Propagating 2D dynamics on the Muller potential using OpenMM. OpenMM runs the dynamics in CUDA or OpenCL, so this is pretty snappy.

figure_1.png

LICENSE

Copyright 2020 Robert T. McGibbon

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openmuller.py

"""Propagating 2D dynamics on the muller potential using OpenMM.

Currently, we just put a harmonic restraint on the z coordinate,
since OpenMM needs to work in 3D. This isn't really a big deal, except
that it affects the meaning of the temperature and kinetic energy. So
take the meaning of those numbers with a grain of salt.
"""
from simtk.unit import kelvin, picosecond, femtosecond, nanometer, dalton
import simtk.openmm as mm

import matplotlib.pyplot as pp
import numpy as np

class MullerForce(mm.CustomExternalForce):
    """OpenMM custom force for propagation on the Muller Potential. Also
    includes pure python evaluation of the potential energy surface so that
    you can do some plotting"""
    aa = [-1, -1, -6.5, 0.7]
    bb = [0, 0, 11, 0.6]
    cc = [-10, -10, -6.5, 0.7]
    AA = [-200, -100, -170, 15]
    XX = [1, 0, -0.5, -1]
    YY = [0, 0.5, 1.5, 1]

    def __init__(self):
        # start with a harmonic restraint on the Z coordinate
        expression = '1000.0 * z^2'
        for j in range(4):
            # add the muller terms for the X and Y
            fmt = dict(aa=self.aa[j], bb=self.bb[j], cc=self.cc[j], AA=self.AA[j], XX=self.XX[j], YY=self.YY[j])
            expression += '''+ {AA}*exp({aa} *(x - {XX})^2 + {bb} * (x - {XX}) 
                               * (y - {YY}) + {cc} * (y - {YY})^2)'''.format(**fmt)
        super(MullerForce, self).__init__(expression)
    
    @classmethod
    def potential(cls, x, y):
        "Compute the potential at a given point x,y"
        value = 0
        for j in range(4):
            value += cls.AA[j] * np.exp(cls.aa[j] * (x - cls.XX[j])**2 + \
                cls.bb[j] * (x - cls.XX[j]) * (y - cls.YY[j]) + cls.cc[j] * (y - cls.YY[j])**2)
        return value

    @classmethod
    def plot(cls, ax=None, minx=-1.5, maxx=1.2, miny=-0.2, maxy=2, **kwargs):
        "Plot the Muller potential"
        grid_width = max(maxx-minx, maxy-miny) / 200.0
        ax = kwargs.pop('ax', None)
        xx, yy = np.mgrid[minx : maxx : grid_width, miny : maxy : grid_width]
        V = cls.potential(xx, yy)
        # clip off any values greater than 200, since they mess up
        # the color scheme
        if ax is None:
            ax = pp
        ax.contourf(xx, yy, V.clip(max=200), 40, **kwargs)


##############################################################################
# Global parameters
##############################################################################

# each particle is totally independent, propagating under the same potential
nParticles = 100  
mass = 1.0 * dalton
temperature = 750 * kelvin
friction = 100 / picosecond
timestep = 10.0 * femtosecond

# Choose starting conformations uniform on the grid between (-1.5, -0.2) and (1.2, 2)
startingPositions = (np.random.rand(nParticles, 3) * np.array([2.7, 1.8, 1])) + np.array([-1.5, -0.2, 0])
###############################################################################

system = mm.System()
mullerforce = MullerForce()
for i in range(nParticles):
    system.addParticle(mass)
    mullerforce.addParticle(i, [])
system.addForce(mullerforce)

integrator = mm.LangevinIntegrator(temperature, friction, timestep)
context = mm.Context(system, integrator)

context.setPositions(startingPositions)
context.setVelocitiesToTemperature(temperature)

MullerForce.plot(ax=pp.gca())

for i in range(1000):
    x = context.getState(getPositions=True).getPositions(asNumpy=True).value_in_unit(nanometer)
    pp.scatter(x[:,0], x[:,1], edgecolor='none', facecolor='k')
    integrator.step(100)

pp.show()