Leiden, 30/5/2008
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 - p53Red -mdm2
N. Geva-Zatorsky et al. Mol. Syst. Bio. 2006, msb4100068-E1
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
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Ultradian oscillations
• Period ~ hours• Periodic - “irregular”• Causes? Purposes?
Ex: p53 system - single cell fluorescence experiment
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The p53 example - geneticsCore modeling - guessing the most relevant
interactions
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The p53 example - time delayed model
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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)
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Spiky oscillations
Spikiness is needed to reduce DNA traffic?
Ex. NfkB Oscillations
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Testing negative feedback loops: the Repressilator
coherent oscillations, longer than the cell division time
MB Elowitz & S. Leibler, Nature 403, 335-338 (2000)
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•Regulatory networks
• dynamical models (rate equations)
• continuous variables xi 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)
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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)
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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
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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)
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The sectors - 2D case
Nullclines can be crossed only in one direction -Only one symbolic pattern is possible for this loop
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The sectors - 3D case
Nullclines can be always crossed in only one direction! How to generalize it?
dx1/dt=s-x3x1/(K+x1)dx2/dt=x1
2-x2
dx3/dt=x2-x3
P53 model:
with S=30, K=.1
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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!
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Rules for crossing sectors - single loop
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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
Leiden, 30/5/2008
One loop - one symbolic sequence
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Example: p53
Rules still apply if there are non-observed chemicals: p53 activates mdm2, mdm2 represses p53
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Circadian oscillations in cyanobacteria
Ken-Ichi Kucho et al. Journ. Bacteriol. Mar 2005 2190-2199
KaiB
KaiC1 KaiA
predicted loop:
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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)
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General case - more loops HP system
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
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Slow timescales
• Transcription regulation is a very slow process• It involves many intermediate steps • Chemistry is much faster!