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Publications of Eduardo D. Sontag jointly with A. Duvall
Conference articles
  1. M. Ali Al-Radhawi, K. Manoj, D. Jatkar, A. Duvall, D. Del Vecchio, and E.D. Sontag. Competition for binding targets results in paradoxical effects for simultaneous activator and repressor action. In Proc. 63rd IEEE Conference on Decision and Control (CDC), 2024. Note: To appear. Preprint in arXiv.[PDF] Keyword(s): resource competition, epigenetics, systems biology, synthetic biology, gene regulatory systems.
    Abstract:
    In the context of epigenetic transformations in cancer metastasis, a puzzling effect was recently discovered, in which the elimination (knock-out) of an activating regulatory element leads to increased (rather than decreased) activity of the element being regulated. It has been postulated that this paradoxical behavior can be explained by activating and repressing transcription factors competing for binding to other possible targets. It is very difficult to prove this hypothesis in mammalian cells, due to the large number of potential players and the complexity of endogenous intracellular regulatory networks. Instead, this paper analyzes this issue through an analogous synthetic biology construct which aims to reproduce the paradoxical behavior using standard bacterial gene expression networks. The paper first reviews the motivating cancer biology work, and then describes a proposed synthetic construct. A mathematical model is formulated, and basic properties of uniqueness of steady states and convergence to equilibria are established, as well as an identification of parameter regimes which should lead to observing such paradoxical phenomena (more activator leads to less activity at steady state). A proof is also given to show that this is a steady-state property, and for initial transients the phenomenon will not be observed. This work adds to the general line of work of resource competition in synthetic circuits.


  2. A. Duvall and E.D. Sontag. A remark on omega limit sets for non-expansive dynamics. In Proc. 63rd IEEE Conference on Decision and Control (CDC), 2024. Note: To appear. Preprint in arXiv.[PDF] Keyword(s): contractive systems, contractions, non-expansive systems.
    Abstract:
    In this paper, we study systems of time-invariant ordinary differential equations whose flows are non-expansive with respect to a norm, meaning that the distance between solutions may not increase. Since non-expansiveness (and contractivity) are norm-dependent notions, the topology of $\omega$-limit sets of solutions may depend on the norm. For example, and at least for systems defined by real-analytic vector fields, the only possible $\omega$-limit sets of systems that are non-expansive with respect to polyhedral norms (such as $\ell^p$ norms with $p =1$ or $p=\infty$) are equilibria. In contrast, for non-expansive systems with respect to Euclidean ($\ell^2$) norm, other limit sets may arise (such as multi-dimensional tori): for example linear harmonic oscillators are non-expansive (and even isometric) flows, yet have periodic orbits as $\omega$-limit sets. This paper shows that the Euclidean linear case is what can be expected in general: for flows that are contractive with respect to any strictly convex norm (such as $\ell^p$ for any $p ot=1,\infty$), and if there is at least one bounded solution, then the $\omega$-limit set of every trajectory is also an omega limit set of a linear time-invariant system.


  3. A. Duvall and E. D. Sontag. Global exponential stability or contraction of an unforced system do not imply entrainment to periodic inputs. In Proc. 2024 Automatic Control Conference, 2024. Note: Preprint in arXiv:2310.03241.
    Abstract:
    It is often of interest to know which systems will approach a periodic trajectory when given a periodic input. Results are available for certain classes of systems, such as contracting systems, showing that they always entrain to periodic inputs. In contrast to this, we demonstrate that there exist systems which are globally exponentially stable yet do not entrain to a periodic input. This could be seen as surprising, as it is known that globally exponentially stable systems are in fact contracting with respect to some Riemannian metric. The paper also addresses the broader issue of entrainment when an input is added to a contractive system.



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Last modified: Fri Sep 20 11:51:28 2024
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