J. theor. Biol. (1976) 59, 107-126

Adaptation in Micro-organisms: Variation in Macromolecular Composition with Growth Rate U. N. SrNGH

Molecular Biology Unit Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400005, India (Received 18 November 1974, and in revised form 11 June 1975) A model of bacterial population maintained in a steady state of sustained exponential growth is developed. The analysis is based essentially on a stochastic model of protein synthesis on labile mRNA's described in earlier communications. Two basic postulates: (1) co-ordinate synthesis of ribosomal proteins and (2) free ribosomal proteins in the pool functioning as a positive control element in the synthesis of rRNA provide the necessary framework for biogenesis of ribosomes. Macromolecular compositions of bacterial cells, with respect to RNA and protein contents at a gross level, are defined in terms of various kinetic parameters related to protein synthesis and maturation of ribosomes. Three alternative modes of regulation operating at the level of (i) functional efficiency of ribosomes, (ii) transcription of rRNA and (iii) differential synthesis rate of ribosomal proteins have been analysed. It is concluded that while the observed variations in macromolecular compositions of bacterial cells with growth rate constant at medium to high growth rates are compatible with (iii), the regulation at the level of functional efficiency of ribosomes (i) may be important at low growth rates.

1. Introduction Free living microbial cells in equilibrium with external milieu constitute self-contained units. The problem of adaptation in such a system is reduced essentially to the question--how the individual cells adjust their multiplication rates to the external environment? Or, in more definitive terms--what is the nature o f regulatory mechanism underlying observed variations in maeromolecular composition with growth rate ? In this paper we have made an attempt to develop a rigorous mathematical framework for a population o f bacteria maintained in a steady state of exponential growth. The model enables us to define, in a purely formal sense, constraints on the kinetic parameters related to the synthesis of proteins on labile templates (Singh, 107

108

u.N.

SINGH

1969, 1973; Singh & Gupta, 1971) and the assembly of ribosomal particles in a cell. The composition of a bacterial cell at the gross level, often expressed as R N A and protein contents, is explicitly described in terms of various functional and metabolic' states of these macromolecules. The behavior of a model system, as predicted for different modes of regulation, is discussed in the light of experimental observations. 2. Basic Postulates and Assumptions

The model analysed in this paper is diagrammatically represented in Fig. 1. Ribosomal proteins are envisaged to comprise two distinct classes-P,, the proteins that get attached to nascent ribosomal R N A (rRNA) before the latter is fully transcribed and Pro, the "maturation" proteins added to subribosomal particles to give rise to mature ribosomes. This classification is based essentially on in vitro reconstitution of ribosomes (Nomura, 1970; Wittmann & StOflier, 1972) and the nature of proteins associated with subribosomal particles isolated from ceils (Mizushima & Nomura, 1970;

IT : ~ ]

/

S

k,

/ \ \

\I

,

~

'J

..

ms ......

j/ I

ISubrib.part.

23s . . . .

50S I

30S I~G. 1. Biogenesis of Ribosomes: The diagram describes the steps involved in the sTnthesis of ribosomal RNA and proteins. The transcription of ]6 S and 23 S RNA,

organized in a single operon, is facilitated by the attachment of ribosomal proteins belonging to the class "P,". Attachment of ribosomal proteins "P,," to subribosomal particles leads to the formation of mature ribosomal subunits--30 S and 50 S. Various kinetic parameters used in the mathematical formulation (section 3) are also indicated. Solid arrows represent the rate-limiting steps.

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109

Homann & Nierhaus, 1971). As a first approximation P, and Pm may be considered to be synonymous with "core" and "split" proteins of ribosomes, respectively. The basic assumptions underlying the mathematical formulation described in the following section may be summarized as follows: (i) All structural proteins of ribosomes are synthesized in a coordinate manner (Gupta & Singh, 1972; Nomura & Engback, 1972; Gullav, Meyenburg & Molin, 1974). (ii) Free ribosomal proteins comprising P, in the pool function as a positive control element in the synthesis of rRNA. This implies that RNA polymerase is not a limiting factor in the synthesis of rRNA, and that one of the proteins in conjunction with RNA polymerase determines the frequency of initiation. Note that a co-ordinate synthesis of these proteins envisaged in (i) ensures a minimal elongation rate of the chain once its transcription is initiated. (iii) The maturation process involving attachment of "maturation" proteins Pm follows a first order kinetics with respect to subribosomal particles. This is based on the observations by Traub & Nomura (1969a,b) who showed that the rate of conversion of 30 s "core" particles to mature subunits in in vitro reconstitution studies is unaffected by an increase in the concentration of "split" proteins, presumably over a minimal value. (iv) All messenger RNA's (mRNA) including those of ribosomal proteins have the same half-life. We have recently determined the average functional half-life of mRNA's of ribosomal proteins in E. coli using an immunochemical technique and found it to be of the same order of magnitude as that of nonribosomal proteins (unpublished observations). Similar conclusions have been recently derived by Molin, Meyenburg, von Gullov & Maalge (1974). (v) Transcription, translation and degradation of mRNA are stringently coupled. (vi) All mRNA's fn a cell are of the same size. This assumption, though necessary to keep the mathematics at a manageable level, will not affect the main conclusions derived in this paper. 3. Mathematical Formulation of the Model (A) S~BOLS

The symbols used in these analyses are summarized below: Pn, P,n and Ps free ribosomal and non-ribosomal proteins in a cell, expressed in molar amounts. Subscripts m and n refer to two classes of ribosomal proteins as indicated in Fig. 1, and subscript s to all non-ribosomal proteins. Rn nascent rRNA (growing chain still attached to the D N A template) expressed in units of fully transcribed rRNA.

110

Rs R~ ( - M 3 Rts M

6 T kl k~

k2 ks

k4 f,,fmandf~

Pn, Pm

if/*,/~*

U. N. SINGH

number of free ribosomes comprising 70 S ribosomes as well as dissociated subunits 30 S and 50 S. number of ribosomes attached to nascent mRNA ( ~ amount of nascent mRNA expressed in units of 6).

Rf+Ri. amount of completed mRNA (free polyribosomes) expressed in units of ¢$. Note that total amount of mRNA (or total number of ribosomes bound to mRNA) is given by ( M + M.,) ~- tiM, where fl = kxT/(1--e-k'r). unit of time, time required by a ribosome to traverse a distance 6 of mRNA. unit of length, defined as the average distance between adjacent ribosomes in polyribosomes. transcription time (--~ translation time in stringently coupled system) of an mRNA. exponential decay constant of mRNA expressed in units of ~. the proportionality constant describing the linear relationship between the rate of protein synthesis and amount of mRNA; k'l = k f f ( 1 - e -k~r) (Singh, 1973). coupling constant, related to the functional efficiency of ribosomes in a stringently coupled system (see text for details). kinetic parameter having the dimension of a first-order rate constant; defines the relationship between the amount of free ribosomal proteins P, and rate of rRNA synthesis. first order rate constant for the maturation of ribosomes, i.e. R, ~ R I. differential synthesis rates of various classes of proteins. Subscripts n and m refer to ribosomal proteins Pn and pro, respectively, and s to non-ribosomal proteins; f, +fm+f~ = 1. number of protein molecules belonging to the class n and m, respectively, in a ribosome. exponential growth rate constant of bacteria. rates of synthesis of mRNA and rRNA, respectively, as determined in pulse-labelling experiments.

(B) NONGROWINCCELLS On the basis of the postulates and assumptions outlined above we can write down the following set of differential equations for a single nongrowing cell: dM dt = k 2 R f - k l M ' (1)

A D A P T A T I O N IN M I C R O - O R G A N I S M S

d R l f - k4R. + k l M dt dR.

k2R.r,

111

(2)

= k a P . - k4R.,

(3)

dt - f.k'~ M - p . k a P .,

(4)

dt

dP. dPm dt = f m k ' ~ M - pmk4R"' -~ = f s k i M .

(5) (6)

The derivation of these equations is quite straightforward and follows directly from the postulates and assumptions described earlier. However, a few remarks about the physical significance of these equations may be in order. Equation (1) is a formal representation of the stringent coupling between transcription and translation processes. The main merit of this expression, as we will see later, lies in the fact that it allows us to relax the constraint dictated by the "constant ribosomal efficiency" hypothesis of Maaloe (I969). Further, in the present context it makes the system of exponentially growing bacteria internally consistent by limiting the analysis to only those mRNA's that are translated in the cell. Needless to say that any extra mRNA synthesized in a cell due to a relaxation in the coupling between transcription and translation processes will be of no consequences as far as the sustenance of exponential growth of bacteria is concerned. The terms p~ andpm appearing in equations (4) and (5) arise from the unit used to define R~--nascent rRNA (including subribosomal particles). ((2) EXPONENTIALLY GROWING CELLS

One must bear in mind that the description of bacterial growth by an exponential function is valid only in a statistical sense and, is strictly applicable to a population of asynchronously dividing bacteria. So also is the notion of invariance of macromolecular composition of a bacterial cell, which only refers to average values obtained from measurements on a large number of bacteria in a steady state of exponential growth. A necessary and sufficient condition for sustained exponential growth is that the amounts of all bacterial components--low or high molecular weights--should also increase exponentially with the same rate constant as that of bacterial multiplication. Thus if iV, = No e~t describes an increase in a bacterial population, where Nt = number of cells at any time t, No = the number at time "0" and

112

u.N.

SllqOI-I

a = the rate constant, then the amount X, o f any chemical component o f a cell in the same volume of culture will be given by: Xt -- x N t = x N o C t, where x is the amount pe r cell. This condition must also be compatible with the functional relationships between various macromolecular components as described by equations (1) to (6) for an isolated bacterium. Considering now a population of growing bacteria, i.e. substituting X, for x (M, R s, R,s, etc.), we obtain the following set of simultaneous equations:

O-

dM dt - k u R $ - ( a + k l ) M ,

(7)

0 = dRis = k,,R. + k i M - k 2 R f - ctR~s, dt dR.

(8) (9)

0 - - - - k 3 P , , - (a + k4)R ., dt 0-

dP,, d~ - f , , k ' x M - ( a + p,,ka)P,,,

(10)

0 =

dPm =f,,,k'~M-p,,k4.R,,-ctP,,, dt

(11)

o -

dP, dt

(12)

-fsk'~M-~P,.

The zero on the left-hand side of these equations indicates that the system is in its steady state, i.e. the cellular composition (average) remains constant. Although equations (7) to (12) do not permit an evaluation of the absolute values of various components, they do enable us to express their relative amounts in terms of various kinetic parameters. Without getting into algebraic details, some expressions which are relevant in our analysis of a model system, are summarized below:

Rt

o~(o:+ kl + i l k 2 )

P~

f sk'l k 2

P.

f.o~

(13)

'

(14)

P,.

f.

p.ct(a + k l + flk2)

P~

f~

f~k'~k2

R.

a2(a+ kl +ilk2)

P~

f~k'lk2k4

'

/~* 45 x k a f . k ' 1 " - ~ = (o~+ p.k3)(o~ + kl)"

(15) (16) (17)

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IN M I C R O - O R G A N I S M S

In defining the macromolecular composition we have chosen P,---nonribosomal proteins--as reference. Rt in equation (13) refers to the total number of mature ribosomes--both free and mRNA-bound. (D) CONSTRAINTS ON PARAMETERS

Equations (7) to (12) also lead to an important relationship which defines formal constraints on the various kinetic parameters related to protein synthesis and biogenesis of ribosomes. For the sake of simplicity we assume P. =Pm (f~ = fro). From equations (7), (8) and (9) we obtain an expression for P,: e,, -

~(~ + k4)(~ + kl + fig2)

(18)

k2k3k,

Also from equation (10) we have f , kl P" - a + pnk 3"

(19)

From equations (18) and (19) we obtain the relationship: oe(a+ k,,)(a + p.ka)(a + k I + ilk2) f. = (20) k'lk2kak * As we recall k~ is not an independent parameter, and is related to k~ and length of mRNA. Thus, at least in theory, equation (20) can be satisfied by appropriate changes in any one of the parameters--k~, k2, ka, k a and f~-for different values of growth rate constant, k a is a first-order rate constant for the maturation of subribosomal particles, presumably determined by conformational changes in these particles (Traub & Nomura, 1969a, b). Assuming that the coordinate synthesis of ribosomal proteins [postulate (i) described in the preceding sbction] is adequate to ensure a minimal level of "maturation" proteins P,, in a cell and that no other additional cofactor is needed, k4 may be considered to have a constant value for all growth rates. We also now have direct experimental evidence which indicates that translational rate (elongation rate of protein chain) and half-life of mRNA are not significantly altered over a wide range of variation in the growth rate of bacteria (Coffman, Norris & Koch, 1971; Engback, Kjelgaard & MaaMe, 1973). In the light of these considerations we will assume kl (k~) and k4 to have constant values and consider only the regulatory mechanisms operating at the remaining three levels defined by the parameters--k 2, k3 andfn. (E) TURNOVER OF RIBOSOMES

Thus far we have tacitly assumed that all cellular proteins are stable. If we consider the two major classes of proteins--ribosomal and nonribosomal--to be equally stable or unstable, the basic conclusions derived T.B.

8

114

u.N.

SINGH

from these analyses will not be affected. However, ribosomes may undergo turnover at the particulate level arising from degradation of rRNA by nucleases. This indeed has been considered by some workers (see reviews by Travers, 1974; Nierlich, 1974) as a possible mechanism for the regulation of ribosomal population in a cell at different growth rates. It is generally accepted that ribosomes in rapidly multiplying bacteria are fairly stable, and the turnover rate at the particulate level under these conditions is minimum. Ribosomes in cells growing at very low growth rate, on the other hand, are known to undergo degradation. The factor or factors responsible for this enhanced susceptibility of rRNA to nucleases, however, are still unknown. In developing a rationale for the turnover of ribosomes we assume that (i) the level of nuclease responsible for the breakdown of rRNA is same under all growth conditions and (ii) only the "idle" ribosomes, i.e. ribosomes in free state undergo turnover. According to this model stability of ribosomes in rapidly multiplying cells is attributed to a relatively small number of free ribosomes present in these cells. Assuming k d as a first-order rate constant for the degradation of ribosomes, equations (8), (10) and (ll) may be written as 0---- dRifdt = k4Rn + k t M - (k 2 + kd)R $ -- ctRif ,

(21 )

dPn = f n k ' x M - (ct + pnka)P~ + p.kdRy , O = dt

(22)

dPm 0 = dt = f m k ' x M - P"k4Rn + P " k a R s - uP,..

(23)

Equations (7), (9) and (12), however, will remain unchanged, f~ for p,, = Pm is now given by: fn =

~(~ + k4)(u + p.ka)(u + kl + ilk2) + ~(~ + kl)kd(u + p.k3 + k4) k'lk2kak 4

,

(24)

Expressions for relative amounts of various macromolecular components (not shown here) can be readily derived from above equations.

4. Behaviour of a Model System

This section includes the results obtained from a quantitative analysis of a hypothetical model system comprising a bacterial population in steady state of exponential growth. Before describing these results it would be appropriate to consider briefly the basis for the values assigned to various

ADAPTATION

IN MICRO-ORGANISMS

115

parameters. Attempts have been made to ensure that these values are compatible with experimental observations on real systems. Assuming c5 to be equivalent to 100 nucleotides and transcription rate of R N A to be 45 nucleotides per see (corresponding to a translational rate of 15 amino acids per see in a stringently coupled system) • is approximately equal to 2 see. If we assume half-life of m R N A as 120 see, then kl - 0.01 and k~ "-" 0.1 for an m R N A containing 1000 nucleotides coding for a protein of molecular weight of about 30,000 daltons. We further assume Pn = P,, = 25, and f , = f m = 0"2. The value of k3 (0.0004) in units of x has been estimated from the observed amount of free ribosomal proteins in E. coli with a doubling period of about 30 min. As regards k4 it can be shown that k4 " 2/T, where T, is the transcription time of rRNA (16 S and 23 S). This provides an upper limit of 0.05 for k4. In our analysis we assign a value of 0.02 to k4. (A) CASE I: REGULATIONAT THE LEVELOF FUNCTIONAL

EFFICIENCYOF RIBOSOMES~k2 AS A VARIABLE The overall rate of protein synthesis in a cell is determined by two factors: (i) fraction of total number of ribosomes actively engaged in protein synthesis and (ii) number of protein molecules synthesized per unit time on m R N A in its steady state concentration. While (i) manifests in the distribution of ribosomes in free and polyribosomal states, (ii) can be readily evaluated from the observed elongation rate of protein chains and the average number of ribosomes attached to an m R N A of a given size, i.e. the average spacing between adjacent ribosomes in a polyribosome. Maaloe (1969) in his empirical formulation considered both these factors to be invariants under different growth conditions. In the present situation this constraint is partly relaxed and k2 varies with ~ in accordance with equation (20). Equation (20) can be rewritten as ~(~ + k4)(~ + p, ka)(~ + k 1) k2 - f,,k'~ k a k 4 - flot(o~ + k4)(ct -t- p,,ka)"

(25)

For very low grow rate, i.e. ~ ~ O, we have P,,kl°:

k2 "f,,k;-~p,,o:

.., P,,kl °~

- f,,k'~ '

(26)

and k 2 will vary linearly with ~. For higher values of ¢t, k2 will rapidly approach oo as the denominator in equation (26) approaches zero. This also defines the maximal growth rate of bacteria corresponding to a limiting situation in which all the ribosomes in a cell are actively engaged in protein synthesis (Fig. 2).

116

u . N . SINGH I000

I0,000

I00

1000

7~_ 1o X

,oo

I

I0

0-1 ),01

0.1

I a x 104

vii I0

I

FxG. 2. Values o f k= calculated from equation (25) are plotted against g r o w t h rate constant (ct) o n a n expanded log-log scale. Curves I a n d II refer to two different sets o f values: I, kx = 0.01, k; = 0.1, k3 = 0.0004, k4 = 0.02, p, (=pro) ----25 a n d f , = 0.2; 11, kl = 0.1, k~ = 1-6, ks = 0.004, k4 = 0.2, p , ( = pro) = 25 a n d f , = 0.2. The arrows indicate the maximal growth rate constants attainable in the two cases.

Figure 3(a) shows variation in the relative amounts of free ribosomal proteins (P,) with growth rate constant for different values of ka. The model predicts an approximately linear increase in the ratio of free ribosomal proteins to non-ribosomal proteins with growth rate constant. Further, the plot P,/Ps versus = extrapolates to the origin at "0" growth rate. In Fig. 3Co) a similar plot of experimental data recently reported from this laboratory (Gupta & Singh, 1972) is included. Note that while the straight line for "split" proteins in Fig. 3(b) does extrapolate to "0", the corresponding linear plot for "core" proteins shows a ftnite positive intercept on the ordinate, which is contrary to the predictions of the model. Figure 4 summarizes changes in macromolecular composition of a bacterial cell with growth rate in terms of a number of experimentally observable quantities. Curves II and III describe variations in the functional states of ribosomes as inferred from the distribution of ribosomes in free and mRNAbound form (polyribosomes). Another important conclusion derived from these analyses is that the ratio of the amount of proteins (or rRNA) in mature ribosomes to that of non-ribosomal proteins in a cell remains constant at low growth rates, but tends to decrease at higher growth rates (curve I,

ADAPTATION

12

/~!o

117

IN M I C R O - O R G A N I S M S

9

ooJ)

"_o

~s 15 j~ /

0

2

0

6

4

8

o

I

exlO 4

FIG. 3. (a) Variation in relative amounts of free ribosomal proteins with growth rate constant (~) calculated from equation (14) for a model system [case I, section 4 (A)] f# ----0.2, f, -----0.6. The numbers in brackets represent values ofp,k3 (p, = 25). (b) Changes in the relative amounts of free "split" ( ~ p,) and "core" (_-__p,) ribosomal proteins with growth rate constant (~) in E. coil Data taken from Gupta & Singh (1972). C) -- "split"; • = "core" proteins,

0-8

0.8

Eo

% ~c

•~ 0.6

06 il

~ 0.4 o

oi

~- o 2

o

0-2

o

I

2

3

4

5

6

z

7

=xlO 4

Fie. 4. Variations in macromolecular composition of a bacterial cell with growth rate constant (~) in a model system [case I, section 4 (A)]. Values of various kinetic parameters are assumed to be the same as those for Curve I in Fig. 2. I, Mature ribosomes; II, free ribosomal proteins; III, mRNA-bound ribosomes (polyribosomes); IV, nascent ribosomes (subribosomal particles including nascent r R N A chains).

118

u.N.

SINGH

Fig. 4). It can be shown that the decrease in the latter case is essentially due to an increase in the relative amounts of free ribosomal proteins [Fig. 3(a)] and subribosomal particles (curve IV, Fig. 4). If we allow for turnover of ribosomes at the particulate level as defined in section 3 (E), the behavionr of the system undergoes profound modifications. It is seen from Fig. 5 (curve 1P) that for a low value ofkn (0.0001) the relative amount of free ribosomal proteins increases linearly with the growth rate constant, with the straight line having a positive intercept on the ordinate 0-08

.

.

.

.

.

. 0.65

0.60

~-" o

E

I

,,';- 0.02 ?

01

0

~

0.55

~

,

I

B

,

2

i

5

,

4

,

5

L

6

---. 0"50 7

cexlO 4

FIo. 5. Behaviour of the model system described in Fig. 4 [case I, section 4 (k)] assuming finite turnover rate of ribosomes. 1R, Mature ribosomes; 1P, free ribosomal proteins; kd = 0.0001.2R and 2P represent corresponding curves for ka = 0.001. Note the inverse correlation between the curves for mature ribosomes and free ribosomal proteins.

instead of passing through the origin. For a high value of k a (0.001) the corresponding plot is still linear but with a negative slope (curve 2P, Fig. 5). Relative amounts of proteins in mature ribosomes (curves 1R and 2R) bear an inverse relationship with those of free ribosomal proteins in the poolcurves 1P and 2P. (B) CASEII : REGULATIONAT THE LEVELOF rRNA TRANSCRIPTION-ka AS A VARIABLE In recent years still another mode of regulation operating at the level of rRNA transcription has been extensively discussed in literature (see review by Nierlich, 1974). Much of the evidences in support of this contention have come from the discovery of two unusual nudeotides (MS, "magic spot"),

119

A D A P T A T I O N IN M I C R O - O R G A N I S M S

ppGpp, and also pppGpp (Cashel & Gallant, 1969; Cashel & Kalbacher, 1970). There appears to be an inverse correlation between the amount of these nucleotides in the ceils and the observed rate of RNA synthesis. These observations have been further substantiated by in vitro studies, in which ppGpp is shown to inhibit EF-TuTs--(original ~k factor of Travers) stimulated rRNA synthesis (Travers, Kamen & Cashel, 1970; Travers, 1973). In a model system considered here we conserve the notion of free ribosomal proteins, or one of these proteins, functioning as a positive control element in the synthesis of rRNA. The coefficient ka which determines the efficiency of the transcription process, however, is now assumed to vary with the growth rate of bacteria. Other parameters including k2 (0.02) are considered to be invariants. The inset in Fig. 6 shows variation in k3 with ~ as calculated from equation (20). If the value of k3 obtaining in a cell under given growth conditions is determined by negative type control exercised by a metabolite like ppGpp then the concentration of this key element should bear a reciprocal

I-0

~" o_ '~ v

0.8 0-6 0.4

/

0.2

o

5 o., '~

~

;

"

;,~

..,o'

~/

0.6

r ~/ / .

3

~o.5 - -

-..

....;

0.4

0

2 *

I

2

3

4

5

6

7

czxl0 4

Fie. 6. 1"he figure describes the behaviou[ of a model system discussed in [case H, section 4 (B)]. In the inset ka is plotted against ~ in accordance with equation (20); values of aI[ other parameters being same as in Fig. 4. [, free ribosomal proteim; ]I, mature ribosomes; I l l , subribosomal particles (hlc]uding nascent rRNA); IV, relative rate (~*/z~r*) of ribosomal and mRNA synthesis.

120

u.N.

SINGH

relationship with k3, i.e. a plot of the metabolite concentration in a cell against growth rate constant should be a mirror image of the curve k 3 v e r s u s shown in the inset of Fig. 6. This is an important conclusion, as attempts have often been made to justify the regulatory role of ppGpp on the basis of an inverse correlation between the level of this nucleotide in the cell and the relative rate of rRNA synthesis (Lazzarini, Cashel & Gallant, 1971; Donini, 1972; Lazzarini & Johnson, 1973; Konrad, Toivoneu & Nierlich, 1972). Curve II in Fig. 6 describes changes in the relative amounts (with respect to non-ribosomal proteins) of mature ribosomes with growth rate constant. An increase in the ratio of mature ribosomal proteins to non-ribosomal proteins in this case is in sharp contrast with the inferences, derived in the previous case (curve 1, Fig. 4). The model also predicts an almost linear increase in the relative rate of rRNA synthesis (curve IV, Fig. 6). Both these predictions are qualitatively consistent with experimental observations in bacteria at medium to high-growth rate (Pato & Meyenburg, 1970; Nierlich, 1972; Bremer, Berry & Dennis, 1973; Dennis & Bremer, 1973). However, a decrease in the relative amount of free ribosomal proteins with growth rate inferred from these analyses is incompatible with the observed increase in the pool-size (Gupta & Singh, 1972; Marvaldi, Pichon, Delaage & MarchisMouren, 1974; Dennis, 1974). One could also see intuitively that if the regulation operates only at the level of rRNA transcription, then an increase in the rate of rRNA synthesis with growth rate will deplete the pool of free ribosomal proteins in the cell--unless of course one invokes additional compensatory mechanism that regulates differential synthesis of ribosomal proteins. The proponents of this mode of regulation often fail to take cognizance of the observed increase in f , with growth rate constant (Schleif, 1967; Dennis, 1974; Zaritsky & Meyenburg, 1974). (C) CASE HI: REGULATION AT THE LEVEL OF DIFFERENTIAL SYNTHESIS OF RIBOSOMAL PROTEINS--fn AS A VARIABLE

In this section we have analysed the third alternative mode of regulation in which exponential growth of a bacterial culture is sustained by variation in f~, i.e. differential synthesis rate of ribosomal proteins. This aspect has been earlier discussed in detail by Maaloe (1969) who derived a simple linear relationship betweenf~ and growth rate constant ~. Maaloe in his derivation, besides assuming a constant and high e~ciency of ribosomes in protein synthesis, has also assumed the amount of free ribosomal proteins in the ceil to be negligibly small. An immediate implication of these two assumptions in the present formulation is that the three parameters k2, k3 and k4 are

121

ADAPTATION IN MICRO-ORGANISMS

infinitely large. Under these conditions equation (20) is reduced to f~ -- - -

kl

-- Tp;x,

or

(27)

f, = 2Tp~,

where f , = f , + f m . Equation (27) is identical to Maalee's formulation (1969). Thus the model discussed by Maalee may be regarded as a limiting case of a more general treatment developed in this paper. Figure 7 summarizes the behaviour of a model system in which a state of sustained exponential growth is achieved by variation in f~, all other kinetic parameters being constant. The inset in Fig. 7 includes a plot off~ v e r s u s

~ 0

of o.a

o

02

/

/

/

I

I

06

,.ore/

/

1

I

o.i

04

~

~ 0.2

0-02

B

~

O

0

I

2

~

3

4 axlO 4

5

6

7

0

F1G. 7. The behaviour of a model system described in case III, section 4 (c). The inset shows variation in fn with ~ as calculated from equation (20); values of other parameters being same as in Fig. 4. I, mature ribosomes; II, free ribosomal proteins; III, subribosomal particles; IV, ratio of free ribosomes/polyribosomes (-~ 0.1). as described by equation (20). Curves I, II and III represent variations in the relative amounts o f mature ribosomal proteins, free ribosomal proteins in the pool and the proteins associated with immature ribosomes (including nascent rRNA), respectively, with growth rate. A constant ribosomal efficiency is indicated by the horizontal line IV in Fig. 7 representing the ratio between free and m R N A - b o u n d ribosomes. The model also predicts an increase in the relative rate of r R N A synthesis with growth rate constant

122

U. N. SINGH

io

r

i

J

,

8

O

6

4 I

a

~

,

~xlO 4

FIG. 8. Relative rates of ribosomal RNA and mRNA synthesis, as inferred from pulselabelling experiments, plotted against ~t. I, the model system described in Fig. 4 (case I, with stable ribosomes); II, the system described in Fig. 5 for ka = 0-001; HI, case IlI [section 4 (c)] in Fig. 7. as described by curve III in Fig. 8. All these predictions, at least in a qualitative sense, are consistent with experimental observations on bacteria at medium to high growth rate. 5. Discussion

Any model that is put forward to account for the observed systematic variations in maeromoleeular composition--RNA and protein contents--of a cell with growth rate constant must necessarily be based on, and internally consistent with, the known mechanisms of proteins synthesis and the processes underlying biogenesis of ribosomes. This contention is valid irrespective of the mechanism involved in the replication of D N A and division of a bacterial cell. The only thing one need to ensure is that the daughter cells produced after cell-division receive a full complement of the genetic material (DNA). In the model analysed here we have tacitly assumed that this condition is satisfied under all growth conditions, presumably by the mechanism that controls temporal relationship between D N A replication and division of a bacterial cell. In addition, besides a full complement of the genetic material, a free living microbial cell must also receive the necessary metabolic machinery from its parent cell to be truly viable. In our analysis

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we have divided the proteins comprising the metabolic machinery of a cell into two major groups--ribosomal and non-ribosomal proteins. Apart from functionally distinct roles of these proteins in a cell, such a classification is particularly relevant in the description of macromolecular composition of a bacterial cell in terms of RNA and proteins. Equation (1) marks a significant departure from the conventional formulation of the synthesis and degradation of mRNA. It may be pointed out that by applying a generalized constraint on the overall synthetic rates of mRNA as determined by kz and the number (R f) of free ribosomes in a cell, we do not necessarily violate the control exercised by regulator genes at the level of individual operons (Jacob & Monod, 1961). The latter may still determine the relative rates of synthesis of various mRNA's, as implicitly defined by f , in reference to the two major classes of proteins. In the present context we have sought justification for the notion of stringent coupling between transcription and translation processes on the grounds that only the amount of mRNA's that could be effectively utilized by the translational machinery available in a cell is of any relevance in the maintenance of sustained exponential growth. The significance of this concept in a broader sense, however, still remains obscure. Recently two sets of observations have appeared, which call for a fresh look at this question. Miller, Beatty, Hamkalo & Thomas (1970) and Hamkalo & Miller (1973) in their electron microscopic studies of "transcription-translational complex" have noted that the first ribosome attached to nascent mRNA is always found to be in close proximity of, and almost in physical contact with, the RNA polymerase on the DNA template. Assuming that this is not an artifact of fixation, this observation certainly deserves serious consideration. The second type of observation refers to the stimulatory effect of EF-TuTs (Elongation Factor-TuTs) on the synthesis of R N A by polymerase in vitro (Travers, 1973). In a model of biosynthesis of ribosomes Travers (1974) ascribes the regulation operating at the level of rRNA transcription [case II, in section 4 (B)] to EF-TuTs as modulated by ppGpp. His contention is primarily based on the observation that in an in vitro system ppGpp specifically antagonizes the stimulatory effect of EF-TuTu on rRNA synthesis. The point we wish to emphasize here is that such in vitro RNA-synthesizing systems have a built-in bias towards the transcription of non-translatable R N A like rRNA and tRNA, and that one should exercise utmost caution in extrapolating these observations to in rive conditions. What appears to be more remarkable is that EF-TuTs, which is known to be intimately associated with functionally active ribosomes, should possess an affinity for R N A polymerase. It is conceivable that EF-TuTs may function as a link between ribosome and R N A polymerase in a stringently coupled system.

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The absence of free rRNA in a bacterial cell provides adequate justification to the notion that some ribosomal proteins get attached to the growing r l ~ A chain before triGlatter is fully transcribed. However, the same kind of uncertainty prevails about the positive role ascribed to free ribosomal proteins in facilitating the transcription of rRNA, as that discussed earlier in the case of stringent coupling between transcription and translation processes. Nonetheless, there is a common principal underlying the two types of coupling that makes these ideas particnlarly attractive. Translational control of mRNA synthesis implies that a bacterial cell synthesizes an mRNA only when it is needed, and in amounts that can be utilized by the translational machinery of the cell. The instability of mRNA's in micro-organisms, including those of ribosomal proteins, is consistent with this viewpoint. In a parallel situation, the role of free ribosomal proteins in the regulation of rRNA synthesis envisaged in the present model ensures against abortive production of rRNA. A major limitation of the present model is that it does not enable us to examine situations arising from multiple regulatory elements affecting more than one step at the same time. Such a possibility obviously cannot be ruled out apriori. However, in extreme situations, such as those existing in bacteria with very high or very low growth rates, it is also likely that one of these alternative modes of regulation dominates over the others. It is in this context that the analysis of simple cases involving single control element presented here may be of practical value. Most of the experimental studies on macromolecular composition of bacterial cells involve four types of measurements: (i) ratio of RNA to protein (or of RNA to DNA); (ii) differential synthesis rates of rRNA and mRNA; (iii) differential synthesis rates of ribosomal proteins as referred to non-ribosomal proteins; and (iv) relative amounts of free ribosomal proteins in the cell. No reliable measurements are available on the variation in the amount of subribosomal particles with growth rate. Further, these measurements do not always refer to the same system under identical conditions. This obviously has made it difficult to test the validity of the temporal relationship between various quantities as predicted in this model. However, a qualitative comparison of the inferences derived from these analyses with experimental observations leads to the conclusion that, at least at moderate and high growth rates of bacteria, the regulation operating at the level of differential synthesis rate of ribosomal proteins [f~; case III, section 4 (c)] provides a minimal model of bacterial growth. A closer examination of the curve I in Fig. 7 (case III) describing variation in the ratio of ribosomal proteins (~- rRNA) to non-ribosomal proteins with growth rate constant reveals the nature of deviation from the simple linear

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relationship inferred by Maal~e (1969). The prediction that the curve should extrapolate to the origin at "0" growth rate is contrary to experimental observations in which the corresponding plots show finite positive intercepts on the ordinate (see Koch, 1970; for a summary of available data). Recently it has been further observed that f~ v e r s u s ot curve also, instead of extrapolating to "0", tends to acquire a finite constant value at lower growth rates (Dennis, 1974). This immediately raises the question whether the regulatory mechanism functioning at the level of differential synthesis rate of ribosomal proteins, which at least qualitatively accounts for most of the observations at higher growth rates, is equally relevant in bacteria growing at very low rates. It may be pointed out that regulation at the level of rRNA transcription [case II, section 4 (B)] which leads to essentially similar conelusions (curve II, Fig. 6), will not be able to account for these deviations. On the other hand, an examination of the consequences arising from a relaxation of the constraint on the functional efficiency of ribosomes discussed in case I [section 4 (A), Fig. 4] strongly indicates that the regulation in very slowly growing bacteria may involve variation in k2. On the basis of the analyses presented here we propose a minimal model •of bacterial growth envisaging regulation at two distinct levels: At medium and high growth rates variation in the differential synthesis rate of ribosomal proteins, presumably brought about by derepression or repression of cistrons of non-ribosomal proteins (Maaloe, 1969; Zaritsky & Meyenburg, 1974), constitutes the primary regulatory mechanism. The functional efficiency of ribosomes in a cell, as defined by k2 in our formulation, remains constant under these conditions. At low growth rates, as the level of derepression of cistrons approaches steady state, the differential synthetic rate of ribosomal proteins (f~) tends to acquire a constant finite value (Dennis, 1974). The regulation of bacterial growth rate at this stage is attributed to variation in the functional efficiency of ribosomes determined by parameter k2 [case I, section 4 (A)]. While the present model provides an adequate explanation for many of the features related to changes in macromoleeular composition of bacteria with growth rate, it also questions the universal validity of Maaloe's "constant ribosomal efficiency" hypothesis (Maaloe, 1969). Needless to say that while the invariance of functional efficiency of ribosome particularly in reference to the fraction of ribosomes actively engaged in protein synthesis, is fairly well supported by experimental observations in bacteria at medium and high growth rate, its extrapolation to bacteria maintained at very low growth rate is at present little more than presumptuous. The model further suggests that the so-called "extra" RNA of Koch (1970) may indeed be a reflection of an accumulation of free ribosomes in slowly growing bacteria. The fact that the "extra" RNA accumulating in

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cells a t low g r o w t h rates is r e a d i l y a v a i l a b l e as f u n c t i o n a l l y m a t u r e r i b o s o m e s , w h e n t h e cells a r e t r a n s f e r r e d to a relatively rich m e d i u m , s u p p o r t s this contention.

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