J. theor. Biol. (1975) !50,501-505

On the Effect of a Small Parameter and the Possibility of Limit Cycle Bebaviour in a Negatively hductive Control System In recent years considerable attention has been focused on the study of systemsof chemical kinetics which show oscillatory behaviour (for a review see Gray, 1975). Particularly good candidates are metabolic pathways with positive or negative feedback. Goodwin (1963, 1965)has designed a scheme of feedback inhibition in which the protein produced under the direction of a single gene repressesthe action of that samegene. He claimed to have produced a limit cycle at the analogue computer but it was later shown by Griflhh (1968a) that undamped oscillations were impossible for this system. The latter author also demonstrated that undamped oscillations were impossible for the analogous systememploying positive feedback (Griffith, 1968b).

It is the purpose of this note to demonstrate the possibility of a limit cycle in a negatively inductive control system-such as the Zucoperon in Escherichia coli-when the so-called constant “pool” chemicals are allowed to vary slightly. In formulating the equations for the model we make the

following assumptions. (i) The rate at which the repressor R binds to the operon 0 is proportional both to the repressor concentration and the probability that the operon is in the &repressed state (averaged over a large number of cells). The attachment is reversible and is described by the equation O+R$OR

(1)

where symbols are used unambiguously to denote both speciesand concentrations; OR is the operon-repressor complex; and kI and k-, are the forward and reverse rate constantsrespectively. (ii) The repressor reacts with the inducer (P) according to the equations P+Rk2sPR+P+bP,R+...k$sP$.

(2)

We will take n = 2 in accordancewith the experiments of Overath (1968). 1 gives no interesting results. Furthermore, if kzb $= kezs l%ecasen= and k -zlr % k,, we can get the sameresult by considering only the one step equation .

2P + R 2 P2R. sol

(3)

502

L.

J.

AARONS

AND

8.

F.

GRAY

(iii) For sake of simplicity assume that the inducer P is produced from a substrate S, which is held in large excess, under the influence of the operon. Thus the rate of production of P will be proportional both to the probability that the operon is in its derepressed state and the concentration of S. Finally, let P be destroyed at a rate proportional to its own concentration. (iv) The total amounts of operon and repressor are conserved-at least they turn over at a much slower rate than any of the other species. 0 T=O+OR (4) RT = R+OR+P2R z R+P2R.

(5)

The last approximation will be valid if both R > 0 and k- 1 > k,. However, we will return to this and the other approximations later. Thus the rate equations for the system become d = -k,O.R+k-,[O,-O]

(64

It = -k10.R+k-,[0,-0]-k,P2R+k-2[R,-R]

t6b)

P = k,S.0-k,P-2k2P2R+2k-2[RT-R]

(64

s = -k,S.O+cp

(64

where cp is a fiux term. By taking S, a “pool” chemical, as constant (i.e. ,$ = 0) we obtain a system of three differential equations which have been studied previously by Edelstein (1972) based on the work of Yagil & Yagil (1971). For certain parameter values these equations have three singularities, two being stable steady states of the system and the third being unstable of the saddle type. What we now wish to investigate is under what conditions can S be held constant and what is the effect of letting it vary? To do this is convenient to change to new dimensionless variables, defined by 0’ = 0100, R’ = R/R,, P’ = P/P, and s’ = S/S,,. (7) Also let Se % 00, Ro, PO such that 0,/S, N P,/S, = l and change to a new time scale, z, defined by t = cr. Substituting in equations (6) and dividing by S,, gives 0’

= -kl

O’.R’

y 0

R’=-kt

“-+

+ 2

CO,--O,,O’] 0

O’.R ’ + k~t[O,-O,O 0

‘]

0

-k2

‘+

% 0 P”.R’ 0

-I- sk-2 CRT-R0 0

R’]

(8b)

LETTERS

TO

0

SO

-k300S.0'+;

I

[

503

EDITOR

P;Ro -P~~.R'++[R,-R,R']

P'=kBO,,S'.O' - ;P,P'-2k, f?=cf

THE

(8~)

0

.

W

It can be seen that equitions (8) will only reduce to the three variable case if < -+ 0 and k,, kwl, k,, k-,, k, and cp are of order l/5 while k3 is of order 1. Under these conditions equations (g) take the form xi=F1(x1,x2,x1;y)...i=l,2,3 Y

=

tG(x,,

~2,

x3;

Y).

(94 @b)

For e -) 0 the solutions of these equations can be divided into two regions (Andronov, Vitt & Khaikin, 1966). (i) The region of “slow” motion where a representative point moves with finite velocity in a small neighbourhood (of order l) of the curve Rx,, x2, x3; Y) = 0. (ii) The region of “rapid” motion where a representative point, outside the neighbourhood of F, moves with a large velocity (.%, + co as C + 0)

7-

5-

P

-

3-

l-

FWJ.1. L.iit CYC~behaviour for the system describc+d by equation (8). ‘I& parameters used were: kl = 104, k-1 = 6 X 103, k, = 1667 x 105, kso = 4.167 x 10, ks = 6, # 1 2. kr = lo’, 0~ = 1, RT = 252, O. = 0.1667, &, = 6, PO = 1, { = 10-4,

.

L. J. AARONS

504

AND

B. F.

GRAY

along a path close to the straight line y E y” = a constant, until it again comes within a small neighbourhood of F or goes off to * co. The curve F(x,, x2, x3; y) = 0 for a particular choice of parameters is drawn as a function of S in Fig. 1. The upper and lower branches are stable whereas the middle branch is unstable. If we now let S vary according to equation (8d), for certain values of cp, it will jump to one of the stable arms of the curve F and move along that curve until it reaches a “jump point” (either A or B) and then jumps to the other stable branch. The motion of point thus describes a limit cycle corresponding to discontinuous oscillations. The existence of the limit cycle was confirmed on a Solatron HS7 analogue computer for a wide range of initial conditions. The variation of P as a function of time is shown in Fig. 2.

Time

FIG. 2. Variation

of P with time corresponding

to the limit cycle of Fig. 1.

The existence of this type of limit cycle requires a hysteresis curve similar to that of Fig. 1, i.e. the existence of multiple steady states. Relaxing the approximations made earlier will still lead to regions of multistability and will not alter our basic conclusions. Also it is necessary for n > 2 in equation (2). It is interesting that experimentally this would appear to be the case.

LETTERS

TO

THE

EDITOR

505

To conclude, two points can be made. Firstly, contrary to what Griffith has said (19684, it is possible to obtain limit cycle behaviour by negative feedback to a single gene, without exotic kinetics. Secondly, and more importantly, certain schemes which cannot oscillate when the “pool” chemicals are held constant may do so when they are allowed to vary. Experimentally this may be difficult to observe, as in the case described here, since the “pool” chemicals may only oscillate by < 5 % whereas other metabolites may vary by 200-300 % or more. An even more interesting case is that of non-oscillatory isothermal schemes which can be made to oscillate by allowing small variations in temperature (Gray & Aarons, 1975). One of us fellowship.

(L.J.A.)

is grateful to ICI.

for the support of a postdoctoral

Department of Physical Chemistry, School of Chemistry, l%e University of Lee&, Leeds LS2 9JT, England

LEON

J. Amor

AND BRUN F. GRAY

(Received14 June 1974) REPERENCECS A. A., Vrrr, A. A. & KIMKIN, S. E. (1966). l%eory of Oscillators. Oxford: Press. B. B. (1972). J. iheor. Biol. 37, 221. GOODWIN, B. C. (1963). Temporal Organization in Cells. London: Academic Press. GOODWIN, B. C. (1965). In Aduances in Enzyme Regulation (0. Weber, ed.). vol. 3, p. 425. Oxford: Pergamon Press. GRAY, B. F. (1975). Specialist Periodical Reports of the Chemical Society (F’. G. Ashmore, cd.), vol. 30. London: The Chemical Society. GRAY, B. F. & hRONS, L. J. (1975). Symp. Faraaky Sot. (in press). GRIFFITH, J. S. (196&z). J. theor. Biol. 20,202. GRUTITH, J. S. (196Sb). J. theor. Biol. 20,209. OVERATH, P. (1968). Molec. gen. Genet. 101, 155. YAGIL, G. & YAGJL, E. (1971). Biophys. J. 11,ll. ANDRONOV, Pergamon EDIUTEIN,