1. Discovery and Assesment of New Target Sites for Anti-HIV Therapies Problem given by: Sanjive Qazi, Gustavus Adolphus College, U.S.A. Working group: Chris Breward, Math. Inst., University of Oxford, U.K. Jane Heffernan, York University, Canada. Robert M. Miura, New Jersey Institute of Technology, U.S.A. Neal Madras, York University, Canada. John Ockendon, OCIAM Math. Inst., University of Oxford, U.K. Mads Peter Sørensen, DTU Mathematics, Tech. Univ. of Denmark. Bob Anderssen, CSIRO, Mathematical and Information Sciences, Australia. Roderick Melnik, Wilfrid Laurier University, Canada. Mark McGuinness, Victoria University, New Zealand. Fields-MITACS Industrial Problem-Solving Workshop August 11 – 15, 2008

2. Introduction The HIV viruses infect cells by endocytosis and takes over parts of the cells reaction pathways in order to reproduce itself and spread the infection. One such pathway is the mammalian inflammatory signaling, which invoke NF-κB as the principal transcription factor. A treatment against HIV could be based on blocking the NF-κB pathway by a suitably designed drug. The aim of the current project is to investigate the feasibility of this idea by using mathematical modelling of the NF-κB pathway. Fields-MITACS Industrial Problem-Solving Workshop August 11 – 15, 2008

3. Outline  Cartoon model of the inflammatory pathway.  How HIV attacks mammalian cells through e.g. TNF signalling.  The role of IKK and the TNF receptor in the cell membrane.  Mathematical model of the NF-κB pathway.  The role of IKK signaling. Fixed points and stability.  Numerical examples.  Extended mathematical model.  Fixed points and stability.  Numerical examples.  Outlook and further work. Fields-MITACS Industrial Problem-Solving Workshop August 11 – 15, 2008

4. 4 HIV Life Cycle

5. 5 Drug Therapy

6. Cartoon of the NF-κB pathway Fields-MITACS Industrial Problem-Solving Workshop August 11 – 15, 2008

7. Reaction scheme

8. Mathematical description Use law of mass action for each of the reactions Assume constant concentration of D, and combine with k 5 We get A n after the fact from A=A*-A c -AB

9. Mathematical description Parameter values come from literature (means that someone else guessed them!)

10. 10 Steady State Has unique physical fixed point for all positive parameter values. Stable at given parameter values (in general: Jacobian at fixed point has positive determinant, negative trace, no positive real eigenvalues).

11. 11 Numerics k 5 =0 2k 5

12. 12 Numerics 2k 5 k 5 =0

13. Cartoon number 2 of the NF-κB pathway

14. Modified reaction scheme

15. Modified reaction scheme cont’d

16. Mathematical description Use law of mass action for each of the reactions Concentration of TNF is rolled up into k 6

17. Mathematical description

18. Steady states and stability Has unique physical fixed point for the given parameter values, as well as for all smaller (nonnegative) values of  k 6 ) and   k 5 ). Stable at given parameter values (other values not checked).

19. 19 Numerics

20. 20 Numerics

21. 21 Numerics

22. 22 Future Work 27 variable model Systematic reduction to see if it corresponds with our 7 variable model Control model Consider problem as optimal control with mu and lambda as the control parameters Unclear what to minimize Sensitivity analysis Vary rates

23. Reference: Cheong et.al. Understanding NF-κB signaling via mathematical modeling, Molecular Sytems Biology 4:192, 2008. Fields-MITACS Industrial Problem-Solving Workshop August 11 – 15, 2008 Reference: Krishna et.al. Minimal model of spiky oscillations in NF-κB signaling, PNAS 103(29), 10840- 10845, 2006. Reference: Chan et.al. Quantitative ianalysis of human immunodeficiency virus type 1-infected CD4+ cell proteome: … Journal of Virology, 7571-7583, 2007. Reference: Lipniacki et.al. Mathematical model of NF-κB regulatory module, Journal of Theoretcal Biology 228, 195-215, 2004.

24. Summary and outlook HIV viruses take over host cellular pathways for their reproduction. One such pathway is the NF-κB pathway. Cartoon modeling of the NF-κB pathway. Mathematical modeling for clearifying the underlying regulatory pathway dynamics and hopefully summarizing abundant experimental observations. Mathematical modling as a tool for rational guided drug targeting. Extended complex models and mode reduction of bio chemical complexity. Fields-MITACS Industrial Problem-Solving Workshop August 11 – 15, 2008