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Analysis An of a Circadian Model Using The Small-Gain Approach to Monotone Systems David Angeli Dip. Sistemi e Informatica University of Florence, 50139 Firenze, Italy angeli@dsi.unifi.it Eduardo D. Sontag Dept. of Mathematics Rutgers University, NJ, USA sontag@hilbert.rutgers.edu Abstract In this note, we show how certain properties of Goldbeter s 1995 model 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 particular 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. vm M vs 6 ks V V vd - P0 1- P1 3- P2 V2 V4 k1 62 k PN Fig. 1. Goldbeter s Model I. INTRODUCTION 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 incorporating 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 transcription 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 monotone systems with inputs and outputs. The theory establishes global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat 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. II. THE MODEL The model is as shown in Figure 1. PER protein is synthesized at a rate proportional to its mRNA concentration. Two phosphorylation sites are available, and constitutive phosphorylation and dephosphorylation occur with saturation dynamics, at maximum rate vi s and with Michaelis constants Ki . Doubly phosphorylated PER is degraded, Supported in part by AFOSR Grant F49620-01-1-0063, NIH Grants R01 GM46383 and P20 GM64375, and Aventis also satisfying saturation dynamics (with parameters vd , kd ), and it is translocated to the nucleus with rate constant k1 . Nuclear PER inhibits transcription of the per gene, with a Hill-type reaction of cooperativity degree n and threshold constant KI , and mRNA is produced. and translocated to the cytoplasm, at a rate determined by a constant vs . Additionally, there is saturated degradation of mRNA (constants vm and km ). The equations for concentrations are as follows: M P0 P1 P2 PN n n n = vs KI /(KI +PN ) vm M/(km +M ) = ks M V1 P0 /(K1 +P0 ) + V2 P1 /(K2 +P1 ) = V1 P0 /(K1 +P0 ) V2 P1 /(K2 +P1 ) V3 P1 /(K3 +P1 ) + V4 P2 /(K4 +P2 ) = V3 P1 /(K3 +P1 ) V4 P2 /(K4 +P2 ) k1 P2 + k2 PN vd P2 /(kd +P2 ) = k1 P2 k2 PN where the subscript i = 0, 1, 2 in the concentration Pi indicates the degree of phosphorylation of PER protein, PN is used to indicate the concentration of PER in the nucleus, and M indicates the concentration of per mRNA. The parameters (in suitable units M or h 1 ) are as in Table I. With these parameters, there are limit cycle oscillations. We leave all xed except vs , and show that there are no oscillations if vs = 0.4, but oscillations exist if vs = 0.5 and there are delays in the negative regulatory loop, either in transcription or in translation (or in both). We choose to view the system as the feedback interconnection of two subsystems, see Figure 2. mRNA System: The rst (M ) subsystem is described by the scalar differential equation n n M = vs KI /(KI +un ) vm M/(km +M ) 1 Parameter k2 V1 V3 vs ks kd K1 K3 KI Value 1.3 3.2 5 0.76 0.38 0.2 2 2 1 Parameter k1 V2 V4 km vd n K2 K4 vm Value 1.9 1.58 2.5 0.5 0.95 4 2 2 0.65 TABLE I PARAMETER VALUES satis es all the constraints. As input space for the mRNA system, we pick U1 = R 0 , and as output space Y1 = [0, vd ). Note that y1 = ks M ks M < vd , by (2), so the output belongs to Y1 . For the second system, the state space is R4 , the input 0 space is U2 = Y1 , and the output space is Y2 = U1 . When looking at the rst system, we view U1 as ordered by the cone R 0 , but U2 , Y1 , Y2 are all ordered in the usual manner (cone R 0 ). III. MONOTONICITY AND CHARACTERISTICS u1 - M y1 y2 Fig. 2. P u2 Systems in feedback The rst system is monotone, and has a well-de ned characteristic, in the sense of [2]. Monotonicity is clear (onedimensional system), and the existence of characteristics is immediate from the fact that M > 0 for M < k1 (u1 ) and < 0 for M > k1 (u1 ), where, for each constant input M u1 , n vs KI km k1 (u1 ) = n + v u n v Kn vm KI m1 s I (which is an element of X1 ). Note that all solutions of the differential equations which describe the M -system, even those that do not start in X1 , enter X1 in nite time (because M (t) < 0 whenever , for any input u1 ( )). The restriction to the M (t) M 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 any trajectory, its asymptotic behavior is independent of 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 Pi the fact that Pj > 0 for all i = j; in fact, this is a strongly monotone tridiagonal system ([6], [7]). We show that (for the parameters in the table, as well as for a larger set of parameters) the system has, for each constant input u, a unique equilibrium, and trajectories are all bounded; it follows then from [6], [7] that the unique equilibrium is globally asymptotically stable, which means that characteristics are well-de ned. Proposition 3.1: Suppose that the following conditions hold: with input u1 and output y1 = ks M . PER System: The second (P ) subsystem is fourdimensional: P0 P1 P2 PN = u2 V1 P0 /(K1 +P0 ) + V2 P1 /(K2 +P1 ) = V1 P0 /(K1 +P0 ) V2 P1 /(K2 +P1 ) V3 P1 /(K3 +P1 ) + V4 P2 /(K4 +P2 ) = V3 P1 /(K3 +P1 ) V4 P2 /(K4 +P2 ) k1 P2 + k2 PN vd P2 /(kd +P2 ) = k1 P2 k2 PN with input u2 and output y2 = PN . Assume from now on that: vs 0.54 (the (1) 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, M ], where vd vs km M < (2) vm vs ks and we assume that vs < vm . Note that the rst inequality implies that vm M vs < (3) km + M and therefore n n vs KI /(KI +un ) 1 vm M /(km + M ) < 0 vd + V2 < V1 V1 + V4 < V2 + V3 0 c < vd V4 + vd < V 3 for all u1 0, so that indeed X1 is forward-invariant for the dynamics. With the parameters shown in the table given earlier (except for vs , which is picked as in (1)), M = 2.45 and that all constants are positive and the input u2 (t) c. Then the P -system has a unique globally asymptotically stable equilibrium. This will be a corollary of the following more general result. Theorem 1: Consider a system of the following form: x0 x1 x2 x3 = = = = c 0 (x0 ) + 0 (x1 ) 0 (x0 ) 0 (x1 ) 1 (x1 ) + 1 (x2 ) 1 (x1 ) 1 (x2 ) 2 (x2 ) 2 (x2 ) + 3 (x3 ) 2 (x2 ) 3 (x3 ) bounded function, and the one involving 3 because x3 is bounded. Thus x2 v(t) 2 (x2 ) , where 0 v(t) k for some constant k. Thus x2 (t) < 1 0 whenever x2 (t) > 2 (k), and this proves that x2 is bounded, as claimed. Now we show that x0 and x1 are bounded as well. For x0 , it is enough to notice that x0 c 0 (x0 ) + 0 ( ), so that 1 x0 (t) > 0 (c + 0 ( )) x0 (t) < 0 evolving on R4 , where c 0 is a constant, and the 0 functions i , i , i : [0, ) [0, ) are all differentiable, with derivatives everywhere positive, and so that i and i are bounded, for each i, and 1 , 2 are unbounded. Furthermore, suppose that the following conditions hold: 2 ( ) + 0 ( ) < 0 ( ) 0 ( ) + 1 ( ) < 1 ( ) + 0 ( ) 2 ( ) + 1 ( ) < 1 ( ) c < 2 ( ) . (4) (5) (6) (7) Then, there is a (unique) globally asymptotically stable equilibrium for the system. Note that (4) and (7) imply also: c + 0 ( ) < 0 ( ) . (8) so (8) shows that x0 is bounded. Similarly, for x1 we have that x1 0 ( ) 0 (x1 ) 1 (x1 )+ 1 ( ) 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 asymptotic stability of the equilibrium. So we show that equilibria exist and are unique. It is convenient to change variables and write y0 := x0 + x1 + x2 + x3 , y1 := x1 + x2 + x3 , y2 := x2 + x3 , y3 := x3 . In terms of these variables, we may set yi = 0, i = 0, 2, 1, 3, so that the equilibria are precisely the solutions of: 2 (x2 ) 1 (x1 ) 0 (x0 ) 3 (x3 ) = = = = c 2 (x2 ) + 1 (x2 ) 2 (x2 ) + 0 (x1 ) 2 (x2 ) . 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 (x0 + x1 + x2 + x3 ) = c 2 (x2 ) (9) dt we conclude there is an estimate xi (t) i xi (t) i xi (0) + tc and hence there are no nite escape times. Moreover, we claim that x( ) is bounded. Since the system is a strongly monotone tridiagonal system, we know (see [6], Corollary 1), that x3 (t) is eventually monotone. That is, for some T > 0, either x3 (t) 0 t T or x3 (t) 0 t T . (11) Hence, x3 (t) admits a limit, either nite or in nite. Assume rst that x3 (t) . Then, case (11) cannot hold, so (10) holds. Looking at the differential equation for x3 , we know that 2 (x2 (t)) 3 (x3 (t)) 0 for all t T , which means that 1 x2 (t) 2 ( 3 (x3 (t))) . Looking again at (9), and using that c 2 ( ) < 0 (propd erty (7)), we conclude that dt (x0 + x1 + x2 + x3 ) (t) < 0 for all t suf ciently large. Thus x0 +x1 +x2 +x3 is bounded (and nonnegative), and this implies that x2 is bounded, a contradiction. So x3 is bounded. Next we examine the equation for x2 . The two positive terms are bounded: the one involving 1 because 1 is a (10) This shows uniqueness (all the functions are strictly increasing), and existence follows from, respectively, (7), (6), (4), and the fact that 3 is unbounded. IV. CLOSING THE LOOP Now we are ready to apply the main theorem in [2]. In order to do this, we need to plot the characteristics. See Figure 3 for the spiderweb diagram (the dotted and dashed curves are the characteristics) that shows convergence of the discrete iteration described in [2] when we pick the parameter vs = 0.4. The theorem implies that no oscillations can happen in that case, even under arbitrary delays in the feedback from PN to M . On the other hand, for a larger value, such as vs = 0.5, the discrete iteration conditions are violated; see Figure 4 for the spiderweb diagram that shows 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 Figure 5. Fig. 3. Stability of spiderweb (vs = 0.4) Fig. 5. Oscillations seen in simulations (vs = 0.5, delay of 100, initial conditions all at 0.2), using MATLAB s dde23 package [8] H.L. Smith, Periodic tridiagonal competitive and cooperative systems of differential equations, SIAM J. Math. Anal.22(1991): 11021109. Fig. 4. Instability of spiderweb (vs = 0.5) REFERENCES [1] D. Angeli, J.E. Ferrell, Jr., and E.D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, Proceedings of the National Academy of Sciences USA 101(2004): 1822 1827. [2] D. Angeli and E.D. Sontag, Monotone control systems, IEEE Trans. Autom. Control 48(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, 51(2004): pp. 185 202. (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 differential equations, SIAM J. Math. Anal. 15(1984): pp. 530 534. [7] H.L. Smith, Monotone Dynamical Systems: an Introduction to the Theory of Competitive and Cooperative systems, Mathematical Surveys and Monographs, Vol. 41, American Mathematical Society, Ann Arbor, 1995.

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