Noise in a minimal regulatory network: plasmid copy number control

Abstract

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 7

2.4.2 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 23

4.3 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

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 49

5.1 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

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.