Stochasticity in NF-κB. Regulation asan 3. Julie Blackwood 1,2. 1,2 and

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1 Stochasticity in NF-κB Julie Blackwood 1,2 Regulation and Shlomo Ta asan asan 3 1,2 and 1 Bioengineering and Bioinformatics Summer Institute, Center for Computational C Biology and Bioinformatics, University of Pittsburgh, Pittsburgh, PA Department of Mathematics and Statistics, Rochester Institute of Technology, Rochester, NY Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213

2 Purpose Convert a deterministic model into a stochastic model Study the effects of noise Analyze qualitative differences Use a recently proposed model of NF-κB Second of two models Focus on different aspects of pathway

3 Importance of Mathematical Models and Including Noise Predictive capability under different circumstances Provide explanations Enable better visualization of the elements in a model Inclusion of noise Attempt to understand unknown factors in a system

4 What is NF-κB? Family of transcription factors Formed through the association of multiple subunits 5 main members: RelA (p65), RelB, crel,, NF-κB1 (p50/p105), NF-κB2 (p52/p100) RelA (p65), RelB and crel carry transcriptional activation domain NF-κB1 and NF-κB2 lack transcriptional activity p50 and p52 are repressive

5 What is NF-κB? Regulates transcription of numerous genes Immune response Autoimmune diseases Inflammation Misregulated in several human cancers NF-κB B remains active and in nucleus Protects cancer cells from apoptotic cell death Often enhances growth activity Better understanding is needed!

6 Inhibitor: IκBI Background Processes: Inhibitors and Activators Family of proteins Most inhibitory potential carried by IκBαI Sequesters NF-κB in cytoplasm, which then remains inactive Activation of NF-κB requires degradation of IκBα Exposes NF-κB nuclear localization sequence NF- κb translocated to nucleus Binds to κb motifs present in promoters of numerous genes NF-κB regulates transcription of IκBαI

7 Background Processes: Inhibitors and Activators Activator: cytoplasmic IκB kinase (IKK) Multiprotein complex When active, phosphorylates IκBαI Results in degradation of IκBαI Activated by extracellular signals Interleukin-1 1 (IL-1) Tumor necrosis factor (TNF) Inhibitor: A20 Cytoplasmic protein Exact mechanism not fully resolved Hypothesis: A20 acts at the level of IKK Binds to IKK, making it inactive NF-κB regulates transcription of A20

8 Model Simplifications NF-κB: only p50/p65 heterodimer considered Main activated form Expressed in various cell types IκB: only IκBαI considered Only TNF considered as extracellular signal IKK complex is assumed to be a single protein IKKn IKKa IKKi

9

10 d IKKn () t = k prod k deg IKKn () t T R k 1 IKKn () t d IKKa () t = T k IKKn R 1 () t k IKKa 3 () t T k IKKa R 2 () t A 20() t k IKKa deg () t a IKKa 2 () t I κ B α() t + t ( IKKa I κbα)( t) a IKKa( t)( IκBα NFκB)( t) + t ( IKKa IκBα NFκB)( t) d IKKi() t = k3ikka() t + TRk2IKKa() t A20() t kdegikki() t d ( IKKa I κ B α)() t = a IKKa 2 () t I κ B α() t t 1( IKKa I κ B α)() t d ( IKKa I κ B α NF κ B )() t = a 3( IKKa )()( t I κ B α NF κ B )() t t 2( IKKa I κ B α NF κ B )() t d NF κ B () t = c 6a ( I κ B α NF κ B )() t a NF B 1 κ () t I κbα() t + t2( IKKa IκBα NFκB)() t i1nfκb() t d NF κ B n() t = i 1 k v NF κ B () t a 1 I κ B αn() t NF κ B n() t d A 20( t ) = c A 4 20 t ( t ) c A 5 20( t ) d A 20() t t = c c NF B κ n() t c A 3 20() t t d I κ B α() t = a IKKa 2 () t I κ B α() t a I B 1 κ α() t NFkB () t + c I B 4a κ αt() t c 5aIκBα() t i1aiκbα() t + e1 aiκbαn() t d I κ B αn() t = a 1 I κ B αn() t NF κ B n() t + i 1a k v I κ B α() t e 1a k v I κ B αn() t d I κ B αt() t = c c NF B 2a + 1a κ n() t c I B 3a κ αt() t d ( I κ B α NF κ B ) = a I B 1 κ α() t NF κ B () t c 6a ( I κ B α NF κ B )() t a IKKa 3 ()( t I κ B α NF κ B )() t + e 2 a( IκBαn NFκBn)( t) d ( I κ B αn NFkB n)() t = a 1 I κ B αn() t NF κ B n() t e 2a k v( I κ B αn NF κ B n)() t d cgen t() t = c c NF B 2c + 1c κ n() t c cgen 3c t() t

11 Sample Equations d I κ B αt() t = c 2a + c NF 1a κ B n() t c I B 3a κ αt() t d A 20() t = c + c NF κ B () t c A 20() t t 2 1 n 3 t Formed at constant rate Produced at rate proportional to amount of another variable Decay at rate proportional to itself

12 Simulations: Deterministic Model

13 Stochastic Differential Equations ODE SDE dx x& = f = f dx= f dx = f + Ddw Implemented as: x f t+ D t N(0,1) Why use this form? One of the only types of noise we know how to analyze

14 Simulations: Stochastic Model D =

15 Simulations: Stochastic Model D = 0.005

16 Pathway Stimulus: Tumor Necrosis Induces necrosis, or death, of tumor cells Proinflammatory Diseases associated with TNF Chronic inflammation Prolonged inflammatory response leads to tissue damage Persistence of TNF stimulation Rheumatoid arthritis Inflammatory disease Painful swelling, deformity and deterioration of joints TNF stimulates proliferation of synovial cells Bacterial septic shock Factor (TNF) Infection leads to low blood pressure, low blood flow, and other symptoms Overproduction of TNF

17 Varying TNF TNF cannot be persistently active Signals usually die within hours 2 methods Leave TNF on for exactly two hours Randomly turn TNF on and off On for periods less than 4 hours for long periods of time Persistent activation can be dangerous

18 Simulations: TNF on for 2 hours No noise

19 Simulations: TNF on for 2 hours D =

20 Simulations: TNF on for 2 hours D = 0.005

21 Simulations: TNF on at random TNF activity

22 Simulations: TNF on at random No noise

23 Simulations: TNF on at random D =

24 Simulations: TNF on at random D = 0.005

25 Conclusions Possible source of noise Outside factors Inaccuracies in deterministic model Simplifications Missing or unknown elements of pathway Inaccurate parameters Figures with TNF varying appear to be less sensitive to noise Stochastic model appears to be a better representation of the NF-κB B pathway Stochastic model includes factors not included in the deterministic model

26 References Lipniacki,, T., P. Paszek,, A. Brasier,, B. Luxon,, M. Kimmel. Mathematical model of NF-κB regulatory module. Journal of Theoretical Biology 228, : 23 December Steuer,, Ralf. Effects of stochasticity in models of the cell cycle: from quantized cycle times to noise-induced induced oscillations. Journal of Theoretical Biology 228, : 21 January Tumor Necrosis Factor. University of Wisconsin Eau Claire. her/tablecon.html

27 Acknowledgements Shlomo Ta asan asan NSF/NIH Carnegie Mellon University University of Pittsburgh Rajan Munshi Everyone at BBSI