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Parameters affecting the rate of synthesis of ribosomes and RNA polymerase in bacteria
I. theor. Biol. (1975) 53, 115-124
Parameters Affecting the Rate of Synthesis of Ribosomes and RNA Polymerase in Bacteria HANS BREMER The University of Texas at Dallas, P.O. Box 688, Richardson, Texas 75080, U.S.A. (Received 12 June 1974, and in revisedform
20 September 1974)
Ribosome synthesis in bacteria is linked to RNA polymerase synthesis; both synthesis rates depend upon the values of six parameters : (1) fraction of total ribosomes that is functioning, (2) fraction of total RNA polymerase
that is functioning, (3) fraction of functioning RNA polymerase engaged in rRNA synthesis, (4) fraction of total protein that is RNA polymerase protein, (5) peptide chain elongation rate, (6) rRNA chain elongation rate. If these parameters are constant in time, then the numbers of both ribosomes and RNA polymerase molecules increase exponentially. It is shown how the rate constant (fractional increase per unit of time) relates to these parameters and how the kinetics of ribosome and RNA poly-
merase synthesis respond to a change in any of these parameters.
1. Introduction While we have detailed knowledge about the control of the synthesis of many inducible and repressible enzymes in bacteria, little is known about the control of ribosome synthesis. The main reason for this ignorance of such an important cellular component is the complexity of the problem: the rate of ribosome synthesis is influenced by many parameters. This makes it difficult to find mutants defective in ribosomal control genes, since a change in any of the parameters affecting ribosome synthesis simulates a control gene mutant. The following analysis shows which parameters are important for ribosome synthesis. When this analysis is applied to recent observations on ribosome and RNA polymerase synthesis following a nutritional shift-up (Dennis & Bremer, 1974a,b; Matzura, Hansen & Zeuthen, 1973) it is found that ribosome synthesis and RNA polymerase function are controlled in a more complex manner than has been suspected on the basis of the observed simplicity of the post-shift ribosome synthesis kinetics. 115
116 (A)
1~.
RELATIONSHIP
BETWEEN
BREMER
RIBOSOME
AND
RNA
POLYMERASE
SYNTHESIS
Under normal growth conditions, essentially all ribosomal RNA is matched by ribosomal proteins, and the increase in the number of ribosomes per unit of time (gr = d2?Jdt) is determined by the rate of ribosomal RNA synthesis :t rRNA r;r, = __
synthesis rate (nut/rib)
where (nut/rib) = rRNA nucleotides per 70s ribosome. The rRNA synthesis rate is the product of several factors, one of which is the number1 of RNA polymerase molecules (NJ, such that fir = aN,.
(1)
The factor being Il/*CrP* a = (nucjrib) where $, = fraction of functioning RNA polymerase engaged in the synthesis of rRNA, c, = rRNA chain elongation rate, and &, = fraction of total RNA polymerase that is functioning. (Note that the product $, . c, . /?, . IV,, is the rate of ribosomal RNA synthesis.) On the other hand, the increase in the number of RNA polymerase molecules per unit of time (p(TP= diV,/dt) is proportional to the number of ribosomes (NJ : I$ = bN, (2) a*cpBr b=------(aalp where a,, = differential rate of RNA polymerase synthesis (= ratio polymerase protein synthesis rate/total protein synthesis rate), cP = peptide chain elongation rate, p, = fraction of total ribosomes that is functioning, and (aalpol) = amino acids per RNA polymerase molecule. (Note that the product cP. /3,.N, is the rate of protein synthesis.) Differentiating equation (1) with respect to time: Nr = afip,
(14
i It is also assumed that the turnover of rRNA is negligible. However, this assumption is not necessary if the parameters are appropriately redefined; in particular (~lr = fraction of RNA polymerase engaged in the synthesis of rRNA that becomes incorporated into stable ribosomes. 8 All numbers and rates per unit of culture volume.
RIBOSOME
substituting
SYNTHESIS
IN
BACTERIA
117
equation (2) for fi,: Ici, = abN,
(3)
and doing the analogous operation starting with equation (2): & = abN,
(4)
gives two symmetrical differential equations [(3) and (4)] which express the relationship between ribosome and RNA polymerase synthesis to be explored here. The validity of these equations does not depend on any assumptions (except that rRNA is matched by ribosomal protein) but follows from the definitions of the parameters used.$ The parameters a and b of these equations are not constants but vary with the bacterial growth rate (Table 1).
(B) GENERAL SOLUTION OF DIFFERENTIAL EQUATIONS DESCRIBING RIBOSOME AND RNA POLYMERASE SYNTHESIS
The solutions of the differential equations (3) and (4) are the kinetics of ribosome and RNA polymerase synthesis N,(t) and NJt). For these kinetics, zero time will be defined as the time at which one or both of the factors a and b of equations (1) and (2) might change, i.e. for t < 0 a = a,, b = b, ; for t > 0 a = a2, b = b2. Furthermore, the number of ribosomes and RNA polymerase molecules at zero time is set equal to N,(O) or N,(O), respectively. The general solutions are then: N,(t) = Nr(0){C.ekp2”‘f-(C-1) N,(t) = NP(0)(c’.ekP2i-(C’-1)
ewkpzf) e-kC2f)
(5) (6)
3 That these equations are in fact definitions which do not require experimental proof is seen as follows. The rRNA chain growth rate c, is defined: rate of rRNA synthesis ” = number of RNA polymerase molecules synthesizing rRNA’ The denominator of this equation is contained in the definition of the fraction yr of functioning RNA polymerase molecules synthesizing rRNA: number of RNA polymerase molecules synthesizing rRNA Yr = total number of functioning RNA poiymerase * The denominator of this latter equation in turn is contained in the definition of the fraction, j?,,,of total RNA polymerase molecules that is functioning: functioning RNA polymerase A = total RNA polymerase (= NJ Combining these definitions gives the rate of rRNA synthesis as the product ~rc,./?pN,, of equation (1).
118
H.
BREMER TABLE
1
Growth parametersfor E. coli B/r growing at three d@Eerentrates in succinate medium (p = 0.67 doublings per hour), glucose medium (CL= 1.36) and glucose-amino acids medium (19 an minus leucine; p = l-85) Parameter
succ. /I = 0.67
-_vsI. wa cr3 CP.h4 %l 5 A” fl? b* NlNp9
0.22 0.19 4400 574 o+lO57 0.38 0.066 089.10-3 8.6
Medium Glut. Jo= 1.36 0.39 0.33 5200 809 0.0078 0.39 0.142 1.7. IO-3 9.1
Glut-aa /l = 1.85 0.49 0.42 5900 809 0.0101 0.39 0.203 2.23. 1O-3 9.6
1 vS = fraction of functioning RNA polymerase engaged in the synthesis of stable RNA; succinate and glucose-value from Bremer, Berry & Dennis (1973); glu-aa value obtained by interpolation from Fig, 3 of Dennis & Bremer (1974c). 2 vZ = fraction of functioning RNA polymerase engaged in the synthesis of rRNA; vr = 0.85~~~ assuming the ratio rRNA/(rRNA + tRNA) to be 0.85 (in the growth rate range considered; see Dennis, 1972, and Discussion in Dennis & Bremer, 1974c). 3 c, = rRNA chain growth rate; from Dennis & Bremer, 1973; the value for ,U= 1.85 obtained by interpolation, in nucleotides/min per chain. * cp.B = product peptide chain growth rate times fraction of active ribosomes. This product has been defined as “ribosome efficiency” (eJ and has been calculated previously from measurements of the differential rate of ribosomal protein synthesis (olr; Dennis & Bremer, 1974a), in amino acid residues/min per ribosome. 5 a, = differential rate of RNA polymerase protein synthesis, from Fig. 3 of Dalbow (1973); the curve with the “background subtracted” was used since recent controls indicated that this is justified (Dalbow, unpublished; a similar background has also been observed and was subtracted in the a, measurements by Matzura et al., 1973). The glut-aa-value for CX~observed by Dalbow is plotted in his Fig. 3 at ,u = 2.1, which corresponds to the growth rate in a medium including leucine; the actual growth rate in the medium he used (minus leucine, for leu-labeling) is only 185 doublings/hr (Dalbow, personal communication), i.e. the same as observed by Dennis & Bremer (1974a) for their measurements of dcr (see footnote 4 above). 6 &, = fraction of total RNA polymerase that is functioning, calculated from the parameters above using equation (7) of the text: (nuc~rib)(aa/pol)ka~z 4720.3660. (In 2/60)2~2 2284.j~~ P, = y/r.C*.Olp.Cp.pp = = p,.c*.u*.c,.jr’ vr..cr.%.c,.~r This relationship, resolved for the product &ap (= differential synthesis rate of functional RNA polymerase), has been derived previously (Bremer et aZ., 1973) 7 a = parameter of equation (1) = vl,c,8,/4720. 8 b = parameter of equation (2) = a,cJ3,/3660. 9 NJN, = quotient ribosomes/RNA polymerase molecules = a/kp = b/b. These relationships are obtained from equations (1) or (2), respectively, by setting & = N,kp or &, = N,kp, respectively.
RIBOSOME
SYNTHESIS
IN
119
BACTERIA
where
c-l -2
%!!L+l [ alp2
I
cf+[~+l]
lp1 = k 2/a, b, (doublings/hr)
= growth rate prior to t,
lcl2 = k da, b2 (doublings/hr)
= growth rate after t,
k = In 2/60 (hr/min)
if the chain elongation rates c, and cP in equations (1) and (2) are given in nucleotides/min, or aa/min, respectively.
(C)
EXPONENTIAL
GROWTH
AS A PARTICULAR
DIFFERENTIAL
SOLUTION
OF
THE
EQUATIONS
First, we explore the case that both parameters a and b of equations (1) and (2) do not change at zero time (a 1 = a,; bl = b,). In this case (Table 2, no. l), C = C’ = 1 for equations (5) and (6) and the terms with the negative exponents disappear; thus, (54 KW = K(O) ekfit for t , o N,(t) = N,(O) ekp’ < ’ @a) These relationships define the exponential growth of the system, i.e. p1 = ,u2 = p: before and after zero time both numbers of ribosomes and RNA polymerase molecules increase at the same exponential rate given (in doubiings/hr) by tk.C,.B,.~,.C,.Br (7) 2 J (nuc/rib)(aa/pol) ’ The steady-state growth rate of a bacterial culture is thus determined by eight parameters, six of which might potentially vary. Under many conditions of growth the main variation occurs for r+krand rxP, which reflect the gene activities for ribosomal RNA genes and for RNA polymerase genes, respectively. Thus, ribosomal RNA synthesis is largely determined by the product $IclP. Equation (7) also expresses the well-known notion that the bacterial growth rate is determined by the rate of ribosome synthesis (Maaloe & Kjeldgaard, 1966). The chain growth rates of ribosomal RNA and protein (cr, c,) increase somewhat with growth rate (c,: Dennis & Bremer, 1973 ; cI, : Young, unpublished), but these are second order phenomena (presumably p+&!?
Ill
120
H.
BREMER
not under the influence of a specific control). The fraction of ribosomes and polymerase that are active (B,, &) seem to vary little with growth rate (/$: Forchhammer & Lindahl, 1971; &: Table 1). The validity of equation (7) depends on the assumption made at the beginning of the preceding section, that the synthesis of ribosomal protein matches the synthesis of ribosomal RNA, which normally is the case unless protein synthesis is severely inhibited (e.g. during amino acid starvation). Besides having a theoretical significance which lies in the fact that it shows which parameters determine the growth rate, equation (7) has a practical significance which lies in the possibility of calculating any of these parameters; for example, & (Table 1). Substituting values observed in Escherichia coli B/r for the parameters in equation (7) shows that a and b increase nearly in proportion to p and that the bacteria contain about nine ribosomes for every RNA polymerase molecule, nearly independent of p (Table 1). (D)
RIBOSOME INCREASE
AND IN
RNA
POLYMERASE
THE
GROWTH
SYNTHESIS PARAMETERS
FOLLOWING
AN
(SHIFT-UP)
Three further solutions of the differential equations (3) and (4) result when, at t = 0, the parameters a or b or both increase (factory; Table 2, cases 24). The kinetics of ribosome and RNA polymerase synthesis for these cases are illustrated in Fig. 1, assuming f = 3. If parameter a increases, then the rate of ribosome synthesis increases immediately (f-fold; case 2 and 4), but if only parameter b increases (case 3), there follows a delayed increase in the ribosome synthesis rate. RNA polymerase synthesis responds oppositely: immediately for b- and after a delay for a-increase [Fig. l(b)]. If both a and b increase simultaneously and by the same factor (case 4), then both ribosome and RNA polymerase synthesis respond with an immediate increase in the exponential rate and the post-shift rate constant remains unchanged. The implications of these relations for the interpretation of a nutritional shift-up are discussed below. 2. Discussion (A)
RIBOSOME
CONTROL
The rate of ribosome synthesis was found to be influenced by two parameters [a and b of equation (3)], each being the product of three variables which are a function of the composition of the growth medium ($,, cI, &; c$, c,,, j$). These six variables are influenced by other cellular parameters: e.g. the fraction of functioning ribosomes (83 may be limited by the amount of mRNA, by ribosomal factors, or by ribosome turnover; the rRNA chain elongation rate (c,) depends upon RNA polymerase properties and RNA
RIBOSOME
SYNTHESIS TABLE
IN
121
BACTERIA
2
Changes of exponential growth rate, ribosome synthesis rate and RNA polymerase synthesisrate as a result of changinggrowth parameters
Case No.
Growth parameters1 for t > 0 a2
ba
1
al
2
fal
3
a1
fbl
4
fa1
fbl
Parameters in equations (5) and (6) 2 c
C’
f($+1)5(4y+1) 1
1
Change Change of of exponential ribosome synth. growth ratea rate 4 at t=o Pz
Change of RNA polym. synth. rate5 att=O
Q/h
1
f
fib
f
f
1 For definition of growth parameters a and b see equations (1) and (2). Before zero time (t < 0) a = aI, b = bl; after zero time (t > 0) a = aZ, b = bz. At zero time neither parameter, or only a, or only b, or both CIand b are assumed to increase by the factor f (case No. 1, 2, 3 or 4, respectively). 2 For definition of C and C’ see equations (5) and (6). These parameters define the kinetics of ribosome and RNA polymerase synthesis after zero time. 8 Exponential growth rate (.LJ~,in doubBngs/hr) for t > 0; for t < 0 the growth rate is fll. * Change in the rate of ribosome synthesis at zero time, defined: fir(+ 0) dN,(+ t -+ 0)/d %gg= dN,( - t -+ O)/dt’ The numerator and denominator of this quotient are obtained by differentiation of equations (5) and (5a), respectively, and setting t = 0. A unity value means there is no immediate change; fmeans f-fold increase in the rate. 5 Change in the rate of RNA polymerase synthesis at zero time, defined: @rl+ 0) zzzd&(-t ____- t --f Wdt 7N,(-0) diV,(- t -+ O)/dt obtained by differentiating equations (6) and (6a), respectively.
precursor metabolism. A change in any of these parameters as a result of mutation can be expected to alter the pattern of ribosome synthesis. Furthermore, a primary change in one parameter may lead to adjustments or secondary changes in other parameters as a result of normal control. For example, a ribosomal protein mutant with a reduced efficiency of ribosomes in protein synthesis may show an increased synthesis rate of ribosomes (increased $, or a,) if the reduced consumption of amino acidsby the mutant
H.
BREMER
Time
FIG. 1. Kinetics of ribosome (a) and RNA polymerase (b) synthesis following a threefold increase at zero time in parameter n (curve 2), b (curve 3) or both a and b (curve 4). For dehnition of parameters a and b see equations (1) and (2). The kinetics were calculated using equations (5) or (9, respectively. The numbers of ribosomes and RNA polymerase molecules at zero time are set at unity; the pre-shift growth rate is set at 1 doubling/unit of time. The curves l-4 correspond to case l-4 of Table 2.
causes an “internal shift-up”. The preceding analysis can be used to characterize such mutants, which are being encountered in the search for ribosome control genes. (B)
NUTRITIONAL
SHIFT-UP
Following a nutritional shift-up (addition of nutritional supplements to the growth medium), the exponential rate of ribosome accumulation (ribosome doublingsihr) increases within a few minutes to the higher, post-shift steady-state value [Maaloe & Kjeldgaard, 1966; Schleif, 1967; Dennis & Bremer, 1974b; illustrated by curve 4 of Fig. l(a)]. Such kinetics result when, at the moment of the shift, both parameters a and b [equations (3), (5)] increase by the same factor (Table 1, case 4; Fig. 1). This implies that, at the shift time, both ribosome and RNA polymerase synthesis increase in a parallel manner (Table 1: case 4). The latter expectation, however, is not supported by observational evidence. While $, and c, of parameter a were found to increase within 5 min after the shift to their new steady-state values (Dennis & Bremer, 1974b), which accounts for the initial increase in the
RIBOSOME
SYNTHESIS
IN
123
BACTERIA
ribosome synthesis rate [curves 2 and 4 of Fig. l(a)], C(~of parameter b was found to increase to the post-shift steady-state value only after a delay of 20-60 min (Matzura et al., 1973). Thus, parameter b is a function of postshift time, such that equations (5) and (6), derived under the assumption that a and b are constants (or vary only by a step function) are not the actual solutions of the differential equations (3) and (4) following a shift-up. If parameter a increases immediately (step-wise), but parameter b increases only gradually after the shift, then the post-shift rate constant for ribosome synthesis can be expected after the initial increase, to temporarily decrease some time after the shift. This is not observed, suggesting that there occurs a compensatory temporary increase in parameter a ($, or p,) some time after the shift. Apparently, further measurements of the post-shift kinetic changes of $, and &,, along with ap and j?, are required for our understanding of the control of ribosome synthesis following a shift-up. This
work
was
supported by P.S.H. grant GM 15142. REFERENCES
BREMER, H., BERRY, L. & DENNIS, P. (1973). J. molec. Biol. 75, 161. DALBOW, D. G. (1973). J. mlec. Biol. 75, 181. DENNIS, P. (1972). J. biol. Chem. 247,2842. DENNIS, P. & BREMER, H. (1973). J. molec. BioZ. 75, 145. DENNIS, P. & BREMER, H. (1974~). J. molec. BioZ. 84,407. DENNIS, P. & BREMER, H. (19743). J. moZec.BZoZ. 89, 233. DENNIS, P. & BREMER, H. (1974~). J. Bacr. 119,270. FORCHHAMMER, J. & LINDAHL, L. (1971). J. molec. BioZ. 55,563. MAALOE, 0. & KJELDGAARD, N. (1966). Control of Macromolecular Synthesis. Benjamin. MATZURA, H., HANSEN, B. S. & ZEUTHEN, J. (1973). J. molec. Biol. 74, 9. SCHLEIF, R. (1967). J. molec. Biol. 26, 41.
New
York
Note added in proof
Recently two papers appeared which are relevant for the calculation of parameters a and b in Table 1. (1) Iwakura, Ito & Ishihama (1974, Molec. gen. Genet. 133, 1) reported that crude lysates of E. coli contain a polypeptide x which electrophoretically co-migrates with the RNA polymerase p’ protein, such that the value for ap measured in crude lysates can be assumed to be overestimated. (2) Oeschger & Berlyn (1975, Proc. natn. Acad. Sci. U.S.A. 12, 911) reported the isolation of a ts mutant of E. coli which does not synthesize RNA polymerase /I- and 8’ subunits at 43”; neither does it synthesize at 43” any other detectable protein, like x, migrating to the /I or /Y-position during electrophoresis. The mutant is closely linked to the rif locus; its nature is not yet known (e.g. whether it is within a structural RNA polymerase gene, or within a regulator gene for RNA polymerase). The two
124
H.
BREMER
reports do not contradict one another if either X-protein is co-regulated with RNA polymerase fi and j?’ protein and the mutant is in a regulator gene for RNA polymerase or if X-protein is actually a modified B or /3’ RNA polymerase subunit. In the first case, the RNA polymerase values in Table 1 are overestimated by a constant factor, i.e. qualitatively the conclusions from those data (e.g. the constancy of the ratio ribosomes: RNA polymerase molecules) would still hold. In the second case, the data in Table 1 need not be corrected; further, the x protein would provide an explanation for the inactivity of a large fraction of RNA polymerase (i.e. if X-protein were an inactive RNA polymerase subunit). Iwakura et al. interpreted the curve describing CQ, (excluding x) as a function of p (their Fig. 9) as a straight line. Actually, their data fit better with the interpretation proposed here, namely that the g-curve parallels the curve described M, as a function of h (Fig. 4 of Dennis & Bremer, 1974a) which implies a constant ratio ribosomes : RNA poly, merase molecules.
Parameters Affecting the Rate of Synthesis of Ribosomes and RNA Polymerase in Bacteria HANS BREMER The University of Texas at Dallas, P.O. Box 688, Richardson, Texas 75080, U.S.A. (Received 12 June 1974, and in revisedform
20 September 1974)
Ribosome synthesis in bacteria is linked to RNA polymerase synthesis; both synthesis rates depend upon the values of six parameters : (1) fraction of total ribosomes that is functioning, (2) fraction of total RNA polymerase
that is functioning, (3) fraction of functioning RNA polymerase engaged in rRNA synthesis, (4) fraction of total protein that is RNA polymerase protein, (5) peptide chain elongation rate, (6) rRNA chain elongation rate. If these parameters are constant in time, then the numbers of both ribosomes and RNA polymerase molecules increase exponentially. It is shown how the rate constant (fractional increase per unit of time) relates to these parameters and how the kinetics of ribosome and RNA poly-
merase synthesis respond to a change in any of these parameters.
1. Introduction While we have detailed knowledge about the control of the synthesis of many inducible and repressible enzymes in bacteria, little is known about the control of ribosome synthesis. The main reason for this ignorance of such an important cellular component is the complexity of the problem: the rate of ribosome synthesis is influenced by many parameters. This makes it difficult to find mutants defective in ribosomal control genes, since a change in any of the parameters affecting ribosome synthesis simulates a control gene mutant. The following analysis shows which parameters are important for ribosome synthesis. When this analysis is applied to recent observations on ribosome and RNA polymerase synthesis following a nutritional shift-up (Dennis & Bremer, 1974a,b; Matzura, Hansen & Zeuthen, 1973) it is found that ribosome synthesis and RNA polymerase function are controlled in a more complex manner than has been suspected on the basis of the observed simplicity of the post-shift ribosome synthesis kinetics. 115
116 (A)
1~.
RELATIONSHIP
BETWEEN
BREMER
RIBOSOME
AND
RNA
POLYMERASE
SYNTHESIS
Under normal growth conditions, essentially all ribosomal RNA is matched by ribosomal proteins, and the increase in the number of ribosomes per unit of time (gr = d2?Jdt) is determined by the rate of ribosomal RNA synthesis :t rRNA r;r, = __
synthesis rate (nut/rib)
where (nut/rib) = rRNA nucleotides per 70s ribosome. The rRNA synthesis rate is the product of several factors, one of which is the number1 of RNA polymerase molecules (NJ, such that fir = aN,.
(1)
The factor being Il/*CrP* a = (nucjrib) where $, = fraction of functioning RNA polymerase engaged in the synthesis of rRNA, c, = rRNA chain elongation rate, and &, = fraction of total RNA polymerase that is functioning. (Note that the product $, . c, . /?, . IV,, is the rate of ribosomal RNA synthesis.) On the other hand, the increase in the number of RNA polymerase molecules per unit of time (p(TP= diV,/dt) is proportional to the number of ribosomes (NJ : I$ = bN, (2) a*cpBr b=------(aalp where a,, = differential rate of RNA polymerase synthesis (= ratio polymerase protein synthesis rate/total protein synthesis rate), cP = peptide chain elongation rate, p, = fraction of total ribosomes that is functioning, and (aalpol) = amino acids per RNA polymerase molecule. (Note that the product cP. /3,.N, is the rate of protein synthesis.) Differentiating equation (1) with respect to time: Nr = afip,
(14
i It is also assumed that the turnover of rRNA is negligible. However, this assumption is not necessary if the parameters are appropriately redefined; in particular (~lr = fraction of RNA polymerase engaged in the synthesis of rRNA that becomes incorporated into stable ribosomes. 8 All numbers and rates per unit of culture volume.
RIBOSOME
substituting
SYNTHESIS
IN
BACTERIA
117
equation (2) for fi,: Ici, = abN,
(3)
and doing the analogous operation starting with equation (2): & = abN,
(4)
gives two symmetrical differential equations [(3) and (4)] which express the relationship between ribosome and RNA polymerase synthesis to be explored here. The validity of these equations does not depend on any assumptions (except that rRNA is matched by ribosomal protein) but follows from the definitions of the parameters used.$ The parameters a and b of these equations are not constants but vary with the bacterial growth rate (Table 1).
(B) GENERAL SOLUTION OF DIFFERENTIAL EQUATIONS DESCRIBING RIBOSOME AND RNA POLYMERASE SYNTHESIS
The solutions of the differential equations (3) and (4) are the kinetics of ribosome and RNA polymerase synthesis N,(t) and NJt). For these kinetics, zero time will be defined as the time at which one or both of the factors a and b of equations (1) and (2) might change, i.e. for t < 0 a = a,, b = b, ; for t > 0 a = a2, b = b2. Furthermore, the number of ribosomes and RNA polymerase molecules at zero time is set equal to N,(O) or N,(O), respectively. The general solutions are then: N,(t) = Nr(0){C.ekp2”‘f-(C-1) N,(t) = NP(0)(c’.ekP2i-(C’-1)
ewkpzf) e-kC2f)
(5) (6)
3 That these equations are in fact definitions which do not require experimental proof is seen as follows. The rRNA chain growth rate c, is defined: rate of rRNA synthesis ” = number of RNA polymerase molecules synthesizing rRNA’ The denominator of this equation is contained in the definition of the fraction yr of functioning RNA polymerase molecules synthesizing rRNA: number of RNA polymerase molecules synthesizing rRNA Yr = total number of functioning RNA poiymerase * The denominator of this latter equation in turn is contained in the definition of the fraction, j?,,,of total RNA polymerase molecules that is functioning: functioning RNA polymerase A = total RNA polymerase (= NJ Combining these definitions gives the rate of rRNA synthesis as the product ~rc,./?pN,, of equation (1).
118
H.
BREMER TABLE
1
Growth parametersfor E. coli B/r growing at three d@Eerentrates in succinate medium (p = 0.67 doublings per hour), glucose medium (CL= 1.36) and glucose-amino acids medium (19 an minus leucine; p = l-85) Parameter
succ. /I = 0.67
-_vsI. wa cr3 CP.h4 %l 5 A” fl? b* NlNp9
0.22 0.19 4400 574 o+lO57 0.38 0.066 089.10-3 8.6
Medium Glut. Jo= 1.36 0.39 0.33 5200 809 0.0078 0.39 0.142 1.7. IO-3 9.1
Glut-aa /l = 1.85 0.49 0.42 5900 809 0.0101 0.39 0.203 2.23. 1O-3 9.6
1 vS = fraction of functioning RNA polymerase engaged in the synthesis of stable RNA; succinate and glucose-value from Bremer, Berry & Dennis (1973); glu-aa value obtained by interpolation from Fig, 3 of Dennis & Bremer (1974c). 2 vZ = fraction of functioning RNA polymerase engaged in the synthesis of rRNA; vr = 0.85~~~ assuming the ratio rRNA/(rRNA + tRNA) to be 0.85 (in the growth rate range considered; see Dennis, 1972, and Discussion in Dennis & Bremer, 1974c). 3 c, = rRNA chain growth rate; from Dennis & Bremer, 1973; the value for ,U= 1.85 obtained by interpolation, in nucleotides/min per chain. * cp.B = product peptide chain growth rate times fraction of active ribosomes. This product has been defined as “ribosome efficiency” (eJ and has been calculated previously from measurements of the differential rate of ribosomal protein synthesis (olr; Dennis & Bremer, 1974a), in amino acid residues/min per ribosome. 5 a, = differential rate of RNA polymerase protein synthesis, from Fig. 3 of Dalbow (1973); the curve with the “background subtracted” was used since recent controls indicated that this is justified (Dalbow, unpublished; a similar background has also been observed and was subtracted in the a, measurements by Matzura et al., 1973). The glut-aa-value for CX~observed by Dalbow is plotted in his Fig. 3 at ,u = 2.1, which corresponds to the growth rate in a medium including leucine; the actual growth rate in the medium he used (minus leucine, for leu-labeling) is only 185 doublings/hr (Dalbow, personal communication), i.e. the same as observed by Dennis & Bremer (1974a) for their measurements of dcr (see footnote 4 above). 6 &, = fraction of total RNA polymerase that is functioning, calculated from the parameters above using equation (7) of the text: (nuc~rib)(aa/pol)ka~z 4720.3660. (In 2/60)2~2 2284.j~~ P, = y/r.C*.Olp.Cp.pp = = p,.c*.u*.c,.jr’ vr..cr.%.c,.~r This relationship, resolved for the product &ap (= differential synthesis rate of functional RNA polymerase), has been derived previously (Bremer et aZ., 1973) 7 a = parameter of equation (1) = vl,c,8,/4720. 8 b = parameter of equation (2) = a,cJ3,/3660. 9 NJN, = quotient ribosomes/RNA polymerase molecules = a/kp = b/b. These relationships are obtained from equations (1) or (2), respectively, by setting & = N,kp or &, = N,kp, respectively.
RIBOSOME
SYNTHESIS
IN
119
BACTERIA
where
c-l -2
%!!L+l [ alp2
I
cf+[~+l]
lp1 = k 2/a, b, (doublings/hr)
= growth rate prior to t,
lcl2 = k da, b2 (doublings/hr)
= growth rate after t,
k = In 2/60 (hr/min)
if the chain elongation rates c, and cP in equations (1) and (2) are given in nucleotides/min, or aa/min, respectively.
(C)
EXPONENTIAL
GROWTH
AS A PARTICULAR
DIFFERENTIAL
SOLUTION
OF
THE
EQUATIONS
First, we explore the case that both parameters a and b of equations (1) and (2) do not change at zero time (a 1 = a,; bl = b,). In this case (Table 2, no. l), C = C’ = 1 for equations (5) and (6) and the terms with the negative exponents disappear; thus, (54 KW = K(O) ekfit for t , o N,(t) = N,(O) ekp’ < ’ @a) These relationships define the exponential growth of the system, i.e. p1 = ,u2 = p: before and after zero time both numbers of ribosomes and RNA polymerase molecules increase at the same exponential rate given (in doubiings/hr) by tk.C,.B,.~,.C,.Br (7) 2 J (nuc/rib)(aa/pol) ’ The steady-state growth rate of a bacterial culture is thus determined by eight parameters, six of which might potentially vary. Under many conditions of growth the main variation occurs for r+krand rxP, which reflect the gene activities for ribosomal RNA genes and for RNA polymerase genes, respectively. Thus, ribosomal RNA synthesis is largely determined by the product $IclP. Equation (7) also expresses the well-known notion that the bacterial growth rate is determined by the rate of ribosome synthesis (Maaloe & Kjeldgaard, 1966). The chain growth rates of ribosomal RNA and protein (cr, c,) increase somewhat with growth rate (c,: Dennis & Bremer, 1973 ; cI, : Young, unpublished), but these are second order phenomena (presumably p+&!?
Ill
120
H.
BREMER
not under the influence of a specific control). The fraction of ribosomes and polymerase that are active (B,, &) seem to vary little with growth rate (/$: Forchhammer & Lindahl, 1971; &: Table 1). The validity of equation (7) depends on the assumption made at the beginning of the preceding section, that the synthesis of ribosomal protein matches the synthesis of ribosomal RNA, which normally is the case unless protein synthesis is severely inhibited (e.g. during amino acid starvation). Besides having a theoretical significance which lies in the fact that it shows which parameters determine the growth rate, equation (7) has a practical significance which lies in the possibility of calculating any of these parameters; for example, & (Table 1). Substituting values observed in Escherichia coli B/r for the parameters in equation (7) shows that a and b increase nearly in proportion to p and that the bacteria contain about nine ribosomes for every RNA polymerase molecule, nearly independent of p (Table 1). (D)
RIBOSOME INCREASE
AND IN
RNA
POLYMERASE
THE
GROWTH
SYNTHESIS PARAMETERS
FOLLOWING
AN
(SHIFT-UP)
Three further solutions of the differential equations (3) and (4) result when, at t = 0, the parameters a or b or both increase (factory; Table 2, cases 24). The kinetics of ribosome and RNA polymerase synthesis for these cases are illustrated in Fig. 1, assuming f = 3. If parameter a increases, then the rate of ribosome synthesis increases immediately (f-fold; case 2 and 4), but if only parameter b increases (case 3), there follows a delayed increase in the ribosome synthesis rate. RNA polymerase synthesis responds oppositely: immediately for b- and after a delay for a-increase [Fig. l(b)]. If both a and b increase simultaneously and by the same factor (case 4), then both ribosome and RNA polymerase synthesis respond with an immediate increase in the exponential rate and the post-shift rate constant remains unchanged. The implications of these relations for the interpretation of a nutritional shift-up are discussed below. 2. Discussion (A)
RIBOSOME
CONTROL
The rate of ribosome synthesis was found to be influenced by two parameters [a and b of equation (3)], each being the product of three variables which are a function of the composition of the growth medium ($,, cI, &; c$, c,,, j$). These six variables are influenced by other cellular parameters: e.g. the fraction of functioning ribosomes (83 may be limited by the amount of mRNA, by ribosomal factors, or by ribosome turnover; the rRNA chain elongation rate (c,) depends upon RNA polymerase properties and RNA
RIBOSOME
SYNTHESIS TABLE
IN
121
BACTERIA
2
Changes of exponential growth rate, ribosome synthesis rate and RNA polymerase synthesisrate as a result of changinggrowth parameters
Case No.
Growth parameters1 for t > 0 a2
ba
1
al
2
fal
3
a1
fbl
4
fa1
fbl
Parameters in equations (5) and (6) 2 c
C’
f($+1)5(4y+1) 1
1
Change Change of of exponential ribosome synth. growth ratea rate 4 at t=o Pz
Change of RNA polym. synth. rate5 att=O
Q/h
1
f
fib
f
f
1 For definition of growth parameters a and b see equations (1) and (2). Before zero time (t < 0) a = aI, b = bl; after zero time (t > 0) a = aZ, b = bz. At zero time neither parameter, or only a, or only b, or both CIand b are assumed to increase by the factor f (case No. 1, 2, 3 or 4, respectively). 2 For definition of C and C’ see equations (5) and (6). These parameters define the kinetics of ribosome and RNA polymerase synthesis after zero time. 8 Exponential growth rate (.LJ~,in doubBngs/hr) for t > 0; for t < 0 the growth rate is fll. * Change in the rate of ribosome synthesis at zero time, defined: fir(+ 0) dN,(+ t -+ 0)/d %gg= dN,( - t -+ O)/dt’ The numerator and denominator of this quotient are obtained by differentiation of equations (5) and (5a), respectively, and setting t = 0. A unity value means there is no immediate change; fmeans f-fold increase in the rate. 5 Change in the rate of RNA polymerase synthesis at zero time, defined: @rl+ 0) zzzd&(-t ____- t --f Wdt 7N,(-0) diV,(- t -+ O)/dt obtained by differentiating equations (6) and (6a), respectively.
precursor metabolism. A change in any of these parameters as a result of mutation can be expected to alter the pattern of ribosome synthesis. Furthermore, a primary change in one parameter may lead to adjustments or secondary changes in other parameters as a result of normal control. For example, a ribosomal protein mutant with a reduced efficiency of ribosomes in protein synthesis may show an increased synthesis rate of ribosomes (increased $, or a,) if the reduced consumption of amino acidsby the mutant
H.
BREMER
Time
FIG. 1. Kinetics of ribosome (a) and RNA polymerase (b) synthesis following a threefold increase at zero time in parameter n (curve 2), b (curve 3) or both a and b (curve 4). For dehnition of parameters a and b see equations (1) and (2). The kinetics were calculated using equations (5) or (9, respectively. The numbers of ribosomes and RNA polymerase molecules at zero time are set at unity; the pre-shift growth rate is set at 1 doubling/unit of time. The curves l-4 correspond to case l-4 of Table 2.
causes an “internal shift-up”. The preceding analysis can be used to characterize such mutants, which are being encountered in the search for ribosome control genes. (B)
NUTRITIONAL
SHIFT-UP
Following a nutritional shift-up (addition of nutritional supplements to the growth medium), the exponential rate of ribosome accumulation (ribosome doublingsihr) increases within a few minutes to the higher, post-shift steady-state value [Maaloe & Kjeldgaard, 1966; Schleif, 1967; Dennis & Bremer, 1974b; illustrated by curve 4 of Fig. l(a)]. Such kinetics result when, at the moment of the shift, both parameters a and b [equations (3), (5)] increase by the same factor (Table 1, case 4; Fig. 1). This implies that, at the shift time, both ribosome and RNA polymerase synthesis increase in a parallel manner (Table 1: case 4). The latter expectation, however, is not supported by observational evidence. While $, and c, of parameter a were found to increase within 5 min after the shift to their new steady-state values (Dennis & Bremer, 1974b), which accounts for the initial increase in the
RIBOSOME
SYNTHESIS
IN
123
BACTERIA
ribosome synthesis rate [curves 2 and 4 of Fig. l(a)], C(~of parameter b was found to increase to the post-shift steady-state value only after a delay of 20-60 min (Matzura et al., 1973). Thus, parameter b is a function of postshift time, such that equations (5) and (6), derived under the assumption that a and b are constants (or vary only by a step function) are not the actual solutions of the differential equations (3) and (4) following a shift-up. If parameter a increases immediately (step-wise), but parameter b increases only gradually after the shift, then the post-shift rate constant for ribosome synthesis can be expected after the initial increase, to temporarily decrease some time after the shift. This is not observed, suggesting that there occurs a compensatory temporary increase in parameter a ($, or p,) some time after the shift. Apparently, further measurements of the post-shift kinetic changes of $, and &,, along with ap and j?, are required for our understanding of the control of ribosome synthesis following a shift-up. This
work
was
supported by P.S.H. grant GM 15142. REFERENCES
BREMER, H., BERRY, L. & DENNIS, P. (1973). J. molec. Biol. 75, 161. DALBOW, D. G. (1973). J. mlec. Biol. 75, 181. DENNIS, P. (1972). J. biol. Chem. 247,2842. DENNIS, P. & BREMER, H. (1973). J. molec. BioZ. 75, 145. DENNIS, P. & BREMER, H. (1974~). J. molec. BioZ. 84,407. DENNIS, P. & BREMER, H. (19743). J. moZec.BZoZ. 89, 233. DENNIS, P. & BREMER, H. (1974~). J. Bacr. 119,270. FORCHHAMMER, J. & LINDAHL, L. (1971). J. molec. BioZ. 55,563. MAALOE, 0. & KJELDGAARD, N. (1966). Control of Macromolecular Synthesis. Benjamin. MATZURA, H., HANSEN, B. S. & ZEUTHEN, J. (1973). J. molec. Biol. 74, 9. SCHLEIF, R. (1967). J. molec. Biol. 26, 41.
New
York
Note added in proof
Recently two papers appeared which are relevant for the calculation of parameters a and b in Table 1. (1) Iwakura, Ito & Ishihama (1974, Molec. gen. Genet. 133, 1) reported that crude lysates of E. coli contain a polypeptide x which electrophoretically co-migrates with the RNA polymerase p’ protein, such that the value for ap measured in crude lysates can be assumed to be overestimated. (2) Oeschger & Berlyn (1975, Proc. natn. Acad. Sci. U.S.A. 12, 911) reported the isolation of a ts mutant of E. coli which does not synthesize RNA polymerase /I- and 8’ subunits at 43”; neither does it synthesize at 43” any other detectable protein, like x, migrating to the /I or /Y-position during electrophoresis. The mutant is closely linked to the rif locus; its nature is not yet known (e.g. whether it is within a structural RNA polymerase gene, or within a regulator gene for RNA polymerase). The two
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H.
BREMER
reports do not contradict one another if either X-protein is co-regulated with RNA polymerase fi and j?’ protein and the mutant is in a regulator gene for RNA polymerase or if X-protein is actually a modified B or /3’ RNA polymerase subunit. In the first case, the RNA polymerase values in Table 1 are overestimated by a constant factor, i.e. qualitatively the conclusions from those data (e.g. the constancy of the ratio ribosomes: RNA polymerase molecules) would still hold. In the second case, the data in Table 1 need not be corrected; further, the x protein would provide an explanation for the inactivity of a large fraction of RNA polymerase (i.e. if X-protein were an inactive RNA polymerase subunit). Iwakura et al. interpreted the curve describing CQ, (excluding x) as a function of p (their Fig. 9) as a straight line. Actually, their data fit better with the interpretation proposed here, namely that the g-curve parallels the curve described M, as a function of h (Fig. 4 of Dennis & Bremer, 1974a) which implies a constant ratio ribosomes : RNA poly, merase molecules.