modeling and control of gene regulatory...
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Modeling and control of gene regulatory networks
Madalena Chaves
BIOCO2RE (Biological control of artificial ecosystems)
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Math 640 Topics in control theory
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Genetic networks: transcription and translation
Transcription(RNApolymerase)
Translation(Ribosomes)
DNA(1-2 copy /cell)
mRNA(103 in E. coli)
Protein(106 in E. coli
109 mammalian)
1 min to transcribe
103 polymerase/cell
2 min to translate
104 ribosomes/cell
2-5 min mRNA lifetime
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Genetic networks: some common interactions
Activation of transcription (A M)
A
Repression (X M)
XX
Translation (M P)
+Signaling event
(eg., MAPK cascade)
Binding event
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Experimental data (“data rich/data poor” Sontag 2005)
Expression of gene wingless,
fly embryo(dark: higly expressed)
Microarrayrelative changes (red: expression
increased)
Cdc2,cyclin B,
Pomerening,Kim & Ferrell,
Cell 2005
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Genetic networks: questions and challenges
Modeling
Understanding the system; dynamics; predictions
Model and experiments: available data
different mathematical formalisms give different information
Parameters
calibration of models; robustness
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(Too) many components: model reduction techniques
Two well-known modules: interconnection of two systems
Control
How to find feedback laws?
How to implement?
Synthetic biology: assembling components; re-wiring a network
State estimation, observers
Genetic networks: questions and challenges
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dMdt
= An
nAn − M M
Genetic networks: how to model
Activation of transcription (A M)
A
Concentration of mRNA in terms of activator
Repression (X M)
XX dMdt
= n
nXn − M M
Concentration of mRNA in terms of repressor
Translation (M P)Concentration of protein
A
dPdt
= M − P P
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Example: drosophila segment polarity network
Model: concentrations of mRNA and proteins, for a group of 5 genes responsible for generating and maintaining the segmented body of the fruit fly
Goal: reproduce the observed pattern of expression for these 5 genes
Expression of gene wingless
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A model using ordinary differential equations
Drosophila segment polarity genesvon Dassow et al, Nature 2000
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Parameters and dynamical behavior
About 180 eqs.Randomly try 200,000
sets of parametersAbout 0.5% yield
“correct” gene pattern
Drosophila segment polarity genesvon Dassow et al, Nature 2000
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Alternative frameworks: qualitative models
Boolean models: logical rules; 0/1 or ON/OFF states
hh k1 = ENk and not CIR k
CIR
EN hh
Robustness of the model to perturbations in the environment?
Fluctuations in the mRNA/protein concentrations;
Different timescales in biological phenomena;
Degradation and synthesis rates
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SLPi=
0, if i∈{1,2}
1, if i∈{3,4}
wgi= CIAi and SLPi and not CIR i or [wgi and CIAi or SLP i and not CIRi ]
WGi= wgi
eni= WGi−1 or WGi1 and not SLP i
EN i = eni
hhi= EN i and not CIRi
HH i = hh i
ptc i= CIAi and not EN i and not CIRi
PTC i= ptc i or PTCi and not HHi−1 and not HH i1
cii= not ENi
CI i = cii
CIAi= CI i and [not PTC i or HH i−1 or HHi1 or hhi−1 or hhi1]
CIRi = CI i and PTCi and not HHi−1 and not HH i1 and not hhi−1 and not hhi1
A Boolean model of the segment polarity network
Albert & OthmerJ Theor Biol 2003
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wgWGenENhhHHptcPTCciCICIACIR
Wild type No segmentationBroad stripes
ptc mutants,heat shocked genes
en mutants(lethal phenotype)
The model exhibits multiple “biological” equilibria
wg expression
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How to study Boolean models?
dhhc
dt= −i hhc Fhh
with: Fhh t = EN t and not CIRt
hh = {0, if hhc0.51, if hhc0.5 {
CIR
EN hhhhhhc
CIRCIRc
ENENc
Dynamics: synchronous or asynchronous algorithms?
Piecewise linear models - Glass type
hhT hhk1
= ENT ENk
and not CIR TCIRk
Chaves, Albert & Sontag, JTB 2005
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Boolean models: updates and dynamicst
Synchronous Tk
All variables simultaneously updated.Deterministic trajectories in a directed graph.⇒
O11
101
110
O01
100
010
111 000
A B
C
Positive loop Synchronous transition graph
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Boolean models: updates and dynamics
Asynchronous
T1k
T Nk
T2k
...
Each variable updated at its own pace: perturbed time unit (1+ r) T , r in [-ε, ε]
NOT deterministic⇒
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Boolean models: updates and dynamics
Asynchronous
T1k
T Nk
T2k
...
Follow one of many possible trajectories in the asynchronous transition graph,
O11
101
110
O01
100
010
111 000A B
C
Each variable updated at its own pace: perturbed time unit (1+ r) T , r in [-ε, ε]
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56%, Wild type
24%, Broad stripes15%, No segmentation
4%, Wild type variant1%, Ectopic and variant
Totally asynchronous and random order updates
Starting from same initial state, percentage of simulations thatconverge to each steady state ----- low robustness...
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Random order updates + Timescale separation
First, update all protein nodes; then, update all mRNA nodesAny permutation among protein nodes followed by any permutation among mRNA nodes
Theorem: Trajectories diverge from the wild type steady state if and only if the first permutation among proteins satisfies the following order, in the third cell
CIR3 CI
3 CIA
3 PTC
3
CI3 CIR
3 CIA
3 PTC
3 [CI-PTC]
CI3 CIA
3 CIR
3 PTC
3
and all other proteins may appear in any of the remaining sites.
Chaves, Albert & Sontag, JTB 2005
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Random order+
Timescale separation
Markov Chainwith two
absorbing states
Increased robustness
87.5%, Wild type
12.5%, Broad stripes
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Piecewise linear systems: Glass-type model
dx i
dt= i F iX1 , , Xn−xi
with: FiX1 , , Xn = Boolean rule for node X i
and: Xi = {0, if xii1, if xii }
Timescale of node X i
Synthesis of gene/protein X i
(ON/OFF)
Based on: Glass& Kauffman, 1973; Edwards and Glass, 2000
Steady states: same as in Boolean model
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Some simulations
Four cells in each parasegment; periodic boundary conditions
Initial Cell 1 Cell 2 Cell 3 Cell 4 Final(stage 8) (stages 9-11)
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Timescale separation: 100% convergence to WT
Assumption I: protein 2mRNA
Assumption II: 1 = i ≤ 0.5
Assumption III: PTC3 CI3
Theorem: Under these assumptions the Glass-type model alwaysconverges to the wild type steady state
Chaves, Sontag & Albert 2006
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Robustness and fragility of Boolean models for genetic regulatory networks, Chaves, Albert and Sontag, 2005:Paper was in JTB top 10 most cited (of the last 5 years)
“Timescale separation” leads to “Priority classes” (Bioinformatics: GINsim software Chaouiya, Thieffry, etc.)
Further work: asynchronous transition graphs and the dynamical behavior of “large” networks
Further work: piecewise linear systems
Analysis of Boolean models and beyond
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Piecewise linear systems: qualitative framework
x = f x− x
x∈ℝ≥0n , f :ℝ≥0
n×ℝ≥0
n , =diag 1, ,n
Thresholds: 0i1⋯i
r iMi Function f is a sum of products of step functions
sX ,
X
Refs: Casey, de Jong & Gouzé, 2006
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Piecewise linear systems
Regular domains: Bk1 ,,kn, ki∈{0,r i}, i
kixiiki1
Switching domains: Dl , xi=il , for some i
Focal points: x= f k1 ,,kn− x=0 ⇒ k1 ,,kn=−1 f k1 ,,kn
Example:
x1 = 1 s−x2 ,2−1 x1
x2 = 2 s− x1 ,1−2 x2
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Measurements and control
Qualitative measurements:
Know only: position of variables with respect to thresholds (either “weakly expressed” or “strongly expressed”)
Qualitative inputs: u piecewise constant (in each regular domain)
Can only implement three values.Inputs can act on degradation or synthesis rates (inducers)
sxi ,i
r ∈ {0,1}
u: ℝ≥0×ℝ≥0n
{umin ,1,umax}
Chaves & Gouzé, Automatica 2011
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Control of simple biological motifs:the bistable switch
x1 = 1 s− x2 ,2−1 x1 , x2 = 2 s
− x1 ,1−2 x2
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Problem: using only qualitative control laws, is it possible to drive the system to either of its stable steady states?
Control: relocate focal points
x1 = u1 s− x2 ,2 − 1 x1 , x2 = u2 s
− x1 ,1 − 2 x2
Control of the bistable switch
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u=umax
u=umin
Control to steady state P1
Theorem: Assume that Φ(ϴ1)<ϴ
2 .
The system with this control law converges to point P1.
ux =
1 x∈B11∪B10
umin x∈B01
umax x∈B00
Chaves & Gouzé, Automatica 2011
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u=umin
Control to steady state P2
ut , x =
1 ∀ t , x∈B11∪B01
umin ∀ t , x∈B01
umin ∀ tT1, x∈B00
umax ∀ t≥T1, x∈B00
Theorem: Assume that Φ(ϴ1)<ϴ
2 ,
and condition on separatrix.The system with this control law converges to point P2.
Chaves & Gouzé, Automatica 2011
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Using Filippov solutions
x1
x2 ∈ co { f Ax − x , f B x− x }
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A synthetic bistable switch
Gardner, Cantor & Collins, Nature 2000
u ≈IPTG
Temperature
umax ≈ Apply IPTG
umin ≈ High temperature
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Conclusions
Experimental data: choose appropriate formalism different formalisms provide complementary information
Qualitative control find feedback laws using only qualitative data (for simple motifs)
easier to implement “add large amount of inducer when expression of X is high”
synthetic biology: assembling components; re-wiring a network
Boolean models: large networks as interconnection of two smaller modules
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Final conclusion
THANK YOU EDUARDO
.... AND CONGRATULATIONS !