Ozgur E. Akman

Peeking into pit fields: a multiple twinning model of secondary plasmodesmata formation in tobacco

In higher plants, plasmodesmata (PD) are major conduits for cell–cell communication. Primary PD are laid down at cytokinesis, while secondary PD arise during wall extension. During leaf development, the basal cell walls of trichomes extend radially without division, providing a convenient system for studying the origin of secondary PD. We devised a simple freeze-fracture protocol for examining large numbers of PD in surface view. In the postcytokinetic wall, simple PD were distributed randomly. As the wall extended, PD became twinned at the cell periphery. Additional secondary pores were inserted at right angles to these, giving rise to pit fields composed of several paired PD. During wall extension, the number of PD increased fivefold due to the insertion of secondary PD. Our data are consistent with a model in which a subset of the original primary PD pores function as templates for the insertion of new secondary PD, spatially fixing the position of future pit fields. Many of the new PD shared the same wall collar as the original PD pore, suggesting that new PD pores may arise by fissions of existing PD progenitors. Different models of secondary PD formation are discussed. Our data are supported by a computational model, Plasmodesmap, which accurately simulates the formation of radial pit fields during cell wall extension based on the occurrence of multiple PD twinning events in the cell wall. The model predicts PD distributions with striking resemblance to those seen on fractured wall faces.

Quantitative analysis of regulatory flexibility under changing environmental conditions

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.

Robustness from flexibility in the fungal circadian clock

BackgroundRobustness 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 clocks 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.ResultsThe 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 FRQs positive feedback, and shows that the positive loop enhances the buffering of conidiation phase against seasonal photoperiod changes. This property is quantified using Kitanos 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.ConclusionsOur 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.

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 clocks 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.

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.

Complementary approaches to understanding the plant circadian clock

Circadian clocks are oscillatory genetic networks that help organisms adapt to the 24-hour day/night cycle. The clock of the green alga Ostreococcus tauri is the simplest plant clock discovered so far. Its many advantages as an experimental system facilitate the testing of computational predictions. We present a model of the Ostreococcus clock in the stochastic process algebra Bio-PEPA and exploit its mapping to di erent analysis techniques, such as ordinary di erential equations, stochastic simulation algorithms and model-checking. The small number of molecules reported for this system tests the limits of the continuous approximation underlying di erential equations. We investigate the di erence between continuous-deterministic and discrete-stochastic approaches. Stochastic simulation and model-checking allow us to formulate new hypotheses on the system behaviour, such as the presence of self-sustained oscillations in single cells under constant light conditions. We investigate how to model the timing of dawn and dusk in the context of model-checking, which we use to compute how the probability distributions of key biochemical species change over time. These show that the relative variation in expression level is smallest at the time of peak expression, making peak time an optimal experimental phase marker. Building on these analyses, we use approaches from evolutionary systems biology to investigate how changes in the rate of mRNA degradation impacts the phase of a key protein likely to a ect fitness. We explore how robust this circadian clock is towards such potential mutational changes in its underlying biochemistry. Our work shows that multiple approaches lead to a more complete understanding of the clock.

Modelling Biological Clocks with Bio-PEPA: Stochasticity and Robustness for the Neurospora crassa Circadian Network

Circadian clocks are biochemical networks, present in nearly all living organisms, whose function is to regulate the expression of specific mRNAs and proteins to synchronise rhythms of metabolism, physiology and behaviour to the 24 hour day/night cycle. Because of their experimental tractability and biological significance, circadian clocks have been the subject of a number of computational modelling studies. In this study we focus on the simple circadian clock of the fungus Neurospora crassa . We use the Bio-PEPA process algebra to develop both a stochastic and a deterministic model of the system. The light on/off mechanism responsible for entrainment to the day/night cycle is expressed using discrete time-dependent events in Bio-PEPA. In order to validate our model, we compare it against the results of previous work which demonstrated that the deterministic model is in agreement with experimental data. Here we investigate the effect of stochasticity on the robustness of the clocks function in biological timing. In particular, we focus on the variations in the phase and amplitude of oscillations in circadian proteins with respect to different factors such as the presence/absence of a positive feedback loop, and the presence/absence of light. The time-dependent sensitivity of the model with respect to some key kinetic parameters is also investigated.

Nested sampling for parameter inference in systems biology: application to an exemplar circadian model

BackgroundModel 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.ResultsWe 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.ConclusionsWe 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.

Light and circadian regulation of clock components aids flexible responses to environmental signals

The circadian clock measures time across a 24 h period, increasing fitness by phasing biological processes to the most appropriate time of day. The interlocking feedback loop mechanism of the clock is conserved across species; however, the number of loops varies. Mathematical and computational analyses have suggested that loop complexity affects the overall flexibility of the oscillator, including its responses to entrainment signals. We used a discriminating experimental assay, at the transition between different photoperiods, in order to test this proposal in a minimal circadian network (in Ostreococcus tauri) and a more complex network (in Arabidopsis thaliana). Transcriptional and translational reporters in O. tauri primarily tracked dawn or dusk, whereas in A. thaliana, a wider range of responses were observed, consistent with its more flexible clock. Model analysis supported the requirement for this diversity of responses among the components of the more complex network. However, these and earlier data showed that the O. tauri network retains surprising flexibility, despite its simple circuit. We found that models constructed from experimental data can show flexibility either from multiple loops and/or from multiple light inputs. Our results suggest that O. tauri has adopted the latter strategy, possibly as a consequence of genomic reduction.

Characterisation of congenital nystagmus waveforms in terms of periodic orbits

Because the oscillatory eye movements of congenital nystagmus vary from cycle to cycle, there is no clear relationship between the waveform produced and the underlying abnormality of the ocular motor system. We consider the durations of successive cycles of nystagmus which could be (1) completely determined by the lengths of the previous cycles, (2) completely independent of the lengths of the previous cycles or (3) a mixture of the two. The behaviour of a deterministic system can be characterised in terms of a collection of (unstable) oscillations, referred to as periodic orbits, which make up the system. By using a recently developed technique for identifying periodic orbits in noisy data, we find evidence for periodic orbits in nystagmus waveforms, eliminating the possibility that each cycle is independent of the previous cycles. The technique also enables us to identify the waveforms which correspond to the deterministic behaviour of the ocular motor system. These waveforms pose a challenge to our understanding of the ocular motor system because none of the current extensions to models of the normal behaviour of the ocular motor system can explain the range of identified waveforms.