For µ >0 bounded and a17(x1)increasing, or non-decreasing, and bounded, if multiple simple2 equilibria exist, then such equilibria are alternatively stable and unstable. In the special case of three equilibria, then the system is bi-stable.
Forµ >0bounded anda17(x1)increasing asymptotically unbounded, then the number of equilibria is necessarily even (typically 0 or 2). Moreover, if we assume that there exists µ∗ > 0 such that the system admits two distinct equilibria for any 0< µ≤µ∗, then one is stable, while the other is unstable.
The proof of this last proposition also shows that multiple equilibria xA, xB, ... have a partial order: ¯xA1 ≤¯xB1 ≤¯xC1 . . ., ¯xA4 ≤¯xB4 ≤¯xC4 . . ., ¯xA7 ≤¯xB7 ≤¯xC7 . . ., while ¯x2 and ¯x5 have the reverse order ¯xA2 ≥¯xB2 ≥¯xC2 . . . and ¯xA5 ≥¯xB5 ≥¯xC5 . . ..
Remark 6 The simplest case of constant a17 has been fully developed in [10] 3 and [121], and it turns out that the system may exhibit bi-stability for suitable values of the feedback strengthµ. Here it was shown that, for constant a17, bi-stability is actually a robust property. These results are consistent with the fact that the MAPK cascade is a monotone system and some of them could be demonstrated with the same tools used in [10, 121]. With respect to such literature, the contribution of this work is that of inferring properties such as number of equilibria and mono- or bi-stability starting from qualitative assumptions on the dynamics of the model, without invoking monotonicity.
Remark 7 Finally, it is necessary to remark that our results on the MAPK pathway robust behaviors hold true given the model (4.19) and its structure. Other work in the literature shows that feedback loops are not required to achieve a bi-stable behavior in the MAPK cascade [79] when the dual phos- phorylation and de-phosphorylation cycles are non-processive (i.e., sites can be phosphorylated/de- phosphorylated independently) and distributed (i.e., the enzyme responsible for phosphorylation/de- phosphorylation is competitively used in the two steps).
Robustness is often tested through simulations, at the price of exhaustive campaigns of numerical trials and, more importantly, with no theoretical guarantee of robustness. We are far from claiming that numerical simulation is useless. It it important, for instance, to falsify “robustness conjectures”
by finding suitable numerical counter-examples. Furthermore, for very complex systems in which analytic tools can fail, simulation appears be the last resort. Indeed a limit of the considered theoretical investigation is that its systematic application to more complex cases is challenging.
However, the set of techniques we employed can be successfully used to study a large class of simple systems, and are in general suitable for the analytical investigation of structural robustness of biological networks, complementary to simulations and experiments.
Chapter 5
Summary and future work
In this last chapter, I will briefly summarize the contributions of each chapter and outline future research plans.
5.1 Flux regulation
In this chapter I proposed two network architectures based on negative and positive feedback, to regulate and match the output flow rate of two interconnected systems. Feedback is implemented through mass action chemical reactions, which down- or up-regulate the activity of the molecules generating the network output. To my knowledge, this design has not been considered elsewhere in the literature. Numerical simulations and data suggest that feedback confers robustness to the system with respect to certain parametric variations and to initial conditions.
The analytical and experimental results presented in Chapter 2 need substantial refinement.
First, the simple model problems 2.3 and 2.7 will be non-dimensionalized in search of key parameter aggregates and nullcline characteristic behaviors. Systematic numerical analysis of the systems will be a useful aid, starting from the results in [42]. Additionally, parameter-free models will be considered, along the lines of those presented in Chapter 4, to explore the structural properties of these feedback schemes. Additional experiments and analysis need to be carried out.
•Negative auto-regulation scheme: The experiments shown for this case will be repeated, focusing on gel-based quantitation of the RNA concentration in solution. The gel electrophoresis data currently available were processed using the DNA ladder as a control for concentration, and they may lack accuracy. It will also be interesting to explore the robustness of the system to larger variations of the template concentrations over time and to external disturbances/load processes.
The data fits will be improved, extending the fitted parameter set and including gel electrophoresis data.
•Cross-activation scheme Further experiments will be run to characterize the unsatisfactory aspects of the current design, with the purpose of understanding which design details should be
improved. In particular, I will focus on the transcription leak and undesired inhibitory pathways.
In the future, I plan to fully re-design this system. I will consider the use of translator gates, or of decoupling genelets, to avoid the self inhibitory reactions structurally present in the current design.
It will be interesting to consider a circuit design incorporating both self-inhibition and cross- activation and compare it to the two described schemes. An additional interesting series of experi- ments will consist of systematically varying the toehold lengths in order to speed up or slow down the feedback loops, and assessing the robustness of the schemes with respect to such rates.
I also plan on exploring the theoretical interconnections between flux matching and consensus problems [109]. This might be useful in finding general feedback schemes to match flows of n interconnected systems.