J. Phys. Chem. 1989, 93. 2755-2759
(a) their need to be. realized as limiting cases of reversible reactions, (b) the survival of the oscillations when lower order autocatalysis occurs, (c) their survival when the uncatalyzed steps occur, and (d) the possibility of replacing them by larger sets of reactions with only second-order steps. It is instructive to sketch the evolution of our model schemes. Oscillatory behavior3v4 involves at least two variables and governing equations that are sufficiently nonlinear. Probably all real chemical systems involve more than two variables, but the general area of oscillatory behavior is sufficiently strange to most chemists, and the two-variable case is so deeply understood that it furnishes the most helpful of introductions. The first two-variable, isothermal scheme to satisfy most of the basic chemical requirements was the one put forward by Lotka in 1920 and known for him and Volterra. It does not, however, furnish stable limit cycles, and nearly 50 years went by before the proposals of Sel’kov in Puschino and Lefever in Brussels invoking alternative termolecular schemes were published’ in 1968. Both of these were rather complex; both have at their hearts the core reactions (or Y 2X 3X) A 2B 3B
+
-
B-
-
+
C
(or X -
E)
that constitute the autocatalator. This is not only the simplest but also the most versatile, chemically satisfactory, two-variable model. In Sel’kov’s case, the heart of his scheme was rather concealed but its connections to enzyme kinetics were made plausible and furnished the starting point for many fruitful advances by B. Hess and his collaborators in Dortmund. Lefever, Prigogine, and Nicolis pursued the reaction set that was aptly named the Brusselator by Tysons They chose to imagine the primitive genesis of the catalyst B (or X) from a precursor, so they had also to postulate a switch reaction to convert species B (their X) into A (their Y). In their symbols this is the additional step B+X-D+Y
(A. 1)
It is this step that has not only made the Brusselator something of a nuisance to study but has also lain at the seat of a good deal of acrimony, not all of it necessary. Step A.l is superfluous and its implications would never have clouded understanding had the
2755
equally valid alternative of generating species A (their Y) been invoked in the first place: (or A Y , in Belgian) P A
-
-
In 1972 and 1973, Hanusse18and Tysons independently stressed the trade-off between the number of independent variables and the need for third-order steps. They proved that kinetic schemes confined to elementary steps of first and second order could not satisfy chemical reasonableness and produce oscillations in a two-variable system; they recognized the possibility of producing oscillations if termolecular steps were allowed. Tyson’s paperss are particularly rewarding: both investigations relate to “pool chemical” models with constant rates of generation of intermediates. They do not lead to multiplicity, but they can be taken over to the CSTR to do this, too. The termolecular step was not confidently invoked even by its first proposers, and it has remained unpopular with those chemists who insist it must be regarded as an elementary step. This is of course not necessary, and the genesis of the termolecular step is the opposite. It represents in a compact manner a set of reactions whose stoichiometry corresponds to A B but for which the rate is catalyzed by the product according to the kinetic rule: rate = kcab2. Both aspects of such behavior can be represented for the purpose at hand by writing A 2B 3B
-
+
-
What is essential is that this representation should satisfyst6 thermodynamic requirements. Thus it must have the same equilibrium constant as the uncatalyzed step: A-B so that (kc/k,) = ( k U / k J . Breaking down the cubic step into constituents is done in two Bz; A + Bz B2 + B) main ways. That adopted here (2B was suggested by Tyson, who ascribess the proposal to B. B. Edelstein. These authors comment on the limiting case when [BZ] = K3[BI2. The alternative writes a sequence of associations-the model for allosteric enzyme catalysis. This speedily produces a multivariable, multiparameter system remote from any general attack save numerical computation.
-
-
(18) Hanusse, P. C. R.Acad. Sci. 1972, 274, 1245.
Stochastic Effects in Front Propagation: The Acetylcholinesterase Reaction Marek Frankowicz Department of Theoretical Chemistry, Faculty of Chemistry, Jagellonian University, Karasia 3, KrakBw. Polmd
and Andrzej L. KawczyLki* Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52. Warsaw, Poland (Received: July 1 1 , 1988; In Final Form: October 7, 1988)
The propagation of concentration profiles for the acetylcholinesterasereaction in a one-dimensional system subject to random perturbations is discussed. In a closed system, irregular frontlike concentration profiles are observed. For open systems in the trigger regime, depending on the strength of perturbations and their correlation time, the evolution pattern changes from the almost deterministic running front propagation to the rapid decay of the spatial region with a less stable state. Between these limiting cases, patterns are observed in which one or more pulses appear ahead of the running front initiated by the initial steplike distribution of acetylcholine. Introduction Nonlinear behavior such as limit cycle oscillations,l multistability,2 and chaos3 has been observed in a growing number of chemically reacting systems. If these reactions proceed in inhoTo whom all correspondence should be directed.
0022-3654/89/2093-2755$01,50/0
mogeneous systems, various spatietemporal patterns can appear;4 one of the simplest is the propagation Of a single wave.’ In Closed (1) Noyes, R. M. Ber. Bunsen-Ges. Phys. Chem. 1980, 84, 295. (2) Pacault, A. Synergetics. A Workshop; Haken, H., Ed.; SpringerVerlag: Berlin, 1977; p 133.
0 1989 American Chemical Society
2756 The Journal of Physical Chemistry, Vol. 93, No. 7, 1989
systems such waves are influenced by the exhaustion of reagents, which causes changes in their shapes and velocities of propagation. Recently, however, a so-called Turing-Nicolis-Prigogine circular one-dimensional open reactor has been constructed6 that ensures conditions such that real running waves with stationary shape and constant propagation velocity can be observed. Stochastic effects may deeply influence the evolution of nonlinear systems, giving rise to noise-induced phase transitions, appearance of transient structures, etc. The influence of noise on propagating fronts has been and it has been found that fluctuations do not change the mean velocity of the front but instead lead to a diffusive random motion of its position with respect to an equivalent unperturbed deterministic front. In the present work we have considered a realistic model of an enzymatic reaction, namely, the hydrolysis of acetylcholine (ACh) to choline and acetic acid catalyzed by acetylcholinesterase. This reaction displays bistability in an open system, while in a closed system it may proceed in an “explosive” fashion in which initial slow evolution is followed by a rapid transition toward the state of total consumption of ACh. We have chosen conditions in which random effects can switch a system from a bistable (trigger) regime to a regime with one stationary state. In this case noise can induce the generation of pulses of concentration ahead of the wave front. We will use the concept of a random medium, Le., a medium with properties changing randomly in space and time. Such changes may be caused by internal fluctuations related to the discrete nature of chemical change as well as to the influence of external disturbances. The problem of wave propagation in random media has been the subject of many studies in radiophysics and continuum mechanic^.^ For chemical systems, effects of diffusion in fluctuating media have been investigated for the linear birth and death processlo and for the Lotka-Volterra model.” It was found that noise-induced phase transitions can take place.
The Model The hydrolysis of acetylcholine (ACh) to choline (Ch) and acetic acid (A) in synaptic clefts is catalyzed by acetylcholinesterase (AChE). It is well-known that this reaction is inhibited by an excess of both ACh and Ch,I2 but details of the mechanism of this inhibition are not completely ~ l e a r . ~ ~Therefore .’~ we will use the experimental expression for the reaction rate of the acetylcholinesterase reaction givenI5 by L‘=
a4(l
+ a,[AChl3)(1 +
%[ACh] [Ch]/(a,(l
Frankowicz and Kawczyfiski A finite system of scaled length 1 with zero-flux boundary conditions will be considered. We assume that the initial distribution of ACh has a steplike form given by [AChIo =
ACh, AChl
for 0 < x < a for a < x < 1
(3)
whereas the initial distribution of Ch is uniform: [Ch], = Cho
for 0 C x
