Concluding, in this Chapter I examined the curious case of the Kinetoplast DNA, the mitochondrial genome of organisms of the class Kinetoplastida. I designed a model which allowed me to test some key features driving the Kinetoplast topological organisation. I found that by modelling the system as a collection of phantom loops and mapping the system to a graph of linked rings, one can observe that the onset of the percolation seem to occur at concentrations around ρ = ρp ∼ 0.0064
σ−3. This value should be interpreted as a rough estimate of the real percolation density ¯ρp. On the other hand, I observed that at around this monomer density, the mean valence of the nodes is found to be around 3, which is in agreement with the experimental findings in in vivo Kinetoplast DNA. Importantly, at the same density value, I compared the results from an in silicodigestion of the network by a restriction enzyme, finding very good quantitative agreement with the experimental data found in Ref. [Chen et al., 1995b]. These results strongly suggest that the Kinetoplast topology can be understood as a system of confined linkable rings whose
5. A Bio-Physical Model for the Kinetoplast DNA 84
density ρ can be tuned near the percolating density ¯ρp. In other words, it can be represented by a network of linked rings near its critical point, i.e. the point at which the network starts to show percolating behaviour.
The key message of this Chapter is that the available experimental findings can be understood in terms of a simpler system, where rings form linked structures by passing through one another and that a good agreement with the available exper- imental observations can be retrieved when the network is neither too poorly, nor too heavily, connected, but near what is seems to be the percolation threshold. The precise value of the latter, ¯ρp, is on the other hand, beyond the precision of the presented results. Nonetheless, it is possible to make some comments regarding this conceptual finding. In fact, being close to the percolation transition may well provide an evolutionary advantage for the Kinetoplast DNA network, as this structure may be favoured over a more heavily connected network, as it facilitates the decatenation during replication, but at the same time ensures that mini-circles are not released by mistake, conferring robustness and increasing the conservation of genetic material across generations. In other words, too heavily linked network would be very good at preserving the genetic material, but would severely slow down the replication stage of the network, where the Kinetoplast has to be taken apart. On the the other hand, networks that are too poorly linked might result in the loss of genetic material during cell mitosis and therefore put the cell viability at risk. Another property of the Kinetoplast-like network is that it is very resistant to digestion by a restriction enzyme, i.e. the digestion has to proceed significantly before large clusters disap- pear (see Fig. 5.7(d)). This feature again appears to be functionally relevant, as it provides a way to preserve genetic material against random breakage and replica- tion mistakes. It is tempting to conjecture therefore, that the evolutionary pressure has pushed the KDNA toward a topological structure resembling a percolating net- work of linked rings at its transition, or critical, point, which offers many non-trivial evolutionary advantages.
A final remark is in order: these organisms have had a very unusual evolutionary path. They show a uniquely structured mitochondrial genome, very distant from anything else in Nature. On the other hand, they seem to have found an (unstable) equilibrium, since small changes in the Kinetoplast structure usually lead to the evolution of novel species which rapidly diverge from their common ancestor [Lai et al., 2008] (in some cases, the network structure seem to have been traded for super- coiling,e.g. in the Pan-KDNA structure ofCryptobia helicis [Lukeˇs et al., 2002]). I speculate that this equilibrium must have been reached via successive “attempts”, i.e. mutations, which slightly modified the topological structure until a balance between several key elements, such as speed of replication (see previous Section and also Appendix C), accuracy of replication (Appendix C) and resistance against mistakes and network robustness (previous Section) had been reached, leading to the modern organisms of the class Kinetoplastida showing a KDNA.
5. A Bio-Physical Model for the Kinetoplast DNA 85
Finally, I would like to stress again that the survival of these organisms depends on the regulation of the Kinetoplast topology; Nonetheless, their genome is consis- tently duplicated with very few, or no, mistakes at every duplication cycle. Such reliable topology regulation is far from being trivial to achieve. During the ’70s de Gennes advanced the idea of an “Olympic gel” [de Gennes, 1979] and 40 years later, the scientific community has not made much progress toward its realisation. Perhaps the answer lies within these organisms, as they have made an Olympic gel out of their own mitochondrial genome (it is fair although to say that they had some millions of years to try and make it!).
A better understanding of how they manage this incredible feat would surely im- prove the current understanding and ability to realise similar topological materials, perhaps using techniques drawn from modern synthetic biology.
The Red Queen said:
“Now, here, you see, it takes all the running you can do, to keep in the same place”
L. Carroll
6
The Role of Topology in DNA Gel
Electrophoresis
Contents
6.1 Gel Electrophoresis of DNA Rings and Strands . . . 89
6.1.1 Getting More from Pushing Less . . . 90 6.1.2 Non-Equilibrium Response Theory . . . 92 6.1.3 Topology can Sense Disorder . . . 95