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)

Leiden, 30/5/2008 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

Leiden, 30/5/2008 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, msb E1

Leiden, 30/5/2008 Ultradian oscillations Period ~ hours Periodic - irregular Causes? Purposes? Ex: p53 system - single cell fluorescence experiment

Leiden, 30/5/2008 The p53 example - genetics Core modeling - guessing the most relevant interactions

Leiden, 30/5/2008 The p53 example - time delayed model

Leiden, 30/5/2008 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)

Leiden, 30/5/2008 Spiky oscillations Spikiness is needed to reduce DNA traffic? Ex. NfkB Oscillations

Leiden, 30/5/2008 Testing negative feedback loops: the Repressilator coherent oscillations, longer than the cell division time MB Elowitz & S. Leibler, Nature 403, (2000)

Leiden, 30/5/2008 Regulatory networks dynamical models (rate equations) continuous variables x i on the nodes (concentrations, gene expressions, firing rates?) arrows represent interactions

Leiden, 30/5/2008 Regulatory networks and monotone systems What mean the above graphs for the dynamical systems ? Deterministic, no time delays Monotone dynamical systems!

Leiden, 30/5/2008 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)

Leiden, 30/5/2008 General monotone feedback loop The g i s are decreasing functions of x i and increasing (A) / decreasing (R) functions of x i-1 Trajectories are bounded SP, S. Krishna, MH Jensen, PNAS (2007)

Leiden, 30/5/2008 The fixed point From the slope of F(x*) one can deduce if there are oscillations!

Leiden, 30/5/2008 Stability analysis and Hopf scenario What happens far from the bifurcation point? By varying some parameters, two complex conjugate eigenvalues acquire a positive real part. Simple case - equal degradation rates at fixed point

Leiden, 30/5/2008 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 (1990)

Leiden, 30/5/2008 The sectors - 2D case Nullclines can be crossed only in one direction - Only one symbolic pattern is possible for this loop

Leiden, 30/5/2008 The sectors - 3D case Nullclines can be always crossed in only one direction! How to generalize it? dx 1 /dt=s-x 3 x 1 /(K+x 1 ) dx 2 /dt=x 1 2 -x 2 dx 3 /dt=x 2 -x 3 P53 model: with S=30, K=.1

Leiden, 30/5/2008 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!

Leiden, 30/5/2008 Rules for crossing sectors - single loop

Leiden, 30/5/2008 The stationary state H = number of mismatches H can decrease by 2 or stay constant H min = 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

Leiden, 30/5/2008 One loop - one symbolic sequence

Leiden, 30/5/2008 Example: p53 Rules still apply if there are non-observed chemicals: p53 activates mdm2, mdm2 represses p53

Leiden, 30/5/2008 Circadian oscillations in cyanobacteria Ken-Ichi Kucho et al. Journ. Bacteriol. Mar KaiB KaiC1 KaiA predicted loop:

Leiden, 30/5/2008 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)

Leiden, 30/5/2008 General case - more loops HP system BHS system SP, S. Khrishna, MH Jensen, in preparation Different basic symbolic dynamics (different kind of control) but same scenarios

Leiden, 30/5/2008 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

Leiden, 30/5/2008 Slow timescales Transcription regulation is a very slow process It involves many intermediate steps Chemistry is much faster!