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.

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.

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 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.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 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.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.

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.

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 Q 10 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.

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.

BackgroundParameter optimisation is a critical step in the construction of computational biology models. In eye movement research, computational models are increasingly important to understanding the mechanistic basis of normal and abnormal behaviour. In this study, we considered an existing neurobiological model of fast eye movements (saccades), capable of generating realistic simulations of: (i) normal horizontal saccades; and (ii) infantile nystagmus – pathological ocular oscillations that can be subdivided into different waveform classes. By developing appropriate fitness functions, we optimised the model to existing experimental saccade and nystagmus data, using a well-established multi-objective genetic algorithm. This algorithm required the model to be numerically integrated for very large numbers of parameter combinations. To address this computational bottleneck, we implemented a master-slave parallelisation, in which the model integrations were distributed across the compute units of a GPU, under the control of a CPU.ResultsWhile previous nystagmus fitting has been based on reproducing qualitative waveform characteristics, our optimisation protocol enabled us to perform the first direct fits of a model to experimental recordings. The fits to normal eye movements showed that although saccades of different amplitudes can be accurately simulated by individual parameter sets, a single set capable of fitting all amplitudes simultaneously cannot be determined. The fits to nystagmus oscillations systematically identified the parameter regimes in which the model can reproduce a number of canonical nystagmus waveforms to a high accuracy, whilst also identifying some waveforms that the model cannot simulate. Using a GPU to perform the model integrations yielded a speedup of around 20 compared to a high-end CPU.ConclusionsThe results of both optimisation problems enabled us to quantify the predictive capacity of the model, suggesting specific modifications that could expand its repertoire of simulated behaviours. In addition, the optimal parameter distributions we obtained were consistent with previous computational studies that had proposed the saccadic braking signal to be the origin of the instability preceding the development of infantile nystagmus oscillations. Finally, the master-slave parallelisation method we developed to accelerate the optimisation process can be readily adapted to fit other highly parametrised computational biology models to experimental data.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-017-0416-2) contains supplementary material, which is available to authorized users.

Network models of brain dynamics provide valuable insight into the healthy functioning of the brain and how this breaks down in disease. A pertinent example is the use of network models to understand seizure generation (ictogenesis) in epilepsy. Recently, computational models have emerged to aid our understanding of seizures and to predict the outcome of surgical perturbations to brain networks. Such approaches provide the opportunity to quantify the effect of removing regions of tissue from brain networks and thereby search for the optimal resection strategy.Here, we use computational models to elucidate how sets of nodes contribute to the ictogenicity of networks. In small networks we fully elucidate the ictogenicity of all possible sets of nodes and demonstrate that the distribution of ictogenicity across sets depends on network topology. However, the full elucidation is a combinatorial problem that becomes intractable for large networks. Therefore, we develop a global optimisation approach to search for minimal sets of nodes that contribute significantly to ictogenesis. We demonstrate the potential applicability of these methods in practice by identifying optimal sets of nodes to resect in networks derived from 20 individuals who underwent resective surgery for epilepsy. Abbreviations(iEEG) intracranial electroencephalographic recordings, (EZ) epileptogenic zone, ( ) Brain Network Ictogenicity, ( ) Node Ictogenicity, ( ) Set Ictogenicity 'Declarations of interest: none'