Fig 1.
Demonstration of chaotic attractor identified by Vano et al. in a four species competitive Lotka-Volterra model [11].
A Shows the parameters used in the chaotic attractor and an illustration of the interaction topology. B Time series of the chaotic attractor. C Posterior parameter distribution for chaotic objective, identified using ABC SMC (red) and the individual chaotic particle identified by Vano et al. (black). Center grid shows 2D parameter distributions, left and top rows 1D parameter distribtuions.
Fig 2.
Overview of the pipeline for identifying chaotic topologies.
A(i) The initial model space is built from different combinations of engineering options. N1, N2, N3 are the three strains being engineered, and can optionally express QS molecules A1, A2 and bacteriocins B1, B2, B3. 4182 models are generated forming our initial model space. A(ii) We then perform ABC SMC for an oscillatory objective, which yielded 117 models that were capable of producing oscillations. A(iii) These form the prior model space for the chaos objective, using a threshold of λ1 > 0.003, we identify models capable of producing chaotic behaviour B The barchart shows the probability of models for the chaotic objective. The error bars represent the standard deviation. C An example time series representative of the chaos objective posterior distribution. Population densities as optical density (OD) show sustained, nonrepetitive oscillatory behaviour for the three species community.
Fig 3.
Topologies and properties associated with chaotic behaviour.
A Shows the models with highest posterior probability when subsetted for number of parts expressed, in order of increasing complexity (4, 5 and 6 expressed parts). The bar chart shows the mean model posterior probability across three experiments, the error bars indicate the standard deviation. B Comparison between average posterior probabilities with different properties. In order from left to right, the barcharts compare: The number of QS systems used, the modes by which QS regulates bacteriocin expression (positive, negative or both), the number of bacteriocins used, and systems containing self-limiting (SL), other-limiting (OL) or SL and OL interactions.
Fig 4.
A Topology of m850 with key parameters labelled. kA1 is the rate of QS molecule production, KBmax1 and KBmax2 are the maximal expression rates of bacteriocins B1 and B2 respectively. B Posterior parameter distributions of m850 for chaos (red) and oscillatory (blue) objectives for key parameters in system design. The borders show 1D posterior distributions for each parameter and the lower-diagonal element the 2D posterior marginals, and the upper-diagonal shows the Pearson correlations. C Feature importance calculated using random forest regression. The information gain (bits) is calculated as an average of the reduction in entropy across all trees in the forest (2000 trees). The error bars indicate the standard deviation of the entropy for each feature across all trees. D Sensitivity analysis of a chaotic input vector with chaotic region in red. Black stars refer to the identified stable steady state. The fixed parameter values are shown in Table 2
Fig 5.
Parameters kA1 and D can be tuned to control transitions between chaotic, oscillatory and stable states.
The fixed parameter values are shown in Table 2. A Map showing how different combinations of kA1 and D change population behaviour. The grid fill colour corresponds to the maximum Lyapunov exponent measured, the grid outlines indicate the approximate classification where green is stable, yellow is oscillatory, red is chaotic and white is extinction. B Bifurcation diagram showing the community states visited for different values of kA1. C Bifurcation diagram showing the community states visited for different values of D. D Real-time ramp down tuning of kA1, moving the system from a chaotic state to a stable steady state. E Real-time ramp up tuning of D, moving the system from a chaotic state to a stable steady state.
Fig 6.
Illustration of dual-orbit algorithm used to calculate the λ1.
Two orbits with an initial state separation of △0 are followed. After each time step measure the separation, △1, is measured. The perturbed orbit (red) is readjusted to prevent excess separation. The average rate of separation between the two orbits corresponds with the λ1.
Fig 7.
Confusion matrix showing accuracy of random forest classifier on test data.
Table 1.
Prior distributions for both two and three strain systems are sampled uniformly between the min and max values listed below.
Constant parameters have the same min and max value.