Time series analysis of the Bacillus subtilis sporulation network reveals low dimensional chaotic dynamics
Lecca P., Mura I., Re A., Barker G., Ihekwaba A.
Journal: Frontiers in Microbiology
Chaotic behaviour refers to a behaviour which, albeit irregular, is generated by an underlying deterministic process. Therefore, a chaotic behaviour is potentially controllable. This possibility becomes practically amenable especially when chaos is shown to be low-dimensional, i.e. to be attributable to a small fraction of the total systems components. In this case, including the major drivers of chaos in a system into the modelling approach allows us to improve predictability of the systems dynamics. Here, we analysed the numerical simulations of an accurate ordinary differential equation model of the gene network regulating sporulation initiation in Bacillus subtilis to explore whether the non-linearity underlying time series data is due to low-dimensional chaos. Low-dimensional chaos is expectedly common in systems with few degrees of freedom, but rare in systems with many degrees of freedom such as the B. subtilis sporulation network. The estimation of a number of indices, which reflect the chaotic nature of a system, indicates that the dynamics of this network is affected by deterministic chaos. The neat separation between the indices obtained from the time series simulated from the model and those obtained from time series generated by Gaussian white and coloured noise confirmed that the B. subtilis sporulation network dynamics is affected by low dimensional chaos rather than by noise. Furthermore, our analysis identifies the principal driver of the networks chaotic dynamics to be sporulation initiation phosphotransferase B (Spo0B). As Spo0A is the master regulator of the sequence of the events regulating B. subtilis sporulation, we then analysed a sub-space of the systems phase space phase spanned defined just by Spo0A and Spo0B dynamics. A characterization of the instability points in this sub-space allowed us to identify the ranges of values assumed by Spo0A and Spo0B for which the whole system is highly sensitive to minimal perturbation. In summary, we described an unappreciated source of complexity in the B. subtilis sporulation network by gathering evidence for the chaotic behaviour of the system, and by suggesting candidate molecules driving chaos in the system. The results from this chaos analysis could help to refine theoretical predictions of the behaviour of the system and to advance the control of the mechanisms underlying B. subtilis sporulation.