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Undamped sinusoidal oscillations in linear chemical reaction systems
J. theor. Biol. (1970) 27, 197-206
UndampedSinusoidalOscillations in Linear Chemical Reaction Systems FRIEDRICH FRANZ SEELIG~ Institute of Physical Chemistry, University of Marburg,
West Germany
(Received 2 June 1969) It is shown both by mathematical analysis and by computer simulation that it is possible to design a chemical analogue to the well-known electronic RC phase-shift oscillator. The system consists of (1) a chemical reaction whose rate is linearly dependent on the concentration of a catalyst, and (2) a chain of successivefirst order reactions with the product of part 1 as input and the catalyst as output. Initial substrates considered as pools with constant concentrations and certain relations between the parameters of the system (rate constants, initial conditions, etc.) obeyed, the system shows undamped purely sinusoidal oscillations in the reaction rates and in the concentrations of the intermediates. The main characteristics of the system are that there is a closed loop in the flux of information, but an open chain in the flux of matter, the catalyst, which at the same time is one of the products, being destroyed by a first-order reaction and in addition by a reaction with constant rate.
1. Introduction Oscillating reactions have gained growing interest in biochemistry and other biological sciences, e.g. ecology, if we do not confine ourselves to purely chemical reactions. The first author to have seen the possibility of periodicities in autocatalytic metabolic reactions seems to be Lotka (1910). In his paper and in contributions by Moore (1949) and Bak (1963) it is the non-linear oscillator that is investigated. Subsequently many experimental data on such systems have been compiled and computer simulations have been carried out. On the other hand, the possibility of oscillation in linear systems seemed to be excluded. Theoretical investigations by Denbigh, Hicks & Page (1948), Meixner (1949) and Hearon (1953) showed that undamped oscillations are impossible in linear systems under certain conditions. Although these results are correct within the scope treated there, they have been generalized in an invalid way in the past so that it was accepted tacitly that non-linearity is a conditio sine qua non for undamped oscillations. t New address: Department of Chemistry, University of Ttibingen, West Germany. 197
198
F. F. SEELIG
It will be shown in this paper that undamped oscillations are possible in linear systems, indeed, and furthermore that these oscillations are even pure sine-functions. The systems showing this property can be designed in close analogy to the well-known electronic feedback amplifier, built from an active element (vacuum tube or transistor) and a feedback network (capacitors and resistors). The system shown here contains some idealizations. It is intended to relax them in subsequent investigations.
2. Outline of the System Since we want to design the chemical reaction system in close analogy to the electronic RC phase-shift oscillator, it is useful to recall briefly the main features of this device, whose circuitry is shown in Fig. 1. A
Eb II
If
Feedback
network
Ampliiier
FIG. 1. Circuit of the electronic RC phase-shift oscillator. (Cathode resistor causing selfbias and by-pass capacitor have been omitted.)
For the sake of simplicity a triode is taken as the active element. Here in the linear part of its characteristic line the anode current Z. is proportional to the grid voltage E, and practically independent from the anode voltage Ei, which property is best achieved in a pentode. Ideally the flux of energy into the grid is zero, actually it is negligible. What is fed essentially into the triode via its grid is not energy, but information. The current Z, causes a voltage drop Z,R, over the load resistor RL against the constant voltage of the power supply, Eb,yielding for the voltage Ei the linear function E,(E,,)in the form E,=AE,(+const.), (1)
UNDAMPED
SINUSOIDAL
OSCILLATIONS
199
A being the amplifier gain. The internal phase-shift is 71,thus A is real and negative. In the most general case E, is a function of time and can be split up in several components, one of which is a constant voltage. In the case of periodicity the other components can be represented by a sum of alternating sinusoidal voltages of various frequencies, amplitudes, and phases by means of Fourier analysis. This mixture applied to a RC-network with the (frequencydependent) feedback factor a, will change the amplitude and phase, but not the frequency, of each component in a way characteristic for the network and for the frequency of the component considered according to E,, =BEi.
(2)
There is at least one component whose phase-shift is just z so that for this component E, inserted in (1) is proportional to EO fed back. If the latter has a greater amplitude than the first one, this component grows very fast until saturation in the non-linear part of the characteristic line of the triode is achieved. In the limiting case we get A/l= 1. (3) All other components that are not in phase are finally suppressed, more or less, by interference. The feedback network can consist of several subunits, each containing one capacitor and one resistor. Since each subunit cannot shift the phase by more than ?r/2 eventually, the minimum number of reasonable subunits is three. In the chemical counterpart a chemical reaction of pseudo first order is introduced S(+S’+
. . . )1;A(+A’. E
, .)
(1)
with at least one substrate S and at least one product A, which is catalysed by a catalyst E. For the sake of simplicity we confine ourselves to the special case with one substrate and one product for the moment. Under certain conditions it is possible to have the following differential equation for the rate of the reaction (4)
the rate (d[A]/dt), being a linear function concentration of the catalyst, [El.
only of the (time-dependent)
If E is an enzyme, according to the well-known theory of Michaelis 8c Menten, in detail we find:S+E&S,
(Ia)
200
F.
F.
SEELIG
a fast-achieved equilibrium with
and a slow sequential reaction ES:A+E
(Ib)
with the differential equation for the rate of reaction = k[ES] = ;[S][E],
(6)
which is primarily of second order. But if now the concentration of the substrate, [5’j, is great compared to that of the enzyme, [El, which is normally the case and if it is in consequence relatively constant, the rate of reaction becomes virtually of first order in [El, and with (7) we get (4). It is easily seen that (4) is not changed, if there are other substrates S’ . . . and if these in turn are in great excess. Now product A may react (possibly also together with other substrates A*, . . . , which are in great excess in turn) in a pseudo first order reaction to product(s) B( + B’ + . . .) A(fA*+
. . .):B(+B’+
. . .).
(11)
Similarly, we have a chain of first-order reactions B(fB*+
. . .)ZC(+C+
. . .)
(III)
C(+C*+
. . . )L(+D’+
.. .)
(IV)
D(+D*+
. . &(+E+
. . .,
(V)
E(+E*+
. . .):Q(+Q’+
. ..)
WI)
in the same sense and with the same additional conditions as in reaction (II), where all rate constants k, through k, refer to first-order reactions. For reaction (II) we get the reaction rate (!!).
= -(%$x1
= k,[A].
Since the total flux of A is the sum of the fluxes, namely (d[A]/dt), and (d[A]/ dt),, we arrive at
UNDAMPED
SINUSOIDAL
201
OSCILLATIONS
‘9
= k,[E] - k&4],
y
= k2[A]-kJ[B],
(10)
‘9
= k,[B] - k4[C],
(11)
‘g
= k4[C] - k,[D],
(12)
‘$
= k,[D] - k,[E] -jE,
(9)
(j, =
(13)
const.).
Now it is claimed that the system of reactions (I) through (VI) [Fig. 2(a)], r
------
0
I -------A Cotolyzed
reachon
ProductIon
and destruction
Ic)
of catalyst
FIG. 2. Circuit of the chemical phase-shift oscillator (The reaction A+P is not essential, but has been included for reasons of analogy), (a) complete circuit, (b) flux of matter, (c) flux of information (signal flow).
[the intermediate steps (Ia) and (Ib) omitted] which are represented by the differential equations (9) through (13) show undamped sinusoidal oscillations in the reaction rates and in the concentrations of all substances appearing in (9) through (13), if certain relations to be derived in the next section between the initial conditions and the parameters k, through k, and j, are obeyed. In addition it is claimed that the minimum number of intermediates is just five as formulated above (A through E) and finally that it is necessary to have a constant rate of destruction of E, j,, in addition to the first-order reaction (VI). This constant rate of destruction can be imagined to be realized
202
F.
F.
SEELIG
by a sort of “titration” reaction with constant rate that is quantitative and so fast that its kinetics does not interfere. Comparing the chemical equations (I) through (VI) which represent the flux of matter, with the differential equations (9) through (13) which represent the flux of information rather (especially in the term k,[E] in (9), while there is a close correspondence of both otherwise), it can be seen that the flux of matter is open, while the flux of information (= signal flow) is a closed loop, typical for feedback. This fact (together with the termj,) is the main difference between our system and those systems analysed by the authors cited above, which do not show undamped sinusoidal oscillations. The flux of matter and the flux of information are sketched separately in Fig. 2(b) and (c), respectively. To make the analogy to the electronic RC phase-shift oscillator, which was the outset, perfect, we can split up the flux of matter in reaction (II) in a main stream over A(+A+ . . . )S(+p/. .. ) (W and the side chain (II) through (VI) as formulated above. In this case the term k,[A] in (9) must be replaced by (k, + k’,)[A]. This possibility has been included in Fig. 2, but it must be emphasized that this branching is not necessary. 3. Mathematical
Analysis
We suppose that there is one undamped sinusoidal oscillation in [E] with the frequency o, amplitude e,, and a phase constant equal to zero by virtue of a suitably chosen time zero. Since concentrations cannot become negative, we have to add a constant concentration e, 2 e, > 0 yielding [E] = e, + e, sin+, = e, + e, sinor. (14) This causes an oscillation in [A] like [A] = a, + a, sin+, = a, + a, sin(wt - a,), (15) where simple analysis by standard methods (namely insertion of (14) and (15) in (9) and equating all terms containing sinot, cos~t, and neither to zero separately) shows that in the case of branching at A
kt a0= k,+k;eoj k, a1= (d+(k2+4)2ff’ 6, = arctan-%.
k,+k;
(18)
It follows that the phase-shift 4, - 4, = - 6, is negative, that its absolute value
UNDAMPED
SINUSOIDAL
203
OSCILLATIONS
cannot exceed n/2, and that the “d.c.” amplifier gain a,/ee=k,/(k,+k’,), is greater than the “a.c.” gain, al/e,, by the factor (1 + (o/(k, + k;)*)* > 1. If now [B]=b,+b, sin&=&+b, sin(wt-6,-J,) (19) with phase-shift & - 4, = - a,, amplitude b,, and constant part of concentration b, is inserted in (lo), we get b, = +, 3
b,=
k2 (co* + kf)+“’
S, = arctan;. 3
Continuing this procedure to (13), the time-dependent concentration of Eat the end of the chain, which is distinguished from the concentration of E at the beginning of the chain by a prime for the moment is obtained [El’ = eb + e; sin 4, = eb + e; sin(wt - 6, - 6, - 6, - 6, - 6,) with
e; = fLSd-Ji k, ’
(23) (24)
k,’
k, e’ = (W2+,&+d1’ 6, = arctan;. 6
In order to get an undamped sinusoidal oscillation with frequency w in all reaction rates and concentrations of the intermediates, it is necessary to get a closed loop in the flux of information, which is equivalent to the equality of [E] and [El’, meaning that the amplitude and constant part of the concentration of catalyst and intermediate E must be equal at the beginning and the end of the loop and that the sum of the absolute values of all phase-shifts 6, through 6, is 27~.Thus by iterative insertion we get the three principal relations
arctan-
w
k,+k;
+ arctan;
+ arctan; 3
+ arctan; 4
+ arctan; 5
= 2n. 6
(29)
204
F. F. SEELIG
From (27) and (28) it is seen that the constant j, > 0 is necessary in order to get o>O. The case discussed by Hearon (1953) is included, where jr= 0, k; =0, and kl =k6 (closed loop in the flux of matter). In this case the left sides of (27) and (28) become unity and in addition (27) reduces to an identity while (28) can only be satisfied for w=O. Thus no undamped oscillations are possible. In our case, however, the three main equations (27) through (29) relate nine quantities (k,, kt, k;, k,, k4, k,, kg, jr, CD),giving six degrees of freedom. 4. Computer Simulation
Although a complete mathematical analysis of the situation, which is possible in the case of purely linear systems, is the most rigorous and satisfying discussion thinkable, it is yet desirable to demonstrate the result in a special example. Even nicer would be to give an experimental case with the sub0 002
/
[El
0.00,
_--
---___---
-----
0.000 Pdu
t
[Al
4.0
-36 ----
----
----
--
.35 LdYl 0.0
2-o
4.0
t--+ FIG. 3. Plot of concentration (in arbitrary units).
vs. time for the intermediates
E (catalyst), A, and C
UNDAMPED
SINUSOIDAL
205
OSCILLATIONS
stances S, A, . . . E given explicit chemical names. Computer simulation of the system of chemical reactions discussed above is the classical realm of analog computers. As there was none available to us and since there exist a number of very elegant programs for the abundant digital computers that simulate analog computers with sophisticated numerical methods in a much higher precision, we made some simulations on the IBM 7094. The program used was MIDAS, distributed by the SHARE organization under number 3101. We desisted from studying the transient behaviour and chose the parameters consistent with (27) through (29) and the initial conditions from (14) through (26) so that according to the theory, undamped sinusoidal oscillations in the concentrations of the intermediates A through E were to be expected. Figure 3 shows some computer plots for the following set of parameters k, through k,, j, and of the initial conditions designated by [A],, through [El,, as comprised in Table 1. TABLE
1
Parameters and initial conditions of the computer simulation (arbitrary units) kl 3 548.85
[Alo 2.610
kz
k’z
h
h
ks
k6
0.1
0.9
1 .o
1 .o
1.0
I.0
[No 17
0.336
958
[Cl0 0.360
425
MO 0.357
655
No 0.001
jE 0.353
885
resulting w 3.078
5. Conclusions
This is the beginning of a project started in our group some months ago in which it is intended not only to transform the highly developed theory of electronic circuitry to chemical reaction systems to design chemical realizations of such things as flip-flops, multivibrators, etc., for which some features in terms of binary logic have already been given by Sugita (1961), but also phenomena like amplitude and frequency modulation. Such work is interesting in the field of theoretical biology as shown by the growing number of publications on related topics. Another possible field of application is chemical technology. It is our intention to investigate the influence of reaction rates with nontrivial time dependence on the yields and purities of chemical syntheses in complicated coupled reaction systems.
206
F.
F.
SEELIG
This work was supported partially by the German “Fonds der Chemischen Industrie”. Computer simulations were executed on the IBM 70% of the “Deutsches Rechenzentrum” in Darmstadt, Germany. REFERENCES T. A. (1963). “Contributions to the Theory of Chemical Kinetics”. New York: W. A. Benjamin, Inc. DENBIOH,K. G., HICKS, M. & PAGE, F. M. (1948). Trans. Faraday Sot. 44, 479.
BAK,
HURON, J. Z. (1953). BuZf. math. Biophys. 15, 121. LOTKA, A. J. (1910).J. phys. Chem. Zthuca 14, 271. MEIXNER, J. (1949). Z. Naturf 4a, 594. MOORE, M. J. (1949). Trans. Farachy Sot. 43, 1088. SIJGITA, M. (1961). J. theor. Biof. 1,415.
UndampedSinusoidalOscillations in Linear Chemical Reaction Systems FRIEDRICH FRANZ SEELIG~ Institute of Physical Chemistry, University of Marburg,
West Germany
(Received 2 June 1969) It is shown both by mathematical analysis and by computer simulation that it is possible to design a chemical analogue to the well-known electronic RC phase-shift oscillator. The system consists of (1) a chemical reaction whose rate is linearly dependent on the concentration of a catalyst, and (2) a chain of successivefirst order reactions with the product of part 1 as input and the catalyst as output. Initial substrates considered as pools with constant concentrations and certain relations between the parameters of the system (rate constants, initial conditions, etc.) obeyed, the system shows undamped purely sinusoidal oscillations in the reaction rates and in the concentrations of the intermediates. The main characteristics of the system are that there is a closed loop in the flux of information, but an open chain in the flux of matter, the catalyst, which at the same time is one of the products, being destroyed by a first-order reaction and in addition by a reaction with constant rate.
1. Introduction Oscillating reactions have gained growing interest in biochemistry and other biological sciences, e.g. ecology, if we do not confine ourselves to purely chemical reactions. The first author to have seen the possibility of periodicities in autocatalytic metabolic reactions seems to be Lotka (1910). In his paper and in contributions by Moore (1949) and Bak (1963) it is the non-linear oscillator that is investigated. Subsequently many experimental data on such systems have been compiled and computer simulations have been carried out. On the other hand, the possibility of oscillation in linear systems seemed to be excluded. Theoretical investigations by Denbigh, Hicks & Page (1948), Meixner (1949) and Hearon (1953) showed that undamped oscillations are impossible in linear systems under certain conditions. Although these results are correct within the scope treated there, they have been generalized in an invalid way in the past so that it was accepted tacitly that non-linearity is a conditio sine qua non for undamped oscillations. t New address: Department of Chemistry, University of Ttibingen, West Germany. 197
198
F. F. SEELIG
It will be shown in this paper that undamped oscillations are possible in linear systems, indeed, and furthermore that these oscillations are even pure sine-functions. The systems showing this property can be designed in close analogy to the well-known electronic feedback amplifier, built from an active element (vacuum tube or transistor) and a feedback network (capacitors and resistors). The system shown here contains some idealizations. It is intended to relax them in subsequent investigations.
2. Outline of the System Since we want to design the chemical reaction system in close analogy to the electronic RC phase-shift oscillator, it is useful to recall briefly the main features of this device, whose circuitry is shown in Fig. 1. A
Eb II
If
Feedback
network
Ampliiier
FIG. 1. Circuit of the electronic RC phase-shift oscillator. (Cathode resistor causing selfbias and by-pass capacitor have been omitted.)
For the sake of simplicity a triode is taken as the active element. Here in the linear part of its characteristic line the anode current Z. is proportional to the grid voltage E, and practically independent from the anode voltage Ei, which property is best achieved in a pentode. Ideally the flux of energy into the grid is zero, actually it is negligible. What is fed essentially into the triode via its grid is not energy, but information. The current Z, causes a voltage drop Z,R, over the load resistor RL against the constant voltage of the power supply, Eb,yielding for the voltage Ei the linear function E,(E,,)in the form E,=AE,(+const.), (1)
UNDAMPED
SINUSOIDAL
OSCILLATIONS
199
A being the amplifier gain. The internal phase-shift is 71,thus A is real and negative. In the most general case E, is a function of time and can be split up in several components, one of which is a constant voltage. In the case of periodicity the other components can be represented by a sum of alternating sinusoidal voltages of various frequencies, amplitudes, and phases by means of Fourier analysis. This mixture applied to a RC-network with the (frequencydependent) feedback factor a, will change the amplitude and phase, but not the frequency, of each component in a way characteristic for the network and for the frequency of the component considered according to E,, =BEi.
(2)
There is at least one component whose phase-shift is just z so that for this component E, inserted in (1) is proportional to EO fed back. If the latter has a greater amplitude than the first one, this component grows very fast until saturation in the non-linear part of the characteristic line of the triode is achieved. In the limiting case we get A/l= 1. (3) All other components that are not in phase are finally suppressed, more or less, by interference. The feedback network can consist of several subunits, each containing one capacitor and one resistor. Since each subunit cannot shift the phase by more than ?r/2 eventually, the minimum number of reasonable subunits is three. In the chemical counterpart a chemical reaction of pseudo first order is introduced S(+S’+
. . . )1;A(+A’. E
, .)
(1)
with at least one substrate S and at least one product A, which is catalysed by a catalyst E. For the sake of simplicity we confine ourselves to the special case with one substrate and one product for the moment. Under certain conditions it is possible to have the following differential equation for the rate of the reaction (4)
the rate (d[A]/dt), being a linear function concentration of the catalyst, [El.
only of the (time-dependent)
If E is an enzyme, according to the well-known theory of Michaelis 8c Menten, in detail we find:S+E&S,
(Ia)
200
F.
F.
SEELIG
a fast-achieved equilibrium with
and a slow sequential reaction ES:A+E
(Ib)
with the differential equation for the rate of reaction = k[ES] = ;[S][E],
(6)
which is primarily of second order. But if now the concentration of the substrate, [5’j, is great compared to that of the enzyme, [El, which is normally the case and if it is in consequence relatively constant, the rate of reaction becomes virtually of first order in [El, and with (7) we get (4). It is easily seen that (4) is not changed, if there are other substrates S’ . . . and if these in turn are in great excess. Now product A may react (possibly also together with other substrates A*, . . . , which are in great excess in turn) in a pseudo first order reaction to product(s) B( + B’ + . . .) A(fA*+
. . .):B(+B’+
. . .).
(11)
Similarly, we have a chain of first-order reactions B(fB*+
. . .)ZC(+C+
. . .)
(III)
C(+C*+
. . . )L(+D’+
.. .)
(IV)
D(+D*+
. . &(+E+
. . .,
(V)
E(+E*+
. . .):Q(+Q’+
. ..)
WI)
in the same sense and with the same additional conditions as in reaction (II), where all rate constants k, through k, refer to first-order reactions. For reaction (II) we get the reaction rate (!!).
= -(%$x1
= k,[A].
Since the total flux of A is the sum of the fluxes, namely (d[A]/dt), and (d[A]/ dt),, we arrive at
UNDAMPED
SINUSOIDAL
201
OSCILLATIONS
‘9
= k,[E] - k&4],
y
= k2[A]-kJ[B],
(10)
‘9
= k,[B] - k4[C],
(11)
‘g
= k4[C] - k,[D],
(12)
‘$
= k,[D] - k,[E] -jE,
(9)
(j, =
(13)
const.).
Now it is claimed that the system of reactions (I) through (VI) [Fig. 2(a)], r
------
0
I -------A Cotolyzed
reachon
ProductIon
and destruction
Ic)
of catalyst
FIG. 2. Circuit of the chemical phase-shift oscillator (The reaction A+P is not essential, but has been included for reasons of analogy), (a) complete circuit, (b) flux of matter, (c) flux of information (signal flow).
[the intermediate steps (Ia) and (Ib) omitted] which are represented by the differential equations (9) through (13) show undamped sinusoidal oscillations in the reaction rates and in the concentrations of all substances appearing in (9) through (13), if certain relations to be derived in the next section between the initial conditions and the parameters k, through k, and j, are obeyed. In addition it is claimed that the minimum number of intermediates is just five as formulated above (A through E) and finally that it is necessary to have a constant rate of destruction of E, j,, in addition to the first-order reaction (VI). This constant rate of destruction can be imagined to be realized
202
F.
F.
SEELIG
by a sort of “titration” reaction with constant rate that is quantitative and so fast that its kinetics does not interfere. Comparing the chemical equations (I) through (VI) which represent the flux of matter, with the differential equations (9) through (13) which represent the flux of information rather (especially in the term k,[E] in (9), while there is a close correspondence of both otherwise), it can be seen that the flux of matter is open, while the flux of information (= signal flow) is a closed loop, typical for feedback. This fact (together with the termj,) is the main difference between our system and those systems analysed by the authors cited above, which do not show undamped sinusoidal oscillations. The flux of matter and the flux of information are sketched separately in Fig. 2(b) and (c), respectively. To make the analogy to the electronic RC phase-shift oscillator, which was the outset, perfect, we can split up the flux of matter in reaction (II) in a main stream over A(+A+ . . . )S(+p/. .. ) (W and the side chain (II) through (VI) as formulated above. In this case the term k,[A] in (9) must be replaced by (k, + k’,)[A]. This possibility has been included in Fig. 2, but it must be emphasized that this branching is not necessary. 3. Mathematical
Analysis
We suppose that there is one undamped sinusoidal oscillation in [E] with the frequency o, amplitude e,, and a phase constant equal to zero by virtue of a suitably chosen time zero. Since concentrations cannot become negative, we have to add a constant concentration e, 2 e, > 0 yielding [E] = e, + e, sin+, = e, + e, sinor. (14) This causes an oscillation in [A] like [A] = a, + a, sin+, = a, + a, sin(wt - a,), (15) where simple analysis by standard methods (namely insertion of (14) and (15) in (9) and equating all terms containing sinot, cos~t, and neither to zero separately) shows that in the case of branching at A
kt a0= k,+k;eoj k, a1= (d+(k2+4)2ff’ 6, = arctan-%.
k,+k;
(18)
It follows that the phase-shift 4, - 4, = - 6, is negative, that its absolute value
UNDAMPED
SINUSOIDAL
203
OSCILLATIONS
cannot exceed n/2, and that the “d.c.” amplifier gain a,/ee=k,/(k,+k’,), is greater than the “a.c.” gain, al/e,, by the factor (1 + (o/(k, + k;)*)* > 1. If now [B]=b,+b, sin&=&+b, sin(wt-6,-J,) (19) with phase-shift & - 4, = - a,, amplitude b,, and constant part of concentration b, is inserted in (lo), we get b, = +, 3
b,=
k2 (co* + kf)+“’
S, = arctan;. 3
Continuing this procedure to (13), the time-dependent concentration of Eat the end of the chain, which is distinguished from the concentration of E at the beginning of the chain by a prime for the moment is obtained [El’ = eb + e; sin 4, = eb + e; sin(wt - 6, - 6, - 6, - 6, - 6,) with
e; = fLSd-Ji k, ’
(23) (24)
k,’
k, e’ = (W2+,&+d1’ 6, = arctan;. 6
In order to get an undamped sinusoidal oscillation with frequency w in all reaction rates and concentrations of the intermediates, it is necessary to get a closed loop in the flux of information, which is equivalent to the equality of [E] and [El’, meaning that the amplitude and constant part of the concentration of catalyst and intermediate E must be equal at the beginning and the end of the loop and that the sum of the absolute values of all phase-shifts 6, through 6, is 27~.Thus by iterative insertion we get the three principal relations
arctan-
w
k,+k;
+ arctan;
+ arctan; 3
+ arctan; 4
+ arctan; 5
= 2n. 6
(29)
204
F. F. SEELIG
From (27) and (28) it is seen that the constant j, > 0 is necessary in order to get o>O. The case discussed by Hearon (1953) is included, where jr= 0, k; =0, and kl =k6 (closed loop in the flux of matter). In this case the left sides of (27) and (28) become unity and in addition (27) reduces to an identity while (28) can only be satisfied for w=O. Thus no undamped oscillations are possible. In our case, however, the three main equations (27) through (29) relate nine quantities (k,, kt, k;, k,, k4, k,, kg, jr, CD),giving six degrees of freedom. 4. Computer Simulation
Although a complete mathematical analysis of the situation, which is possible in the case of purely linear systems, is the most rigorous and satisfying discussion thinkable, it is yet desirable to demonstrate the result in a special example. Even nicer would be to give an experimental case with the sub0 002
/
[El
0.00,
_--
---___---
-----
0.000 Pdu
t
[Al
4.0
-36 ----
----
----
--
.35 LdYl 0.0
2-o
4.0
t--+ FIG. 3. Plot of concentration (in arbitrary units).
vs. time for the intermediates
E (catalyst), A, and C
UNDAMPED
SINUSOIDAL
205
OSCILLATIONS
stances S, A, . . . E given explicit chemical names. Computer simulation of the system of chemical reactions discussed above is the classical realm of analog computers. As there was none available to us and since there exist a number of very elegant programs for the abundant digital computers that simulate analog computers with sophisticated numerical methods in a much higher precision, we made some simulations on the IBM 7094. The program used was MIDAS, distributed by the SHARE organization under number 3101. We desisted from studying the transient behaviour and chose the parameters consistent with (27) through (29) and the initial conditions from (14) through (26) so that according to the theory, undamped sinusoidal oscillations in the concentrations of the intermediates A through E were to be expected. Figure 3 shows some computer plots for the following set of parameters k, through k,, j, and of the initial conditions designated by [A],, through [El,, as comprised in Table 1. TABLE
1
Parameters and initial conditions of the computer simulation (arbitrary units) kl 3 548.85
[Alo 2.610
kz
k’z
h
h
ks
k6
0.1
0.9
1 .o
1 .o
1.0
I.0
[No 17
0.336
958
[Cl0 0.360
425
MO 0.357
655
No 0.001
jE 0.353
885
resulting w 3.078
5. Conclusions
This is the beginning of a project started in our group some months ago in which it is intended not only to transform the highly developed theory of electronic circuitry to chemical reaction systems to design chemical realizations of such things as flip-flops, multivibrators, etc., for which some features in terms of binary logic have already been given by Sugita (1961), but also phenomena like amplitude and frequency modulation. Such work is interesting in the field of theoretical biology as shown by the growing number of publications on related topics. Another possible field of application is chemical technology. It is our intention to investigate the influence of reaction rates with nontrivial time dependence on the yields and purities of chemical syntheses in complicated coupled reaction systems.
206
F.
F.
SEELIG
This work was supported partially by the German “Fonds der Chemischen Industrie”. Computer simulations were executed on the IBM 70% of the “Deutsches Rechenzentrum” in Darmstadt, Germany. REFERENCES T. A. (1963). “Contributions to the Theory of Chemical Kinetics”. New York: W. A. Benjamin, Inc. DENBIOH,K. G., HICKS, M. & PAGE, F. M. (1948). Trans. Faraday Sot. 44, 479.
BAK,
HURON, J. Z. (1953). BuZf. math. Biophys. 15, 121. LOTKA, A. J. (1910).J. phys. Chem. Zthuca 14, 271. MEIXNER, J. (1949). Z. Naturf 4a, 594. MOORE, M. J. (1949). Trans. Farachy Sot. 43, 1088. SIJGITA, M. (1961). J. theor. Biof. 1,415.