Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
Systems and models in chronobiology
Systems and models in chronobiologyA delay model for the circadian rhythm
M. DechesneR. Sepulchre
Department of Electrical Engineering and Computer ScienceMonte�ore InstituteUniversity of Liège
24th Benelux Meeting on Systems and ControlHou�alize, 22-24 March 2005
Systems and models in chronobiology
Outline
Introduction
State of the art in mathematical modeling of the circadian rhythmBiological modeling for the circadian oscillatorResults
Systems viewpoint: Why is it useful?
Application to the analysis of the circadian rhythm
Conclusion
Systems and models in chronobiologyIntroduction
Outline
Introduction
State of the art in mathematical modeling of the circadian rhythmBiological modeling for the circadian oscillatorResults
Systems viewpoint: Why is it useful?
Application to the analysis of the circadian rhythm
Conclusion
Systems and models in chronobiologyIntroduction
Circadian rhythms (1)De�nition and roles
De�nitionBiological rhythms with a period τ ∼ 24h
RolesCircadian rhythms control
I Sleep
I Muscular activity and metabolism
I Food ingestion
I ...
Moreover, there are possible links with various pathologies
Systems and models in chronobiologyIntroduction
Circadian rhythms (2)Properties
Property Comment
Ubiquitous All eukaryotes and some prokaryotesEntrainment Zeitgeber=light/temperature cyclesGenetic Single-gene clock
mechanisms mutants isolatedPrecision ∆τ < 0.1%Robustness T◦, IC...
A cell-autonomous circadian oscillatorCellular nature mechanism exists and appears to be a fundamental
unit even among multicellular organisms
Systems and models in chronobiologyState of the art in mathematical modeling of the circadian rhythm
Outline
Introduction
State of the art in mathematical modeling of the circadian rhythmBiological modeling for the circadian oscillatorResults
Systems viewpoint: Why is it useful?
Application to the analysis of the circadian rhythm
Conclusion
Systems and models in chronobiologyState of the art in mathematical modeling of the circadian rhythm
Biological modeling for the circadian oscillator
Simple modelBiochemical principle
(P2)PER2PER1
(P1)PER0
(P0)
V1
vd
k2 k1
����������� (M)ks
vm
V2 V4
V3
vs
������������������������������� � !�#"�$#���&%('(� (PN)
from Goldbeter, Biochemical oscillations and cellular rhythms, CambridgeUniversity Press, 1996
Systems and models in chronobiologyState of the art in mathematical modeling of the circadian rhythm
Biological modeling for the circadian oscillator
Simple modelEquations
M = vsKn
I
Kn
I+ Pn
N
− vmM
km + M
P0 = ksM − V1P0
K1 + P0+ V2
P1K2 + P1
P1 = V1P0
K1 + P0− V2
P1K2 + P1
− V3P1
K3 + P1+ V4
P2K4 + P2
P2 = V3P1
K3 + P1− V4
P2K4 + P2
− k1P2 + k2PN − vdP2
kd + P2
PN = k1P2 − k2PN
I 5 nonlinear ODEsI 17 parameters
Systems and models in chronobiologyState of the art in mathematical modeling of the circadian rhythm
Biological modeling for the circadian oscillator
Evolutions of the model
Drosophilia
I 10 nonlinear ODEs
I 34 parameters
Systems and models in chronobiologyState of the art in mathematical modeling of the circadian rhythm
Biological modeling for the circadian oscillator
Evolutions of the model
Drosophilia
I 10 nonlinear ODEs
I 34 parameters
Mammals
I 19 nonlinear ODEs
I 59 parameters
Systems and models in chronobiologyState of the art in mathematical modeling of the circadian rhythm
Results
Results (1)Oscillations and limit cycle
Oscillations Limit Cycle
Systems and models in chronobiologyState of the art in mathematical modeling of the circadian rhythm
Results
Results (2)Entrainment
Systems and models in chronobiologyState of the art in mathematical modeling of the circadian rhythm
Results
Results (3)Bifurcation diagram
Systems and models in chronobiologySystems viewpoint: Why is it useful?
Outline
Introduction
State of the art in mathematical modeling of the circadian rhythmBiological modeling for the circadian oscillatorResults
Systems viewpoint: Why is it useful?
Application to the analysis of the circadian rhythm
Conclusion
Systems and models in chronobiologySystems viewpoint: Why is it useful?
Limitations of current mathematical modeling
I Very complicated systems (strongly nonlinear, several variablesand parameters)→ restricted to numerical simulations
I Impossible to test all parameter combinations
I Mathematically (and computationally) di�cult to studyinterconnections
I Exhibit some properties, but NOT explain them
Systems and models in chronobiologySystems viewpoint: Why is it useful?
System approachPrinciple
Developing or using system models and tools for the analysis ofbiological system (and particularly oscillatory systems)
Systems and models in chronobiologySystems viewpoint: Why is it useful?
System approachLinks with system question
By nature, cells are open systems, i.e. with inputs and outputs
Many unexplained properties are related to fundamental systemsquestions:
I Why is there an entrainment?
I Why is the system robust to certain parameters variation (T◦,initial conditions...)? Why is it robust to molecular noise? Andto external disturbance?
I How can we guess from observed properties, the values ofunknown parameters?
I Why is there a synchronization in networks of oscillators?What kind of synchronized behaviour may we expect given aparticular network con�guration?
Systems and models in chronobiologyApplication to the analysis of the circadian rhythm
Outline
Introduction
State of the art in mathematical modeling of the circadian rhythmBiological modeling for the circadian oscillatorResults
Systems viewpoint: Why is it useful?
Application to the analysis of the circadian rhythm
Conclusion
Systems and models in chronobiologyApplication to the analysis of the circadian rhythm
Recent approaches
Classical methods for the analysis of limit cycles
Limitations: either useless in high dimension (and so forinterconnections) or unable to handle global treatment
Abstract modelsLimitations: Acts more as a �black box� i.e. it is possible to obtainqualitative information, but no quantitative one.
Development of new methods
I Monotone Systems (P. de Leenheer, D. Angeli, E. Sontag)
I Piecewise Linear Systems - PLS (J. Goncalves)
I Dissipative Systems (R. Sepulchre, G.B. Stan)
= dedicated I/O approaches
Systems and models in chronobiologyApplication to the analysis of the circadian rhythm
Recent approaches
Classical methods for the analysis of limit cycles
Limitations: either useless in high dimension (and so forinterconnections) or unable to handle global treatment
Abstract modelsLimitations: Acts more as a �black box� i.e. it is possible to obtainqualitative information, but no quantitative one.
Development of new methods
I Monotone Systems (P. de Leenheer, D. Angeli, E. Sontag)
I Piecewise Linear Systems - PLS (J. Goncalves)
I Dissipative Systems (R. Sepulchre, G.B. Stan)
= dedicated I/O approaches
Systems and models in chronobiologyApplication to the analysis of the circadian rhythm
Recent approaches
Classical methods for the analysis of limit cycles
Limitations: either useless in high dimension (and so forinterconnections) or unable to handle global treatment
Abstract modelsLimitations: Acts more as a �black box� i.e. it is possible to obtainqualitative information, but no quantitative one.
Development of new methods
I Monotone Systems (P. de Leenheer, D. Angeli, E. Sontag)
I Piecewise Linear Systems - PLS (J. Goncalves)
I Dissipative Systems (R. Sepulchre, G.B. Stan)
= dedicated I/O approaches
Systems and models in chronobiologyApplication to the analysis of the circadian rhythm
A delay model for the circadian rhythmVariation on Goldbeter's model: 2 variables
����������� ������
�������
� ����� �"!#�
����������� ������
$�&%'�&(θ
NL
NL
vm
vd
vs
M = vsKn
I
Kn
I+ Pn
− vmM
P = vPM(t − τ)m − vdP
from T. olde Scheper and al., A Mathematical Model for the IntracellularCircadian Rhythm Generator, J. of Neuroscience, 1999
Systems and models in chronobiologyApplication to the analysis of the circadian rhythm
A delay model for the circadian rhythmFurther simpli�cation : 1 variable
+
H
e−sθ
HNL
u
uNL
����������� ���� �����
v = 0
φ(uNL)
1
τs+1
�����
y
yNL
K
V
V2
M = vsKn
I
Kn
I+ M(t − τ)n
− vmM
This model is equivalent to the famous Mackey and Glass' model
for Red Blood Cell regulation
Systems and models in chronobiologyConclusion
Outline
Introduction
State of the art in mathematical modeling of the circadian rhythmBiological modeling for the circadian oscillatorResults
Systems viewpoint: Why is it useful?
Application to the analysis of the circadian rhythm
Conclusion
Systems and models in chronobiologyConclusion
Conclusion and perspectives
There remains many unsolved questions in the exploding �eld ofmathematical biology.Most of these are natural in the framework of systems modeling:synchronization, robustness, entrainment...
Our research group currently develops methods based on aninput/output approach to answer (some of) those questions andclassify basic oscillatory mechanisms.
We are tempting to analyze a new mechanism based on a delay
and a speci�c regulatory nonlinearity in the feedback loop.
Systems and models in chronobiologyConclusion
Thank you
Thank you for your attention... :-)