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In conclusion, Type III networks can form a pattern even when the diffusing signals have the same range due to the capacitor effect of immobile reactants and associated reaction terms. Remarkably, this can happen even when the activator diffuses more than the inhibitor ( Appendix 3—figure 13 ). This in an intrinsic property of Type III topologies, where the immobile reactant acts as a buffer to amplify any small advantages or disadvantages over the inhibitor diverging quickly from the equilibrium state. In agreement with this finding, if the fast divergence of the immobile reactant is limited by a negative self-regulation, most networks are ofType II, meaning that their ability to compensate for the range of the mobile reactants depends on the relative speed at which each reactant grows. For example, in the Type II network shown in Appendix 3—figure 3 , this can happen only when the growth of the activator is slowed down sufficiently by the inhibitor through the negative cycle ${\text{c}}_{3}$ .

Appendix 3—figure 13

In this case, there is an even larger flow of X from the first to the second cell. Initially, the immobile reactant Z is not able to compensate for this larger flow (see time point 0.15, where the …

The examples presented in this Appendix highlight that in classical two-reactant Turing models thedifferential diffusivity destabilizes the equilibrium state by maintaining an imbalance between reactants, which drives a further deviation from equilibrium. Importantly, the reaction terms of the Turing system guarantee that the deviation happens simultaneously above and below the equilibrium state. In the activator-inhibitor model, for example, the differential diffusivity not only gives an advantage to the self-enhancement of the activator but simultaneously to the auto-inhibition of the inhibitor. Therefore, in accordance with a recent proposal ( Klika et al., 2012 ), we suggest that the negative self-regulation of the inhibitor has a more important role than previously assumed.

In agreement with these observations, one-dimensional simulations like the one shown in Appendix 3—figure 8 reveal that the periodic patterns of Turing systems are formed with a simultaneous appearance of activation and inhibition peaks and that the patterning dynamics do not follow the sequence of events described by the LALI mechanism based on local auto-activation and lateral-inhibition. The periodic patterns therefore do not reflect a longer range of the inhibitor to limit the auto-activation. Instead, we propose that the periodic patterns of both the activator and the inhibitor reflect only one range, usually referred to as the wavelength, which is determined by the differential diffusion but also by reaction terms. According to this view, the periodic patterns formed in the activator-inhibitor model reflect different amplitudes of activator and inhibitor levels - rather than a difference in the ranges - that depend both on differential diffusivity and reaction terms.

We therefore propose that the role of immobile reactants in Type II and Type III networks is not to implement an effective difference in the ranges of local auto-activation and long-range inhibition, but rather to help the system to diverge from equilibrium, which is normally achieved by differential diffusivity in classical two-component Turing systems. We find that immobile factors can help to destabilize the system, since they are not subjected to the equilibrating effect of diffusion and therefore fulfill a role as 'capacitors' that can integrate the effect of diffusing reactants to destabilize the reaction-diffusion system by quickly amplifying perturbations.

The networks presented in Figure 5—figure supplement 1 are all alternative implementations of synthetic reaction-diffusion systems obtained by addition of negative feedbacks to an existing synthetic circuit that implements a positive feedback. In contrast to classical activator-inhibitor models, these networks show that many realistic reaction-diffusion systems do not require differential diffusivity. In addition, given the explicit representation of cell-autonomous factors, these networks also suggest at which level of the signaling pathways the new feedbacks should be introduced. On theone hand, these predictions help to bridge the gap between theoretical models and real systems, and on the other hand they present engineers with new challenges for the implementation of specific synthetic network designs. The high-throughput results of RDNets can be used to choose the network design that better fits the available synthetic toolkits.

Laursen PB
Hurley BF
Balabinis CP

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