SDEL

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SDEL Trajectories Approach the Steepest Descent Path... 

IV. SDEL TRAJECTORIES APPROACH THE STEEPEST DESCENT PATH... 

The question we address in the present section is the asymptotic behavior of a trajectory when the step becomes quite large. We are unable to answer that question in a useful way for the SDET algorithm however, an intriguing result is obtained for SDEL. In the present section we will show a connection between SDEL trajectories and the usual definition of the reaction coordinate, the steepest descent path (SDP). 

In the SDEL formulation the most probable trajectory is the solution of the following finite difference formula [Eq. (33)]... 

Figure 7. Bond energy distribution along the path for four different sizes of length steps. The data are extracted from SDEL calculations of valine dipeptide. Note the significant reduction of bond energies (filtering) as the step size increases.

Figure 8. Equatorial to axial transition for glycine dipeptide using increasing length size with SDEL. Note that the spatial distribution of configurations along the trajectories remains similar at all resolutions. The main difference for a refined trajectory is the significantly larger density of configurations near minima associated with incubation periods.

In this review a variant of the SDET algorithm is summarized. In this more recent formulation called SDEL (for stochastic difference equation in length) the trajectory is parameterized as a function of its arc length and a unique path is obtained connecting the two boundary conformations [45,54,55]. In the next section, we will describe the algorithm and details of its numerical implementation to obtain conformational changes of peptides and folding mechanisms of protein systems. 

As it was mentioned previously the use of straightforward MD simulations to study slow processes is limited by the size of the time step that is required to obtain a stable trajectory. The SDEL algorithm allows the computation of atomically detailed trajectories connecting two known conformations of the molecule over long time scales. In contrast to normal MD, step sizes can be easily increased by factors of thousands (or more) without significative changes in many properties of the trajectory. The trade-off is that trajectories obtained with such a large step size are approximate molecular motions that occur on a scale shorter than the... 

The Onsager-Machlup action mefhodology has a disadvanfage the need to know the total time of fhe frajecfory in advance. Also, low resolution trajectories do not approach a physical limit when the step size increases, in contrast to SDEL as we are going to see below. 

Similarly to the Onsager-Machlup action method the SDEL algorithm is based on the classical action. However, in this case the starting point is the action S parameterized according to the length of the trajectory ... 

Equation (4) generates a path in which the force is minimized in all directions but the direction of the path. This is one of the definitions of the Steepest Descent Path (SDP) [57]. This suggests that SDEL provides a physically meaningful result even at low resolution (large step sizes). 

The SDEL algorithm has been efficiently parallelized using MPI libraries. In the parallelization scheme each node of a cluster of computers calculates the energy and derivatives for a particular segment of the path [60]. Inter-node communication is not heavy and the computation scales favorable with cluster size. 

There are several advantages of SDEL compared to the methods mentioned previously ... 

The SDEL formulation is very general. It is not limited to processes with large energy barriers, single barriers or with exponential kinetics. 

The length formulation makes it difficult to estimate the timescale of the process. Because of this limitation SDEL can provide information about the relative sequence of events but not absolute times. 

A simpler way to include the effect of water interactions entails the extraction of configurations snapshots from SDEL trajectories and to immerse each one of these structures into a box with explicit water molecules. Then MD simulations can be performed until equilibration is reached [61]. A variant of this procedure is to use constant pressure and temperature MD to extract volume and enthalpy changes during the molecular event. These properties are probed by photothermal techniques. Space and Larsen demonstrated the application of this algorithm for a small f sheet peptide [62]. 

SDEL was also used to study the coil-helix transition of an alanine-rich peptide [68], the conformational transition of sugar puckering in deoxyadenosine [69], polymerase P [70], and the B-Z DNA transition [71,72]. The coil to helix study [68] demonstrated several properties of SDEL trajectories, like the filtering of high frequency modes and the preservation of thermodynamic properties from slow degrees of freedom when the trajectory resolution is decreased. 

The SDEL algorithm has also been applied to more complicated systems, such as the wild type human Cu, Zn superoxide dismutase (SOD) dimer. SOD is a 153-residue, homodimeric, anti-oxidant enzyme that dismutates superoxide ion to hydrogen peroxide and oxygen [87]. It is an eight-strand, flattened, beta-barrel protein with one copper and one zinc ion per monomer [88]. There are over 100... 

These applications demonstrate the potential of the SDEL algorithm as a tool to study conformational dynamics of large molecular systems such as peptides and proteins. This is the only algorithm from the methods discussed in the introduc-... 

The SDEL algorithm allows the computation of atomically detailed trajectories connecting two known conformations of the molecule over long time scales. In contrast to normal and MTS molecular dynamics algorithms, step sizes can be increased easily by two or three orders of magnitude without... 

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