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.

1. Introduction 62

2. Channel structure 63

2.1 Isoforms, primary structure and topology of Ca2+-release channels 63

2.2 Identification of ligand binding sites within the primary sequence of RyR 65

2.2.1 Calcium 66

2.2.2 Calmodulin 66

2.2.3 FK506-binding proteins 66

2.2.4 L-type Ca2+ channel 66

2.2.5 Ryanodine 67

2.3 The three-dimensional structure of the RyR channel 68

3. Channel function 70

3.1 RyR channel gating 70

3.2 Ion translocation and discrimination 71

3.2.1 Monovalent inorganic cations 71

3.2.2 Divalent inorganic cations 74

3.2.3 Organic monovalent cations 75

3.2.4 Permeant ion translocation can be blocked 75

3.3 Summary of ion handling in RyR 76

3.4 Where is the pore and what components of RyR are involved in its formation? 76

3.5 The mechanisms underlying ion translocation and discrimination in RyR 79

3.6 Does RyR employ ion–ion repulsion to attain high unitary conductance? 82

3.6.1 Conductance–activity relationships 83

3.6.2 Concentration dependence of reversal potential 83

3.6.3 The dependence of unitary conductance on mole-fraction 83

3.6.4 Effective valence of channel-blocking cations 84

3.6.5 Modelling ionic conduction 84

3.7 Factors influencing maximum conductance 85

3.8 Factors influencing ion entry 86

3.9 Theoretical design for the pore of RyR 88

4. What do we know about the structure of the conduction pathway in RyR? 88

4.1 The narrowest region of the conduction pathway – the ‘selectivity filter’ of RyR 89

4.2 The voltage drop across RyR – the length of the ‘pore’ 89

4.3 Mechanisms for ion discrimination in RyR 93

4.4 Musings on the structure of the RyR/InsP3R pore 95

5. Summary 98

6. Acknowledgements 98

7. References 98

The purpose of this article is to provide a description of the current state of our understanding of certain aspects of the relationship between the structure and function of the ryanodine receptor (RyR). RyR is an ion channel found in the membranes of the intracellular Ca2+ storage organelles, the endoplasmic reticulum (ER) and its counterpart in muscle cells the sarcoplasmic reticulum (SR), where it provides a regulated pathway for the release of stored Ca2+ during Ca2+ signalling processes such as fertilization and muscle contraction. RyR possesses both high- and low-affinity binding sites for the plant alkaloid ryanodine; however, whilst this ligand gives the channel its name and is of toxicological and pharmacological interest, physiological regulation of channel gating is mediated by the binding of cytosolic ligands (primarily Ca2+, ATP, Mg2+) and in some cases by direct coupling with a surface membrane voltage sensor.