Oscillation patterns in biological networks

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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)
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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

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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
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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 - genetics
Core 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
Ex. NfkB Oscillations
Spikiness is needed to reduce DNA traffic?
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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

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Regulatory networks and monotone systems
What mean the above graphs for the dynamical
systems ?
Deterministic, no time delays
Monotone dynamical systems!
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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 gis 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!
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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?
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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
P53 model
dx1/dts-x3x1/(Kx1) dx2/dtx12-x2 dx3/dtx2-x3
with S30, K.1
Nullclines can be always crossed in only one
direction! How to generalize it?
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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
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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
predicted loop
KaiB KaiC1 KaiA
Ken-Ichi Kucho et al. Journ. Bacteriol. Mar 2005
2190-2199
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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
Different basic symbolic dynamics (different kind
of control) but same scenarios
BHS system
SP, S. Khrishna, MH Jensen, in preparation
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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!