Reducing the complexity of mathematical models for the plant circadian clock by distributed delays

Tokuda, Isao T.; Akman, Ozgur E.; Locke, James

doi:10.1016/j.jtbi.2018.12.014

Cited by 24 publications

(26 citation statements)

References 46 publications

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“…In the future it will be interesting to understand if the performance increase in predicting a pathway's input-output dynamics will hold also when a linear pathway is part of a larger dynamical system, including multiple interactions and feedbacks. Recently, Tokuda et al (43) showed that introducing distributed delays (fixed-rate assumption) between nodes in circadian clock models allowed for a parameters reduction. Given that they did not allow for fully dynamic changes of pathway lengths (n), it would be interesting to see if this can improve the model predictability.…”

Section: Discussionmentioning

confidence: 99%

“…This approach models the output of the linear pathway as the convolution between the input to the first pathway step and the probability density function of the gamma distribution. This convolution has been used to describe delays in a diverse set of processes, including: drug uptake (39,40), circadian clocks (41)(42)(43), population dynamics (44) and even traffic jams (45). Similar to the fixed-delay approach, it is commonly used as a method to introduce a delay without explicit regard to what the underlying cause of that delay is.…”

Section: Introductionmentioning

confidence: 99%

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It’s about time: Analysing an alternative approach for reductionist modelling of linear pathways in systems biology

Korsbo

Jönsson

2019

Preprint

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Thoughtful use of simplifying assumptions is crucial to make systems biology models tractable while still representative of the underlying biology. A useful simplification can elucidate the core dynamics of a system. A poorly chosen assumption can, however, either render a model too complicated for making conclusions or it can prevent an otherwise accurate model from describing experimentally observed dynamics.Here, we perform a computational investigation of linear pathway models that contain fewer pathway steps than the system they are designed to emulate. We demonstrate when such models will fail to reproduce data and how detrimental truncation of a linear pathway leads to detectable signatures in model dynamics and its optimised parameters.An alternative assumption is suggested for simplifying linear pathways. Rather than assuming a truncated number of pathway steps, we propose to use the assumption that the rates of information propagation along the pathway is homogeneous and instead letting the length of the pathway be a free parameter. This results in a three-parameter representation of arbitrary linear pathways which consistently outperforms its truncated rival and a delay differential equation alternative in recapitulating observed dynamics.Our results provide a foundation for well-informed decision making during model simplifications. Author summaryMathematical modelling can be a highly effective way of condensing our understanding of biological processes and highlight the most important aspects of 1 them. Effective models are based on simplifying assumptions that reduce complexity while still retaining the core dynamics of the original problem. Finding such assumptions is, however, not trivial.In this paper, we explore ways in which one can simplify long chains of simple reactions wherein each step is linearly dependent on its predecessor. After generating synthetic data from models that describe such chains in explicit detail, we compare how well different simplifications retain the original dynamics. We show that the most common such simplification, which is to ignore parts of the chain, often renders models unable to account for time delays. However, we also show that when such a simplification has had a detrimental effect, it leaves a detectable signature in its optimal parameter values. We also propose an alternative assumption which leads to a highly effective model with only three parameters. By comparing the effects of these simplifying assumptions in thousands of different cases and for different conditions we are able to clearly show when and why one is preferred over the other.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

It’s about time: Analysing an alternative approach for reductionist modelling of linear pathways in systems biology

Korsbo

Jönsson

2019

Preprint

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“…In related work, distributed delays were used to represent TF production pathways in three established clock models. This approach replaces the set of parameters governing the delays in each pathway with a pair of parameters that control the mean and variance of the delay distribution (assumed to be a gamma function), leading to simplified differential equation models with markedly reduced parameterisations [16].…”

Section: Reducing Model Complexity In Plant Clock Modelsmentioning

confidence: 99%

“…This apparent redundancy, together with a desire for reduced parametrisations, may partly have driven the subsequent models to jettison separate compartments. It would therefore be of interest to extend our extended S-System framework to integrate distributed delay-based models of protein pathways [19,72,73], as these yield tunable transfer functions of arbitrary order, whilst maintaining a compact parametrisation [16]. In addition, although our modification to the original S-System framework extends the range of transcriptional regulation mechanisms that can be modelled beyond the multi-input AND gate of the original formulation (cf.…”

Section: Future Directionsmentioning

confidence: 99%

A simplified modelling framework facilitates more complex representations of plant circadian clocks

2020

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The circadian clock orchestrates biological processes so that they occur at specific times of the day, thereby facilitating adaptation to diurnal and seasonal environmental changes. In plants, mathematical modelling has been comprehensively integrated with experimental studies to gain a better mechanistic understanding of the complex genetic regulatory network comprising the clock. However, with an increasing number of circadian genes being discovered, there is a pressing need for methods facilitating the expansion of computational models to incorporate these newly-discovered components. Conventionally, plant clock models have comprised differential equation systems based on Michaelis-Menten kinetics. However, the difficulties associated with modifying interactions using this approach-and the concomitant problem of robustly identifying regulation types-has contributed to a complexity bottleneck, with quantitative fits to experimental data rapidly becoming computationally intractable for models possessing more than �50 parameters. Here, we address these issues by constructing the first plant clock models based on the S-System formalism originally developed by Savageau for analysing biochemical networks. We show that despite its relative simplicity, this approach yields clock models with comparable accuracy to the conventional Michaelis-Menten formalism. The S-System formulation also confers several key advantages in terms of model construction and expansion. In particular, it simplifies the inclusion of new interactions, whilst also facilitating the modification of regulation types, thereby making it well-suited to network inference. Furthermore, S-System models mitigate the issue of parameter identifiability. Finally, by applying linear systems theory to the models considered, we provide some justification for the increased use of aggregated protein equations in recent plant clock modelling, replacing the separate cytoplasmic/nuclear protein compartments that were characteristic of the earlier models. We conclude that as well as providing a simplified framework for model development, the S-System formalism also possesses significant potential as a robust modelling method for designing synthetic gene circuits. PLOS COMPUTATIONAL BIOLOGYPLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1007671 March 16, 2020 1 / 34 OPEN ACCESS Citation: Foo M, Bates DG, Akman OE (2020) A simplified modelling framework facilitates more complex representations of plant circadian clocks. PLoS Comput Biol 16(3): e1007671. https://doi. Author summaryMathematical models have been widely employed as a complement to experimental work in elucidating the underlying mechanistic behaviour of the plant circadian clock. In this study, we investigate the use of a simplified modelling strategy, the S-System framework, to reduce the computational complexity of such models. We test the efficacy of our approach by constructing S-System versions of five established plant clock models, which we fit to synthetic and experimental gene exp...

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“…For this, more computationally tractable models of the clock network are necessary. Reduced models of the network have already been constructed that capture many of the features of the single cell clock dynamics [29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

A spatial model of the plant circadian clock reveals design principles for coordinated timing under noisy environments

Greenwood

Tokuda

Locke

2020

Preprint

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Individual plant cells possess a genetic network, the circadian clock, that times internal processes to the day-night cycle. Mathematical models of the clock network have driven a mechanistic understanding of the clock in plants. However, these models are typically either whole plant models that ignore tissue or cell type specific clock behavior, or phase only models that do not include clock network components explicitly. It is increasingly clear that clock network models must address spatial differences, as complex spatial behaviors have been observed in tissues and cells in plants, including period and phase differences between cells and spatial waves of gene expression between organs. Here, we implement an up to date clock network model on a spatial template of the plant. In our model, the sensitivity to light inputs varies across the plant, and cells communicate their clock timing locally via the levels of core clock mRNA levels by cell-to-cell coupling. We found that differences in sensitivities to environmental input in the model can explain the experimentally observed differences in clock periods in different organs, and we show using the model that a plausible coupling mechanism can generate the experimentally observed waves in clock gene expression across the plant. We then examined what features of the plant circadian system allow it to keep time under noisy light-dark (LD) cycles. We found that differences in sensitivity to light can allow regional flexibility in phase even under LD cycles, whilst local cell-to-cell coupling minimized variability in clock rhythms in neighboring cells. Thus, local sensitivity to environmental inputs combined with cell-to-cell coupling allows for flexible yet robust circadian timing under noisy environments.

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Reducing the complexity of mathematical models for the plant circadian clock by distributed delays

Cited by 24 publications

References 46 publications

It’s about time: Analysing an alternative approach for reductionist modelling of linear pathways in systems biology

It’s about time: Analysing an alternative approach for reductionist modelling of linear pathways in systems biology

A simplified modelling framework facilitates more complex representations of plant circadian clocks

A spatial model of the plant circadian clock reveals design principles for coordinated timing under noisy environments

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