A global stability criterion for simple control loops

“…On mathematical analysis, a linear stability analysis was performed in [3, 30] for a class of cyclic systems and a cellular control process with positive feedback; a global stability analysis was carried out for monotone cyclic systems [34] using the Poincaré-Bendixon Theorem in multi-dimension based on the discrete Lyapunov function method [33]. In particular, global stability may be studied using ω–limit set [2], Lyapunov function method [3] or standard Poincaré-Bendixon Theorem in two-dimension [14]. …”

Section: Introductionmentioning

confidence: 99%

Feedback regulation in multistage cell lineages

Chou

Gokoffski

et al. 2009

Mathematical Biosciences and Engineering

106

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Studies of developing and self-renewing tissues have shown that differentiated cell types are typically specified through the actions of multistage cell lineages. Such lineages commonly include a stem cell and multiple progenitor (transit amplifying; TA) cell stages, which ultimately give rise to terminally differentiated (TD) cells. In several cases, self-renewal and differentiation of stem and progenitor cells within such lineages have been shown to be under feedback regulation. Together, the existence of multiple cell stages within a lineage and complex feedback regulation are thought to confer upon a tissue the ability to autoregulate development and regeneration, in terms of both cell number (total tissue volume) and cell identity (the proportions of different cell types, especially TD cells, within the tissue). In this paper, we model neurogenesis in the olfactory epithelium (OE) of the mouse, a system in which the lineage stages and mediators of feedback regulation that govern the generation of terminally differentiated olfactory receptor neurons (ORNs) have been the subject of much experimental work. Here we report on the existence and uniqueness of steady states in this system, as well as local and global stability of these steady states. In particular, we identify parameter conditions for the stability of the system when negative feedback loops are represented either as Hill functions, or in more general terms. Our results suggest that two factors – autoregulation of the proliferation of transit amplifying (TA) progenitor cells, and a low death rate of TD cells – enhance the stability of this system.

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“…Using a similar argument as in Proposition 5.1, we can show that the system (5.4) has a unique equilibrium point, which is denoted as a D .a 1 ; a 2 ; : : : ; a n /. With these definitions we are ready to state Theorem 1 of [51].…”

Section: Stability Conditions For Grns Under Negative Feedbackmentioning

confidence: 95%

“…In order to derive the main result of this section, Theorem 1 of [51] will be used. The ODE model studied in [51] is given as: • T j is a non-negative constant.…”

Section: Stability Conditions For Grns Under Negative Feedbackmentioning

confidence: 99%