Methods for Constrained and Unconstrained Simulations

Several techniques are available to calculate dJf /dQ. Ciccotti and coworkers [17, 19-27] have developed a technique, called blue-moon ensemble method or the method of constraints, in which a simulation is performed with fixed at some value. This can be realized by applying an external force, the constraint force, which prevents from changing. From the statistics of this constraint force it is possible to 

An advantage of this technique is that it allows for getting as many sample points as needed at each location along the interval of interest. In particular, it is possible to obtain very good statistics even in transition regions which are rarely visited otherwise. This leads in general to an efficient calculation and small statistical errors. Nevertheless, despite its many successes, this method has some difficulties. First, the system needs to be prepared such that has the desired value (at which dA/d needs to be computed) and an equilibration run needs to be performed at this value of . Second, it is not always obvious to determine how many quadrature points are needed to calculate the integral (deff j 9 ) d . 

Finally, it may be difficult to sample all the relevant conformations of the system with fixed. This problem is more subtle, but potentially more serious, as illustrated by Fig. 4.2. Several distinct pathways may exist between A and B. It is usually relatively easy for the molecule to enter one pathway or the other while the system is close to A or B. However, in the middle of the pathway, it may be very difficult to switch to another pathway. This means that, if we start a simulation with fixed inside one of the pathway, it is very unlikely that the system will ever cross to explore conformations associated with another pathway. Even if it does, this procedure will likely lead to large statistical errors as the rate-limiting process becomes the transition rate between pathways inside the set = constant. 

These difficulties can be circumvented by using the adaptive biasing force (ABF) method of Darve, Pohorille, and coworkers [18, 28, 29], which is based on unconstrained molecular dynamics simulations. This is a very efficient approach which begins by establishing a simple formula to calculate d,4/d from regular molecular dynamics in which is not constrained. This derivative represents the mean force acting on . Therefore if we remove this force from the system we obtain 

Importantly, in contrast to constrained simulations, the system is allowed to evolve freely and in particular to explore the various pathways connecting A and B, see Fig. 4.2. This is one of the reasons why ABF can converge much faster than the method of constraints. 

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