There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

Cloning and sequence analysis of cDNA for the Electrophorus electricus electroplax sodium channel indicate that this protein, consisting of 1,820 amino acid residues, exhibits four repeated homology units, which are presumably oriented in a pseudosymmetric fashion across the membrane. Each homology unit contains a unique segment with clustered positively charged residues, which may be involved in the gating structure, possibly in conjunction with negatively charged residues clustered elsewhere.

Primary structure of Electrophorus electricus sodium channel deduced from cDNA sequence.

Cl− channels reside both in the plasma membrane and in intracellular organelles Their functions range from ion homeostasis to cell volume regulation, transepithelial transport, and regulation of e

https://www.leibniz-fmp.de/fileadmin/user_upload/Physiology_and_Pathology_of_Ion_Transport/documents/PhysRev2002.pdf

Molecular Structure and Physiological Function of Chloride Channels

Cyclic nucleotide-gated (CNG) channels are nonselective cation channels first identified in retinal photoreceptors and olfactory sensory neurons (OSNs). They are opened by the direct binding of cyclic nucleotides, cAMP and cGMP. Although their activity shows very little voltage dependence, CNG channels belong to the superfamily of voltage-gated ion channels. Like their cousins the voltage-gated K+ channels, CNG channels form heterotetrameric complexes consisting of two or three different types of subunits. Six different genes encoding CNG channels, four A subunits (A1 to A4) and two B subunits (B1 and B3), give rise to three different channels in rod and cone photoreceptors and in OSNs. Important functional features of these channels, i.e., ligand sensitivity and selectivity, ion permeation, and gating, are determined by the subunit composition of the respective channel complex. The function of CNG channels has been firmly established in retinal photoreceptors and in OSNs. Studies on their presence in other sensory and nonsensory cells have produced mixed results, and their purported roles in neuronal pathfinding or synaptic plasticity are not as well understood as their role in sensory neurons. Similarly, the function of invertebrate homologs found in Caenorhabditis elegans, Drosophila, and Limulus is largely unknown, except for two subunits of C. elegans that play a role in chemosensation. CNG channels are nonselective cation channels that do not discriminate well between alkali ions and even pass divalent cations, in particular Ca2+. Ca2+ entry through CNG channels is important for both excitation and adaptation of sensory cells. CNG channel activity is modulated by Ca2+/calmodulin and by phosphorylation. Other factors may also be involved in channel regulation. Mutations in CNG channel genes give rise to retinal degeneration and color blindness. In particular, mutations in the A and B subunits of the CNG channel expressed in human cones cause various forms of complete and incomplete achromatopsia.

Cyclic Nucleotide-Gated Ion Channels

The cyclic GMP-dependent cation channel from bovine rod outer segments has been purified to greater than 90% homogeneity by a rapid two-step chromatographic procedure. The purified channel has an apparent molecular mass of 63 kDa as determined by NaDodSO4/gel electrophoresis. When incorporated into the membrane of liposomes, the purified protein mediates the cyclic GMP-dependent efflux of entrapped Ca2+. The reconstituted channel protein exhibits properties similar to the cyclic GMP-dependent channel observed in excised patches of the plasma membrane and in disk membranes. Cyclic GMP activated the channel cooperatively (Hill coefficient n = 3.1) with an apparent Michaelis constant of approximately 11 microM. After reconstitution of the purified protein into a planar lipid bilayer, we recorded cyclic GMP-stimulated single-channel activity. The single-channel conductance at physiological salt concentrations and in the absence of divalent cations was 26 pS. The drug l-cis-diltiazem, shown to block the cyclic GMP-dependent channel in excised patches of the plasma membrane and in isolated disks of rod outer segments, was ineffective against the purified channel.

Identification, purification, and functional reconstitution of the cyclic GMP-dependent channel from rod photoreceptors.

Single-channel fluctuations of a chloride-specific channel from Torpedo californica electroplax were studied with high current and time resolution. Channels were incorporated into virtually solvent-free planar bilayer membranes formed from phospholipid monolayers, and the substructure of the open channel was analyzed. The single channel displays three well-defined substates of conductances 0, 10, and 20 pS in 200 mM Cl-. These three substates are interpreted in terms of a dimeric channel complex composed of two identical "protochannels" gating independently in parallel on a time scale of milliseconds, but coupled together by a bursting process on a time scale of seconds. The probability of forming an open protochannel is voltage dependent and is increased strongly as aqueous pH is lowered. Variations of pH are effective only on the same side of the bilayer as the addition of electroplax vesicles. The dependence of single-channel kinetics on pH and voltage lead to a minimal four-state model in which both open and closed states can be protonated on a residue that changes its pK from 6 to 9 upon opening of the protochannel.

/pdf/single-chloride-channels-from-torpedo-electroplax-activation-4xus6xf9j0.pdf

Single chloride channels from Torpedo electroplax. Activation by protons.

The voltage-dependency of alamethicin pore formation is explained by a flip-flop gating mechanism of single alamethicin molecules. The energetically preferred aggregate structure is changed from antiparallel to parallel molecule orientation by membrane voltage application. The electrical field is sensed by the permanent dipole of the α-helical molecule part which spans the hydrophobic membrane core. Ion conducting pore and pore states result from electrostatic repulsion of a varying number of parallel dipoles which arrange circularly. This model is consistent with published data and with two additional experimental facts, that pore state distributions are ionic strength dependent and pore state conductances depend on ionic current direction.

Alamethicin pore formation: Voltage-dependent flip-flop of α -helix dipoles

Melittin and a chemically modified trichotoxin form alamethicin-type multi-state pores

Using mixed-chain lipids, we have recorded cooling and heating curves of planar bilayer membranes while they passed the lipid phase transition range. With unmodified planar bilayers, spontaneous current fluctuations are observed near the lipid phase transition temperature (tc approximately 29 degrees C). This effect coincides with the expected and measured decrease in membrane capacitance. Carrier (valinomycin)-modified planar bilayers exhibit near tc an abrupt change from a high-conducting state above tc to the state of bare membrane conductance below tc. In contrast to this behavior, planar bilayers modified by pore-forming antibiotics (gramicidin A, alamethicin) do not show any peculiar effect at tc. However, at 22--23 degrees C a pronounced maximum in pore-induced conductance is seen. Whereas the gramicidin A pore abruptly stops stepwise fluctuations below approximately 16 degrees C, with alamethicin a few long-lasting pore and pore state fluctuations persist down to 10 degrees C. It is suggested that the carrier may freeze out into the membrane/water interface. The effects observed with pore-forming substances, on the other hand, are interpreted in terms of lateral phase separation into pure lipid and lipid/antibiotic domains.

https://www.pnas.org/content/pnas/77/6/3403.full.pdf

Wolfgang Hanke

Papers

Identification, purification, and functional reconstitution of the cyclic GMP-dependent channel from rod photoreceptors.

Single chloride channels from Torpedo electroplax. Activation by protons.

Alamethicin pore formation: Voltage-dependent flip-flop of α -helix dipoles

Melittin and a chemically modified trichotoxin form alamethicin-type multi-state pores

Lipid phase transition in planar bilayer membrane and its effect on carrier- and pore-mediated ion transport.