Mathematical modelling plant signalling networks

During the last two decades, molecular genetic studies and the completion of the sequencing of the Arabidopsis thaliana genome have increased knowledge of hormonal regulation in plants. These signal transduction pathways act in concert through gene regulatory and signalling networks whose main components have begun to be elucidated. Our understanding of the resulting cellular processes is hindered by the complex, and sometimes counter-intuitive, dynamics of the networks, which may be interconnected through feedback controls and cross-regulation. Mathematical modelling provides a valuable tool to investigate such dynamics and to perform in silico experiments that may not be easily carried out in a laboratory. In this article, we firstly review general methods for modelling gene and signalling networks and their application in plants. We then describe specific models of hormonal perception and cross-talk in plants. This sub-cellular analysis paves the way for more comprehensive mathematical studies of hormonal transport and signalling in a multi-scale setting

Quantitative inference of dynamic regulatory pathways via microarray data

BACKGROUND: The cellular signaling pathway (network) is one of the main topics of organismic investigations. The intracellular interactions between genes in a signaling pathway are considered as the foundation of functional genomics. Thus, what genes and how much they influence each other through transcriptional binding or physical interactions are essential problems. Under the synchronous measures of gene expression via a microarray chip, an amount of dynamic information is embedded and remains to be discovered. Using a systematically dynamic modeling approach, we explore the causal relationship among genes in cellular signaling pathways from the system biology approach. RESULTS: In this study, a second-order dynamic model is developed to describe the regulatory mechanism of a target gene from the upstream causality point of view. From the expression profile and dynamic model of a target gene, we can estimate its upstream regulatory function. According to this upstream regulatory function, we would deduce the upstream regulatory genes with their regulatory abilities and activation delays, and then link up a regulatory pathway. Iteratively, these regulatory genes are considered as target genes to trace back their upstream regulatory genes. Then we could construct the regulatory pathway (or network) to the genome wide. In short, we can infer the genetic regulatory pathways from gene-expression profiles quantitatively, which can confirm some doubted paths or seek some unknown paths in a regulatory pathway (network). Finally, the proposed approach is validated by randomly reshuffling the time order of microarray data. CONCLUSION: We focus our algorithm on the inference of regulatory abilities of the identified causal genes, and how much delay before they regulate the downstream genes. With this information, a regulatory pathway would be built up using microarray data. In the present study, two signaling pathways, i.e. circadian regulatory pathway in Arabidopsis thaliana and metabolic shift pathway from fermentation to respiration in yeast Saccharomyces cerevisiae, are reconstructed using microarray data to evaluate the performance of our proposed method. In the circadian regulatory pathway, we identified mainly the interactions between the biological clock and the photoperiodic genes consistent with the known regulatory mechanisms. We also discovered the now less-known regulations between crytochrome and phytochrome. In the metabolic shift pathway, the casual relationship of enzymatic genes could be detected properly

Stochastic neural network models for gene regulatory networks

Recent advances in gene-expression profiling technologies provide large amounts of gene expression data. This raises the possibility for a functional understanding of genome dynamics by means of mathematical modelling. As gene expression involves intrinsic noise, stochastic models are essential for better descriptions of gene regulatory networks. However, stochastic modelling for large scale gene expression data sets is still in the very early developmental stage. In this paper we present some stochastic models by introducing stochastic processes into neural network models that can describe intermediate regulation for large scale gene networks. Poisson random variables are used to represent chance events in the processes of synthesis and degradation. For expression data with normalized concentrations, exponential or normal random variables are used to realize fluctuations. Using a network with three genes, we show how to use stochastic simulations for studying robustness and stability properties of gene expression patterns under the influence of noise, and how to use stochastic models to predict statistical distributions of expression levels in population of cells. The discussion suggest that stochastic neural network models can give better description of gene regulatory networks and provide criteria for measuring the reasonableness o mathematical models

When kinases meet mathematics: the systems biology of MAPK signalling

The mitogen activated protein kinase/extracellular signal regulated kinase pathway regulates fundamental cellular function such as cell proliferation, survival, differentiation and motility, raising the question how these diverse functions are specified and coordinated. They are encoded through the activation kinetics of the pathway, a multitude of feedback loops, scaffold proteins, subcellular compartmentalisation, and crosstalk with other pathways. These regulatory motifs alone or in combination can generate a multitude of complex behaviour. Systems biology tries to decode this complexity through mathematical modelling and prediction in order to gain a deeper insight into the inner works of signalling networks

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

This is the final version. Available on open access from Public Library of Science via the DOI in this recordData Availability: All MATLAB files used to generate the results presented in the study are available from https://github.com/mathiasfoo/essystemplantcircadian.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.Royal SocietyEngineering and Physical Sciences Research Council (EPSRC)Biotechnology and Biological Sciences Research Council (BBSRC

Digital clocks: simple Boolean models can quantitatively describe circadian systems

Copyright © 2012 The Royal Society This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.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