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\documentclass[11pt]{article} |
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\chapter{\label{chapt:oopse}OOPSE: AN OPEN SOURCE OBJECT-ORIENTED PARALLEL SIMULATION ENGINE FOR MOLECULAR DYNAMICS} |
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\begin{document} |
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\lstset{language=C,float,frame=tblr,frameround=tttt} |
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\renewcommand{\lstlistingname}{Scheme} |
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\title{{\sc oopse}: An Open Source Object-Oriented Parallel Simulation |
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Engine for Molecular Dynamics} |
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\author{Matthew A. Meineke, Charles F. Vardeman II, Teng Lin, Christopher J. Fennell and J. Daniel Gezelter\\ |
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Department of Chemistry and Biochemistry\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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%% \begin{abstract} |
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%% We detail the capabilities of a new open-source parallel simulation |
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%% package ({\sc oopse}) that can perform molecular dynamics simulations |
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%% on atom types that are missing from other popular packages. In |
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%% particular, {\sc oopse} is capable of performing orientational |
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%% dynamics on dipolar systems, and it can handle simulations of metallic |
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%% systems using the embedded atom method ({\sc eam}). |
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%% \end{abstract} |
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|
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\begin{abstract} |
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We detail the capabilities of a new open-source parallel simulation |
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package ({\sc oopse}) that can perform molecular dynamics simulations |
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on atom types that are missing from other popular packages. In |
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particular, {\sc oopse} is capable of performing orientational |
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dynamics on dipolar systems, and it can handle simulations of metallic |
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systems using the embedded atom method ({\sc eam}). |
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\end{abstract} |
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\lstset{language=C,frame=TB,basicstyle=\small,basicstyle=\ttfamily, % |
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xleftmargin=0.5in, xrightmargin=0.5in,captionpos=b, % |
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abovecaptionskip=0.5cm, belowcaptionskip=0.5cm} |
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\newpage |
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\section{\label{oopseSec:foreword}Foreword} |
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\section{\label{sec:intro}Introduction} |
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In this chapter, I present and detail the capabilities of the open |
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source simulation package {\sc oopse}. It is important to note, that a |
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simulation package of this size and scope would not have been possible |
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without the collaborative efforts of my colleagues: Charles |
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F.~Vardeman II, Teng Lin, Christopher J.~Fennell and J.~Daniel |
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Gezelter. Although my contributions to {\sc oopse} are major, |
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consideration of my work apart from the others would not give a |
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complete description to the package's capabilities. As such, all |
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contributions to {\sc oopse} to date are presented in this chapter. |
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\begin{itemize} |
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Charles Vardeman is responsible for the parallelization of the long |
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range forces in {\sc oopse} (Sec.~\ref{oopseSec:parallelization}) as |
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well as the inclusion of the embedded-atom potential for transition |
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metals (Sec.~\ref{oopseSec:eam}). Teng Lin's contributions include |
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refinement of the periodic boundary conditions |
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(Sec.~\ref{oopseSec:pbc}), the z-constraint method |
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(Sec.~\ref{oopseSec:zcons}), refinement of the property analysis |
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programs (Sec.~\ref{oopseSec:props}), and development in the extended |
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system integrators (Sec.~\ref{oopseSec:noseHooverThermo}). Christopher |
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Fennell worked on the symplectic integrator |
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(Sec.~\ref{oopseSec:integrate}) and the refinement of the {\sc ssd} |
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water model (Sec.~\ref{oopseSec:SSD}). Daniel Gezelter lent his |
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talents in the development of the extended system integrators |
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(Sec.~\ref{oopseSec:noseHooverThermo}) as well as giving general |
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direction and oversight to the entire project. My responsibilities |
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covered the creation and specification of {\sc bass} |
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(Sec.~\ref{oopseSec:IOfiles}), the original development of the single |
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processor version of {\sc oopse}, contributions to the extended state |
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integrators (Sec.~\ref{oopseSec:noseHooverThermo}), the implementation |
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of the Lennard-Jones (Sec.~\ref{sec:LJPot}) and {\sc duff} |
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(Sec.~\ref{oopseSec:DUFF}) force fields, and initial implementation of |
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the property analysis (Sec.~\ref{oopseSec:props}) and system |
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initialization (Sec.~\ref{oopseSec:initCoords}) utility programs. {\sc |
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oopse}, like many other Molecular Dynamics programs, is a work in |
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progress, and will continue to be so for many graduate student |
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lifetimes. |
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|
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\item Need for package / Niche to fill |
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\section{\label{sec:intro}Introduction} |
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|
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\item Design Goal |
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When choosing to simulate a chemical system with molecular dynamics, |
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there are a variety of options available. For simple systems, one |
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might consider writing one's own programming code. However, as systems |
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grow larger and more complex, building and maintaining code for the |
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simulations becomes a time consuming task. In such cases it is usually |
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more convenient for a researcher to turn to pre-existing simulation |
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packages. These packages, such as {\sc amber}\cite{pearlman:1995} and |
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{\sc charmm}\cite{Brooks83}, provide powerful tools for researchers to |
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conduct simulations of their systems without spending their time |
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developing a code base to conduct their research. This then frees them |
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to perhaps explore experimental analogues to their models. |
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\item Open Source |
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Despite their utility, problems with these packages arise when |
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researchers try to develop techniques or energetic models that the |
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code was not originally designed to simulate. Examples of uncommonly |
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implemented techniques and energetics include; dipole-dipole |
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interactions, rigid body dynamics, and metallic embedded |
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potentials. When faced with these obstacles, a researcher must either |
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develop their own code or license and extend one of the commercial |
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packages. What we have elected to do, is develop a package of |
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simulation code capable of implementing the types of models upon which |
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our research is based. |
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\item Discussion of Paper Layout |
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In developing {\sc oopse}, we have adhered to the precepts of Open |
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Source development, and are releasing our source code with a |
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permissive license. It is our intent that by doing so, other |
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researchers might benefit from our work, and add their own |
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contributions to the package. The license under which {\sc oopse} is |
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distributed allows any researcher to download and modify the source |
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code for their own use. In this way further development of {\sc oopse} |
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is not limited to only the models of interest to ourselves, but also |
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those of the community of scientists who contribute back to the |
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project. |
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|
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\end{itemize} |
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We have structured this chapter to first discuss the empirical energy |
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functions that {\sc oopse } implements in |
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Sec.~\ref{oopseSec:empiricalEnergy}. Following that is a discussion of |
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the various input and output files associated with the package |
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(Sec.~\ref{oopseSec:IOfiles}). Sec.~\ref{oopseSec:mechanics} |
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elucidates the various Molecular Dynamics algorithms {\sc oopse} |
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implements in the integration of the Newtonian equations of |
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motion. Basic analysis of the trajectories obtained from the |
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simulation is discussed in Sec.~\ref{oopseSec:props}. Program design |
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considerations are presented in Sec.~\ref{oopseSec:design}. And |
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lastly, Sec.~\ref{oopseSec:conclusion} concludes the chapter. |
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\section{\label{sec:empiricalEnergy}The Empirical Energy Functions} |
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\section{\label{oopseSec:empiricalEnergy}The Empirical Energy Functions} |
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\subsection{\label{sec:atomsMolecules}Atoms, Molecules and Rigid Bodies} |
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\subsection{\label{oopseSec:atomsMolecules}Atoms, Molecules and Rigid Bodies} |
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The basic unit of an {\sc oopse} simulation is the atom. The |
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parameters describing the atom are generalized to make the atom as |
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atoms of an element, or be used for collections of atoms such as |
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methyl and carbonyl groups. The atoms are also capable of having |
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directional components associated with them (\emph{e.g.}~permanent |
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dipoles). Charges on atoms are not currently supported by {\sc oopse}. |
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dipoles). Charges, permanent dipoles, and Lennard-Jones parameters for |
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a given atom type are set in the force field parameter files. |
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\begin{lstlisting}[caption={[Specifier for molecules and atoms] A sample specification of the simple Ar molecule},label=sch:AtmMole] |
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\begin{lstlisting}[float,caption={[Specifier for molecules and atoms] A sample specification of an Ar molecule},label=sch:AtmMole] |
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molecule{ |
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name = "Ar"; |
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nAtoms = 1; |
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} |
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\end{lstlisting} |
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Atoms can be collected into secondary srtructures such as rigid bodies |
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Atoms can be collected into secondary structures such as rigid bodies |
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or molecules. The molecule is a way for {\sc oopse} to keep track of |
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the atoms in a simulation in logical manner. Molecular units store the |
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identities of all the atoms associated with themselves, and are |
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responsible for the evaluation of their own internal interactions |
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(\emph{i.e.}~bonds, bends, and torsions). Scheme \ref{sch:AtmMole} |
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shws how one creates a molecule in the \texttt{.mdl} files. The |
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position of the atoms given in the declaration are relative to the |
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origin of the molecule, and is used when creating a system containing |
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the molecule. |
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identities of all the atoms and rigid bodies associated with |
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themselves, and are responsible for the evaluation of their own |
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internal interactions (\emph{i.e.}~bonds, bends, and torsions). Scheme |
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\ref{sch:AtmMole} shows how one creates a molecule in a ``model'' or |
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\texttt{.mdl} file. The position of the atoms given in the |
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declaration are relative to the origin of the molecule, and is used |
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when creating a system containing the molecule. |
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As stated previously, one of the features that sets {\sc oopse} apart |
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from most of the current molecular simulation packages is the ability |
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to handle rigid body dynamics. Rigid bodies are non-spherical |
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particles or collections of particles that have a constant internal |
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potential and move collectively.\cite{Goldstein01} They are not |
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included in most simulation packages because of the requirement to |
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propagate the orientational degrees of freedom. Until recently, |
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integrators which propagate orientational motion have been lacking. |
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included in most simulation packages because of the algorithmic |
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complexity involved in propagating orientational degrees of |
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freedom. Until recently, integrators which propagate orientational |
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motion have been much worse than those available for translational |
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motion. |
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Moving a rigid body involves determination of both the force and |
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torque applied by the surroundings, which directly affect the |
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first be calculated for all the internal particles. The total force on |
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the rigid body is simply the sum of these external forces. |
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Accumulation of the total torque on the rigid body is more complex |
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than the force in that it is the torque applied on the center of mass |
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that dictates rotational motion. The torque on rigid body {\it i} is |
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than the force because the torque is applied to the center of mass of |
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the rigid body. The torque on rigid body $i$ is |
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\begin{equation} |
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\boldsymbol{\tau}_i= |
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\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} |
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+ \boldsymbol{\tau}_{ia}, |
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\sum_{a}\biggl[(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} |
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+ \boldsymbol{\tau}_{ia}\biggr] |
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\label{eq:torqueAccumulate} |
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\end{equation} |
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where $\boldsymbol{\tau}_i$ and $\mathbf{r}_i$ are the torque on and |
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position of, and torque on the component particles of the rigid body. |
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The summation of the total torque is done in the body fixed axis of |
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the rigid body. In order to move between the space fixed and body |
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each rigid body. In order to move between the space fixed and body |
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fixed coordinate axes, parameters describing the orientation must be |
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maintained for each rigid body. At a minimum, the rotation matrix |
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(\textbf{A}) can be described by the three Euler angles ($\phi, |
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systems.\cite{Evans77} |
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{\sc oopse} utilizes a relatively new scheme that propagates the |
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entire nine parameter rotation matrix internally. Further discussion |
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on this choice can be found in Sec.~\ref{sec:integrate}. An example |
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definition of a riged body can be seen in Scheme |
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entire nine parameter rotation matrix. Further discussion |
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on this choice can be found in Sec.~\ref{oopseSec:integrate}. An example |
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definition of a rigid body can be seen in Scheme |
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\ref{sch:rigidBody}. The positions in the atom definitions are the |
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placements of the atoms relative to the origin of the rigid body, |
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which itself has a position relative to the origin of the molecule. |
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\begin{lstlisting}[caption={[Defining rigid bodies]A sample definition of a rigid body},label={sch:rigidBody}] |
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\begin{lstlisting}[float,caption={[Defining rigid bodies]A sample definition of a rigid body},label={sch:rigidBody}] |
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molecule{ |
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name = "TIP3P_water"; |
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nRigidBodies = 1; |
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} |
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\end{lstlisting} |
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\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
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\subsection{\label{sec:LJPot}The Lennard Jones Force Field} |
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The most basic force field implemented in {\sc oopse} is the |
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Lennard-Jones potential, which mimics the van der Waals interaction at |
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Lennard-Jones force field, which mimics the van der Waals interaction at |
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long distances, and uses an empirical repulsion at short |
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distances. The Lennard-Jones potential is given by: |
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\begin{equation} |
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Where $r_{ij}$ is the distance between particles $i$ and $j$, |
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$\sigma_{ij}$ scales the length of the interaction, and |
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$\epsilon_{ij}$ scales the well depth of the potential. Scheme |
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\ref{sch:LJFF} gives and example partial \texttt{.bass} file that |
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shows a system of 108 Ar particles simulated with the Lennard-Jones |
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force field. |
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\ref{sch:LJFF} gives and example \texttt{.bass} file that |
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sets up a system of 108 Ar particles to be simulated using the |
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Lennard-Jones force field. |
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\begin{lstlisting}[caption={[Invocation of the Lennard-Jones force field] A sample system using the Lennard-Jones force field.},label={sch:LJFF}] |
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\begin{lstlisting}[float,caption={[Invocation of the Lennard-Jones force field] A sample system using the Lennard-Jones force field.},label={sch:LJFF}] |
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/* |
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* The Ar molecule is specified |
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* external to the.bass file |
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*/ |
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#include "argon.mdl" |
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nComponents = 1; |
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nMol = 108; |
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} |
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/* |
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* The initial configuration is generated |
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* before the simulation is invoked. |
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*/ |
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initialConfig = "./argon.init"; |
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forceField = "LJ"; |
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Because this potential is calculated between all pairs, the force |
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evaluation can become computationally expensive for large systems. To |
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keep the pair evaluations to a manageable number, {\sc oopse} employs |
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a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be |
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a cut-off radius.\cite{allen87:csl} The cutoff radius can either be |
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specified in the \texttt{.bass} file, or left as its default value of |
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$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest Lennard-Jones |
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length parameter present in the simulation. Truncating the calculation |
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at $r_{\text{cut}}$ introduces a discontinuity into the potential |
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energy. To offset this discontinuity, the energy value at |
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$r_{\text{cut}}$ is subtracted from the potential. This causes the |
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potential to go to zero smoothly at the cut-off radius. |
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energy and the force. To offset this discontinuity in the potential, |
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the energy value at $r_{\text{cut}}$ is subtracted from the |
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potential. This causes the potential to go to zero smoothly at the |
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cut-off radius, and preserves conservation of energy in integrating |
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the equations of motion. |
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|
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Interactions between dissimilar particles requires the generation of |
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cross term parameters for $\sigma$ and $\epsilon$. These are |
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\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field} |
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\subsection{\label{oopseSec:DUFF}Dipolar Unified-Atom Force Field} |
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|
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The dipolar unified-atom force field ({\sc duff}) was developed to |
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simulate lipid bilayers. The simulations require a model capable of |
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forming bilayers, while still being sufficiently computationally |
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efficient to allow large systems ($\approx$100's of phospholipids, |
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$\approx$1000's of waters) to be simulated for long times |
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($\approx$10's of nanoseconds). |
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efficient to allow large systems ($\sim$100's of phospholipids, |
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$\sim$1000's of waters) to be simulated for long times |
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($\sim$10's of nanoseconds). |
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|
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With this goal in mind, {\sc duff} has no point |
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charges. Charge-neutral distributions were replaced with dipoles, |
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while most atoms and groups of atoms were reduced to Lennard-Jones |
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interaction sites. This simplification cuts the length scale of long |
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range interactions from $\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us |
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to avoid the computationally expensive Ewald sum. Instead, we can use |
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neighbor-lists, reaction field, and cutoff radii for the dipolar |
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interactions. |
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range interactions from $\frac{1}{r}$ to $\frac{1}{r^3}$, and allows |
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us to avoid the computationally expensive Ewald sum. Instead, we can |
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use neighbor-lists and cutoff radii for the dipolar interactions, or |
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include a reaction field to mimic larger range interactions. |
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|
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As an example, lipid head-groups in {\sc duff} are represented as |
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point dipole interaction sites. By placing a dipole of 20.6~Debye at |
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the head group center of mass, our model mimics the head group of |
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phosphatidylcholine.\cite{Cevc87} Additionally, a large Lennard-Jones |
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site is located at the pseudoatom's center of mass. The model is |
| 308 |
< |
illustrated by the dark grey atom in Fig.~\ref{fig:lipidModel}. The |
| 309 |
< |
water model we use to complement the dipoles of the lipids is our |
| 310 |
< |
reparameterization of the soft sticky dipole (SSD) model of Ichiye |
| 304 |
> |
point dipole interaction sites. By placing a dipole at the head group |
| 305 |
> |
center of mass, our model mimics the charge separation found in common |
| 306 |
> |
phospholipids such as phosphatidylcholine.\cite{Cevc87} Additionally, |
| 307 |
> |
a large Lennard-Jones site is located at the pseudoatom's center of |
| 308 |
> |
mass. The model is illustrated by the red atom in |
| 309 |
> |
Fig.~\ref{oopseFig:lipidModel}. The water model we use to complement |
| 310 |
> |
the dipoles of the lipids is our reparameterization of the soft sticky |
| 311 |
> |
dipole (SSD) model of Ichiye |
| 312 |
|
\emph{et al.}\cite{liu96:new_model} |
| 313 |
|
|
| 314 |
|
\begin{figure} |
| 315 |
< |
\epsfxsize=\linewidth |
| 316 |
< |
\epsfbox{lipidModel.eps} |
| 315 |
> |
\centering |
| 316 |
> |
\includegraphics[width=\linewidth]{lipidModel.eps} |
| 317 |
|
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
| 318 |
|
is the bend angle, $\mu$ is the dipole moment of the head group, and n |
| 319 |
|
is the chain length.} |
| 320 |
< |
\label{fig:lipidModel} |
| 320 |
> |
\label{oopseFig:lipidModel} |
| 321 |
|
\end{figure} |
| 322 |
|
|
| 323 |
|
We have used a set of scalable parameters to model the alkyl groups |
| 328 |
|
equilibria using Gibbs ensemble Monte Carlo simulation |
| 329 |
|
techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that |
| 330 |
|
it generalizes the types of atoms in an alkyl chain to keep the number |
| 331 |
< |
of pseudoatoms to a minimum; the parameters for an atom such as |
| 331 |
> |
of pseudoatoms to a minimum; the parameters for a unified atom such as |
| 332 |
|
$\text{CH}_2$ do not change depending on what species are bonded to |
| 333 |
|
it. |
| 334 |
|
|
| 335 |
|
TraPPE also constrains all bonds to be of fixed length. Typically, |
| 336 |
|
bond vibrations are the fastest motions in a molecular dynamic |
| 337 |
|
simulation. Small time steps between force evaluations must be used to |
| 338 |
< |
ensure adequate sampling of the bond potential to ensure conservation |
| 339 |
< |
of energy. By constraining the bond lengths, larger time steps may be |
| 340 |
< |
used when integrating the equations of motion. A simulation using {\sc |
| 341 |
< |
duff} is illustrated in Scheme \ref{sch:DUFF}. |
| 338 |
> |
ensure adequate energy conservation in the bond degrees of freedom. By |
| 339 |
> |
constraining the bond lengths, larger time steps may be used when |
| 340 |
> |
integrating the equations of motion. A simulation using {\sc duff} is |
| 341 |
> |
illustrated in Scheme \ref{sch:DUFF}. |
| 342 |
|
|
| 343 |
< |
\begin{lstlisting}[caption={[Invocation of {\sc duff}]Sample \texttt{.bass} file showing a simulation utilizing {\sc duff}},label={sch:DUFF}] |
| 343 |
> |
\begin{lstlisting}[float,caption={[Invocation of {\sc duff}]Sample \texttt{.bass} file showing a simulation utilizing {\sc duff}},label={sch:DUFF}] |
| 344 |
|
|
| 345 |
|
#include "water.mdl" |
| 346 |
|
#include "lipid.mdl" |
| 362 |
|
|
| 363 |
|
\end{lstlisting} |
| 364 |
|
|
| 365 |
< |
\subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions} |
| 365 |
> |
\subsection{\label{oopseSec:energyFunctions}{\sc duff} Energy Functions} |
| 366 |
|
|
| 367 |
|
The total potential energy function in {\sc duff} is |
| 368 |
|
\begin{equation} |
| 369 |
|
V = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
| 370 |
< |
+ \sum^{N}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}} |
| 370 |
> |
+ \sum^{N-1}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}} |
| 371 |
|
\label{eq:totalPotential} |
| 372 |
|
\end{equation} |
| 373 |
|
Where $V^{I}_{\text{Internal}}$ is the internal potential of molecule $I$: |
| 394 |
|
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
| 395 |
|
\end{equation} |
| 396 |
|
Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
| 397 |
< |
(see Fig.~\ref{fig:lipidModel}), $\theta_0$ is the equilibrium |
| 397 |
> |
(see Fig.~\ref{oopseFig:lipidModel}), $\theta_0$ is the equilibrium |
| 398 |
|
bond angle, and $k_{\theta}$ is the force constant which determines the |
| 399 |
|
strength of the harmonic bend. The parameters for $k_{\theta}$ and |
| 400 |
|
$\theta_0$ are borrowed from those in TraPPE.\cite{Siepmann1998} |
| 407 |
|
+ c_3[1 + \cos(3\phi)] |
| 408 |
|
\label{eq:origTorsionPot} |
| 409 |
|
\end{equation} |
| 410 |
< |
Here $\phi$ is the angle defined by four bonded neighbors $i$, |
| 367 |
< |
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). For |
| 368 |
< |
computational efficiency, the torsion potential has been recast after |
| 369 |
< |
the method of CHARMM,\cite{charmm1983} in which the angle series is |
| 370 |
< |
converted to a power series of the form: |
| 410 |
> |
Where: |
| 411 |
|
\begin{equation} |
| 412 |
+ |
\cos\phi = (\hat{\mathbf{r}}_{ij} \times \hat{\mathbf{r}}_{jk}) \cdot |
| 413 |
+ |
(\hat{\mathbf{r}}_{jk} \times \hat{\mathbf{r}}_{kl}) |
| 414 |
+ |
\label{eq:torsPhi} |
| 415 |
+ |
\end{equation} |
| 416 |
+ |
Here, $\hat{\mathbf{r}}_{\alpha\beta}$ are the set of unit bond |
| 417 |
+ |
vectors between atoms $i$, $j$, $k$, and $l$. For computational |
| 418 |
+ |
efficiency, the torsion potential has been recast after the method of |
| 419 |
+ |
{\sc charmm},\cite{Brooks83} in which the angle series is converted to |
| 420 |
+ |
a power series of the form: |
| 421 |
+ |
\begin{equation} |
| 422 |
|
V_{\text{torsion}}(\phi) = |
| 423 |
|
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
| 424 |
|
\label{eq:torsionPot} |
| 449 |
|
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 450 |
|
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 451 |
|
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model |
| 452 |
< |
(Sec.~\ref{sec:SSD}). Note that not all atom types include all |
| 452 |
> |
(Sec.~\ref{oopseSec:SSD}). Note that not all atom types include all |
| 453 |
|
interactions. |
| 454 |
|
|
| 455 |
|
The dipole-dipole potential has the following form: |
| 458 |
|
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
| 459 |
|
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
| 460 |
|
- |
| 461 |
< |
\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) % |
| 462 |
< |
(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) } |
| 413 |
< |
{r^{2}_{ij}} \biggr] |
| 461 |
> |
3(\boldsymbol{\hat{u}}_i \cdot \hat{\mathbf{r}}_{ij}) % |
| 462 |
> |
(\boldsymbol{\hat{u}}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr] |
| 463 |
|
\label{eq:dipolePot} |
| 464 |
|
\end{equation} |
| 465 |
|
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 466 |
|
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
| 467 |
|
are the orientational degrees of freedom for atoms $i$ and $j$ |
| 468 |
|
respectively. $|\mu_i|$ is the magnitude of the dipole moment of atom |
| 469 |
< |
$i$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
| 470 |
< |
vector of $\boldsymbol{\Omega}_i$, and $\boldsymbol{\hat{r}}_{ij}$ is |
| 471 |
< |
the unit vector pointing along $\mathbf{r}_{ij}$. |
| 469 |
> |
$i$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector |
| 470 |
> |
of $\boldsymbol{\Omega}_i$, and $\boldsymbol{\hat{r}}_{ij}$ is the |
| 471 |
> |
unit vector pointing along $\mathbf{r}_{ij}$ |
| 472 |
> |
($\boldsymbol{\hat{r}}_{ij}=\mathbf{r}_{ij}/|\mathbf{r}_{ij}|$). |
| 473 |
|
|
| 474 |
+ |
To improve computational efficiency of the dipole-dipole interactions, |
| 475 |
+ |
{\sc oopse} employs an electrostatic cutoff radius. This parameter can |
| 476 |
+ |
be set in the \texttt{.bass} file, and controls the length scale over |
| 477 |
+ |
which dipole interactions are felt. To compensate for the |
| 478 |
+ |
discontinuity in the potential and the forces at the cutoff radius, we |
| 479 |
+ |
have implemented a switching function to smoothly scale the |
| 480 |
+ |
dipole-dipole interaction at the cutoff. |
| 481 |
+ |
\begin{equation} |
| 482 |
+ |
S(r_{ij}) = |
| 483 |
+ |
\begin{cases} |
| 484 |
+ |
1 & \text{if $r_{ij} \le r_t$},\\ |
| 485 |
+ |
\frac{(r_{\text{cut}} + 2r_{ij} - 3r_t)(r_{\text{cut}} - r_{ij})^2} |
| 486 |
+ |
{(r_{\text{cut}} - r_t)^2} |
| 487 |
+ |
& \text{if $r_t < r_{ij} \le r_{\text{cut}}$}, \\ |
| 488 |
+ |
0 & \text{if $r_{ij} > r_{\text{cut}}$.} |
| 489 |
+ |
\end{cases} |
| 490 |
+ |
\label{eq:dipoleSwitching} |
| 491 |
+ |
\end{equation} |
| 492 |
+ |
Here $S(r_{ij})$ scales the potential at a given $r_{ij}$, and $r_t$ |
| 493 |
+ |
is the taper radius some given thickness less than the electrostatic |
| 494 |
+ |
cutoff. The switching thickness can be set in the \texttt{.bass} file. |
| 495 |
|
|
| 496 |
< |
\subsubsection{\label{sec:SSD}The {\sc duff} Water Models: SSD/E and SSD/RF} |
| 496 |
> |
\subsection{\label{oopseSec:SSD}The {\sc duff} Water Models: SSD/E and SSD/RF} |
| 497 |
|
|
| 498 |
|
In the interest of computational efficiency, the default solvent used |
| 499 |
|
by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water |
| 553 |
|
can be found in the original SSD |
| 554 |
|
articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03} |
| 555 |
|
|
| 556 |
< |
Since SSD is a single-point {\it dipolar} model, the force |
| 556 |
> |
Since SSD/E is a single-point {\it dipolar} model, the force |
| 557 |
|
calculations are simplified significantly relative to the standard |
| 558 |
|
{\it charged} multi-point models. In the original Monte Carlo |
| 559 |
|
simulations using this model, Ichiye {\it et al.} reported that using |
| 560 |
|
SSD decreased computer time by a factor of 6-7 compared to other |
| 561 |
|
models.\cite{liu96:new_model} What is most impressive is that these savings |
| 562 |
|
did not come at the expense of accurate depiction of the liquid state |
| 563 |
< |
properties. Indeed, SSD maintains reasonable agreement with the Soper |
| 563 |
> |
properties. Indeed, SSD/E maintains reasonable agreement with the Head-Gordon |
| 564 |
|
diffraction data for the structural features of liquid |
| 565 |
< |
water.\cite{Soper86,liu96:new_model} Additionally, the dynamical properties |
| 566 |
< |
exhibited by SSD agree with experiment better than those of more |
| 565 |
> |
water.\cite{hura00,liu96:new_model} Additionally, the dynamical properties |
| 566 |
> |
exhibited by SSD/E agree with experiment better than those of more |
| 567 |
|
computationally expensive models (like TIP3P and |
| 568 |
|
SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate depiction |
| 569 |
< |
of solvent properties makes SSD a very attractive model for the |
| 569 |
> |
of solvent properties makes SSD/E a very attractive model for the |
| 570 |
|
simulation of large scale biochemical simulations. |
| 571 |
|
|
| 572 |
|
Recent constant pressure simulations revealed issues in the original |
| 577 |
|
of a reaction field long-range interaction correction is desired, it |
| 578 |
|
is recommended that the parameters be modified to those of the SSD/RF |
| 579 |
|
model. Solvent parameters can be easily modified in an accompanying |
| 580 |
< |
{\sc BASS} file as illustrated in the scheme below. A table of the |
| 580 |
> |
\texttt{.bass} file as illustrated in the scheme below. A table of the |
| 581 |
|
parameter values and the drawbacks and benefits of the different |
| 582 |
< |
density corrected SSD models can be found in reference |
| 583 |
< |
\ref{Gezelter04}. |
| 582 |
> |
density corrected SSD models can be found in |
| 583 |
> |
reference~\cite{Gezelter04}. |
| 584 |
|
|
| 585 |
< |
\begin{lstlisting}[caption={[A simulation of {\sc ssd} water]An example file showing a simulation including {\sc ssd} water.},label={sch:ssd}] |
| 585 |
> |
\begin{lstlisting}[float,caption={[A simulation of {\sc ssd} water]An example file showing a simulation including {\sc ssd} water.},label={sch:ssd}] |
| 586 |
|
|
| 587 |
|
#include "water.mdl" |
| 588 |
|
|
| 597 |
|
forceField = "DUFF"; |
| 598 |
|
|
| 599 |
|
/* |
| 529 |
– |
* The reactionField flag toggles reaction |
| 530 |
– |
* field corrections. |
| 531 |
– |
*/ |
| 532 |
– |
|
| 533 |
– |
reactionField = false; // defaults to false |
| 534 |
– |
dielectric = 80.0; // dielectric for reaction field |
| 535 |
– |
|
| 536 |
– |
/* |
| 600 |
|
* The following two flags set the cutoff |
| 601 |
|
* radius for the electrostatic forces |
| 602 |
|
* as well as the skin thickness of the switching |
| 609 |
|
\end{lstlisting} |
| 610 |
|
|
| 611 |
|
|
| 612 |
< |
\subsection{\label{sec:eam}Embedded Atom Method} |
| 612 |
> |
\subsection{\label{oopseSec:eam}Embedded Atom Method} |
| 613 |
|
|
| 614 |
< |
Several other molecular dynamics packages\cite{dynamo86} exist which have the |
| 614 |
> |
There are Molecular Dynamics packages which have the |
| 615 |
|
capacity to simulate metallic systems, including some that have |
| 616 |
|
parallel computational abilities\cite{plimpton93}. Potentials that |
| 617 |
|
describe bonding transition metal |
| 618 |
< |
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a |
| 618 |
> |
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have an |
| 619 |
|
attractive interaction which models ``Embedding'' |
| 620 |
|
a positively charged metal ion in the electron density due to the |
| 621 |
|
free valance ``sea'' of electrons created by the surrounding atoms in |
| 622 |
< |
the system. A mostly repulsive pairwise part of the potential |
| 622 |
> |
the system. A mostly-repulsive pairwise part of the potential |
| 623 |
|
describes the interaction of the positively charged metal core ions |
| 624 |
|
with one another. A particular potential description called the |
| 625 |
|
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has |
| 626 |
|
particularly wide adoption has been selected for inclusion in {\sc oopse}. A |
| 627 |
< |
good review of {\sc eam} and other metallic potential formulations was done |
| 627 |
> |
good review of {\sc eam} and other metallic potential formulations was written |
| 628 |
|
by Voter.\cite{voter} |
| 629 |
|
|
| 630 |
|
The {\sc eam} potential has the form: |
| 632 |
|
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
| 633 |
|
\phi_{ij}({\bf r}_{ij}) \\ |
| 634 |
|
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
| 635 |
< |
\end{eqnarray}S |
| 573 |
< |
|
| 635 |
> |
\end{eqnarray} |
| 636 |
|
where $F_{i} $ is the embedding function that equates the energy required to embed a |
| 637 |
|
positively-charged core ion $i$ into a linear superposition of |
| 638 |
|
spherically averaged atomic electron densities given by |
| 639 |
|
$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 640 |
|
between atoms $i$ and $j$. In the original formulation of |
| 641 |
< |
{\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 641 |
> |
{\sc eam}\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 642 |
|
in later refinements to EAM have shown that non-uniqueness between $F$ |
| 643 |
|
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
| 644 |
|
There is a cutoff distance, $r_{cut}$, which limits the |
| 645 |
|
summations in the {\sc eam} equation to the few dozen atoms |
| 646 |
|
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
| 647 |
< |
interactions. Foiles et al. fit EAM potentials for fcc metals Cu, Ag, Au, Ni, Pd, Pt and alloys of these metals\cite{FDB86}. These potential fits are in the DYNAMO 86 format and are included with {\sc oopse}. |
| 647 |
> |
interactions. Foiles et al. fit EAM potentials for fcc metals Cu, Ag, Au, Ni, Pd, Pt and alloys of these metals\cite{FBD86}. These potential fits are in the DYNAMO 86 format and are included with {\sc oopse}. |
| 648 |
|
|
| 649 |
|
|
| 650 |
< |
\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
| 650 |
> |
\subsection{\label{oopseSec:pbc}Periodic Boundary Conditions} |
| 651 |
|
|
| 652 |
|
\newcommand{\roundme}{\operatorname{round}} |
| 653 |
|
|
| 654 |
< |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 655 |
< |
macroscopic systems with a relatively small number of particles. The |
| 656 |
< |
simulation box is replicated throughout space to form an infinite lattice. |
| 657 |
< |
During the simulation, when a particle moves in the primary cell, its image in |
| 658 |
< |
other boxes move in exactly the same direction with exactly the same |
| 659 |
< |
orientation.Thus, as a particle leaves the primary cell, one of its images |
| 660 |
< |
will enter through the opposite face.If the simulation box is large enough to |
| 661 |
< |
avoid \textquotedblleft feeling\textquotedblright\ the symmetries of the |
| 662 |
< |
periodic lattice, surface effects can be ignored. Cubic, orthorhombic and |
| 663 |
< |
parallelepiped are the available periodic cells In OOPSE. We use a matrix to |
| 664 |
< |
describe the property of the simulation box. Therefore, both the size and |
| 603 |
< |
shape of the simulation box can be changed during the simulation. The |
| 604 |
< |
transformation from box space vector $\mathbf{s}$ to its corresponding real |
| 605 |
< |
space vector $\mathbf{r}$ is defined by |
| 654 |
> |
\textit{Periodic boundary conditions} are widely used to simulate bulk properties with a relatively small number of particles. The |
| 655 |
> |
simulation box is replicated throughout space to form an infinite |
| 656 |
> |
lattice. During the simulation, when a particle moves in the primary |
| 657 |
> |
cell, its image in other cells move in exactly the same direction with |
| 658 |
> |
exactly the same orientation. Thus, as a particle leaves the primary |
| 659 |
> |
cell, one of its images will enter through the opposite face. If the |
| 660 |
> |
simulation box is large enough to avoid ``feeling'' the symmetries of |
| 661 |
> |
the periodic lattice, surface effects can be ignored. The available |
| 662 |
> |
periodic cells in OOPSE are cubic, orthorhombic and parallelepiped. We |
| 663 |
> |
use a $3 \times 3$ matrix, $\mathbf{H}$, to describe the shape and |
| 664 |
> |
size of the simulation box. $\mathbf{H}$ is defined: |
| 665 |
|
\begin{equation} |
| 666 |
< |
\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}% |
| 666 |
> |
\mathbf{H} = ( \mathbf{h}_x, \mathbf{h}_y, \mathbf{h}_z ) |
| 667 |
|
\end{equation} |
| 668 |
+ |
Where $\mathbf{h}_j$ is the column vector of the $j$th axis of the |
| 669 |
+ |
box. During the course of the simulation both the size and shape of |
| 670 |
+ |
the box can be changed to allow volume fluctations when constraining |
| 671 |
+ |
the pressure. |
| 672 |
|
|
| 673 |
< |
|
| 674 |
< |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
| 675 |
< |
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the |
| 676 |
< |
simulation box respectively. |
| 677 |
< |
|
| 678 |
< |
To find the minimum image of a vector $\mathbf{r}$, we convert the real vector |
| 679 |
< |
to its corresponding vector in box space first, \bigskip% |
| 673 |
> |
A real space vector, $\mathbf{r}$ can be transformed in to a box space |
| 674 |
> |
vector, $\mathbf{s}$, and back through the following transformations: |
| 675 |
> |
\begin{align} |
| 676 |
> |
\mathbf{s} &= \mathbf{H}^{-1} \mathbf{r} \\ |
| 677 |
> |
\mathbf{r} &= \mathbf{H} \mathbf{s} |
| 678 |
> |
\end{align} |
| 679 |
> |
The vector $\mathbf{s}$ is now a vector expressed as the number of box |
| 680 |
> |
lengths in the $\mathbf{h}_x$, $\mathbf{h}_y$, and $\mathbf{h}_z$ |
| 681 |
> |
directions. To find the minimum image of a vector $\mathbf{r}$, we |
| 682 |
> |
first convert it to its corresponding vector in box space, and then, |
| 683 |
> |
cast each element to lie on the in the range $[-0.5,0.5]$: |
| 684 |
|
\begin{equation} |
| 618 |
– |
\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}% |
| 619 |
– |
\end{equation} |
| 620 |
– |
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
| 621 |
– |
\begin{equation} |
| 685 |
|
s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
| 686 |
|
\end{equation} |
| 687 |
< |
where |
| 688 |
< |
|
| 626 |
< |
% |
| 627 |
< |
|
| 687 |
> |
Where $s_i$ is the $i$th element of $\mathbf{s}$, and |
| 688 |
> |
$\roundme(s_i)$is given by |
| 689 |
|
\begin{equation} |
| 690 |
< |
\roundme(x)=\left\{ |
| 691 |
< |
\begin{array}{cc}% |
| 692 |
< |
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
| 693 |
< |
\lceil{x-0.5}\rceil & \text{otherwise}% |
| 694 |
< |
\end{array} |
| 634 |
< |
\right. |
| 690 |
> |
\roundme(x) = |
| 691 |
> |
\begin{cases} |
| 692 |
> |
\lfloor x+0.5 \rfloor & \text{if $x \ge 0$} \\ |
| 693 |
> |
\lceil x-0.5 \rceil & \text{if $x < 0$ } |
| 694 |
> |
\end{cases} |
| 695 |
|
\end{equation} |
| 696 |
< |
|
| 697 |
< |
|
| 698 |
< |
For example, $\roundme(3.6)=4$,$\roundme(3.1)=3$, $\roundme(-3.6)=-4$, $\roundme(-3.1)=-3$. |
| 699 |
< |
|
| 700 |
< |
Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
| 641 |
< |
transforming back to real space,% |
| 696 |
> |
Here $\lfloor x \rfloor$ is the floor operator, and gives the largest |
| 697 |
> |
integer value that is not greater than $x$, and $\lceil x \rceil$ is |
| 698 |
> |
the ceiling operator, and gives the smallest integer that is not less |
| 699 |
> |
than $x$. For example, $\roundme(3.6)=4$, $\roundme(3.1)=3$, |
| 700 |
> |
$\roundme(-3.6)=-4$, $\roundme(-3.1)=-3$. |
| 701 |
|
|
| 702 |
+ |
Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
| 703 |
+ |
transforming back to real space, |
| 704 |
|
\begin{equation} |
| 705 |
< |
\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
| 705 |
> |
\mathbf{r}^{\prime}=\mathbf{H}^{-1}\mathbf{s}^{\prime}% |
| 706 |
|
\end{equation} |
| 707 |
+ |
In this way, particles are allowed to diffuse freely in $\mathbf{r}$, |
| 708 |
+ |
but their minimum images, $\mathbf{r}^{\prime}$ are used to compute |
| 709 |
+ |
the interatomic forces. |
| 710 |
|
|
| 711 |
|
|
| 712 |
< |
\section{Input and Output Files} |
| 712 |
> |
\section{\label{oopseSec:IOfiles}Input and Output Files} |
| 713 |
|
|
| 714 |
|
\subsection{{\sc bass} and Model Files} |
| 715 |
|
|
| 716 |
< |
Every {\sc oopse} simuation begins with a {\sc bass} file. {\sc bass} |
| 716 |
> |
Every {\sc oopse} simulation begins with a {\sc bass} file. {\sc bass} |
| 717 |
|
(\underline{B}izarre \underline{A}tom \underline{S}imulation |
| 718 |
|
\underline{S}yntax) is a script syntax that is parsed by {\sc oopse} at |
| 719 |
|
runtime. The {\sc bass} file allows for the user to completely describe the |
| 723 |
|
Fig.~\ref{fig:bassExample}. |
| 724 |
|
|
| 725 |
|
\begin{figure} |
| 662 |
– |
|
| 726 |
|
\centering |
| 727 |
|
\framebox[\linewidth]{\rule{0cm}{0.75\linewidth}I'm a {\sc bass} file!} |
| 728 |
|
\caption{Here is an example \texttt{.bass} file} |
| 729 |
|
\label{fig:bassExample} |
| 730 |
|
\end{figure} |
| 731 |
|
|
| 732 |
< |
Within the \texttt{.bass} file it is neccassary to provide a complete |
| 732 |
> |
Within the \texttt{.bass} file it is necessary to provide a complete |
| 733 |
|
description of the molecule before it is actually placed in the |
| 734 |
|
simulation. The {\sc bass} syntax was originally developed with this goal in |
| 735 |
|
mind, and allows for the specification of all the atoms in a molecular |
| 736 |
|
prototype, as well as any bonds, bends, or torsions. These |
| 737 |
|
descriptions can become lengthy for complex molecules, and it would be |
| 738 |
< |
inconvient to duplicate the simulation at the begining of each {\sc bass} |
| 738 |
> |
inconvenient to duplicate the simulation at the beginning of each {\sc bass} |
| 739 |
|
script. Addressing this issue {\sc bass} allows for the inclusion of model |
| 740 |
|
files at the top of a \texttt{.bass} file. These model files, denoted |
| 741 |
|
with the \texttt{.mdl} extension, allow the user to describe a |
| 742 |
|
molecular prototype once, then simply include it into each simulation |
| 743 |
|
containing that molecule. |
| 744 |
|
|
| 745 |
< |
\subsection{\label{subSec:coordFiles}Coordinate Files} |
| 745 |
> |
\subsection{\label{oopseSec:coordFiles}Coordinate Files} |
| 746 |
|
|
| 747 |
|
The standard format for storage of a systems coordinates is a modified |
| 748 |
|
xyz-file syntax, the exact details of which can be seen in |
| 754 |
|
There are three major files used by {\sc oopse} written in the coordinate |
| 755 |
|
format, they are as follows: the initialization file, the simulation |
| 756 |
|
trajectory file, and the final coordinates of the simulation. The |
| 757 |
< |
initialization file is neccassary for {\sc oopse} to start the simulation |
| 757 |
> |
initialization file is necessary for {\sc oopse} to start the simulation |
| 758 |
|
with the proper coordinates. It is typically denoted with the |
| 759 |
|
extension \texttt{.init}. The trajectory file is created at the |
| 760 |
|
beginning of the simulation, and is used to store snapshots of the |
| 763 |
|
file at an interval specified in the \texttt{.bass} file. The |
| 764 |
|
trajectory file is given the extension \texttt{.dump}. The final |
| 765 |
|
coordinate file is the end of run or \texttt{.eor} file. The |
| 766 |
< |
\texttt{.eor} file stores the final configuration of teh system for a |
| 766 |
> |
\texttt{.eor} file stores the final configuration of the system for a |
| 767 |
|
given simulation. The file is updated at the same time as the |
| 768 |
|
\texttt{.dump} file. However, it only contains the most recent |
| 769 |
|
frame. In this way, an \texttt{.eor} file may be used as the |
| 770 |
|
initialization file to a second simulation in order to continue or |
| 771 |
|
recover the previous simulation. |
| 772 |
|
|
| 773 |
< |
\subsection{Generation of Initial Coordinates} |
| 773 |
> |
\subsection{\label{oopseSec:initCoords}Generation of Initial Coordinates} |
| 774 |
|
|
| 775 |
< |
As was stated in Sec.~\ref{subSec:coordFiles}, an initialization file |
| 775 |
> |
As was stated in Sec.~\ref{oopseSec:coordFiles}, an initialization file |
| 776 |
|
is needed to provide the starting coordinates for a simulation. The |
| 777 |
|
{\sc oopse} package provides a program called \texttt{sysBuilder} to aid in |
| 778 |
|
the creation of the \texttt{.init} file. \texttt{sysBuilder} is {\sc bass} |
| 779 |
|
aware, and will recognize arguments and parameters in the |
| 780 |
|
\texttt{.bass} file that would otherwise be ignored by the |
| 781 |
< |
simulation. The program itself is under contiunual development, and is |
| 781 |
> |
simulation. The program itself is under continual development, and is |
| 782 |
|
offered here as a helper tool only. |
| 783 |
|
|
| 784 |
|
\subsection{The Statistics File} |
| 787 |
|
file records such statistical quantities as the instantaneous |
| 788 |
|
temperature, volume, pressure, etc. It is written out with the |
| 789 |
|
frequency specified in the \texttt{.bass} file. The file allows the |
| 790 |
< |
user to observe the system variables as a function od simulation time |
| 790 |
> |
user to observe the system variables as a function of simulation time |
| 791 |
|
while the simulation is in progress. One useful function the |
| 792 |
|
statistics file serves is to monitor the conserved quantity of a given |
| 793 |
|
simulation ensemble, this allows the user to observe the stability of |
| 794 |
|
the integrator. The statistics file is denoted with the \texttt{.stat} |
| 795 |
|
file extension. |
| 796 |
|
|
| 797 |
< |
\section{\label{sec:mechanics}Mechanics} |
| 797 |
> |
\section{\label{oopseSec:mechanics}Mechanics} |
| 798 |
|
|
| 799 |
< |
\subsection{\label{integrate}Integrating the Equations of Motion: the Symplectic Step Integrator} |
| 799 |
> |
\subsection{\label{oopseSec:integrate}Integrating the Equations of Motion: the Symplectic Step Integrator} |
| 800 |
|
|
| 801 |
|
Integration of the equations of motion was carried out using the |
| 802 |
|
symplectic splitting method proposed by Dullweber \emph{et |
| 836 |
|
\ref{timestep}. |
| 837 |
|
|
| 838 |
|
\begin{figure} |
| 839 |
< |
\epsfxsize=6in |
| 840 |
< |
\epsfbox{timeStep.epsi} |
| 839 |
> |
\centering |
| 840 |
> |
\includegraphics[width=\linewidth]{timeStep.eps} |
| 841 |
|
\caption{Energy conservation using quaternion based integration versus |
| 842 |
|
the symplectic step method proposed by Dullweber \emph{et al.} with |
| 843 |
|
increasing time step. For each time step, the dotted line is total |
| 875 |
|
{\sc oopse} implements a |
| 876 |
|
|
| 877 |
|
|
| 878 |
< |
\subsubsection{\label{sec:noseHooverThermo}Nose-Hoover Thermostatting} |
| 878 |
> |
\subsubsection{\label{oopseSec:noseHooverThermo}Nose-Hoover Thermostatting} |
| 879 |
|
|
| 880 |
|
To mimic the effects of being in a constant temperature ({\sc nvt}) |
| 881 |
|
ensemble, {\sc oopse} uses the Nose-Hoover extended system |
| 903 |
|
set to 1 ps using the {\tt tauThermostat = 1000; } command. |
| 904 |
|
|
| 905 |
|
|
| 906 |
< |
\subsection{\label{Sec:zcons}Z-Constraint Method} |
| 906 |
> |
\subsection{\label{oopseSec:zcons}Z-Constraint Method} |
| 907 |
|
|
| 908 |
< |
Based on fluctuatin-dissipation theorem,\bigskip\ force auto-correlation |
| 908 |
> |
Based on fluctuation-dissipation theorem,\bigskip\ force auto-correlation |
| 909 |
|
method was developed to investigate the dynamics of ions inside the ion |
| 910 |
|
channels.\cite{Roux91} Time-dependent friction coefficient can be calculated |
| 911 |
< |
from the deviation of the instaneous force from its mean force. |
| 911 |
> |
from the deviation of the instantaneous force from its mean force. |
| 912 |
|
|
| 913 |
|
% |
| 914 |
|
|
| 915 |
|
\begin{equation} |
| 916 |
|
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T |
| 917 |
|
\end{equation} |
| 855 |
– |
|
| 856 |
– |
|
| 918 |
|
where% |
| 919 |
|
\begin{equation} |
| 920 |
|
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle |
| 944 |
|
resetting the coordinate, we reset the forces of z-constraint molecules as |
| 945 |
|
well as subtract the total constraint forces from the rest of the system after |
| 946 |
|
force calculation at each time step. |
| 947 |
< |
\begin{verbatim} |
| 948 |
< |
$F_{\alpha i}=0$ |
| 949 |
< |
$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{i}M_{_{\alpha i}}}$ |
| 950 |
< |
$F_{\alpha i}=F_{\alpha i}-\frac{M_{_{\alpha i}}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}\sum\limits_{\beta}F_{\beta}$ |
| 951 |
< |
$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}$ |
| 952 |
< |
\end{verbatim} |
| 947 |
> |
\begin{align} |
| 948 |
> |
F_{\alpha i}&=0\\ |
| 949 |
> |
V_{\alpha i}&=V_{\alpha i}-\frac{\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{i}M_{_{\alpha i}}}\\ |
| 950 |
> |
F_{\alpha i}&=F_{\alpha i}-\frac{M_{_{\alpha i}}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}\sum\limits_{\beta}F_{\beta}\\ |
| 951 |
> |
V_{\alpha i}&=V_{\alpha i}-\frac{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}} |
| 952 |
> |
\end{align} |
| 953 |
|
|
| 954 |
|
At the very beginning of the simulation, the molecules may not be at its |
| 955 |
|
constraint position. To move the z-constraint molecule to the specified |
| 969 |
|
Worthy of mention, other kinds of potential functions can also be used to |
| 970 |
|
drive the z-constraint molecule. |
| 971 |
|
|
| 972 |
< |
\section{\label{sec:analysis}Trajectory Analysis} |
| 972 |
> |
\section{\label{oopseSec:props}Trajectory Analysis} |
| 973 |
|
|
| 974 |
< |
\subsection{\label{subSec:staticProps}Static Property Analysis} |
| 974 |
> |
\subsection{\label{oopseSec:staticProps}Static Property Analysis} |
| 975 |
|
|
| 976 |
|
The static properties of the trajectories are analyzed with the |
| 977 |
|
program \texttt{staticProps}. The code is capable of calculating the following |
| 1099 |
|
between the frames contained within the two blocks are |
| 1100 |
|
calculated. Once completed, the memory blocks are incremented, and the |
| 1101 |
|
process is repeated. A diagram illustrating the process is shown in |
| 1102 |
< |
Fig.~\ref{fig:dynamicPropsMemory}. As was the case with \texttt{staticProps}, |
| 1103 |
< |
multiple properties may be calculated in a single run to avoid |
| 1104 |
< |
multiple reads on the same file. |
| 1102 |
> |
Fig.~\ref{oopseFig:dynamicPropsMemory}. As was the case with |
| 1103 |
> |
\texttt{staticProps}, multiple properties may be calculated in a |
| 1104 |
> |
single run to avoid multiple reads on the same file. |
| 1105 |
|
|
| 1045 |
– |
\begin{figure} |
| 1046 |
– |
\epsfxsize=6in |
| 1047 |
– |
\epsfbox{dynamicPropsMem.eps} |
| 1048 |
– |
\caption{This diagram illustrates the dynamic memory allocation used by \texttt{dynamicProps}, which follows the scheme: $\sum^{N_{\text{memory blocks}}}_{i=1}[ \operatorname{self}(i) + \sum^{N_{\text{memory blocks}}}_{j>i} \operatorname{cross}(i,j)]$. The shaded region represents the self correlation of the memory block, and the open blocks are read one at a time and the cross correlations between blocks are calculated.} |
| 1049 |
– |
\label{fig:dynamicPropsMemory} |
| 1050 |
– |
\end{figure} |
| 1106 |
|
|
| 1107 |
< |
\section{\label{sec:ProgramDesign}Program Design} |
| 1107 |
> |
|
| 1108 |
> |
\section{\label{oopseSec:design}Program Design} |
| 1109 |
|
|
| 1110 |
< |
\subsection{\label{sec:architecture} OOPSE Architecture} |
| 1110 |
> |
\subsection{\label{sec:architecture} {\sc oopse} Architecture} |
| 1111 |
|
|
| 1112 |
< |
The core of OOPSE is divided into two main object libraries: {\texttt |
| 1113 |
< |
libBASS} and {\texttt libmdtools}. {\texttt libBASS} is the library |
| 1114 |
< |
developed around the parseing engine and {\texttt libmdtools} is the |
| 1115 |
< |
software library developed around the simulation engine. |
| 1112 |
> |
The core of OOPSE is divided into two main object libraries: |
| 1113 |
> |
\texttt{libBASS} and \texttt{libmdtools}. \texttt{libBASS} is the |
| 1114 |
> |
library developed around the parsing engine and \texttt{libmdtools} |
| 1115 |
> |
is the software library developed around the simulation engine. These |
| 1116 |
> |
two libraries are designed to encompass all the basic functions and |
| 1117 |
> |
tools that {\sc oopse} provides. Utility programs, such as the |
| 1118 |
> |
property analyzers, need only link against the software libraries to |
| 1119 |
> |
gain access to parsing, force evaluation, and input / output |
| 1120 |
> |
routines. |
| 1121 |
|
|
| 1122 |
+ |
Contained in \texttt{libBASS} are all the routines associated with |
| 1123 |
+ |
reading and parsing the \texttt{.bass} input files. Given a |
| 1124 |
+ |
\texttt{.bass} file, \texttt{libBASS} will open it and any associated |
| 1125 |
+ |
\texttt{.mdl} files; then create structures in memory that are |
| 1126 |
+ |
templates of all the molecules specified in the input files. In |
| 1127 |
+ |
addition, any simulation parameters set in the \texttt{.bass} file |
| 1128 |
+ |
will be placed in a structure for later query by the controlling |
| 1129 |
+ |
program. |
| 1130 |
|
|
| 1131 |
+ |
Located in \texttt{libmdtools} are all other routines necessary to a |
| 1132 |
+ |
Molecular Dynamics simulation. The library uses the main data |
| 1133 |
+ |
structures returned by \texttt{libBASS} to initialize the various |
| 1134 |
+ |
parts of the simulation: the atom structures and positions, the force |
| 1135 |
+ |
field, the integrator, \emph{et cetera}. After initialization, the |
| 1136 |
+ |
library can be used to perform a variety of tasks: integrate a |
| 1137 |
+ |
Molecular Dynamics trajectory, query phase space information from a |
| 1138 |
+ |
specific frame of a completed trajectory, or even recalculate force or |
| 1139 |
+ |
energetic information about specific frames from a completed |
| 1140 |
+ |
trajectory. |
| 1141 |
|
|
| 1142 |
< |
\subsection{\label{sec:programLang} Programming Languages } |
| 1142 |
> |
With these core libraries in place, several programs have been |
| 1143 |
> |
developed to utilize the routines provided by \texttt{libBASS} and |
| 1144 |
> |
\texttt{libmdtools}. The main program of the package is \texttt{oopse} |
| 1145 |
> |
and the corresponding parallel version \texttt{oopse\_MPI}. These two |
| 1146 |
> |
programs will take the \texttt{.bass} file, and create then integrate |
| 1147 |
> |
the simulation specified in the script. The two analysis programs |
| 1148 |
> |
\texttt{staticProps} and \texttt{dynamicProps} utilize the core |
| 1149 |
> |
libraries to initialize and read in trajectories from previously |
| 1150 |
> |
completed simulations, in addition to the ability to use functionality |
| 1151 |
> |
from \texttt{libmdtools} to recalculate forces and energies at key |
| 1152 |
> |
frames in the trajectories. Lastly, the family of system building |
| 1153 |
> |
programs (Sec.~\ref{oopseSec:initCoords}) also use the libraries to |
| 1154 |
> |
store and output the system configurations they create. |
| 1155 |
|
|
| 1156 |
< |
\subsection{\label{sec:parallelization} Parallelization of OOPSE} |
| 1156 |
> |
\subsection{\label{oopseSec:parallelization} Parallelization of {\sc oopse}} |
| 1157 |
|
|
| 1158 |
< |
Although processor power is doubling roughly every 18 months according |
| 1159 |
< |
to the famous Moore's Law\cite{moore}, it is still unreasonable to |
| 1160 |
< |
simulate systems of more then a 1000 atoms on a single processor. To |
| 1161 |
< |
facilitate study of larger system sizes or smaller systems on long |
| 1162 |
< |
time scales in a reasonable period of time, parallel methods were |
| 1163 |
< |
developed allowing multiple CPU's to share the simulation |
| 1164 |
< |
workload. Three general categories of parallel decomposition method's |
| 1165 |
< |
have been developed including atomic, spatial and force decomposition |
| 1075 |
< |
methods. |
| 1158 |
> |
Although processor power is continually growing month by month, it is |
| 1159 |
> |
still unreasonable to simulate systems of more then a 1000 atoms on a |
| 1160 |
> |
single processor. To facilitate study of larger system sizes or |
| 1161 |
> |
smaller systems on long time scales in a reasonable period of time, |
| 1162 |
> |
parallel methods were developed allowing multiple CPU's to share the |
| 1163 |
> |
simulation workload. Three general categories of parallel |
| 1164 |
> |
decomposition method's have been developed including atomic, spatial |
| 1165 |
> |
and force decomposition methods. |
| 1166 |
|
|
| 1167 |
|
Algorithmically simplest of the three method's is atomic decomposition |
| 1168 |
|
where N particles in a simulation are split among P processors for the |
| 1198 |
|
favorably then spatial decomposition up to 10,000 atoms and favorably |
| 1199 |
|
competes with spatial methods for up to 100,000 atoms. |
| 1200 |
|
|
| 1201 |
< |
\subsection{\label{sec:memory}Memory Allocation in Analysis} |
| 1201 |
> |
\subsection{\label{oopseSec:memAlloc}Memory Issues in Trajectory Analysis} |
| 1202 |
|
|
| 1203 |
< |
\subsection{\label{sec:documentation}Documentation} |
| 1203 |
> |
For large simulations, the trajectory files can sometimes reach sizes |
| 1204 |
> |
in excess of several gigabytes. In order to effectively analyze that |
| 1205 |
> |
amount of data+, two memory management schemes have been devised for |
| 1206 |
> |
\texttt{staticProps} and for \texttt{dynamicProps}. The first scheme, |
| 1207 |
> |
developed for \texttt{staticProps}, is the simplest. As each frame's |
| 1208 |
> |
statistics are calculated independent of each other, memory is |
| 1209 |
> |
allocated for each frame, then freed once correlation calculations are |
| 1210 |
> |
complete for the snapshot. To prevent multiple passes through a |
| 1211 |
> |
potentially large file, \texttt{staticProps} is capable of calculating |
| 1212 |
> |
all requested correlations per frame with only a single pair loop in |
| 1213 |
> |
each frame and a single read through of the file. |
| 1214 |
|
|
| 1215 |
< |
\subsection{\label{openSource}Open Source and Distribution License} |
| 1215 |
> |
The second, more advanced memory scheme, is used by |
| 1216 |
> |
\texttt{dynamicProps}. Here, the program must have multiple frames in |
| 1217 |
> |
memory to calculate time dependent correlations. In order to prevent a |
| 1218 |
> |
situation where the program runs out of memory due to large |
| 1219 |
> |
trajectories, the user is able to specify that the trajectory be read |
| 1220 |
> |
in blocks. The number of frames in each block is specified by the |
| 1221 |
> |
user, and upon reading a block of the trajectory, |
| 1222 |
> |
\texttt{dynamicProps} will calculate all of the time correlation frame |
| 1223 |
> |
pairs within the block. After in block correlations are complete, a |
| 1224 |
> |
second block of the trajectory is read, and the cross correlations are |
| 1225 |
> |
calculated between the two blocks. this second block is then freed and |
| 1226 |
> |
then incremented and the process repeated until the end of the |
| 1227 |
> |
trajectory. Once the end is reached, the first block is freed then |
| 1228 |
> |
incremented, and the again the internal time correlations are |
| 1229 |
> |
calculated. The algorithm with the second block is then repeated with |
| 1230 |
> |
the new origin block, until all frame pairs have been correlated in |
| 1231 |
> |
time. This process is illustrated in |
| 1232 |
> |
Fig.~\ref{oopseFig:dynamicPropsMemory}. |
| 1233 |
|
|
| 1234 |
+ |
\begin{figure} |
| 1235 |
+ |
\centering |
| 1236 |
+ |
\includegraphics[width=\linewidth]{dynamicPropsMem.eps} |
| 1237 |
+ |
\caption[A representation of the block correlations in \texttt{dynamicProps}]{This diagram illustrates the memory management used by \texttt{dynamicProps}, which follows the scheme: $\sum^{N_{\text{memory blocks}}}_{i=1}[ \operatorname{self}(i) + \sum^{N_{\text{memory blocks}}}_{j>i} \operatorname{cross}(i,j)]$. The shaded region represents the self correlation of the memory block, and the open blocks are read one at a time and the cross correlations between blocks are calculated.} |
| 1238 |
+ |
\label{oopseFig:dynamicPropsMemory} |
| 1239 |
+ |
\end{figure} |
| 1240 |
|
|
| 1241 |
< |
\section{\label{sec:conclusion}Conclusion} |
| 1241 |
> |
\subsection{\label{openSource}Open Source and Distribution License} |
| 1242 |
|
|
| 1243 |
< |
\begin{itemize} |
| 1121 |
< |
|
| 1122 |
< |
\item Restate capabilities |
| 1243 |
> |
\section{\label{oopseSec:conclusion}Conclusion} |
| 1244 |
|
|
| 1245 |
< |
\item recap major structure / design choices |
| 1245 |
> |
We have presented the design and implementation of our open source |
| 1246 |
> |
simulation package {\sc oopse}. The package offers novel |
| 1247 |
> |
capabilities to the field of Molecular Dynamics simulation packages in |
| 1248 |
> |
the form of dipolar force fields, and symplectic integration of rigid |
| 1249 |
> |
body dynamics. It is capable of scaling across multiple processors |
| 1250 |
> |
through the use of MPI. It also implements several integration |
| 1251 |
> |
ensembles allowing the end user control over temperature and |
| 1252 |
> |
pressure. In addition, it is capable of integrating constrained |
| 1253 |
> |
dynamics through both the {\sc rattle} algorithm and the z-constraint |
| 1254 |
> |
method. |
| 1255 |
|
|
| 1256 |
< |
\begin{itemize} |
| 1257 |
< |
|
| 1258 |
< |
\item parallel |
| 1259 |
< |
\item symplectic integration |
| 1260 |
< |
\item languages |
| 1256 |
> |
These features are all brought together in a single open-source |
| 1257 |
> |
development package. This allows researchers to not only benefit from |
| 1258 |
> |
{\sc oopse}, but also contribute to {\sc oopse}'s development as |
| 1259 |
> |
well.Documentation and source code for {\sc oopse} can be downloaded |
| 1260 |
> |
from \texttt{http://www.openscience.org/oopse/}. |
| 1261 |
|
|
| 1132 |
– |
\end{itemize} |
| 1133 |
– |
|
| 1134 |
– |
\item How well does it meet the primary goal |
| 1135 |
– |
|
| 1136 |
– |
\end{itemize} |
| 1137 |
– |
\section{Acknowledgments} |
| 1138 |
– |
The authors would like to thank espresso for fueling this work, and |
| 1139 |
– |
would also like to send a special acknowledgement to single malt |
| 1140 |
– |
scotch for its wonderful calming effects and its ability to make the |
| 1141 |
– |
troubles of the world float away. |
| 1142 |
– |
\bibliographystyle{achemso} |
| 1143 |
– |
|
| 1144 |
– |
\bibliography{oopse} |
| 1145 |
– |
|
| 1146 |
– |
\end{document} |
