1. Hybrid Functional Petri Net model of the Canonical Wnt Pathway Koh Yeow Nam, Geoffrey

2. Modeling the Wnt Signaling Pathway The Wnt signaling pathway has been a focus of intense research as it is one of the main pathways of both cell growth/development as well as genetically linked diseases such as human cancer. The more understood pathway that involves Wnt proteins is known as the canonical Wnt pathway, and in this pathway, the protein β-catenin plays an important part. However, in the recent years, other pathways that involved the Wnt proteins are being discovered, including those that control the aspects of gastrulation movements etc. These pathways are collectively known as the non- canonical pathways.

3. Modeling the Wnt Signaling Pathway Although the canonical pathway has been studied extensively, quantitative studies are less common. Here we will attempt to recreate a model based on “The roles of APC and Axin Derived from Experimental and Theoretical Analysis of the Wnt Pathway” by Lee et al. using a model known as a Hybrid Functional Petri Net, which is a variant of the Petri Net model, used traditionally to model computer, telecommunication systems etc.

4. Canonical Wnt Pathway The canonical Wnt pathway has three major components – the extracellular Wnt protein, intracellular β-catenin and the Glycogen Synthase Kinase 3β (GSK-3β) Although there are variations by different groups on the exact details on how the components interact, the main idea, however, remains the same

5. Canonical Wnt Pathway Wnt proteins are a secreted protein which trigger off cell activities by first binding with a seven-membrane- spanning superfamily of cell surface receptors – Frizzled (Fz) Recent evidence also suggests that co-receptors LRP5/6 are also required in binding of Wnt proteins Note that for our model, some of the other components such as the DKK and WIF-1 are not added in as there is not enough information about them. However in subsequent work, as more and more information are obtained, they will be added in

6. Canonical Wnt Pathway When the Wnt protein (or ligands in some text) binds to the Fz receptors, it recruits the intracellular Dishevelled Protein (Dsh) to the plasma membrane where it is phosphorylated.

7. Canonical Wnt Pathway The degradation complex consist of the three proteins GSK-3β, Axin and APC. Initially, GSK-3β will phosphorylate Axin and APC within the complex. In this form, the degradation complex can bind to another protein – β-catenin and phosphorylate it, marking it for degradation by proteasomes (by ubiquitination).

8. Canonical Wnt Pathway Dsh regulates this degradation negatively by inhibiting the phosphorylation of Axin and APC in the degradation complex (Possibly by relocating the Axin from the cytoplasm to the plasma membrane), preventing β-catenin from binding to the degradation complex. β-catenin is an important component in this pathway as it is a required co-factor in the expression of certain genes that is needed in cell growth and development, such as WISP1 – 3 family and the NOV gene. (Similarly, uncontrolled action of β-catenin will lead to mass expression of certain genes, leading to Cancer)

9. Canonical Wnt Pathway β-catenin exists in the cytoplasm (attached to cadherin, a protein involved in cell adhesion) in the unphosphorylated form In the presence of the degradation complex, it gets phosphorylated and subsequently marked for degradation by proteasomes However, if it remains unphosphorylated, it will get translocated into the nucleus, where it acts as a co-factor (together with other factors like TCF/LEF) to trigger off gene expression

10. Modeling the pathway In modeling the pathway, we use the Hybrid Functional Petri Net, which is a variant of Petri net Like the Petri net, it has places and transitions but it has continuous and discrete versions of it to capture discrete events such as whether a gene is ‘on’ or ‘off’ as well as continuous features such as protein concentrations Also, it has, in addition to the normal arc, test arcs and inhibition arcs

11. Modeling the pathway The model that we are adopting from (with slight modifications) is as below

12. Modeling the pathway The entire pathwaypathway

13. Simulation This model is then input into a commercial program – Cell Illustrator TM by Gene Networks Inc Simulation is then conducted using that program [Demonstration]

14. Parameters One of the main concerns in any modeling endeavor is the availability of parameters, such as initial constants, association constants etc. For this modeling, most of the parameters were obtained from the paper by Lee et al. but some had to be guesses from the constants that are given or are obtained from the Internet

15. Parameters The equation equivalent for the HFPN model is based on the mass action law form of the kinetic model

16. Results For the results, we are interested in the steady state values of the Wnt pathway with and without Wnt protein stimulation (Dynamics will be considered at a later stage)

17. Results Some of the results, (especially the ones with Wnt stimulation) do not really correspond to the results given in the paper. A possible reason is that the parameters are not accurately captured (those missing parameters). Also the inability of the model to capture constraints can be another factor in the inaccuracy. But still, we can see the potential of using such a model to simulate bio-pathways such as the Wnt signaling pathway

18. Conclusions For this modeling exercise, we are trying to access the suitability of using a model such as HFPN as well as to see if we could generate any interesting results In terms of ease of modeling and visual appeal, the HFPN is definitely superior to equation based modeling but further work has to be established to handle other factors such as its relationship with the already established equation based laws such as the mass-action laws, rate laws and power laws as well as to handle other factors such as constraints and also the role genetic switches play (in this example, the discrete portion the HFPN is not exploited as we stop short just before gene expression)

19. Conclusions In future works, we will definitely look at some other models as HFPN is just a starting platform. Possible candidates include the Hybrid Automata, which is closer to control theory and can capture constraints in a natural way The long term goal is to develop a modeling framework in which all pathways associated with Wnt signaling can be handled in a uniform and integrated manner