Oscillations birhythmicity

G. Houart, G. Dupont, and A. Goldbeter, Bursting, chaos and birhythmicity originating from self-modulation of the inositol 1,4,5-triphosphate signal in a model for intracellular Ca + oscillations. Bull. Math. Biol. 61, 507-530 (1999). 

Fig. 3.1. Model of an enzyme reaction with positive feedback and recycling of product (P) into substrate (S). This model serves as a two-variable prototype for the study of birhythmicity and multiple domains of oscillations (Moran Goldbeter, 1984).

Let us now see how the nullcline deformation due to recycling gives rise to birhythmicity. Shown in fig. 3.6 are the bifurcation diagrams obtained as a function of parameter v for eight increasing values of the maximum rate of recycling, a , for a fixed value of constant K. In each part, the steady-state value of the substrate ao is indicated, as well as the maximum value qim reached by the substrate during oscillations. Solid lines denote stable steady-state or periodic solutions both types of solution are indicated by dashed lines when unstable. The stability properties of the steady state were determined by linear stability analysis of eqns... 

Fig. 3.6. Effect of the recycling reaction on birhythmicity. A series of bifurcation diagrams are represented for increasing values of the maximum rate of product recycling, o- (in s ) (a) 0 (b) 0.5 (c) 0.6 (d) 1.2 (e) 1.3 (f) 1.4 (g) 1.5 (h) 2. Each diagram shows the steady-state concentration of substrate, o, and the mtiximum concentration of substrate in the course of oscillations, < m> s a function of parameter (qvlk ) equal to the steady-state concentration of product. The curve yielding the steady-state level of substrate is therefore identical with the product nullcline. The sohd and dashed lines denote, respectively, stable and unstable branches of periodic or steady-state solutions. Parameter values are <7=10s, L = 5xl0, /iC=10, m = 4, q = l, k = 0.06s. Periodic re mes were obtained by numerical integration of eqns (3.1). The stability properties of the steady state were determined by Unear stabUity analysis. Birhythmicity is apparent in (d)-(f), while in (h) two distinct instabiUty domains appear as a function of parameter v (Moran Goldbeter, 1984).

A typical example of birhythmicity in the phase plane is illustrated in fig. 3.7. The large-amplitude limit cycle encloses a limit cycle of more reduced amplitude these two stable cycles are separated by an unstable limit cycle. Birhythmicity is of interest because it allows the existence of two distinct oscillations in the same conditions, i.e. for a given set of parameter values. Moreover, the passage from one stable rhythm to the other can be effected by means of the same type of perturbation. Thus,... 

The qualitative agreement between the biochemical system and the behaviour of thalamic cells suggests that birhythmicity, predicted by the model, could also occur in these neurons. To demonstrate such a phenomenon, each type of oscillation should be perturbed by depolarization or hyperpolarization, in order to determine whether the perturbation can elicit the switch from one rhythm to the other. Alternatively, an increase followed by a decrease in the applied current could reveal the existence of hysteresis two different rhythms would then be observed at a given value of the membrane potential. Such a procedure, suggestive of birhythmicity, has been followed in another neuronal system (Hoimsgaard et ai, 1988). 

Fig. 3.23. Scheme of another two-variable biochemical model admitting birhythmicity. Substrate S, injected at a constant rate, is transformed into product P by two isozymes that differ in their allosteric properties and in their catalytic activity. This model represents one of the simplest examples of coupling between two biochemical oscillators (Li Goldbeter, 1989a). 

In the two-variable models studied for glycolytic oscillations and birhythmicity, periodic behaviour originates from a unique instability mechanism based on the autocatalytic regulation of an allosteric enzyme by its reaction product. The question arises as to what happens when two instabiUty-generating mechanisms are present and coupled within the same system can new modes of dynamic behaviour arise from such an interaction ... 

An example of such a situation was considered at the end of the preceding chapter the system with two oscillatory isozymes (fig. 3.23) contains two instability mechanisms coupled in parallel. Compared with the model based on a single product-activated enzyme, new behavioural modes may be observed, such as birhythmicity, hard excitation and multiple oscillatory domains as a function of a control parameter. The modes of dynamic behaviour in that model remain, however, limited, because it contains only two variables. For complex oscillations such as bursting or chaos to occur, it is necessary that the system contain at least three variables. 

Figure 4.2 is the bifurcation diagram obtained when k varies over some five orders of magnitude. Equations (4.1) admit a single steady state the value of the substrate concentration in this state, oq, is indicated as a function of k, as well as the maximum amplitude reached by the substrate in the course of oscillations. The steady state is unstable (dashed line) for most values of k, except those ranging from 0.792 to 1.584 s (see also table 4.1). In this domain, the stable steady state coexists with a stable limit cycle, denoted LCl, which appears at low values of k. When k exceeds the value 1.584 s a Hopf bifurcation occurs, beyond which a new stable limit cycle, LC2, arises. As limit cycle LCl has not yet disappeared, a phenomenon of birhythmicity takes place (see chapter 3) as a result of the coexistence of two stable rhythms for the same set of parameter values. 

The ereation of a homoclinic orbit thus gives rise to a sharp decline in the munber of peaks within a burst. The abrupt transition from the bursting pattern with 11 peaks per period, ir(ll), to the pattern with 7 peaks, tt(7), in table 4.4 originates from the appearance of such a homoclinic orbit. It should be noted that the situation depicted in fig. 4.19e creates conditions suitable for the coexistence between bursting oscillations and a limit cycle that would be stabilized around a value of a between a and a. Such a situation is indeed close to that described further in chapter 6 for the origin of birhythmicity of a similar kind in a model for the intercellular communication system of Dictyostelium amoebae. 

The largest domain in this parameter space is clearly that of simple periodic oscillations. Second in importance is the domain of complex periodic oscillations. Then comes the domain of coexistence between a stable limit cycle and a stable steady state (dotted zone), which situation is associated with the phenomenon of hard excitation. Just below the latter domain are two regions of birhythmicity corresponding to the coexistence of limit cycles LCl and LC2 on the one hand and LC2 and LC3 on the other. These two domains of birhythmicity partly overlap their intersection defines the domain of trirhythmicity, where the three stable limit cycles LCl, LC2 and LC3 coexist. Near the domains of birhythmicity are three distinct regions of chaos (dark zones), whose size is relatively reduced. 

The comparison of the different behavioural domains in parameter space shows that simple periodic oscillations remain, by far, the most common type of dynamic behaviour. Complex periodic oscillations of the bursting type are also rather frequent, but much less than simple oscillations. The coexistence between a steady state and a limit cycle comes third by virtue of the importance of the domain in which such behaviour occurs in the v-k plane. Birhythmicity and chaos come next... 

Fig. 4.30. Diagram showing the various modes of dynamic behaviour of the multiply regulated biochemical system in the parameter space v-k. The indications stable and unstable relate to the stability properties of the unique steady state admitted by eqns (4.1). The domain of coexistence of such a stable state with a stable lirnit cycle is represented by the dotted area. Two domains of birhythmicity are observed their overlap gives rise to trirhythmicity. Three regions of chaos are represented in black, while the domains of simple or complex periodic oscillations occupy the rest of the space. The diagram is established for the parameter values of fig. 4.2 (Decroly Goldbeter, 1982 Goldbeter, Decroly Martiel, 1984).

As indicated by the analysis of the multiply regulated biochemical system, birhythmicity occurs in a domain of v values close to those producing chaos. Of interest, in this respect, is the preliminary evidence obtained by Markus Hess (1990) for birhythmicity in glycolysing yeast extracts. These authors showed the existence of a hysteresis loop between two types of oscillations upon increasing and then decreasing the value of the substrate injection rate. A direct test of the transition between two different rhythms at the same value of the substrate input remains to be performed. 

Fig. 6.10. Origin of bursting and birhythmicity. The bifurcation diagram for the two-variable system (6.5), estabUshed as a function of a considered as a parameter, shows the steady state y and the mean value y) of cAMP in the course of one period of the oscillations, on the two branches of periodic solutions Fj, Fj. The particular values of a indicated on the abscissa are defined in the text. For the sake of clarity, the diagram, obtained by means of the program AUTO (Doedel, 1981), is presented in a schematic manner the precise values of the critical points are a - 1.074 a, = 1.084 a = 1.101 a j = 1.126 a j = 1.287 2 3.934 (Martiel Goldbeter, 1987b).

The preceding discussion accounts for the mechanism of large-amplitude oscillations of the bursting type in figs. 6.7 and 6.9b. What then about the origin of the birhythmicity demonstrated in fig. 6.8 As in fig. 6.7, the passage to the small limit cycle occurs when an adequate amount of cAMP is added at the appropriate phase of the large-amplitude oscillations, namely just after the last peak of the active phase of... 

Similar complex oscillatory phenomena have been observed in a closely related model containing two regulated enzyme reactions coupled in a different manner (Li, Ding Xu, 1984). An additional indication of the generality of the results obtained in the multiply regulated biochemical system is given by the study of the model for the synthesis of cAMP in Dictyostelium amoebae. In addition to simple periodic oscillations and excitability (see chapter 5), this realistic model based on experimental observations also predicts the appearance of more complex oscillatory phenomena in the form of birhythmicity, bursting and chaos (chapter 6). 

In each of the models studied here, the domains of chaos and birhythmicity are much more reduced than the domain of complex periodic oscillations, which is itself smaller than the domain where simple periodic oscillations occur. This observation, which is corroborated by results obtained on the occurrence of complex oscillations in multi-looped negative feedback systems (Glass Malta, 1990), accounts for... 

Alamgir, M. I.R. Epstein. 1983. Birhythmicity and compound oscillations in coupled chemical oscillators Chlorite-Bromate-Iodide system. J. Am. Chem. Soc. 105 2500-1. 

Goldbeter, A., O. Decroly J.L. Martiel. 1984. From excitability and oscillations to birhythmicity and chaos in biochemical systems. In Dynamics of Biochemical Systems. J. Ricard A. Cornish-Bowden, eds. Plenum Press, New York, pp. 173-212. 

Martiel, J.L. A. Goldbeter. 1987b. Origin of bursting and birhythmicity in a model for cyclic AMP oscillations in Dictyostelium cells. Lect. Notes Biomath. 71 244-55. 

Volkov, E.I. M.N. Stolyarov. 1991. Birhythmicity in a system of two coupled identical oscillators. Phys. Lett. A 159 61-66. 

Linear stability analysis of model for birhythmicity, 95 of model for cAMP oscillations, 180 of model for glycolytic oscillations,... 

Model for oscillatory isozymes, birhythmicity, 115-17 Models, usefulness, 2,3,491,493 Molecular bases of oscillations, 356,359,381 of cAMP oscillations, 209 for efficiency of pulsatile signals, 305, 327... 

---