Translational Regulation (case 1)

CHAPTER II. Translational Regulation of rpoB Expression

5. Translational Regulation (case 1)

The simplest model of autogenous translational regulation is illustrated sche- matically in Figure 4. RN A polymerase binds specifically to a site in the vicinity of the Shine-Dalgarno sequence on rpoBC mRNA, thereby preventing translation of the message. Binding is assumed to have no effect on message stability. The equations governing expression must now be modified to reflect the different com-

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plexation states of rpoBC mRNA. Equation (2) becomes:

(7)

and equation (3) is replaced by:

(8)

and

(9)

where

[ri]o

and

[ri]i

are, respectively, the concentrations of unbound and bound rpoBC mRNA in length class i, and k1 (M-¹sec-¹⁾ and r1 (sec-^{1 )} are the rate constants for association and dissociation. Equation (9) applies to all length classes

(S.,.o

is taken to be zero), whereas equation (8) applies to all but the first. The equation governing [P]:

(10)

differs from equation (4) only in that fJ2 is now implicit due to the fact that the summation excludes bound mRN A.

Physical limitations should be considered before attempting to optimize the values of k1 and r1 • A theoretical upper bound on the rate constant for RN A polymerase/promoter association has been estimated by assuming the interaction to be controlled entirely by diffusion (von Hippel et al., 1984). The calculated value (10⁸ M-¹sec-^{1 )} is two orders of magnitude lower than experimentally determine~

apparent rate constants for strong promoters (Bujard et al., 1982; Chamberlin et al., 1982), suggesting that a process other than free diffusion, such as linear dif- fusion along DNA, is involved in promoter recognition. Linear diffusion of RNA

polymerase along rpoBC mRNA seems unlikely in that it would imply a general in- teraction between RN A polymerase and mRN A. Since the estimate of von Rippel et al. would not be greatly different if the binding site were on RNA instead of DNA, we will consider association rate constants with values exceeding (10⁸ M-¹sec-1)

to be physically unrealistic.

The rate constant for dissociation is a function of the binding energy, tight binding corresponding to a low r1 value. Rate constants for dissociation of RN A polymerase from strong promoters can be as low as 10-⁵ sec-¹ (Cech and McClure, 1980). An upper bound on r1 can be estimated from the velocity of translational motion of freely diffusing RNA polymerase and the approximate distance over which binding occurs. We estimate the nonbinding limit of r1 to be on the order of 10¹⁰ sec-1 • The value of r1 would have to be much lower than this for the interaction to be termed "binding" in the usual sense. We will use a conservative upper limit of 3 x 10

1

sec-

1

for r

1.

For the purpose of selecting optimal k1 and r1 values, we will initially assume that binding is in rapid equilibrium. H it proves impossible to approach the max- imum '1i value while satisfying this condition, the steady state equations can be used. ·when binding is equilibrated,

[r.i]1

is related to

[ri]o

by:

(11)

where K 1 is the equilibrium constant for binding. Defining fJ2 as the fraction of rpoBC mRNA that is unbound, we have:

(12) which can be substituted into equation (4) to describe [P]. The form of control function represented by equation (12) has only one similarity class because the only

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adjustable parameter, K 1 , is a coefficient of [ P1]. For this class W has the form:

(13) Here we see by inspection that for a given value of K1 , W asymptotically approaches 1 as [ P1] approaches infinity. For a given [ P1]

*,

optimal response would then be obtained when K1 is infinite.

The values of k1 and :r1 are restricted by the rapid equilibrium condition in addition to physical limitations. Binding equilibrium will be approached if both ki[P,]* and :r1 are much greater than the total degradative rate constant for the full-length transcript, 1.2 x 10-² sec-^{1 •} If k1 assumes its maximum value, 1 x 10⁸ M-¹sec-1 , then ki[P₁]*

=

^{1.3 x 101 ~} 1.2 x 10-² sec-1 . Conflicting demands arce, being placed on :r1• Rapid equilibrium requires that :r1 be large while optimal response requires that it be small, but since "large" and "small" are to be judged relative to different numbers we can attempt a compromise. If :r1 is taken to be 0.5 sec-¹ (substantially greater than 1.2 x 10-² sec-1 ), the equilibrium constant, K1 ,

has the value 2.0 x 10⁸ M-1 • Setting [P1] equal to [Pi]* in equation (13), we obtain W

=

0.96, which indicates that these k1 and :r₁ values result in a control stringency that approaches the theoretical maximum for equilibrium binding.

It is worth considering whether nonequilibrium binding might yield an even better response. Under steady state conditions, equation (9) can be written as:

(14)

where

[rh

is the total concentration of bound mRNA, and

61

is the average rate constant for degradation of bound mRNA. By rearranging we obtain:

(15)

This equation reduces to the form of the equilibrium relation (equation (11)) if r1

is the dominant term in the denominator. Comparison of these equations reveals that ifµ, and 61 are not negligible, they effectively reduce the value of K1 . That is, at steady state 82 and 'Ill' are of the same forms as those derived for equilibrium binding (equations (12) and (13)), the only difference being that K₁ is reduced by the factor rif (r1

+

^µ,

+ 6

1 ). This will necessarily reduce the value of 'Ill' vis-a-vis the rapid equilibrium value. Hence the maximum possible value of 'Ill' is 'i, and the kinetic values determined above can be used to compute the optimal response of the current model.

The rapid-equilibrium approximation was used only to optimize kinetic param- eters. Having accomplished this, we can solve the full set of equations ((7)-(10)) to obtain response curves. The results (Figure 5) indicate that addition of transla- tional regulation improves the control response significantly. Combined regulation results in a 50% reduction in [ P] displacement after one generation of recovery, and a 75% reduction after two generations. This is reflected in a diminished [r]

overshoot (Figure 5B).