Modeling Macromolecular Complexes: A Journey Across Scales - Modeling in Computational Biology and Biomedicine - page 11

accurate results, e.g. [ 62 ]. The relevant particles whose coordinates are required
in order to define a configuration X are thus limited to the set of atoms forming
the macromolecule and the solvent included in the system. The forcefield approach
provides an empirical energy model that is parameterized by referring to molecules
whose structures and physical properties are known to a high degree of accuracy.
The applicability of the forcefield to, say, a protein depends on the transferability of
the parameters. The potential energy is usually formulated as a sum of many terms,
which is designed to take into account as many physical interactions as possible.
Non-covalent interactions such as the van der Waals interaction are typically treated
using a Lennard-Jones potential. In this function the interaction energy is negative at
distances on the order of the sum of the vdW radii but climbs steeply as the distance
of approach of the two atoms is reduced, while at longer distances the energy
tends towards zero. For covalently bonded atoms, the parameters are often simple
Hooke's-law style force constants associated with deviation of a bond distance or
angle from a reference value; this value is itself another parameter. Dihedral angles
are used to model the energy of rotation about a covalent bond, for which a term is
constructed having the number of minima chosen as a function of the bond order.
Functional forms can be found in texts, such as [ 41 ].
Electrostatic interactions between charged atoms or groups present somewhat of
a special case. The simplest approach is also the most demanding computationally:
the protein and the solvent are both represented explicitly, using a large number of
molecules of water together with dissolved ions, in order to model as faithfully as
possible an entire region of aqueous solution containing the protein. For accurate
energy calculations this approach requires on the order of 10

more atoms of
solvent than of the macromolecule itself. A second approach treats the solvent as
a continuum dielectric and resorts to solving the Poisson-Boltzmann equation to
obtain solvation energies. It is then necessary to define the dielectric boundary
between the bulk and the protein itself, and to ignore the molecular nature of the
water itself. The dielectric boundary may be defined as the union of vdW spheres
representing the atoms in the macromolecule. The vdW volume for the molecule as
a whole is smaller than the sum of the volumes of its atoms, because atoms joined
by a covalent bond lie closer together than the sum of their vdW radii. However,
there are inevitably a large number of empty spaces (packing defects) in the interior
of the macromolecule as well. In the corresponding electrostatic calculation, these
interstitial volumes are formally added to the bulk solvent—even if they provide
too little space for a physical solvent molecule to lodge [ 21 ]. Other approaches rely
on the solvent accessible surface (SAS) to define the dielectric boundary, as we
shall detail in Sect. 1.2.3 . In short, the SAS consists of enlarging the atoms, so as
to guarantee that only truly solvent-accessible cavities in the macromolecule are
counted for the water contribution (Fig. 1.2 a).

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All-atom simulations. Once one has a suitable potential energy function, together
with a set of initial positions of the atoms of the macromolecule (e.g., the PDB
structure) and a set of pseudo-random starting atom velocities consistent with a
given temperature, the equations of motion for the system can be solved numerically.