Model selection in gene
regulation and prediction of
oscillations
Tomas Gedeon
Montana State University
Jeff Heys
Montana State University
Graham Cummins
Tulane University
Vsetko najlepsie k narodeninam, Eduardo.
Wednesday, May 25, 2011
Main Question
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Neuroscience has a fundamental equation: HodgkinHuxley
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Cellular/gene processes modeling does not have such an
equation.
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Selection of ad-hoc models, nonlinearities.
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Can we justify making model-independent conclusions
based on analysis of particular models?
Wednesday, May 25, 2011
Conclusions are often worded in general terms: adding
positive feedback to a negative feedback circuit makes it
more robust; negative feedback oscillator is less robust
than slow-fast oscillator.
Robustness with respect to network
structure perturbation
•
Robustness in broader sense, not just parameters and
non-linearities.
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A particular problem: Given a network, do we only
model proteins, or include both proteins and mRNA?
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Adding delay?
Wednesday, May 25, 2011
Test case: prediction of stability of
equilibria
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•
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Wednesday, May 25, 2011
Simple feedback loop
General case
Numerical simulation of the “Composite
regulatory oscillator” of Yang et. al. 2009
Simple negative feedback loop
ẋ1
ẋi
= −d1 x1 − a1 f1 (xn )
= −di xi + ai fi (xi−1 ), i = 2, . . . , n
dfi
dxi−1
di , ai
= 1 for all i
> 0
Theorem. (Othmer, Arcak-Sontag)
Equilibrium 0 is stable, if
a1 a2 . . . , an
1
.
<
n
d1 d2 . . . dn
(cos(π/n))
This condition is necessary, when
d 1 = d 2 = . . . dn
Wednesday, May 25, 2011
Simple feedback loop with mRNA
ẏ1
ẋ1
y˙i
ẋi
= −b1 y1 + δa1 f1 (xn )
= −d1 x1 + c1 y1
= −bi yi + ai fi (xi−1 ) i = 2, . . . , n
y=mRNA
x=protein
= −di xi + ci yi , i = 2, . . . , n.
Theorem. (Othmer, Arcak-Sontag)
Equilibrium 0 is stable, if
c1 . . . cn a1 a2 . . . , an
1
.
<
2n
b 1 . . . bn d 1 d 2 . . . d n
(cos(π/2n))
Wednesday, May 25, 2011
Relationship between big and small
systems
If
c1 . . . cn
≥ 1 then stability in the large system implies the stability
b 1 . . . bn
in the small system.
“Longer loops are less stable”
Proof:
1
a1 a2 . . . an
c1 . . . cn a1 a2 . . . , an
1
<
≤
<
2n
d1 d2 . . . dn
b1 . . . bn d 1 d 2 . . . d n
(cos(π/2n))
(cos(π/n))n
If production rate of proteins ci too small the implication does not hold.
Wednesday, May 25, 2011
General case
Two linear systems:
ẋ = Ax − Dx
Protein only
protein and mRNA
ẋ = Cy − Dx
ẏ
= Ax − By,
(Small)
(Large)
B, C, D are diagonal matrices
A is mRNA production and has all combinatorial control in it
Is there a correspondence between eigenvalues of small and large
system?
Wednesday, May 25, 2011
Small system is more stable
Simplify: Assume B = bI, C = cI, D = dI, where I is the identity matrix
b=d=c=1
Any eigenvalue for the small system
in the black region will yield an
unstable eigenvalue of the large
region.
Wednesday, May 25, 2011
Larger translation rate yields more
instability
b = d = 1, c = 3
b
Instability region
d( − 1) < x < 0
meets real axis in
c
Wednesday, May 25, 2011
b = d = 1, c = 10
Nonempty when
c
>1
b
Case study: Kuznetzov oscillator
1. Hysteresis based relaxation oscillators:
difficult to synchronize but support a pattern formation
perhaps more robust to noise
II. repressilator (cyclic feedback) oscillators:
easier to synchronize, no pattern formation.
Yang, Lee, Kuznetzov (2009):combine two types of oscillators in one model
� = small, α2 = O(1)
relaxation oscillator
� = O(1), α2 = small
repressilator (cyclic feedback
system)
Wednesday, May 25, 2011
u̇
v̇
ẇ
α1
=
−u
n
1+v
α1
α2
=
+
−v
n
n
1+w
1+u
α1
= �(
− w)
n
1+u
Two oscillators
u
�
α2
w
v
u
u,v proteins, w small signaling molecule
α1
u̇ =
−u
n
1+v
α1
α2
v̇ =
+
−v
n
n
1+w
1+u
α1
ẇ = �(
− w)
n
1+u
�
w
v
Relaxation: Bistability in u-v network and small �
� = small, α2 = O(1)
u
α2
v
Wednesday, May 25, 2011
w
Repressilator: three negative feedbacks
� = O(1), α2 = small
Observations
u
�
α2
v
w
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No oscillations for small α2 , �
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for small � oscillations limited by
saddle-node on invariant circle.
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no oscillations for large �
for small α2 oscillations limited by
Hopf bifurcations
Are these conclusions persistent,
or model dependent?
Wednesday, May 25, 2011
Extended model with mRNA
u1
u2
�
α2
v2
v1
u
�
α2
v
w
Protein only model
Wednesday, May 25, 2011
w
u˙1
=
u˙2
=
v˙1
=
v˙2
=
ẇ
=
α1
− u1
n
1 + v2
u1 − u2
α1
α2
− v1
+
n
n
1+w
1 + u2
v1 − v2
α1
�(
− w)
n
1 + u2
Same equilibria structure for both models.
Structure of periodic orbits?
Numerical integration in parameter space
Heat map: amplitude of the periodic solution for two models
protein model
Wednesday, May 25, 2011
protein and mRNA model
Bifurcation diagrams
Protein model
Saddle-node
Protein and mRNA model
Hopf
Saddle-node
Hopf
Saddle-node
Hopf
Hopf
Qualitative or quantitative differences?
Wednesday, May 25, 2011
Saddle-node
Qualitative difference!
Protein model
Protein and mRNA model
Top Hopf curve:
Wednesday, May 25, 2011
Oscillations for all � at small α2
Top Hopf curve
has a vertical asymptote
Wednesday, May 25, 2011
Bifurcation diagrams quantitatively and
qualitatively different
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Significant differences for large epsilon
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Broader question: How do we make conclusions from
models that are model independent?
Wednesday, May 25, 2011
Region of no oscillation for small �, α2 much smaller
Boundaries of bifurcation region different - implication
for length of the periodic orbit
Conclusions
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How do we make responsibly model-independent
conclusions from models?
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Is it even possible?
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For cyclic feedback systems longer system less stable
(assumption - sufficiently strong translation rates)
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General problem: adding mRNA can destabilize the
system
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Particular problem repressilator & relaxation oscillator.
Conclusions different for different models.
Wednesday, May 25, 2011
Analyzed particular dilemma: do we use protein only, or
protein and mRNA models for a given network.
Thanks
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Wednesday, May 25, 2011
NSF-DMS 081878
DARPA
NSF CMMI 0849433