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!  The JAK (Janus kinase)-STAT (Signal transducer and activator of transcription) signal transduction pathway is a cascade of downstream cellular events initiated from outside of the cell through the cell surface to the DNA in the nucleus, causing transcription.

!  The conventional modeling approach for signal transduction pathways involves solving ordinary differential equations (ODEs).

!  We study here a computational alternative. We build two models of 46 reactions in the JAK-STAT pathway and compare the results.

!  We implement a deterministic mathematical model using the ODEs solver COPASI, and we build a stochastic computational model using the Stochastic Pi Machine (SPiM).

!  Since dysregulation in the functionality of JAK-STAT pathway results in immune deficiency syndrome and cancers, like lymphomas, leukemia and breast cancer, we believe that models like this have the potential to contribute to cancer research.

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!  Figure 1 depicts the 46 reactions of the JAK-STAT signal transduction pathway.

1. Receptor-JAK binding 2. Interferon-Receptor binding 3. IFN-Receptor complex dimerization 4. IFN-Receptor complex activation 5. Activated IFNRJ2-STAT1c binding 6. STAT1c activation 7. Activated IFNRJ2-STAT1c binding 8. Activated STAT1c dimerization 9. SHP2 binding 10. IFNRJ2 dephosphorylation 11. Phosphorylated STAT1c-PPX binding 12. STAT1c dephosphorylation 13. PPX binding 14. STAT1c dimmer phosphorylation 15. STAT1c-phosphorylated 16. STAT1c binding 17. STAT1c-nuclear transport 18. Phosphorylated STAT1n dimerization 19. PPN binding 20. STAT1n dephosphorylation 21. PPN binding 22. STAT1n-phosphorylated STAT1n dimerization 23. STAT1n-phosphorylated STAT1n 24. STAT1n transport to cytoplasm dimerization 25. Transcription 26. mRNA transport to cytoplasm

!  The reactions used to model the JAK-STAT signal transduction pathway and the corresponding rate constants (rate parameters) for each reaction are listed in Table 1.

!  The association reaction rate constants and dissociation reaction rate constants are represented as k and kd, respectively.

27. SOCS1 synthesis 28. mRNAc degradation 29. SOCS1 degradation 30. Phosphorylated IFNRJ2-SOCS1 binding 31. STAT1c binding 32. SHP2 binding 33. IFNRJ2 dephosphorylation 34. SOCS1 unbinding 35. SHP2 binding 36. STATc binding 37. SHP2 binding 38. IFNRJ2*-SHP2-STAT1c complex disassembly 39. SOCS1 unbinding 40. SOCS1 unbinding 41. IFNRJ2 dephosphorylation 42. SOCS1 unbinding 43. SOCS1 binding 44. SOCS1 binding 45. SOCS1 binding Interferon- 46. IFNR-JAK binding Receptor binding

! Process model generated by SPiM for the reaction 45 (Interferon-Receptor Binding)

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No. Reactions and Rate Parameters

1. [R] + [JAK] (k01 = 0.1) (kd01 = 0.05) [RJ]

2. [RJ] + [IFN] (k02 = 0.02) (kd02 = 0.02) [IFNRJ]

7.0BC45DEF64BDC0!  The Stochastic Pi Machine2 (SPiM) is a programming language based on the Stochastic Pi- Calculus, and used to model large biological systems incrementally. SPiM is used to run in

silico simulations, modeling the number of species over a period of time.

!  We focus on COPASI3 and SPiM to compare deterministic and stochastic results, respectively.

!  For instance, a bidirectional reaction, such as Receptor R and Ligand L binding to form the Receptor Ligand complex RL, where k

1 is the rate constant for forward reaction and

k-1 is the rate constant for reverse reaction, can be written as:

R + L RL

ODEs for the above reaction will be: i. d[R]/dt = -k1[R][L] + k-1[RL] ii. d[L]/dt = -k1[R][L] + k-1[RL] iii. d[RL]/dt = k1[R][L] – k-1[RL]

SPiM agents for the above reaction are: new r1@k1:chan let R() = !r1; () let L() = ?r1; RL() and RL() = delay@k-1; (RL() | L())

!  The JAK-STAT signaling pathway, an intracellular signaling pathway that leads from cell surface cytokine receptors, provides the fastest track to the nucleus causing alteration in gene transcription.

!  The JAK activation is responsible for cell differentiation, proliferation, cell migration and apoptosis. The cascade of cellular events involved in the JAK-STAT signaling pathway are critical to immune development, hematopoiesis, mammary gland development and lactation, adipogenesis and sexually dimorphic growth.

!  Computational models of the JAK-STAT signal transduction pathway have the potential

to contribute to biomedical research.

k1!

k-1!

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Figure 1: JAK-STAT Signal Transduction Pathway

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44. [IFNRJ2*-SHP2-STAT1c] + [SOCS1] (k44 = 0.02) (kd44 = 0.1) [IFNRJ2*-SHP2-SOCS1-STAT1c]

45. [IFN] + [R] (k45 = 0.02) (kd45 = 0.02) [IFNR]

46. [IFNR] + [JAK] (k46 = 0.1) (kd46 = 0.05) [IFNRJ]

Table 1: Reactions and Rate Parameters4

N.06DH>F414BDC1J0HDEIJBCO0BC034D6L134B60>B<61J6FJF30!  In process algebra, binding of two proteins can be explained as the communication between two proteins via a communication channel.

!  Consider reaction 45, Interferon-Receptor binding in Table 1.

[IFN] + [R] [IFNR]; k45 = 0.02, kd45 = 0.02

!  SPiM code for reaction 45 (Interferon-Receptor binding)

directive sample 28.8e+3 1000 Time Units directive plot R(); IFN(); IFNR() Processes Declaration

to be plotted during simulation. val k45 = 0.02 val kd45 = 0.02 val r45 = k45 val s45 = kd45

new c45@r45:chan Channel Declaration

let R() = !c45; () and IFN() = ?c45; IFNR() !c45(Send) and ?c45(Receive) and IFNR() = delay@s45; (R()|IFN())

run 100 of (R() | IFN()) Initial Concentrations

! Simulation result produced using SPiM for binding of two proteins, Receptor, R (red) and Interferon, IFN (green) to form protein complex, IFNR (blue).

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!  In this study, we have used COPASI, COmplex PAthway Simulator and SPiM to produce our deterministic and stochastic simulation results.

!  Simulation results of deterministic ODEs model in COPASI and non-deterministic stochastic model in SPiM. Population of mRNAc (blue), STAT1c (red), SOCS1 (magenta) and STAT1n*-STAT1n* (green).

!  Signal Transducer and Activator of Transcription in cytoplasm (STAT1c): Due to the interferon signaling, the STAT1c is phosphorylated in the cytoplasm and phosphorylated STAT1c travel to nucleus. The phosphorylated STAT1n activates transcription, which leads to the synthesis of SOCS1. !  Phosphorylated dimers of STAT1 in nucleus (STAT1n*-STAT1n*): The phosphorylated dimers of STAT1 increase in the nucleus due to the JAK signaling. !  mRNA in cytoplasm (mRNAc): The output of JAK-STAT signal transduction pathway is mRNA, produced during transcription. mRNA are transported from nucleus to cytoplasm and are responsible for the synthesis of SOCS1 gene. !  Suppressor of cytokine signaling-1(SOCS1): SOCS1 inhibits the JAK signaling in the cells. !  Figure 6 depicts the variations in the results produced using SPiM due to its stochastic nature.

Figure 2: Process Model Figure 3: Simulation Result (SPiM)

Figure 4: Simulation Results (COPASI) Figure 5: Simulation Results (SPiM)

Figure 6: Simulation result for the population of mRNA in the cytoplasm mRNAc, using COPASI (red) and SPiM (green and blue).

!  We build a stochastic computational model using the Stochastic Pi Machine (SPiM).

!  We implement a deterministic mathematical model using the ODEs solver COPASI, and compared our results.

!  We highlight difference between stochastic (SPiM) and non-stochastic (ODE-COPASI) modeling.

!  We developed a model that can be used to predict the behavior of the JAK-STAT pathway in the presence of inhibitory agent, creating a platform to assist in the development of new drugs.

1.  Vishakha Sharma and Adriana Compagnoni. Computational and Mathematical models of the JAK-STAT Signal Transduction Pathway. To appear in Proceedings of SCSC 2013. July 7-10, Toronto, Canada. 2.  Andrew Phillips and Luca Cardelli. Efficient, Correct Simulation of Biological Processes in the Stochastic Pi-calculus. In Computational Methods in Systems Biology, pages 184-199, September 2007. 3.  Stefan Hoops, Sven Sahle, Ralph Gauges, Christine Lee, Jurgen Pahle, Natalia Simus, Mudita Singhal, Liang Xu, Pedro Mendes and Ursula Kummer.

COPASI – a COmplex PAthway SImulator. Bioinformatics, 22(24):3067-3074, October 2006. 4.  Satoshi Yamada, Satoru Shiono, Akiko Joo and Akihiko Yoshimura. Control mechanism of JAK/STAT signal transduction pathway. FEBS Letters, 534

(16):190-196, 2003.