noise in gene networks

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Quarterly Reviews of Biophysics 34 , 1 (2001), pp. 1–59. Printed in the United Kingdom

2001 Cambridge University Press

1

Noise in a minimal regulatory network:

plasmid copy number control

Johan Paulsson* and Ma ns Ehrenberg

Department of Cell and Molecular Biology, Biomedical Center Box 596, SE 75124 Uppsala, Sweden

1. Introduction 2

2. Plasmid biology 3

2.1 What are plasmids? 3

2.2 Evolution of CNC: cost and benefit 4

2.3 Plasmids are semi-complete regulatory networks 6

2.4 The molecular mechanisms of CNC for plasmids ColE1 and R1 6

2.4.1 ColE1 72.4.2 R1 7

2.5 General simplifying assumptions and values of rate constants 9

3. Macroscopic analysis 11

3.1 Regulatory logic of inhibitor-dilution CNC 11

3.2 Sensitivity amplification 12

3.3 Plasmid control curves 13

3.4 Multistep control of plasmid ColE1: exponential control curves 14

3.5 Multistep control of plasmid R1: hyperbolic control curves 16

3.6 Time-delays, oscillations and critical damping 18

4. Mesoscopic analysis 20

4.1 The master equation approach 20

4.2 A random walker in a potential well 234.3 CNC as a stochastic process 24

4.4 Sensitivity amplification 26

4.4.1 Single-step hyperbolic control 26

4.4.2 ColE1 multistep control can eliminate plasmid copy number variation 28

4.4.3 Replication backup systems – the Rom protein of ColE1 and CopB of R1 29

4.5 Time-delays 30

4.5.1 Limited rate of inhibitor degradation 30

4.5.2 Precise delays – does unlimited sensitivity amplification always reduce plasmid losses? 32

4.6 Order and disorder in CNC 33

4.6.1 Disordered CNC 34

4.6.2 Ordered CNC : R1 multistep control gives narrowly distributed interreplication times 34

4.7 Noisy signalling – disorder and sensitivity amplification 37

4.7.1 Eliminating a fast but noisy variable 38

4.7.2 Conditional inhibitor distribution: Poisson 39

4.7.3 Increasing inhibitor variation I: transcription in bursts 40

* Author to whom correspondence should be addressed. Present address: Department of Molecular

Biology, Princeton University, NJ 08544, USA. E-mail: paulssonprinceton.edu



2 Johan Paulsson and Ma ns Ehrenberg

4.7.4 Increasing inhibitor variation II: duplex formation 41

4.7.5 Exploiting fluctuations for sensitivity amplification: stochastic focusing 44

4.7.6 A kinetic uncertainty principle 45

4.7.7 Disorder and stochastic focusing 46

4.7.8 Do plasmids really use stochastic focusing? 47

4.8 Metabolic burdens and values of in vivo rate constants 48

5. Previous models of copy number control 495.1 General models of CNC 49

5.2 Modelling plasmid ColE1 CNC 49

5.3 Modelling plasmid R1 CNC 52

6. Summary and outlook: the plasmid paradigm 53

7. Acknowledgements 56

8. References 56

1. Introduction

This work is a theoretical analysis of random fluctuations and regulatory efficiency in genetic

networks. As a model system we use inhibitor-dilution copy number control (CNC) of the

bacterial plasmids ColE1 and R1. We chose these systems because they are simple and well-

characterised but also because plasmids seem to be under an evolutionary pressure to reduce

both average copy numbers and statistical copy number variation: internal noise.

It is often hard to guess whether noise in concentrations of cellular components is

advantageous or disadvantageous, but for plasmids we believe the picture is unusually clear.

The net growth rate of plasmid-containing cells depends on the metabolic burden that

plasmids confer on their hosts as well as the frequency with which plasmids are lost at cell

division. Since cells with a lower than average copy number can have a drastically increased

loss probability, the rate of loss in a cell population can increase greatly with random copy

number variation (Nordstro m & Austin, 1989). The problem is brought to a head by the fact

that plasmids are replicators: when the synthesis rate increases with copy number, the ability

to check random deviations requires that the elimination rate increases even more.

Elimination of plasmids occurs at cell divisions when the individual copies are distributed

between the two daughters. This is qualitatively similar to a first-order degradation rate and

combined with first-order synthesis it makes for die-hard random deviations that only are

limited by extinctions and reduced host resources. Both effects are highly unfavourable for

plasmids and, to our knowledge, all plasmids studied indeed code for CNC systems that

down-regulate the replication frequency as a response to an increase in copy number. Here

we address some putative designs of the negative feedback loops used by ColE1 and R1.

The suggestion that cells use negative feedback to neutralise internal noise was recently

supported by forward engineering experiments on a minimal network where a protein

inhibits its own expression (Becskei & Serrano, 2000). Using a popular analogy, the protein

concentration behaves like a random walker in a bowl. The negative feedback makes the ratiobetween its elimination and formation rates more sensitive to concentration changes. In other

words, it increases the steepness of the bowl walls and thereby restricts the random walk more

efficiently.

This article looks into (i) how sensitivity amplification in the response to changes in



3Noise in a minimal regulatory network

inhibitor concentration is used to check internal noise; (ii) how some types of amplification

scheme are designed to give high sensitivity far from steady state and therefore cannot be

described by linear perturbation theory; (iii) how delays due to e.g. long inhibitor half-lives

make the response more sluggish; (iv) how the impact of noise in the rate constants

(dynamical disorder) depends on the time-scales of the system; (v) how some kinetic designs

can exploit correlations between the noise of two different rates to neutralise dynamical

disorder; (vi) how noise can be attenuated by decreasing the statistical variation in the timebetween two subsequent replications of the same copy; (vii) how inhibitors only represent

plasmids in a probabilistic sense; (viii) how a probabilistic representation can induce

dynamical disorder as well as affect sensitivity amplification; and (ix) how an increased

sensitivity amplification due to inhibitor noise (stochastic focusing) means that noisy

signalling may be required to reduce internal noise in the plasmid copy number.

Some basic concepts in parts (i)–(iii) can be understood from a macroscopic perspective,

while (iv)–(ix) are either invisible or show how macroscopic approaches are flawed even for

describing averages. Stochastic analysis is then necessary to understand why the kinetic

circuitry is wired as it is. Here we use simple birth-and-death master equations, solved either

analytically, by numerical integration or, to a limited extent, using Monte-Carlo simulations

(Gillespie, 1977).

Since plasmids use host resources for replication, transcription and translation, and these

processes also determine plasmid copy numbers and loss rates, the biochemistry of CNC

could in principle be linked to evolution. This could quantitatively point at the trade-off

between regulatory efficiency and metabolic burden. However, little is known about how

growth rates respond to small changes in the available amount of, for example, RNA

polymerase. The conditions also change continuously as plasmids co-evolve with their hosts.

This requires more experiments to back up theory and here we therefore only model

internal noise as a function of the kinetic parameters that in turn represent non-quantified

metabolic burdens.

2. Plasmid biology

2.1 What are plasmids?

Plasmids are extrachromosomal, non-virulent gene clusters ubiquitously found in pro-

karyotes. The first steps towards their discovery were taken in the 1940s when E. coli cells

were observed to mate and horizontally transfer genetic material. The fertility factor F

received much attention in the following decades. One reason was its usefulness in genetic

analyses and another that it started epidemics of bacteria with multiple resistances to

antibiotics. Infamous for its connections to sex and drugs, F was isolated in 1961 (Marmur

et al . 1961) and shown to be an extrachromosomal DNA molecule.

Because plasmids are extrachromosomal, they can be lost at cell division if all copies by

chance end up in one of the two daughters. The loss rate largely depends on (i) the number

of plasmid copies at cell division; (ii) the extent of plasmid multimerisation; and (iii) thedivision of plasmid copies between the two daughter cells. Copy numbers are regulated by

plasmid-carried CNC genes (for reviews see Nordstro m & Austin, 1989; Summers, 1998;

Chattoraj, 2000; del Solar & Espinosa, 2000); multimers are resolved by a partly

plasmid-encoded molecular machinery (Summers & Sheratt, 1984; Summers et al . 1985, 1993)




and many low-copy plasmids code for an active partition mechanism ( par +) (Austin & Abeles,

1983; Nordstro m & Austin, 1989). Knocking out any of these systems can increase plasmid

loss rates by orders of magnitude.

Plasmids provide numerous traits that allow their hosts to exploit new niches. Once

established in a cell population, they can also try to make themselves indispensable. One

example is the diversity of plasmid coded bacterial toxins and their corresponding antidotes

(for a review see Riley, 1998). As a reply to SOS response, the plasmid may force the cell tocommit suicide by overproducing the toxin and releasing it into the environment. This can

provide a selective pressure for antidote-producing plasmids in the population. A similar

poison–antidote strategy is involved in the post-segregational killing (PSK) of plasmid-free

segregants (Koyama & Yura, 1975). These plasmids code both for stable toxins and their

unstable antidotes. A cell that loses the plasmid soon loses the antidote and is killed by the

toxin.

Plasmids can help cells to rapidly acquire exotic genes when these are needed. However,

plasmid-containing cells compete both with cells that do and do not code for the same genes

on their chromosomes. This means that their stable existence on an evolutionary time-scale

cannot be taken for granted. The conditions required for plasmid existence thus deserve close

inspection (Bergstrom et al . 2000).

2.2 Evolution of CNC : cost and benefit

The evolutionary success of a plasmid–host system will, for a given set of plasmid carried

genes, be determined by the plasmid loss rate and the metabolic burden required to reduce

losses (Stewart & Levin, 1977; Wouters et al . 1980; Cooper et al . 1987; Chiang & Bremer,

1988; Proctor, 1994; Paulsson & Ehrenberg, 1998). When plasmids are essential for the

survival of their hosts, selection favours maximisation of the net-growth rate of the

plasmid–host system. However, under non-selective conditions, plasmid losses result in

competing plasmid-free cells. Since these cells are not burdened by plasmid metabolism they

may grow faster, which can result in a rapid wash-out of plasmid-containing cells (Wouters

et al . 1980; Modi et al . 1991). If selection favours plasmids only intermittently, it may thus

be in their long-term interest to delay the emergence of the first plasmid-lacking competitorfor as long as possible. Accordingly, it may be selectively advantageous to keep plasmid losses

extremely low, even if the price in terms of an impaired growth rate due to metabolic

costs is quite high.

The probability that plasmids are lost at a particular cell division depends on the number

of plasmid copies present in the mother cell. This number will vary from cell cycle to cell

cycle in a way that is largely determined by the CNC system. Since large random copy

number variation can drastically increase the plasmid loss rate in the cell population (the low

loss rate from cells with high copy number does not compensate for the high loss rate from

cells with low copy number), it is of particular interest to relate CNC to the copy number

distribution in cells about to divide. Two principal strategies for reducing the loss rate can

immediately be identified: increasing the average copy number and reducing copy number

A similar argument can perhaps be made for metabolic burdens: the increased growth rate of cells

with low copy number cannot compensate for the decreased growth rate of cells with high copynumber. Although this is pure speculation, it is not unlikely since many synthetic pathways in cells are

saturated so that an increase in a certain concentration may have less effect than a correspondingdecrease.




variation. Maintaining a higher average copy number requires more of the host’s resources

and thus imposes a larger metabolic burden. Reducing copy number variation requires

efficient CNC systems. High efficiency often requires high turnover of regulatory molecules

and thus also impairs host growth.

The above suggests that there is an optimal point of operation where plasmid losses are

sufficiently low and metabolic burdens not too high. Cells containing plasmids that reduce the

loss rate too much would grow very slowly. By contrast, cells with plasmids that spend verylittle of the cell’s resources on reducing loss rates would grow more rapidly but also more

frequently lose their plasmids. The optimal point of operation, where plasmid–host systems

will out-grow their competitors, is somewhere between these extremes.

If the regulatory machinery of cells swiftly compensated for minor shortages in certain

resources, the cell economy could roughly be described in terms of, for example, ATP

consumption. However, it may also be the case that one or a few processes are rate-

limiting under any given conditions and that what is rate-limiting differs from environment

to environment. This puts the metabolic burden of plasmids in perspective: plasmids may

try to avoid even a small drainage of the cell’s resources not because they normally

would reduce host fitness, but to reduce the risk that they at some point would affect

the rate-limiting process. The picture is further complicated by the fact that plasmids may

interfere with intracellular processes in obscure ways (Summers, 1996).

Regardless of these complications, the optimal strategy generally depends on the costs

associated with plasmid copy numbers (replication and gene expression) in relation to the

costs associated with efficient CNC (turnover of regulatory components). For large plasmids

it is favourable with few plasmid copies per cell in combination with an efficient but perhaps

costly CNC, while for small plasmids it is better to increase the average copy number in

combination with a simpler and less costly CNC. A quantitative cost–benefit analysis could

make it possible to create kinetic fitness landscapes where the net growth rate of the

plasmid–host system is expressed in terms of the rate constants of CNC. Such an analysis

could explain how the in vivo rate constants have evolved, but could also be used to evaluate

mechanisms related to CNC. It could for instance show under what conditions partitioning

systems that actively distribute plasmids between the daughter cells are cost-effective.

Although extensive experimental data on CNC have become available for many different

plasmid systems, computation of realistic fitness landscapes will perhaps always be out of

reach. In this study, we will therefore quality-rank two control designs, CNC

and CNC,

simply by comparing their corresponding plasmid loss rates for a given average copy number.

In some cases we will also calculate how much the average copy number must be adjusted

up or down for CNC

to have the same loss rate as CNC. In this way the quality of CNC

is always described in the same unit.

The relation between function and evolution is far from straight-forward. For illustration,

assume that a plasmid–host system operates under optimal growth conditions as defined

above. Any mutation to a higher copy number would then bring it away from its optimum

and reduce its fitness in relation to the wild type. However, plasmid mutants that partially

escape the constraints of CNC outreplicate the wild type copies inside single cells. CNC thusintroduces an additional level of selection and the CNC designs observed in nature are shaped

by two opposing selective forces: competition between plasmid copies in single cells and

competition between hosts in cell populations. Selection on the intracellular level can be

expected to cause genetic drift to higher average copy numbers and will in general be a




driving force generating CNC variation and speeding up the evolution of CNC incompatibility

groups.

2.3 Plasmids are semi-complete regulatory networks

Many intracellular control systems are entangled in a complex web of cellular networks,

which obscures their experimental and theoretical analyses. By contrast, although plasmids

rely on the intracellular environment of their hosts, their basic control elements are plasmid

specific and they are therefore easier to experimentally study and theoretically describe with

the host cell as a background. In addition, plasmids only code for a few signal molecules,

sometimes just one, and CNCs of many plasmids have been thoroughly analysed on a

molecular level. This makes them important model systems for intracellular regulation in

general and replication control in particular.

Selective forces work on CNC to reduce plasmid copy number variation and thereby the

plasmid loss rate, but where does this variation come from in the first place? From a common

quasi-deterministic perspective, it might be suspected that fluctuations are due to infrequent

external disturbances or rare failures in the steps that lead to the initiation of replication.

However, the cause of fluctuations is much more fundamental than occasional system

failures; it is a manifestation of the intrinsically random nature of all chemical reactions. Forpractical reasons this is often neglected in kinetic modelling, but when low numbers of

reactions and molecule copies are considered, such simplifications are both inaccurate and

misleading. In fact, if plasmids did not regulate initiation of their replication, variation in

copy numbers would inevitably become very large. Even in a hypothetical extreme (and

biochemically bizarre) situation where initiation of plasmid replication were so regular that

every plasmid copy replicated exactly once per cell cycle, copy number variation would slowly

but surely accumulate until it totally overshadowed the average, as verified experimentally

with mini-chromosomes (Løbner-Olesen, 1999). Such an accumulation of variation is not

standard in kinetics but is due to the fact that every plasmid molecule has the capacity to

initiate its own replication. Regular replication could delay the emergence of variation from

some historic initial value, but without the capacity to check deviations (CNC), fluctuations

would inevitably increase all the way to the point where plasmids are very frequently lostfrom some host cells and impose an intolerable metabolic burden on others. Both these effects

are strongly selected against, constituting an evolutionary niche for CNC. In short, for

plasmids to live in reasonable harmony with their hosts, plasmid replication should not be

limited by depletion of the host’s resources, but be determined by the internal dynamics of

CNC. Within broad limits the effective restrictions on plasmid replication come from the

regulatory reactions that constitute CNC, i.e. from the plasmid itself rather than from its

intracellular environment. In a regulatory sense, plasmid CNC is thus semi-complete.

2.4 The molecular mechanisms of CNC for plasmids ColE1 and R1

The present paper deals exclusively with ColE1, R1 and other plasmids where CNC is based

on the principle of inhibitor-dilution. The regulatory logic of these plasmids is that the rate-limiting step(s) in the initiation of plasmid replication is negatively regulated by a plasmid-

encoded inhibitor. The inhibitor is produced at a constant rate per plasmid and spontaneously

An incompatibility group is a group of plasmids that cannot coexist in the same host cell due tointerference between their CNCs.




degraded with short half-life. An increase in the number of plasmids in a cell then results in

an increase in the number of inhibitors, thereby reducing the plasmid replication frequency

per plasmid and vice versa. Therefore the quality of CNC depends both on how the current

number of inhibitors represents the current number of plasmids and how sensitively the per

plasmid replication frequency responds to changes in the number of inhibitors.

2.4.1 ColE1

ColE1 is a small (6–7 kb) high-copy number plasmid (about 10–40 copies per cell depending

on physiological conditions) that seems to lack an active partition system ( par −). ColE1

replication is regulated by the inhibition of the maturation of a cis -acting RNA replication

primer precursor (RNA II) by a trans -acting antisense RNA (RNA I) (Tomizawa &

Itoh, 1981; Tomizawa, 1984; Masukata & Tomizawa, 1986) (Fig. 1(a)). Cis -action means

that RNA II can only activate replication of the plasmid from which it is transcribed and

trans -action means that RNA I can inhibit primer precursors on all plasmids in the cell.

Transcription of RNA II is initiated 555 base pairs (bp) upstream of the origin of plasmid

replication (ori ). RNA I is about 110 nucleotides (nt) long and is transcribed from the

complementary strand, starting 445 bp upstream of the ori (Morita & Oka, 1979; Lacatena

& Cesareni, 1981; Tomizawa & Itoh, 1981; Tomizawa et al . 1981). RNA II may act as aprimer for plasmid replication unless it is inhibited by RNA I. Inhibition is only effective

when transcription proceeds through a so called inhibition window, approximately extending

from base 110 to 360 in the RNA II coding region (Tomizawa, 1986). During this time, the

RNAs may form a reversible initial ‘ kissing’ complex (Tomizawa, 1990a) that can be

converted into a stable duplex. When RNA II is shorter than 110 nt, RNA I binds inefficiently

and when it is longer than 360 nt, RNA I can still bind to RNA II but has very little effect

on primer formation (Tomizawa, 1986). The dimeric form of a small plasmid encoded

polypeptide, the Rom protein, stabilises the RNA I–RNA II interaction, increasing the

probability of duplex formation from an initial ‘ kissing ’ complex (Cesareni et al . 1982;

Lacatena et al . 1984; Tomizawa & Som, 1984; Tomizawa, 1990b). If RNA I does not bind

to RNA II in the inhibition window, RNA II transcription through ori results in the

formation of a stable DNA–RNA hybrid. The RNA II part of the hybrid is subsequentlycleaved by RNase H, creating a mature replication primer that can be recognised by DNA

polymerase I (Itoh & Tomizawa, 1980). Constitutive synthesis and rapid degradation of RNA

I makes its concentration approximately proportional to plasmid concentration (Bremer &

Lin-Chao, 1986; Brenner & Tomizawa, 1991; Merlin & Polisky, 1995). Both synthesis of

Rom mRNA and its subsequent translation are believed to be constitutive, but the Rom half-

life is to our knowledge unknown. If Rom concentration followed changes in plasmid

concentration and the changes in Rom concentration affected the interactions between

RNA I and RNA II in a way that significantly affected the probability of inhibition,

Rom could work as a second inhibitor. Such a double layer control system could increase

the efficiency of CNC and reduce plasmid loss rates (see Section 4.4.3; Ehrenberg, 1996).

2.4.2 R1

R1 (see Summers, 1996; Nordstro m & Wagner, 1994, and references therein) is a low copy

number plasmid of the incFII group (about 3–6 copies per cell depending on physiological

conditions, see Engberg & Nordstro m, 1975; Gustafsson & Nordstro m, 1980; Light &




Fig. 1. Molecular mechanisms of ColE1 and R1. (a) Schematic overview of the replication primingprocess of plasmid ColE1. Both the preprimer RNA II (bases 0–555) and the inhibitor RNA I (bases

2–110 from theopposite strand) areconstitutivelyexpressed. RNAI canbind to RNAII andform a stableduplex. If binding occurs during RNA II synthesis in an ‘inhibition window’, stretching roughly from

base 110 to 360 in the RNA II gene, conformational changes in RNA II are triggered and subsequentprimer formation is inhibited. (b) Schematic overview of the CNC region of plasmid R1. The P

copB




Molin, 1982) that carries a partition system ( par +). Ideally this ensures that one plasmid copy

enters each daughter cell in all cases where the mother cell contains at least two copies. The

presence of a very effective partition system means that the major contributions to the plasmid

loss rate come from cells with a single plasmid at the end of the cell cycle. The primary task

of CNC is then to reduce the probability for this event. Traditionally one separates between

pair-site partitioning and equipartitioning. Equipartitioning means that plasmids are divided

as evenly as possible between the daughters (e.g. 41 copies in the mother become 20 and 21in the daughters). Pair-site partitioning means that one copy is actively partitioned to each

daughter, while the rest segregate randomly.

The rate-limiting step in the initiation of plasmid R1 replication is the synthesis of the

replication protein RepA (Masai et al . 1983; Masai & Arai, 1988). RepA molecules bind to

the ori of the plasmid copy from which the RepA mRNA was synthesised (cis -action). After

a large enough number (Masai et al . 1983) of RepA molecules have accumulated at ori ,

conformational changes are triggered that allow the replication machinery to initiate

replication of that plasmid copy. Following replication, the RepA molecules are removed and

a new set of RepA molecules must be expressed from each of the two daughter copies before

they can replicate again. RepA synthesis is inhibited by the plasmid encoded antisense RNA

(CopA), binding to a target sequence (CopT) upstream of RepA mRNA (Stougaard et al .

1981). Synthesis of RepA mRNA is initiated from two different plasmid promoters, PcopB

and

PrepA

(Light et al . 1985). Transcripts from the constitutively active PcopB

include CopB

mRNA, CopT and RepA mRNA. The tetrameric configuration of the CopB gene product

represses the downstream promoter PrepA

. PrepA

transcripts lack CopB mRNA but are

otherwise identical to PcopB

transcripts (Fig. 1(b)). CopB seems to be present in saturating

concentration in the sense that PrepA

is almost completely repressed at steady state. However,

analysis of the kinetics of the PrepA

control loop indicates that if the concentration of CopB

follows changes in plasmid concentration, CopB could, as the Rom protein for ColE1,

increase plasmid replication frequency at very low plasmid concentrations (see Section 4.4.3;

Paulsson et al ., work in progress).

2.5 General simplifying assumptions and values of rate constantsIn this study, we have made a number of deliberate idealisations concerning CNC and related

processes.

(1) We have assumed that host cells divide into daughter cells of identical size after a precisely

defined generation time with a precisely defined division volume, ignoring the statistical

variation in these three parameters.

(2) The partition mechanism of R1 is assumed to work perfectly so that plasmid losses only

promoter is constitutive and its transcript includes mRNA for CopB and RepA. Promoter PrepA

is

down-regulated by the tetrameric form of the CopB protein and is almost completely repressed at steadystate. Its transcript does not include CopB mRNA, but is in other respects similar to P

copB transcripts.

CopA is constitutively expressed from PcopA

and acts in trans by binding to a target sequence on PrepA

and PcopB transcripts, inhibiting RepA synthesis. RepA molecules bind in cis to oriR1 of the plasmidsfrom which the RepA mRNA was expressed. When a sufficient number of RepAs have bound at oriR1,

replication can be initiated. The total effect of the control system is an inverse relation between RepAsynthesis and plasmid concentration, resulting in an inverse relation between replication frequency per

plasmid copy and plasmid concentration. It is possible that PrepA

is strongly derepressed at very lowplasmid concentrations (work in preparation).




occur when there is a single plasmid at the end of the cell cycle. This is an idealisation

since experiments where the more efficient partition system from plasmid F has been

introduced in R1, report a decrease in plasmid loss rate by an order of magnitude (Boe

et al . 1987). For ColE1, we assume that all copies segregate independently (binomially).

This is also an idealisation. Plasmids can form clusters so that the number of

independently segregating units is much lower than the number of plasmid copies. Such

‘worse-than-random’ segregation can increase the loss rate greatly. A similar effect ariseswhen plasmid copies form multimers through homologous recombination.

(3) Simplifications regarding replication control differ between different parts of the study,

but some of them are made throughout the analysis. Generally, only fluctuations arising

from reactions that concern plasmids are considered. However, the compounded rate

constants used in the analysis generally depend on concentrations that are not included

as variables in the model. For instance, the RNA transcription frequencies depend on the

concentration of free RNA polymerase which may fluctuate randomly, or change

systematically during the cell cycle. Some of these simplifications are partially relieved in

Section 4.6.1.

(4) Plasmids ColE1 and R1 are assumed to regulate their replication by genuinely trans -acting

inhibitors with homogeneous concentrations throughout the cytoplasm, neglecting that

the inhibitors are produced very near to the site where they can bind and inhibit

replication. For instance, if RNA II is transcribed at about the same time as RNA I (see

Fig. 1) from the same plasmid copy, that particular RNA I molecule might be expected

to have a greater probability of inhibiting replication than an RNA I molecule that is

transcribed from another plasmid copy. The inhibitor would then be semi- trans -acting.

However, this complication may be of minor importance since for 3D-diffusion processes

the probability of binding to a site a few reaction radiuses away is almost identical to the

probability of binding to sites at any location in the cell. A dynamically similar

complication that will also be neglected is that the replication frequency of plasmids

seems to be composed of a small constant term and a much larger regulated term

(Dasgupta et al . 1987).

(5) We have ignored complications due to the fact that diffusion-limited reactions in

inhomogeneous media can behave qualitatively differently from reactions in

homogeneous media. We have also ignored the observation that DNA polymerase is

concentrated at the centre of the cell. This can be motivated by the fact that for a well-

working CNC system, the rate limiting step in replication is production of an initiation

complex. However, it could very well introduce an eclipse time between committing to

replication and actually replicating. Eclipse times also arise from plasmid replication. Too

long an eclipse time could have devastating consequences for CNC (see Section 4.5).

(6) Plasmids may sometimes affect other cellular events in cryptic ways, which in extreme

cases may result in cell death. Such phenomena are mostly observed for plasmid mutants

or plasmids that are newly introduced into a host strain. We assume throughout the

analysis that these effects can be ignored on a shorter time-scale and that the growth rate

of a cell is a weak function of the number of plasmid copies that a particular cell by chancecontains.

The above-mentioned complications may all increase random fluctuations. Our estimates

of plasmid copy number variation and loss rates are therefore lower limits.

The goal of this analysis is not to reproduce single experimental results, but to identify and




inspect properties that otherwise may elude experiments and non-quantitative reasoning. In

particular we will show that a number of critically important properties of CNC are invisible

from a macroscopic perspective. To do this, we minimise the number of concentration

variables included, but instead systematically analyse the impact of selected rate constants.

The reasons we did not use experimentally measured parameters are that (1) the rate constants

are known to change greatly with physiological conditions; (2) they are not measured under

the same conditions; (3) there are many types of plasmids that are governed by the same basiccontrol principles; and (4) analysing a single set of parameters gives no insight into their

impact on the system. Experimental data was therefore only used to infer reasonable but

generous parameter regions.

3. Macroscopic analysis

Macroscopic analyses are useful since they are simpler and more transparent than their

mesoscopic counterparts. This property of macroscopic descriptions will in this section be

used to open a quantitative discussion of CNC for ColE1 and R1 plasmids. However, the

validity of macroscopic approaches cannot be taken for granted even for describing

concentration averages over very large cell populations. Macroscopic analyses insteaddescribe hypothetical systems so large that fluctuations are insignificant, e.g. when all cells in

a population have been merged into one giant cell.

3.1 Regulatory logic of inhibitor-dilution CNC

A simple mechanism, with strong experimental support, for the dynamics of inhibitor-

dilution CNC, can be represented by the coupled differential equations

1

23

4

y r (s ) y y

s β yαs ,(1)

where y is the plasmid and s is the inhibitor (s ignal) concentration. The plasmid production

rate per plasmid, r (r esponse or r eplication frequency), is a function of the inhibitor

concentration. Plasmids are also continuously diluted as all intracellular components in a

growing host. The rate constant for exponential host cell growth has been put equal to one

and used for normalising the time scale in Eq. (1). In this time unit the dilution rate per

plasmid is 1 and the host generation time is ln(2). Inhibitors are constitutively synthesised

with rate constant β per plasmid and are degraded and diluted according to first order kinetics

with total rate constant α.

We suggest that the following properties generally hold for the replication frequency per

plasmid, r

1

23

4

r (0) 1

r (s ) 1

dr

ds 0 .

(2)

The first property means that when there are no inhibitors, the replication frequency per

plasmid is higher than the dilution rate per plasmid. When this is not the case, plasmids will

be outgrown by their hosts and y will decrease indefinitely. The second property means that




there exists a steady state inhibitor concentration (indicated by over-bar) at which plasmid

replication and dilution balance each other. The third property means that the replication

frequency per plasmid is a monotonically decreasing function of the inhibitor concentration.

Not all feedback mechanisms are monotonically inhibiting or activating (Ptashne, 1992).

However, the molecular basis for inhibitor-dilution CNC includes competition between the

inhibitory pathway and a pathway that leads to plasmid replication so that monotonicity is

a reasonable assumption.Inhibitor-dilution CNC is a straight-forward case of signal and response. The value of y is

the original signal and the synthesis rate r is the final response. The inhibitor concentration

s acts as a kinetic link between y and r so that the individual plasmid copies can sense the entire

pool of plasmid copies in a cell. When α in Eq. (1) is high, as in vivo (Stougaard et al . 1981;

Bremer & Lin-Chao, 1986; Brenner & Tomizawa, 1991; Merlin & Polisky, 1993; So derbom

et al . 1997; So derbom & Wagner, 1998), a change in y rapidly leads to a proportional change

in s . The value of s can thus be seen as a linear deputy for the original signal y and is the actual

quantity that r responds to.

Under assumptions (2) above, the non-zero steady state of the system can be calculated as

1

2

3

4

y α

β

r −(1)α

β

s

s r −(1),(3)

where r − is the inverse function of r . Interestingly, the steady state inhibitor concentration

is determined only by the dynamics for initiation of plasmid replication, not by the synthesis

or degradation of inhibitor. The steady state plasmid concentration, in contrast, depends

both on inhibitor turnover and the steady state inhibitor concentration. A change in α

or β affects s temporarily, but subsequently results in a change in y that allows s to return

to its previous steady state. This somewhat counterintuitive effect is a common signature

of negative feedback.

3.2 Sensitivity amplification

A critical property of inhibitor-dilution is how sensitively the replication frequency per

plasmid, r , responds to changes in inhibitor concentration, s . High sensitivity amplification

reflects the ability to transform a small percentage change in the signal to a large percentage

change in the response. A commonly used measure of sensitivity amplification is the slope,

ar,s

, of the response as function of signal in log–log scale.

ar,s

d ln r

d ln s

dr

ds

s

r . (4)

This slope is known as the amplification factor, response coefficient, control coefficient, net

sensitivity, reflection coefficient or logarithmic sensitivity (Fell, 1996) and corresponds to the

normalised change in the response for a small normalised change in the signal. It is central

in both Metabolic Control Analysis (Fell, 1996) and Biochemical Systems Theory (Savageau,1971, 1976) and used to formulate complicated reaction schemes in terms of power-laws.

Ultra- and subsensitivity could be defined by ar,s 1 and a

r,s 1, respectively.

However, it is convenient to take the degree of saturation into account. A perhaps more

suitable definition (Koshland et al . 1982) when analysing kinetic mechanisms that can be




saturated, is therefore to compare the ar,s

-value of a mechanism with the ar,s

-value of the

hyperbolic standard curve r 1(1s ) at the same degree of saturation. The latter gives ar,s

(1r ) so that the criterion for ultra- and subsensitivity would be ar,s 1r and a

r,s

1r , respectively.

3.3 Plasmid control curves

In the initial formulations of inhibitor-dilution replication control (Ycas et al . 1965 ; Pritchard

et al . 1969), it was assumed that small changes in inhibitor concentration would greatly affect

the replication frequency, i.e., that ar,s 1. Nordstro m et al . (1984) noted that such large

sensitivity amplification constitutes an extreme and instead introduced control curves that

measure efficiency of CNC through the relation between r r and y y . This measure is of great

importance for CNC since changes in plasmid concentration is the original signal that the

replication frequency should respond to. However, how the replication frequency responds

to plasmid concentration also depends on how the signal molecule concentration s is related

to the plasmid concentration y. Unless this functional dependence between s and y is specified,

the control curve is undetermined. Initially we will therefore inspect control curves that relate

r r to s s . In the stochastic analysis in Section 4, we define generalised control curves that

relate the replication frequency to the number of plasmid copies so that all aspects of CNCfor plasmids in growing and dividing cells are included.

Hyperbolic inhibition is a convenient starting point for analysing inhibitor-dilution CNC

since the mechanism is simple and has been used in several theoretical analyses of plasmids R1

and ColE1 (see Section 5). Hyperbolic inhibition describes the probability that a replication

trial commits to plasmid replication in the scheme:

ktr I

kas

inhibition

kp commitment to replication. (5)

The rate constant ktr

determines the frequency of trials to initiate replication, kp

how rapidly

the intermediate state I proceeds to a state where the plasmid is committed to replication, and

the inhibition rate ka

s is determined by the inhibitor concentration s multiplied by an

association rate constant ka. According to this scheme, the probability q

h of committing to

replication from the intermediate state I depends hyperbolically on s :

q h

kp

kpk

as

1

1s K , (6)

where K kpk

a is the inhibition constant. The sensitivity of the mechanism according to

Eq. (4) is aqh,s 1q

h. For small values of s K (q

h close to one) the sensitivity is small and

increases to its maximal value of 1 when s K increases (q h

tends to zero). This principle is

rather general ; low probabilities avoid saturation and allow for larger sensitivity amplification.

One of the questions addressed in this study is how plasmids can modify CNC to achieve

higher sensitivity than the naturally arising hyperbolic control curve (Eq. (6)). Sensitivity

amplification is relevant far beyond plasmid CNC, since sensitive regulation is requiredalso in the co-ordination of the cell cycle, chromosome replication, metabolic control,

morphogenesis, chemotaxis, neural recognition and basically any other life process. Another

question we address (in relation to plasmid R1) is if control curves and sensitivity

amplification cover all important aspects of CNC precision.




3.4 Multistep control of plasmid ColE1: exponential control curves

The mechanism for initiation of ColE1 replication is very similar to the branching reaction

(5). A replication preprimer, RNA II, (see Section 2.4.1 and Fig. 1 (a)) is synthesised

constitutively with rate constant ktr

per plasmid. When the transcription machinery enters the

inhibition window, replication priming is sensitive to attack from RNA I until transcription

has proceeded through the window. If kp

is identified as the effective rate constant with which

the system leaves the inhibition window, and an effective association rate kas is assumed for

the inhibition, then scheme (5) and Eq. (6) are recovered. However, not all processes can be

described by a single rate constant and since the inhibition window contains about 250

transcription steps, the kinetics deserves careful inspection. A truly elementary-step time-

constant reaction has an exponentially distributed reaction time. If this were the case for

ColE1, the duration of the inhibition window would vary greatly from time to time. This

randomisation of the inhibition time is automatically included in Eq. (6), which was derived

from a simple probabilistic argument. This can be seen as follows. If inhibition of RNA II

occurs with intensity kas , the probability of not inhibiting during a time interval ∆t is e−kas∆t.

If the time interval ∆t is exponentially distributed with average 1kp, the expression (6) for

the probability q h

is recovered

q h&

e−kas∆t f (∆t ) d∆t &

e−kas∆t kp

e−kp∆t d∆t k

p

kpk

as

1

1s K . (7)

In the perspective of Eq. (7) the low sensitivity of hyperbolic inhibition is caused by statistical

variation in the time interval ∆t during which inhibition can occur. By contrast, if the

statistical variation in the inhibition time ∆t is negligible (so that f (∆t ) is zero everywhere

except in a small interval around its average 1kp

), it follows from the first integral in (7) that

the probability of replication would instead be exponential

q e e−kas/kp e−s/K (8)

Exponential inhibition can alternatively be derived from reaction schemes where inhibition

can occur at multiple steps along the pathway

( no

primer*ktr I ka,s

kp, I

ka,s

kp,

kp,n− n1ka,ns

kp,n (ready to

replicate*. (9)

The arrows pointing downward correspond to the return to no primer. Since ktr

is rate-

limiting by almost two orders of magnitude (see Section 2.4.1), this scheme can be

condensed to

( no

primer*ktrn

i=

1

1s K i (ready to

replicate*, (10)

where the inhibition constants are K ik

p,ik

a,i. Since every step responds to changes in s ,

a multistep mechanism can result in very high sensitivity amplification (Eq. (4)). The qualityof such a multistep mechanism increases with the number of steps that contribute significantly

to the overall inhibition. For a given number, n, of steps and a given overall probability, q ,

of inhibition, sensitivity amplification is highest when all steps have identical inhibition

properties. If only one step contributed, scheme (9) would be reduced to (5) and the control




would be hyperbolic. For n identical steps the replication probability associated with each trial

is

q 0 1

1s K n

1n. (11)

Taking the limit of infinitely many steps with K K nn recovers the exponential expression

(8)

limn

q limn

0 1

1s K n

1n e−s/K q e. (12)

The equality between Eqs (8) and (12) reflects the fact that the multiple steps remove the

uncertainty in the time duration of the inhibition window. Since the inhibition process for

ColE1 indeed comprises a very large number of transcription steps, it is not unlikely that the

replication probability q depends on the signal s approximately according to (11) with n

significantly greater than one.

At steady state, replication balances dilution (Eqs (1) and (3)), which means that

r ktr

q (s ) 1 independently of mechanism. Due to CNC feedback, a high value of ktr

automatically results in a very low steady state replication probability (q h

or q e) and

consequently in a very high value of s K . By the same logic, a low value of ktr is automaticallyassociated with a small value of s K . It is therefore the frequency k

tr by which replication

trials occur, rather than the inhibition constant K , that determines the sensitivity of the

mechanism. The inhibition constant K only defines the characteristic concentration scale. For

trial frequencies ktr

in the biologically relevant range (approximately 5ktr 100 for

ColE1), there is little functional difference between a three-step mechanism and the limit, Eq.

(12). For most purposes, it is therefore sufficient to compare exponential inhibition (Eqs (8)

or (12)) with hyperbolic inhibition (Eqs. (6) or (7)).

The equations that specify the control curves (see Fig. 2) for hyperbolic and exponential

control are given by

1

23

4

r h

ktr

1s K 1

r h

r h

k

tr

1(ktr1)s s

r ek

tre−s /K 1

r e

r e

k−s/s tr

.

(13)

At low ktr

, the two mechanisms behave similarly since e−x 1(1x) 1x for small

values of x. However, q h

and q e become radically different when k

tr increases. This divergence

in function is dramatically illustrated in the limit of infinite frequency of replication trials

1

23

4

limktr

r h

r h

s

s

limktr

r e

r e

1

23

4

0, s s 1

, s s 1.

(14)

Erratum : Some previously published figures unfortunately reported wrong parameter values. A

figure similar to Fig. 2 was included in Paulsson & Ehrenberg, 1998 (repeated in Paulsson et al . 1998).

The parameter value ktr10 reported in that figure legend should have been k

tr 270. The same goes

for Fig. 3(a), and Fig. 3(b) has the x-axis shifted a factor two. The value presented as 4 should thus have

been 8 etc. This correction makes little differencefor thefigures andthe qualitative results are unchanged,but we deeply regret the incident.




Fig. 2. Control curves for hyperbolic and exponential inhibition. When the two mechanisms work withfull capacity (k

tr), the normalised replication frequency r r is inversely proportional to s s for

hyperbolic inhibition, while exponential inhibition results in a switch. Exponential inhibition workssignificantly better also when k

tr 10 or k

tr 3.

Hyperbolic control can thus, at best, result in an inverse relationship between signal and

response, while, in this limit, the exponential control curve has zero replication rate below

and infinite replication rate above the steady state signal.

It is also instructive to compare the steady state sensitivity amplification factors for the two

cases. These are given by

1

23

4

a rh,s(11k

tr)

a re,sln (k

tr).

(15)

For hyperbolic inhibition the sensitivity increases with increasing ktr but saturates at1. Thegain in sensitivity when k

tr increases from 1 to 10 is substantial, but further increase in k

tr

only gives a marginal improvement. For exponential inhibition, sensitivity has no upper

bound, but ktr

must be squared to double the sensitivity amplification.

3.5 Multistep control of plasmid R1 : hyperbolic control curves

As outlined in Section 2.4.2, R1 replication requires de novo synthesis of a large number of

replication proteins, RepA. RepA has been shown to work in cis , meaning that it binds to the

origin of replication of the plasmid copy from which the RepA mRNA was synthesised. It

has been shown both that RepA synthesis is the rate-limiting step in the initiation of replication

and that it is hyperbolically inhibited by CopA (Eq. (6)). This implies that initiation of R1

replication is regulated at many sequential steps.Even if R1 CNC has multiple hyperbolically regulated steps, the number of steps may

vary from time to time. For a discussion about the factors that contribute or reduce the

effective number of rate-limiting steps, as well as other potential complications, see Paulsson

& Ehrenberg (2000a). Here we reduce the model to its essence and assume a number n of




identical (rate-limiting) hyperbolically regulated steps between each round of replication of

a given plasmid copy. The reaction scheme describing the initiation process of a single

plasmid copy is then given by

( newly

replicated*ktr

kas

I kp 1 …

ktr

kas

I kp n1

ktr

kas

I kp (ready to

replicate*. (16)

A RepA mRNA (state I ) is synthesised with rate constant k tr, translated with rate constantkp

and inhibited with effective rate constant kas . Every single step is thus hyperbolic. In vivo,

plasmid and inhibitor concentrations are almost perfectly proportional and, furthermore, the

relation kas k

pk

tr seems to hold even at comparatively low s . Therefore

ktr

kp

kpk

as

ktr

kp

kpk

a yβ α

κ

y. (17)

The compounded rate constant κ thus depends on the rate constants of RepA mRNA

synthesis, translation and inhibition as well as on the rate constants for inhibitor turnover

(Ehrenberg & Sverredal, 1995). Using Eq. (17), scheme (16) can then be simplified to

( newly

replicated*κ/y

1κ/y

… …κ/y

n1κ/y

(ready to

replicate*. (18)

This scheme is principally different from that of ColE1 (scheme (10)), although it may

superficially look similar since both schemes are based on multiple hyperbolically regulated

steps. The important differences are as follows.

(1) If initiation of ColE1 replication is inhibited at any of the steps, the process starts from

the beginning and must synthesise a new RNA II molecule. For R1, by contrast,

inhibition of a certain RepA mRNA does not remove the previously synthesised RepA

molecules from the origin of replication.

(2) The duration of ColE1 multistep inhibition is only about five seconds (Tomizawa, 1986)

and the rate-limiting step is the initiation of RNA II synthesis. R1 multistep initiation may

instead stretch over the whole cell cycle and includes synthesis of a number of RepA

mRNA.

For ColE1 (Section 3.4), the effective inhibition probability is the product of a number of

hyperbolic steps (Eqs. (10)–(12)), leading to potentially very large sensitivity amplification

(Eqs (14) and (15)). For R1, the multiple hyperbolic steps in CNC do not contribute to

sensitivity amplification because inhibition of a RepA mRNA does not force R1 to restart

with zero RepA molecules. In fact, when y remains constant throughout the chain of events

in scheme (18), the total average time between newly replicated and ready to replicate

is nyκ. The average replication frequency in a cell population (the inverse of the total

interreplication time) is thus inversely proportional to plasmid concentration, as has been

established experimentally (Gustafsson & Nordstro m 1980; Nielsen & Molin, 1984). This

can also be shown by using macroscopic equations for every state in scheme (18): R1

multistep CNC is hyperbolic.As was shown in Sections 3.3–3.4, hyperbolic control generally corresponds to

exponentially distributed reaction times. However, if y were constant in scheme (18), the

standard deviation divided by the mean of the time from newly replicated and ready to

replicate would be 1n.




In summary, ColE1 multistep CNC results in sensitivity amplification, but the inter-

replication time is approximately exponentially distributed. R1 multistep CNC does not lead

to sensitivity amplification but instead to reduced random variation in the interreplication

time.

3.6 Time-delays, oscillations and critical damping

Sections 3.4 and 3.5 exclusively dealt with the response in the per plasmid replication

frequency to changes in the inhibitor concentration. Here we return to the equally important

relation between the inhibitor concentration s and the plasmid concentration y.

The rate constants β and α (Eq. (1)) are both compounded. For α we have α 1kd

corresponding to dilution by volume expansion with rate constant 1 and degradation with

rate constant kd. We have chosen to model inhibitor degradation with a single rate constant

kd

although it is known to involve several steps. This simplification is motivated from

experiments (Tomizawa, 1984; Tamm & Polisky, 1985; Lin-Chao & Cohen, 1991) that show

how intermediate forms of RNA I are rapidly degraded and cannot inhibit primer formation.

RNA I can therefore be treated as a single chemical species, where kd

is interpreted as the first-

order rate constant for the first step in the degradation pathway. Single-step first-order

degradation kinetics means that the life-time of RNA I is exponentially distributed, as inradioactive decay. Consequently, the current inhibitor concentration is determined by the

whole history of plasmid concentrations.

The rate constant β includes both a positive term, ks (the synthesis frequency per plasmid

of RNA I or CopA), and a term for degradation of inhibitor through duplex formation with

its target (RNA II or CopT). Since duplex formation between inhibitor and target takes

place very shortly after target transcription (Tomizawa, 1986), even at fairly low inhibitor

concentrations, the target concentration can be excluded as a concentration variable with little

loss in generality. The rate of degradation of inhibitor through duplex formation can instead

be well approximated by the rate of target production, ktr y, implying effectively zero-order

kinetics for the duplex formation pathway.

Returning to the full system (1), it is convenient to normalise the concentrations with their

respective steady state values, yN y y and s

N s s . System (1) can then be expressed as

1

23

4

y N (r (s

Nr −(1))1) y

N

s Nα( y

Ns

N).

(19)

Parameter β has no impact on the normalised dynamics of the system. This means that

how rapidly the relative inhibitor concentration s N

follows changes in the relative plasmid

concentration yN

does not in any way depend on the rate constant β for inhibitor

synthesis (as long as β 0). For instance, if the plasmid concentration is at steady state but

the inhibitor concentration is at half its steady state, the rate with which the latter increases

and adjusts to the plasmid concentration is only determined by rate constant α, and not by

β . This somewhat surprising behaviour is a direct consequence of the fact that an increase in

β results in a corresponding decrease in y

(Eq. (3)) so that the effective steady state rate, β y ,

remains constant. A twofold increase in α, on the other hand, results in a twofold increase

in both rates β y and αs (Eq. (3)). It is well known in biochemistry that the half-life of a

molecule, rather than rate constants for its synthesis, generally determines how rapidly its

concentration responds to perturbations. The reason is that the degradation pathway






low, s N

initially lags behind, allowing for a more rapid approach to steady state in yN

. When

yN

reaches steady state, s N

still lags behind, causing an over-shoot in yN

. The phenomenon

repeats itself when s N

is above steady state and the relaxation to steady state oscillates. When

α is very high, there is a perfect proportionality between yN

and s N

and adjustment to steady

state does not over-shoot. In an intermediate region, α can be high enough to avoid over-

shooting, but still be so low that s N1 y

N1. Since the rate of adjustment is critically

dependent on s N1, limited inhibitor turnover could be utilised for more efficient CNC.The linearised model predicts critical damping at α 4a

r,s. However, the phenomenon is

dependent on initial conditions. If, for instance, yN 05 and s

N 1, then a low α would

initially give the opposite effect, i.e., it would reduce the adjustment rate to steady state. The

impact of inhibitor half-life on CNC must therefore be inspected mesoscopically where initial

conditions can be dealt with appropriately (see Section 4.5.1).

4. Mesoscopic analysis

The term mesoscopic implies different things in different contexts. Here it is used to denote

an intermediate level of description where individual molecules and random events are taken

into account, but not the fact that the molecules actually have structures that must followphysical laws. In spite of the limitations, this broadens the scope of kinetic modelling greatly

and many kinetic design principles can only be understood from a mesoscopic perspective.

However, though macroscopic descriptions treat concentrations as continuously and

deterministically changing entities, they can be modulated to incorporate some aspects of

random fluctuations. For example, it has been shown how noise may allow for symmetry-

breaking in the spatial self-organisation of amoebae (Pa lsson & Cox, 1996). Such qualitative

effects are intelligible by adding noise to a macroscopic system, while a mesoscopic

formulation might be more confusing than useful. In the present work we only use ‘ purely ’

macroscopic (Section 3) or mesoscopic (Section 4) kinetic models. However, one may keep

in mind that macro- and mesoscopic analyses are always intertwined in any given application:

macroscopic models make implicit assumptions about the underlying stochastics (see e.g.

Section 3.4) and finite-dimensional mesoscopic models involve parameters that are treated as

deterministic.

4.1 The master equation approach

Rates of intracellular processes often depend on concentrations of molecular species that are

present in low copy numbers per cell. Average rates and concentrations then lose their

dominant role and the dynamics must instead be understood in terms of fluctuations. In

addition, when rates depend nonlinearly on randomly fluctuating components, macroscopic

rate equations may be far off the mark even in their estimates of averages. True averages over

cell populations can then only be found from probabilistic single-cell descriptions. This

somewhat strange fact can be understood as follows: When fluctuations are insignificant, onlydynamics close to a stable steady state, or near a trajectory leading to it, affect the system’s

behaviour. By contrast, significant random fluctuations in concentrations render relevance

In this case critical damping corresponds to the value of α that allows yN

to most rapidly return toits steady state without oscillations.




also to the dynamics far from steady state or from a macroscopically identified trajectory to

it. Fluctuations thus globalise a system so that other nonlinearities than those seen

macroscopically may come into play. Even averages can then be greatly affected by

fluctuations since the system behaviour at deviations below the average may not compensate

for deviations above. This calls for mesoscopic descriptions where conventional rate

equations are exchanged for stochastic birth and death processes: chemical master equations.

Chemical master equations (for very readable and clear introductions to birth and deathprocesses and master equations, see van Kampen, 1992, and Taylor & Karlin, 1998) are time-

continuous Kolmogorov equations that determine the evolution of probability distributions.

The randomness included in master equations simply reflects the intrinsically random and

discrete nature of all chemical reactions.

The macroscopic assumption that sets of states (all possible combinations of numbers of

molecules) can be replaced by scalar states (guessed averages) is only motivated when

significant random deviations from the scalar states are rare. The size of fluctuations in the

number of a certain molecule type is often assumed to be of the order of the square root of

the average, implying that mesoscopic approaches are only necessary for very small systems.

This argument is valid when the noise is e.g. Poisson distributed, as is often approximately

the case at thermodynamic equilibrium (see van Kampen, 1992, for a discussion of the grand

canonical Poisson distribution). Cells, by contrast, operate far-from-equilibrium and there are

therefore no reasons to expect Poisson fluctuations. In fact, there are no concentrations in cells

that are so high that fluctuations can be ignored a priori .

To introduce master equations we start with an equilibrium-type example: independent

synthesis and degradation. Assume that the probability that a particular molecule is degraded

in a sufficiently short time interval ∆t is proportional to ∆t , i.e., P(degradation)

kd∆t o(∆t )k

d∆t . For n independent molecules, the probability that exactly one of them

is degraded during ∆t is then kd

n∆t o(∆t ) kdn∆t where the approximation is exact in the

limit where ∆t 0. Since kdn is the probability per time unit for a degradation event given

that there are n molecules, the actual intensity of the probability flow from a state with n to

a state with n1 molecules is kd

npn

where pn

is the time-dependent probability for having

n molecules. If also synthesis events are independent (rather than occurring in clusters or with

randomly varying intensities) and the synthesis rate is time-constant, the birth and death

process can be represented by the state diagram:

n1k

kdn

n k

kd(n+)

n1. (21)

This corresponds to the following master equation:

p nkp

n−k

d(n1) p

n+(kk

dn) p

n (k(E−1)k

d(E1)n) p

n. (22)

We will not give separate master equations for the boundary conditions (e.g. n 0) when

these are obvious. The ‘step operator’ E (van Kampen, 1992), defined by E j f (n) f (n j ),

makes for easier notation since every type of event then only must be written once. Subscripts

on E are used when it can operate on more than one index.The master equation is a gain-loss equation. For systems like (21), a state loses probability

mass to its neighbours that in turn contribute with probability. When a stationary state exists,

the net probability flow must be zero over all steps (Eq. (21)) so that all probabilities pn

can

be expressed in terms of p

through iteration. Since the probability sums to one it follows




directly that the stationary distribution for Eq. (22) is a Poissonian with an average and

variance of nσnkk

d. The success of this simple and mechanical way of calculating

a stationary distribution depends on the fact that scheme (21) is one-step and one-

dimensional. There is no generic method for obtaining stationary distributions for arbitrary

master equations.

When first starting to model mesoscopic kinetics, some things can be good to keep in

mind.

A constant transition probability per time unit corresponds to an exponentially distributed

reaction time. However, a macroscopically constant synthesis rate does not necessarily

mean that the process is truly single-step, which means that the reaction times are not

necessarily exponentially distributed. The intensity for synthesis may for instance be

determined by the concentration of an enzyme or precursor that fluctuates randomly,

which would mean that the process in (21) is a birth-and-death process in a randomly

fluctuating environment (i.e., there can be more dimensions in the master equation than

there are rate equations). Also, a series of rate-limiting reactions can sometimes be exactly

described macroscopically as a single step, but mesoscopically one must take into account

that the reaction time for the total process can be more narrowly distributed (Section

4.5.2). Transitions between two states in a master equation do not necessarily correspond to

elementary-step reactions in homogenous media. Complicated reaction schemes with many

types of states can often be compressed into descriptions with fewer types of states but

more complicated transition probabilities between them. Such compressions are present

also in the simple elementary-step applications: all master equations and all rate equations

necessarily rely on idealisations, such as approximating fast events as immediate. The

common belief that such simplifications can only be made for elementary-step but not for

complicated reactions is simply not correct.

Autonomous rate equations may correspond to inhomogeneous master equations, i.e., the

latter can show explicit time-dependence even when the former do not. To account for cell

growth, rate equations simply include a dilution term (Eqn. (1)), but if the master

equations are formulated in terms of numbers of molecules (which is convenient thoughnot a necessity), the transition probabilities per time unit become time-dependent (e.g.

Eq. (29)).

The master equation is usually linear in pn

. The term nonlinear is then instead used to

denote systems where transition probabilities depend nonlinearly on n. However, in some

cases it is practical to formulate master equations so that also the probabilities pn

enter

nonlinearly (Paulsson & Ehrenberg, 2000b).

In conclusion, single macroscopic rate constants must sometimes be represented by an entire

spectrum of mesoscopic transition probabilities, but a large number of mesoscopic transition

probabilities can sometimes be compounded to reduce the dimensionality of the system. The

only rule when using master equations is to choose the states in such a way that those

transition probabilities that are relevant for system behaviour in the time scale of interest canbe formulated using all available experimental and theoretical insight into the system.

Mesoscopic transition probabilities are intrinsically different from macroscopic rate

constants. However, to make for easier comparison with the macroscopic analysis in Section

3, we will use the same notations in this section.




Fig. 4. A qualitative description of a random walker in a potential well. The steeper the walls, the moreunlikely to deviate from the average. The analogy covers multistable systems when the potential hasmultiple local minima.

4.2 A random walker in a potential well

So what determines when internal noise is so large that it must be explicitly accounted for?

This completely depends on the nonlinearities of the system. Very few statements about the

significance or effect of noise hold true universally. In spite of that, the noise-determining

factors may be illustrated by a popular analogy: that the number of molecules, n, of a certain

species behaves like a random walker in a potential well. The significance of stationary

fluctuations depends on the ‘frenzy’ of the walker, the ‘random wobbling’ of the potential

well itself and the steepness of the well walls (Fig. 4). The frenzy of the walker corresponds

to the number of molecules added or removed per reaction event and the random wobbling

is due to random fluctuations in the rate constants. Finally, the steepness of the well is

determined by how the rates of formation and elimination respond to changes in copy

number. Here we briefly inspect the last property of the well before turning to plasmid CNC.

In the Poissonian example above (scheme (21)), the probabilities that the next jump is to

the left or right are P(L)kd

n(kkdn) and P(R)k(kk

dn) respectively. The system

thus has a tendency to return to a preferred value where P(L)P(R). Accordingly, the

probability flows appear to be hyperbolically regulated by n. The sensitivity of regulation is

illustrated by the slope of the well walls and can be approximated by the difference in apparent

kinetic orders between the elimination and degradation reactions when these are formulated

as power-laws (Savageau, 1976). For one-dimensional one-step processes with birth and death

intensities J +

(n) and J −

(n), respectively, the sensitivity with which the ratio Rn J

−(n) J

+(n1)

responds to changes in n can be estimated using a discrete version of sensitivity amplification

factors (Eq. (4)), AR,n

:

AR,n

(RnR

n−)

Rn

n

(n(n1)) n 01

Rn−

Rn

1. (23)

Stationary Poissonians arise when Rn

is proportional to n, giving AR,n 1. In fact, with A

as the value of AR,n

around the stationary average n, the linear noise approximation of the

master equation (van Kampen, 1992) can be used to approximate the stationary copy number

variance σn

by

σnnA−. (24)

The linear noise approximation can also be used to describe more complicated situations, like

when a large and random number of molecules is added or removed in each reaction. This

analysis illustrates that there is a direct relation between macroscopic homeostasis principles

We do not use the term ‘potential well’ rigorously but as an analogy only.




and levels of internal noise (Paulsson et al . in preparation). Some systems, like zero-order

ultrasensitive control of averages (Goldbeter & Koshland, 1981; Berg et al . 2000) or

microtubules with dynamical instability (Dogterom & Leibler, 1993), can have extremely

small A-values. Equation (24) appears to suggest that σnn is independent of n.

However, this is misleading since A, n, rate fluctuations etc. all depend on the underlying

kinetics.

4.3 CNC as a stochastic process

To describe CNC stochastically, we generally model changes in the numbers of plasmid (m)

and inhibitor (i ) copies in individual cells of an exponentially growing cell population. We

use a master equation to calculate the copy number distribution at the end (t τ) of the j th

cell cycle, P jτ (m, i ), from the distribution over the newborn (t 0) cells, P j

(m, i ). This master

equation is combined with a rule for how the plasmid and inhibitor molecules are distributed

between the two daughters at cell division, determining how P j+

(m, i ) depends on

P jτ (m, i ). Repeating the procedure does not generate stationary probability distributions.

Rather, the copy number distribution among plasmid-containing cells converges to an

asymptotic cell cycle development: the cyclic state. For the processes studied here, the cyclic

state is approached within a few cell cycles regardless of initial conditions. The copy numberdistributions discussed in this paper always correspond to the cyclic state.

The probability that a plasmid-containing cell gives rise to a plasmid-free progeny at cell

division, P(loss ), is calculated from the cyclic-state distribution of plasmid copies at the end

of the cell cycle, Pτ(m). If the plasmid lacks an active partition system ( par −) and all copies

can be treated as independently segregating units (see Section 2.5), they are simply binomially

distributed between the two daughters. When plasmids instead carry a perfectly working

partition system ( par +), plasmid losses only occur when there is a single plasmid copy at the

end of the cell cycle (see Section 2.5). This means that P(loss ) is given by:

P(loss )

1

23

4

2

m=

(12)m Pτ(m), par −

Pτ(1), par +

. (25)

Changes in the parameters of CNC will not only affect plasmid copy number variation but

also the average copy number. To study effects of variation separate from effects of averages

it is necessary to introduce a change in an additional parameter so that the average remains

constant. In the macroscopic analysis (Section 3.1) we showed how the parameters β and K

only determine the concentration scale and not the dynamics of the system. In a mesoscopic

analysis, such characteristic concentration scales cannot be introduced because the difference

between, for example, 10 and 100 copies of a molecule cannot be compensated for by a change

in a rate constant. However, some of the master equations used in this study only include two

parameters. When one parameter is given, the average plasmid copy number uniquely

determines the other. The parametric dependence of P(loss ) shown in the figures below is thus

not a description of what happens when a given parameter changes. It is a comparisonbetween parameter combinations for a constant average copy number. The alternative to this

approach is to let a change in a parameter affect both average copy number and copy number

variation. Although such an analysis is much less demanding computationally, it would give

little insight into the CNC. The details of the compensation are discussed in each section




separately, but normally we make the compensatory changes in the compounded inhibition

constant K (Eq. (6)).

The distribution Pτ(m) determines the plasmid loss rate in a cell population (Eq. (25)). A

narrow Pτ(m) requires a high sensitivity amplification in the response to changes in m. For

instance, Pτ(m) will be infinitely narrow around the average mτ if ratem for mmτ

and ratem 0 for mmτ in the state diagram

m1ratem−(t) m

ratem(t) m1. (26)

By contrast, if plasmids were not regulated then ratemkm except at very high copy

numbers, where limited host resources would reduce the intensity of plasmid replication. To

see more clearly what this implies, one may approximate cell growth and division by an

intrinsic instability of plasmids. That is, replace the combination of a birth scheme (26) and

a mechanism for distributing plasmids between daughter cells by a homogenous birth and

death process

m1 ratem−

µm

m ratem

µ(m+)m1. (27)

When ratem µm in Eq. (27), then A 0 (Eq. (24)). This corresponds to a completely flat

potential well and copy number variation accumulates indefinitely. The time-dependent

average and variance can be calculated analytically both for scheme (27) and for a modified

scheme where only plasmid-containing cells are considered (i.e., the distribution is conditioned

on m 0) (Harris, 1963; Berg, 1995).

Generally, Pτ(m) cannot be calculated from a one-dimensional state diagram like (26).

Simply knowing that state m is reached at time t is insufficient for specifying the effective rate

ratem

(t ) because the conditional distribution of inhibitors as well as other features must be

known. However, by using multidimensional master equations, a cyclic-state version of

diagram (26) can be calculated, where all such features are accounted for. The corresponding

averaged intensities ratem(t ) then uniquely specify the plasmid cyclic state distribution at alltimes t in the cell cycle. This representation, which depends on an already existing cyclic-state

solution of the multivariate process, generalises the control curves in chapter (3.3) and can

be used to illustrate the quality of CNC also for the more realistic mesoscopic models. An

example is given in Section 4.5.1.

Continuing the analogy of a random walker in a potential well, Sections 4.4–4.7 are

organised as follows:

Section 4.4 inspects the shape of the potential well and how some designs of CNC only

affect sensitivity amplification far from steady state. Eq. (24) is then inadequate.

Section 4.5 inspects time-delays. Long inhibitor half-lives are shown to result in more

sluggish responses so that the effective shape of the potential well is affected.

Section 4.6 inspects the frenzy of the walker and random wobbling of the well. Multistepcontrol of R1 is shown to reduce the statistical variation in the reaction times. The effect

of noisy rates is shown to depend critically on time scales.

Section 4.7 shows how random inhibitor fluctuations can affect the shape of the potential

well and how they can be used to reduce plasmid copy number variation.




Fig. 5. (a) Plasmid copy number distributions for exponential and hyperbolic inhibition for ktr5 and

ktr 100. (b) P(loss ) as a function of k

tr for exponential inhibition without partition system (binomial

distribution of the plasmids present in the mother cell at division) and for hyperbolic inhibition withand without an equipartition mechanism. The lower limits correspond to Eqs. (33) and (36). The

average copy number is 16 in both (a) and (b).

4.4 Sensitivity amplification

As can be seen from the simplified Eq. (24), sensitivity amplification in the negative feedback

loop generally decreases random fluctuations. Here we inspect this aspect of CNC under the

assumption that the inhibitor immediately follows changes in plasmid concentration without

significant random fluctuations. Also disregarding other possible noise sources, plasmid CNC

is treated as a one-dimensional birth process where the transition probabilities can be

postulated in terms of m and t .

4.4.1 Single-step hyperbolic control

One of the simplest mesoscopic models of CNC, that still has some claims to realism is single-

step hyperbolic control described by the following birth process state diagram:

m1

ktr(m−)

+(m−)/(etK) m

ktrm

+m/(etK) m1. (28)

The plasmid concentration at time t in the cell cycle in an exponentially growing host with

m copies of the plasmid and volume 1 at t 0 is met. The transition intensity from state m

to m1 equals the frequency ktr

m of replication trials multiplied by the probability

1(1m(et K )) that a given trial is successful. The corresponding master equation is:

dP(m, t )

dt (E−

1)

ktr

m

1m(et K ) P(m, t ). (29)

For a given mτ, this process is fully determined by ktr

since K is uniquely specified by these

two parameters. Examples of Pτ(m) and P(loss ) calculated from numerical integration of Eq.

(29) are given in Fig. 5. These graphs show that plasmid copy number fluctuations and loss




rates only can be efficiently reduced if ktr

is high, as expected from the macroscopic analysis

(Section 3).

In the limit where ktr, Eq. (29) can be simplified to:

dP (m, t )

dt κet (E−1) P(m, t ) (30)

where κktr

K . This is a good approximation when ktr 10, as in vivo, and has the

advantage that the mechanism is fully specified by the average copy number. Since Eq. (30)

is a Poisson process that evolves in exponential time (due to exponential volume increase),

the probability Pτ(m j ) of having m plasmid copies at the end of the cell cycle, given j copies

in the newborn cell, is simply:

Pτ(m j ) κm− j

(m j ) !e−κ. (31)

This means that the number of plasmids made per cell cycle is Poisson distributed with an

average that is independent of how many copies that were present when the cell was born.

At high average copy numbers, so that plasmids are rarely lost, κ is approximately equal to

the average number of plasmids in the newborn cell, κm. When plasmids arebinomially divided between the two daughters at cell division, copy numbers are nearly

Poisson distributed at all time points t in the cell cycle:

Pt(m)

mmt

m !em (32)

Here mt is the average copy number at time t in the cell cycle, approximatelym

t

κ2t/τ. This result comes directly from the fact that the binomial partitioning conserves

the Poisson distribution, i.e., if (X M m) Bin(m, p) and M Po(λ) then X Po( pλ).

Furthermore, the sum of two Poisson distributed random variables is again Poisson

distributed with an average of the sum of the two averages, i.e., if X Po(λ

) and X

Po(λ

)

then X X

Po(λ

λ

). A more intuitive explanation of the result in Eq. (32) can be found

along the same lines as for the paradigmatic birth-and-death process in Eq. (21). When thebirth rate is independent of the number of molecules present, the birth process is not affected

by when (or if) the deaths appear. Furthermore, the removal of a binomially distributed

number of molecules can (in this case) be seen as independent degradation. The only

difference is that all molecules are removed at the same time rather than continuously during

the cell cycle.

At a first glance it may seem strange that plasmids that negatively regulate their own

synthesis with the control mechanism above at best can reduce fluctuations down to a

Poissonian limit. However, it is only in the limit where hyperbolic control works perfectly

that negative feedback fully compensates for the fact that every plasmid copy can initiate its

own replication (see Section 3.2): the sensitivity amplification factor A in Eq. (24) approaches

the value one (Eq. (15)) first in the limit ktr.

When copy numbers of par − plasmids are approximately Poisson distributed (Eq. (32)), the

probability P(loss ) that a plasmid-containing cell produces a plasmid-free progeny at cell

division takes the simple form:

HLbin 2e−mτ/ 2(12)mτ/(ln()) 2e−κ. (33)




Fig. 6. Plasmid loss rate as a function of average copy number at the end of the cell cycle for par + (a)and par − (b) plasmids. Both curves in (a) correspond to perfectly working single-step hyperbolic control.In (b), one of the curves is instead perfect exponential control (zero copy number variation).

HLbin(H yperbolic Limit) is the lowest P(loss ) possible for par −

plasmids with single-stephyperbolic CNC. The approximations (32) and (33) are nearly exact for large average copy

numbers, but break down when mτ is so small that plasmid losses become large. More exact

estimates of HLbin

and HLpart

, the corresponding limit for par + plasmids, are given as

functions of the average copy number in Fig. 6. HLpart

can be calculated analytically, at least

for pair-site partitioning, but the expression is a very messy function of the average.

4.4.2 ColE1 multistep control can eliminate plasmid copy number variation

Using the same assumptions as in Section 4.4.1, exponential control (defined in Section 3.4)

is described by the master equation:

dP(m, t )

dt k

tr(E−1) e−m/(etK) mP(m, t ). (34)

For a given value of ktr

, exponential control reduces the dispersion in Pτ(m) compared with

hyperbolic control (Fig. 5). Since the sensitivity of the mechanism in principle is unlimited:

Pτ(m)

1

23

4

1 for mmτ0 otherwise

(35)

in the limit where ktr. However, the sensitivity increases only logarithmically with k

tr,

as discussed in Section 3.4. This roughly means that ktr

must be squared to reduce the

plasmid variance two-fold (for a given average) (Eq. (24)).

Since ColE1 is par −, the effect of exponential control is only analysed when plasmid copiesare binomially distributed between the two daughter cells. The lowest possible loss rate EL

(Exponential Limit) is then given by:

EL 2(12)mτ . (36)




Exponential CNC can in principle reduce P(loss ) indefinitely for par + plasmids.

A comparison of Eqs. (33) and (36) shows that when plasmids are binomially distributed

between the two daughter cells, a perfectly working hyperbolic CNC must compensate with

a 2ln2 14 times higher average copy number to give the same P(loss ) as a perfectly working

exponential CNC (Fig. 6). Although exponential control requires higher ktr

to work with full

potential, it is superior to the hyperbolic mechanism also at low values of ktr

(Fig. 5).

The results shown in Sections 4.4.1–4.4.2 largely support the results in Eq. (24) (seeEq. (15)): the significance of fluctuations can be predicted from steady state sensitivity

amplification analysis.

4.4.3 Replication backup systems – the Rom protein of ColE1 and CopB of R1

Up to this point the analysis has been confined to the simplest possible formulations of

inhibitor-dilution CNC and only included plasmids and inhibitors as variables. However,

both ColE1 and R1 additionally code for one regulatory protein each, Rom and CopB

respectively. The analyses in Sections 3 and 4 implicitly assume that Rom and CopB

concentrations influence the dynamics of CNC in a trivial way and that their effects could be

included in the rate and binding constants. There is some experimental support for thesesimplifications (Gustafsson & Nordstro m, 1980; Nielsen & Molin, 1984) but it is not unlikely

that Rom and CopB take a more active part in CNC dynamics, at least under some

physiological conditions. These possibilities will be briefly addressed here although giving

full justice to the complicated kinetics of Rom and CopB would require a separate analysis.

Plasmid ColE1 constitutively transcribes an mRNA encoding a small protein: Rom. A

single copy of the dimer configuration of Rom, DR

, binds to the initial RNA I–RNA II

kissing complex and accelerates duplex formation (Tomizawa & Som, 1984; Tomizawa

1990b). A shortage of DR

consequently reduces the efficiency of inhibition of the replication

preprimer RNA II and results in a higher per plasmid replication frequency.

The probability, Pci

, that an initial RNA I–RNA II kissing complex inhibits plasmid

replication can be written (Ehrenberg, 1996):

Pci

K [D

R]

K [D

R]

, (37)

where K K

. Experiments (Tomizawa, 1990a, b) indicate that normally [D

R]K

so that

Pci

is constant as assumed in Sections 3 and 4. If this were the case at all plasmid copy

numbers, the role of the Rom protein would only be to assure that RNA I–RNA II

interactions lead to inhibition. [DR

] could then be included in the effective binding constant

for RNA I–RNA II interactions (unless it fluctuates slowly and significantly). However, if K

[DR

]K , then P

ci [D

R] and the rate of inhibition would be directly proportional to

[DR

]. If [DR

] also responds to changes in plasmid copy number, as is reasonable since the

Rom gene seems to be constitutively expressed, Rom could take a much more active role in

CNC. This may drastically increase the sensitivity of control and for a given average copynumber reduce the plasmid loss rate (Ehrenberg, 1996). The lowest loss rates would be

obtained when Rom has a short half-life and the monomers dominate over the dimers. In this

scenario, a two-fold change in plasmid copy number would rapidly result in a four-fold

change in [DR

]. As far as we know, neither the half-life of Rom nor the relative occurrences




Fig. 7. A potential well that is rather flat around the minimum but where the steepness increasesdrastically at some point.

of monomers and dimers have been measured, though standard assays would suffice. This is

quite remarkable considering all the work that went into solving its structure.

It has been suggested (Summers, 1996) that the CopB protein of R1 could play a role

analogous to Rom by boosting the rate of RepA mRNA transcription from PRepA

at low

plasmid copy numbers (see Fig. 1). A the same time, quantitative experimental (Gustafsson

& Nordstro m, 1980; Nielsen & Molin, 1984; Nordstro m et al . 1980) and theoretical

(Rosenfeld & Grover, 1993) studies have led to the view that the PRepA

activity is negligible

at all plasmid copy numbers. However, a more recent theoretical analysis (work in progress)

of PRepA

and CopB shows that they could, in fact, have major impacts on CNC. The two

major resultsfromthis analysis are that (1) PRepA

could be regulated by threshold-type kinetics,

and (2) the plasmid loss rate would only be reduced significantly by loss of CopB regulation

if the plasmid carries a partition function, as does wild type R1. This analysis indicates that

previous conclusions may be due to artefacts: the experiments measuring plasmid number

adjustments to steady state (Gustafsson & Nordstro m, 1980; Nielsen & Molin, 1984;

Nordstro m et al . 1984) were made with plasmid mutants where the CopB concentration was

artificially increased and always above the threshold. Furthermore, experiments recording the

influence of CopB on P(loss ) were made with plasmids where the partition system had been

removed (Nordstro m et al . 1980). Finally, the previous theoretical analysis (Rosenfeld &

Grover, 1993) implicitly assumed, without experimental support, conditions under which

CopB could not contribute to CNC.If the effects of Rom and CopB only are significant at deviations to low copy numbers, they

may reduce the plasmid loss rate greatly without affecting the local sensitivity parameter A

(Eq. (24)) at all. This shows the limitations of linear noise approximations and the power-law

formalism: In systems with large relative fluctuations, sensitivity is important in a greater part

of phase space (Fig. 7).

4.5 Time-delays

4.5.1 Limited rate of inhibitor degradation

In the macroscopic analysis it was shown how a limited degradation rate (kd) of inhibitors

introduces time-delays that may cause oscillations in the CNC system (Section 3.6). For themesoscopic analysis of how low k

d values affect CNC we have chosen situations where the

average number of inhibitors is so high that random inhibitor variation for a given number

of plasmids is insignificant. The exact stochastic mechanisms for inhibitor synthesis and

degradation are then unimportant and one could, in principle, use a mixture of stochastic and




Fig. 8. (a) P(loss ) as a function of the inhibitor degradation rate constant kd

for exponential inhibitionwith k

tr 10 and insignificant inhibitor variation. (b) The normalised number of inhibitors as a function

of the normalised number of plasmids for the same parameters as in ( a). When kd, there is a perfect

proportionality, but for lower values of kd

, the relative deviation from steady state is smaller in inhibitor

than in plasmid. Both (a) and (b) are from data collected at the end of the cell cycle where the averagecopy number was 16 plasmids.

deterministic modeling. Here we instead use a two-dimensional master equation for the

probability of m plasmids and i inhibitors

p m,i (k

tre−i/K(E−

m1) mk

sm(E−

i 1)k

d(E

i1)i ) p

m,i. (38)

This system was integrated for different values of kd

for a fixed average plasmid copy number.

For high average inhibitor copy numbers, so that fluctuations are small, neither ks nor K affect

the plasmid copy number dispersion around a given average (tested but not shown), as is

expected from a macroscopic analysis. When CNC is analysed for a fixed average plasmid

copy number but different kd-values, the only system parameter that must be specified is thus

ktr

. The value ktr 10 was used throughout this section and this is a representative in vivo

value as judged from experimental data (see Section 4.8). However, the qualitative behaviour

shown in Fig. 8 is rather insensitive to the value of ktr

in the entire realistic region 5 ktr

200.

In Fig. 8 (a), P(loss ) is given as a function of kd

for a constant average copy number. When

kd 0, the time lag between plasmid and inhibitor concentrations is limited only by dilution

due to volume increase and P(loss ) approaches an upper plateau. Increasing kd

reduces P(loss )

until a lower plateau is reached where the inhibitor follows plasmid concentration perfectly.

However, there is no optimum corresponding to critical damping, as found in the

macroscopic analysis (Section 3.6). This is because critical damping requires special initial

conditions and improves CNC by making the inhibitor deviate slightly more from steady

state than the plasmid. In the mesoscopic case, plasmid copy number deviations areinstead likely to have been closer to steady state in the recent history. This means that for small

values of kd

, the current inhibitor concentration is often closer to its steady state than the

current plasmid concentration (Fig. 8(b)). Low kd

will thus generally slow down, rather than

speed up, adjustments in plasmid concentration. Though this phenomenon is different from




Fig. 9. Control curves for ktr 40. A generalised, mesoscopic control curve at the end of the cell cycle

for an exponential mechanism with kd 0 and an average of four plasmid copies per cell is compared

with macroscopic exponential and hyperbolic control for which kd.

low sensitivity amplification in every single cell, the average behaviour will be more sluggishand in some sense reduces the effective sensitivity.

Another intriguing difference between the meso- and macroscopic descriptions concerns

Eq. (20). From this inequality one would expect that a higher sensitivity of inhibition would

require a higher kd

for CNC to work well. By contrast, the mesoscopic analysis showed that

the choice of kd

-value to stay near minimal P(loss )-values is surprisingly independent of

inhibition mechanism as well as of the value of ktr

(data not shown). The simple explanation

is that even if sensitive control means that plasmid concentration changes rapidly when out

of steady state, it also means that plasmid concentration rarely is out of steady state. These

two effects tend to neutralise each other and there is no reason to a priori assume that sensitive

control would result in more situations where a high kd

is required.

Mesoscopic analysis of CNC allows for construction of generalised control curves (Section

4.3) that include not only traditional sensitivity amplification, but time-lags and all other

aspects of CNC. Figure 9 exemplifies a generalised control curve with a highly sensitive

inhibition mechanism but low kd

. It is seen from the figure that a combination of sharp

control and slow inhibitor degradation may result in similar generalised control curves as

insensitive control and rapid inhibitor degradation.

Long inhibitor half-lives can make regulation more sluggish and must be taken into

account when assessing the significance of internal noise.

4.5.2 Precise delays – does unlimited sensitivity amplification always reduce plasmid losses ?

Judging by the previous results, unlimited sensitivity will cause the plasmid copy number

distributions to collapse into their averages. However, this is an idealisation depending onapproximations that may become invalid when CNC becomes very sensitive. For instance, the

eclipse time after an initiation of replication, during which the next replication cannot take

place, may be important when CNC is very sensitive. If no inhibitors are produced during

the eclipse, they transiently increase in numbers. This may trigger further replications




with further reductions in inhibitor numbers. CNC systems with high sensitivity are

particularly vulnerable to this type of cascades, which may lead to runaway replication

and possibly to cell death. Both plasmid ColE1 and R1 are organised so that the CNC genes

are the last to be replicated. This means that even if the template plasmid produces

inhibitor, there is an inevitable time-delay between committing to replication and increasing

the inhibitor production.

One can expect that ColE1, owing to its much smaller size, is less sensitive to this problemthan R1. ColE1 is small (approximately 6–7 kb), which suggests a short eclipse time that

in turn would allow sensitive control without cascade effects. Plasmid R1 with its 90 kb has a

longer eclipse that may forbid ultrasensitive CNC. Interestingly, the multistep CNC discussed

for R1 (Sections 3.5 and 4.5.2) is an elegant solution to the problem of reducing P(loss ) to

very low values without causing runaway replication. A temporary shut-down in inhibitor

synthesis would not increase the replication frequency greatly for R1 due to its hyperbolic

CNC. The R1 multistep mechanism in combination with a partition function (Section 4.5.2)

may thus be a way to reduce plasmid loss rates when sharp control curves cannot be

implemented due to the size of the plasmid.

4.6 Order and disorder in CNC

Apart from a deterministic volume expansion, the cell has so far been treated as a constant

environment in which plasmid replication behaves as an independent random process.

However, the kinetic parameters used reflect concentrations that also may fluctuate randomly.

Plasmid replication should then be described as a random process in a random environment.

This is quite general: if one component of a chemical network fluctuates randomly it may

force all other components to follow. In the literature this phenomenon is filed under external

noise (van Kampen, 1992), Markovian rates (Eisen & Tainter, 1963; Peccoud & Ycart, 1995;

Taylor & Karlin, 1998) or dynamical disorder (Vlad et al . 1996).

If a parameter fluctuates very slowly, a concentration that it affects will display a quasi-

stationary distribution around an average determined by the changing parameter. When this

parameter changes, new quasi-stationary distributions for the concentration are continuously

approached. Even when quasi-stationarity is never reached, such time-correlations are the

reason why noisy rate parameters invoke noise in affected reactions. If, in contrast, the rates

fluctuate very rapidly then the correlations become insignificant and the noise in the rates does

not carry over into the noise in affected concentrations.

Another way of looking at disorder is to consider the long-run distribution of sojourn

times (Taylor & Karlin, 1998). For illustration, assume a pure birth process where the

random walker sojourns at position m for a time T m

. If the transition between m and m1

occurs with a memory-lacking intensity λm

, T m

is exponentially distributed. If λm

instead

fluctuates randomly, the long-run distribution of T m

becomes a mixture of exponentials and

can thus be much broader (disorder). This directly points at the possibility of also having

‘ordered’ systems: If the transition from m to m1 includes many rate-limiting elementary

steps, the sojourn time distributions can become narrower than exponential. In some casesthis can reduce the internal noise.

Here we exemplify how dynamical disorder in CNC can increase plasmid copy number

variation and how R1 multistep regulation can narrow the sojourn time distributions

(increased order) and thereby lower the plasmid loss rates.




4.6.1 Disordered CNC

We first inspect the limit of single-step hyperbolic control (Eq. (30)) where κ fluctuates

randomly. Plasmid replication is thus assumed to be an inhomogenous Poisson process with

a Markov intensity: a Cox process (Taylor & Karlin, 1998). For simplicity we assume that

κ is proportional to n in the simple birth and death process (21) so that stationary fluctuations

are Poissonian with average nkkd

and a molecule half-life of τn/ ln(2)k

d. To

simplify matters further we replace host growth and division by a normalised plasmid

degradation rate m (see Section 4.3) corresponding to a plasmid half-life of τm/ ln(2) time

units. These four birth and death reactions form a homogenous two-dimensional master

equation, where the significance of fluctuations can be shown to exactly follow:

σm

m

1

m

1

n

τn/

τn/τm

/

. (39)

The first term shows the Poissonian plasmid copy number fluctuations in the absence of rate

fluctuations. The second term reflects dynamical disorder. Its first factor is set by the

significance of rate fluctuations, σκκ 1n, and the second by their relative

permanence. Rapid rate fluctuations (τn/ τm

/) do not increase plasmid copy number

fluctuations significantly from its Poisson limit, simply because they do not induce significant

time-correlations in the replication frequency.

The reason why dynamical disorder has such a straightforward effect in (39) is that the

system is linear. In general, regulatory circuits like the CNC systems described not only are

affected differently by internal noise, they also differ in their susceptibility to noisy rates. For

instance, a well-working (ktr 1) exponential control mechanism is insensitive to fluctuations

in ktr

simply because the average is roughly logarithmic in ktr

(Eqs (3) and (8)). However,

both exponential and hyperbolic control have averages that are approximately proportional

to the ratio between the rate constants for formation and elimination of inhibitors (Eq. (3)).

A complication that cannot be overlooked when trying to make predictions about real

systems is that the noise in different rate constants can be strongly correlated. In fact,

regulatory systems may exploit correlations to eliminate dynamical disorder. For instance, arandom change in the free RNA polymerase concentration affects the intensity for making

RepA mRNA or RNA II (Fig. 1), but the effect on the replication frequency is counteracted

by a corresponding change in the intensity for making their respective antisenses, CopA and

RNA I. In other words, periods when the individual plasmid copies attempt to replicate

more frequently are also characterised by higher inhibition probabilities. For well-working

hyperbolic control (ktr 1), these effects may in fact perfectly cancel each other so

that the system can rid itself from dynamical disorder. Of course, this is just another example

of the general principle that robustness to a parameter means robustness to random variation

in the same.

4.6.2 Ordered CNC: R1 multistep control gives narrowly distributed interreplication times

As was shown in Section 3.5, R1 multistep control does not contribute to sensitivity

amplification (Section 3.2). Here we show that the multiple steps still may reduce the plasmid

loss rate greatly by increasing the regularity of the individual reactions.




The birth and death process assumed in this section is simplified to only include two

parameters: the number of hyperbolically regulated steps, n, and the average plasmid copy

number. We used an average copy number of six in dividing cells (in the in vivo region

(Engberg & Nordstro m, 1975)), but similar results were obtained in a broad range of average

copy numbers. A master equation formulation of scheme (18) is:

dP(m, m

, … m

n−, t )

dt κet

A

B

mn−

1

0 n− j=

m j11

P(m2, m, … mn−1, t )

n−

i=

E

F

mi1

n−

j=

m j

P(… mi1, m

i+1 …, t )

G

H

P(m, m

,…, m

n−, t )

C

D

. (40)

The index m j

denotes the number of plasmids with j RepA molecules bound at the origin of

replication and et is the cell volume at time t . The master equation thus simply states that

every plasmid copy has a rate of RepA synthesis that is inversely proportional to the total

plasmid concentration in the cell. With n 1, it simplifies to Eq. (30). All data shown are

generated for par + (pair-site partitioning, Section 2.4), unless stated differently. The com-

pounded rate constant κ is determined by n and the average copy number and is excludedfrom the figures. Owing to its high dimensionality, Eq. (40) was solved through a

combination of numerical integration, exact Gillespie-type (Gillespie, 1977) Monte-Carlo

simulations and analytical solutions (see Paulsson & Ehrenberg, 2000a, for a more precise

description). The precision obtained with this combined method is sufficiently high to

make error-bars redundant.

To better understand the dynamics of Eq. (40), we conditioned the simulated data on mstart

n−i=

mi at time t 0, i.e., the number of plasmid copies that by chance were present at the

start of the cell cycle. This was done by simulating Eq. (40) for a very large number of

subsequent cell cycles, recording all cell cycle characteristics and sorting them according to

the value of mstart

. We did thus not artificially start a cell cycle with a certain mstart

and certain

numbers of RepAs bound.

The macroscopic analysis (Section 3.5) indicated that the control curve for R1 is hyperbolic

regardless of the number of regulated steps in scheme (18). The average number of

replications per cell cycle conditioned on mstart

, repl mstart, would then be constant. This

result is not supported by the mesoscopic analysis for exceptionally low average copy numbers

(between 1 and 2 at the start of the cell cycle). For multistep control, repl mstartm

start as

a function of mstart

is then sharper than hyperbolic (not shown). However, in the in vivo range

of copy numbers, the control curve difference between single- and multistep CNC is

negligible (Fig. 10(a)) and the macroscopic prediction is consistent with the mesoscopic

averages.

The relative standard deviation in the number of replications per cell cycle conditioned on

mstart

, σ(repl mstart

)repl mstart, is given as a function of m

start in Fig. 10(b) for different

values of n. Since repl mstart is almost independent of mstart (Fig. 10(a)), the interpretationof Fig. 10(b) is straightforward. Variation in the number of replications per cell cycle can

be reduced by multistep control and there is much less variation for low mstart

.

R1 is par + in vivo and when the partitioning works perfectly, plasmids are only lost when

mstart 1 and this single plasmid copy additionally fails to replicate. The probability for




start start

r e p l

r e p l

Fig. 10. (a) The control curve (symbols) showing the average number of replications per plasmid andcell cycle, repl m

startm

start as a function of m

start for step number n 6. The solid line corresponds

to the macroscopic hyperbolic control curve. (b) The relative standard deviation in the number of replications per plasmid and cell cycle, σ(repl m

start)repl m

start, as a function of m

start for different

values of n. (c ) The probability density for the waiting time for the first replication event conditionedon m

start1, f

tr(t m

start 1) for n 1 and n 6. The shape of the dashed area was calculated as if

cell division did not occur. The average copy number in newborn cells was three in all subfigures.

plasmid loss given that mstart 1, P(loss m

start 1), thus depends on the conditional

probability density for the waiting time for the first replication, f r(t m

start 1). As shown

in Fig. 10(c ), this distribution is significantly narrower for the multistep mechanism. This

corresponds to the discussion on sojourn times above. The statistical variation in the time the

system sojourns at m 1 decreases with n.

The probability for making zero replications in a cell cycle, conditioned on mstart

, decreases

significantly with increasing n, but also with decreasing mstart

(Fig. 10(a)). The simple reason

for the latter effect is that even if every single plasmid copy has a clock-like behaviour, the

individual copies will be out of phase since they have different numbers of RepA molecules

bound at ori . The replication behaviour will then not be clock-like. By studying the collective

behaviour of a large number of plasmids, it is thus hard to distinguish this kind of

multistep mechanism from its single-step counterpart.

To compare the impact of R1 multistep CNC for par + and par − plasmids, we look at two

different ways of designing CNC systems, CNC

and CNC.

CNC

:Fixed average copy number of mτ,CNC 6 and varying n.

CNC

:Varying average copy number mτ,CNCand fixed n 1.

We then define an efficiency ratio RE

as the ratio between mτ,CNCand mτ,CNC

where

P(loss CNC )P(loss CNC ). RE thus reflects how much higher the average number of plasmids must be for a mechanism with single-step hyperbolic control to give the same

stability as multistep control. Figure 11(c ) shows RE

as a function of n. Clearly, a par + plasmid

can utilise the multiple steps much better than a par − plasmid. For instance, for a par + plasmid,

an increase from n 1 to n 3 would almost reduce plasmid losses as much as a twofold




start

Fig. 11. (a) Probability of zero replication in the cell cycle as a function of mstart

for different numbers

of rate-limiting steps. (b) Probability of zero replications when mstart1 as a function of the number

of rate-limiting steps in the initiation of replication. The average copy number in newborn cells was

three in all simulations. (c ) Trade-off of multistep initiation for plasmids with and without a partitionsystem. The relative increase in copy number (R

E on y-axis) required to give the same increase in

segregational stability as an increased number of rate limiting steps (n on x-axis). The average copy

number in newborn cells was three.

increase in the average copy number (Fig. 11(c )). For a par − plasmid, it would only reduce

losses as much as a 15% increase in the average copy number. Simulations for other average

copy numbers gave similar results.

For a given average copy number, plasmid loss rates can only be reduced to 2 −mτ (Eq.

(25)) in the absence of partition system. The maximal RE

is therefore 1.4 (see Section 4.4.2)

for par − plasmids regardless of the choice of CNC, while R

E is unlimited for par + plasmids.

However, the R1 multistep mechanism would never work this well and as soon as otherpotential sources of fluctuations in plasmid copy number are included, its shortcomings

become even more pronounced. Plasmids without partition system must efficiently correct for

copy number deviations and would not benefit significantly from multistep mechanisms of

the R1 type.

4.7 Noisy signalling – disorder and sensitivity amplification

The basic logic of inhibitor-dilution CNC is that a higher plasmid copy number m

corresponds to a higher inhibitor number i and thus a lower replication frequency per

plasmid. However, because individual reactions are random, i is only determined by m in a

probabilistic sense. This implies a loss of information in the sense that a sample of i is notsufficient for determining m. One may ask how CNC can work well if the signal level that

regulates the plasmid replication frequency is significantly randomised. Would not significant

inhibitor noise cause CNC to make the wrong decisions and allow replication when m is

already high and vice versa? Here we show how the effect of noisy signalling can be the direct




opposite. The key to understanding this is to separate between effective (average) reaction

intensities and dynamical disorder.

4.7.1 Eliminating a fast but noisy variable

In the macroscopic analysis, the second-order system (1) can be reduced to a first-order system

when the inhibitor concentration follows plasmid concentration closely. In the mesoscopicapproach, proper elimination of a fast variable must also take noise into account.

A first complication is related to dynamical disorder: even though an average over a cell

population responds rapidly, the inhibitor concentration may fluctuate slowly in each single

cell as a result of dynamical disorder. In other words, a short inhibitor half-life makes it adjust

quickly to the combination of the current plasmid concentration and the current rate constants

for inhibitor turnover, not to the plasmid concentration alone. Assume a constant number of

m plasmids and that the number of inhibitors, i , changes as in the state diagram:

i 1ksm

kdi

i ksm

kd(i+)

i 1. (41)

If ks

fluctuates randomly in the same way as κ in Section 4.6.1, the quasi-stationary

conditional average number of inhibitors given m plasmids, i m

, is still proportional to m,

but the corresponding variance can be greatly increased:

σim

i m

1

i m

1

n

τn/

τn/τi

/

(42)

as in Eq. (39). When m changes, this equation only holds when τn/

is very short compared

to the length of the cell cycle. Otherwise the quasi-stationarity of Eq. (42) is never established

and the three-dimensional master equation for m, i , and n must be used. If both τn/

and τi/

are very short compared to the length of the cell cycle, the three-dimensional master equation

can be reduced to a one-dimensional master equation for m, and a separate two-dimensional

master equation for i and n.

When inhibitors fluctuate rapidly in individual cells, it is possible to study themseparately from the plasmids. The one-dimensional master equation for the number of

plasmid copies is:

p mk

tr(E−1)q

m,tmp

m. (43)

Here q m,t

is the effective probability that an attempt to replicate the plasmid is successful.

This must be calculated from quasi-stationary conditional probability of i inhibitors, given

m plasmids at time t , pim,t

, as:

q m,t

i=

pim,t

q i,t

. (44)

This assumes that the replication trial has a negligible duration so that i stays constant during

the time for the chemical reactions condensed into the single intensity q m,t. For ColE1, theinhibition window seems to stretch over five seconds (Brenner & Tomizawa, 1991) and R1

seems to operate on a similar time-scale. If fluctuations are so exceptionally rapid that the

inhibitor distribution is reshuffled during these five seconds, then the fluctuations will have

no impact on the regulatory reaction at all (q m,t q im,t

). In intermediate cases, the




effective probabilities must be calculated through summation over an integral (Paulsson et al .

2000), including not only the stationary signal distributions but also the underlying stochastic

process. However, even if degradation of both CopA and RNA I is rapid (half-lives ranging

from 30 s to a couple of minutes depending on conditions), they are still slow compared to

the duration of the inhibition windows so that Eq. (44) may be a good approximation.

For linear reaction intensities q i abi , the average q coincides with the reaction

intensity for the average signal, q i. However, this is generally not the case for nonlinearmechanisms that instead receive a disproportional contribution from the tails of distributions.

The general principle is very clear for bimolecular reactions (Renyi, 1953). If a reaction occurs

with rate r nkn(n1), where the probabilities for n are p

n then the average is not r

k(nn) but instead r k(nσnn). Since the variance can decrease even

though the average increases, r can actually locally behave as a decreasing function of n.

From a first glance, q m,t

appears to be a mere average and it may seem that the variation

in q i,t

must have an impact on the process. However, that a quantity can be defined as

an average over an accessory signal distribution does not mean that the variation over that

distribution is at all relevant. This can be illustrated with an example where the fate of an

attempt to replicate a new plasmid is determined by throwing dice. Assume random variables

X

and X

for the number of eyes of two normal six-sided dice. Assume further that a trial

results in plasmid replication if X X

10. This corresponds to six outcomes out of 36

possible, i.e. a probability of 16, and is in direct analogy with Eq. (44). Assume that pi is

the probability that X i and q

i is the conditional probability that X

X

10 given that

X i . This would result in the following possible cases

i pi

q i

1 16 02 16 03 16 04 16 165 16 266 16 36

From this perspective, q i=

pi

q i 16 is just an average and the conditional

probability q i also has a variance over distribution p

i. Instead having p

1, corresponding

to insignificant signal noise, would give the same average q 16, but zero variation in

q i. Clearly, only the value 16 is important and not how many die that were used.

In summary, rapid fluctuations do not increase the disorder of a system but may still affect

effective intensities.

4.7.2 Conditional inhibitor distribution: Poisson

The simplest possible realistic birth and death process for the number of inhibitors i , giventhe number of plasmids m, follows from the state diagram:

i 1ksm

kdi

i ksm

kd(i+)

i 1. (45)




Fig. 12. Conditional inhibitor distributions for different numbers of plasmids. (a) Poissonian pi

, pi

and pi

when i 10. (b) Poissonian p

i, p

i and p

i when i

100. (c ) Negative binomial (Eq.

48)) pi, pi and pi when i 100 with b 10 and λ 1. (d ) Displaced Poissonian (case 1) (Eq.(52)) p

i, p

i and p

i with k

sk

d 90 and k

trk

d80.

When kd

is high, the conditional quasi-stationary inhibitor distribution for a given number,

m, of plasmids, corresponds to the stationary distribution of the conditional master equation:

p im (k

sm(E−

i 1)k

d(E

i1)i ) p

im. (46)

The stationary distribution of this process in a Poissonian with conditional average i m

ks

mkd

. The average number of inhibitors is thus always proportional to the current number

of plasmids, but there is also variation around the average. In the terminology of Haken

(1978) and Gardiner (1985), the inhibitor is a noisy slave of the plasmid. In Fig. 12, we show

some conditional inhibitor distributions for ksk

d 1 (Fig. 12(a)) and k

sk

d 10 (Fig.

12 (b)).

4.7.3 Increasing inhibitor variation I : transcription in bursts

Although the inhibitor genes for RNA II and CopA are transcribed constitutively from R1

and ColE1 (Section 2.4) respectively, this does not necessarily mean that the individual

transcription events are independent. If the inhibitor gene shifts between an active and an

inactive state with first order kinetics, the time it stays active is exponentially distributed.

Such random jumps between different states of the gene implies a dynamically disordered

transcription process. If the transcription intensity is constant (Poisson process) in the active

time interval, the total number of transcripts made in the interval, j , is geometrically

distributed, G jφ j(1φ ) (Berg, 1978). When the active state is short-lived, the RNAs are

synthesised in instantaneous, geometrically distributed bursts. When ks denotes the intensity

for the transition from inactive to active gene, the conditional master equation (assumptions

otherwise the same as for Eq. (46)) is given by:

p imk

sm0i

j=

G j pi− jmφ p

im1kd(E

i1) ip

im. (47)




Using moment generating functions it is straightforward to show that the stationary

distribution of this process is the Negative Binomial (NB):

pimφ i(1φ )λm

Γ (λmi )

Γ (λm)i !, (48)

where Γ is the gamma function and λksk

d (Paulsson et al . 2000). The average burst size

is bφ (1φ ), the conditional average number of inhibitors is i mλbm and the

variance is σimλmb(b1). Fluctuations can thus be considerably larger than for the

Poisson distribution (Fig. 12(c )), which is recovered from the NB in the limit b 0 where

every synthesis event is independent.

4.7.4 Increasing inhibitor variation II : duplex formation

In the macroscopic analysis (Section 3.6) it was shown how degradation of an antisense

inhibitor through duplex formation with its sense counterpart could be compounded in an

effective synthesis rate constant β per plasmid. It was further shown that β had no dynamic

effect on CNC as long as β 0. Mesoscopically, however, the jump processes :

i 1 (ks−ktr)m

kdi

i (ks−ktr)m

kd(i+)

i 1 (49)

and

i 1 ksm

kdi+ktrm

i ksm

kd(i+)+ktrm

i 1 (50)

can display dramatically different behaviour. Transitions in two opposing directions can not

be replaced by a net-transition in one direction.

To take duplex formation into account one must know what happens when the number

of antisense inhibitors is small and, for sure, when i 0. Here we consider both the situation

when the sense target molecules do not (case 1) and do (case 2) accumulate in the absence of

antisense molecules. The difference between the two cases is only important if i in scheme (50)

frequently takes very low values.

Case 1. The master equation for the conditional number of inhibitors (assumptions

otherwise the same as for Eq. (46)) is now :

1

23

4

p im (k

sm(E−

i 1)k

d(E

i1) i k

trm(E

i1)) p

im for i 0

p m (k

trmk

d) p

mk

smp

m

. (51)

The master equation for i 0 is given separately because the degradation rate ktr

m due to

duplex formation then must be zero. The stationary solution to this process is:

pim

(mksk

d)i+mktr/kd

Γ (i mktrkd1)

C N

, (52)

To be exact, the rate of degradation through duplex formation depends on i at very low values of i . To get around this problem, one can include the target molecule as a variable T and use k

TiT as the

bimolecular rate of duplex formation. The approximate analytical solution of Eq. (52) corresponds tothe limit when k

T.




Fig. 13. Conditional inhibitor averages i m

as functions of m. The broken line is i m

for the displacedPoisson distribution in Eq. (52) with k

sk

d 25 and k

trk

d2. The solid line corresponds to perfect

proportionality with the same i

as the displaced Poissonian and the dotted line corresponds toperfect proportionality with the approximate formula i

mm(k

sk

tr)k

d.

where C N is a normalisation constant. This distribution is a displaced Poissonian (Johnson &Kotz, 1969), i.e. a Poissonian where the average has been displaced mktrk

d units to the left,

but the shape is unchanged. Some examples of these conditional distributions are given in

Fig. 12(d )). When the probabilities for very low numbers of inhibitors are insignificant, so

that the normalisation constant C N

differs insignificantly from the corresponding Poisson

distribution, duplex formation gives a standard deviation of mksk

d, i.e. that of a

Poissonian with average mksk

d, but the actual average is now only i

mm(k

sk

tr)k

d.

When the probabilities for very low numbers of inhibitors are significant, then i m

m(ksk

tr)k

d so that the conditional average i

m is no longer proportional to m (Fig. 13).

The increased random variation in inhibitor numbers caused by duplex formation can be

understood from sensitivity amplification arguments (see Section 4.2). If one term of the total

rate of degradation of inhibitors is constant rather than proportional to the number of

inhibitors, the rate of degradation is less sensitive to changes in the number of molecules.The ratio between degradation from position i and synthesis from position i 1 is R

i

(ktr

mkd

i )(ks

m) so that the discrete sensitivity amplification factor (Eq. (23)) is given

by :

AR,i

i

i ktr

mkd

. (53)

The sensitivity amplification factor thus varies between 0 and 1 depending on i . Evaluating

AR,i

at the approximate average i mm(k

sk

tr)k

d gives:

A 1ktr

ks

. (54)

As ktr approaches ks, the sensitivity amplification factor decreases and fluctuations maybecome very large. The increased variation is obviously not due to zero-order degradation

in itself but is instead caused by opposing fluxes with rates of approximately the same kinetic

The probability mass that falls in the tail with negative states is zero and instead increases thenormalisation constant C

N.




orders. Adding a first-order synthesis term (autocatalysis) to scheme (45) has a similar effect.

This was shown in 1957 with respect to maser amplification (Shimoda et al . 1957) and has

been theoretically applied to autocatalytic reaction schemes (Gardiner, 1985).

Case 2. If target molecules accumulate when there are zero antisense molecules, the process

can be formulated in terms of the difference j between the numbers of antisense and target

molecules. This situation is similar to social pair formation. For instance, if the cell is a dance

hall and sense and antisense are boys and girls respectively, rapid duplex formation means thateveryone dances if possible. If both boys and girls enter the dance hall with constant

intensities and every individual leaves again with a constant intensity (both intensities may

differ between the sexes), there will be a randomly sized same-sex crowd waiting to dance.

Expressed in terms of the difference j , j 4 corresponds to four boys waiting and j 7

corresponds to seven girls waiting. This situation is described by scheme (4.30) (by replacing

i by j ) for j 0, but for j 0 one instead gets the mirror-turned scheme:

j 1 ksm−kd( j−)

ktrm

j ksm−kd j

ktrm

j 1 … …1 ksm+kd

ktrm

0 , (55)

where kd

is the rate constant for degradation of targets in the absence of antisense molecules.

This distribution has a peculiar property: it can be formed by taking the right tail of onePoisson distribution and mirror-turning it so that it starts in j 0 and goes to . The

right-hand side is formed by taking the right tail of another Poisson distribution and shifting

it so that it starts in j 0 and ends in . When kdk

d the distribution of j is almost a

displaced Poissonian with average j mm(k

sk

tr)k

d (that can take negative values) and

variance mkkd, where k is the maximal value of k

s and k

tr. The approximation is close to

exact unless k is small.

If one instead is only interested in antisense molecules, i , the probabilities for i 0 again

take the shape of a displaced Poissonian, but the probability for zero molecules is much higher

and is instead the sum of the probabilities for j from to 0. We will refer to this

distribution for i as the special displaced Poissonian. Again there is a nonlinear relation between

i m

and m, but the effects are less pronounced than in case 1 (Fig. 13).

Duplex formation between inhibitors and targets seems to have many potential impact onCNC.

It can greatly increase the random variation in the number of inhibitors (Fig. 12). This can

be understood in terms of the uncertainty in the difference between two independent

random variables (Paulsson & Ehrenberg, 1998). If X and Y are independent and Z

X Y , then ZX Y and σzσ

Xσ

Y so that fluctuations in Z in relation to

its mean may be very large. The reason why this analogy works for degradation through

duplex formation, but not for first-order degradation, is that the numbers of molecules

synthesised and degraded during a longer time period are approximately independent (and

Poisson distributed) in the former (given that i rarely is zero) but not in the latter.

A relative change in m may result in a smaller relative change in i m

(Fig. 13). This is

quite interesting from a modeling perspective. The corresponding macroscopic rateequation is exactly linear all the way down to (identically) zero concentrations.

However, the fact that there cannot be diffusion to negative numbers of molecules

introduces a mesoscopic nonlinearity that means that the macroscopic description is

insufficient also for describing averages.




Fig. 14. Plasmid control curves constructed so that when m 10 the replication frequency per plasmidis 1 and a replication trial has 1% chance of being successful (k

tr100). The distributions used for

noisy signals are derived in Sections 4.7.2–4.7.4. Upper left: Poisson fluctuations. Lower left: NBbinomial fluctuations ( p

im taken from Eq. (48)). Middle: Displaced Poisson fluctuations ( p

im taken

from Eq. (52)). Right: Special Displaced Poisson (calculated from schemes (50) and (55)).

In case 2 above (special displaced Poissonian) it can increase the probability for zero

molecules enormously. When ktrks, CNC is kinetically impossible and plasmidsdisplay runaway replication, as shown experimentally with plasmid mutants (Nordstro m

& Uhlin, 1992).

Since i m

becomes very sensitive to changes in ktr

or ks

when these are close to each

other, slow random fluctuations in any of these rate constants can result in extreme

dynamical disorder (not shown).

Under normal conditions the effects of duplex formation are probably negligible in vivo where

it appears that ksk

tr (see Section 4.8), but they may come into play under exotic conditions.

4.7.5 Exploiting fluctuations for sensitivity amplification: stochastic focusing

For threshold control, fluctuations inevitably reduce sensitivity since the signal concentration

then randomly jumps back and forth across the threshold. By contrast, more realisticbiochemical mechanisms have random responses also to noise-free signals and the effect of

noise on sensitivity amplification can then be the opposite. Figure 14 shows control curves

calculated from Eq. (44) using the conditional inhibitor distributions derived in Sections

4.7.2–4.7.4.




As can be seen in Fig. 14, Poisson fluctuations can enhance sensitivity greatly. In fact, when

fluctuations are very significant (low average), the hyperbolic control curve approaches the

exponential control curve because pm e−ksm/kd. Exponential control is unaffected by the

fluctuations. This is because

i=

e−i/Kxi

i !

e−x e−x(−e−/K) .

The same goes for the NB distribution; hyperbolic CNC can exploit fluctuations for higher

sensitivity, but the exponential CNC is unaffected since:

i=

e−i/Kφ i(1φ )x Γ (xi )

Γ (xi )i ! (1φ e−/K)−x .

The displaced Poissonian results in a more complicated behaviour. In this case inhibitor

fluctuations can both enhance and reduce the quality of hyperbolic control. It may also reduce

the quality of exponential control greatly. This is due to the qualitative change in dynamics

at very low values of i where the nonlinearity introduced by the lower boundary comes into

play (see Section 4.7.4). For the special displaced Poissonian, the situation is similar but a lot

better. Hyperbolic control can become much more sensitive and the sensitivity of exponentialcontrol is only reduced in extreme cases.

The increased sensitivity can be explained in the same terms as the multiple steps that

lead to exponential control curves for plasmid ColE1. Hyperbolic control receives a

disproportional contribution from the tail of the inhibitor distribution where i is low. A small

change in m affects the rate of inhibitor synthesis at every step in the jump process that

determines i . Consequently, a small change in m can invoke a great change in the probability

mass of the tail that contributes disproportionally to q . This is a quite general effect; tails

of distributions generally respond sensitively to changes in averages. SF is thus equivalent to

multistep control where the states correspond to the discrete numbers of molecules rather

than to special molecular configurations. In other words, significant signal noise allows for

sensitivity amplification that is not possible when all these states can be lumped into an

average.

4.7.6 A kinetic uncertainty principle

In Section 4.4.1 it was shown how single-step hyperbolic control due to its limited sensitivity

only can reduce copy number fluctuations down to (approximately) a Poissonian limit.

However, this result relied on the assumption that the conditional inhibitor fluctuations were

negligible, i.e., that the number of inhibitors represented the number of plasmids without

statistical variation. As shown above, these fluctuations may be exploited to increase a

kinetic mechanism’s capacity for sensitivity amplification. When inhibitor fluctuations are

significant, plasmid copy number fluctuations can thus be further reduced. In other words, to

reduce the uncertainty in the copy numbers of one chemical species in a regulatory network, itmay be necessary to increase the (conditional) uncertainty in the other. Kinetic principles differ

from system to system, as opposed to the more constant laws of physics, but this ‘kinetic

uncertainty principle’ is applicable to a wide variety of systems that display either stochastic

focusing or its opposite, stochastic defocusing.




Fig. 15. (a) Plasmid copy number distributions with averages m 10 calculated from Eqs (43), (44)and (48). The inhibition constant K differs between the three cases to keep the average m constantthroughout. (b). The conditional inhibitor distributions given that there are 10 plasmids.

To separate the impact of signal noise on effective reaction intensities from correlations

between outcomes of subsequent signal dependent reactions, we first assume that inhibitor

fluctuations are very rapid (Eq. (43)). In the example given in Fig. 15, ktr 100, the average

plasmid copy number is fixed at m 10 and the conditional average number of inhibitors

is fixed at i 50. The conditional inhibitor distributions are assumed to be NB (Eq. (48))

and the signal noise level is changed by increasing the average burst size b for a given average

(Fig. 12 c )). The kinetic uncertainty principle shown is similar in all cases that give rise to SF

in Fig. 14.

4.7.7 Disorder and stochastic focusing

To study the combined impact of inhibitor fluctuations on dynamical disorder and sensitivity

amplification, we formulate a two-dimensional master equation that takes changes in both

plasmid and inhibitor copy numbers into account. Assuming hyperbolic CNC and

independent inhibitor synthesis and degradation events, the master equation is

p m,i

0 k

tr

1i K (E−

m1) mk

sm(E−

i 1)k

d(E

i1) i

1 pm,i

. (56)

The cyclic-state plasmid copy number distributions at the end of the cell cycle were calculated

and the results are displayed in Fig. 16. For high kd

(and ks), CNC is more efficient when

fluctuations are large, but for low kd, CNC is much more efficient when fluctuations are




Fig. 16. Plasmid copy number distributions at the end of the cell cycle for an average of eight plasmids.

The solid lines correspond to the solution of Eq. (56) for ktr 40 and kskd 1, i.e. where randomfluctuations are very significant (eight inhibitors on average). The dotted lines correspond toinsignificant fluctuations.

insignificant. The reduction in performance for low kd

is not related to sensitivity in itself.

Conventional sensitivity amplification, such as exponential control, does better than

hyperbolic irrespective of kd

(not shown). For CNC to work well, a high value of kd

is thus

not only important to allow rapid adjustment of the average number of inhibitors to the

current number of plasmids. The value of kd

must also be sufficiently high to ensure that there

are no persistent rate-fluctuations.

4.7.8 Do plasmids really use stochastic focusing?

For plasmids to utilise SF it is necessary both that the inhibition mechanisms are fairly

insensitive without fluctuations and that inhibitor numbers fluctuate significantly around

their conditional averages (see Section 4.7.1–4.7.4). Experiments report that both RNA I and

CopA are present in a few hundred copies per cell, though the synthesis and degradation rates

vary greatly with physiological conditions. If these numbers obey Poisson statistics, SF is not

possible since the relative fluctuations would be too small to induce SF for biologically

realistic values of the rate constant ktr

for initiation of transcription of the genes for RNA II

and RepA mRNA (see Sections 2.4 and 4.8). Plasmid R1 is perhaps more likely to benefit

from SF. Partly because it appears to use hyperbolic control, but also because it is par + and

requires sensitive CNC especially when there is only a single plasmid copy in the cell. The

conditional average number of inhibitors is then much lower. However, it is also possible that

conditional CopA and RNA I pools display enormous fluctuations. Some principles werediscussed here, but there are many other factors that might contribute to the statistical

variation.

The system comprises three time-scales. If fluctuations are exceptionally rapid, they affect

neither correlations nor average rates (Paulsson et al . 2000). If fluctuations are slow compared




with the duration of the sampling process (e.g. inhibition window) but rapid compared with

the characteristic time for changes in plasmid concentration, the average rates may change but

correlations are still insignificant. If fluctuations are very slow, signal noise will affect average

rates and also introduce correlations between subsequent signal dependent reactions. Both

CopA and RNA I have half-lives ranging from 30 s to a couple of minutes depending on

conditions. These signal turnover rates seem to be high enough to avoid significant time

correlations. Furthermore, the lengths of the inhibition windows seem to be a few secondsboth for ColE1 (Brenner & Tomizawa, 1991) and R1 (Nordstro m & Wagner, 1994).

Inhibitor numbers then change insignificantly within the window and the average rates will

be affected by inhibitor noise.

In conclusion, the impact of internal signal noise depends on the regulatory mechanisms,

the noise distribution and the involved time-scales. The only thing that can be said with

certainty is that inhibitor-dilution plasmids R1 and ColE1 may have general regulatory design

principles and time-scales that permit SF in their CNC.

4.8 Metabolic burdens and values of in vivo rate constants

In this work all rate constants are expressed in normalised form, i.e., relative to the growth

rate of the host which has been put equal to one. This not only avoids excessive notations,

it also reflects the inherent dynamics of CNC since the copy numbers on average double in

one cell cycle. Typical values may be ktr 40, k

d 30 and k

s 300 (Lin-Chao & Bremer,

1986, 1987; Brenner & Tomizawa, 1991; Nordstro m & Wagner, 1994) but these numbers

should be taken with a pinch of salt since they seem to vary manifold with physiological

conditions (Engberg & Nordstro m, 1975; Gustafsson & Nordstro m, 1980; Light

& Molin, 1982).

An RNA II primer that is not inhibited only appears to have a 50% chance of ever

initiating replication. This efficiency parameter can be included in the rate constant, as has

been done here. However, ktr

is not only the per plasmid replication trial frequency, it is also

the per plasmid rate of inhibitor degradation due to duplex formation with target. The rate

of duplex formation is thus twofold higher than the trial rate. Here we represented them with

the same parameter for shorter notation. The only result in the paper where this difference

is important is in Fig. 14, but that figure only illustrates first principles.

To evaluate the efficiency of a certain CNC in an evolutionary context, the associated

reduction in growth rate must be considered. A quantitative cost–benefit analysis of plasmids

R1 and ColE1 is beyond the scope of the present work, but qualitative features that emerge

from the computations may give insight in the problem of optimal CNC design. A striking

feature in Figs 5 and 8 is that there are diminishing returns in CNC efficiency for increasing

metabolic load. While ktr 10 gives a much better CNC than k

tr 2, random variation and

P(loss ) are not so much reduced by further increasing ktr

from 10 to 100 and almost nothing

more is gained by an additional increase to 1000. Since a high ktr

means a high turnover of

both the initiator and antisense molecule, ktr

is proportional to a metabolic burden. From this

perspective, the estimated in vivo value seems to be in the optimal region. A similar argument

can be made for the inhibitor degradation rate constant kd : the in vivo value indicates that CNCworks close to its full capacity (Fig. 8) without an excessive metabolic burden. Since k

s is

significantly higher than ktr

, inhibitor degradation due to duplex formation is probably

negligible. In a previous paper (Paulsson & Ehrenberg, 1998) we suggested that it in fact

seemed to be in an optimal region. However, as this extended analysis has shown, that




conclusion was premature. The qualitative impact of duplex formation sensitively depends on

the exact conditions, such as inhibitor half-life, inhibition mechanism etc.

5. Previous models of copy number control

Plasmid copy number control has attracted a large number of modellers, most of which have

worked with ColE1. The present analysis is mostly based on our previous papers, especiallyPaulsson & Ehrenberg (1998, 2000a, b) and Paulsson et al . (2000).

5.1 General models of CNC

In 1983, Seneta and Tavare presented a paper on ‘ Some stochastic models for plasmid copy

number’, that was followed by a paper in 1997 by Hoppe, resolving some mathematical

questions raised by Seneta and Tavare . Replication during the cell cycle and segregation at

cell division were formulated as stochastic processes. Segregation was assumed to be

binomial, and three models for replication were suggested: (1) a multiplicative model (M)

where every plasmid copy replicates once or zero times each cell cycle; (2) an additive model

(A) where a total constant number of replication attempts are made and a constant fraction

of these are successful; and (3) an equilibrium model (E) where a cell that starts with mplasmids makes a total of N m plasmid replication attempts with success probability p where

N and p are constants. They did not attempt to explain CNC in itself.

As was stated by the authors, the M-model is not plausible because it allows for

unrestricted replication and results in both high average and variance. However, this kind of

model can also be discarded because it would be very difficult to implement kinetically due

to the general random nature of the events that lead to replication. A slightly more realistic

model, where a plasmid also could replicate more than once per cell cycle, would not do much

better. Unless replication is regulated, plasmid copy number variation is only limited by host

cell limitations and plasmid losses. Plasmids also enter new cells in single copies. Failing to

adjust to a higher steady state will then result in great losses. The E-model is almost as bizarre

and, to our knowledge, does not even have remote principles in common with any known

plasmids. The A-model on the other hand has some features in common with plasmid CNC.It was assumed that there are exactly N replication attempts during a generation time and that

each is successful with probability p. If they instead had assumed a Poisson distributed

number of attempts, the probability p would only have resulted in a thinning of the Poisson

distribution. A Poisson distributed number of replications per cell cycle corresponds to

perfect single-step hyperbolic control as in Eq. (30), which was derived from kinetics of CNC

where the intensity of replication trials is proportional to copy number, and where the success

probability of every trial is inversely proportional to copy number due to hyperbolic antisense

control. As shown in Eq. (31) this may lead to a Poisson distributed number of replications

per cell cycle.

5.2 Modelling plasmid CoLE1 CNC

Many of the differences between macroscopic models of ColE1 CNC originate from different

implicit assumptions about essential features of the inhibition mechanism. The first model

was made by Ataai & Schuler (1986) where they numerically investigated average copy

numbers (but not dynamics or sensitivity amplification) and compared with plasmid mutants.




The model was derived using an inhibition window of ∆t during which replication priming

is sensitive to inhibition by RNA I. Statistical variation in this time was not considered and

exponential inhibition (Section 3) was consequently used. They inspected the influence of

Rom on the average copy number and also predicted runaway replication when RNA II is

transcribed at a higher frequency than RNA I. Later the same year, Bremer & Lin-Chao

(1986) presented an experimental paper which included coupled differential equations for

inhibitor and plasmid concentrations (like Eq. (1)), assuming hyperbolic inhibition. Theynumerically looked at the dynamics of plasmid replication for in vivo rate constants and

compared it to experiments of adjustment to steady state after a physiological shift in

conditions. In 1989, Keasling & Palsson (not Paulsson), made a time constant analysis of

the system. This was a major step forward in the theory of CNC in general. Not only because

of its high quality, but because they made a symbolic parametric analysis of CNC dynamics.

They derived hyperbolic control from a differential equation for RNA II where it was

implicitly assumed that the inhibition time window is exponentially distributed. They also

made a careful analysis to inspect if Rom could have a dynamic effect on copy number control,

but came to the conclusion that the measured concentrations of Rom, together with values of

binding constants, strongly suggest that Rom is saturating and that it can be included in an

efficient binding constant between RNA I and RNA II. However, as was also stated in their

paper, the individual parameters for Rom synthesis and degradation were not known. In

1991, Brenner & Tomizawa published an experimental paper including some equations ex-

pressing the relation between inhibitor concentration and plasmid replication. As in the

Ataai & Shuler (1986) paper, they assumed that a fixed time-increment∆t could be used for the

inhibition window, and thus also arrived at exponential control. Brenner & Tomizawa (1991)

then used their minimal model to estimate a binding constant from experiments.

The most complete analysis, in the sense that it included the highest number of pathways

and concentration variables, came in 1993 from Brendel & Perelson. They analysed CNC as

a problem of obtaining the right average copy number and argued that inhibitor degradation

should neither be too low nor too high, because that would result in a too low or too high

average copy number. However, obtaining the right copy number is not something that is

kinetically intricate, but designing efficient CNC is . Furthermore, even if they took very many

states into account, a number of critical assumptions were made implicitly. These for instance

include (1) that inhibition is hyperbolic; (2) that Rom dimerisation can be ignored (see

Section 4.4.3); and (3) that the half-life of Rom is 5 min (see Section 4.4.3). The model of

Brendel & Perelson was warranted in that it pointed at many simplifications in previous

work, but the predicted behaviour was in most aspects close to the previous simple models

of hyperbolic control.

In 1995, Merlin & Polisky compared the 1991 Brenner & Tomizawa (B&T) and the 1993

Brendel & Perelson (B&P) analyses. Their conclusion was that the former was flawed in

various ways while the latter had predictive powers and could be tested against experiments.

One of the arguments raised against the B&T analysis was that the exponent in the expression

f e−∆tq was not necessarily dimensionless since it excluded the time unit. However,

dimension (such as time) and unit (such as seconds) are different things and B&T did in factstate that the unit was seconds for ∆t and per second for q as should be. Rather than reflecting

logical inconsistencies, the difference between exponential (B&T) and hyperbolic (B&P)

inhibition came from different implicit assumptions about the statistical variation in the time

spent in the inhibition window. Merlin & Polisky further claimed that a key assumption in




the B&T model is that the per plasmid replication rate equals dilution at steady state. This

is not an assumption but instead the definition of steady state and is included in all the other

models as well. They also claim that the simplified B&T model ‘merely summarizes self-

consistent relationships’ between parameters and cannot be predictive. However, it is the

nature and purpose of steady state analyses that they result in such relations and holds equally

true for the B&P analysis. As a consequence, Merlin & Polisky state that B&T, in contrast

to B&P, do not express steady states in terms of independent variables. This is not so. B&Tused the assumed inhibition mechanism and known values of steady state concentrations and

parameters to predict the inhibition constant. This does not mean that the variables are

dependent, only that the model contains an experimentally unknown parameter value.

Furthermore, B&P used three of the critical parameters in their model as free parameters to

obtain consistency with steady state concentrations, picked seven of them from 25 C in vitro

experiments and one from thin air.

The ColE1 analysis presented here was to a large extent based on our previous papers, even

if many of the results have not been published before. In the first paper, Ehrenberg (1996)

showed the difference between exponential and hyperbolic control, analysed sensitivity in

terms of RNA II transcription frequency and adopted the master equation approach to study

the impact of sensitivity on plasmid loss rates. He also showed conditions when Rom can take

a more active part of CNC and how it could reduce the dispersion of plasmid copy number

distributions. However, mostly due to the computational resources available, Ehrenberg’s

stochastic analysis of ColE1 CNC was made in a hypothetical case where ColE1 was present

in extremely low copy numbers and carried a partition mechanism, i.e. in a situation typical

for the E. coli chromosome. In a follow-up paper by Paulsson et al . (1998) we took a

deterministic approach to study the impact on adjustment rates to steady state as well as the

effect of limited inhibitor half-life. The major contribution of that paper was that it addressed

experiments, for instance plasmid dimerisation. In a second paper in 1998 by Paulsson &

Ehrenberg we stochastically inspected the impact of hyperbolic and exponential control for

high average copy numbers without a partition mechanism. The analysis showed the limit-

ations of hyperbolic and exponential CNC, effects of limited RNA I turnover mesoscopically,

the increase in random RNA I fluctuations due to zero-order degradation and finally addressed

mesoscopic experiments. To our knowledge, this was the first analysis of the drastic

mesoscopic consequences of intracellular zero-order degradation.

In two recent papers, Goss & Peccoud (1999a, b) also analysed the ColE1 system using the

proper master equation formulation of stochastic kinetics to find the cyclic state distributions.

In the first of these papers (Goss & Peccoud, 1999a), they used ColE1 CNC to exemplify an

alternative method for calculating master equations numerically (Stochastic Petri Nets) and

refrained themselves from making any kinetic analysis of CNC. In their second paper, they

looked at the impact of Rom on cyclic state copy number distributions (Goss & Peccoud,

1999b). It is commendable to see if other reactions in the CNC system may have a surprising

effect, but to really study the effect of Rom, it is necessary to extend their analysis so that

dimerisation, degree of saturation and Rom half-life are inspected systematically (see Section

4.4.3). Their conclusions were that Rom reduced the width of the distribution, but usedparameter values from Brendel & Perelson (1993) were it was assumed that Rom has a short

half-life and therefore follows changes in plasmid concentration rather closely. To our

knowledge, the Rom half-life has not been measured yet. There are also other possible CNC

effects of Rom that should be taken into account (Ehrenberg, 1996; Paulsson et al . 1998). We




are also convinced that Rom has an effect on the dispersion of copy number distributions

(Ehrenberg, 1996; Paulsson et al . 1998; Paulson & Ehrenberg, 1998), but the question is still

in what way.

The part on stochastic focusing is based on two recent papers about fluctuations in genetic

networks (Paulsson et al . 2000; Paulsson & Ehrenberg, 2000b). The first of these introduced

the notion of stochastic focusing and the second showed how random copy number

fluctuations in one component of a regulatory network may reduce random fluctuations inanother component. This network was close to identical to hyperbolic plasmid CNC.

5.3 Modelling plasmid R1 CNC

First out to model R1 copy number control kinetically were Kurt Nordstro m and co-

workers. It was also this group that did much of the experimental characterisation of R1. In

a review by Nordstro m et al . from 1984, they discuss hyperbolic as well as switch-onswitch-

off CNC. They also present a kinetic model where they deduce hyperbolic inhibition from the

reactions of CNC and compare it to inhibition of enzymatic reactions. In a previous analysis

(Nordstro m et al . 1980), they had assumed that n plasmids were produced each cell cycle,

regardless of the number in the dividing cell. In the 1984 paper, they expand this model and

introduce a Poisson distributed number of replications with average n. In a later paper from

1984, Nordstro m & Aagaard-Hansen make three assumptions: (1) that the replication

probability per time unit is independent of the number of plasmids; (2) that it is the same

throughout the cell cycle; and (3) that individual replication events are independent. They

then refer to the theory of stochastic processes and conclude that this means that the number

of replications per cell cycle is Poisson distributed. Even if assumption (2) does not take into

account that the inhibition event is volume-dependent, their result is sound. Taking volume

dependence into account only changes it to a Poisson process in exponential time (Eq. (30))

and the number of replications will still be Poisson distributed. However, as was shown in

Section 4.6.2, these are not the only assumptions that are supported by experiments and lead

to hyperbolic control on the average. The model by Nordstro m and co-workers thus

corresponds to a perfectly working single-step hyperbolic control mechanism. In a second

review in 1994, Nordstro m & Wagner present a more thorough macroscopic derivation of

hyperbolic control from the underlying kinetics of R1 CNC.

In 1986, Womble & Rownd presented a kinetic model of plasmid NR1, a close relative to

plasmid R1, that inspects the effects on CNC of the NR1 counterparts to CopB, RepA

and CopA. They use what they call a probabilistic model based on an ‘integration counter’.

This means that they have macroscopic, deterministic, equations for time evolution of

concentration variables. The macroscopic equations are then used to calculate ‘accumulated

probability’ and, when the value of the counter reaches unity, it is reset to zero and the

number of molecules produced in the reaction is increased with one molecule. They used the

same approach in at least two other theoretical papers concerning plasmid CNC. This

procedure is a discretisation of a continuous variable, but does not include probabilities. To

introduce chance in the model, they used their deterministic discrete model to calculate theaverage number of RepA mRNA synthesised per cell cycle as a function of the number of

plasmids the cell started with. They then assume that the number of transcripts made is

Poisson distributed around this average and proceed to calculate copy number distributions.

This approach is unrelated to the stochastic mesoscopic foundation of kinetics and would not




properly simulate the kinetic mechanism they presented, but instead something without

obvious relevance for real kinetic schemes.

In 1992, Rosenfeld & Grover published a computer simulation of R1 CNC. This

is to our knowledge the first time that single cells in an entire cell population were

simulated including the molecular details of plasmid CNC. The analysis mainly concerns

the possible influence of CopB, and the authors make a number of retrodictions that very

closely mimic experimental data without making any parameter adjustments. They alsochecked that their results were robust to small changes in all of the parameters. In spite of

these virtues, we believe that most of the conclusions concerning CopB are due to invalid

assumptions (unpublished results). In the idealised situation of perfect hyperbolic control, the

average number of replications per cell cycle was, surprisingly, not found to be independent

of the number of plasmid copies in the newborn cell. As was shown here, constant average

is a direct logical consequence of perfect hyperbolic control (Eq. (30)). The authors motivate

their numerical results with symbolic calculations, defining a replication probability that is

inversely proportional to plasmid concentration. When there are g plasmids, they say that this

probability is P g and when there are h plasmids it is Ph. Carrying out this calculation, they

first convert the transition probability per time unit to a probability P. This procedure is only

sound for very short times, i.e. when P 1. They then proceed to calculate the probability

of no replications as (1P g )g and (1Ph)h respectively and use the difference between

these two probabilities to explain their aberration. However, the difference disappears in the

very limit where their approach becomes valid. When P is sufficiently small, then (1P g )g

(1Ph)h 1P. In short, they did something equivalent to approximating the time-

dependent probability F 1e−kt with its first order Taylor expansion, kt , for too long

times t .

The first report, to our knowledge, where plasmid CNC was formulated in terms of master

equations, was in the 1995 paper by Ehrenberg & Sverredal where they introduced the

concept of R1 multistep kinetics. They showed, for instance, that it could result in cell cycle-

specific replication and reduce plasmid losses greatly when the plasmid is present at an

average copy number of 15 per cell. For in vivo R1 average copy numbers, they showed that

replications were scattered throughout the cell cycle. In their analysis they considered burst

production of RepA from a single mRNA and showed how multistep control requires that

every mRNA only translates a single, or at least very few, RepA molecules. They also

analysed kinetic eclipse times for different average burst sizes. We recently followed up this

analysis (Paulsson & Ehrenberg, 2000a), showing that when the average copy number is in

the in vivo range, control would be hyperbolic on the average, but multistep control would

still reduce losses greatly. The results from this analysis are presented in Section 4.6.2.

6. Summary and outlook : the plasmid paradigm

Regulatory networks are complex systems. However, even if there is no simple path we can

take in formulating one part of a system in terms of another, kinetic explanations of design

principles must be attempted. The majority of network analyses are non-quantitative andinstead focus on mechanistic aspects. When they are quantitative, they often rely on Boolean

threshold logic, again ignoring biochemical function and kinetics. In the study of large

networks this may be the only practical alternative, but our understanding of networks so

small that they only contain a few chemical species has only started to develop. Much has been




done macroscopically but these analyses can be misleading since molecule copy numbers tend

to be low and the reactions can be far from equilibrium. The highly stochastic nature of

intracellular processes has been anticipated for quite some time (see e.g. Benzer, 1953;

Mcquarrie et al . 1964; Mcquarrie, 1967; Spudich & Koshland, 1976; Berg, 1978) but never

received much recognition from mainstream macroscopic kineticists. This is about to change

as progress in the field has accelerated in the last decade (see e.g. Ko, 1992; Dogterom &

Leibler, 1993; Guptasarma, 1995 ; Peccoud & Ycart, 1995; McAdams & Arkin, 1997; Arkinet al . 1998; Cook et al . 1998; Morton-Firth & Bray, 1998; Ross & Vlad, 1999; Smolen et al .

1999; Wang & Wolynes, 1999; Hasty et al . 2000 ; Paulsson et al . 2000; Berg et al . 2000; Barkai

& Leibler, 2000).

The present study is a review and extension of our efforts to use master equations to

identify and inspect properties of one of the simplest and most studied genetic networks:

inhibitor-dilution plasmid copy number control (CNC). A number of putative properties

are suggested, some of which are invisible or completely obscured from a macroscopic

viewpoint.

(1) Sensitivity amplification in the negative feedback loop can be utilised to reduce copy

number variation. When plasmids use standard single-step hyperbolic inhibition, it is

hard to reduce copy number fluctuations below a (approximately) Poissonian limit. ColE1

may have developed multistep inhibition to increase sensitivity amplification. The

sensitivity of both hyperbolic and exponential control critically depends on a parameter

choice such that regulation operates far from saturation. The principal results from a

stochastic analysis of these mechanisms can be anticipated macroscopically.

(2) Both ColE1 and R1 have been suggested to use replication backup systems. These may

have a negligible effect around steady state, but boost the replication frequency at very

low plasmid concentrations. They could thus decrease segregational losses without

changing the steady state sensitivity amplification.

(3) When the inhibitors are slowly degraded, they do not follow changes in plasmid

concentration closely. From a macroscopic viewpoint, this can lead to oscillations or

critical damping, but when the probability distributions have converged to their cyclic

state (corresponding to stationarity), limited turnover means that the inhibitor most of

the time is closer to steady state than the plasmid. This leads to slower adjustment of copy

number deviations, higher plasmid copy number variation and greater plasmid loss rates.

It is also shown how the time-lag between plasmid and inhibitor only is determined by

the latter’s intrinsic half-life, not its synthesis rate or its rate of degradation due to duplex

formation.

(4) Plasmid replication is a random process in a random environment. The effect of random

fluctuations in the intracellular environment critically depends on the involved time-

scales. It is also possible to design regulation so that fluctuations in different rates cancel

each other.

(5) Just as for ColE1, initiation of R1 replication depends on a number of hyperbolically

regulated steps but, by contrast, the multiple steps can not amplify sensitivity beyond thehyperbolic limit. From a macroscopic perspective, the multiple steps make no difference,

but by formulating CNC mesoscopically it becomes clear that conventional sensitivity

amplification does not cover all types of regulatory precision. The multiple steps reduce

statistical variation in the time between two replications of the same plasmid. This may




have a great impact on the plasmid loss rate for plasmids that carry a partition system

since it drastically reduces the probability that a plasmid present in a single copy in a new-

born cell fails to replicate before the next cell division.

(6) Even when the average number of inhibitors follows the number of plasmids very

closely, the inhibitor copy numbers still fluctuate randomly around their conditional

averages. In other words, the inhibitors only represent the number of plasmids in a

probabilistic sense. This is a quite general phenomenon. Original stimuli generally affectsecondary stimuli that in turn determine the response. The relation between original and

secondary stimulus is inevitably blurred by random fluctuations due to the probabilistic

nature of all chemical reactions. Here we analytically calculated probability distributions

for some different mechanisms that all correspond to macroscopic constitutive

synthesis and first order degradation.

(7) Significant random inhibitor fluctuations could be exploited by CNC for stochastic

focusing (Paulsson et al . 2000). In fact, the intrinsic upper limit on the quality of CNC for

single step hyperbolic inhibition is only valid when inhibitor fluctuations are negligible.

When random fluctuations are large, sensitivity amplification is unlimited also for

hyperbolic inhibition. In other words, plasmid copy number fluctuations can not be

reduced below a certain limit with hyperbolic inhibition unless the conditional number

of inhibitors displays significant random fluctuations. This is quite general. To reduce

dispersion in some components in regulatory networks, it may be necessary to increase

dispersion in others, i.e., if regulatory couplings are inefficient, intrinsic noise can be used

to attenuate intrinsic noise. When the inhibitor has a long half-life, fluctuations may also

introduce correlations between individual replication trials. This drastically reduces the

quality of CNC and increases plasmid loss rates. Inhibitor fluctuations can thus both

decrease and increase plasmid loss rates depending on inhibition mechanism, inhibitor

copy number distributions and how rapidly the inhibitor is degraded.

(8) To encompass the complete dynamics of CNC, we suggest a type of generalised control

curves generated in the mesoscopic analysis. These control curves include time-lags,

stochastic focusing, correlations and all other aspects of CNC. However, they only

provide a means of comparison with macroscopic description and rely on pre-existing

numerical or analytical solutions to the full stochastic process.

Some models of plasmid copy numbers do not take CNC into account, while others include

as many as 18 reactions and 10 concentration variables. We took a middle way. We do

consider molecular details of CNC, but simplify our models until they are stripped to their

bare essence. It is obvious how lumping parameters together introduces limits on the scopes

of models and how splitting makes them harder to inspect. We believe there is a fundamental

difference between these two types of short-comings. In a model that is simple enough to be

understood, the relation between results and assumptions is explicit. It is then easier to

uncover critical splits that were not thought of initially. With this kind of model, we cannot

predict any actual numbers until more detailed kinetic data become available. However, we

can suggest experiments to analyse properties that so far have received little or no attention(Ehrenberg & Sverredal, 1995; Paulsson et al . 1998; Paulsson & Ehrenberg, 1998, 2000a, b).

The theory can also be used to anticipate experimental problems so that these experiments

stand a greater chance of being conclusive.

In view of the recent progresses in forward engineering (Becksei & Serrano, 2000; Elowitz




& Leibler, 2000; Gardner et al . 2000), one may also envision a new type of plasmid

experiments. With support from master equation analysis, it should be possible experimentally

to design plasmids that more efficiently eliminate internal noise, albeit at a higher metabolic

burden. Such plasmids could turn out to be very useful as cloning vectors, allowing for

stricter control of gene dosages in single cells. However, from a basic research perspective

it would perhaps be more interesting to restrict the extent of the changes: By trying to

improve existing CNC systems through minor but well-planned changes, one could getquantitative information about the trade-off between the cost and efficiency of regulation.

Plasmid CNC received much attention 10 to 20 years ago, but the activity in the field

declined as the molecular map was being completed. A few experimental groups have

continued the work, trying to draw the kinetic map. Such work makes it possible to relate

molecular compositions to function and evolution. In fact, due to their simplicity, diversity

and the strong selective pressure for efficient CNC, plasmids have all essentials for the making

of a regulatory paradigm.

7. Acknowledgements

This work was supported by the National Graduate School of Scientific Computing, the

Swedish Natural Science Research Council and the Swedish Research Council for Engineering

Sciences. We thank Dr O. G. Berg for interesting discussions.

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