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- Anael Verdugo, P. K. Vinod, John J. Tyson, Bela Novak
- Open biology
- 2013

Progression through the eukaryotic cell cycle is characterized by specific transitions, where cells move irreversibly from stage i-1 of the cycle into stage i. These irreversible cell cycle transitions are regulated by underlying bistable switches, which share some common features. An inhibitory protein stalls progression, and an activatory protein promotes… (More)

We study a system of three limit cycle oscillators which exhibits two stable steady states. The system is modeled by both phase-only oscillators and by van der Pol oscillators. We obtain and compare the existence, stability and bifurcation of the steady states in these two models. This work is motivated by application to the design of a machine which can… (More)

This work presents an explicit formula for determining the radius of a limit cycle which is born in a Hopf bifurcation in a class of first order constant coefficient differential-delay equations. The derivation is accomplished using LindstedtÕs perturbation method.

We analyze a model of gene transcription and protein synthesis which has been previously presented in the biological literature. The model takes the form of an ODE (ordinary differential equation) coupled to a DDE (delay differential equation), the state variables being concentrations of messenger RNA and protein. Linear analysis gives a critical time delay… (More)

We use center manifold theory to analyze a model of gene transcription and protein synthesis which consists of an ordinary differential equation (ODE) coupled to a delay differential equation (DDE). The analysis involves reformulating the problem as an operator differential equation which acts on function space, with the result that an infinite dimensional… (More)

- J. Bridge, L. Mendelowitz, S. Sah, A. Verdugo
- 2008

The dynamics of a ring of three identical relaxation oscillators is shown to exhibit a variety of periodic motions, including clockwise and counterclockwise wave-like modes, and a synchronous mode in which all three oscillators are in phase. The model involves individual oscillators which exhibit sudden jumps, modeling the relaxation oscillations of van der… (More)

- Anael Verdugo
- J. Computational Applied Mathematics
- 2016

- Anael Verdugo
- 2008

This paper presents an analytical study of the stability of the steady state solutions of a gene regulatory network with time delay. The system is modeled as a continuous network and takes the form of a nonlinear delay differential-integral equation coupled to an ordinary differential equation. Two examples are given in which the critical delay causing… (More)

We analyze a model of gene transcription and protein synthesis which has been previously presented in the biological literature. The model takes the form of an ODE (ordinary differential equation) coupled to a DDE (delay differential equation), the state variables being concentrations of messenger RNA and protein. The delay is assumed to depend on the… (More)

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