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Rigorous proof of convergence of MD simulations cannot be performed [31]. Efforts can be made, however, to ensure that various properties calculated from a simulation have reached satisfactory levels of convergence. First, the simulations should be analyzed to determine if global (1) energetic and (2) structural properties have stabilized. Energetic stability is typically investigated by monitoring the potential energy versus time. During most simulations the potential energy initially relaxes, after which it fluctuates around a constant value for the remainder of the simulation. In NVE simulations the ratio of RMS fluctuations of the potential energy and kinetic energy can be monitored, with RMSPE/ RMSke = 0.01 indicating proper energy conservation. Analysis of structural properties versus time (e.g., RMS difference with respect to the starting structure) will also typically show an initial relaxation followed by fluctuations around an average value for the remainder of the simulation. In certain cases these fluctuations may be relatively large, indicating the sampling of alternative conformations that may be biologically relevant [16]. From the analysis of structural and energetic properties versus time, the initial portion of the production simulation during which relaxation of these properties occurs is discarded. Additional convergence tests are performed only on the remainder of the production simulation.
Tests of convergence on the remaining portion of the simulation can most readily be performed by calculating the desired property (i.e., the property that is of most interest to the simulator) over different simulation time lengths and monitoring the change in the average value as a function of simulation time. Convergence is indicated by the lack of significant change in the average value of the property as the simulation time increases. Another method involves separating the total trajectory into independent blocks (e.g., a 500 ps production simulation may be separated into five 100 ps blocks), calculating average values for each block, and comparing the average values from the individual blocks. These block averages can also be used to calculate overall averages and standard errors for individual properties [151]. This is an excellent method for obtaining the statistical significance of results from a simulation.
For an entire MD simulation, as well as for separate blocks from a simulation, determination of convergence of a property can be most rigorously carried out by calculating the time series of a property and determining its autocorrelation function and accompanying relaxation time. For adequate convergence the total MD simulation time or the block time should be approximately four times as long as the relaxation time. This test is also appropriate for determining the length of blocks that can be considered independent. It should be emphasized that the relaxation time and, accordingly, the amount of simulation time required for convergence are dependent on the property being investigated. Thus, the total required simulation time is dependent on the type of information that is to be obtained from a simulation. If it is determined that the properties of interest have not converged, then the production simulation should be extended, as indicated by loop II of Figure 1.
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