Chapter 9 - Free energy landscapes of DNA and its assemblies: perspectives from coarse-grained modelling
Introduction
Free energy landscapes depict the free energy of a system as a function of a limited number of coordinates (often termed order parameters or reaction coordinates). These landscapes can provide a simplified representation of the main features of configuration space, for example, the (meta)stable states of the system, the pathways between these states in this reduced order parameter space, and the size of the associated free energy barriers.
These landscapes provide a much more coarse-grained view of a system than a potential energy surface (PES). For example, the number of potential energy wells that contribute to a particular point on the free energy landscape may be enormous, particularly for large systems and those where some of the states involve structural disorder. As such the free energy landscapes naturally capture the effects of entropy and, unlike a PES, free energy landscapes are specific to the temperature of interest. As the projection of the system's configurations onto a limited number of order parameters inevitably leads to a loss of information, the choice of these order parameters is particularly important if the free energy landscapes are to provide a useful representation of the system; some of the problems associated with bad choices are well documented [1]. Even with a good choice, however, the properties of the landscapes are still intrinsically dependent on the order parameter.
Here, we will illustrate the utility of analysing systems in terms of their free energy landscapes for a series of systems made out of DNA. As the systems of interest would often involve an extremely large number of atoms if all were explicitly represented, we instead represent the DNA using a coarse-grained model of DNA at the nucleotide level, thus allowing us to access much larger systems and much longer time scales. Specifically, we use the oxDNA model [2], [3], [4], which models the DNA as a set of rigid nucleotides with a set of interactions representing the bonding along the backbone, Watson–Crick base-pairing, stacking, coaxial stacking, cross-stacking, and electrostatics, as illustrated in Fig. 9.1. One of the dangers of coarse-graining is that the removed degrees of freedom are in some way coupled to the process of interest and cannot simply be ‘integrated out’. In the case of oxDNA, it is probably reasonable to assume that the intramolecular degrees of freedom of the nucleotide do not change significantly in most of the processes in which we are interested, but the potential effects of only including the solvent degrees of freedom implicitly are somewhat less clear.
In the development of the oxDNA model a particular focus was on the accurate representation of the thermodynamics of hybridisation and a good description of single-stranded DNA and double-stranded DNA mechanics (i.e., the thermodynamic cost of structural deformations). This makes the model particularly well suited for the computation of free energy landscapes associated with the biophysical properties of DNA and for systems of interest to DNA nanotechnology. Although certain features of these landscapes are relatively straightforward to predict from simpler models (e.g., the SantaLucia model for hybridisation free energies [5] and the worm-like chain model for DNA mechanics [6]), the oxDNA model naturally captures the interplay of configurational entropy (particularly of single-stranded sections), geometric effects resulting from the double-helical structure of double-stranded DNA, and DNA mechanics that go into determining the size of free energy barriers for intramolecular processes, which would be otherwise hard to estimate accurately.
We should note that the oxDNA model has a number of variants. In particular a second version of the model (sometimes called oxDNA2) was developed that introduced explicit electrostatic interactions and fine-tuned structural parameters to enable the structure of large DNA nanostructures, such as DNA origami, to be accurately described [4], [7]. Although this second model is now generally the version of choice, the predictions of the two models will be very similar for many systems, and the results reported here will be for both models. In addition, one has the option to use a sequence-averaged or sequence-dependent [3] parameterisation, where in the former the strengths of the interactions are independent of the identity of the nucleotides involved. Most of the results reported here are for the sequence-averaged parameterisation, as it allows the generic properties of systems to be elucidated, whereas the sequence-dependent parameterisation tends to be reserved for detailed comparisons to specific experimental systems.
Section snippets
Computing free energy landscapes
Free energy landscapes are simply related to equilibrium probability distributions through where is a set of order parameters that are being used to characterise the system and c is an arbitrary constant. To facilitate visualisation of the free energy landscape often only one or two order parameters are used.
The major difficulty in computing such free energy landscapes is that one is often interested in high free energy regions of the landscape – say because one
Hybridisation
Hybridisation, the association of two strands to form a double helix, represents the most fundamental process of DNA self-assembly. Consequently, this process has been well characterised experimentally with the thermodynamics well understood [5], [15], but with open questions still remaining concerning the hybridisation kinetics [16], [17].
The most basic measure of the degree of assembly is the number of base pairs formed. The free energy profile as a function of this order parameter is shown
Conclusions
This chapter provides some examples of how free energy landscapes computed using a coarse-grained DNA model can provide important insights into the biophysics of DNA and the properties of the DNA assemblies used in DNA nanotechnology. Furthermore, given that the model used (oxDNA) has been parameterised to reproduce accurately the thermodynamics and mechanics of DNA both in its double-stranded and single-stranded forms, these landscapes can provide quantitative insights into relative rates.
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