Vetores de prueba para PRESENT

them are Fi=−∂V vdw ∂ri =48εi j r2_{i j σ} i j ri j 12 −1 2 σi j ri j 6 ri j (1.31) and Fj=−Fi (1.32)

In a 6–12 potential, ther−6_{term describes the attractive long-range interaction and is physi-}

cally well justified. Ther−12_{term provides the short range repulsion but has no physical justifi-}

cation. It approximates Pauli repulsion, but was originally used as it is easy to obtain from the r−6term, even though a 6–exp function would provide a better representation of the potential.38 Use of the 6–12 functional form has continued even as computer power has increased markedly over the years since force fields were originally developed.

The second component of the non-bonded terms are the electrostatics. The electrostatic term of equation 1.9 can be partitioned into three separate terms; asurfaceterm, apairwiseterm and aself interaction term, as given by

Velec_{(r;q) =V}srf_{(r;q) +V}pws_{(r;q) +V}self_{(r;q) (1.33)}

1.2. Force Fields

The form of these terms varies depending on the scheme used to evaluate the long range electro- static interactions. Such schemes include reaction-field,39,40_{Ewald summation,}41 _{and Particle-}

mesh Ewald (PME) summation.42The GROMOS force fields were parameterised for use with the reaction-field scheme, which is explained here.

The surface term arises from the definition of the medium surrounding an infinite periodic system such as when periodic spatial boundary conditions are utilised. It is defined as

Vsrf_(r;q):= M2

2πε0(2εR+1)V

(1.34) whereε0is the permittivity of vacuum,εRis the relative permittivity of the medium in which the

simulation is performed,V is the volume of the simulation box, andMis the dipole moment of the box.

The pairwise term is the most expensive and is only evaluated for atom pairs which are not excluded, i.e. 1–2 and 1–3 atom pairs, and if the distance between the atoms is less than the cut-off distance. It is given by

Vpws_{(r;q) =} 1 4πε0εR N−1

∑

∑

whereqiis the charge on atomiandψi j(r)is theelectrostatic influence functionassociated with

the atom pairi j ψi j(r):= 1 ri j− Cr_{i j}2 2R3_F− 1−1₂C RF (1.36) C:=(2εR−2εF)(1+κRF)−εF(κRF) 2 (εR+2εF)(1+κRF) +εF(κRF)2 (1.37) whereεFis the permittivity of the reaction-field,κis the inverse Debye screening length andRF

is the radius of the reaction-field. Finally, the self interaction term is given by Vself_{(r;q) =} 1 8πε0εR N

∑

With all terms combined together, the force on atomsiand jdue to the reaction-field electro- static interaction between them is given by

Fi= qiqj 4πε0εR 1 r3 i j +Cri j RF ri j (1.39)

1. Introduction

and Fj=−Fi (1.40)

Compared with the more realistic PME scheme for calculating long range electrostatic interac- tions, the reaction-field scheme has some deficiencies. PME is fundamentally a computationally efficient means to approximate a solution to the Ewald summation problem, which gives the ex- act electrostatics but is slow to converge. As such, PME describes the electrostatics of heteroge- neous systems much more realistically than a reaction-field, which assumes a constant dielectric continuum, does. However, this comes at a much greater computational cost. Additionally, it has been shown that small changes in the force field terms have a much greater effect on simulations than any difference due to treating the long range electrostatics differently.Poger2012

Additional non-bonded components can be included within the force field. For example, AMBER was originally developed with an additional 10–12 term to allow greater control over the hydrogen bonding interactions within systems.21,22 This term was later removed by Cornell et al.as it was no longer necessary due to improvements in the electrostatic and van der Waals parameter values.23

Parameter values for the non-bonded components of force fields are the most difficult, and in some ways the most important, values to define. Electronic point charges are relatively simple to obtain. Quantum mechanical calculations can be performed on the molecule, at an appropriate level of theory, and the electronic distribution obtained collapsed into point charges for each of the atoms. This approach does have a number of drawbacks.

Firstly, it is well known that charge distributions are highly conformationally dependent.43–46 These charge variances are not small. Stouch and Williams noted variation of nearly one elec- tronic unit in the charge on a methylene carbon in glycerylphosphorylcholine, depending on the conformation.43_{Such a dependence indicates that point charges derived from a single con-}

formation may not represent the molecule sufficiently accurately during a molecular dynamics simulation. The obvious solution to this is to fit the charges based on a number of conforma- tions. Cornellet al. have shown that such multi-conformational derived charges do have im- proved results over single conformational charges.47This will not always be the case though as the choices of which conformations to use will have an effect on the results obtained. Unwisely chosen source conformations will give charges just as poorly transferable as single conformation charges. In general, all of conformational space is too large to sample efficiently, so informed decisions on which conformations to use need to be made.

Secondly, the symmetry of a molecule plays an important role in the charge distribution. Quantum mechanical calculations on an isolated benzene molecule in implicit water would be expected to produce similar charges on all the hydrogen atoms. Figure 1.3 shows that there are dramatic differences between the charges assigned to the benzene hydrogen atoms when no

1.2. Force Fields

Figure 1.3: The partial atomic charges of hydrogen atoms in benzene derived by fitting to the electro- static potential at B3LYP/6-31G* level of theory optimised assumingC1and D6h symmetry using the Kollmann-Singh method.

symmetry is enforced (C1) compared with whenD6h symmetry is enforced. Symmetry, both

local to certain areas of the molecule and global, must be considered when assigning point charges to atoms.

Finally, the point charges obtained are highly dependent on the level of quantum theory used to calculate the electronic density, in particular the basis set used. Pople’s 6-31G* basis set48is a popular choice, and is used in the derivation of a number of force fields.18,21,23,49,50The 6-31G* basis set is known to uniformly overestimate molecular polarity in the gas-phase. However, molecules in the condensed phase are expected to be more polarised than those in the gas phase. As such, Kuyperet al. suggested that the 6-31G* basis set is the logical choice for deriving partial charges.51

Transferability is another consideration. OPLS is developed so that functional group sub- units are neutrally charged overall, making derivation of charges for novel molecules trivial. In contrast, AMBER has charges derived on a case-by-case basis.23 _{Which option is chosen is a}

matter of preference, though transferability of functional group charges between different kinds of molecules may not always be applicable.

Van der Waals parameter values are more difficult to define than the electrostatic term. Rappe et al.developed a means to systematically derive van der Waals parameter values in their Univer- sal Force Field (UFF), based mainly on the element type with some empirical data included.52_In

general though, van der Waals parameter values are derived by an initial estimate of reasonable values, then refined through empirical means until selected reference data can be reproduced.

There are a number of potential sources for reference data against which to parameterise the van der Waals terms. Hagler et al. derived parameters for sp2 atoms from fits of lattice energies and crystal structures in amides,53_{Dunfieldet al.determined 6–12 parameters for UA}

1. Introduction

CH, CH₂ and CH₃ parameters based on crystal packing calculations of hydrocarbons,54 and Jorgensen calculated parameters for OPLS from Monte Carlo liquid simulations of ethers and alcohols.25 With a wide range of experimental data available, there is no single right type of data to aim to reproduce. AMBER is parameterised to fit experimental conformational and vibrational energy profiles,23whereas GROMOS is parameterised to reproduce liquid densities, heats of vaporisation and free energies of hydration.33,34,36 The fact that different force fields exist and can generate results consistent with their designed purpose shows that there is no one right philosophy when it comes to force field parameterisation.

1.3. Other Force Field Types

The majority of force fields mentioned above follow the same functional form as shown in equation 1.9. Several ways to improve upon this functional form are available. Firstly, the intramolecular terms of standard force fields can be improved upon. As mentioned earlier, a number of force fields include improper torsional terms which aid in maintaining chirality and out-of-plane vibrational modes. Such terms can be defined between any arbitrary set of atoms, though generally they are defined between non-interconnected atoms around the same centre.

Some force fields include additional bonded terms to those shown in equation 1.9. For in- stance, CHARMM has an additional Urey-Bradley term for in-plane deformations, and sepa- rating symmetric and asymmetric bond stretching.18,19 Only the quadratic portion of the Urey- Bradley function is used as the linear term can be excluded.55MMFF94 goes even further. The bond stretch term is expanded to include cubic and quartic components as well as the general quadratic portion, the angle bend term is expanded to include a cubic portion or, in the case of near-linear bond angles, replaced with a sinusoidal term.56Additional terms for representing the coupling between thei–jandk–jbond stretches and thei–j–kangle bend as well as out-of-plane bending at tricoordinate centres are included. Non-bonded terms are also modified, employing a “buffered” form for both the van der Waals and electrostatic terms.57_{Such improvements come}

at a computational cost, so as always, using a force field that better represents the energy of a system has to be traded off with the ability to simulate larger systems for longer time periods.

In the majority of standard force fields, a major limitation of the intermolecular terms is the use of fixed atomic point charges. Electron distributions are inherently anisotropic, for example with lone pairs of electrons and delocalisedπ-bond clouds. The isotropic nature of fixed atomic point charges means they lack the mathematical flexibility to describe certain features of molec- ular charge distributions, such as the anisotropic nature, and are unable to respond to changes in the molecular environment, for example through conformational changes. No amount of repa- rameterisation can change this basic fact, meaning fixed atomic point charge model force fields

1.3. Other Force Field Types

will never be able to describe the electrostatics of general polar molecules to within chemical accuracy. Polarisable force fields extend the fixed atomic point charge concept to allow for some degree of polarisation of the electron distribution.

There are a number of existing methods which add polarisation effects to a force field. Fluc- tuating charges/charge equilibration allows individual atomic partial charges to change over the course of a simulation. By treating the charges as additional degrees of freedom, charges are allowed to flow between atoms until instantaneous electronegativities are neutralised.58 _The

CHEQ force field of Bauer and Patel implements this method of polarisation, and is devel- oped within the CHARMM program.59,60 A fluctuating charges implementation has very little additional computational overhead when compared with a fixed point charge force field, though as charges flow along bonds only, it cannot easily represent polarisation orthogonal to bonds, as is required to correctly represent planar molecules such as benzene. In principle, this could be alleviated by adding additional point charges to represent charge density not localised to specific atoms.61

Drude oscillators represent charges on each atomic centre as a pair of point charges, the sum of which gives the total charge on the atom. The first charge is centred on the nucleus and the second is a massless particle (Drude particle) attached to the nucleus by a spring.62,63 _{This ap-}

proach is relatively easy to incorporate into existing force fields but because of the large number of extra charges added, computational cost is increased markedly. MacKerell and Roux’s groups have developed a force field based on the Drude model.64–66In this model, polarisability is de- termined solely by the charge on the Drude particle (as the force constant on the springs is the same across all atom types), and additional point charges with fixed magnitude and location are included to allow for better representation of hydrogen bonding.

The AMBER ff02 force field includes polarisation through the use of inducible dipoles.67 Fixed atomic charges are retained, with additional inducible point dipoles added, generally to the atomic nuclei,68 _{but occasionally to the bonds between atoms.}69 _{The induced dipole at a}

particular site is determined by the electric field at that site. Use of inducible dipoles means that extra terms to account for the various interactions between dipoles and charges have to be introduced into the force field. This makes implementation a challenging process, though the pa- rameterisation of such a force field is relatively straight forward61. The Polarisable Simulations with Second order Interaction Model (POSSIM) also employs an inducible dipoles model.70–73

Software developed along with the force field allows for a speed up of the polarisable component of the calculations by around an order of magnitude, without loss of accuracy, when compared with traditional evaluation procedures.

Finally, a more rigorous approach is to do away with point charges all together. By using multipole moments that include terms up to hexadecapoles, electronic charge density can be

1. Introduction

modelled in a way that naturally captures the anisotropic and non-spherical nature of the den- sity.74,75_{AMOEBA is the most widely used force field that includes multipole electrostatics.}76–78

Terms up to quadrupole moments are included. AMOEBA has been used to generate incredibly accurate peptide electrostatic properties,79results which would be impossible to obtain with a non-polarisable force field. Of course, this accuracy comes at a cost. AMOEBA is a computa- tionally intensive force field, meaning a choice must be made between the accuracy it provides, and the simulation system sizes and timescales attainable with a simpler model.