(272g) Multicanonical Simulations Based On Contact Maps for Square-Well Heteropolymers Using Event Driven Molecular Dynamics

The folding of proteins into a unique, specific 3-dimensional
structure is driven primarily by the hydrophobic effect, with
hydrophobic groups buried in the interior of the molecule, away from
contact with water, and hydrophilic groups on the surface of the
molecule, solvated with water.  It is the ability of the protein to
fold into its native conformation that allows it to function (e.g.,
catalyze a biochemical reaction).  While the hydrophobic effect drives
protein folding, it also can lead to protein misfolding (trapped in a
non-native conformation) and, in protein solutions, to aggregation,
where the hydrophobic groups from distinct proteins adhere to each
other.  The misfolding and aggregation of protein solutions is thought
to be the basis of Alzheimer's and Parkinson's diseases.  They also
lead to difficulties for bioprocessing, such as in the recovery and
purification of biopharmaceuticals.

The delicate competition between intramolecular and intermolecular
interactions determines whether the proteins in a solution will remain
folded or aggregate.  While simulations of detailed, fully atomistic
models have significantly advanced our understanding of protein
behavior, they are too computationally intensive to directly apply to
the aggregation problem.  Coarse-grained protein models, where the
protein is represented by a collection of bonded hard spheres that
interact with each other with a square-well potential, are simple
enough to offer the possibility of simulating bulk protein solutions
but are sufficiently complex to capture the essential physics of
protein behavior (e.g., folding) [1].  By varying the monomer size,
bond length, and range of interaction, these off-lattice models can
exhibit intricate three-dimensional structures, such as helices [2,3]
and other more complicated configurations.

The use of a discontinuous interaction potential (i.e., hard sphere
plus a square-well attraction) greatly simplifies the integration of
the equation of motion.  We have recently developed an extremely
efficient event driven molecular dynamics simulation package,
DynamO [4], which allows the study of large systems and sufficiently
long time scales needed to describe the different stages of protein
folding and aggregation on relatively modest computing hardware.
These large-scale simulations are necessary, in order to properly
investigate the coupling between the intra and intermolecular
interactions.

In order to determine the equilibrium behavior of these systems, it is
crucial that the important configurations of the system can be rapidly
and properly sampled.  Even these relatively simple coarse-grained
models possess a fairly rugged free energy landscape, and conventional
canonical Monte Carlo and molecular dynamics simulations have limited
applicability.  Simulation methods have been developed to try to
overcome the difficulties associated with rough free energy
landscapes, such as the replica exchange method and multicanonical
simulations.

For monomers which interact through square-well potentials, the
structure of the molecule can characterized by its contact map, a list
of monomer pairs that are located within each other's interaction
well.  In this work, we develop a multicanonical simulation technique
where distinct contact maps are sampled with equal likelihood by
associating different multicanonical weights to each contact map.
This has the advantage over conventional multicanonical simulations,
where different energies are sampled with equal probability, because
the use of contact maps gives the ability to distinguish between
different folded states with the same energy and allows us to label
and follow the folding pathway of the molecules.  In addition, it
allows the determination of the free energy associated with a
particular contact map.  These multicanonical contact map simulations
were performed using event driven molecular dynamics by altering the
collision rules between monomers.  This has been implemented within
DYNAMO.  In addition, an algorithm has been develop to iteratively
determine value of the multicanonical weights.

To test this method, we first determine the density of states of
homopolymer models, which exhibit a coil-helix transition, and compare
the efficiency with other simulation techniques.  Then we examine the
structure and folding of a series of protein sequences, focusing on
characterizing the folded state of the proteins, with particular
attention paid to its entropy.  Finally, the interaction between pairs
of protein molecules are examined to understand the influence of
intermolecular interactions on the folding of the individual proteins.

[1] JE Magee, J Warwicker, and L Lue, "Freezing and folding behavior
    in simple off-lattice heteropolymers," J. Chem. Phys. 120, 11285
    (2004).
[2] JE Magee, VR Vasquez, and L Lue, "Helical structures from an
    isotropic homopolymer model," Phys. Rev. Lett. 96, 207802 (2006).
[3] MN Bannerman, JE Magee, and L Lue, "Structure and stability of
    helices in square-well homopolymers," Phys. Rev. E 80, 021801
    (2009).
[4] MN Bannerman, R Sargant, and L Lue, "DynamO: A free O(N) general
    event-driven molecular dynamics simulator," J. Comput. Chem. 32,
    3329 (2011).