2.2 Classical methods
2.2.1 Sampling the phase space Molecular Dynamics
All three quantities needed to establish the heme-to-heme ET rates kET - reac-
tion free energies ∆A, reorganization free energies λ and electronic coupling elements
D
|HAB|2E require the evaluation of some ensemble average. Two
decisions have to be made in this regard:
1. The computational model used to describe the system and its Hamilto-
2.2. CLASSICAL METHODS
2. The method used to sample the system’s configuration space in order to obtain ensemble averages.
In regard to the first point, the configurations to be sampled belong to the equilibrium ensemble of a protein system without any chemical changes, i. e. formation or breaking of chemical bonds. Therefore, costly electronic struc- ture methods can be avoided in favor of the computationally much cheaper Molecular Mechanics methods [57]. In constrast to electronic structure meth- ods, the approach of Molecular Mechanics does not deal with the electronic degrees of freedom explicitly; the Hamiltonian of a molecular system is ex- pressed as a function of nuclear coordinates and momenta, possibly aug- mented by additional terms describing electronic degrees of freedom in a very simplified manner, like induced point dipoles [58]. In the absence of such ad- ditional terms (referred to as the “nonpolarizable” case in the following due to the absence of electronic polarization), the Hamiltonian is generally given by: H ~R, ~P= KP~+V~R=
∑
N i=1 |~pi|2 2mi +V~R (2.8)i. e. it can be expressed as a sum of kinetic energy K~P
and potential energy V~R, with ~Rand ~Pdenoting the total of all position and momentum
vectors, respectively, and ~pi and mi momentum and mass of atom i, respec-
tively; the sum runs over all N atoms in the system.
Different functional forms for the potential energy V~R in this purely
classical model Hamiltonian exist; they all contain parameters that need to be determined by fitting to either experimental observables (e.g. vibrational fre- quencies) or quantum mechanical calculations (on simple model compounds) [59]. Functional form and parameters together then constitute a so-called force field.
A common functional form for a nonpolarizable molecular mechanical potential energy (and indeed the form used for the sampling of configuration space in this work) is given by [60]:
V~R =
∑
bonds ki 2(li− li,0) 2 +∑
angles κi 2 (θi− θi,0) 2 +∑
torsions M∑
n Vn 2 (1 + cos (nω − γ)) + N∑
i=1 N∑
j=i+1 4εi j σi j ri j 12 − σi j ri j 6! + qi· qj 4πε0ri j ! (2.9)In this equation, the total potential energy of the molecular system is de- composed into bonding terms (first three summations) and nonbonding terms (last double-summation). The bonding terms assume that each covalent bond
2.2. CLASSICAL METHODS
length li and angle θi between covalent bonds fluctuates around an equilib-
rium value (li,0 for bonds and θi,0 for angles, respectively) and that for values
close to the respective equilibrium value (for which the force field is designed only), the associated potential energy can be approximated to be harmonic with a force constant ki(for bonds) or κi(for angles), respectively. Torsional
terms (third summation) represent interactions between atoms separated by three bonds in dependence on rotation around the second (central) bond; they thus constitute four-body interaction terms of these atoms and the two atoms connecting them and are used to model steric and, to some extent, nonbonded (see below) interactions between the two outer atoms. Torsional terms are modelled as a sum of M cosine terms (with M often just equal to 1) of dif-
ferent multiplicity n, “barrier height” Vn and phase shift γ; ω denotes the
torsional angle, i.e. the angle of rotation around the bond between the two inner atoms.
The last double-summation runs over all unique pairs of atoms in the sys- tem that are at least three bonds apart from each other (or not connected by bonds at all, respectively); this sum represents electric Coulomb interactions (second addend) as well as Van der Waals interactions (first addend), the lat- ter being interactions that are even present in completely noncharged and un- polar systems. Coulomb interactions are calculated by assigning effective
(partial) atomic charges qi to each atom and calculating the interaction us-
ing Coulomb’s law, with ri j as the interatomic distance and ε0as the electric
constant; Van der Waals interactions are modelled by the so-called Lennard- Jones potential which consists of an attractive long-range dispersion interac- tion term −σi j
ri j
6
as well as a short-range repulsion term σi j ri j
12
, both to- gether resulting in a potential well of depth εi j; σi j is the interatomic distance
for which the two parts cancel out. As mentioned above, torsional interac- tion terms already partially account for nonbonded interactions as well; thus Van der Waals and Coulomb interactions are typically scaled by a scaling fac- tor smaller than one for atoms separated by three bonds. As bond and angle terms are supposed to fully describe the interactions between the atoms in- volved, nonbonded interactions are not calculated at all for atoms separated by only one or two bonds.
In order to avoid artificial boundary effects at the system’s boundary, peri- odic boundary conditions [61] are frequently used and were used in this work as well. In this approach, the simulation box is surrounded by an infinite set of identical copies (“image cells”), filling out the space entirely. If an atom leaves the central cell, its image enters on the opposite side (i. e. the atom is projected back into the central cell again). Electrostatic interactions are then calculated not only within the central cell but also between atoms in the
2.2. CLASSICAL METHODS
central cell and all image cells:
Velec ~R= 1 2 0
∑
~I N∑
i=1 N∑
j=1 qi· qj 4πε0 r~i j+~I (2.10) where Velec~Rdenotes the electrostatic potential energy; the sum ∑0 ~I
runs over all cells, i.e. central cell as well as image cells, where ~I is the transla- tion vector of a given image cell (~0 for the central cell itself); the interatomic distance ri j is substituted by r~i j+~I
, the distance between an atom i in the
central cell and atom j’s image in the image cell specified by ~I. The apos- trophe above the summation sign indicates that the interaction of an atom with itself in the central cell is still ignored, but atoms do interact with their own periodic images. To calculate this infinite sum, Ewald summation is fre- quently used [62, 63]. Van der Waals interactions are also computed between atoms in the central cell and surrounding neighbour images; however, a cut- off is usually applied for Van der Waals interactions to prevent an atom from interacting with any other atom twice [64].
This description of the electrostatic energy however disregards any polar- ization effects on individual atoms from their environment. Several simple procedures exist to incorporate such effects at least partially, like fluctuating atomic charges and the aforementioned induced dipoles [58]. In the latter
approach, each atom is assigned an isotropic atomic polarizability αi and a
polarization energy term Epol is added to the total energy, defined as [58]
Epol = −1
2
∑
i αi~E(0)
i ~Ei (2.11)
where ~Eidenotes the total electric field acting on atom i (from both static
monopole charges and induced dipoles) and ~Ei(0) denotes the field due to
monopole charges only.
Coming to the choice on how to sample the phase space, there are two par- ticularly popular computational approaches: Molecular Dynamics [65] and Monte Carlo methods [66]. Monte Carlo methods are based on randomly generating new configurations of atoms in a system and accepting or reject- ing these new configurations with a probability depending on their energy, thereby obtaining a Boltzmann (or any other desired) distribution. However, for molecular systems most random coordinate changes will sharply increase the system’s energy by e.g. severely stretching bonds and thus will lead to configurations with small Boltzmann weights; this renders the application of Monte Carlo methods to molecular systems rather difficult [67]. Molecular Dynamics, on the other side, are based on computing actual dynamics of a
2.2. CLASSICAL METHODS
system in order to obtain a trajectory in phase space along which the quan- tities of interest can be computed; the ergodic hypothesis can then be used to equate a quantity’s time average as obtained from these dynamics with its ensemble average. (Provided that the average from these dynamics of finite length can be assumed to come sufficiently close to the average that infinitely long dynamics would yield.) The modelling of a system via Molecular Me- chanics allows to describe the system’s dynamics by the classical Newtonian dynamics of its atoms; the equation to be integrated is then Newton’s equation of motion for each atom:
d2~ri
dt2 =
~Fi
mi (2.12)
where ~ri denotes the position vector of atom i, mi denotes its mass, d
2 dt2
denotes the second derivative with respect to time and ~Fiis the external force
acting on atom i; this force is in turn obtained from the potential energy’s gradient with respect to atom i’s coordinates:
~Fi= −~∇
~riV
~
R (2.13)
As for a large system with a molecular mechanical force field solving these equations analytically becomes unfeasible, numerical finite difference methods have to be employed which are based on discretizing time into tiny steps. The algorithm used in this work is the so-called velocity Verlet al-
gorithm [68]; in this algorithm, positions are updated based on positions ~ri,
velocities ~vi and accelerations ~aiat the current time t, whereas velocities can
be updated once the forces and hence accelerations at the next step t + δ t are known: ~ri(t + δ t) =~ri(t) + δ t ·~vi(t) + 1 2δ t 2·~a i(t) (2.14) ~vi(t + δ t) =~vi(t) + 1 2δ t · (~ai(t) +~ai(t + δ t)) (2.15) where δ t denotes the discrete time step of integration.
Incorporation of polarizability via induced point dipoles increases the com- putational cost of force field calculations considerably; in this work, only a nonpolarizable force field was used in sampling the phase space.