measuring conformational fluctuations/disorder in proteins at an atomistic level, and therefore approximating their associated entropy. The parameters apply only to the bond vector analysed, which assumes independent motion for each vector, and captures motions only on the ps-ns timescale. Nonetheless, the common use of these values to represent flexibility changes and calculate entropy changes is supported by examples of close agreement with independent measurements.46
1.3.3 Molecular Dynamics (MD)
Molecular dynamics (MD) solves Newton’s equations of motion for an atomic resolution system, generating a time evolution of a system as a sequence of atomic positions and velocities termed the trajectory. The energy of the system configuration sampled at each step is calculated using a potential energy function.
Figure 1.8 CHARMM MD potential energy function. The energy of the system is calculated using bonded and non-bonded terms. k values are force constants and subscript 0 refers to equilibrium values generated by parameterisation. All terms with equilibrium values are harmonic functions. Figure provided by Dr Emanuele Paci (personal communication).
The potential energy function represents a compromise between the speed and accuracy of the calculation, as terms that could be accurately calculated quantum mechanically at greater computational expense are approximated into classical forms. Contributions from both bonded and nonbonded terms are summed over a number of terms to calculate the potential energy. These terms include bond lengths and angles, dihedral angles, and van der Waals and electrostatic interactions. For nonbonded terms, the first is a Coulombic term for point charges, and the second a Lennard Jones term for calculation of vdW attraction and atomic repulsion. The various K values are force constants, and the use of subscript 0 is to indicate the equilibrium value of the
relevant parameter. All bonded terms are evaluated as harmonic functions, with the energy minimum at the equilibrium value.
Equilibrium values are contained in the parameter files generated for each molecule. The generation of parameter files, namely parameterisation, can be a lengthy process involving quantum mechanical calculations: fortunately efforts are being made to parameterise molecules on a faster timescale.47 Combined, the potential energy
function and parameters are referred to as the forcefield.
MD trajectories are maximally temporally and spatially resolved, presenting an ideal method for investigating the structure and dynamics of biomolecules. Almost any parameter or experimental observable is potentially calculable from a trajectory using MD software itself, or external trajectory analysis tools such as wordom for CHARMM48. Some observables that can be calculated are NMR S2, radius of gyration,
average structures, and rmsd or rmsf values between frames of the trajectory. Assuming sufficent accuracy of the forcefield, as simulation length tends toward the experimental timescale an increasingly representative set of system configurations should be sampled. Accordingly, extending the practicable limit of simulation timescales has received much attention.49
However, the forcefield is an approximation, optimised for the native states of proteins. The requirement for accurate sampling of non-native states, for example in protein folding simulations, resulted in recent modifications of the backbone torsional energy term in both the major MD packages, CHARMM and AMBER.50-53 Further
approximations are available that reduce the number of particles in the system and thus the computational cost of MD. These include implicit solvent models that replace the need to include explicit solvent atoms by representing solvent as a continuous medium,54 and coarse-grained protein models wherein entire amino acid residues are
represented as single particles with averaged characteristics.55 Therefore MD of all
types contain degrees of approximation, and for every new purpose, new approximation or new timescale, need to be corroborated against experimental measurements to be used with confidence.
Fortunately, average parameters extracted from short timescale all-atom MD have been robustly corroborated by NMR, the only experimental technique that can measure at atomic resolution on timescales easily accessible by MD, i.e. picosecond- nanosecond.50
1.4 Aims and scope of this thesis
Though techniques exist for the computational estimation of ∆H°b to a reasonable
degree56, a better understanding of dynamic entropic contributions to binding
thermodynamics is required for truly rational manipulation and optimisation of interactions.
This thesis addresses four important questions regarding the dynamics and thermodynamics of protein-ligand interactions, using MUP as a model system and employing a wide range of biophysical techniques. Some work directly utilises the perturbation-decomposition approach described herein. Other work capitalises on avenues of inquiry that have arisen due to the wealth of data regarding protein-ligand interactions in MUP.
Chapter 2 is an investigation into the proposed entropic solvation of the MUP binding pocket, and the possibility of engineering a new binding profile through manipulation of sidechains and solvation in the binding pocket. Chapter 3 is the first systematic assessment of the widely predicted and presumed benefits of minimising ligand conformational entropy loss by removing and restricting ligand bonds. The effects of such modifications on intrinsic entropy are considered across a wide panel of ligands. Chapter 4 investigates the biggest entropic loss, that of ligand rotational and translational entropy, an under-investigated question addressed for the first time using a combination of NMR approaches to assess MD predictions of significant residual translation and rotational motion when bound. Finally, Chapter 5 constitutes a preliminary investigation into a potentially promising novel technique for probing site- specific changes in protein dynamics upon ligand binding.