Systems and models in chronobiology · 2005. 6. 10. · Systems and models in chronobiology Systems...

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




Conclusion


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


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




Conclusion


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



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



Evolutions of the model

Drosophilia

I 10 nonlinear ODEs

I 34 parameters



Evolutions of the model

Drosophilia

I 10 nonlinear ODEs

I 34 parameters

Mammals

I 19 nonlinear ODEs

I 59 parameters


Results

Results (1)Oscillations and limit cycle

Oscillations Limit Cycle


Results

Results (2)Entrainment


Results

Results (3)Bifurcation diagram

Systems and models in chronobiologySystems viewpoint: Why is it useful?

Outline

Introduction




Conclusion


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


System approachPrinciple

Developing or using system models and tools for the analysis ofbiological system (and particularly oscillatory systems)


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




Conclusion


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


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


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




Conclusion


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.


Thank you

Thank you for your attention... :-)

Systems and models in chronobiology · 2005. 6. 10. · Systems and models in chronobiology Systems...

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