guanosine tetraphosphate as a global regulator of bacterial rna synthesis: a model involving rna...

ELSEVIER Biochimica et Biophysica Acta 1262 (1995) 15-36

BB Biochi~ic~a et Biophysica A~ta

R e v i e w

Guanosine tetraphosphate as a global regulator of bacterial RNA synthesis" a model involving RNA polymerase pausing and queuing

Hans Bremer a,*, Mfins Ehrenberg b

a Program in Molecular and Cell Biology, University of Texas at Dallas, Richardson, TX 750831, USA b Department ofMolecularBiology, Uppsala University, Biomedical Center, Box 590, S-751 24 Uppsala, Sweden

Received 30 September 1994; accepted 19 January 1995

Keywords: ppGpp; rRNA synthesis; Bacterium; Growth rate control; (E. coli)

Contents

1.

2.

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1. Relationship between ribosome synthesis and growth rate . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2. Maximizat ion of growth rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3. Amino acid feedback control of ribosome synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4. Two guanosine tetraphosphate synthetases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5. Properties of stable RNA promoters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6. Determinants of stable RNA and mRNA synthesis rates . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7. Expression of stable RNA and mRNA genes at different levels of ppGpp . . . . . . . . . . . . . . . . 19

3. Global control of RNA synthesis rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1. Outline of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2. Implications of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3. Effect of RNA polymerase queuing on RNA polymerase activity . . . . . . . . . . . . . . . . . . . . . 21 3.4. Alternative explanation for increased mRNA synthesis in ppGpp-less bacteria . . . . . . . . . . . . . . 22 3.5. ppGpp-mediated transcriptional pausing in vitro and in vivo . . . . . . . . . . . . . . . . . . . . . . . 22 3.6. Control of stable RNA synthesis by depletion of the free RNA polymerase pool through ppGpp-in-

duced transcriptional pausing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4. Appendix: theory of RNA polymerase queuing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1. A. Visualization of RNA polymerase queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1. Definition of terms and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.2. Visualization of RNA polymerase pausing and queuing . . . . . . . . . . . . . . . . . . . . . . 25

4.1.3. Promoter blockage increases with increasing RNA polymerase concentration . . . . . . . . . . . 26

4.1.4. Promoter saturation and inhibition of gene expression . . . . . . . . . . . . . . . . . . . . . . . 26 4.1.5. Gene activity and polymerase idling increase as pause site is moved away from promoter . . . . 28 4.1.6. Combined effects of varying ppGpp and RNA polymerase concentrations . . . . . . . . . . . . 29

4.2. B. Queuing theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1. Partitioning between mRNA and stable RNA initiation as a function of ppGpp concentration . . 30 4.2.2. RNA polymerase pausing downstream of mRNA promoters may reduce transcription initiation

by a queuing effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.3. Evaluation of in vivo observations involving ppGpp . . . . . . . . . . . . . . . . . . . . . . . . 34

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

* Corresponding author.

0 1 6 7 - 4 7 8 1 / 9 5 / $ 0 9 . 5 0 © 1995 Elsevier Science B.V. All rights reserved SSDI 01 6 7 - 4 7 8 1 ( 9 5 ) 0 0 0 4 2 - 9

16 H. Bremer, M. Ehrenberg / Biochimica et Biophysica Acta 1262 (1995) 15-36

1. Summary

A recently reported comparison of stable RNA (rRNA, tRNA) and mRNA synthesis rates in ppGpp-synthesizing and ppGpp-deficient (ArelA AspoT) bacteria has sug- gested that ppGpp inhibits transcription initiation from stable RNA promoters, as well as synthesis of (bulk) mRNA. Inhibition of stable RNA synthesis occurs mainly during slow growth of bacteria when cytoplasmic levels of ppGpp are high. In contrast, inhibition of mRNA occurs mainly during fast growth when ppGpp levels are low, and it is associated with a partial inactivation of RNA polymerase. To explain these observations it has been proposed that ppGpp causes transcriptional pausing and queuing during the synthesis of mRNA. Polymerase queuing requires high rates of transcription initiation in addition to polymerase pausing, and therefore high concentrations of free RNA polymerase. These conditions are found in fast growing bacteria. Furthermore, the RNA polymerase queues lead to a promoter blocking when RNA polymerase molecules stack up from the pause site back to the (mRNA) promoter. This occurs most frequently at pause sites close to the promoter. Blocking of mRNA promoters diverts RNA polymerase to stable RNA promoters. In this manner ppGpp could indirectly stimulate synthesis of stable RNA at high growth rates. In the present work a mathematical analysis, based on the theory of queuing, is presented and applied to the global control of transcription in bacteria. This model predicts the in vivo distribution of RNA polymerase over stable RNA and mRNA genes for both ppGpp-synthesizing and ppGpp-deficient bacteria in response to different environmental conditions. It also shows how small changes in basal ppGpp concentrations can produce large changes in the rate of stable RNA synthesis.

2. Introduction

2.1. Relationship between ribosome synthesis and growth rate

When bacteria are in their logarithmic (exponential) phase of growth, the increase in protein mass per minute (dP/d t ) , normalized to the total amount of amino acid residues in protein (P), is a constant k/x:

ktz = ( d P / d t ) . ( l / P )

/x is commonly given in doublings/h, in which case the normalization constant k equals ln2/60 min/h. The rate of protein synthesis, d P /d t , is determined by the number of ribosomes in the cell population, Nr, and by the average activity of single ribosomes, ( d P / d t ) / N r. The ribosome activity, in turn, is defined by the fraction, fir, of ribosomes translating mRNA in a given moment, and by the average protein chain elongation rate, %:

( d P / d t ) / N r = [3 r " Cp

so that k# can be written as the following product:

kt z = ( N J P ) . fir" Cp

As long as turnover of proteins can be neglected [1], this relation defines the bacterial growth rate in terms of the fraction of ribosomes per protein (Nr/P), the fraction of active ribosomes (fir), and the rate of peptide chain elongation (Cp). This relation is always valid, irrespective of how it is implemented by the particular control systems of the bacterium.

In a rich, amino acid-containing environment, a large proportion of the cell mass consists of ribosomes (Nr/P is large) and the rate of protein chain elongation (Cp) is also comparatively high (see Fig. 1). In a poor medium, in contrast, both Nr/P and Cp are significantly reduced (Fig. 1). Thus, the quality of the medium determines the growth rate by regulating the concentration and function of ribosomes [2]. A major point of the present review is to explain how this is achieved by a global control system that involves guanosine tetraphosphate.

2.2. Maximization of growth rate

Ehrenberg and Kurland [3] have investigated how bacteria may grow as fast as possible in any particular medium, and how cell components can be adjusted to generate a maximal growth rate. In particular, the concentration of ribosomes ( N J P ) and the protein chain elongation rate (%) must have values that, in each medium, are associated

u t O O

.7.. .o 2 k.

v

11.

Z

.~, 20

0 o o .

o 1 0

Ribosomes/protein

j O

j °

o~O j °

I I I I I

Peptlde chain growth rate

e j e j "

0 0 1 2 3

Growth rate (doubl./h)

Fig. 1. Ribosome concentration (number of ribosomes per amount of protein, N r / P ) and ribosome activity (peptide chain elongation rate, c o) as a function of growth rate. Values for E. coli B/r , from Bremer and Dennis [29].

H. Bremer, M. Ehrenberg / Biochimica et Biophysica Acta 1262 (1995) 15-36 17

with the highest possible k~ value. Their results, indicate that, in a poor medium, a large proportion of the metabolic activities in the cell is devoted to synthesis of amino acids and charging of tRNAs, so that only a small fraction of the cell's protein can be ribosomal protein. Here both Nr/P and the rate of growth are small. In contrast, in a rich medium, amino acids can be rapidly transported into the cell so that the cell can invest more in ribosomes to increase Nr/P and thereby k/x.

Another, somewhat counterintuitive prediction from this analysis is that a maximal growth rate in poor media is associated with comparatively small peptide chain elongation rates, Cp. This follows by taking into account that high values of Cp require high concentrations of the ternary complexes consisting of EF-Tu, aminoacyl tRNA, and GTP. Especially the production of high levels of EF-Tu is expensive for the cell [3]. These predictions are in accor- dance with observations, as can be seen from Fig. 1. This further implies that the growth rate associated with a certain medium cannot be derived from physical arguments based on energy flows or by identifying a single, rate-limiting step. The unique relation between medium and growth rate comes from the cell's ability to optimally balance the production of all its components for fastest possible growth. In such a system the steady-state growth rate is ultimately determined by the rate constants of all enzymes in the cell, and by the number of amino acids required for their structures [3].

2.3. Amino acid feedback control of ribosome synthesis

From the preceding arguments it follows that one of the major regulatory tasks of the bacterial cell is to optimally partition its metabolic activities between synthesis of ribosomes on the one hand, and components related to genera- tion of amino acids and energy-rich molecules like ATP and GTP on the other. Bacterial strains that are most successful in accomplishing this in different environments will outgrow their competitors. For this purpose, bacteria have evolved a feedback mechanism that senses the availability of amino acids to adjust ribosome synthesis to its optimal value. This feedback involves control of the accumulation of the effector nucleotide guanosine tetraphosphate, ppGpp [4,5]. In the case of shortage of amino acids during a nutritional shift-down, ppGpp-synthesizing enzymes are stimulated so that ppGpp accumulates and interacts with RNA polymerase. Thereby transcription is directed away from ribosomal RNA (rRNA) towards mes- senger RNA (mRNA) promoters. In contrast, during a nutritional shift-up to a rich medium, ppGpp levels drop, and RNA polymerase is directed towards rRNA promoters. In this manner, different levels of amino acid supply produce different ribosome concentrations. Since different ribosome concentrations are associated with different rates of amino acid consumption, this generates a feedback loop

so that a steady-state is reached when amino acid supply and consumption, as well as ppGpp levels and ribosome concentrations, are all in balance.

Nomura and coworkers have proposed that the feedback control of rRNA synthesis involves free or translating ribosomes [6,7]. No evidence for a direct effector role of free or translating ribosomes has been found. However, translating ribosomes, as consumers of amino acids, indirectly affect the activity of enzymes responsible for ppGpp synthesis and degradation, consistent with the amino acid feedback mechanism outlined above.

2.4. Two guanosine tetraphosphate synthetases

In Escherichia coli and other bacteria, ppGpp is produced by two ppGpp synthetases, PSI and PSII, whose activities are controlled by the growth medium [8]. PSI (relA gene product; [9-11]) is activated by amino acid starvation [12]. This leads to an accumulation of high levels of ppGpp and a reduction in stable RNA synthesis. These effects are known as stringent response to amino acid starvation, or stringent control of stable RNA synthesis [13]. During exponential growth, PSI shows little or no activity [12,14]. PSII (spoT gene product; [15,16]) is active during exponential growth, especially during growth in poor media, and is partly responsible for the growth medium control of stable RNA synthesis. In contrast to PSI, PSII is inactive during amino acid starvation [14,17]. This amino acid control of PSII activity, we assume, provides the feedback required to adjust ribosome synthesis to the availability of amino acids (see preceding section).

2.5. Properties of stable RNA promoters

With regard to control, rRNA and tRNA will here be considered together as stable RNA, in contrast to the metabolically unstable mRNA. The promoters of rRNA and tRNA genes in E. coli have similar features. They consist of two differently controlled, tandem promoters, P1 and P2, about 120 bp apart [18], and an upstream activator region, UAR [19]. The latter contains binding sites for the FIS factor which increases the strength of the P1 promoter [20-22] without affecting the relative increase in P1 promoter activity with increasing growth rate [19]. In contrast to the P1 promoters, expression from P2 promoters is not under growth medium-dependent control [19,23], meaning that their expression is not inhibited by ppGpp. A major determinant of the control of P1 is the 'discriminator' sequence bordering the - 10 (TATAAT) region of stable RNA promoters. This sequence is GCGC in the case of P1, and GCAC in the case of P2. When the A residue in the P2 sequence is changed into a G residue, as in the P1 promoter, the P2 promoter assumes the same growth medium- and ppGpp-dependent control as P1 [24].

18 H. Bremer, M. Ehrenberg/Biochimica et Biophysica Acta 1262 (1995) 15-36

1 O0

80

ZR 6 0 v

) 40

2 0

. - . 0 O o

u~

o 4

._c E 3 o

E v 2 2 o

E

u~ 0 0

Stable RNA eynth, rate total RNA synth. 0 / 0 per

. / . / ' /

I I I I I

RNA synth, rote • per amount of prot. / /

Stable RNA

mRNA ~ ; ~ a o ~ o ~ o

I i i i i

I 2 Growth rate (doub l . / h )

20

40

60 E

80

I O0

Fig. 2. Rates of stable RNA (rRNA, tRNA) and mRNA synthesis per total RNA synthesis rate ( r ~ / q , r m / q ; top panel, left and right ordinate scale) and per amount of total protein ( r ~ / P , r m / P ; bottom panel) for a wild-type, ppGpp-synthesizing strain of E. coli B/r. Values from Bremer and Dennis [29]. In the lower panel, mRNA synthesis rates were calculated from the r ~ / r t values in the top panel. Since this calculation becomes inaccurate for r~/6--* 1.0, it is possible that the mRNA synthesis rates per protein are somewhat underestimated at high growth rates.

2.6. Determinants o f stable RNA and mRNA synthesis rates

The rate of stable RNA synthesis per protein ( r J P ) depends on four independently controlled factors: (1) the total amount of RNA polymerase per protein (RNAP/P) ; (2) the RNA polymerase activity (fraction, tip, of RNA

polymerase elongating RNA chains); (3) the distribution of active RNA polymerase over stable RNA and mRNA genes (fraction, ~0~, of active RNAP engaged in stable RNA synthesis), and (4) RNA polymerase function (stable RNA chain elongation rate, c~):

r J P = R N A P / P . flp . q4~ . c~

Similarly, the rate of mRNA synthesis per protein ( r m / P ) is given by:

r m / P = R N A P / P " flp " ~m "era

where 0m = 1 - q4,- These rates and the factors that determine them have been measured under different growth condi t ions ( [25-28] ; reviewed by Bremer and Dennis [29]). Whereas the rate of stable R N A synthesis per protein increases parabolical ly with increasing growth rate, the

rate of m R N A synthesis per protein is nearly constant (Fig. 2). For the rate of stable R N A synthesis, all factors with the exception of c~ increase with increasing growth rate and thereby contr ibute to the parabolic increase in r~ seen

in Fig. 2. The approximate constancy of the rate of m R N A synthesis per protein is due to a decrease in ~b m that compensates for increases in R N A P / P , tip, and c m. Val- ues of q(, have been calculated from the rate of stable R N A synthesis, measured as fraction of total R NA synthe-

sis, r J r t [27], taking into account differences in c~ and c m. Like q4,, r s / r t increases with increasing growth rate (Fig. 2).

The factors determining r~ and r m depend on the cytoplasmic concentrat ion of ppGpp. R N A P / P and tip increase with increasing growth rate differently in the absence [30] and in the presence of ppGpp [27,28]. This indicates that these factors are influenced, but not fully determined, by the cytoplasmic levels of ppGpp. In contrast, ~, and 0m are strictly determined by the level of ppGpp and become nearly growth ra te- independent in the absence of ppGpp (see below; Fig. 3). The R NA chain

1.0

0.8

0.6

o.4

0.2

0,0

½ B/r

I I I l

e - - e e o •

i zz I i I I I 0 I00 0 200 400 600 800

Conc. of ppGpp ( pmo les /OD46 o)

l_ i.O

0.8

0.6

0.4

• : 0.2

0.0 I000

Fig. 3. Distribution of transcribing RNA polymerase over stable RNA and mRNA genes, defined by rs/r t = 6/(r~ + r m) (left ordinate scale) or t~, defined as the fraction of total transcribing RNA polymerase molecules engaged in stable RNA synthesis (right ordinate scale), as a function of cytoplasmic ppGpp concentration. Values from Bremer et al. [5] (filled circles) and from Hernandez and Bremer [30] (open circle at zero ppGpp concentration). Each point represents a culture growing under particular conditions (see text). The curve drawn through the points (except at zero ppGpp concentration) was calculated according to the hypothesis of RNA polymerase partitioning by ppGpp [14]. The new RNA polymerase queuing model proposed here predicts the whole curve, including the large changes near zero ppGpp concentration.


elongation rates c~ and Cm are reduced at very high levels of ppGpp, but low, basal levels of ppGpp as observed during exponential growth in different media have no significant effect on c~ and c m [31].

2.7. Expression o f stable R N A and m R N A genes at differ-

ent levels o f p p G p p

Of particular significance in the present work is the ratio r J r t = r J ( r ~ + rm). In the following we develop a model that explains the complex dependence of r J r t on the concentration of ppGpp and free RNA polymerase.

Among the factors above determining stable RNA and mRNA synthesis rates, r~ / r t (or the related ~,) measures the combined strength of stable RNA promoters relative to all active mRNA promoters. In Fig. 3, r J r t (left ordinate scale) and q(~ (right ordinate scale) are shown as functions of the intracellular concentration of ppGpp. Each of the data points in Fig. 3 represents measurements from a single culture chosen from cultures grown in different media, either exponentially or subjected to amino acid starvation, and the strains contained different relA and spoT alleles (data from [5,14,32]). All points obtained in a broad range of conditions fit a single curve. This indicates that ppGpp is the major factor controlling how RNA polymerase is distributed between stable RNA and mRNA genes. It is noteworthy that the unique relationship between cytoplasmic ppGpp levels and r J r t does not depend on the enzyme catalysing ppGpp synthesis; that is, the relationship is unaffected by mutations in the genes for the two ppGpp synthetases, reIA and spoT. In contrast, the relationship is affected by mutations in RNA polymerase subunit genes ( rpoB, C or D [4,5,33,34]). This strongly suggests that the relative strengths of promoters for stable RNA and mRNA synthesis depend on a specific interaction between RNA polymerase and ppGpp.

The decrease of r J r t with increasing cytoplasmic levels of ppGpp has previously been ascribed mainly to an inhibition of transcription from stable RNA promoters by ppGpp [14,35]. This is consistent with results from in vitro experiments with purified RNA polymerase and DNA templates carrying rRNA or tRNA genes [18,36-40]. These experiments show that ppGpp preferentially inhibits stable RNA gene expression. In vivo, r J r t approaches a minimum level (0.25) at high ppGpp concentrations (Fig. 3); here the P1 promoters are 'silent' and stable RNA synthesis mainly reflects transcription from the P2 promoters of stable RNA genes, which are not under stringent control (see above).

A striking feature of the curve in Fig. 3 is its behavior near zero ppGpp concentration: it begins with an r J r t

value of 0.55 at [ppGpp] = 0. Then it abruptly increases to r J r t values near 1.0 at ppGpp levels only slightly above zero. The sudden change in the r~ / r t curve was discovered only recently [30], when strains with deletions of both relA

and spoT genes became available [16]. Such strains lack

both ppGpp synthetases, PSI and PSII, and do not accumu- late detectable levels of ppGpp ([ppGpp] = 0). The low r J r t value at zero ppGpp level is caused by an increased rate of mRNA rather than by a decreased rate of stable RNA synthesis [30]. This suggests that ppGpp inhibits the synthesis of mRNA during growth in rich media when it is present at very low, but still greater than zero, concentrations. It seemed paradoxical that the ppGpp-dependent inhibition of mRNA synthesis in (ppGpp-synthesizing) wild-type bacteria is strongest at low rather than high concentrations of ppGpp.

r s / r t is independent of growth conditions in ppGpp-less mutants (approx. 0.5; Fig. 3, point at [ppGpp] = 0). This suggests that, in the absence of ppGpp, the combined strength of repressable mRNA promoters is small compared to the combined strength of constitutive (stable RNA and mRNA) promoters. It should be noted that r J r t

depends not only on the promoter strengths, but also on the average length of transcripts and on differences in mRNA and stable RNA chain elongation rates (see Appendix B, section 4.2.1).

In the following we explain how the relationship between ppGpp level and r J r t, including the low r s / r t

value at [ppGpp] = 0, results from two effects of ppGpp. The first is a ppGpp-dependent reduction in transcription initiation at stable RNA promoters as observed in vitro (see above). The second effect is ppGpp-dependent transcriptional pausing at specific sites on the DNA template during the synthesis of mRNA. These sites, we postulate, exist proximal to and downstream of mRNA promoters. Also ppGpp-dependent transcriptional pausing has been observed in vitro [41,42], but the relevance of this effect for the in vivo control of transcription has been questioned [43]. Our present analysis is summarized as a model for the global control of transcription in bacteria by ppGpp.

It has been argued that the relation between ppGpp and stable RNA synthesis in vivo (Fig. 3) is only a correlation and not a causal relationship. This conclusion is not justified since it ignores the effect of mutations in the ppGpp- synthetase genes, relA and spoT. These mutations cause

changes in the level of ppGpp and corresponding changes in r J r t [14,30,44]. Thus, if the altered levels of ppGpp in these mutants were not causing the concomitant changes in r J r t, then relA and spoT gene products must have addi-

tional functions as regulators of stable RNA synthesis, different from, but correlated with, their ppGpp synthesis activities. This appears to be unlikely. The argument above also ignores the reported in vitro effects of ppGpp on the transcription of stable RNA genes (see above).

More recently, it has been claimed that "ppGpp is not required for growth rate-dependent control of rRNA synthesis" [45]. This assertion is based on the observation that ppGpp-less mutants grow at different rates in different media with different rates of stable RNA synthesis, and in this resemble ppGpp-synthesizing wild-type strains. How- ever, this argument does not take into account that both

20 H. Bremer, M. Ehrenberg / Biochimica et Biophysica Acta 1262 (1995) 15 36

RNA polymerase synthesis and activity (factors R N A P / P and /3p in the equation above) increase with increasing growth rate, even in the absence of ppGpp [30]. In other words, the global ppGpp-dependent growth rate control is normally operating together with other, local feedback loops in the control of RNA polymerase synthesis and activity. The important point here is that, without the additional specific control of stable RNA synthesis by ppGpp, ppGpp-less bacteria are unable to optimize ribosome concentration and activity to achieve a maximal growth rate. As a result, ppGpp-less bacteria have reduced growth rates compared to wild-type bacteria in a given

medium [30].

p p G p p - s y n t h e s i z i n g bacteria

Promoters ~ r R N A m R N A Totl. Fract. RNAP ppGpp-RNAP

Slow growth ~ ~ 2 0.5 P

rJ / " t = 0,5

r - " r - * Fast growth cx-~cx-x-~ c~a 10 0.1

P r , / r t = 0.7

Slow growth

r , / r , = 0.5

p p G p p - d e f i c i e n t bac te r i a

r R N A m R N A

¢3 ¢3 2 0

3. Global control of RNA synthesis rates

3.1. Outline o f the model

The principle underlying the model is illustrated in Fig. 4. Four cells are schematically shown, two representing a ppGpp-synthesizing (wild-type) strain, and two a ppGpp- less (mutant) strain. One cell for each strain represents slow growth with a low RNA potymerase concentration (two RNA polymerase molecules indicated by circular symbols), the other fast growth with a high RNA polymerase concentration (10 RNA polymerase molecules shown). For the purpose of illustration, each cell contains only two genes: an rRNA gene, and an mRNA gene with a ppGpp-dependent transcription pause site (P). RNA polymerase which has bound ppGpp is indicated by a filled circle. The numbers of RNA polymerase molecules on the genes are assumed to represent rates of transcription. The rationale for this is that every RNA polymerase that has initiated transcription will eventually produce a finished transcript. For ppGpp-synthesizing bacteria, the example shows a 7-fold increase in rRNA synthesis, a 3-fold increase in mRNA synthesis, and an increase in r J r t from 0.5 to 0.7, when the number of total RNA polymerases increases 5-fold (from 2 to 10) with increasing growth rate. At the same time the fraction of pausing (ppGpp-bound) RNA polymerase decreases from 0.5 to 0. 1, and some RNA polymerase is transiently inactivated in queues gen- erated by the pausing RNA polymerase. In the ppGpp-deficient bacteria (lower two cells), no queues form, and rRNA and mRNA syntheses increase equally with growth rate, so that r J r t remains constant at 0.5, as observed in vivo (Fig. 3). For the same 5-fold increase in RNA polymerase concentration with increasing growth rate, rRNA synthesis increases less, and mRNA synthesis more, than in ppGpp-synthesizing bacteria. All RNA polymerase remains active, i.e., neither pauses nor queues, in ppGpp-less bacteria.

This scheme is oversimplif ied for the sake of illustration; a more precise evaluation of the model requires a quantitative analysis. The major feature of this model is

Fast growth ~r-~c-,n c,o nr-~ c~ 10 0 r, lr, = 0.5

Fig. 4. Schematic illustration of the model of global transcription control by ppGpp-mediated RNA polymerase pausing and queuing during the synthesis of mRNA. ~ , promoter (direction of transcription); ©, ppGpp-free RNA polymerase; O, ppGpp-RNA polymerase (pausing) complex; P, pause site downstream of promoter. For both ppGpp-synthesizing and ppGpp-deficient bacteria, it is assumed that the RNA polymerase concentration increases 5-fold (from 2 to 10 RNA polymerase molecules illustrated) with growth rate. In ppGpp-synthesizing bacteria, the ppGpp concentration decreases with increasing growth rate, reflected in a decreasing fraction of RNA polymerase which has ppGpp bound (0.5 and 0.1, respectively). In this schematic illustration, only one stable RNA (rRNA) and one mRNA gene (with ppGpp-dependent pause site) are shown. During fast growth, a queue of 3 RNA polymerase molecules has formed at the pause site that blocks further initiation at the mRNA promoter and diverts RNA polymerase to stable RNA promoters, so that the proportion of stable RNA relative to mRNA synthesis increases (r~/r t increases from 0.5 to 0.7). RNA polymerase with ppGpp bound is only seen to initiate transcription at mRNA promoters due to the inhibition of transcription initiation at stable RNA promoters by ppGpp. In the ppGpp-deficient bacteria, stable RNA and mRNA synthesis increase equally with growth due to the absence of ppGpp-dependent RNA polymerase pausing. These illustrations are oversimplified and do not show the dynamic aspects of the transcription process.

the queuing of RNA polymerase at ppGpp-dependent transcription pause sites, where the queues can extend back to the promoters of mRNA genes. When this happens, mRNA promoters are blocked and RNA polymerase is directed toward stable RNA promoters. For such queues to form, a high concentration of free RNA polymerase is required, as is found at high growth rates. In addition, the initiation frequency at mRNA promoters, the duration of pauses, and the location of pause sites with respect to promoters, are important parameters. This is explained in mathematical detail in the Appendix.

3.2. Implicat ions o f the model

By taking RNA polymerase queuing into account and by choosing appropriate parameter values, the relationship between cytoplasmic levels of ppGpp and r J r t has been derived (Appendix B, section 4.2.3). Three major assump-


1.0

0.8

0.6

0.4 c

0.2

0.0 0

l'ppGpp] (pMoi/OD460)

Fig. 5. r, / r t as a function of cytoplasmic ppGpp concentration, calculated according to the original RNA polymerase partitioning hypothesis ([ 14]; solid line) and according to the new model that takes ppGpp-mediated RNA polymerase pausing and queuing into account (see Appendix B, section 4.2.3 for details; open circles and dashed line).

tions were used in this model: (1) ppGpp specifically inhibits transcription from stable RNA promoters according to the RNA polymerase partitioning hypothesis [14]. (2) In addition, we postulate that ppGpp causes RNA polymerase to pause downstream of, and proximal to, mRNA promoters. (3) Free RNA polymerase concentrations are negatively correlated with ppGpp levels. The resulting (theoretical) relationship between ppGpp level and r~/r t is graphically illustrated in Fig. 5 (open circles and dashed line). The curve begins at r J r t = 0.5 for [ppGpp] = 0, rises sharply to values near 1.0 at [ppGpp] = 3 pMol/OD, and then drops again, approaching a minimum value of 0.25 at high levels of ppGpp. The calculated curve in Fig. 5 closely matches the observed data in Fig. 3.

For comparison, a theoretical curve based on the earlier hypothesis of RNA partitioning by ppGpp as the only regulatory feature [14] is also shown in Fig. 5 (solid line). The new, predicted relationship between ppGpp level and rs//rt (dashed line) comes from a revision and extension of this previous RNA polymerase partitioning hypothesis. To explain the experimental observations, we now also consider ppGpp-dependent transcriptional pausing of RNA polymerase at sites close to mRNA promoters. A detailed re-evaluation of the RNA polymerase partitioning hypothesis can be found below (Appendix B, section 4.2.1).

Low (but not zero) levels of ppGpp in wild-type bacteria are maintained by a low ppGpp-synthetic activity of PSII during fast growth in rich media [14]. This gives r J r t

values greater than 0.5 and approaching 0.9 (Fig. 5). According to the present hypothesis, these high values of r J r t are caused by a specific inhibition of mRNA synthesis by ppGpp-dependent RNA polymerase pausing and queuing at sites close to mRNA promoters. In other words, we postulate that mRNA promoters become saturated in the presence of ppGpp, provided that the concentration of free RNA polymerase is sufficiently high.

High levels of ppGpp can be produced either by PSII during slow bacterial growth [14,44] or by PSI during the stringent response [12,14]. These high levels of ppGpp

give r J r t values less than 0.5 (Fig. 3). We suggest that these low values of r J r t are due to inhibition of transcription initiation at stable RNA, but not mRNA promoters by ppGpp. Interestingly, the analysis suggests that ppGpp can inhibit both stable RNA and mRNA synthesis. The inhibition of stable RNA synthesis is relieved with increasing growth rate because of decreasing ppGpp concentration. At the same time, inhibition of mRNA synthesis increases. This is because the increasing RNA polymerase concentration leads to a higher fraction of blocked mRNA promoters by queuing, despite the decreasing ppGpp concentration. This may seem paradoxical at first sight, but the quantitative evaluation of the model shows that such a mechanism is perfectly reasonable (Appendix B, section 4.2.3). The combined inhibitory effects of ppGpp on either stable RNA or mRNA synthesis produce the observed increase in r J r t in wild-type, ppGpp-synthesizing bacteria from values below 0.5 during slow growth to values above 0.5, approaching 1.0, during fast growth (Fig. 2). The unex- pected experimental finding of a drastic change in r s / r t

between zero and near zero levels of ppGpp (Fig. 3) is in accord with theory (Fig. 5).

3.3. Effect o f RNA polymerase queuing on RNA polymerase activity

According to the queuing model, the absence of guanosine tetraphosphate in ppGpp-less bacterial mutants elimi- nates those RNA polymerase queues that exist in the presence of ppGpp, provided that RNA polymerase concentrations are high. As a consequence, removal of ppGpp during growth in rich media should lead to a significant increase in total RNA polymerase activity ( r t / P ; related to RNAP/P; see Introduction), i.e., because removal of the queues liberates more RNA polymerase for transcription. In agreement with this prediction, an up to three-fold increased RNA polymerase activity has been observed in ppGpp-less bacterial mutants relative to ppGpp-synthesizing wild-type bacteria during growth in rich media [30]. Conversely, when ppGpp levels are raised during the stringent response or by overexpression of relA, ppGpp- mediated transcriptional pausing should greatly increase, which should lead to reduced RNA polymerase activity. Also this prediction is confirmed by experiments: drasti- cally reduced RNA polymerase activity was observed under these conditions [14,34].

In addition to the (few) strong ppGpp-dependent pause sites that we postulate to exist close to mRNA promoters, there may also be minor ppGpp-dependent pause sites scattered throughout mRNA and stable RNA genes. Such sites could lead to frequent transcriptional pausing at high levels of ppGpp and thereby to a general reduction in RNA chain elongation rates, as observed by Vogel et al. [31]. Such frequent transcriptional pausing would contribute to the low RNA polymerase activity observed during the stringent response [46].


According to our model, the plateau of rm/P in Fig. 2 at high growth rates is caused by a saturation of mRNA promoters in wild-type bacteria due to RNA polymerase queuing and promoter blocking. In ppGpp-less bacterial mutants there are, by hypothesis, no queues and removal of pausing liberates more RNA polymerase for transcription. As a result, the RNA polymerase activity in fast growing ppGpp-less bacteria should equally increase mRNA and stable RNA synthesis (unchanged r~/r t ratio). Again, this prediction agrees with observations [30]. This explains why in rich growth media, the rates of stable RNA synthesis (per protein) are similar for ppGpp-less and wild-type bacteria, despite the lower r J r t values in ppGpp-less bacteria.

3.4. Alternative explanation for increased mRNA synthesis in ppGpp-less bacteria

It is a priori possible that the ppGpp-dependent reduced mRNA synthesis in fast growing wild-type bacteria results from a stimulation of synthesis of some specific mRNAs by the absence of ppGpp. To be consistent with observed data [30], one has to postulate that this stimulation occurs mainly in rich media. In this alternative hypothesis, lack of ppGpp stimulates transcription from a new set of genes rather than relieves promoters blocked by queuing RNA polymerases. This idea may be rationalized as follows. It is known that ppGpp-less mutants have multiple amino acid auxotrophies that can be suppressed by mutations in RNA polymerase subunit genes [16]. This indicates that promoters of genes for amino acid biosynthetic enzymes require ppGpp for efficient transcription initiation, consistent with the reported positive control of the his operon by ppGpp [47]. It is conceivable that other essential genes also require ppGpp for efficient transcription, so that the absence of ppGpp might cause physiological stress. As a consequence, the increased rate of mRNA synthesis in ppGpp- less mutants might result from an activation of special stress-relief genes, rather than from the absence of polymerase queuing.

This hypothesis appears unlikely for a number of reasons: (1) protein patterns in SDS electrophoresis gels give no indication of specific, highly expressed polypeptides in the absence of ppGpp (unpublished observations from this laboratory); (2) such physiological stress would be expected to occur during growth in poor media, rather than in rich media [30]; and (3) in the absence of ppGpp, both mRNA synthesis and RNA polymerase activity are en- hanced during growth in rich media [30]. Whereas the stress hypothesis, predicts an increase only in mRNA synthesis, the queuing model predicts increases in the synthesis of both mRNA and stable RNA by removal of ppGpp, exactly as observed.

Since there are no plausible alternative explanations for the increased mRNA synthesis and RNA polymerase activity in fast-growing ppGpp-less bacteria, and because

ppGpp-dependent RNA polymerase pausing has been observed in vitro [41], we suggest that RNA polymerase pausing and queuing does, in fact, occur in vivo during the synthesis of mRNAs, but not during stable RNA chain elongation. For stable RNA synthesis, antitermination mechanisms might prevent or reduce the pausing [48-50].

3.5. ppGpp-mediated transcriptional pausing in vitro and in L, ivo

Previously, transcriptional pausing has mostly been studied in vitro [41,42,51,52]. Transcriptional pausing in vivo has been observed mainly in connection with transcription attenuation control of amino acid biosynthetic enzymes [53,54], or with transcription termination [55]. Only few instances of in vivo transcriptional pausing, not associated with transcription termination or attenuation, have been reported [56]. As mentioned, ppGpp-dependent transcriptional pausing has also been observed in vitro. In vivo, a ppGpp-dependent reduction in RNA chain elongation rates has been observed and ascribed to transcriptional pausing [31]. However, frequent short pauses that reduce the RNA chain elongation rate have to be distinguished from few long pauses near promoters that can give rise to RNA polymerase queues extending all the way back to the promoter. There is no experimental evidence for such queue-generating pause sites, neither in vivo nor in vitro, and neither ppGpp-dependent nor ppGpp-independent. Ob- servations from ppGpp-less bacteria [30], together with our new model (Appendix, below) suggest that this kind of ppGpp-dependent RNA polymerase pausing occurs in vivo and that it is responsible for the low rate of mRNA synthesis in fast-growing wild-type bacteria (Fig. 2).

It would be important to find direct evidence for this kind of pausing in the future. A first step in this direction would be the identification of a gene with a transcriptional pause site that gives rise to polymerase queues. Since it was estimated that 75% of all mRNA synthesized in ppGpp-less bacteria during growth in rich media originates from genes whose activity is normally reduced by polymerase queues, such genes may be found. The criterion to search for is a gene whose expression is negatively affected by changes in RNA polymerase concentration, and this dependence should be correlated with a site a few 100 bp downstream of the promoter.

3.6. Control of stable RNA synthesis by depletion of the .flee RNA polymerase pool through ppGpp-induced transcriptional pausing

The present description of transcriptional growth control has a superficial resemblance with a previous attempt by Jensen and Pedersen [57] to model how RNA polymerase chooses between promoters for stable RNA and mRNA, in a way that depends on the cytoplasmic concentration of ppGpp. Their model is built on the assumption


that promoters for stable RNA have small and mRNA promoters relatively large (kcat/Km)-values for initiation of transcription. They further assume that the kcat-values are higher for initiation of transcription at stable RNA than at mRNA-promoters. When these conditions are fulfilled, initiation at mRNA-promoters is favored at low and initiation at stable RNA-promoters is favored at high concentrations of free RNA polymerase [57].

Jensen and Pedersen postulate that elevated levels of ppGpp induce frequent pausing of RNA polymerase during transcription of both mRNA and stable RNA genes, and that, in this way, ppGpp effectively reduces the rate of RNA-elongation. They consider the pool of total RNA polymerase to be essentially the same under different conditions, so that the main effect of increasing the ppGpp level is to shift a certain amount of RNA polymerase from a free to a gene-bound state, thereby decreasing the concentration of free RNAP. In this way, initiation at mRNA is favored over initiation at stable RNA promoters when the ppGpp level increases. Similarly, a decrease in the ppGpp level induces a relative increase in transcription of stable RNA genes.

The model of Jensen and Pedersen [57] differs from the present theory for transcription control in a number of ways.

(1) When Jensen and Pedersen formulated their theory, the experimentally observed sharp reduction in r J r t, which occurs as the concentration of ppGpp decreases from near to identically zero (Fig. 3 above; [30]), was not known. It is therefore not surprising that their model fails to account for this behavior. In contrast, this observation is, fully accounted for by the present model (Fig. 5).

(2) According to Jensen and Pedersen, the ppGpp-dependent pause sites are frequent and scattered throughout the genes, so that ppGpp reduces the average RNA elongation rate. In their model, therefore, pausing only indirectly affects the saturation of promoters by tuning the level of free RNA polymerase. In contrast, we postulate here that there exist strong ppGpp-dependent pause sites close to mRNA-promoters. When the rate at which RNA polymerase molecules arrive at these promoters is sufficiently high, a queue may form and extend all the way up to the promoter and block further initiation. Therefore, in the present model it is vital where pausing occurs, while the location of the pause sites is irrelevant in the mechanism proposed by Jensen and Pedersen.

Reduced mRNA-chain elongation rates have been found at elevated levels of ppGpp during the stringent response and in mutants deficient in ppGpp-degradation [31,58]. However, at the much lower basal levels of ppGpp that are typical for exponential growth (Fig. 3), the observed changes in RNA-elongation rates observed by Vogel and Jensen [31] are far too small to account for a growth medium control in terms of the model of Jensen and Pedersen.

(3) Jensen and Pedersen have postulated a saturation

with RNA polymerase as an intrinsic property of mRNA promoters. In wild-type bacteria, mRNA synthesis rates were found to be approximately constant at different growth rates (Fig. 2), which might be interpreted as a saturation of mRNA promoters with RNA polymerase. However, in ppGpp-less bacteria, the rate of mRNA synthesis increases with increasing growth rate as much as, and in parallel with, the increasing rate of stable RNA synthesis [30]. This gives no indication of a saturation of mRNA promoters. Instead, we suggest here that the saturation of mRNA promoters in wild-type bacteria results from ppGpp-dependent RNA polymerase queues blocking the mRNA promoters.

(4) Jensen and Pedersen [57] assume that the 'intrinsic' promoter strength is unchanged by the action of ppGpp. According to them, ppGpp acts indirectly through a modu- lation of the concentration of free RNA polymerase by transcriptional pausing. If the concentration of free RNA polymerase were constant, there would be no effect of ppGpp neither on mRNA nor on stable RNA promoters. Their model discounts in vitro observations that indicate a specific inhibition of transcription initiation at stable RNA promoters by ppGpp [18,36-40]. Our model, like the previous RNA polymerase partitioning model [14], assumes that ppGpp also inhibits initiation of transcription at stable RNA promoters directly, as a result of changed ppGpp-bound RNA polymerase-promoter interactions. This assumption is supported by the in vitro observations mentioned above, and also by the in vivo observations that the relationship between ppGpp and r s / r t (Fig. 3) is affected by mutations in rpoD ([5]; personal communications from J. Hernandez and K. Tedin), the gene encoding the o- subunit of RNA polymerase. The o- subunit is required only during initiation of transcription. In addition, a single nucleotide change in the rrn P2 promoter was found to make the promoter activity ppGpp-dependent, like the rrn PI promoter [24].

In summary, a reduction in the RNA chain elongation rate by ppGpp does, indeed, occur [31,58]. At the moderate ppGpp concentrations prevalent during exponential growth, this rate reduction, which may be caused by frequent pausing, is too small to significantly affect the concentration of free RNA polymerase. Therefore, it cannot explain the observed global distribution of transcriptional activities or the rate of stable RNA synthesis (Fig. 3). In contrast, the RNA polymerase queuing hypothesis correctly predicts this distribution, as a comparison of Fig. 3 with Fig. 5 demonstrates.

4. Appendix: theory of RNA polymerase queuing

This Appendix has two parts, A (section 4.1) and B (section 4.2). In part A, the formation of RNA polymerase queues at transcription pause sites, and the effects of such queues on gene activity, are visualized. The idea is to


Pause duration: 5 time units (TU); Site: 160 bp downstream of promoter 1/3 of RNAP pauses

(a) 1 initiation/4 TU: pausing has no effect on rate of transcription

t = 0 •

t = 1 •

t = 2 •

t = 3 m

t = 4

t = 5

t = 6

t = 7

t = 8 0

t = 9

t = i0

t = ii

t = 12 a

1st RNAP-ppGpp binds to promoter

1st RNAP pauses 160 bp downstream of promoter

1st RNAP still pauses

1st RNAP still pauses

2nd RNAP binds to promoter; 1st RNAP still pauses

1st RNAP has cleared pause site, both RNAPs have finished transcript

Promoter remains free, no new RNAP binds


Third RNAP binds to promoter

3rd RNAP has finished transcript



Another RNAP-ppGpp binds promoter

(b) 1 initiation/2 TU: pausing reduces transcription by 14% (1/7)

t = 0 •

t = 1 •

t = 2 0 m

t = 3 _0 e

t = 4 --00•--

t = 5

t = 6

t = 7 m


1st RNAP pauses

2nd RNAP binds to promoter; 1st RNAP pauses

2nd RNAP trapped behind pausing RNAP

3rd RNAP binds, all three RNAPs smiled

1st RNAP clears pause site, all three finish transcript


Another RNAP-ppGpp binds, 2 t ime units after promoter became free

(C) 1 initiation/TU: pausing reduces transcription by 50% (3/6)

t = 0 •

t = 1 0 m

t = 2

t = 3

t -- 4

t = 5

t = 6 •


2nd RNAP binds to promoter; 1-st RNAP pauses

3rd RNAP binds, 1st pauses, 2rid trapped

2nd and 3rd RNAP trapped behind pausing RNAP

2nd and 3rd RNAP trapped behind pausing RNAP

1st RNAP clears pause site, all three f'mish transcript

Another RNAP-ppGpp binds promoter

Fig. 6. Visualization of the effect of increasing RNA polymerase concentration, given by the rate of transcription initiation when the promoter is free, on the expression of a gene with a transcription pause site. • indicates an RNA polymerase that has ppGpp bound and pauses, in this example, for 5 time units; C) indicates an RNA polymerase molecule without ppGpp bound that does not pause. The pause site is assumed to be located two RNA polymerase diameters downstream of the promoter. For more details see text and Table 1.


show individual frames like a time lapse film to provide an intuitive understanding of the model. In the second part B, the queuing theory is derived quantitatively, which is necessary for evaluating the effects of RNA polymerase queuing in vivo.

4.1. A. Visualization of RNA polymerase queues

4.1.1. Definition of terms and assumptions To visualize the effects of RNA polymerase pausing

and queuing according to our model, the following terms and conditions are defined:

(1) A fraction, p, of initiating RNA polymerase has ppGpp bound and pauses at a given pause site for an average time tp.

(2) Only the first pause site at a distance n downstream of the promoter is considered; n is given in RNA polymerase diameters on the DNA (1 unit approx. 80 bp). It includes the length of both promoter and pause site, so that n corresponds to the maximum number of polymerases that can be in a queue from the pause site back towards the promoter.

(3) When the promoter is free, RNA chains are initiated at a given promoter at time intervals t 0, which may be greater or smaller than the pause time (tp). The initiation frequency u 0 is defined as the inverse of t0:u 0 = 1/ t o.

(4) It is assumed that ppGpp-free RNA polymerase does not pause, ppGpp-independent pausing at sites close to promoters would cause a promoter saturation with RNA polymerase in the absence of ppGpp, which has not been observed: despite a decreasing number of active mRNA promoters with increasing growth rate, the synthesis rates of both mRNA and rRNA increase in the absence of ppGpp in direct proportion to the concentration of active RNA polymerase. Therefore, ppGpp-independent pausing is disregarded in the following.

(5) The time for the RNA polymerase to move to the pause site is assumed to be negligible compared to the duration of pauses. This is consistent with in vitro observations that nascent transcripts spend much longer time at pause sites than at intermediate stages.

4.1.2. Visualization of RNA polymerase pausing and queuing

RNA polymerase molecules binding to the promoter of a gene are pictured, using a filled circle ( O ) for an RNAP-ppGpp complex that pauses, or an open circle ( O ) for ppGpp-free polymerases that do not pause. First, we assume that RNA polymerase molecules bind cyclically to the promoter. A cycle consists of c RNA polymerase molecules only one of which contains ppGpp and pauses. When one RNA polymerase-ppGpp complex initiates transcription, it is followed by (c - 1) ppGpp-free RNA polymerases, which completes the first cycle. In the next cycle an RNA polymerase-ppGpp complex binds again, and so on. During one cycle of duration t c, c RNA polymerase

molecules initiate and terminate transcription, c is equal to 1/p, where p is the fraction of initiating RNA polymerase that has bound ppGpp. For visualization we assume that there are no statistical variations in the cyclical order or average pause times. This simplification is omitted in part B of the Appendix.

We will now examine how long it takes to complete one cycle (cycle period t c) and for how much time every RNA polymerase idles, either at a pause site, or arrested in a queue. As an example, we assume the following values to be fixed: (i) p = fraction of initiating RNA polymerase that pauses

---- 1 /3 ( p - - l / c ) ( i i ) tp = time of RNA polymerase pausing = 5 (arbitrary)

time units The following parameters will be varied: (i) t o = time intervals between transcription initiations

when the promoter is free (ii) n = distance from promoter to pause site, which equals

the maximum length of queues The parameters to be determined are:

(i) t c = cycle period, time required for c polymerase molecules to finish their transcripts

(ii) tc/C = average time interval between two initiation events

(iii) u = c / t c = average rate of transcription initiation (in the absence of inhibitory queuing t i c = t o and u =

u0) ( i v ) tavg = average idling time of RNA polymerase

molecules in queues First, we assume that transcripts are initiated at intervals

of four time units (t o = 4) and that the distance between pause site and promoter is n = 3 (Fig. 6a). The three RNA polymerase molecules in a single cycle bind in regular time intervals at t = 0, t = 4, and t = 8 units. The first (ppGpp-bound) RNA polymerase pauses at the site two RNA polymerase diameters downstream of the promoter until t = 5. At t = 12 a polymerase-ppGpp complex binds again. Here the situation is the same as at t = 0, and the cycle repeats. The three transcripts of one cycle are made during a period of 12 time units, so that the cycle period t c equals 12 (t c does not necessarily equal the time of the first recurrence of an earlier situation since transcription is initially not in a steady-state). In this example, ppGpp-dependent transcriptional pausing has no effect on the rate of gene expression, although the initiation intervals are shorter (4 units) than the pause time (5 units). The first RNA polymerase pauses for 5 time units, the second for 1 unit, the third does not pause at all. The average pause time (tavg) is (5 + 1 + 0 ) / 3 = 2 time units.

A superficial inspection of Fig. 6a might suggest that the first RNA polymerase pauses only for 4 time units rather than 5, and that the second does not pause at all, rather than for 1 unit. However, the visualization represents a series of discontinuous events. At t = 4, three situations occur nearly simultaneously: at t = 4 - dt, the


promoter is still free, at t = 4, a polymerase binds, and at t = 4 + dt, the just-bound RNA polymerase has moved and stacks behind the pausing RNA polymerase. Of these three situations, only the middle one corresponding to t = 4, is shown. This is the time point at which RNA polymerase has just bound to the promoter, but not yet moved. Similarly, at t = 5 - dt the second RNA polymerase is still stacked against the pausing RNA polymerase; at t = 5 both RNA polymerases finish their transcripts and move away; at t = 5 + dt, no further change has occurred; the promoter is free to accept a new RNA polymerase.

4.1.3. Promoter blockage increases with increasing RNA polymerase concentration

We will now examine what happens when the RNA polymerase concentration increases. Figs. 6b and c show the effects of doublings of the RNA polymerase concentration with other parameters unchanged. This corresponds to a factor of two shorter initiation intervals t 0. Fig. 6b, at t = 5, shows the situation immediately after the promoter has been cleared. Therefore, the next RNA polymerase will bind 2 time units later, at t = 7, rather than at t = 6. It now takes 7 time units for three transcripts to be initiated in the presence, but only 6 time units in the absence of ppGpp. In this case, gene expression is reduced by queuing to 6 / 7 ( = 86%), corresponding to a 14% inhibition of transcription. The pause times of the three RNA polymerase molecules of one cycle are 5, 3, and 1 unit. This means that doubling of the initiation rate by doubling the RNA polymerase concentration increases the average idling time of RNA polymerase by 50% (tavg = (5 + 3 + 1) /3 = 3, compared to tavg = 2 in the previous example).

Another doubling in the RNA polymerase concentration (reduction in initiation intervals, t o, to 1 unit) further

reduces the promoter activity in comparison to a situation without pausing (Fig. 6c). Between t = 2 and 5 no transcript is initiated, because the promoter is blocked until t = 5. At this time the first RNA polymerase clears the pause site and allows all three polymerases to finish their transcripts. The cycle period, t c, is now 6 time units. Since it would have taken 3 time units to make three transcripts in the absence of pausing, ppGpp reduces the promoter activity by 50%. Again, some RNA polymerase is transiently inactivated while waiting for the pause site to be cleared. The first RNA polymerase pauses for 5, the second for 4, and the third for 3 units. The average idling t i m e , tavg, has now increased to (5 + 4 + 3 ) / 3 = 4 units.

4.1.4. Promoter saturation and inhibition of gene expression

The initiation rate (v = c / t c) in the presence of ppGpp is an average between the rate v o when the promoter is free, and zero rate when it is blocked. The ratio v / v o is the probability that the promoter is free, and the comple- mentary fraction ( 1 - v / v o) is the probability that the promoter is blocked by a queue of length n from the pause site all the way back to the promoter. In the following, the rate u 0 will be used as a measure for the concentration of free RNA polymerase. In addition, the probability p that an initiating RNA polymerase has ppGpp bound will be used as a measure for the cytoplasmic concentration of ppGpp. The in vivo concentrations of free RNA polymerase and ppGpp are generally not independent, since ppGpp affects the RNA polymerase activity: more pausing implies less free RNA polymerase.

Table 1 summarizes the visualizations in Fig. 6 as well as other examples not illustrated; all for p = 1/3 , n = 3, and 5-time-unit pauses, ordered according to increasing RNA polymerase concentrations (~'0 values). With in-

Table 1 Effect of RNA polymerase concentration on expression of a gene with pause Site a

t o t c t c /C v o p lnh tavg /2 0 /~ tavg (TU) (TU) (TU- ~) (%) (TU) (min- ~ ) (s)

5.0 15 5.00 0.20 0.20 0 (5) 1.67 2.0 2.0 10

4.0 b 12 4.00 0.25 0.25 0 (8) 2.00 2.5 2.5 12

3.0 9 3.00 0.33 0.33 0 (14) 2.33 3.3 3.3 14 2.0 b 7 2.33 0.50 0.43 14 (26) 3.00 5.0 4.3 18 1.5 6.5 2.17 0.67 0.46 31 (37) 3.50 6.7 4.6 21 1.0 b 6 2.00 1.00 0.50 50 (52) 4.00 10 5.0 24 0.5 5.5 1.83 2.00 0.55 73 (73) 4.50 20 5.5 27

Assuming a pause duration of 5 time units (TU) and a pause site 2 RNA polymerase diameters downstream of the promoter (maximum length of queue

n = 3 polymerases); every third RNA polymerase has ppGpp bound and pauses (polymerases per cycle c = 3). t o = time interval between consecutive RNA chain initiations when the promoter is free; tp = time required for c transcripts of one cycle to be made; v 0 = rate of RNA chain initiation when the promoter is free ( = 1/to), reflecting the RNA polymerase concentration; v = r a t e of RNA chain initiation in the presence of paus ing=(C/ tp ) ; lnh = inhibition of transcription due to pausing = percentage of time the promoter is blocked by RNA polymerase molecules stacking up from the pause site toward the promoter = (1 - V / V o ) . 100%; values in parentheses were calculated under the assumption of random RNA polymerase binding to the promoter and random variation of pause times (see Appendix A, section 4.1.2); tavg = pause duration averaged over all RNA polymerases, including those that are only transiently trapped behind a pausing RNA polymerase. The last three columns show the rate of gene expression and average pause time in

absolute units after setting 1 TU = 6 s = 0.1 min, so that the 5-TU pause becomes a 30-s pause. b Examples shown in Fig. 6.


creasing v 0, the rate of gene expression in the presence of ppGpp ( v ) approaches a plateau, corresponding to a pro-

moter saturation (Vma x = C/tp; here Vma x = 3 / 5 = 0.6 initi-

a t i ons / t ime unit). At the highest v 0 value shown in Table 1, v has reached 90% of Vma x (v = 0.55). The inhibition of gene expression due to RNA polymerase pausing ( 1 -

Pause duration: 5 time units (TU); 1/3 of RNAP pauses

(a) Pause at promoter: 62% inhibition

t = 0 •

t = 1 •

t = 2 •

t = 3 •

t = 4 •

t = 5

t = 6 0

t = 7 0

t = 8 •

1st RNAP-ppGpp binds to promoter and pauses

1st RNAP-ppGpp pauses




1st RNAP clears pause site (promoter) and finishes transcript

2rid RNAP binds to promoter and finishes transcript

3rd RNAP binds to promoter and finishes transcript

Another RNAP-ppGpp binds to promoter

( b ) D i s t a n c e = 1 R N A P d i a m e t e r s (80 b p ) f r o m p r o m o t e r : 5 7 % inhibition

t = 0 •

t = 1

t = 2

t = 3

t = 4

t = 5

t = 6 0

t = 7 •



2nd RNAP trapped; 1st RNAP still pauses

2rid RNAP still trapped; 1st RNAP still pauses

2nd RNAP still trapped; 1st RNAP still pauses

1st RNAP clears pause site, both RNAPs finish transcript

3rd RNAP binds to promoter and finishes transcript

Another RNAP-ppGpp binds to promoter

(c) Distance = 5 RNAP diameters (400 bp) from promoter: 40% inhibition

t = 0 •

t = 1 O •

t = 2 0 0 •

t -- 3

t -- 4

t = 5 0 0 •

t = 6

t = 7

t = 8

t = 9

t = i0 O0 •

t = 11



3rd RNAP binds, 2nd trapped behind pausing RNAP

4th RNAP binds first 3 trapped behind pausing RNAP

5th RNAP binds 3 RNAPs trapped behind pausing RNAP

6th RNAP binds 3 finish transcript, 4th moves to pause site

7th RNAP binds 2 RNAPs trapped, 4th RNAP pauses

8th RNAP binds 3 trapped behind pausing RNAP

9th RNAP binds 5 trapped behind pausing RNAP

All 6 RNAPs idle

3 RNAPs finish transcript, 7th RNAP moves to pause site

10tb RNAP binds, RNAP 8 and 9 trapped behind pausing RNAP

Fig. 7. Visualization of the effect of increasing the distance of the transcription pause site from the promoter. Same symbols and conditions as in Fig. 6; see text and Table 2 for further details.

28 H. Bremer. M. Ehrenberg / Biochimica et Biophysica Acta 1262 (1995) 15 36

v / v o) increases from zero to 73% when v o increases 10-fold from 0.2 to 2.0 initiations per time unit. 'Inhibi- tion' here means how v relates to v 0 when v 0 increases and v approaches Uma × The average idling time of RNA polymerase (pauses averaged over all RNA polymerase molecules) nearly triples from 1.67 to 4.5 units in this range of polymerase concentrations.

To obtain a realistic range of mRNA initiation rates, a time unit of 0. 1 min was chosen, so that the 5-time-unit pause becomes a 0.5-min pause and the maximum rate of 2 initiations per time unit becomes 20 initiations/min (Table 1, last row). The last three columns in Table 1 show the rates and pause times after conversion with this 0.l-min time unit.

The examples in Fig. 6 and Table 1 demonstrate that ppGpp-induced RNA polymerase queuing depends not only on the ppGpp concentration, which normally decreases with increasing growth rate, but also on the RNA polymerase concentration which normally increases with increasing growth rate. At low RNA polymerase concentrations (small v0), pausing has little effect on the rate of transcription (zero inhibition), no matter how large the fraction (p ) of RNA polymerase that pauses. In fact, the combined effects of ppGpp and RNA polymerase concentration in growing bacteria can lead to more inhibition of gene expression and inactivation of RNA polymerase by queuing at high growth rates (low ppGpp concentrations) than at low growth rates (high ppGpp concentrations). In rich media only a small fraction of RNA polymerase has ppGpp bound and stops at pause sites. However, the much larger fraction of ppGpp-free RNA polymerases may be trapped in queues for long times.

4.1.5. Gene activity and polymerase idling increase as pause site is moved away from promoter

The location of the pause site relative to the promoter is also important. To see this, an idealized gene is visualized as before with the same parameter values: 1,/3 of the RNA polymerase molecules have ppGpp bound and pause for 5 time units. The rate of initiation in the absence of ppGpp (v o) will be kept at 1 initiation per time unit (Fig. 7). If the RNA polymerase pauses at the promoter itself (n = 1), it is blocked for 5 units in a cycle period (t c) of 8 units (Fig. 7a). In this case the promoter blockage is maximal, whereas the average RNA polymerase idling time is minimal: the 1st, 2nd, and 3rd RNA polymerases halt for 5, 0 and 0 units, respectively (tavg = (5 + 0 + 0 ) / 3 = 1.67 units). When the pause site is moved one RNA polymerase diameter away from the promoter (n = 2), the cycle period is shortened (t n = 7 units; Fig. 7b), which means that the promoter activity increases. At the same time, the average RNA polymerase activity decreases: the second (ppGpp- free) RNA polymerase, which binds at t = 1, is now forced to pause almost as long as the preceding RNA polymerase-ppGpp complex (tavg = (5 + 4 + 0 ) / 3 = 3 units). When n = 6, it takes 5 time units to reach a

Table 2 Effect of increasing distance of pause site from promoter '~

n (- 80 bp) u 0 v (ini/min) lnhib (%) t~vg (s)

1 10 3.8 62 16 2 10 4.3 57 18 3 10 5.0 50 24 4 10 6.0 40 28 5 10 6.0 40 34 6 10 6.0 40 48

Same conditions and notation as described for Table 1 : every third RNA polymerase pauses for 30 s. The values shown were obtained by visualizations as shown in Fig. 7, assuming non-random, cyclical binding and fixed, non-random pause durations.

steady-state pattern, so that the first recurrent cycle begins at t = 6 and lasts until t = 1 1 ; it then repeats every 5 time units (Fig. 7c; t,. = 5 units and tavg = (9 + 8 + 7 ) / 3 = 8 units). The DNA template between promoter and pause site fills up quickly with idling RNA polymerase molecules, because the free RNA polymerase concentration has been set so high that the initiation intervals in the absence of ppGpp are shorter (1 unit) than the pause time (5 units). When the RNA polymerase concentration is reduced to the point where the initiation intervals are longer than the pauses, then there are no RNA polymerase queues.

After conversion with the 0.l-rain time unit, the results of these and additional visualizations are summarized in Table 2. As the pause site moves away from the promoter, inhibition of transcription decreases to a minimum level while the rate of transcription (v) approaches its saturation value (//'max = C/tp = 3/0 .5 = 6 initiations/min), and the number of RNA polymerase molecules transiently trapped increases. In summary, we find:

(1) The greater the distance between pause site and promoter, the greater the number of RNA polymerase molecules that must queue behind a pausing RNA polymerase to block the promoter. Therefore, only those pause sites that are near the promoter can, in practice, limit the promoter activity. If, in the case of n = 6 (Fig. 7c), the fraction of initiating RNA polymerase with ppGpp bound (p ) is reduced from 1 /3 to 1 /5 , initiation at the promoter may take place in regular 0. l-min (1-time unit) intervals, although transcripts would be finished only in groups of five every 0.5 min ( = pause time, tp), corresponding to 10 transcripts (two groups of 5) per rain. This means that gene expression is not reduced ( v = v0), despite the 0.5-min pauses.

(2) The average idling time of RNA polymerase (tavg) continues to increase without a limit value if both v 0 and n increase. This time may become longer (0.8 min in the example of Fig. 7c) than the obligatory pause (0.5 min) of the minority of RNA polymerase that has ppGpp bound (Table 2). This happens when the queues are sufficiently long so that ppGpp-free RNA polymerases may have more than one ppGpp-bound RNA polymerase in front of them.


4.1.6. Combined effects of varying ppGpp and RNA polymerase concentrations

For a quantitative evaluation of the effects of transcriptional pausing, the theory of RNA polymerase queuing has been derived (Appendix B, section 4.2.2). This will now be used to examine in more detail the effects of changes in RNA polymerase and ppGpp concentrations on the activity of a given (hypothetical) gene with a pause site. The

calculations were made for a gene with a ppGpp-dependent pause site located 160 bp (two RNA polymerase diameters) downstream of the transcription start (n = 3), and for an average pause duration of 0.5 min (Fig. 8). The left panels in Fig. 8 show the effects of varying RNA polymerase concentrations. The initiation frequency v o is varied between 0 and 60 initiations/min, assuming fixed ppGpp concentrations (given as fixed probability p that an

~ 1 0 0

a. 411

20

0

' ~ 15 E m

;~ ~ 10 o

.c 5

0 3

~ 2 r -

1.~

E 1.0

13.

(] " p-,'l..O. . . . . : . . . . . . . : . . . . . , ] b . . . . . . . . . . . . . . . . . . . . .

... " " ... - - I . " . . . . . . . . . . . .

/ . . :o.:- r e / /

' V _ Z ' - % - - . - - at= i a i t , i v . . . . . i

c

f /

p - 0 . 1 . , .

p - 0.33

...#

" p - 1.0

I I I I

e . . • -p.:.. . . . . . . . . . . . . . . . . . . . . . . . , 1.0

/

f / p - 0 . 1

s

i d

"..U= - 6 0

1,/o = 1 I I I I

f .. . . . . ~,~= . . . . . . . . . . . . . . . . . . . . . . . . . . ~ .

"" / 1 / _ - 10

r e - 1

g . . . " i , ' ; ' i : 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

L/ 0.5 p - 0.~1

~X,~ p - 0 . 1 0.0 , "7

0 10 20 30 40 5 0 v= (initiations/min) [RNA polym, conc.]

• h "" , ° , "

. . ~ V= = 10

. . . ~ , , , , , / v . - 1 ,... , .- " "

0.0 0.2 0.4. 0.6 O.B 1.0 p (Fraction of pauslng RNAP)

[ppGpp cont.]

Fig. 8. Expected results of RNA polymerase pausing from the theory of RNA polymerase queuing; results calculated using the equations derived in Appendix B, section 4.2.2, and assuming a pause site located 2 RNA polymerase diameters downstream of the promoter (maximum length of queues, n = 3). Left panels (a,c,e,g) show the effects of increasing RNA polymerase concentration (% = rate of transcription initiation when the promoter is free); right panels (b,d,f,h) show the effects of increasing ppGpp concentration ( p = probability an RNA polymerase molecule initiating at the promoter has ppGpp bound and pauses; p = 1.0 means that every RNA polymerase has ppGpp bound and pauses). Closed circular symbols in panel (a) represent the values from Table 1, obtained by assuming non-random promoter binding and pausing of RNA polymerase, as in the visualizations of Figs. 6 and 7.


initiating RNA polymerase has ppGpp bound and pauses). The right panels in Fig. 8 show the effects of increasing ppGpp concentration (p varied between 0 and 1.0) at fixed ~'0. The effects on the following parameters were examined: probability of promoter blocking ( 1 - V/Vo; panels a and b), promoter saturation (u; panels c and d), average length of queues (panels e and f), and average polymerase idling time (tavg; panels g and h).

The results in Fig. 8a confirm that inhibition of transcription, due to ppGpp-dependent transcriptional pausing and promoter blocking, may increase without a change in ppGpp concentration, merely as a result of increasing RNA polymerase concentration. Even when only 1 /10 of the initiating polymerases pause (p = 0.1), inhibition may reach values close to 80%, provided that the polymerase concentration (v 0) is sufficiently high. In contrast, if the polymerase concentration is low (e.g., u 0 = 1 initiation/min), inhibition is negligible, even at saturating ppGpp concentrations (p = 1 ; lower curve in Fig. 8b). Fig. 8c shows how initiation of transcription saturates with increasing concentration of free RNA polymerase (v 0) at different concentrations of ppGpp (p). The saturation value of v ( l tmax) decreases with increasing ppGpp concentration. As before, at low RNA polymerase concentrations (v 0 = 1), ppGpp has no effect on the activity of the gene with transcription pause site ( v = u0; Fig. 8d). At high RNA polymerase concentrations, ppGpp-mediated transcriptional pausing severely reduces transcription (Fig. 8d, dotted curve). Figs. 8e and f show how the average length of polymerase queues approaches its maximum as the concentration of either RNA polymerase, or ppGpp, increases. This maximum is determined by the distance between the pause site and the promoter (in this example n = 3). The lengths of queues (panels e and f) roughly correspond to the inhibition of transcription (panels a and b). The inactivation of RNA polymerase by pausing and queuing is more accurately reflected in the average idling time tavg), which reaches a plateau when the RNA polymerase concentration increases (Fig. 8g). tavg also increases with increasing ppGpp concentration (Fig. 8h).

4.2. B. Queuing theory

This part B of the Appendix covers three topics. The first (B 1) describes how RNA polymerase (RNAP) chooses between stable RNA and mRNA promoters. Here the model is built on the idea that RNAP in complex with ppGpp prefers mRNA promoters, while ppGpp-free RNAP prefers stable RNA promoters. The present calculation considers the kinetic properties and response to ppGpp of every type of promoter, whereas Ryals et al. [14] considered only the two classes of stable RNA and mRNA promoters in bulk. The new analysis also gives us an opportunity to correct an error in the previous assumptions. The second topic (B2) describes how queuing RNA polymerases at pause sites downstream of RNA promoters

influence the initiation frequency at these promoters. In the third part (B3) we combine the first two topics in a new model for the global role of ppGpp in growth regulation of stable RNA synthesis. Here a new, important parameter is the concentration of free RNAP in addition to the partitioning of RNAP by ppGpp.

4.2.1. Partitioning between mRNA and stable RNA initiation as a,function of ppGpp concentration

Following Ryals et al. [14], we assume that free RNAP can either be free of ppGpp (type I) or bound to ppGpp (type II). Type I RNAPs prefer stable RNA and type II prefer mRNA promoters. The initiation frequency at the stable RNA promoter i is i~ and at the mRNA promoter j it is imj events per rain per cell volume. These frequencies are given by:

isi = y[ p~i](k~i(I)L + k~i(II)fm) (B1.1)

i~j = y[ Pmj] (kmj(I)f~ + kmj(II)fm) (B1.2)

The symbols used are: y, concentration of free RNAP; [Psi], [ Pmj], concentration of free stable RNA promoters of type i, or of mRNA promoters of type j, respectively; ksi(I), ksi(II), second order rate constants for RNAP of type I, or type II, respectively, to initiate at stable RNA promoter i; kmj(I), kmj(II), second order rate constants for RNAP of type I, or type II, respectively, to initiate at mRNA promoter j; f, , fm, probability for free RNAP to be in configuration I, or II, respectively.

Since free RNAP can be either in configuration I or II, and in no other state, it follows that:

L +fro = 1 (B1.3)

Eqs. (BI.1) and (B1.2) imply that the frequency of initiation increases linearly with the concentration of free RNAP; i.e., the promoters are not saturated with RNAP. This assumption is supported by the observation that RNA synthesis rates per cell volume unit are unchanged in a mutant bacterial strain with reduced DNA concentration [59]. In addition, rRNA and (bulk) mRNA synthesis rates per gene were observed in ppGpp-less bacteria and found to increase in direct proportion to the concentration of active RNAP, without an indication of a promoter saturation [30].

The use of separate k values for type I and II RNAP implies that the promoter strength does not have a unique value; e.g., a promoter that is strong in the absence of ppGpp (like rrn P1) could be a weak promoter in its presence.

The total initiation rates at stable RNA (i s) and mRNA (i m) promoters follow from (B 1.1, 2) by summations:

is = Y'~ isi (B1.4) i

i m = y ' imj J

14. Bremer, M. Ehrenberg / Biochimica et Biophysica Acta 1262 (1995) 15-36 31

The total rate of stable RNA synthesis per volume is given by:

r~ = ~ (i.~i "m~i) (B1.5) i

and the total rate of mRNA synthesis similarly:

r m = E( imj .mmj) (BI.6) J

m~i and mmj are the lengths in bases of stable RNA transcript i and mRNA transcript j, respectively. The transcription rates r~ and r m are available from experiments [14]. We will be particularly concerned with the ratio:

E ( i m j "mmj) rm J ( B 1 . 7 )

r~ ~-~ (isi "msi) i

Using Eqs. (BI.1, 2) we get the more explicit expression:

E Qmj(kmj(I ) ' f , + kmj(II) "fm) rm j - - = ( B 1 . 8 )

r, E Q~i(k,i(I) "X + k,i(II) "fm) i

where we have simplified the notation by setting:

Omj = [ emj] "mmj; Osi = [ Psi] • msi

When [ppGpp] = 0 all RNAP must be in configuration I, and in the limit [ppGpp] ~ oz all RNAP must be in configuration II. In these two limits, the ratio r m / r s will be denoted r I = ( r m / r s ) 1 and r i i=(rm/rs ) t l , respectively. For f , = 1, it follows from Eqs. (B 1.1, 2, 6, 7, 8):

E Qmj kmj(I)

q = J (B1.9)

i

and for f~ = 0:

E Qmj kmj(II)

rll = J (BI.10) E asi ksi(II)

i

We have made the simplifying assumption that the concentrations of stable RNA and mRNA promoters, [Psi] and [ Pmj ], do not change as the system moves from [ppGpp] = 0 to [ppGpp] ~ oz. This is probably correct for stable RNA promoters since most stable RNA comes from rrn genes close to the chromosomal origin of replication (or iC) which is relatively constant, reflecting the constancy of the initiation mass [29,60]. Therefore, the concentration of stable RNA promoters is approximately growth rate- and ppGpp-invariant. The concentration of active mRNA promoters, however, might change with growth conditions, since many mRNA genes are repressed in rich growth

media associated with low ppGpp concentrations, and derepressed in poor media at higher ppGpp concentrations. In addition, the chromosome branching pattern, and with it, the concentrations of all genes except those close to oriC, change with the growth rate. Nevertheless, the assumption of constant [ Pmj ] may be justified for the following reasons: (1) in ppGpp-less bacteria, rm//r s changes very little with growth rate despite the repression of many mRNA genes in rich growth media. (2) Elevated levels of ppGpp can be obtained either by the use of a poor growth medium or by mild amino acid starvation of relA + bacteria. In the first case many mRNA genes are expected to be active, but in the second case these mRNA genes should be repressed because of rising internal pools of unused amino acids and other metabolites [44]. Despite these expected differences in mRNA gene activities, r m / r s was experimentally found to depend only on the level of ppGpp [321.

We now introduce the parameters amj and asi that describe the effects of ppGpp on initiation at particular mRNA promoters j, or stable RNA promoters, i, respectively:

amj = k m j ( I ) / k m j ( I I ) (B1.11)

%i = k~i ( l ) /k~i ( l I )

If, for instance, amj < 1, it means that an RNAP of type II (i.e., in complex with ppGpp) initiates more effectively at the mRNA promoter j than does an RNAP of type I.

The experimentally determined ratios r I and ril are related to the 'microscopic' parameters amj and asi by the expression:

r i / r l l = a m / a s (B 1.12)

where a m and a s are the averages:

E Qmj" kmj(I) ~ Qmj" amj" kmj(II)

J J (Bl .13) a m ~ emj" kmj(II) ~ emj" kmj(II)

J J

a s =

Q~i" k,,i(I) Y'~ Q~i" %i" k~i(II) i i

Q~i" ksi(II) ~ Q~i" ksi(II) i i

In the present model, we assume that a m and a~ do not change when the concentration of ppGpp varies. In the present section, we also do not take ppGpp-dependent saturation of mRNA promoters (due to RNA polymerase queuing) into account. When this is done in section 4.2.3, a suitable redefinition of a m is introduced.

For stable RNAs, only two kinds of promoters, i.e., P1 and P2 of rrn and tRNA genes, have to be considered (i = 1 or i = 2, respectively). The P1 promoters are under stringent control and seem to be nearly inactive at high ppGpp concentrations [44], whereas the P2 promoters are insignificantly affected by ppGpp [23]. This means that

32 14. Bremen M. Ehrenberg / Biochimica et Biophysica Acta 1262 I1995) 15-36

% > 1. For simplicity, c~ is presumed to be independent of factors other than [ppGpp] that might, in principle, affect stable RNA promoter activity. For example, the concentration of the factor FIS, which binds to the upstream activator region of stable RNA promoters [21], is assumed not to be limiting for stable RNA promoter activity.

The constancy of o~ m is more difficult to assess because some mRNA promoters are repressed by ppGpp, like the two dnaA promoters [61]. Others are stimulated by ppGpp, like the his promoter [47], and again others seem unaffected, like r-protein gene promoters [62]. The relative proportions of these different promoters might change as growth conditions vary, but for simplicity, we assume that they do not. It is also not known whether the bulk of mRNA promoters is stimulated by ppGpp, or not. In other words, it is not clear whether c% < or > 1.

If we introduce our new parameters a m and % according to Eq. (Bl .13) and use Eq. (Bl.12), then r m / r ~ in Eq. (B1.8) takes the general form, valid for all ppGpp concentrations:

rm ~m" (1 - - fm) +fm - - = ( B 1 . 1 4 ) r~ OL m • (1 - f m ) / r , q-fm/r l i

In this relationship r m / r ~, r I, and r,i, are available from observations. The factor c~, i.e., the increase in stable RNA promoter strength after removal of saturating concentrations of ppGpp, is potentially observable, but presently it can only be indirectly estimated. With an estimate of c~ s, (Bl .14) can be used to estimate the fraction fm of free (or initiating) RNA polymerase that has ppGpp bound. (Note that f ~ = 1 - fm)"

Updating the RNA polymerase partitioning hypothesis - We will now update the RNA partitioning model of Ryals et al. [14] without taking RNAP queuing effects into account. This will be done below in section 4.2.3.

Instead of r m / r s, Ryals et al. [14] observed r J r , where r t is the total rate of RNA synthesis, given by the sum r t = r s + r m. Instead of the limit values rº and rll for r m / r ~ at low and high ppGpp concentrations, these authors used the limit values of r J r t (or 0s, a related parameter; see below), a s and a m. The parameters a s and a m are related to r I and r n by a s = 1 / ( r I + 1 )and a m = l / ( r n + 1). Introducing these parameters into Eq. (Bl .14) we obtain:

rs Olsfs +fm - - = ( B 1 . 1 5 ) r, OZsf~/a ~ + f m / a m

Ryals et al. [14] derived the formula:

r s / r t =f~a~ +fmam (B l .16 )

(again, they used q,~ instead of r J r t ; tO~ is the fraction of total active RNAP that synthesizes stable RNA; q's = r J r t if the chain elongation rates for stable RNA and mRNA are the same). Ryals et al. [14] have determined experi-

mentally r J r t, a s, and a m, and used Eq. (BI.16) to determine the fractions of free ppGpp-bound (fro), as well as ppGpp-lacking (f ,) , RNA polymerase. Comparison of Eqs. (Bl .15) and (Bl.16) shows that this calculation is valid only if % = (a J a m ) . At very low ppGpp concentrations, r J r t is close to 1.0, meaning that very little mRNA is synthesized compared to stable RNA, so that % = 1. At high ppGpp concentrations r i g approaches its minimum value 0.25, so that a m = 0.25. Thus a J a m = 4. This means that Eq. (Bl .16) is exactly valid only if the combined strength of the tandem P1 and P2 promoters of rRNA and tRNA genes is four-times greater in the absence than in the presence of ppGpp. Therefore, the parameter f~ calculated by Ryals et al. [14] from Eq. (BI.16) should generally differ from the correct f , in Eq. (B 1.15).

The basic assumption of Ryals et al. [14] underlying their Eq. (Bl .16) was that the fraction of ppGpp-free RNAP, f , , decreases linearly from 1.0 to 0 when r J r t decreases from its maximum at [ppGpp]--* 0 to its minimum at [ppGpp] ~ oc. In that case f~ = 0.5 at the ppGpp concentration at which r J r t has a value halfway between its maximum and minimum. Although this seemed plausible, the more rigorous derivation of Eq. (B 1.15) shows that the relationship between f , and r J r t is likely to be nonlinear; only in the special case that a s / a m = 4 it is linear.

4.2.2. RNA polymerase pausing downstream o f mRNA promoters may reduce transcription initiation by a queu-

ing effect Consider an mRNA promoter with a downstream tran-

scription pause site at distance n from the promoter, n is measured in RNA polymerase diameters, and we will assume that transcription initiation cannot occur when n RNA polymerase molecules have piled up at the pause site towards the promoter. Two forms of RNA polymerase are considered, one without (I) and one with ppGpp bound to it (II), as discussed in the preceding section. We assume further that an RNA polymerase of type I will always pass through the pause site, whereas a polymerase of type II will always pause. A polymerase in the pause site leaves it with a first order rate constant kp The reciprocal, 1 /kp, is the average pause time tp used for the visualizations in Figs. 2 and 3. The probability that an RNA polymerase is of type I is f~ and of type II f m = 1 --fs. The initiation frequency v 0 at a particular mRNA promoter and at a particular ppGpp concentration is given by:

P0 = Y(km(I) "fs + km(II) "fm) (B2.1)

Here, y is the concentration of free RNA polymerase, so that y "fs and y "fm are the concentrations of free RNA polymerase of type I and II, respectively. The expressions y . km(I)"f~ and y . km(II) ' fm define the frequencies by which RNA polymerase of type I or II, respectively, initiate transcription at the open promoter. Accordingly,


the current initiation frequency v is given by v 0 times the probability that the promoter is open (free) Pf:

/" = /}0 " P f (B2.2)

The calculation of Pf is the main topic of the present section.

The probability p that an RNA polymerase initiating at a given mRNA promoter is of type II is given by:

km(II) "fro (B2.3)

P = km(I) ' L + km(II) "Urn

Using Eq. (B 1.11), we find that km(II) cancels so that we obtain:

fm P = Cem(l - - f m ) + f m ( B 2 . 4 )

cr m may be different for different promoters. When mRNA promoters are unaffected by ppGpp, then o/m = 1 and p =frn The parameter p reflects the ppGpp concentration, so that when [ppGpp] = 0, then p = 0, and when [ppGpp]

~ , then p o l . We can now rewrite Eq. (B2.2), using Eqs. (B2.1) and

(BI . l l ) :

v = Y " (km(II)" (C~m(1 --fm) +frn ) ) 'P f (B2.5)

where (krn(II). (am(l --fm) +fm)) is the promoter strength averaged for both types of RNA polymerase at a given ppGpp concentration, and the product y . (km(II)" (Crm(1 --fro) +fro)) is the initiation frequency v 0 for open promoters.

Now, let Pl(t) be the probability that a queue of length l has piled up at a given pause site at time t. The number 1 counts polymerases in the pause site itself (zero or one), as well as in an upstream queue. If l = n (= maximum number of polymerases in a queue), initiation of transcription is blocked. When l < n, initiation occurs at rate v 0. As before, we will assume that the time to transcribe from the promoter to the pause site is negligible in relation to the time the polymerase pauses. The probability P~(t) that n polymerases queue at time t determines the probability Pf(t) that the promoter is free:

P f ( t ) = 1 - - P, ( t ) (B2.6)

P,(t) obeys the differential equation:

d P ~ ( t ) / d t = - k p . P n ( t ) + v o . P , _ l ( t ) (B2.7)

The first term on the right side is the probability per time unit that the polymerase at the pause site leaves it. The second term gives the contribution to P,(t) from a queue with length l = n - 1 by initiation. The probability P ,_ j ( t ) , where l < j < ( n - 1 ) , obeys the differential equation:

J d P ~ _ j ( t ) / d t = kp E p(1 - p ) j - i p n + l _ i ( t )

i=1

- - ( v o + kp )P ,_ j ( t ) + v o e , _ j _ l ( t ) (B2.8)

On the right side of this expression are a sum and two simpler terms. The term v 0 Pn ~- ~(t) is the transition rate from a state with n j 1 to a state with n - j polymerases by initiation. The term (v 0 + kp) .Pn_ / ( t ) is the transition rate from queues of n - j polymerases, by initiation (v 0) or by dissociation from the pause site (kp) The sum describes transitions to queues of n - j polymerases from all queues with more than n - j polymerases. How the terms in the sum arise may be under- stood from the special case where n - j = 1, where there is a polymerase at the pause site, but none in the queue. The transition rate from a queue with n polymerases to one pausing polymerase is p ( 1 - p ) n - 2 " k p ' P n ( t ) , because such a transition is conditional on that the first n - 2 polymerases lack ppGpp (probability 1 - p ) , and that the last one in the queue has ppGpp (probability p). To this pathway must be added the transition rates from queues with n - 1 polymerases, n - 2, etc.

In steady-state, these equations are simpler. Here all time derivatives are zero, so that the differential equations become algebraic, recursive relations for the probabilities of queues with 0, 1, 2 . . . n polymerases. The first equation is obtained from Eq. (B2.7):

dPn(t) /dt=O= -kp .P , + v 0 "P, 1 (B2.9)

or after solving for Pn:

Pn = (vo/kp)P~_ , = a . P , 1 (B2.9a)

In this and the following equations, v 0 and kp always appear as the quotient (v0/kp) This implies that any change, e.g., a doubling, of the pause duration has the same effect as a corresponding change in the initiation frequency. The recursive relation for P,_ 1 becomes:

P , _ , = ( V o / ( V o ( 1 - p ) + k p ) ) P , _ 2 = b . P . _ 2 (B2.10)

To simplify notation, we have introduced the parameter b = Vo/(Vo(1 - p ) + kp) = a/(1 + a(1 - p ) ) . The general expression valid for queues of any length l between 1 and n - 1 (1 < l < n) is given by the same type of relation as in Eq. (B2.10):

Pt = b . P i - l (B2.11)

For l = 1, the probability P1 is found to be:

P, = p . bP o (B2.12)

These relationships lead to:

P, = p . b' . P o (B2.13)

Eq. (B2.13) is valid for queues of any length between 1 and n - 1. The probability for the maximum queuing length, l = n, is found by combining Eqs. (B2.9) and (B2.13):

Pn = a . p . b " - l "P0 (B2.14)

34 H. Bremer, M. Ehrenberg / Biochimica et Biophysica Acre 1262 (1995) 15-36

The probability P0 of no queue is found from the condi- tion that the sum of all probabilities must be one, so that:

Po= I - ( P, + P2 + ... P,) = I - ~ P, (B2.15) 1 = 1

Substituting Eqs. (B2.12-14), we obtain:

P0 = 1/(1 + p . B ) (B2.16)

where B is given by:

n - 1

B = ~ b J + a . b n-1 j=l

Substituting P0 into Eq. (B2.14) and using Eq. (B2.6), we obtain for the probability Pf that the promoter is free:

P f = 1 - P , = 1/(1 + a ) (B2.17)

where A is defined as:

A = a p b " - l / ( 1 + p ( b " - b ) / ( b - 1))

Here we have explicitly calculated the geometric sums. In the case when n = 1 (polymerase pauses at the promoter), one should have an ordinary Michaelis-Menten saturation curve for the promoter. Setting n = 1, we get from Eq. (B2.17):

P f = 1/(1 +ap) = 1/(1 +P(Vo/kp) )

= 1 / ( 1 + p ' y ' k a / k p )

If the identifications p . y = [S] and (ka/k p) = 1 / K M are made, then the well known saturation curve for enzyme kinetics is recovered.

4.2.3. Evaluation of in vivo observations involving ppGpp The original observations by Ryals et al. [14] of the

ratio r J r t at different ppGpp concentration in the cyto- plasm could be described by a simple model where the presence of ppGpp on the RNA polymerase guides it toward mRNA promoters (see section 4.2.1 above). This model correctly predicts how r J r t varies from 0.25 at high to about 0.9 at low ppGpp concentrations. However, it fails to predict the decrease in r J r t as ppGpp goes from low to zero concentration [30]. This discrepancy between theory and observation has led us to suggest that there is a ppGpp-dependent probability for RNA polymerase to pause downstream of mRNA promoters (section 4.2.2). In the present section we combine the results in sections 4.2.1 and 4.2.2 in a model that reproduces the experimental relationship between [ppGpp] and r J r t (Fig. 3).

With experimental support, we assume that initiation of transcription at stable RNA promoters does not saturate, even at the high RNA polymerase concentrations associated with very rich growth media. Initiation from mRNA promoters, in contrast, is inhibited by queuing as discussed in section 4.2.2.

Saturation of mRNA promoters by transcriptional pausing and RNA polymerase queuing depends on the concen-

trations of both ppGpp and RNA polymerase. To include these effects, Eq. (B 1.4) can be generalized by substituting the Qmj'S in Eq. (B1.8) with the products Qrnj " Pmj' Qmj is now defined as the concentration of the derepressed mRNA promoter j, irrespective of its degree of saturation by polymerase queuing, multiplied by the number of bases mj of the gene (see Eq. B I.8). Pq is the probability that an unrepressed mRNA promoter is unblocked by queuing polymerases, so that the product Qmj " P0 is the concentration of open mRNA promoters of type j times the length mj of that gene. With these definitions, Eq. (B1.8) is now generalized to:

Qmj (kmj(I) 'L + kmj(II) "fro )" PrY rm j

- - = ( B 3 . 1 )

rs Ees (k i(O f , + k i(II) "fro) i

If we make the further simplification that the degree of inhibition by queuing is uniform among all mRNA promoters, so that Pfj = Pf for all j, Eq. (B3.1) takes the form:

rm am(1 --frn) +fm - - "Pf (B3.2)

r s O ~ m ( l - - f m ) / / r l + f m / r t ,

where am, fro' rI and ri1 are defined in Eqs. (B1.13, 1, 9, 10), respectively. The probability Pf that an mRNA promoter is free and not blocked by queuing RNA polymerase molecules, is given in Eq. (B2.17). Pf can be written as:

n - - 1

l + p . ~ b j j=l

e l = n-I (B3.3) 1 + p . ~ b J + p . a . b n-l

j=l

The parameters a and b, introduced in the previous section, are defined as

a = (vo/kp) (B3.4)

b = a / ( 1 + a ( 1 - p ) ) (B3.5)

The initiation frequency v 0 at a free mRNA promoter is given by (B2.1). The probability p that an initiating RNA polymerase has ppGpp bound is related to the probability fm that a free RNA polymerase is in complex with ppGpp according to Eq. (B2.4). In the present model the ratio rm/r s, or rs/r t = 1 / ( 1 - rm/r~), is determined by two factors that are related to the concentration of ppGpp and of free RNA polymerase. The first factor, discussed in section 4.2.1 above, describes bow an RNA polymerase molecule to chooses between free mRNA and stable RNA promoters. It varies from r I = 1 at zero (fro = 0) to r . = 3 at high ppGpp concentrations (fro = 1) in a way that depends on the parameter o% defined by the relationship Eq. (B 1.12).

An important property of our model is that P~. tends to one when the free RNA polymerase concentration y tends


to zero, irrespective of p and thus of the ppGpp concentration. Another crucial feature is that Pf also tends to one when p goes to zero, irrespective of the value of the free RNA polymerase concentration y.

The observed curve r J r t vs. [ppGpp] in Fig. 3 can now be explained as follows: at zero ppGpp concentration there is no blocking of mRNA promoters since only RNA polymerase molecules in complex with ppGpp can pause. Here Pf = 1, so that r s / r t = 1/(1 + r I) = 0.5.

As the ppGpp concentration also increases, p increases rapidly. Since y is high, there will be substantial blocking of mRNA promoters by queuing. This leads to a drastic decrease in rm/ / r s so that r s / r t increases from 0.5 to its maximum value near 0.9. When the concentration of ppGpp increases further, the concentration of free RNA polymerase decreases and reduces queuing. This increases rm/r,~ so that r , J r t decreases from its maximum value in Fig. 3. At very high ppGpp concentrations the RNA polymerase concentration is, by assumption, so low that there is no inhibition by queuing. Here r s / r t = 1/(1 + r l l )

= 0.25. To show quantitatively how this works we first assume

that fro, the fraction of free RNA polymerase that has ppGpp bound, is related to the ppGpp concentration by a simple binding isotherm, like

fm = X / ( X + K ) (B3.6)

where x = [ppGpp] and K is the binding constant for the interaction between ppGpp and RNA polymerase. Alterna- tively, we may assume with Ryals et al. [14] that fm corresponds to the zero term of a Poisson distribution, in which case

fm = 1 - 2 -x/K (B3.6a)

where K is the concentration of ppGpp at which half of the free RNA polymerase has interacted with ppGpp. It follows from Eq. (B2.4) that

p = x / ( Of. m " g -~- x ) (B3.7)

Using Eq. (B3.1, 6) we obtain:

r m of. m " K + x - - = "P t (B3.8) r~ a m • K / r ~ + x / r n

From Eqs. (B2.5) and (B3.3) we get further

u 0 = y " km(I I ) • ( o/m • g -I- x ) / ( g -1-" x ) ( B 3 . 9 )

How the concentration of free RNA polymerase, y, varies with ppGpp in the cell is not known. We will make the assumption that

y = c 2 / ( x + Cl) (B3.10)

where c~ and c 2 are two constants. When this relation is introduced in Eq. (B3.9), and setting km(II)= kp, we get the expression:

a = (v0 /kp) : ( c 2 / ( x + C 1 ) ) " (( a m " K + x ) / ( K + x ) )

(B3.11)

where c l, c 2, O/m, and K are adjustable constants. Using Eq. (B3.7) to calculate p and Eq. (B3.11) to calculate a, we can now also calculate the parameter b in Eq. (B3.5), and with a, b, and p, obtain the probability, Pf, that a mRNA promoter is free (not blocked by polymerase queues), using the relationship in Eq. (B3.3). This requires to choose a further free parameter for the location of the pause site, given by n, the maximum number of RNA polymerase molecules that can stack up in a queue. Finally with Pf, we can calculate r m / r ~ from Eq. (B3.8), using the additional observed parameters r I and r u. After conversion of r m / r ~ into r J r t = 1 / ( 1 - rm/rs) , the simu- lated curve in Fig. 5 (dashed line and open circles) was obtained with the parameter values c I = 1 pMol/OD, c 2 = 70, O~ m = 2, K = 3 pMol/OD, n = 4, r I = 1 [30] and rli = 3 [14].

Acknowledgements

This work was supported by grants from the NIH and from the Swedish Natural Science Research Council. We thank Erik Boye, Gordon Churchward, Karsten Tedin, and Ryland Young for helpful suggestions during the prepara- tion of this manuscript.

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