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Oscillation patterns in biological networks

Oscillation patterns in biological networks. Simone Pigolotti (NBI, Copenhagen) 30/5/2008. In collaboration with: M.H. Jensen, S. Krishna, K. Sneppen (NBI) G. Tiana (Univ. Milano). Outline. Review of oscillations in cells - examples

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Oscillation patterns in biological networks

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  1. Oscillation patterns in biological networks Simone Pigolotti (NBI, Copenhagen) 30/5/2008 In collaboration with: M.H. Jensen, S. Krishna, K. Sneppen (NBI) G. Tiana (Univ. Milano)

  2. Outline • Review of oscillations in cells • - examples • - common design: negative feedback • Patterns in negative feedback loop • - order of maxima - minima • - time series analysis • Dynamics with more loops

  3. Complex dynamics p53 system - regulates apoptosis in mammalian cells after strong DNA damage Single cell fluorescence microscopy experiment Green - p53 Red -mdm2 N. Geva-Zatorsky et al. Mol. Syst. Bio. 2006, msb4100068-E1

  4. Ultradian oscillations • Period ~ hours • Periodic - “irregular” • Causes? Purposes? Ex: p53 system - single cell fluorescence experiment

  5. The p53 example - genetics Core modeling - guessing the most relevant interactions

  6. The p53 example - time delayed model

  7. Many possible models Not all the interactions are known - noisy datasets, short time series Basic ingredients: negative feedback + delay (intermediate steps) Negative feedback is needed to have oscillations! G.Tiana, S.Krishna, SP, MH Jensen, K. Sneppen, Phys. Biol. 4 R1-R17 (2007)

  8. Spiky oscillations Ex. NfkB Oscillations Spikiness is needed to reduce DNA traffic?

  9. Testing negative feedback loops: the Repressilator coherent oscillations, longer than the cell division time MB Elowitz & S. Leibler, Nature 403, 335-338 (2000)

  10. Regulatory networks • dynamical models (rate equations) • continuous variables xi on the nodes (concentrations, gene expressions, firing rates?) • arrows represent interactions

  11. Regulatory networks and monotone systems What mean the above graphs for the dynamical systems ? Deterministic, no time delays Monotone dynamical systems!

  12. Regulatory networks - monotonicity • Interactions are monotone (but poorly known) • Models - the Jacobian entries never change sign • Theorem - at least one negative feedback loop is needed to have oscillations - at least one positive feedback loop is needed to have multistability (Gouze’, Snoussi 1998)

  13. General monotone feedback loop • The gi‘s are decreasing functions of xi and increasing (A) / decreasing (R) functions of xi-1 • Trajectories are bounded SP, S. Krishna, MH Jensen, PNAS 104 6533-7 (2007)

  14. The fixed point From the slope of F(x*) one can deduce if there are oscillations!

  15. Stability analysis and Hopf scenario Simple case - equal degradation rates at fixed point By varying some parameters, two complex conjugate eigenvalues acquire a positive real part. What happens far from the bifurcation point?

  16. No chaos in negative feedback loops Even in more general systems (with delays): monotonic only in the second variable, chaos is ruled out Poincare’ Bendixson kind of result - only fixed point or periodic orbits J. Mallet-Paret and HL Smith, J. Dyn. Diff. Eqns 2 367-421(1990)

  17. The sectors - 2D case Nullclines can be crossed only in one direction - Only one symbolic pattern is possible for this loop

  18. The sectors - 3D case P53 model: dx1/dt=s-x3x1/(K+x1) dx2/dt=x12-x2 dx3/dt=x2-x3 with S=30, K=.1 Nullclines can be always crossed in only one direction! How to generalize it?

  19. Rules for crossing sectors • A variable cannot have a maximum when its activators are increasing and its repressors are decreasing • A variable cannot have a minimum when its activators are decreasing and its repressors are increasing Rules valid also when more loops are present!

  20. Rules for crossing sectors - single loop

  21. The stationary state H = number of mismatches H can decrease by 2 or stay constant Hmin = 1 Corresponding to a single mismatch traveling in the loop direction! - defines a unique, periodic symbolic sequence of 2N states Tool for time series analysis - from symbols to network structure

  22. One loop - one symbolic sequence

  23. Example: p53 Rules still apply if there are non-observed chemicals: p53 activates mdm2, mdm2 represses p53

  24. Circadian oscillations in cyanobacteria predicted loop: KaiB KaiC1 KaiA Ken-Ichi Kucho et al. Journ. Bacteriol. Mar 2005 2190-2199

  25. General case - more loops Hastings - Powell model Blausius- Huppert - Stone model Different symbolic dynamics - logistic term Hastings, Powell, Ecology (1991) Blausius, Huppert, Stone, Nature (1990)

  26. General case - more loops HP system HP system Different basic symbolic dynamics (different kind of control) but same scenarios BHS system SP, S. Khrishna, MH Jensen, in preparation

  27. Conclusions • Oscillations are generally related to negative feedback loops • Characterization of the dynamics of negative feedback loops • General network - symbolic dynamics not unique • but depending on the dynamics

  28. Slow timescales • Transcription regulation is a very slow process • It involves many intermediate steps • Chemistry is much faster!

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