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Digital clocks: simple Boolean models can quantitatively describe circadian systems.

Akman OE, Watterson S, Parton A, Binns N, Millar AJ, Ghazal P - J R Soc Interface (2012)

Bottom Line: Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable.We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging.Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations.

View Article: PubMed Central - PubMed

Affiliation: Centre for Systems, Dynamics and Control, College of Engineering, Computing and Mathematics, University of Exeter, Exeter, UK. o.e.akman@ex.ac.uk

ABSTRACT
The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day-night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock's oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of more complex clocks, as well as other circuits with different qualitative dynamics. In particular, we anticipate that the ability of logic models to provide a computationally efficient representation of system behaviour could greatly facilitate the reverse-engineering of large-scale biochemical networks.

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Comparing the photoperiodic behaviour of the Boolean and differential equation (DE) versions of each model. For Boolean models, the phase of each species is taken as the time within the LD cycle of the ON to OFF transition (downward triangles). Analogously, the DE model phases are defined as the times at which species decrease below the thresholds yielding the optimal fit of the corresponding Boolean circuit to data (upward triangles). Shaded areas of plots, darkness; open areas, light. (a) 1-loop Neurospora. The phase–photoperiod profiles are coincident, indicating that the Boolean model exactly reproduces the photoperiodic behaviour of its DE counterpart: FRQ transcript and protein are both locked to dusk across the photoperiod range. (b) 2-loop Neurospora. The phase plots are almost exactly equal, except for shorter photoperiods where they differ by the data sampling interval. As for (a), all components are locked to dusk. (c) 2-loop Arabidopsis. The Boolean and DE models exhibit very similar patterns of dawn- and dusk-locking across genes. The two Y phase–photoperiod profiles reflect the double peak observed in this component (figure 6e) which gives rise to a (dawn-locked) light-induced peak and a (dusk-locked) circadian peak [57]. (d) 3-loop Arabidopsis. The phase plots are similar, with all components predominately dawn-locked. As for 2-loop Arabidopsis, the two Y profiles reflect the double peak observed for this gene (figure 6g).
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RSIF20120080F7: Comparing the photoperiodic behaviour of the Boolean and differential equation (DE) versions of each model. For Boolean models, the phase of each species is taken as the time within the LD cycle of the ON to OFF transition (downward triangles). Analogously, the DE model phases are defined as the times at which species decrease below the thresholds yielding the optimal fit of the corresponding Boolean circuit to data (upward triangles). Shaded areas of plots, darkness; open areas, light. (a) 1-loop Neurospora. The phase–photoperiod profiles are coincident, indicating that the Boolean model exactly reproduces the photoperiodic behaviour of its DE counterpart: FRQ transcript and protein are both locked to dusk across the photoperiod range. (b) 2-loop Neurospora. The phase plots are almost exactly equal, except for shorter photoperiods where they differ by the data sampling interval. As for (a), all components are locked to dusk. (c) 2-loop Arabidopsis. The Boolean and DE models exhibit very similar patterns of dawn- and dusk-locking across genes. The two Y phase–photoperiod profiles reflect the double peak observed in this component (figure 6e) which gives rise to a (dawn-locked) light-induced peak and a (dusk-locked) circadian peak [57]. (d) 3-loop Arabidopsis. The phase plots are similar, with all components predominately dawn-locked. As for 2-loop Arabidopsis, the two Y profiles reflect the double peak observed for this gene (figure 6g).

Mentions: The phase–photoperiod relationships computed in this fashion are shown for the Neurospora models in figure 7a,b. For both networks, the photoperiodic behaviour of the Boolean and DE models is very close: indeed for the 1-loop network, they are exactly equivalent. Figure 7c,d plots the photoperiod simulations obtained with the Arabidopsis circuits. Here too, the phase–photoperiod profiles are very similar, with the addition of the LHY–PRR loop to the 2-loop model causing a transition from a predominately dusk-locked system to a dawn-locked one [57]. In particular, the Boolean 2-loop Arabidopsis circuit exactly reproduces the dual light response in the Y gene, in which the acute peak tracks dawn, and the circadian peak tracks dusk. This suggests that the logic circuits possess sufficient dynamic flexibility to perform the complex integration of environmental signals that is a central property of circadian systems.Figure 7.


Digital clocks: simple Boolean models can quantitatively describe circadian systems.

Akman OE, Watterson S, Parton A, Binns N, Millar AJ, Ghazal P - J R Soc Interface (2012)

Comparing the photoperiodic behaviour of the Boolean and differential equation (DE) versions of each model. For Boolean models, the phase of each species is taken as the time within the LD cycle of the ON to OFF transition (downward triangles). Analogously, the DE model phases are defined as the times at which species decrease below the thresholds yielding the optimal fit of the corresponding Boolean circuit to data (upward triangles). Shaded areas of plots, darkness; open areas, light. (a) 1-loop Neurospora. The phase–photoperiod profiles are coincident, indicating that the Boolean model exactly reproduces the photoperiodic behaviour of its DE counterpart: FRQ transcript and protein are both locked to dusk across the photoperiod range. (b) 2-loop Neurospora. The phase plots are almost exactly equal, except for shorter photoperiods where they differ by the data sampling interval. As for (a), all components are locked to dusk. (c) 2-loop Arabidopsis. The Boolean and DE models exhibit very similar patterns of dawn- and dusk-locking across genes. The two Y phase–photoperiod profiles reflect the double peak observed in this component (figure 6e) which gives rise to a (dawn-locked) light-induced peak and a (dusk-locked) circadian peak [57]. (d) 3-loop Arabidopsis. The phase plots are similar, with all components predominately dawn-locked. As for 2-loop Arabidopsis, the two Y profiles reflect the double peak observed for this gene (figure 6g).
© Copyright Policy - open-access
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3405750&req=5

RSIF20120080F7: Comparing the photoperiodic behaviour of the Boolean and differential equation (DE) versions of each model. For Boolean models, the phase of each species is taken as the time within the LD cycle of the ON to OFF transition (downward triangles). Analogously, the DE model phases are defined as the times at which species decrease below the thresholds yielding the optimal fit of the corresponding Boolean circuit to data (upward triangles). Shaded areas of plots, darkness; open areas, light. (a) 1-loop Neurospora. The phase–photoperiod profiles are coincident, indicating that the Boolean model exactly reproduces the photoperiodic behaviour of its DE counterpart: FRQ transcript and protein are both locked to dusk across the photoperiod range. (b) 2-loop Neurospora. The phase plots are almost exactly equal, except for shorter photoperiods where they differ by the data sampling interval. As for (a), all components are locked to dusk. (c) 2-loop Arabidopsis. The Boolean and DE models exhibit very similar patterns of dawn- and dusk-locking across genes. The two Y phase–photoperiod profiles reflect the double peak observed in this component (figure 6e) which gives rise to a (dawn-locked) light-induced peak and a (dusk-locked) circadian peak [57]. (d) 3-loop Arabidopsis. The phase plots are similar, with all components predominately dawn-locked. As for 2-loop Arabidopsis, the two Y profiles reflect the double peak observed for this gene (figure 6g).
Mentions: The phase–photoperiod relationships computed in this fashion are shown for the Neurospora models in figure 7a,b. For both networks, the photoperiodic behaviour of the Boolean and DE models is very close: indeed for the 1-loop network, they are exactly equivalent. Figure 7c,d plots the photoperiod simulations obtained with the Arabidopsis circuits. Here too, the phase–photoperiod profiles are very similar, with the addition of the LHY–PRR loop to the 2-loop model causing a transition from a predominately dusk-locked system to a dawn-locked one [57]. In particular, the Boolean 2-loop Arabidopsis circuit exactly reproduces the dual light response in the Y gene, in which the acute peak tracks dawn, and the circadian peak tracks dusk. This suggests that the logic circuits possess sufficient dynamic flexibility to perform the complex integration of environmental signals that is a central property of circadian systems.Figure 7.

Bottom Line: Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable.We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging.Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations.

View Article: PubMed Central - PubMed

Affiliation: Centre for Systems, Dynamics and Control, College of Engineering, Computing and Mathematics, University of Exeter, Exeter, UK. o.e.akman@ex.ac.uk

ABSTRACT
The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day-night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock's oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of more complex clocks, as well as other circuits with different qualitative dynamics. In particular, we anticipate that the ability of logic models to provide a computationally efficient representation of system behaviour could greatly facilitate the reverse-engineering of large-scale biochemical networks.

Show MeSH