Lattice Structures for Attractors II

Kalies, William D.; Mischaikow, Konstantin; Vandervorst, Robert

doi:10.1007/s10208-015-9272-x

Cited by 30 publications

(72 citation statements)

References 20 publications

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“…To address the lack of robustness of invariant sets with respect to parameters, another change in perspective is needed. Initiated by C. Conley [37] and developed over the last 40 years [38][39][40][41] the emphasis shifts from invariant sets to positively attracting sets.…”

Section: Discussionmentioning

confidence: 99%

Multistability in the epithelial-mesenchymal transition network

2020

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Background: The transitions between epithelial (E) and mesenchymal (M) cell phenotypes are essential in many biological processes like tissue development and cancer metastasis. Previous studies, both modeling and experimental, suggested that in addition to E and M states, the network responsible for these phenotypes exhibits intermediate phenotypes between E and M states. The number and importance of such states is subject to intense discussion in the epithelial-mesenchymal transition (EMT) community. Results: Previous modeling efforts used traditional bifurcation analysis to explore the number of the steady states that correspond to E, M and intermediate states by varying one or two parameters at a time. Since the system has dozens of parameters that are largely unknown, it remains a challenging problem to fully describe the potential set of states and their relationship across all parameters. We use the computational tool DSGRN (Dynamic Signatures Generated by Regulatory Networks) to explore the intermediate states of an EMT model network by computing summaries of the dynamics across all of parameter space. We find that the only attractors in the system are equilibria, that E and M states dominate across parameter space, but that bistability and multistability are common. Even at extreme levels of some of the known inducers of the transition, there is a certain proportion of the parameter space at which an E or an M state co-exists with other stable steady states. Conclusions: Our results suggest that the multistability is broadly present in the EMT network across parameters and thus response of cells to signals may strongly depend on the particular cell line and genetic background.

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Section: Discussionmentioning

confidence: 99%

Multistability in the epithelial-mesenchymal transition network

2020

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“…The collection of all attractors in 𝒱 under ℱ is denoted by Att(ℱ) and as discussed in [17, Section 2] is a bounded distributive lattice where 0 := ∅ and 1 = max {𝒜 | 𝒜 ∈ Att(ℱ)}. Furthermore, given 𝒜 0 , 𝒜 1 ∈ Att(ℱ) the lattice operations are defined by…”

Section: Conley Theorymentioning

confidence: 99%

Global dynamics for steep nonlinearities in two dimensions

Gedeon

Harker

Kokubu

et al. 2017

Physica D: Nonlinear Phenomena

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“…We show that Inv + (·, ϕ) is spacious (Proposition 5.12) even though in general Inv(·, ϕ) is not (Example 5.11). An important benefit of this approach, that is exploited in future work [11], is that if we develop numerical methods under which Inv + (·, ϕ) or Inv(·, ϕ), restricted to computable neighborhoods, is spacious, then we can guarantee that our numerical methods are capable of capturing the desired algebraic structure of the dynamics. A version of Theorem 1.2 for invertible systems is proved in Robbin and Salamon [14].…”

Section: Throughout This Paper We Make the Following Assumptionmentioning

confidence: 99%