1. Introduction 270

1.1 Overview of nucleosome structure 271

2. Relative equilibrium stability (affinity) of histone–DNA interactions in nucleosomes 272

2.1 Relative affinity equals relative equilibrium stability 272

2.2 Competition assays for relative free-energy measurements 273

2.3 Technical issues in relative free-energy measurements 275

2.4 Range of affinities 278

3. Relation of nucleosome stability to nucleosome positioning 279

3.1 Translational nucleosome positioning 279

3.2 Rotational positioning 280

3.3 Unfavorable positioning 281

3.4 Experiments 281

4. Physical basis of DNA sequence preferences 282

4.1 Free-energy cost of DNA bending 283

4.2 Molecular mechanics of DNA bending and bendability 284

4.3 Bent and bendable DNA sequences 286

4.4 Parameter sets for prediction of DNA bending and bendability 288

4.5 DNA twisting 290

4.6 Energetics of nucleosomal DNA packaging 291

5. DNA sequence motifs for nucleosome packaging 292

5.1 Natural and designed nucleosomal DNAs 293

5.2 New rules and reagents from physical selection studies 294

5.3 Molecular basis of DNA sequence preferences 299

5.4 Special properties of the TA step 300

5.5 Unfavorable sequences 302

5.6 Natural genomes 303

5.7 Evolutionary approach toward an optimal sequence 305

5.8 Optimization by design 305

6. Dynamic nucleosome instability 308

6.1 Site-exposure equilibria 308

6.2 DNA sequence-dependence to site-exposure equilibria 312

6.3 Nucleosome translocation 315

6.4 Action of processive enzymes 319

7. Conclusions 319

8. Acknowledgements 320

9. References 320

The nucleosome core particle is the fundamental repeating subunit of chromatin. It consists of two molecules each of the four ‘core histone’ proteins, H2A, H2B, H3 and H4, and a 147 bp stretch of DNA. The lowest level of chromatin organization consists of a repeated array of nucleosome core particles separated by variable lengths of ‘linker DNA’. In many, but not all, cases, each core particle plus its linker DNA is associated with one molecule of a fifth ‘linker’ histone protein, H1. The complex of the core particle plus its linker DNA and H1 (when present) is called a ‘nucleosome’.

1. Introduction 475

1.1 Ion channels 475

1.1.1 Gramicidin 476

1.1.2 Helix bundle channels 477

1.1.3 K channels 480

1.1.4 Porins 483

1.1.5 Nicotinic acetylcholine receptor 483

1.1.6 Physiological properties 483

1.2 Simulations 484

1.2.1 Atomistic versus mean-field simulations 484

2. Atomistic simulations 485

2.1 Modelling of ion-interaction parameters 485

2.1.1 Interatomic distances and the problem of ionic radii 486

2.1.2 Solvation energy 487

2.1.3 Hydration shells and coordination numbers 489

2.1.4 Parameters in common use and transferability 491

2.1.5 Summary 491

2.2 Water in pores versus bulk 491

2.2.1 Simple pore models 494

2.2.2 gA 495

2.2.3 Alm 496

2.2.4 LS36 (and LS24) 496

2.2.5 Nicotinic receptor M2δ5 497

2.2.6 Influenza A M2 497

2.2.7 K channels 497

2.2.8 nAChR 498

2.2.9 Porins 498

2.2.10 Relevance 499

2.2.11 Problems with simulations 501

2.3 Dynamics of ions in pores 503

2.3.1 Simple pore models 503

2.3.2 Helix bundles 504

2.3.3 gA and KcsA 505

2.4 Energetics of permeation and ion selectivity 509

2.4.1 Potential and free energy profiles 509

2.4.2 gA 510

2.4.3 α-Helix bundles 511

2.4.4 KcsA 512

2.4.5 Ion selectivity 514

2.4.6 Problems of estimating energetic profiles 515

2.5 Conformational changes 516

2.5.1 gA 516

2.5.2 Alm and LS3 516

2.5.3 KcsA 517

2.6 Protonation states 523

3. Coarse-grained simulations 524

3.1 Introduction 524

3.1.1 Predicting conductance magnitudes 525

3.2 Electro-diffusion: the Nernst–Planck approach 526

3.2.1 Calculating the potential profile from Poisson and PB theory 528

3.2.2 Calculating the potential profile from BD simulations 530

3.2.3 Combining Nernst–Planck and Poisson: PNP 530

3.3 Beyond PNP 532

3.4 BD simulations 532

3.4.1 Basic theory in ion channels 532

3.4.2 Incorporating the environment 533

3.5 Applications 535

3.5.1 Model systems 535

3.5.1.1 Solving the Poisson and PB equation for channel-like geometries 535

3.5.1.2 Comparing PB, PNP and BD 536

3.5.2 Applications to known structures 537

3.5.2.1 gA 537

3.5.2.2 Porin 539

3.5.2.3 LS3 540

3.5.2.4 Alm 542

3.5.2.5 nAChR 542

3.5.2.6 KcsA 543

3.6 pKa calculations 543

3.7 Selectivity 544

3.7.1 Anion/cation selectivity 545

3.7.2 Monovalent/divalent ion selectivity 545

4. Problems 546

4.1 Atomistic simulations 546

4.1.1 Problems 546

4.1.2 Parameters 548

4.2 BD 549

4.3 Mean-field simulations 549

5. Conclusions 550

5.1 Progress 550

5.2 The future 550

6. Acknowledgements 551

7. References 551

Ion channels are proteins that form ‘holes’ in membranes through which selected ions move passively down their electrochemical gradients. The ions move quickly, at (nearly) diffusion limited rates (ca. 107 ions s−1 per channel). Ion channels are central to many properties of cell membranes. Traditionally they have been the concern of neuroscientists, as they control the electrical properties of the membranes of excitable cells (neurones, muscle; Hille, 1992). However, it is evident that ion channels are present in many types of cell, not all of which are electrically excitable, from diverse organisms, including plants, bacteria and viruses (where they are involved in functions such as cell homeostasis) in addition to animals. Thus ion channels are of general cell biological importance. They are also of biomedical interest, as several dizeases (‘channelopathies’) have been described which are caused by changes in properties of a specific ion channel (Ashcroft, 2000). Moreover, passive diffusion channels for substances other than ions are common (porins, aquaporins), as are active membrane transport processes coupled to ion gradients or ATP hydrolysis. An understanding of ion channels may also provide a gateway to understanding these processes.

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.

0. Abbreviations 106

1. Introduction: overview of forces in biology 108

1.1 Subtleties of biological forces and interactions 108

1.2 Specific and non-specific forces and interactions 113

1.3 van der Waals (VDW) forces 114

1.4 Electrostatic and ’double-layer‘ forces (DLVO theory) 122

1.4.1 Electrostatic and double-layer interactions at very small separation 126

1.5 Hydration and hydrophobic forces (structural forces in water) 131

1.6 Steric, bridging and depletion forces (polymer-mediated and tethering forces) 137

1.7 Thermal fluctuation forces: entropic protrusion and undulation forces 142

1.8 Comparison of the magnitudes of the major non-specific forces 146

1.9 Bio-recognition 146

1.10 Equilibrium and non-equilibrium forces and interactions 150

1.10.1 Multiple bonds in parallel 153

1.10.2 Multiple bonds in series 155

2. Experimental techniques for measuring forces between biological molecules and surfaces 156

2.1 Different force-measuring techniques 156

2.2 Measuring forces between surfaces 161

2.3 Measuring force–distance functions, F(D) 161

2.4 Relating the forces between different geometries: the ‘Derjaguin Approximation’ 162

2.5 Adhesion forces and energies 164

2.5.1 An example of the application of adhesion mechanics of biological adhesion 166

2.6 Measuring forces between macroscopic surfaces: the surface forces apparatus (SFA) 167

2.7 The atomic force microscope (AFM) and microfiber cantilever (MC) techniques 173

2.8 Micropipette aspiration (MPA) and the bioforce probe (BFP) 177

2.9 Osmotic stress (OS) and osmotic pressure (OP) techniques 179

2.10 Optical trapping and the optical tweezers (OT) 181

2.11 Other optical microscopy techniques: TIRM and RICM 184

2.12 Shear flow detachment (SFD) measurements 187

2.13 Cell locomotion on elastically deformable substrates 189

3. Measurements of equilibrium (time-independent) interactions 191

3.1 Long-range VDW and electrostatic forces (the two DVLO forces) between biosurfaces 191

3.2 Repulsive short-range steric–hydration forces 197

3.3 Adhesion forces due to VDW forces and electrostatic complementarity 200

3.4 Attractive forces between surfaces due to hydrophobic interactions: membrane adhesion and fusion 209

3.4.1 Hydrophobic interactions at the nano- and sub-molecular levels 211

3.4.2 Hydrophobic interactions and membrane fusion 212

3.5 Attractive depletion forces 213

3.6 Solvation (hydration) forces in water: forces associated with water structure 215

3.7 Forces between ‘soft-supported’ membranes and proteins 218

3.8 Equilibrium energies between biological surfaces 219

4. Non-equilibrium and time-dependent interactions: sequential events that evolve in space and time 221

4.1 Equilibrium and non-equilibrium time-dependent interactions 221

4.2 Adhesion energy hysteresis 223

4.3 Dynamic forces between biomolecules and biomolecular aggregates 226

4.3.1 Strengths of isolated, noncovalent bonds 227

4.3.2 The strengths of isolated bonds depend on the activation energy for unbinding 229

4.4 Simulations of forced chemical transformations 232

4.5 Forced extensions of biological macromolecules 235

4.6 Force-induced versus thermally induced chemical transformations 239

4.7 The rupture of bonds in series and in parallel 242

4.7.1 Bonds in series 242

4.7.2 Bonds in parallel 244

4.8 Dynamic interactions between membrane surfaces 246

4.8.1 Lateral mobility on membrane surfaces 246

4.8.2 Intersurface forces depend on the rate of approach and separation 249

4.9 Concluding remarks 253

5. Acknowledgements 255

6. References 255

While the intermolecular forces between biological molecules are no different from those that arise between any other types of molecules, a ‘biological interaction’ is usually very different from a simple chemical reaction or physical change of a system. This is due in part to the higher complexity of biological macromolecules and systems that typically exhibit a hierarchy of self-assembling structures ranging in size from proteins to membranes and cells, to tissues and organs, and finally to whole organisms. Moreover, interactions do not occur in a linear, stepwise fashion, but involve competing interactions, branching pathways, feedback loops, and regulatory mechanisms.

1. Introduction 326

1.1 What? Terminology and general properties 327

1.2 Why? Reasons for biophysical study 329

1.3 How? Special issues for study of connexin channels 330

2. Molecular and structural context 331

2.1 Biochemical features 331

2.2 Structures 334

2.2.1 Junctional channels 335

2.2.2 Hemichannels 338

2.2.3 Heteromeric channels 342

2.2.4 Junctional plaques 347

3. Experimental approaches and issues specific to study of connexin channel physiology 349

3.1 Macroscopic currents 349

3.1.1 Junctional channels 349

3.1.2 Hemichannels 354

3.2 Single-channel currents 355

3.2.1 Junctional channels 355

3.2.2 Hemichannels 358

3.3 Molecular permeability 361

3.3.1 A selection of tracers 361

3.3.2 Junctional channels 362

3.3.3 Hemichannels 366

3.4 Other 367

4. Structural issues 368

4.1 What lines the pore? 368

4.2 Docking between hemichannels 373

4.2.1 Structural and molecular basis 374

4.2.2 Determinants of specificity of interaction 380

5. Permeability and selectivity 381

5.1 Among the usual ions 383

5.1.1 Unitary conductance 383

5.1.2 Selectivity 384

5.1.3 Nonlinear single-channel I–V relations and their molecular determinants 386

5.2 Among large permeants 391

5.2.1 Uncharged molecules 392

5.2.2 Charged molecules 393

5.2.3 Cytoplasmic/signaling molecules 396

6. Voltage sensitivity 399

6.1 Macroscopic transjunctional voltage sensitivity 404

6.2 Microscopic voltage sensitivity – Vj-gating 407

6.2.1 Molecular basis – voltage sensor 407

6.2.2 Molecular basis – transduction and/or state stability 409

6.3 Microscopic voltage sensitivity – loop gating 412

6.4 Vm-gating 414

7. Direct chemical modulation 415

7.1 Phosphorylation 417

7.2 Cytoplasmic pH and aminosulfonates 419

7.3 Calcium ion 424

7.4 Lipophiles 424

7.4.1 Long chain n-alkyl alcohols 425

7.4.2 Fatty acids and fatty acid amides 426

7.4.3 Halothane 426

7.5 Glycyrrhetinic acid and derivatives 427

7.6 Cyclic nucleotides 428

7.7 Other candidates 430

8. Connexinopathies 431

9. Summary 435

10. Acknowledgements 438

11. References 438

Connexins are the proteins that form the intercellular channels that compose gap junctions in vertebrates. Connexin channels mediate electrotonic coupling between cells and serve important functions as mediators of intercellular molecular signaling. Convincing demonstration of the latter function has been elusive, as have the experimental tools required for detailed functional study of the channels. Recently, substantial progress has been made on both fronts. Connexin channels are now known to be dynamic, multifunctional channels intimately involved in development, physiology and pathology, and amenable to study by state-of-the-art approaches. A host of developmental and physiological defects are caused by defects in connexin channels, and therefore in the intercellular molecular movement they mediate. The channel structure has been determined to 7·5 Å resolution within the plane of the membrane. Experimental paradigms have been developed that enable application of the tools of modern channel biophysics to study connexin channel structure–function. As a result, the biophysical mechanisms and biological functions of connexin channels now enjoy a vigorous and expanding experimental interest. This article focuses on the former, but with attention to issues likely to have biological consequences.

1. Introduction 563

2. Obtaining rate constants from molecular-dynamics simulations 564

2.1 General relationships between quantum electronic structures and reaction rates 564

2.2 The transition-state theory (TST) 569

2.3 The transmission coefficient 572

3. Simulating biological electron-transfer reactions 575

3.1 Semi-classical surface-hopping and the Marcus equation 575

3.2 Treating quantum mechanical nuclear tunneling by the dispersed-polaron/spin-boson method 580

3.3 Density-matrix treatments 583

3.4 Charge separation in photosynthetic bacterial reaction centers 584

4. Light-induced photoisomerizations in rhodopsin and bacteriorhodopsin 596

5. Energetics and dynamics of enzyme reactions 614

5.1 The empirical-valence-bond treatment and free-energy perturbation methods 614

5.2 Activation energies are decreased in enzymes relative to solution, often by electrostatic effects that stabilize the transition state 620

5.3 Entropic effects in enzyme catalysis 627

5.4 What is meant by dynamical contributions to catalysis? 634

5.5 Transmission coefficients are similar for corresponding reactions in enzymes and water 636

5.6 Non-equilibrium solvation effects contribute to catalysis mainly through Δg[Dagger], not the transmission coefficient 641

5.7 Vibrationally assisted nuclear tunneling in enzyme catalysis 648

5.8 Diffusive processes in enzyme reactions and transmembrane channels 651

6. Concluding remarks 658

7. Acknowledgements 658

8. References 658

Obtaining a detailed understanding of the dynamics of a biochemical reaction is a formidable challenge. Indeed, it might appear at first sight that reactions in proteins are too complex to analyze microscopically. At room temperature, even a relatively small protein can have as many as 1034 accessible conformational states (Dill, 1985). In many cases, however, we have detailed structural information about the active site of an enzyme, whereas such information is missing for corresponding chemical systems in solution. The atomic coordinates of the chromophore in bacteriorhodopsin, for example, are known to a resolution of 1–2 Å. In addition, experimental studies of biological processes such as photoisomerization and electron transfer have provided a wealth of detailed information that eventually may make some of these processes classical problems in chemical physics as well as biology.

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.