Glycogen synthase kinase-3 (GSK3) is a protein kinase involved in many physiological processes including cell structure, metabolism, gene expression and apoptosis. There are two forms of GSK3 - GSK3α and GSK3β both of which are ubiquitously expressed and constitutively active. Many of GSK3's substrates require pre-phosphorylation (priming) and so activity of priming kinases may limit GSK3 activity. GSK3 activity is also inhibited by insulin and Wnt signalling.

GSK3 has been implicated in both sporadic and familial forms of Alzheimer's disease (AD) and this recently led Hooper et al. to put forward the 'GSK3 hypothesis of AD' 1 . This hypothesis proposes that over activity of GSK3 accounts for tau hyper-phosphorylation, increased production of Aβ, inflammatory responses, reduction in acetylcholine synthesis, and memory impairment, all typical features of AD. GSK3 is also a key mediator of apoptosis but some studies show opposing effects on apoptosis (e.g. 2 3 ). A solution to this paradox is that GSK3 has a pro-apoptotic effect on the intrinsic pathway but an anti-apoptotic effect on the extrinsic pathway 4 .

The tumour suppressor protein, p53, is well known for its role in cell cycle arrest, DNA repair and apoptosis and is the most frequently mutated gene in human cancer. It is also plays an important role in ageing 5 . A role for p53 in neurodegeneration is less well known but several studies have reported an increase in p53 immunoreactivity in sporadic AD (cited in Hooper et al) 6 especially in subpopulations of cortical neurons undergoing neurofibrillary degeneration. It has recently been demonstrated that expression of p53 is partly mediated by the intracellular, transcriptionally active domain of the Amyloid Precursor Protein (APP) 7 and also that Aβ, especially Aβ42 binds to the p53 promoter and enhances transcription 8 . An interesting observation by Hooper et al (2007) is that p53 induces tau phosphorylation but that the effects are indirect since they observed p53 in the nucleus and tau in the cytoskeletal compartment 6 . Such effects could either be due to a p53 target gene or a kinase which is downstream in a p53 signalling pathway. One possibility is that GSK3β is the kinase responsible for the p53-induced tau phosphorylation, since p53 affects GSK3β activity. This paper focuses on this possible link which is described further below.

AD is characterised by the presence of extracellular amyloid plaques and intracellular tangles of which tau is the principal component. There is controversy over whether these plaques and tangles are toxic to cells or whether they are in fact a protective mechanism. The main consensus is that the intermediate soluble oligomers are the most harmful as these interfere with the cellular machinery. In particular they may bind to proteasomes and inhibit proteasomal function. Proteasomes are required for the turnover of regulatory short-lived proteins such as p53 and so any decline in proteasomal function may have severe consequences for cellular homeostasis. Oxidised and damaged/misfolded proteins are also degraded by proteasomes, so less efficiency of this system may result in aggregation of other misfolded proteins, apart from Aβ and tau. Aggregated protein inhibits proteasomal function with small soluble oligomers being the most toxic. For example, Tseng et al (2008) 9 show that small oligomers but not monomers of Aβ inhibit proteasomes. Similarly, it has been shown that paired helical filaments of tau inhibit the proteasome 10 .

Tau protein binds and stabilises microtubules (MT) but it also has to disassociate to allow trafficking of other molecules along MTs and so the interaction is dynamic. A widely held view is that phosphorylation of tau induces aggregation since tau tangles often contain hyperphosphorylated tau. However, it has been shown that phosphorylation does not directly increase the propensity of tau to form aggregates 11 12 but rather that the effect is due to increased pools of unbound tau since phosphorylated tau is unable to bind to microtubules. Tau is phosphorylated by several different kinases but the Cdk5 and GSK3β kinases phosphorylate the sites which are implicated in AD 13 . Normal tau is also turned over by the 20S proteasome with a half-life of about 12-14 hours 14 , whereas phosphorylated tau may be ubiquitinated and degraded by the 26S proteasome.

To build a quantitative model of the system it is necessary to specify all the molecular components of the system and the interactions between the different components. A mathematical model does not need to include every component and reaction in a system as this would make the model too complex to aid understanding. However, the model needs to contain enough detail to make it biologically realistic. It is generally best to start by keeping the model as simple as possible and then to add more detail if required. Using the Systems Biology Markup Language (SBML) to build the model is ideal for this approach 18 . SBML is a computer-readable format for representing biochemical reaction networks and complies to modelling standards allowing models to be portable and shared by the scientific community. There are many open source tools available for building and simulating SBML models. We have developed our own system at Newcastle University, known as BASIS (Biology of Ageing e-Science Integration and Simulation system) 19 20 . We used SBML shorthand to code the model and then converted it to full SBML using a Python script 21 . The model is publicly available on the BASIS website 22 and at the Biomodels database 23 24 . (Biomodel ID: MODEL0910130002). The SBML code is also available as supplementary material (See additional file 1: SBML code for GSK3-p53 model with IR event).

Most computer simulation models of biological systems use a deterministic approach which involves solving a set of differential equations to find the steady state of the system. However, this approach is not suitable for this model, since we are interested in the variability of cellular outcomes. For example, the time at which aggregates start to form in cells will be very variable, and the destination of aggregates will be affected by random processes. Therefore, we use stochastic simulation based on the Gillespie algorithm 25 . This means that the output of each simulation run is different for the same set of initial conditions and parameter values, unlike a deterministic simulation which always produces the same results.

The model of the p53/Mdm2/ATM circuit (Proctor & Gray 2008) 26 was extended to include details of GSK3β, Aβ and tau. The earlier model 26 was developed to explain the oscillatory behaviour in MCF7 cells with data supplied by Uri Alon 27 . It contained details of p53 and Mdm2 turnover but the reactions for protein degradation were simplified and did not include details of ubiquitination. Since GSK3β is involved in p53 degradation via phosphorylation of Mdm2 which is in complex with polyubiquitinated p53, it is necessary to add these steps. A model of the ubiquitin-proteasome has already been developed (Proctor et al 2007) 28 and so parts of this model were incorporated into the p53 model. Figure 2 shows a diagram of the overall reaction system with the important parts of the network shown in red. Table 1 lists all the species in the model, Table 2 shows all the reactions concerning GSK3β, Aβ and tau and Table 3 shows the reactions involved in p53 and Mdm2 turnover. The previous p53/Mdm2 model included a module for DNA damage to mimic the experimental system of irradiating cells and the emergence of oscillations in p53 and Mdm2 after damage. We considered two mechanisms which led to these oscillations: sequestering of Mdm2 from the p53/Mdm2 complex by the oncoprotein ARF; or activation of ATM which then phosphorylated both p53 and Mdm2 and so inhibits their binding. Either mechanism leads to an increase in levels of p53 which is followed by an increase in Mdm2 levels. As Mdm2 levels rose, complexes of p53/Mdm2 are able to reform leading to an increase in p53 degradation and so levels decrease again (see Proctor & Gray 26 for full details of modelling the oscillatory behaviour of p53, and Lahav et al 29 for the experimental details). This negative feedback loop leads to oscillations, which persist as long as DNA damage remains, keeping cells in an arrested state while DNA is repaired. The oscillations also prevent p53 levels reaching levels which would trigger apoptosis.

GSK3β has a relatively long half-life (about 48 hours) 17 and like p53, it is turned over by the proteasome. It has been shown that Hsp90 maintains the stability of GSK3β 30 . This study also showed that Hsp90 inhibition results in proteasome-dependent degradation of GSK3β, and subsequently reduces tau phosphorylation. However, we chose not to include GSK3β turnover in this model because it would be stable under the conditions we are simulating and we aim to keep the model as simple as possible. GSK3β binds to p53 and we assume that it binds to either phosphorylated or unphosphorylated p53 but that it does not bind to p53/Mdm2 complexes. GSK3β has higher activity when bound to p53. p53 bound by GSK3β also has increased transcriptional activity, as shown by Watcharisit et al. (2003) 31 whereby inhibition of GSK3β leads to 50% reduction in Mdm2 mRNA.

We assume that GSK3β phosphorylates tau with increased activity if GSK3β is bound to p53. We also include de-phosphorylation of tau by a phosphatase. We also assume that GSK3β_p53 produces Aβ, however, under normal conditions Aβ is removed by degradation and does not accumulate. We expect that under conditions of stress, levels of GSK3β_p53 would increase and that there would also be increased Aβ production. Aβ may form small aggregates which could either inhibit proteasomes or form plaques. We assume that plaques, being outside the cell, do not interfere with proteasomes. We also assume that Aβ increases the transcription of p53. We assume that tau is normally bound to microtubules but that binding is prevented when tau is phosphorylated. We also include tau turnover in the model and assume that unphosphorylated tau is degraded by the proteasome without the need for ubiquitination.

The parameter values were taken from experimental data where possible. For example we set the degradation rates for p53, Mdm2 and tau based on their half-life of 20 minutes, 30 minutes and 12 hours respectively and then set the synthesis rates so that the proteins stayed at basal levels under unstressed conditions. We set aggregation rates to be low so that no aggregation would take place under unstressed conditions. It is known that phosphorylation reactions take place on time scales in the order of minutes so we assumed that these reactions are fast. Further details of the parameters involved in the DNA damage response are given in 26 . Where parameter values are unknown, the values were adjusted so that the system was at steady state under unstressed conditions. The model was validated by checking the model predictions for levels of p53 and Mdm2 after the irradiating event.

We extended the model of Proctor & Gray (2008) 26 to include details of p53 and Mdm2 ubiquitination, the interaction of p53 with GSK3β and the activity of GSK3β. As we had used SBML to build the models, it was straight forward to make the necessary modifications. As in our previous model, we find that a sudden increase in DNA damage leads to oscillations of p53 and Mdm2. Our new model shows that the disruption of the Mdm2/p53 complex, allows the formation of GSK3β/p53 complexes which results in increased transcriptional activity of p53 and increased kinase activity of GSK3β. The result is an increase in Aβ production, an increase in Mdm2 mRNA and an increase in tau phosphorylation. Under normal conditions, the model predicts that Aβ is cleared from cells and so does not accumulate, and tau is dephosphorylated to maintain the correct balance of phosphorylated and unphosphorylated tau. However, after a stress event, the DNA damage response leads to increased activity of p53 and GSK3β which results in increased production of Aβ and increased phosphorylation of tau. However, the parameter for DNA repair was set so that most DNA damage is repaired by 24 hours, and then Aβ is cleared and tau is dephosphorylated so that aggregates do not accumulate. In the ageing brain we would expect that DNA damage might sometimes persist either due to a decline in repair mechanisms or an increase in ROS production. We used our model to examine the outcome of a deficiency in DNA repair. In this case, we find that aggregates are much more likely to accumulate which in turn lead to increased ROS production and further DNA damage which leads to further activation of p53 and GSK3β and even more aggregation.

We also used the model to examine cellular outcomes which would be more typical of ageing, where the accumulation of DNA damage and other types of cellular damage is more gradual. As stochastic computer simulations take a long time to run, we increased the aggregation rates so that the model could be run for a reasonably short time period. Without a major damaging event, stochastic events become much more important and we see considerable variation in the time at which damage starts to accumulate. We also noted that our model predicted that tau tangles start to form before Aβ plaques but that levels of these plaques soon overtake those of tau tangles due to their more rapid increase at the elongation stage. Our results might seem contradictory to experimental data and the amyloid cascade hypothesis 34 . However, aggregates need to reach a certain size to be detectable whereas our model output can detect aggregates as soon as formation begins. If we also imposed a threshold of detection on our model output, we would see Aβ plaques before tau tangles in some simulations, depending on the threshold chosen. Most importantly, our model suggests that the formation of plaques and tangles are independent events, but that they share a common cause, namely GSK3β overactivity.

The output from our model showed that even when p53 levels rose after DNA damage, they were prevented from rising indefinitely by the action of the negative feedback loop involving Mdm2. However, due to random fluctuations in protein levels, it is possible that p53 levels would sometimes rise to levels which would exceed the threshold required to activate apoptotic pathways. In this case neuronal death would take place. Our model predicted that this was much more likely to occur when protein aggregates start to accumulate and the proteasome becomes inhibited due to less efficient turnover of p53. Interestingly studies have shown that AD patients have a lower incidence of several types of cancer 35 . So although high levels of p53 are detrimental in the ageing brain due to neuronal loss, an increase in apoptosis prevents cancer.

We used the model to test the effects of deleting either p53 or GSK3β to examine whether the interaction between GSK3β and p53 was necessary to obtain aggregation to Aβ and tau. The model shows that GSK3β is required for aggregation to take place when p53 is present but that it was possible for aggregates to form in the absence of p53, providing that GSK3β was highly expressed. A number of experiments are planned to test the model predictions. If the vicious cycle hypothesis is correct, the self-amplifying cycle of GSK3β/p53 activation should be critically dependent on both entities and should not occur when one or both are inactivated or deficient. It should be possible to monitor the status of protein aggregates, tau phosphorylation, and proteasome inhibition in cultured cells irradiated to induce DNA damage either in the presence of functional GSK3β and p53 or in conditions where a component is inhibited (for example by the GSK3β inhibitor alsterpaullone) or missing (as in cells derived from p53 null mice). The predicted role of accumulating DNA damage in triggering the GSK3β/p53 vicious cycle can be tested in cells that are genetically deficient in DNA repair components or in normal cells chronically exposed to DNA damaging agents. It will also be informative to carry out experiments with GSK3β and p53 over-expression together to see if there is enhanced Aβ production or accumulation of tau. The outcome of these experiments will be reported in a future publication.

This model only examines a very small part of the cellular system. In reality there are many pathways and many other components that we have not considered here. For example, the chaperone system is very important in maintaining protein homeostasis by binding to misfolded protein and assisting in either refolding or elimination of the damaged protein. Mitochondria also play an important role as they produce the ATP required for many cellular processes such as protein degradation. They also produce ROS as a by-product of respiration. Mitochondria, themselves are susceptible to oxidative damage and this can lead to increased levels of ROS. Other models are being developed where the emphasis is on using a systems biology approach.

When we simulated the effects of an increased aggregation rate without any initial DNA damaging event, we in effect speeded up the ageing process. Care is needed in interpreting these results as they could give the false impression that aggregation might occur at an early age. However, in reality, it is unlikely that protein homeostasis would be disturbed until old age, which is why only people over a certain age get Alzheimer's disease. However, there is considerable variation in the actual age of disease onset which may be mainly due to stochastic effects as our model suggests.

The model was kept as simple as possible but included enough detail to make some meaningful predictions. We only included degradation of unphosphorylated tau by the 20S proteasome in a ubiquitin-independent manner. Although phosphorylated tau is resistant to degradation by proteases 36 , it has been shown that phosphorylated tau is degraded by ubiquitin-dependent proteasome pathway and the heat shock proteins Hsp70, Hsp70, the ubiquitin ligase CHIP and the kinase Akt are all involved 37 38 39 . More detailed models of Aβ and tau turnover which also contain more detail of the aggregation process are being developed and these could be easily incorporated in the current GSK3β/p53 model. We did not include turnover of GSK3β since GSK3β is relatively stable but over long time periods it could start to accumulate if the proteasome became inhibited and so it may be desirable to add detail of GSK3β turnover. Interestingly Hsp90 is also involved in maintaining the stability of GSK3β 30 . Furthermore, there are many other pathways leading to activation of GSK3β, such as deregulation of the insulin and Wnt pathways. Our current model could be used to examine cellular outcomes by assuming that if p53 reached above a certain threshold, apoptosis would take place. Alternatively we could further extend the model to include pathways leading to cell death. This would be a very useful addition to the model as neuronal loss is a common feature of neurodegenerative disorders but it occurs only in subsets of neurons in specific regions of the brain.

Our model predicts that GSK3β overactivity leads to an increase in levels of Aβ plaques and tau tangles by independent processes suggesting that the observed correlation between plaques and tangles may not be due to a causal relationship. The interaction of GSK3β with p53 leads to increase activity of both proteins. Since p53 and GSK3β are both involved in the apoptotic pathway, and GSK3β overactivity leads to increased levels of plaques and tangles, our model might explain the link between protein aggregation and neuronal loss in neurodegeneration. Therefore modulating the interaction of GSK3β and p53 may be a useful therapeutic strategy.

The authors declare that they have no competing interests.

CJP built the model, ran simulations, plotted results and drafted the manuscript. DAG advised on model construction, and helped to draft the manuscript. All authors read and approved the final manuscript.