# MOLDY Algorithm Development

This website describes our investigation of the Omelyan integrator [1]. The Omelyan integrator is an integration technique which is designed to be more stable than the more commonly used Verlet and Leapfrog integrators. Because of its stability it should allow longer timesteps to be used in molecular dynamics simulations, reducing the computational cost of long timescale MD.

In order to test the Omelyan integrator a testbed system was created. We will detail the results obtained from this testbed under simple forces. The results show that the Omelyan is indeed more stable in these scenarios. In order to see how the Omelyan integrator performs on more practical problems an MD code, Protomol [2], was modified to include the Omelyan integrator. We describe the results obtained from Protomol first using a very small and then a more representative biomolecular simulation. We also analyse the runs to ensure that the Omelyan integrator is exploring the correct configuration space. The results show that the Omelyan integrator is an important algorithm for molecular dynamics simulation and warrants further exploitation.

#### The Omelyan Integrator

In this section we shall briefly introduce the Omelyan integrator in terms of its implementation, we shall not discuss the derivation of this algorithm, details of which can be found in [1]. The Omelyan 2-step integrator [1] is a relatively recent development and is summarised in equation 1 (reproduced from [1]). For comparison a Verlet Velocity algorithm using similar notation is shown in equation 2.

Where the optimal value of is given as:

#### Integrator Testbed

As previously mentioned the testbed tool is designed to simulate small numbers of bodies under simple forces using a variety of integration algorithms. Our purpose here is partly to gain an understanding of the techniques involved, but also to give us an initial indication of the improvements that can be gained with the Omelyan integrator. At present the following integration algorithms have been added to the testbed system: Euler, Leapfrog, Verlet, Verlet Velocity, Omelyan and Nomotion.

The Nomotion integrator is used when you require a body to be fixed within a simulation. The system is designed to allow different force and integrator types to be mixed within the testbed system.

Figure 1: Integrator testbed OpenGL rendering showing system time, and energy.

#### A Comparison of Integration Methods for a Simple Harmonic System

As a first step in comparing how effective the integrators implemented in the testbed system were, tests were carried out using a simple harmonic force. The energy of the system was measured and the standard deviation of the energy over the course of the simulation was measured. Figure 2 shows, the standard deviation of simulations at differing timestep. The Omelyan algorithm far out performed all other integrators in this scenario.

Figure 2: Standard deviation of energies against varying timestep for a simple harmonic system (composed of one body). The simulation was run for 5000 timesteps.

#### Extending the Analysis to Larger Systems

Results similar to those shown for the Simple Harmonic force have been obtained for the integration of systems using Inverse Square and Biomolecular force somulations. For Biomolecular simulations a 30,000 atom simulation of DHFR in water was used (the starting structure is shown in figure 3). The Inverse Square and Biomolecular simulations show a similar improvement to those obtained with using the simple harmonic force. We hope to publish these results in the near future.

Figure 3: The DHFR starting structure used in our simulations, composed of 31 168 atoms.

#### References

[1] I.P. Omelyan, I.M. Mryglod and R. Folk. Optimized verlet-like algorithms for molecular dynamics simulations. Phys. Rev. E, 65(5):056706, May 2002

[2] Protomol website, http://protomol.sourceforge.net