The Goodwin Oscillator

The quintessential example of a biochemical oscillator based on negative feedback alone (Figure 9.3C) was invented by Brian Goodwin ([Goodwin, 1965, Goodwin, 1966]; see also [Griffith, 1968a]). The kinetic equations describing this mechanism are d[Xi] _ vo dt 1 + ([Xs]/Km)p d[X2]

Here [Xi], [X2], and [X3] are concentrations of mRNA, protein, and end product, respectively; v0, v1, and v2 determine the rates of transcription, translation, and catalysis; k1, k2, and k3 are rate constants for degradation of each component; 1/Km is the binding constant of end product to transcription factor; and p is a measure of the cooperativity of end product repression.

Next we introduce dimensionless variables:

k3Km

X3 = [X3]/Km, t' = at, where a = (voViV2)/(Kmk2k3).

In terms of these new variables, the dynamical system becomes dxi 1

Furthermore, to make the example easier, we shall assume that bi = b2 = b3. In this case, the dynamical system has a steady state at xi = x2 = x3 = £, where £ is the unique real positive root of 1/(1 + £p) = b£. The Jacobian matrix at this steady state is J_, with a = P = y = b,Ci = c2 = b, and 0 = (p£p-i)/(1 + £p)2 = bp(1 — b£) > 0. Hence, the characteristic equation is (b + A)3 + b20 = 0, whose roots are

A2,3 = —b + b^p(1 - b£) [cos(n/3) ± i sin(n/3)].

The steady state of Goodwin's model is unstable if Re (A2,3) > 0, i.e., if — b + (b/2) 3p(1 — b£) > 0. This condition is equivalent to p(1 — b£) > 8, or b£ < (p — 8)/p. Hence, if p (the cooperativity of end product repression) is greater than 8, then we can choose k small enough to destabilize the steady—state solution of Goodwin's equations. At the critical value of k, when Re (A2,3) = 0, the steady state undergoes a Hopf bifurcation, spinning off small-amplitude periodic solutions with period close to 2n/lm (A2,3) = 2n/(bV3).

In Exercise 8 you are asked to generalize this derivation to negative feedback loops with an arbitrary number n of components. You will find that the steady state is unstable when b£ < (p — pmin)/p, where pmin = secn(n/n). Notice that pmin ^ 1+ as n ^ to, i.e., the minimum cooperativity of endproduct repression required for oscillations becomes small as the length of the feedback loop increases.

The analysis of Hopf bifurcations in Goodwin's model uncovers a number of problems with his negative—feedback mechanism for biochemical oscillations [Griffith, 1968a]. In a three-variable system (mRNA, protein, end product), the cooperativity of feedback must be very high, p > 8. Also, it is necessary, in this case, for the degradation rate constants of the three components to be nearly equal. If not, pmin increases dramatically; e.g., if one of the ki's is tenfold larger than the other two, then pmin = 24. The value of pmin can be reduced by lengthening the loop, but one must still ensure that the k's are nearly equal.

Bliss, Painter and Marr [Bliss et al., 1982] fixed these problems by a slight modification of Goodwin's equations:

d,X2

dt dx

Notice that the feedback step is no longer cooperative (p = 1), and the uptake of end product is now a Michaelis—Menten function. The steady state of this system is x\ = a/(bi(1 + £)), x*2 = a/(b2(1 + £)), Xg = where £ is the unique real positive root of a/(1 + £) = c£/(K + £). The stability of this steady state is determined by the roots of the characteristic equation (b1 + A)(b2 + A)(/ + A) + bjb20 = 0, where / = cK/(K + £)2 and 0 = a/(1 + £)2.

The characteristic equation is hard to solve in this completely general case. In order to get a start on it, we make some simplifying assumptions. First, suppose that K = 1, so £ = a/c. Next, suppose b1 = b2 < c, and choose a = c c/b1 — , so that P = b 1 as well. In this case, 0 = b1 a/c, and the characteristic equation becomes (A + b)3 + \f\a/c = 0. The solutions of this characteristic equation are A1 = —b1 + 3a/c^ ,

Figure 9.8 Locus of Hopf bifurcations in the Bliss— Painter-Marr equations given in system (9.18), for bi = b2 = 0.2. We also plot loci of constant period (18 — 27) within the region of limit cycle oscillations.

A2,3 = —b1 +b1 3a/c[cos(n/3)±i sin(n/3)]. The dynamical system has a Hopf bifurcation when Re (A2,3) = 0, i.e., when a = 8c. Hence, at the Hopf bifurcation, c = 81b1 and a = 8c = 648b1. If we set b1 = 0.2, then the Hopf bifurcation occurs at c = 16.2, a = 129.6. At this Hopf bifurcation the period of oscillation is close to 2n/Im (A2,3 ) = 2n/(biV3) = 18. Starting at this point, we can trace out the locus of Hopf bifurcations numerically as a and c vary at fixed b1. See Figure 9.8.

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