network model as it is used to build the MSM in patches. Occasionally, states are chosen from one part of the network for SC-MD calculations, and later, another part may be selected based on the calculated leakage flux. When the largest leakage is from core stateSto a periphery stateS0, state-constrained MD is performed inS0 untilS0 becomes a core state. In summary, in PSC-MD method, one identifies configurations where kinetic information appears inadequate as starting states for subsequent MD.
While ad−hoc SC-MD similar to such approaches can result in some improvement in computational efficiency, an impressive speedup is expected from PSC-MD where the objective is to determine the core/peripheral state that currently has the largest leakage rate and perform MD in that state for a chosen duration. In this way PSC-MD extends the validity time in an efficient manner. Hence, PSC-MD is more efficient than regular MD.