Proc. Nati. Acad. Sci. USA Vol. 86, pp. 142-146, January 1989 Biophysics Limit-cycle oscillations and chaos in reaction networks subject to conservation of mass (nonlinear dynamics/free-energy transduction/linkage) ENRICO Di CERA*t, PAUL E. PHILLIPSON*, AND JEFFRIES WYMAN§ *Istituto di Fisica, UniversitA Cattolica, Largo F. Vito 1, 00168 Rome, Italy; tDepartment of Physics, Box 390, University of Colorado, Boulder, CO 80309- 0390; and §Department of Chemistry and Biochemistry, University of Colorado, Boulder, CO 80309-0215. Contributed by Jeffries Wyman, August 1, 1988 ABSTRACT A cyclic network of autocatalytic reactions conservative oscillations, they can be used as skeletons in involving an unbuffered cofactor and a number of components other reaction mechanisms to generate complex dynamical subject to conservation of mass displays a surprising richness behavior. In this paper we demonstrate with a particular of dynamical behaviors. Limit-cycle oscillations are possible model that under suitable conditions the conservative nature over a wide range of parameter values. Additionally, a cascade of the oscillations is lost and these networks can display a of period-doubling bifurcations leading to chaos can coexist surprising richness of dynamical behaviors, from limit-cycle with a multiplicity of stable steady states. These results draw oscillations to a cascade of period-doubling bifurcations attention to the role of unbuffering as a feedback in biochemi- leading to chaos. A peculiar feature of the model is the possi- cal systems. bility of either periodic oscillations or chaos coexisting with a multiplicity of stable steady states. Periodic patterns of temporal self-organization may appear in several reactions and biochemical regulatory processes THE MODEL when the system under consideration obeys certain general The particular model considered here is a four-component properties (1): (i) it is open to mass or energy transfer; (ii) it autocatalytic network, such as Eq. 1, driven from the out- is far from equilibrium whereby the individual reaction steps side by a constant energy source. The basic assumption is are mostly irreversible; and (iii) the chemical reaction mech- that, in addition to the components X1, X2, X3, and X4, the anism is nonlinear. Consistent with these criteria we have reaction network now includes a cofactor that is not degrad- drawn attention (2) to the existence of stable, conservative ed and is present in a fixed amount. For a chemical reaction, chemical oscillations in cyclic, autocatalytic reaction net- an example of such a cofactor is provided by a proton taken works of n components obeying the general scheme up at a given step and released in another one. The reaction scheme is Xi + Xi+, -K 2Xi+l [1] Y + X1 + X2 -* 2X2 [3a] where Xi denotes the components, Ki is a kinetic constant, i = 1 ... n, and Xi+, = Xi. The network is driven far from X2 + X3 - 2X3+Y [3b] equilibrium by a constant energy source included in the ki- netic constants. The change in the concentration xi of Xi is [3c] X4 + X1 --4 2X1, [3d] dxi/dt = xi(Ki-lxil -Kixi+,), [2] where Y denotes the cofactor that binds to Xi to give X2 and which indicates that Xi is formed by interaction with the pre- is released from X2 to give X3. The differential equations for ceding component and destroyed by interaction with the suc- the above system are cessive one. The sum of the values of x is independent of time, so that the system is closed with respect to the compo- dxl/dt = x1(K4x4- [4a] nents X, and open to external free-energy sources that drive KjYx2) the unidirectional circulation of the network. The topology dx2/dt = x2(Klyxi - K2X3) [4b] of these networks and their properties dependent upon n dX3/dt = x3(K2x2 -K3X4) [4c] have a close connection to neural networks (3) and to com- peting populations in closed systems (4). Conservation ofthe dx4/dt = x4(K3x3 -K4X), [4d1 components is a feature of particular interest when dealing with biological macromolecules. In this case the components can be considered as different conformational or ligation where xi is the concentration of the various components and states of a biological macromolecule circulating in a reaction y is the cofactor activity. The sum of Eq. 4 a-d is always network, as in the case of the turning-wheel model (5) or the zero, independent of time, because the system is closed to linear networks extensively analyzed by Hill (6). Conserva- the various components of X. tion of mass corresponds to the fact that the macromolecule Ifthe cofactor is present in large excess with respect to the is present in fixed amounts in a fixed volume, and Eq. 2 components, then it is buffered and its activity does not underlies the autocatalytic nature of the interconversions change during circulation of the reaction network. In this among the states of the macromolecule. case the effect of the cofactor can be included in the kinetic Although the networks described by Eq. 1 allow only for constants, and the system shows conservative oscillations if KjK3y = K2K4, or decays toward a stable steady state other- wise (2). If the cofactor is not buffered then its activity may The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact. tTo whom reprint requests should be addressed. 142 Downloaded by guest on September 27, 2021 Biophysics: Di Cera et aL Proc. NatL AcadJ Sci USA 86 (1989) 143 drastically change and affect the rate ofthe reaction in which 1.5 it is involved. The activity y is written as y = m - x2, where m is the total concentration of cofactor and x2 gives the amount of cofactor bound to the component X2. As a conse- quence, nonlinear terms of the third order appear in Eq. 4. 1.2 For the sake of simplicity, but without loss of generality, we normalize the total concentration of the components to 1, so that the steady-state values for the components are given by .9 1 - (m - K2K4/K1K3)(1 + K2/K3) E (1 + K4/K3) [Sal .6 X2 = m - K2K4/K1K3 [Sb] = K4- .3 =K X X3 K3 [Sc] _ 0 = K2- X4 -K3 X2- [Sd] 0 .3 .6 .9 1.2 1.5 K2 Unbuffering the system introduces constraints among the FIG. 1. Dynamical properties of the system as a function of the autocatalytic constants that define a unique steady state. kinetic constant K2 and the concentration of cofactor m. The other The positivity of the values of x leads to the requirement kinetic constants are equal to 1. Region A is the region where oscil- lations decay exponentially, and region B is the region of limit-cycle oscillations. The boundary across these regions is the Hopf bifurca- (K2K4) + (K3 )> m > K2K4 [6] tion. In regions C no steady state is possible. The boundaries be- tween the regions C and those where a steady state exists (A and B) which indicates that a steady state is only possible when the are characterized by a power-law decay of the oscillations. At the total concentration of cofactor lies within this range defined upper boundary the concentrations of components X1 and X3 are by the kinetic constants, with the total mass of the macro- zero. The same holds for the components X2 and X4 at the lower molecule normalized to 1. boundary. The stability of solutions to Eq. 4 is obtained by linearizing about the values of x and investigating the eigenvalues W of and m on the dynamical properties of the system. The Hopf normal mode analysis (7). Three eigenvalues are given by the bifurcation separates the region A of exponential damping of roots of the secular equation the oscillations from the region B where limit cycles appear. The region characterized by the presence of a limit cycle 3 + K1lX2W2 + (K2K4/K3)XlX2 covers a wide range of parameter values. The boundaries A and B the C set the x (2K2K4/K3 + K2 + K3 + K4 - Klm)w separating the regions from regions domain wherein the system has a steady state. At the upper + (K1K2K4/K3)(K3 + K4)#T = 0. [7] boundary xl = X3 = 0 and at the lower boundary X2 = X4 = 0. Both boundaries correspond to the unusual dynamical condi- The fourth eigenvalue is precisely zero due to conservation tion that all eigenvalues of Eq. 8 are precisely zero. An inter- of mass. One eigenvalue is always real and negative and the esting consequence is that on these boundaries the steady remaining two have a common real part that becomes posi- state is approached according to an inverse power law simi- tive when lar to that found for the four-component hypercycle (8). This dynamical behavior is an example ofquasistability in that the Klm > K2(K3 + 2K4), [8] system mimics a limit cycle over long time periods and then K3 ultimately, or adiabatically, decays toward the steady state. However, as distinct from the hypercycle, the present pow- which follows from the Hurwitz criterion (7). The equality er-law decay is accompanied by oscillations that increase in sign yields a Hopf bifurcation. Combination of Eqs. 6 and 8 period. gives a relation among the kinetic constants for the neces- sary, although not sufficient, condition for existence of a Second, we study the effect of changing K1 and m on the limit cycle as dynamical behavior ofthe system keeping all the other kinet- ic constants equal to 1. The results are shown in Fig. 2. Again, the regions where the system has a steady state (A, K32(K- K2) > K2(K2K4 + K3K4 + K2K3), [9] B, and B') are limited by two boundaries, and the Hopfbifur- cation separates the region A of exponential damping of os- which is independent of m.