Nutrient response signalling pathways are activated in response to feeding and control such aspects of an organism’s response to feeding as satiety and the generation, consumption and storage of energy. They also control the allocation of energy and biological substrates to somatic growth, maintenance and repair, and so implicitly underlie theories of ageing such as the Disposable Soma Theory [1] which are based on the idea of trade-offs between these processes. In the animal kingdom the insulin-signalling (IS) pathway is of particular importance [2]. Dietary sugars trigger the production of insulin or insulin-like peptides which are sensed by Insulin receptors at the cell surface and initiate a kinase cascade, leading to the activation of the downstream kinase Akt (PKB), the translocation of GLUT4 glucose transporters to the cell surface, and ultimately glucose uptake. Longer-term adaptive effects are controlled through transcription factors such as FOXO, which is deactivated (via cytoplasmic translocation) by Akt-mediated phosphorylation when IS is active [3]. When active, FOXO transcribes genes regulating cell cycle arrest, apoptosis, metabolism and maintenance [4, 5]. Animal lifespan may be increased by nutrient restriction in many animals including nematodes, flies and mice [6], at least in part through the decrease in activity of IS [2, 7, 8]. In addition to its role in ageing, dysregulation of IS is central in metabolic diseases such as Type II Diabetes (T2D).

Oxidative stress is an inescapable concomitant of the life of aerobic organisms, given that reactive oxygen species (ROS) are produced by energy generation by oxidative phosphorylation in mitochondria. ROS can damage all main categories of biomolecule including DNA, protein and lipids. Accordingly, organisms protect themselves by the production of many classes of antioxidant enzymes, including superoxide dismutase and catalase (that catalyse reactions that lead ultimately to the conversion of ROS to water), and thioredoxins and peroxiredoxins (that use reversible oxidation of sulphydryl groups to remove ROS) [9]. The extent of cellular resistance to oxidative stress varies between organisms and closely correlates with lifespan [10], and in the free radical theory of ageing [11] the damage caused by ROS is hypothesised to be the primary cause of age-related degeneration [12]. Notwithstanding this, recent studies have tended to emphasise that the levels of antioxidant enzymes do not correlate with lifespan [13], and the effects of overexpression or knockout of antioxidants, or of dietary supplementation, are not always as might be expected [14, 15]. In addition ROS activate several stress-activated protein kinases (SAPKs) such as JNK, IKK [16] and p38 MAPK [17]. The response of IS to oxidative stress is complicated by the fact that ROS has been co-opted into the active regulation of the pathway, ROS produced by NOX enzymes affecting IS immediately through the reversible oxidation and inhibition of the catalytic cysteine residues of the protein tyrosine phosphatases and lipid phosphatases that deactivate IS [18–21]. The effect of ROS on IS has also been strongly implicated in the development of metabolic disease such as T2D [22–24] and neurodegenerative disease [25].

Given the complicated nature of these two important processes and their interaction, especially the multiple countervailing effects of ROS, a quantitative model is essential to understand the biological consequences. In particular, the ability of a computational model to quantify multiple effects and study the effects of variation of experimentally-inaccessible parameters may be expected to lead to some insight. Motivated by this, we here present a kinetic computational model of IS in rodent adipocytes including FOXO and some FOXO-mediated outputs, and including interactions with ROS and some oxidative stress activated kinases. Previous modelling work on IS has focused on receptor binding, the activation of Akt and Glucose uptake [26, 27]. In recent years it has expanded to include a more detailed description of Insulin-InR binding [28], integration with connected signalling pathways such as mTOR [29, 30], EGF [31], MAPK [32] and whole-body insulin/glucose dynamics [33], though these aspects are not included in the model presented here. We have used work by Sedaghat et al. [26] as the basis of our modelling of IS and integrated it with a model we previously developed of the regulation of the FOXO transcription factor by multiple post-translational modifications (PTMS) [34]. The role of stress-activated kinases JNK and IKK in regulating FOXO was already included in this model; here we have extended this to include their regulation of IS through phosphorylation of IRS. ROS is included in the model and leads to activation of SAPKs while also having a direct effect on IS through the phosphatase-dependent mechanism described above. In the model, insulin stimulation leads to signalling-induced ROS through NOX but not additional mitochondrially generated ROS resulting from glucose uptake (this would be an oversimplification in other tissues such as muscle, where mitochondrially generated ROS makes up a larger fraction of the total). Aspects of a recent model of the antioxidant system [35] describing the transport of oxidants between the extracellular and cytoplasmic compartment were included; this enables connection with experiments in which oxidants (hydrogen peroxide) are externally applied. We develop a model integrating multiple experimental findings on the IS and oxidative stress systems and study its behaviour in scenarios of nutrient withdrawal, the simultaneous applications of Insulin and ROS, and the effects of low and high oxidative stress on insulin response.

By integrating and extending previous work, we have produced an integrated model of insulin signalling and oxidative stress and shown that it can reproduce multiple experimental observations of the response to these stimuli in isolation and together.

With regard to the modelling approach adopted here, the treatment of the multiple PTMs in the system, especially of FOXO and IRS1, would be better treated in the rule-based modelling/kappa calculus framework currently being intensively developed [77–79] than by SBML. This would also enable treatment of the many FOXO-interacting partners. Nevertheless, we reproduced some of the flexibility of rule-based modelling by using a scripting approach to generate reactions in SBML (see Methods). In addition, FOXO mediated transcription would be better modelled within a stochastic framework, which would allow the appreciable degree of variability observed in the system on a cell-to-cell basis to become apparent. However, our emphasis was on using an integrative framework for all processes and therefore the established ODE approach was the best option.

Only a few key aspects of ROS production and detoxification have been treated in the current work. There are several chemical species of importance, particularly superoxide, produced by complexes I and III of the electron transport chain; it is a highly reactive species that is converted by superoxide dismutase (SOD) in mitochondria and cytoplasm to hydrogen peroxide, which is itself converted to water by catalases in mitochondria, cytoplasm and peroxisomes. In addition, superoxide and hydrogen peroxide can be detoxified by reacting with SH groups in GSH and Prx. Hydroxyl radicals can be produced by reaction of superoxide with Iron (II); they are extremely reactive and will react with a nearby molecule before they can be detoxified. Accordingly they are handled by keeping Iron II and superoxide levels under control and keeping iron sequestered in complexes [80]. A detailed model of the antioxidant system featuring more of these components awaits future work. In our current model, only a “generic” ROS species representing superoxide or hydrogen peroxide is represented. Correspondingly, a single detoxifying enzyme, SOD2, is included, which removes this in one step, rather than the pair of an SOD and catalase (not to mention the numerous enzymes that use dithiol-disulphide conversions to remove ROS). Recent modelling work has described interesting effects of over/underexpression of particular enzymes in this detoxification pathway [81, 82]. However the approach adopted here can be justified at least in part by the fact that both SOD2 and catalase seem to be regulated in parallel by FOXO [54, 83].

ROS can react with proteins, lipids and nucleic acids to form a variety of adducts which are deleterious to a greater or lesser extent. The process may or may not be reversible. This general molecular damage is not explicitly represented in this work: more detailed models of the effects of molecular damage and its resolution by protein turnover [84–86] or the DNA damage response [87, 88] have been developed; in future work it would be desirable to combine these and explicitly represent the different damage inducing species superoxide, peroxide and hydroxyl radical.

Although the production of ROS may appear to be a process that is simply deleterious for an organism, the situation is complicated by the intricacy of the antioxidant system and the co-opting of ROS into signalling pathways such as IS. This underlies the complicated countervailing effects of ROS in the regulation of nutrient uptake and related pathologies, such as diabetes [22, 55, 89]; the present model is a first step to provide a computational framework to address this. Moreover, ROS signalling is important in the regulation of cell growth and division in general: it is becoming clear that changes in intracellular redox state coupled to ROS production are important in the regulation of almost all phases of the cell cycle, including maintenance of a quiescent state and the G1/S transition [90]. FOXO, JNK and PTP1B/PTEN all have important roles here, along with many other proteins not treated in the current work such as NRF2, p53, the CKIs, PP2A. SHP-2, LAR and Cdc25. In yeast at least, the redox potential normally cycles roughly in synchrony with the cell cycle [91]. Moreover, cellular senescence, apparently an anticancer adaptation, proceeds through a feedback loop of ROS generation [87]. Other essential processes in which ROS have a functional role include inflammation and the killing of pathogens by the macrophage oxidative burst.

The activation of SAPKs occurs by similar mechanisms to those outlined above for phosphatases: reversible cysteine oxidation leads to release of upstream activating enzymes from sequestering complexes and activates the kinases after a phosphorylation cascade; cysteine oxidation also leads to the blocking of their inactivating phosphatases (DUSPs) [59]. A more detailed model of the JNK activation cascade was made by Ferrell and co-workers [92], and it was found that in most cell types activation led to an ultrasensitive activation kinetics. Activation of IKK through TNFα was included in several models of NFκB activation [93].

Although lack of correlation of antioxidant levels in different organisms with lifespan, and the unexpected effects of knockouts and dietary antioxidant supplementations have led some authors to suggest that the free radical theory needs to be revised or rejected [14, 15, 94], it remains plausible that the complexity and multifaceted nature of the system could account for the observed behaviour [95]. Exploration of this will require much further development, but it has been begun in the recent work on catalase/SOD [81], dithiol antioxidants and transport of ROS [35] and antioxidant response through KEAP/NRF2 [96, 97], as well as that reported here, with the introduction of the phosphatase inhibition, the interaction with SAPKs and the feedback regulation of FOXO.

The dose–response curve of FOXO-mediated antioxidant regulation is at first puzzling. Upregulation of antioxidant defences in the face of weak external oxidative stress (through a JNK-mediated mechanism) is as expected; however, high oxidative stress then leads to a downregulation again (through an IKK mediated mechanism). Here we have concentrated only on describing this effect; however evolutionarily it is hard to see what its cause could be. It may simply be that the hydrogen peroxide concentrations required to observe it (an extracellular concentration of about 20 μM) are higher than would ever be experienced by a cell in vivo, or it may be that the mechanism is cell type specific. The IKK-mediated mechanism could be activated in vivo by other, more specific, activators of NFκB signalling such as certain cytokines.

The work carried out here has been largely literature based. Better constraints could be put on the model with more detailed and more fully quantified timecourses carried out in a single (or a few) cell types. Rather than adhering to the minimal modelling approach [33, 98], we have not considered multiple model architectures, and the architecture used is fairly complex, so we accept that some parameters will not be uniquely determined. Nevertheless, we have tried to avoid overfitting by extensive human intervention in the fitting, to prevent sparse data driving parameters to particularly large or small values. The sensitivity analysis indicates that all parameters have an effect on at least some species of the model and so are likely to be at least in principle identifiable, although due to the model architecture there may be some unidentifiable parameter combinations; for example the rate of FOXO synthesis is chosen to be, not a zero-order process, but the product of an E2F1 transcription factor species, whose number is assigned and does not otherwise vary, and a per-molecule rate.

We have also found it necessary to use data sources from multiple organisms and multiple tissues. Most of them are adipocyte or adipocytes like (eg 3T3-L1 cells) or skeletal muscle, the two most strongly insulin-responsive tissues, from rodent (rat/mouse) or human sources. Particularly in the FOXO part of the model, however, other cell types have been used, such as human colon and liver carcinoma (see text and Additional file 7). Many mechanisms, particularly of core insulin signalling, are strongly conserved among mammals and indeed between mammals other vertebrates and even invertebrates, but this feature of the model certainly emphasises the need for independent experimental validation.

In this model we have begun to address transcriptional feedbacks, but the simple nature of the treatment here belies the extreme difficulty of handling this: antioxidants are under the control of multiple transcription factors, but the measurements of total protein or transcript level can give no indication of which TF produced it. Hence we expect our long-time (FOXO-transcription-mediated) results will be qualitative only. The situation is further complicated by the regulation of protein synthesis at the translational level, the physical interaction of TFs with cofactors or each other, and the transcription of one TF by another [99]. For example FOXO exerts positive feedback on its own synthesis [100], and there are additional mutual regulations of FOXO, Akt and PP2A [101, 102]. Achieving biological realism here is a formidable task, but could lead to a better understanding of the homeostatic and adaptive behaviours of the pathway.

What, aside from chemical detail of the ROS, is missing in the model? Clearly the system as it stands is much simplified. Other signalling pathways (p38 MAPK etc.) and TFs (NFkB [103], NRF2 [96]) may also be activated by ROS, and there are other effects, for example secretion of inflammatory cytokines. This may lead to reinforcement of inflammation by positive feedback generation of ROS, a process which the current model could be extended to describe. Moreover, a long exposure to anything more than mild oxidative stress may induce apoptosis [104]. With regard to the nutrient sensing and response pathways, a full description would require the inclusion of the mTOR (amino acid sensing) pathway along with AMPK (total energy) and possibly sensing of other nutrients, such as fatty acids, which give rise to inflammatory responses and activate PKC. Modelling of fatty acid metabolism was included in a model of glycolysis by Dash et al. [105]. Other work has already been done, which could be combined with that reported here, including for example insulin signalling combined with mTOR [29, 106], AMPK [107], and the ERK pathway [31]. Significantly, other recent models of insulin signalling [33], as well as including mTOR signalling, have combined the cell signalling model with a whole body model of insulin/glucose dynamics [108].

Uptake of glucose and its transfer to mitochondria will lead to an increase in the energy production of the cell, but also the ROS output of the mitochondria. This has been neglected in the current model for which mitochondrially generated ROS is a constant. This approximation is likely to be acceptable for a cell type such as adipocytes, where the majority of ROS is signalling-associated, but would certainly need to be revised in order to extend the model to other cell types such as muscle for which mitochondrially produced ROS constitutes a larger fraction of the total. The energy demand and the ratio of glucose to other fuel sources, (particularly fatty acids, which may compete with carbohydrates [109]), will also have an influence on mitochondrially-generated ROS; see Fridlyand and Philipson for a review in the context of diabetes [110]. This may be expected to occur on a rather slower timescale than the signalling-related ROS production included in the current model, but, (even in adipocytes), will complicate the long-time effects which lead to the activation of SAPKs and the induction of insulin resistance. Hence, detailed modelling of mitochondria may be required to fully understand the role of chronic oxidative stress in the system. Mechanistic models of glycolysis [111], oxidative phosphorylation [112] and ROS production [113] in mitochondria have been produced and will be combined with the current model in future extensions.

We have adapted an existing model of insulin signalling, integrated it with a model of FOXO regulation, and expanded it by introducing regulations through oxidative stress, to produce a model able to reproduce several quite complex observed effects: oxidative stress is able to activate insulin signalling, endogenous ROS increase the sensitivity of insulin signalling, and weak oxidative stress or fasting upregulates antioxidants through a homeostatic feedback mechanism, that increases the level of oxidative shock that can be withstood for a short time. Nevertheless long term high oxidative stress leads to a collapse of the oxidative stress resistance mechanism.

As outlined in the discussion, the current work addresses only a part of this very complex system and much more development, and extra experimental data, will be required to make a truly biologically realistic model of IS, FOXO and oxidative stress. Nevertheless the current work makes it apparent how valuable such an integrated model will be. A model including signalling pathways from all three food groups, as outlined above, combined with a whole body model of nutrient flux, and the interactions with oxidative stress outlined here, would be a powerful tool to address the progress of insulin resistance to metabolic syndrome and T2D [114], and how fuel use depends on overall nutrient availability and energy expenditure [115]. FOXO is known to contribute to the regulation of this [5], for example by controlling the switching between gluconeogenesis and glycolysis through transcription of PDK4, G6Pase and PEPCK; the model could be extended to include these genes downstream. It is apparent that the modelling of nutrient signalling and oxidative stress, to which the current work contributes, is coming to a level of detail that enables it to address interesting questions of life history, metabolic regulation and disease.

Simulations were carried out with Copasi version 4.6 [116, 117], using the particle number representation in deterministic mode, and analysis was done with R version 2.9.2 which was also used to produce the figures. Particle numbers were converted to concentrations for presentation. Fitting was done using Copasi’s Parameter Estimation task, and parameter sensitivities calculated with the Sensitivities tasks, using time series not steady states. The model was developed using SBML shorthand [118] in separate files corresponding to the modules of Figure 1, which were converted into SBML prior to importing into Copasi. Python scripts were used to combine the individual module files and, in the case of the FOXO module, to generate all species and reactions corresponding to the different possible combinations of PTMs and localization, as previously [34]. In essence, this results in a simulation and analysis pipeline Python - > SBML-shorthand - > SBML - > Copasi - > R. These scripts are available in supplementary material as Additional files 8 and 9, and on GitHub (https://github.com/graham1034/Smith2012_insulin_signalling), together with the model files (SBML) of each module (Additional file 10), and the complete model in two forms: initially (with particle numbers as in Table 1), which was used to produce the majority of the results (Additional file 11); and after equilibration (taken from the end of the simulation in Additional file 2: Figure S2, then further equilibrated at constant low insulin of 0.2 nM for two more days), which was used to produce the results in Figure 4 (Additional file 12). The first of these models. with particle numbers at their initial levels, has been uploaded to BIOMODELS (https://www.ebi.ac.uk/biomodels-main; identifier MODEL1212210000). The differential equations of the model are also available as Additional file 1.