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 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.