An Analysis of a Circadian Model Using The Small-Gain Approach to Monotone Systems

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An Analysis of a Circadian Model Using The Small-Gain Approach to Monotone Systems David Angeli Dip.Sistemi e Informatica University of Florence,50139Firenze,Italy angeli@dsi.unifi.it Eduardo D.Sontag ?Dept.of Mathematics Rutgers University,NJ,USA sontag@https://www.wendangku.net/doc/b36514384.html, Abstract In this note,we show how certain properties of Goldbeter’s 1995model for circadian oscillations can be proved mathematically,using techniques from the recently developed theory of monotone systems with inputs and outputs.The theory establishes global asymptotic stability,and in partic-ular no oscillations,if the rate of transcription is somewhat smaller than that assumed by Goldbeter.This stability persists even under arbitrary delays in the feedback loop.1Introduction The molecular biology underlying the circadian rhythm in Drosophila is the focus of a large amount of both experimental and theoretical work.Goldbeter proposed a simple model for circadian oscillations in [4](see also his book [5]).Although by now several more realistic models are available,in particular incor-

porating other genes,this simpler model exhbits many realistic features,such as a 24-hour period.The key to the model is the inhibition of per gene transcrip-tion by its protein product PER,forming an autoregulatory negative feedback loop.

In this note,we show how certain properties of the model can be proved mathematically,using techniques from the recently developed theory of mono-tone systems with inputs and outputs.The theory establishes global asymptotic

stability,and in particular no oscillations,if the rate of transcription is some-what smaller than that assumed by Goldbeter.This stability persists even under arbitrary delays in the negative feedback loop.On the other hand,a larger–but still smaller than Goldbeter’s–strength,in the presence of delays,results in oscillations.

The terminology and notations are as given in[2,3],and are not repeated here.

2The Model

The model is as shown in Figure1.PER protein is synthesized at a rate pro-

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Value Parameter

k2 1.9 V1 1.58 V3 2.5 v s0.5 k s0.95 k d4 K12 K32 K I0.65

Assume from now on that:

v s≤0.54(1) (the remaining parameters will be constrained below,in such a manner that those in the previously given table will satisfy all the constraints).

As state-space for the?rst system,we will pick a compact interval X1= [0,ˉM],where

v s k m

k s

(2) and we assume that v s

v s<

v mˉM

v m K n

I +v m u1n?v s K n

I

(which is an element of X1).

Note that all solutions of the di?erential equations which describe the M-system,even those that do not start in X1,enter X1in?nite time(because ˙M(t)<0whenever M(t)≥ˉM,for any input u

1(·)).The restriction to the state space X1(instead of using all of R≥0)is done for convenience,so that one can view the output of the M system as in input to the P-subsystem.(Desirable properties of the P-subsystem depend on the restriction imposed on U2.)Given

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any trajectory,its asymptotic behavior is independent on the behavior in an initial?nite time interval,so this does not change the conclusions to be drawn. (Note that solutions are de?ned for all times–no?nite explosion times–because the right-hand sides of the equations have linear growth.)

Monotonicity of the second system is also clear,from the fact that?˙P i

Note that(4)and(7)imply also:

c+β0(∞)<α0(∞).(8) Proof.We start by noticing that solutions are de?ned for all t≥0.Consider any maximal solution x(t)=(x0(t),x1(t),x2(t),x3(t)).From

d

dt (x0+x1+x2+x3)(t)<0for all t su?ciently large.Thus x0+x1+

x2+x3is bounded(and nonnegative),and this implies that x2is bounded,a contradiction.So x3is bounded.

Next we examine the equation for˙x2.The two positive terms are bounded: the one involvingα1becauseα1is a bounded function,and the one involving γ3because x3is bounded.Thus

˙x2≤v(t)?α2(x2),

where0≤v(t)≤k for some constant k.Thus˙x2(t)<0whenever x2(t)>γ?12(k),and this proves that x2is bounded,as claimed.

Now we show that x0and x1are bounded as well.For x0,it is enough to notice that˙x0≤c?α0(x0)+β0(∞),so that

x0(t)>α?10(c+β0(∞))?˙x0(t)<0

so(8)shows that x0is bounded.Similarly,for x1we have that˙x1≤α0(∞)?β0(x1)?α1(x1)+β1(∞)so(5)provides boundedness.

Once that boundedness has been established,if we also show that there is a unique equilibrium then the theory of strongly monotone tridiagonal systems ([6,7])will ensure global asymtotic stability of the equilibrium.So we show

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that equilibria exist and are unique.It is convenient to change variables are write

y0:=x0+x1+x2+x3,y1:=x1+x2+x3,y2:=x2+x3,y3:=x3.

In terms of these variables,we may set˙y i=0,i=0,2,1,3,so that the equilibria are precisely the solutions of:

α2(x2)=c

α1(x1)=α2(x2)+β1(x2)

α0(x0)=α2(x2)+β0(x1)

γ3(x3)=γ2(x2).

This shows uniqueness(all the functions are strictly increasing),and existence follows from,respectively,(7),(6),(4),and the fact thatγ3is unbounded.

s

divergence of the discrete iteration.Thus,and one may expect periodic orbits in this case.Indeed,simulations show that,for large enough delays,such periodic orbits arise,see Figure5.

References

[1]D.Angeli,J.Ferrell,and E.D.Sontag,“Detection of multi-stability,bi-

furcations,and hysteresis in a large class of biological positive-feedback systems,”submitted.

[2]D.Angeli and E.D.Sontag,“Monotone control systems,”IEEE Trans.

Autom.Control48(2003):1684–1698.(Summarized version appeared as “A remark on monotone control systems,”in Proc.IEEE Conf.Decision and Control,Las Vegas,Dec.2002,IEEE Publications,Piscataway,NJ, 2002,pp.1876-1881.)

[3]D.Angeli and E.D.Sontag,“Multi-stability in monotone Input/Output

systems,”Systems and Control Letters,in press.(Summarized version:”A note on multistability and monotone I/O systems,”in Proc.IEEE Conf.

Decision Control,Maui,2003.)

[4]Goldbeter,A.,“A model for circadian oscillations in the Drosophila period

protein(PER),”Proc.Royal Soc.Lond.B.261(1995):319–324.

[5]Goldbeter,A.Biochemical Oscillations and Cellular Rhythms,Cambridge

Univ.Press,Cambridge,1996.

[6]J.Smillie,“Competitive and cooperative tridiagonal systems of di?erential

equations,”SIAM J.Math.Anal.15(1984):pp.530–534.

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Figure5:s initial condi-tions all at0.2),using MATLAB’s dde23package

[7]H.L.Smith,Monotone Dynamical Systems:an Introduction to the The-

ory of Competitive and Cooperative systems,Mathematical Surveys and Monographs,Vol.41,American Mathematical Society,Ann Arbor,1995.

[8]H.L.Smith,“Periodic tridiagonal competitive and cooperatibe systems of

di?erential equations,”SIAM J.Math.Anal.22(1991):1102-1109.

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