Implicit Solvent Methods

Water molecules were included in every simulation performed in the course of this project. However, later in the project, two implicit solvent methods, where the solvent is modelled as a continuum166(p592), were used to estimate binding affinities. The explicit water was removed from the recorded trajectories prior to analysis.

1.4.3.2.1. The MM-PBSA Method

In the molecular mechanics-Poission-Boltzmann surface area (MM-PBSA) implicit solvent method (figure 3)194, a thermodynamic cycle is employed; the free energy of solvation for the ligand, protein and complex are used to convert a binding energy predicted in vacuo to one for in solution.

Figure 1.15: The Bordwell thermodynamic cycle used in the MM-PBSA method. The free energy of

binding in solution (∆𝑮_{𝒃𝒊𝒏𝒅}𝒂𝒒 ) is determined from more easily calculable energies, namely, the free energy

of binding in the gas phase (∆𝑯𝒃𝒊𝒏𝒅𝒈𝒂𝒔 − 𝑻∆𝑺𝒃𝒊𝒏𝒅𝒈𝒂𝒔), and the free energies of solvation of the ligand, the

receptor and the complex, ∆𝑮_{𝒔𝒐𝒍𝒗}𝑳𝒊𝒈𝒂𝒏𝒅,, ∆𝑮𝒔𝒐𝒍𝒗𝑷𝒓𝒐𝒕𝒆𝒊𝒏, and ∆𝑮𝒔𝒐𝒍𝒗

𝒄𝒑𝒙

, respectively. This diagram has been adapted

from a published figure195.

The total free energy change of solvation can be modelled as the sum of two components, an electrostatic component, ΔGelec, and a non-polar component, ΔGnonpolar.

87 When a species becomes solvated, the dielectric permittivity of the region surrounding the

atoms changes. In the MM-PBSA method, the Poisson-Boltzmann equation is used to calculate the resulting change in electrostatic potential around the species. This is then used to determine the change in electrostatic energy, ΔGelec.

∆𝐺_{𝑒𝑙𝑒𝑐} =1

2∑ 𝑞𝑖(r𝑖)∆𝜙(r𝑖)

r

1.71 Where qi is the charge of atom i at position ri experiencing a change in electrostatic potential of Δφ(ri)165(p292,p607). Calculation of the change in electrostatic potential is discussed in more detail below.

The solvent accessible surface area of the species is used to estimate the non-polar component of solvation, ΔGnonpolar.

∆𝐺_{𝑛𝑜𝑛𝑝𝑜𝑙𝑎𝑟} = 𝛾𝐴 + 𝑏 1.72

Where A is the solvent accessible surface area and γ and b are empirically derived constants (which can be used with any molecule).

The solvent accessible surface area indicates the size of this non-polar component because it is roughly proportional to the number of water molecules in the first solvation shell around the species166(p609). Two factors contribute to the non-polar component of the free energy of solvation, an energetically unfavourable cavitation component, ΔGcav and an energetically favourable, van der Waals component, ΔGvdW

196

, both of which are proportional to the size of this first solvation shell.

∆𝐺_{𝑛𝑜𝑛𝑝𝑜𝑙𝑎𝑟} = ∆𝐺_𝑐𝑎𝑣+ ∆𝐺_𝑣𝑑𝑊 1.73

ΔGcav accounts for the decrease in entropy of solvent molecules as they are forced out of the volume to be occupied by the solute and into a more dense and regular arrangement in solvent shells around the solute197. The volume of a complex is roughly equal to the sum of the unbound protein and unbound ligand volumes so any dependence of ΔGnonpolar on species volume should be largely cancelled out when ΔGnonpolar is used to calculate the binding energy of a ligand for a protein.

ΔGvdW is the free energy change due to van der Waals forces (instantaneous dipole-induced dipole attractions and Pauli exclusion principle-related repulsions) the form between the surface of the solute and the first solvation shell of solvent molecules.

Returning to the electrostatic component from Equation 1.70, ΔGelec, calculation of the

electrostatic potential, φ(ri), requires a partial differential equation to be solved. If the dielectric constant of the medium were the same throughout the system (independent of r), then the following form of the Poisson differential equation166(p603) could be used.

88 ∇2_{𝜙(r) ≡ ∇ ∙ ∇𝜙(r) = −}4𝜋𝜌(r)

𝜀 , 1.74

Where ρ(r) is the charge density and ε is the permittivity. 2 indicates application of the del- squared operator (Equation 1.24) which is equivalent to finding the gradient in the x, y and z directions ( operator) and then finding the gradient of each of these gradients and adding these second derivative results (·).

It is assumed when using Equation 1.74 that the change in dielectric constant across the system is uniform when solvation occurs. In practice, the dielectric constant is position dependent due to the motion of mobile charges down the electrostatic potential gradient when the permittivity changes. A term based on the Boltzmann distribution must be introduced to account for this effect.

∇ ∙ ε(r)∇𝜙(r) − 𝜀𝜅 sinh 𝜙(r) = −4𝜋𝜌(r), 1.75

where κ is known as the Debye-Hückel inverse length166(p604).

𝜅 = √ 𝜋𝑁_125𝑘𝐴𝑒2 _𝑇𝐼 1.76

Where I is the ionic strength, T is the temperature and e is a constant, the charge of a proton. Equation 1.75 can be simplified further, when the electrostatic potentials are very small (𝑒|𝜙(r)| ≪ 𝑘𝑇) because sinh 𝑥 =1 2(𝑒𝑥− 𝑒−𝑥) = ∑ 𝑥2𝑛+1 (2𝑛 + 1)! ∞ 𝑛=0 = [𝑥 +𝑥3 6 + 𝑥5 120+ 𝑥7 5040+ ⋯ ] ≈ 𝑥 1.77

Where e is the base of natural logarithms.

The result is the linearized Poisson-Boltzmann equation.

∇ ∙ ε(r)∇𝜙(r) − 𝜀𝜅𝜙(r) = −4𝜋𝜌(r) 1.78

This linearized equation often gives good results even when 𝑒|𝜙(r)| is of a similar size to 𝑘𝑇198 and it is the form of the Poisson-Boltzmann equation used to predict MM-PBSA binding energies by the mmpbsa.py program of the Amber suite, which was used in this project. 1.4.3.2.2. The MM-GBSA Method

The Molecular Mechanics-Generalised Born Surface Area (MM-GBSA) method of predicting the free energy of binding in a solvent is similar to that of the MM-PBSA method with the difference that the free energy of solvation is calculated using the Generalised Born equation rather than the Poisson-Boltzmann equation.

In Born’s model of solvation199

, charges are treated as ions surrounded by a spherical solvent cavity of equal and opposite charge.

89 For a single ion,

𝐺𝑒𝑙𝑒𝑐 = −

𝑞2

2𝜀𝑅 1.79

Where Gelec is its electronic free energy, q is the charge of the atom, ε is the relative permittivity of its environment and R is the radius of the cavity in the model, which corresponds to the van der Waals radius of the charged particle166(p594).

For a system containing many particles (represented using i and j below), the electrostatic energy comprises this Born solvation component (from Equation 1.79) plus the energy of the Coulomb interaction between the particles.

𝐺_{𝑒𝑙𝑒𝑐} =1 𝜀∑ ∑ 𝑞_𝑖𝑞_𝑗 𝑟𝑖𝑗 𝑁 𝑗=𝑖+1 𝑁 𝑖=1 − 1 2𝜀∑ 𝑞_𝑖2 𝑅𝑖 𝑁 𝑖=1 1.80

Where rij is the distance between particles i and j and Ri is the effective Born radius of the particle.

In the model, solvation of an ion is likened to bringing the particle from a vacuum (ε=1) into the solvent sphere, which has relative dielectric ε.

∆𝐺_{𝑒𝑙𝑒𝑐} = −𝑞2 2𝑅(1 −

1

𝜀) 1.81

When multiple particles are considered166(p598), ∆𝐺_{𝑒𝑙𝑒𝑐} = − (1 −1 𝜀) (∑ ∑ 𝑞_𝑖𝑞_𝑗 𝑟𝑖𝑗 𝑁 𝑗=𝑖+1 𝑁 𝑖=1 +1 2∑ 𝑞_𝑖2 𝑅𝑖 𝑁 𝑖=1 ). 1.82

The Born radius must be continually recalculated for particles in close proximity to others. 1

𝑅_𝑖 ≈ 1

𝑎_𝑖− 𝐼 1.83

Where I is estimated using the function 𝐼 = 1 4𝜋∫ 1 |r|4𝑑r ∞ |𝑟|=𝑎𝑖 . 1.84

Where the integral is limited to the solute-occupied volume and the atom i is at position |𝒓| = 0200

.

Equation 1.83 leads to underestimation of the Born radii of atoms buried in a macromolecule. To account for this, other functions have been proposed. In this project the following function of Onufriev et al.200 was tested (p169) to see if it improved results.

𝐼 = 1

𝑎𝑖tanh(𝛼Ψ − 𝛽Ψ

2_{+ 𝛾Ψ}3_{) 1.85}

Where α, β and γ are empirically derived constants and

Ψ = 𝐼(𝑎 − 0.09) _1.86

90