BULLETIN OF

MATHEMATICAL BIOLOGY

VOLUM~ 39, 1977

ANALYTICAL THEORY OF THE CONTROL EQUATIONS FOR PROTEIN SYNTHESIS IN THE GOODWIN MODEL

�9 VIRENDRA SINGH Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

The analytical expressions, and their convergent series expansions, for the time period and various time averages of the dynamical quantities are derived for the Goodwin model. The radius of convergence of the series expansions is also calculated. The changes necessary to fully take into account the restrictions on dynamical variable, following from the positivity of physical quantities of the model, for the statistical mechanical treatment of the system are also discussed.

1. Introduction. An interesting model of a biochemical metabolic oscillator was given by B. C. Goodwin (1963). This and similar models have been repeatedly investigated (Goel, Maitra and Montroll, 1971; Pavilidis, 1973; Maynard Smith, 1968). Due to essential nonlinearities present in these problems the exact analytical results have been rather few. In the present paper we shall present a method which allows us to obtain explicit convergent expressions for various quantities of interest such as time period of oscillation and time averages of various moments of dynamical variables.

In section 2, which discusses the dynamics of a single oscillator in the Goodwin model, are contained the main results of this paper. Section 2.1 deals with equations of motion and other preliminaries and primarily serves to define the notation. In section 2.2 we investigate the allowed range of dynamical variables which follow from the positivity of physical variables involved. We then give in section 2.3 analytical expressions for the time period of Goodwin oscillator and also express it as convergent series. The radius of convergence is

565

566 VIRENDRA SINGH

calculated in section 2.4. We also show in section 2.5 that the technique introduced in section 2.3 allows us to calculate various time averages. Finally in section 3 we point out the necessary changes which one has to introduce in the treatment of the statistical mechanical of an assembly of Goodwin oscillators to take into account the restrictions on the allowed range of values of the dynamical variables found earlier in section 2.2.

2. Single Biochemical Oscillator 2.1. Equations of motion. The final equations describing the time evolution

of Goodwin model of a metabolic feedback control cycle of protein synthesis are given by

d X a b

dt - A + k Y

dY = ~ X - f l (2.1.1)

dt

where a, b, A, k, a, fl are positive constants. The X and Y variables measure m-RNA and protein concentrations respectively.

Let p and q denote the steady values of X and Y respectively i.e.

a - b ( A + kq)=O,

~ p - / ~ = 0 . (2.1.2)

Define new variables x and y, which measure the departure from steady state, by

x = X - p

A + k Y l + y - A+kq " (2.1.3)

The equations of motion (2.1.1) then take the form

dx - - = b 1 dt

dy - - -~- C X dt

where (A + kq)c =ek. (2.1.4)

CONTROL EQUATIONS FOR PROTEIN SYNTHESIS IN THE GOODWIN MODEL

There exists a constant of motion G (x, y) of (2.1.4) given by

567

G(x, y) =-�89 2 + b[y - In(1 + y)] = G (2.1.5)

where G is a positive constant. We shall sometimes refer to G as the "energy" of the biochemical oscillator.

2.2 Allowed range of values of the dynamical variables. Since X, Y represent concentrations and p, q their equilibrium values we have

Clearly then, using (2.1.3).

where

X, Y,,p,q>O. (2.2.1)

x > - p (2.2.2a)

y > - z (2.2.2b)

kq >0. (2.2.3) l > Z = A + k q

Consider the critical values G (~) and G (2) of G given by

G(l) = cp2

G (2) = b [ - z - In (1 - z)].

Note that G (1), G (2) >0. If G > G (1) then, during the time development, x would assume values, for some times, violating the positivity constraint (2.2.2a). Similarly if G > G (2) then the constraint (2.2.2b) would be violated during the time development for some times. Such values of G would thus be clearly unphysical. We thus have the limitation

F > G

where r =min {�89 2, - b[z +In (1 -z)]}. (2.2.4)

Notice that (2.2.4), if satisfied, automatically implies (2.2.2). Besides, (2.2.4) imply not only the lower bounds on the variables X and y but also upper bounds. The existence of the upper bounds was not noticed earlier and will have important consequences.

568 VIRENDRA SINGH

2.3. Period of oscillation. For a given value of G (2.1.5) describes a closed curve in (x,y) space enclosing the origin. For all G > 0 this curve lies in the region of the (x,y) plane given by ~ > x > - ~ , ~ > y > - 1. The equation of motion (2.1.4) thus describe a periodic system for all positive G. Let the period of oscillation be denoted by T(G). Of course as noted in section 2.2 only the G values satisfying (2.2.4) are of physical interest.

Combining (2.1.4) and (2.1.5) we obtain

1 + b [ y - l n (1 +y) ] - G. I2.3.1)

We therefore have, following the standard classical dynamics,

T(G) =~c2c ~ x/G_b[ydYln(1-.,y)] (2.32)

where the integration is over a complete period. The sign of the square root in (2.3.2) is to be appropriately chosen for different values of y. More precisely

T ( G ) = ~ Jr n (1 + y))] 1/2

1 I r~''t~ dy (2.3.3) x~ ~ 3r [G - b(y - I n (1 +y))]1/2

jym~G ) [G - b ( y - In (1 + y))]~/'~" ' 4 ' :

(2.3.4)

The square roots involved in (2.3.3) and 2.3.4) are taken to be positive. Here y~2~(G ) > ym (G) are the two roots of the equation

b[y "~ - In (1 + y~O)] = G (i=1,2).

The expression (2.3.4), while exact, is not very useful. Since it is hard to obtain ym(G) and yt2)(G) and then to perform the necessary integration involved. We will thus have to transform (2.3.4) into something useful.

Using

6(bE-aE)=l [6(b-a)+6(b+a)], a>O,

CONTROL EQUATIONS FOR PROTEIN SYNTHESIS IN THE GOODWIN MODEL 569

where 6(...) is the Dirac delta function, we can reexpress the right hand side of (2.3.4) as the following double integral:

T(G)= dx clyOk-~-+b[y-ln(l + y)]-G oo 1

(2.3.5)

The limits of integration require some comment as it might appear that we allow "unphysical" x, y values, i.e. - p >__ x, - z > y > - 1, to contribute to the integral in (2.3.5). However, since the integrand is a delta function, the contribution from such "unphysicar' x, y values to the integral vanishes for F >=G.

In order to proceed further we use the following representation of the Dirac delta function,

6(z) = ~ - n d~e gz , - - O 0

to obtain

T(G) = d( dx -oo - m - 1

Letting ~ = ip,

. CX 2 d y e x p { l ( [ ~ - - + b ( y - l n ( 1

T(G)=2_~ ; o ~ dpe+Pe [ f S ~ dxe-P~2/z][ f T t

21r _f G\

q dy e- PbY(1 + y)pbj

(2.3.6)

Evaluating the x and y integrals in (2.3.6) we obtain

f (z)=2~z I+i~dpePZ[r(p).(2rt)-a/Zp-{P+ '/2)e+p] j - ioo

=~e- 1It'(p); z]. (2.3.7)

Thusf(z) is the inverse Laplace transform, 2 ' - 1, of the function

fo ~ dt tan- 1 (t/p)~ F(p)e +p 1 exp 2 e TM 1 J r ' ( p ) = , , / ~ - v " + ~/~ - p - " (2.3.8)

570 VIRENDRA SINGH

The power series expansion of f (z) around z = 0 can be easily worked out by using the asymptotic (p ~ oo) expansion of ~'(p) given by

PF(P) = exp [.L_- 1 2n(2-n 2i)-p ~ - - ' (2.3.%)

where B2. are the Bernoulli numbers given by

2(2n)[ ~ 1 Bo = 1, B 2 . - (2zr)2" k=l k2""

�9 Expanding the exponential in (2.3.9a) we obtain

p['(p) 1 ' a-L ' a_2 . a3 . = -I- P tp2tp3t...

where 1 1 139 571

a l = ] 2 ' a 2 = 2 - - ~ ' a 3 = 51840 ' a 4 - -2488320" '" (2.3.9b)

Calculating the inverse Laplace transform of i~(p) we obtain

�9 a l - - a 2 2 - - a 3 2.3 f ( z ) = l + ~ . z + ~ . z -t-~. + . . .

1 1 2 139 3 (2.3.10) = 1 + ] - ~ z + ~ z 31T0-40 z ...

The radius of convergence of this series expansion will be calculated in the next subsection (2.4) and will be shown to be

Iz[ < 2rt (2.3.11 )

2.4. Radius of convergence of the series expansion for f ( z ) around z =0. The radius of convergence of the series expansion forf(z) 'around z =0 would be determined by the position of singularity off(z) , regarded as an analytic function of the complex variable z, nearest to z =0. Now from the expression (2.3.2), on making the change of variables

b[y - In (1 + y)] - G sin 2 0,

CONTROL EQUATIONS FOR PROTEIN SYNTHESIS IN THE GOODW1N MODEL 571

we obtain

where

/~2n sin 0 (2.4.1)

y (0) - In (1 + y (0)) = z sin 20. (2.4.2)

Since y(O)-~x//~ sin 0 as z ~ O , f ( z ) has no singularity at z =0. Consider

- l n (1 + ~)=b/2 (2.4.3)

where ~ and r/are complex variables. We then have

_d(~) = +~/(1+~) (2.4.4) dn

We thus see that 4, regarded as a function of r/, is singular, whenever r = 0 (and ~@0)

i.e. �89 = _ In I = - 2nni, n = +_ 1, +_ 2 , . . .

We thus see that the singularity nearest to ~/= 0 for ~ is at

rl 2 = - 4 i n . (2.4.5)

Applying this result to the function y(O) we see that the series expansion of (1/y(O)) as a series in ~ sin 0 will converge as long as

Iz sin 01 < 2n.

Since 0 = n/2 is included in the range of integration the radius of convergence of the series expansion fo r f (z ) a round z = 0 is given by

Iz l<2n. (2.4.6)

For Iz} > 2n one has to evaluate the needed inverse Laplace transform of r'(p) differently using s tandard analysis.

572 VIRENDRA SINGH

2.5. Time averages. The technique, used in section 2,3 to calculate the time period T(G) for the biochemical oscillator is also convenient for calculating time averages. Consider, for example,

1 f~ (G) I""=(xm(1 +Y)") = T(G) Ix(t)]"[1 +y(t)]"dt. (2.5.1)

I n exactly the same fashion we can express I , . , . as the following phase space average, for F > G,

I m , . = r ( G ) -~o dx - ,

where

" r G) (2.5.2) dy x"(1 +y) fi\-~- + b[y - In (1 +y)] -

m + l o l + (_ l ),, r + i~ dp [- f 2 "~ [ 2b "~(,,-1)/2 F ( - ~ - - ) F (n + l + p )e - [

-- ~(~)~_,~ Lt~)t c ) ~<-+~,,~+.+, j~<o,~,

l + ( - t ) " 2

~ _ 1//i ~ , G\ t (2.5.3)

r, F(n+ l +p)e p

=i01( '+;)r(~) �9 (2.5.4)

By repeated use of the inverse Laplace transform relations given by

and

1 f+io~ = A(p)ePZdp, a(z)=~L~'-1[A(P)'Z] ~ -io~

CONTROL EQUATIONS FOR PROTEIN SYNTHESIS IN THE GOODWIN MODEL 573

One can express all lm,,'s in terms off(z) . For example

{'<~Ib) I f (z)dz

Io 1 = 1 + . ] o ' f(G/b)

i.e. (Y}=fS /b f ( z )dz / f (G/b ) (2.5.5)

3. Statistical mechanics of an assembly of biochemical oscillators. We shall now discuss the statistical mechanics of an assembly of oscillator each of which is described by equations of mot ion (2.1.1). A subscript will be used to distinguish variables referring to different oscillators.

The Gibbs ensemble distribution law is given by, for an assembly of v oscillators,

PGibb~ = e {~'- G,o0/O (3.1)

where

Gtot= L Gi(xi, Yl), (3.2) i=l

and

e- #/0 ~ Z ~-- f e - Gtot/~ (3.2)

dv=dxl . . .dxvdyl . . .dy~.

The statistical mechanics was considered by Goodwin himself. An account of the lower-bounds x i > -p i , y~ > - h was taken in his treatment by restricting the integration range of the variables to this region. However more powerful restriction on (xi, y~), following from a detailed consideration of the dynamics given earlier in section 2.2, and given by

Fi -> Gi(xi; Yi)

where

Fi = min 1 2 {~cipi, - bi[zi + In (1 - zi)]} (3.3)

574 VIRENDRA SINGH

were not taken into account in his treatment. Thus he had allowed the variables x~ and y~ to go to infinity though he also noted that such a procedure is not realistic. Because of the restriction (3.3), which subsume the conditions xi > -P i and Yi > -ri, the variables xi and Yi never become infinitely large. We now proceed to give the statistical mechanics of the system taking (3.3) into account.

We have

Z=Z1Z2,...Zv

where

fr e-a~tx~, ro/o dxi dyi (3.4) Zi= i~Gi(x,;yl)

= dxi dyie-Gitxi'Y~176 Gi(xi, Yi)) ,:r - - 1

(3.5)

where H(x) is the step function i.e. H(x)= 1 for x>O, and H(x)=O for x<O. Using

dH(x)_b(x) dx

we obtain

dZidFi - ff

=e-r,/0. T(Fi)

r,/o 2re , . { r A

It follows therefore that

f:'e-r/~ (F/b3dF (3.6)

CONTROL EQUATIONS FOR PROTEIN SYNTHESIS IN THE GOODWIN MODEL 575

We shall now evaluate Z; in various limits Case a: low 0 region. Consider the region 0 ,~ rain [F~, F z . . . . , Fv] = F~m.

In this case

~ c ~ [ o e-r/~f(F/b~tdr+O(e-V"~176

-,/b~. ~ 288 (b~? + " +O(e-~~176 (3.7)

Case b: high 0 region.

We then have, using (3.6),

Consider now the region

0~> max [F1, . . . , F~,]

Jof(r/b,)dr-~ fo Ff(r/b,)dr+... } One can similarly calculate various other quantities of interest.

(3.8)

L I T E R A T U R E

Goel, N. S. S. C. Maitra and E. W. Montroll, 1971. "On the Volterra and Other Nonlinear models of Interacting Populations." Rev. Mod. Phys., 43, 231 -276.

Goodwin, B. C. 1963. Temporal Organization in Cells, London and New York: Academic Press. Pavlidis, T. 1973. Biological Oscillators: Their Mathematical Analysis, London and New York:

Academic Press. Maynard Smith, J. 1968. Mathematical Ideas in Biology, London: Cambridge University Press.

RECEIVED 6-3-76