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1.  Nested sampling for parameter inference in systems biology: application to an exemplar circadian model 
BMC Systems Biology  2013;7:72.
Background
Model selection and parameter inference are complex problems that have yet to be fully addressed in systems biology. In contrast with parameter optimisation, parameter inference computes both the parameter means and their standard deviations (or full posterior distributions), thus yielding important information on the extent to which the data and the model topology constrain the inferred parameter values.
Results
We report on the application of nested sampling, a statistical approach to computing the Bayesian evidence Z, to the inference of parameters, and the estimation of log Z in an established model of circadian rhythms. A ten-fold difference in the coefficient of variation between degradation and transcription parameters is demonstrated. We further show that the uncertainty remaining in the parameter values is reduced by the analysis of increasing numbers of circadian cycles of data, up to 4 cycles, but is unaffected by sampling the data more frequently. Novel algorithms for calculating the likelihood of a model, and a characterisation of the performance of the nested sampling algorithm are also reported. The methods we develop considerably improve the computational efficiency of the likelihood calculation, and of the exploratory step within nested sampling.
Conclusions
We have demonstrated in an exemplar circadian model that the estimates of posterior parameter densities (as summarised by parameter means and standard deviations) are influenced predominately by the length of the time series, becoming more narrowly constrained as the number of circadian cycles considered increases. We have also shown the utility of the coefficient of variation for discriminating between highly-constrained and less-well constrained parameters.
doi:10.1186/1752-0509-7-72
PMCID: PMC3735395  PMID: 23899119
Model selection; Parameter inference; Nested sampling; Circadian rhythms
2.  SBSI: an extensible distributed software infrastructure for parameter estimation in systems biology 
Bioinformatics  2013;29(5):664-665.
Summary: Complex computational experiments in Systems Biology, such as fitting model parameters to experimental data, can be challenging to perform. Not only do they frequently require a high level of computational power, but the software needed to run the experiment needs to be usable by scientists with varying levels of computational expertise, and modellers need to be able to obtain up-to-date experimental data resources easily. We have developed a software suite, the Systems Biology Software Infrastructure (SBSI), to facilitate the parameter-fitting process. SBSI is a modular software suite composed of three major components: SBSINumerics, a high-performance library containing parallelized algorithms for performing parameter fitting; SBSIDispatcher, a middleware application to track experiments and submit jobs to back-end servers; and SBSIVisual, an extensible client application used to configure optimization experiments and view results. Furthermore, we have created a plugin infrastructure to enable project-specific modules to be easily installed. Plugin developers can take advantage of the existing user-interface and application framework to customize SBSI for their own uses, facilitated by SBSI’s use of standard data formats.
Availability and implementation: All SBSI binaries and source-code are freely available from http://sourceforge.net/projects/sbsi under an Apache 2 open-source license. The server-side SBSINumerics runs on any Unix-based operating system; both SBSIVisual and SBSIDispatcher are written in Java and are platform independent, allowing use on Windows, Linux and Mac OS X. The SBSI project website at http://www.sbsi.ed.ac.uk provides documentation and tutorials.
Contact: stg@inf.ed.ac.uk
Supplementary information: Supplementary data are available at Bioinformatics online.
doi:10.1093/bioinformatics/btt023
PMCID: PMC3582266  PMID: 23329415
4.  Digital clocks: simple Boolean models can quantitatively describe circadian systems 
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.
doi:10.1098/rsif.2012.0080
PMCID: PMC3405750  PMID: 22499125
systems biology; circadian gene networks; Boolean logic; photoperiodism; Arabidopsis thaliana
5.  Quantitative analysis of regulatory flexibility under changing environmental conditions 
Day length changes with the seasons in temperate latitudes, affecting the many biological rhythms that entrain to the day/night cycle: we measure these effects on the expression of Arabidopsis clock genes, using RNA and reporter gene readouts, with a new method of phase analysis.Dusk sensitivity is proposed as a simple, natural and general mathematical measure to analyse and manipulate the changing phase of a clock output relative to the change in the day/night cycle.Dusk sensitivity shows how increasing the numbers of feedback loops in the Arabidopsis clock models allows more flexible regulation, consistent with a previously-proposed, general operating principle of biological networks.The Arabidopsis clock genes show flexibility of regulation that is characteristic of a three-loop clock model, validating aspects of the model and the operating principle, but some clock output genes show greater flexibility arising from direct light regulation.
The analysis of dynamic, non-linear regulation with the aid of mechanistic models is central to Systems Biology. This study compares the predictions of mechanistic, mathematical models of the circadian clock with molecular time-series data on rhythmic gene expression in the higher plant Arabidopsis thaliana. Analysis of the models helps us to understand (explain and predict) how the clock gene circuit balances regulation by external and endogenous factors to achieve particular behaviours. Such multi-factorial regulation is ubiquitous in, and characteristic of, living systems.
The Earth's rotation causes predictable changes in the environment, notably in the availability of sunlight for photosynthesis. Many biological processes are driven by the environmental input via sensory pathways, for example, from photoreceptors. Circadian clocks provide an alternative strategy. These endogenous, 24-h rhythms can drive biological processes that anticipate the regular environmental changes, rather than merely responding. Many rhythmic processes have both light and clock control. Indeed, the clock components themselves must balance internal timing with external inputs, because circadian clocks are reset daily through light regulation of one or more clock components. This process of entrainment is complicated by the change in day length. When the times of dawn and dusk move apart in summer, and closer together in winter, does the clock track dawn, track dusk or interpolate between them?
In plants, the clock controls leaf and petal movements, the opening and closing of stomatal pores, the discharge of floral fragrances, and many metabolic activities, especially those associated with photosynthesis. Centuries of physiological studies have shown that these rhythms can behave differently. Flowering in Ipomoea nil (Pharbitis nil, Japanese morning glory) is controlled by a rhythm that tracks the time of dusk, to give a classic example. We showed that two other rhythms associated with vegetative growth track dawn in this species (Figure 5A), so the clock system allows flexible regulation.
The relatively small number of components involved in the circadian clockwork makes it an ideal candidate for mathematical modelling. Molecular genetic studies in a variety of model eukaryotes have shown that the circadian rhythm is generated by a network of 6–20 genes. These genes form feedback loops generating a rhythm in mRNA production. A single negative feedback loop in which a gene encodes a protein that, after several hours, turns off transcription is capable of generating a circadian rhythm, in principle. A single light input can entrain the clock to ‘local time', synchronised with a light–dark cycle. However, real circadian clocks have proven to be more complicated than this, with multiple light inputs and interlocked feedback loops.
We have previously argued from mathematical analysis that multi-loop networks increase the flexibility of regulation (Rand et al, 2004) and have shown that appropriately deployed flexibility can confer functional robustness (Akman et al, 2010). Here we test whether that flexibility can be demonstrated in vivo, in the model plant, A. thaliana. The Arabidopsis clock mechanism comprises a feedback loop in which two partially redundant, myb transcription factors, LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1), repress the expression of their activator, TIMING OF CAB EXPRESSION 1 (TOC1). We previously modelled this single-loop circuit and showed that it was not capable of recreating important data (Locke et al, 2005a). An extended, two-loop model was developed to match observed behaviours, incorporating a hypothetical gene Y, for which the best identified candidate was the GIGANTEA gene (GI) (Locke et al, 2005b). Two further models incorporated the TOC1 homologues PSEUDO-RESPONSE REGULATOR (PRR) 9 and PRR7 (Locke et al, 2006; Zeilinger et al, 2006). In these circuits, a morning oscillator (LHY/CCA1–PRR9/7) is coupled to an evening oscillator (Y/GI–TOC1) via the original LHY/CCA1–TOC1 loop.
These clock models, like those for all other organisms, were developed using data from simple conditions of constant light, darkness or 12-h light–12-h dark cycles. We therefore tested how the clock genes in Arabidopsis responded to light–dark cycles with different photoperiods, from 3 h light to 18 h light per 24-h cycle (Edinburgh, 56° North latitude, has 17.5 h light in midsummer). The time-series assays of mRNA and in vivo reporter gene images showed a range of peak times for different genes, depending on the photoperiod (Figure 5C). A new data analysis method, mFourfit, was introduced to measure the peak times, in the Biological Rhythms Analysis Software Suite (BRASS v3.0). None of the genes showed the dusk-tracking behaviour characteristic of the Ipomoea flowering rhythm. The one-, two- and three-loop models were analysed to understand the observed patterns. A new mathematical measure, dusk sensitivity, was introduced to measure the change in timing of a model component versus a change in the time of dusk. The one- and two-loop models tracked dawn and dusk, respectively, under all conditions. Only the three-loop model (Figure 5B) had the flexibility required to match the photoperiod-dependent changes that we found in vivo, and in particular the unexpected, V-shaped pattern in the peak time of TOC1 expression. This pattern of regulation depends on the structure and light inputs to the model's evening oscillator, so the in vivo data supported this aspect of the model. LHY and CCA1 gene expression under short photoperiods showed greater dusk sensitivity, in the interval 2–6 h before dawn, than the three-loop model predicted, so these data will help to constrain future models.
The approach described here could act as a template for experimental biologists seeking to understand biological regulation using dynamic, experimental perturbations and time-series data. Simulation of mathematical models (despite known imperfections) can provide contrasting hypotheses that guide understanding. The system's detailed behaviour is complex, so a natural and general measure such as dusk sensitivity is helpful to focus on one property of the system. We used the measure to compare models, and to predict how this property could be manipulated. To enable additional analysis of this system, we provide the time-series data and experimental metadata online.
The circadian clock controls 24-h rhythms in many biological processes, allowing appropriate timing of biological rhythms relative to dawn and dusk. Known clock circuits include multiple, interlocked feedback loops. Theory suggested that multiple loops contribute the flexibility for molecular rhythms to track multiple phases of the external cycle. Clear dawn- and dusk-tracking rhythms illustrate the flexibility of timing in Ipomoea nil. Molecular clock components in Arabidopsis thaliana showed complex, photoperiod-dependent regulation, which was analysed by comparison with three contrasting models. A simple, quantitative measure, Dusk Sensitivity, was introduced to compare the behaviour of clock models with varying loop complexity. Evening-expressed clock genes showed photoperiod-dependent dusk sensitivity, as predicted by the three-loop model, whereas the one- and two-loop models tracked dawn and dusk, respectively. Output genes for starch degradation achieved dusk-tracking expression through light regulation, rather than a dusk-tracking rhythm. Model analysis predicted which biochemical processes could be manipulated to extend dusk tracking. Our results reveal how an operating principle of biological regulators applies specifically to the plant circadian clock.
doi:10.1038/msb.2010.81
PMCID: PMC3010117  PMID: 21045818
Arabidopsis thaliana; biological clocks; dynamical systems; gene regulatory networks; mathematical models; photoperiodism
6.  Robustness from flexibility in the fungal circadian clock 
BMC Systems Biology  2010;4:88.
Background
Robustness is a central property of living systems, enabling function to be maintained against environmental perturbations. A key challenge is to identify the structures in biological circuits that confer system-level properties such as robustness. Circadian clocks allow organisms to adapt to the predictable changes of the 24-hour day/night cycle by generating endogenous rhythms that can be entrained to the external cycle. In all organisms, the clock circuits typically comprise multiple interlocked feedback loops controlling the rhythmic expression of key genes. Previously, we showed that such architectures increase the flexibility of the clock's rhythmic behaviour. We now test the relationship between flexibility and robustness, using a mathematical model of the circuit controlling conidiation in the fungus Neurospora crassa.
Results
The circuit modelled in this work consists of a central negative feedback loop, in which the frequency (frq) gene inhibits its transcriptional activator white collar-1 (wc-1), interlocked with a positive feedback loop in which FRQ protein upregulates WC-1 production. Importantly, our model reproduces the observed entrainment of this circuit under light/dark cycles with varying photoperiod and cycle duration. Our simulations show that whilst the level of frq mRNA is driven directly by the light input, the falling phase of FRQ protein, a molecular correlate of conidiation, maintains a constant phase that is uncoupled from the times of dawn and dusk. The model predicts the behaviour of mutants that uncouple WC-1 production from FRQ's positive feedback, and shows that the positive loop enhances the buffering of conidiation phase against seasonal photoperiod changes. This property is quantified using Kitano's measure for the overall robustness of a regulated system output. Further analysis demonstrates that this functional robustness is a consequence of the greater evolutionary flexibility conferred on the circuit by the interlocking loop structure.
Conclusions
Our model shows that the behaviour of the fungal clock in light-dark cycles can be accounted for by a transcription-translation feedback model of the central FRQ-WC oscillator. More generally, we provide an example of a biological circuit in which greater flexibility yields improved robustness, while also introducing novel sensitivity analysis techniques applicable to a broader range of cellular oscillators.
doi:10.1186/1752-0509-4-88
PMCID: PMC2913929  PMID: 20576110
7.  Isoform switching facilitates period control in the Neurospora crassa circadian clock 
Correction to: Molecular Systems Biology 4:164. doi:10.1038/msb.2008.5; published online 12 February 2008
doi:10.1038/msb.2008.28
PMCID: PMC2387236
8.  Isoform switching facilitates period control in the Neurospora crassa circadian clock 
A striking and defining feature of circadian clocks is the small variation in period over a physiological range of temperatures. This is referred to as temperature compensation, although recent work has suggested that the variation observed is a specific, adaptive control of period. Moreover, given that many biological rate constants have a Q10 of around 2, it is remarkable that such clocks remain rhythmic under significant temperature changes. We introduce a new mathematical model for the Neurospora crassa circadian network incorporating experimental work showing that temperature alters the balance of translation between a short and long form of the FREQUENCY (FRQ) protein. This is used to discuss period control and functionality for the Neurospora system. The model reproduces a broad range of key experimental data on temperature dependence and rhythmicity, both in wild-type and mutant strains. We present a simple mechanism utilising the presence of the FRQ isoforms (isoform switching) by which period control could have evolved, and argue that this regulatory structure may also increase the temperature range where the clock is robustly rhythmic.
doi:10.1038/msb.2008.5
PMCID: PMC2267733  PMID: 18277380
circadian clocks; isoform switching; mathematical models; oscillations; temperature compensation

Results 1-8 (8)