The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment

Woller, Aurore; Gonze, Didier; Erneux, Thomas

doi:10.1088/1478-3975/11/4/045002

Cited by 44 publications

(37 citation statements)

References 58 publications

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“…5), the major difference between HT and PS models has not been reported or investigated, to our knowledge. Future work can also investigate whether PS models follow the entrainment properties [37,45,47,142,[156][157][158] or temperature compensation mechanisms [38,40,42,[159][160][161][162][163][164][165][166] identified with HT models. Furthermore, stochastic simulations of HT models commonly indicate that circadian clocks can maintain rhythms even with low numbers of molecules [167][168][169][170].…”

Section: Resultsmentioning

confidence: 99%

Protein sequestration versus Hill‐type repression in circadian clock models

Kim¹

2016

IET syst. biol.

107

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Circadian (~24hr) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptionaltranslational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, we discuss how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modeling of transcriptional repression mechanisms in molecular circadian clocks.

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Section: Resultsmentioning

confidence: 99%

Protein sequestration versus Hill‐type repression in circadian clock models

Kim¹

2016

IET syst. biol.

107

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“…Analytical amplitude estimates for specific systems typically require a separate treatment of different parameter regimes, e.g., regions close to the Hopf bifurcation [33] or the limit of strong feedback [34]. Hence, it is often elusive how the details of the nonlinear feedback govern the oscillation amplitude.…”

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confidence: 99%

Amplitude bounds for biochemical oscillators

Jörg¹

2017

EPL

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-We present a practical method to obtain bounds for the oscillation minima and maxima of large classes of biochemical oscillator models that generate oscillations through a negative feedback. These bounds depend on the feedback nonlinearity and are independent of explicit or effective feedback delays. For specific systems, we obtain explicit analytical expressions for the bounds and demonstrate their effectiveness in comparison with numerical simulations.

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“…Stochastic simulations of this circuit revealed parameters that resulted in oscillatory behavior (Figures 4A and S6A; Methods). In addition, deterministic modeling and bifurcation analysis confirmed that this circuit is capable of oscillations (limit cycle oscillations or damped oscillations) (Figure S6B and Methods) ( 29, 30 ). When stochasticity is considered, oscillations (due to “stochastic resonance”) can occur even under parameter regimes that are predicted to not oscillate according to deterministic models ( 31–33 ), thereby suggesting that in this circuit, stochasticity together with the appropriate coupling (i.e., governing the PoV) between genes X and Y can give rise to oscillations beyond classic “limit cycle” mechanisms.…”

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confidence: 82%

Machine learning of stochastic gene network phenotypes

Park

Prüstel

et al. 2019

Preprint

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15A recurrent challenge in biology is the development of predictive quantitative models 16 because most molecular and cellular parameters have unknown values and realistic models 17 are analytically intractable. While the dynamics of the system can be analyzed via 18 computer simulations, substantial computational resources are often required given 19uncertain parameter values resulting in large numbers of parameter combinations, 20 especially when realistic biological features are included. Simulation alone also often does 21 not yield the kinds of intuitive insights from analytical solutions. Here we introduce a 22A major goal of systems biology is to develop quantitative models to predict the behavior 36 of biological systems (1, 2). However, most realistic molecular and cellular models have a 37 large number of parameters (e.g., reaction rates, cellular proliferation rates, extent of 38 physical interactions among cells or molecules), whose values remain unknown and are 39 often challenging to measure or infer quantitatively (3, 4). While some biological 40 phenotypes are robust to parameter variations (5), most are "tunable" by parameters (6). 41Therefore, analyzing the behavior of a system over the entire plausible space of parameters 42 is needed to study the phenotypic range of a biological system and its parameter-phenotype 43 relationships (7-9). A case in point concerns a contemporary problem in single cell biology: 44Despite the increasing availability of single-cell gene expression data enabled by rapid 45 technological advances (10), an important unanswered question is how cell-to-cell 46 expression variation and gene-gene correlation among single cells are regulated by the 47 computational resources are required to analyze the parameter space, given the large 59 uncertainty in parameter values and the complex correlation structure among parameters. 60Plus, simulation analysis alone often does not automatically yield intuitive understanding. 61Here, we combine computational simulation of full-feature stochastic models and 62 machine learning (ML) to develop a framework, called MAchine learning of Parameter-63Phenotype Analysis (MAPPA), for constructing, exploring, and analyzing the mapping 64 between parameters and quantitative phenotypes of a stochastic dynamical system (Figures 65 1A and S1; Supplementary text). Our goal is to take advantage of the large amounts of data 66that can be generated from bottom-up, mechanistic computational simulation of dynamical 67 systems and the ability of modern machine learning approaches to "compress" such data 68 to generate computationally efficient and interpretable models. MAPPA thus builds 69 efficient, predictive, and interpretable ML models that capture the nonlinear mapping 70 between parameter and phenotypic spaces (parameter-phenotype maps). The ML models 71 can be viewed as "phenomenological" solutions of the SME that can predict the system's 72 quantitative behavior from parameter combinations, thus bypassing computationally 73 expensive simulatio...

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The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment

Cited by 44 publications

References 58 publications

Protein sequestration versus Hill‐type repression in circadian clock models

Protein sequestration versus Hill‐type repression in circadian clock models

Amplitude bounds for biochemical oscillators

Machine learning of stochastic gene network phenotypes

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