We have introduced a multitude of simulation techniques available to the community that aid us in understanding the function of complex biomolecules. We have seen that many currently available mesoscopic simulation techniques treat systems as networks of spherical beads connected by some form of pair-wise potential. The only real differ- ence between these models is the form of the potential. Yet we have also seen that the mathematics of the macroscale is fundamentally different to that of the microscale, and so that there may be some limit to how far these bead-spring models can help us as we look at larger and larger systems, and the beads no longer represent distinguishable particles within the material. The natural reaction as computational power increases, then, is to rely more on high-resolution simulation techniques and understand meso- scopic dynamical properties as emergent properties. But, a recent review by Gray et al. emphasises that “the ‘computational microscope’ of biomolecular simulation is not infinitely powerful” [47]. They show that the availability of computational resources is insufficient to meet the national demand for high-resolution simulations, and so phys- ical models that are appropriate to the length and time-scales of interest are not only scientifically desirable, but economically essential as well.
CG MD takes us from the ‘bottom-up’ to the upper limit beyond which bead-spring models are no longer effective. DPD then introduces the concept of overall shape, and populates that shape with loosely connected beads to coincide with macroscale observations. But due to the nature of these beads, the potentials connecting them are somewhat arbitrary. The alternative is to develop a ‘top-down’ view of the mesoscale, starting from the fluid dynamics and continuum mechanics that already have a fully developed mathematical framework for us to work from. To that end, this thesis presents continued development of a relatively new technique which begins fully in the continuum regime and takes this ‘top-down’ view of the mesoscale [71].
Chapter 1. Introduction 27
In this thesis, we present the further development of Fluctuating Finite Element Anal- ysis (FFEA), a novel simulation technique and software package that began develop- ment in 2013 with the work of Oliver et al. . The technique implements a continuum mechanical approach to model the dynamics of globular biomolecules, but with the inclusion of stochastic thermal effects [71]. Chapter 2gives an introduction to FFEA and its functionality up to the developmental stage as it was prior to the work pre- sented in this thesis. Chapter3then describes the different modifications I have made to the model to allow the simulation of longer time-scales, as well as different forms of simulation. Chapter 4 then looks at an entirely new form of simulation within the FFEA framework which we have called Kinetic FFEA, allowing the modelling and real time simulation of kinetic events in parallel with underlying dynamical models. Fi- nally, Chapter5 presents an initial study of the molecular motor cytoplasmic dynein, to which we apply Kinetic FFEA in combination with empirical knowledge derived from available experimental evidence in an attempt to determine how, and why, the motor is able to function as a cargo transporter.
Chapter 2
Fluctuating Finite Element
Analysis
We saw in Chapter 1 that the current range of mesoscale simulation technologies are mostly based on coarse-grained particle methods that do not fit naturally with the way we describe the macroscale. To make full use of the emerging experimental tech- niques that allow structure identification at the upper limit of the mesoscale [72], we require a method that maps smoothly onto continuum mechanics as the length scale increases. Those current methods that are available look specifically at viscous fluids. However, Oliver et al. (2013) developed a method based on finite element analysis to model proteins as visco-elastic solids [71]. The technique was further developed by Richardson [73] who developed a C++ implementation of the algorithm called Fluctu- ating Finite Element Analysis (FFEA), a software package designed specifically for the continuum mechanical simulation of large, globular proteins and protein assemblies. These types of protein can have molecular weights ranging from ∼10kDa, approxi- mately that of the hemoglobin molecule we saw in Chapter 1 [74], all the way up to ∼1MDa, approximately the total weight of a fully assembled GroEL chaperonin molecule [75]. System sizes towards the higher end of this range are effectively out of scope for fully atomistic simulations at current computational speeds, at least for readily accesible computational resources [47]. Consequently, many such large proteins have been studied using discrete CG methods. FFEA, though, was not developed as yet another coarse-grained method for simulating atomistic systems. Rather, FFEA originated from the paradigm that the complex properties emerging from large collec- tions of atoms conform more to the mathematical framework of continuum mechanics than discrete Newtonian dynamics. As shown in Chapter1, these kinds of systems and their interactions with the environment are often not spatially discrete at the relevant
Chapter 2. Fluctuating Finite Element Analysis 30
length-scales, i.e. they are no longer atomistic in nature, and so we avoid treating them as such.
The mathematics of FFEA is derived from the continuum mechanics formalism, which was introduced in Chapter1. We progress from there with the idea of our biomolecules of interest being continuum objects.