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Insights into Biological Network Oscillations

Explore oscillation patterns in biological networks, focusing on negative feedback loops, complex dynamics, and regulatory networks. Gain insights from examples like the p53 system and learn about the role of feedback loops in generating oscillations.

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Insights into Biological Network Oscillations

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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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