L A RECONSTRUCCIÓN IDEALIZADA DEL MERCADO

7.1 Molecular dynamics

7.1.1 Molecular simulation techniques

The termmolecular modellingdescribes a number of computer aided tech- niques and algorithms that are used to determine properties of molecules with and without time evolution. The most common methods are quantum mechanics (QM), molecular mechanics (MM) and their joint application (QM/MM).

Molecular mechanics describes methods that treat atoms or groups of atoms as classical objects that can be described by Newtonian mechanics. The most prominent examples are Monte Carlo (MC) simulations[400]and molecular dynamics (MD) simulations[401,402]. MCandMDdiffer in that in

MCtime does not exist and only energies but no forces are computed while

MDexplicitly propagates time and in each step energies as well as forces are

calculated.

Molecular dynamicsis a step wise technique that takes a set of particles where for each particle the3D coordinates and velocities are known, determ- ines forces on each particle according to a force field, i.e., a set of parameters that define potentials between particles, and calculates new positions and velocities of each particle after a time stepΔt. The energy at any given step is the sum of the potential energy, which is given by the force field, and the kinetic energy, which is explicitly known through the velocities, of all particles.

A variation ofMDis stochastic dynamics (SD).[403]Here, in addition to theMDprocedure, atoms of the solute are subject to a random displacement

in each step and velocities are scaled down by a factor to simulate friction. The idea behindSDis, that it allows for simulating a solute molecule without

explicitly treating the solvent, but instead implicitly approximating the effects of the solvent.

MDplays an important role among the molecular modelling techniques

as it is the computationally cheapest method that simulates every atom and includes a concept of time. As a result,MDallows simulations of systems of 105–106atoms over a period ofns–ms, carrying out 106–1012integration steps.[402,404]

At the same time, the accuracy of MDsimulations is lower than semi-

empirical or even quantum mechanical methods. Most force fields can only operate on a fixed molecular graph; breaking or forming bonds is not possible. Despite these limitations to electronically simple and non-reacting systems there is a wide range of systems—especially bio-macromolecules— that are successfully modelled usingMD. This is particularly true for large systems that are characterised by their dynamic behaviour which requires simulations over long timespans to explore the energy landscape and per- form statistical thermodynamics on the trajectory.

7.1.2 Energies and Forces

Force fields divide the total energy of a system and the force on each particle (typically atom) into specific and easy to calculate energy/force terms (eq.7.2).

Vtotal=Vbonded+Vnonbonded+Vspecial (7.1)

=V_bonds+V_angles+· · ·+V_{electrostatic}+V_VdW+· · ·+VXRay+VNMR

(7.2) Here,V_bonded is the energy of bonded interactions (bonds, angles, dihedrals etc.),Vnonbondedis the energy by physical interactions that do not act through

bonds (typical examples are Coulombic interactions and Van der Waals interactions) and V_special denotes auxiliary energy terms that are used to impose restraints on a system, e.g. to take experimental data into account. Each of these terms is a sum over interactions between a subset of the simulated particles: Very few particles (e.g. two atoms per bond and four atoms per dihedral) in the case of bonded interactions, or all particles within a chosen radius from a given particle in the case of non-bonded interactions.

Interactions are typically approximated as simple potentials like a har- monic oscillator for bonds (eq.7.3) and angles or a periodic potential for dihedral angles. More complex potentials can be chosen but increase the computational cost. For example, the bond term is typically expressed as

Vbonds= bonds

∑

where the indexiruns over all bonds, and the values of the force constantsk_i

and the equilibrium valuesr(_i0)are specific to the bond type andri is the

current value of the bond length in the simulated system. The set of these values is fitted to measured or calculated values or vibrational frequencies and is collectively described as the force field.

Forces on atomsi inMDsimulations are obtained by deriving the po- tentials that are given in the force field according to the coordinates of the atoms (eq.7.4).

f_i =−∂_∂V_r

i (7.4)

In general, the forces on one atom in a system are affected by the co- ordinates of all other atoms in that system. Taking only two-body inter- actions into account, there are 1₂N(N−1) interactions (short range and long range) that contribute, whereNis a measure for the system size, i.e., the number of simulated particles. Using a naive approach, the force field would scale O(N2). There are algorithms for calculating short range as well as long range interactions that reduce the computational complexity toO(N32)(Ewald summation[405]),O(Nlog(N))(particle-particle/particle-

mesh summation[406], particle-mesh Ewald summation[407]) or evenO(N)

(fast multipole method[408]). The more efficient algorithms only pay off for very large systems.[409]_{Nevertheless, the calculation of forces is the most}

expensive step in a molecular modelling simulation (and among those, the non-bonded interactions dominate).

7.1.3 Integration schemes

Given the coordinatesrt, the velocity vt and the mass m of a particle at timetas well as the forces on the particle (which are generally a function of the coordinates of some or all other particles), there are different algorithms to compute its coordinates and velocitiesrt+Δtandvt+Δtat timet+Δt.

A good integration algorithm must fulfil a number of requirements: The energy (or a derived extensive thermodynamic potential, if a thermodynamic ensemble other than the microcanonical is chosen[410]) of the system needs to be conserved, preferably both on a short and on a long time scale. Integration algorithms can be stable with respect to the energy for a small number of cycles but have a long term energy drift. Alternatively, the energy can fluctuate on a short time scale while the energy is conserved on a long time scale. Long term energy conservation is more important than short term energy conservation.

Closely related to the requirement of energy conservation is the demand that the simulation be time reversible, i.e., the trajectories of all particles should be reversed if at one step all velocities are reversed and the simulation continued.

The overall momentum of a simulated system should stay constant. Generally, this means that the sum over the momenta of all particles is zero after each time step. The same requirement exists for the angular momentum.

The previous two points on time reversibility and conservation of (angu- lar) momentum are frequently summarised by demanding algorithms to be area preserving. If one plots the trajectory of any particle in the simulated ensemble in anr–vdiagram, the area (i.e.,6dimensional volume) the particle traverses should be constant after a number of initial simulation cycles. For some algorithms the trajectory is increasingly smeared out, which is an indicator for violated time reversibility.

As in any numerical procedure,t the integration of the equations of motion is subject to noise. Even worse, all but the smallest molecular

s_{This subsection is predominantly based on ref. [409]}

t_{Analytical integration methods cannot be used here because the time evolution of a system}

that consists of three or more particles that act on each other cannot be expressed in a closed form (three body problem[411]_).

systems behave chaotically with the result that two trajectories that differ exponentially drift apart from each other. Small changes in the starting positions or the integration scheme lead to huge differences in the obtained trajectories. We are, however, less interested in the trajectories themselves than in derived properties and the populated regions of the phase space. These thermodynamic properties (such as energy, free energy, temperature and pressure) are relatively insensitive to changes of the trajectory. The obtained thermodynamic properties are only weakly affected by numerical noise. The relative population of different geometries in the phase space of a converged simulation is ideally not affected by diverging trajectories during the simulation or the exact starting point. Strong divergence may, however, lead to qualitatively wrong geometries.

One of the most commonly used step wise integration algorithms is the

Verlet algorithm. It can easily be derived by adding the Taylor expansions (aroundt) ofr(t+Δt)andr(t−Δt), r(t±Δt) =r(t)± dr(t) dt Δt+ d2r(t) dt2 Δt2 2 ± d3r(t) dt3 Δt3 6 +O(Δt 4_{), (7.5)}

yielding an expression forr(t+Δt)that is accurate to orderΔt4and does not explicitly depend onv(r),

r(t+Δt) =2r(t)−r(t−Δt) + d 2_r(t)

dt2 Δt

2_+O(Δt4_{). (7.6)}

The difference between the Taylor expansions ofr(t+Δt)andr(t−Δt)

yields the velocity att,

r(t+Δt)−r(t−Δt) =2dr(t) dt Δt+2 d3r(t) dt3 Δt3 6 , (7.7) i.e., v(t) =r(t+Δt)−r(t−Δt) 2Δt + d3r(t) dt3 Δt2 6 O(Δt2₎ . (7.8)

Equivalent to the Verlet scheme is the Leap Frog algorithm. It differs from the former in that positions r and velocitiesv are not given at the