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)

Leiden, 30/5/2008

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)

Leiden, 30/5/2008

•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!

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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)

Leiden, 30/5/2008

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

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

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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)

Leiden, 30/5/2008

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

Leiden, 30/5/2008

Slow timescales

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