Lyotropic Ion Channel Current Model Compared with Ising Model

Trans-membrane currents in ligand-gated ion channels are calculated in a non-equilibrium, chemically open whole cell system. The model is lyotropic in the sense that dynamics and parameters such as ligand concentration for half-maximal response (scale of response), and threshold for firing in neurons, are nonlinear functions of the reactant concentrations. The derived total current fits recorded data significantly better than those derived from mass action, Ising, and other equilibrium type models, in which the derived response can be displaced from the assessed response by several orders in the ligand concentration. A comparison of the model obtained with an Ising-like model provides a methodology to obtain the non-equilibrium scaling dependence of Ising-like models on the reactant concentrations.     

Single-channel dose-response studies in single, cell-attached patches.

A method for carrying out dose-response studies of ion channel currents in cell-attached patches has been devised. Patch pipettes are filled at the tip with a solution containing one concentration of ligand and then backfilled with another. The concentration of ligand at the membrane is described as a function of time by the equation for diffusion in a cone, allowing response vs. time data to be transformed into a dose-response curve. For Xenopus myocyte cholinergic receptors, examples of the use of this method are given for several concentration-dependent reactions including blockade by the local anesthetic QX-222, activation by acetylcholine, and modulation of current amplitude by sodium ions. Several methods of analyzing the nonstationary channel kinetics are presented, including a pseudo-stationary approach that uses interval likelihood maximization.     

Simulation of Association Curves and ‘Scatchard’ Plots of Binding Reactions where Ligand and Receptor are Degraded or Internalized

Abstract

Frequently during the course of binding to a receptor, ligand is degraded. In some preparations receptor is degraded. And with isolated cell preparations, ligand and/or receptor are internalized. Here we present a mathematical model for the combined binding and other reactions which gives useful information about the behaviour of such systems. The set of differential equations is solved numerically to simulate association curves and the resulting values of bound and free ligand are used to construct Scatchard plots. Where non-ideal conditions exist, the Scatchard plots are generally curvilinear. Dependence of this curvilinearity on time of measurement of free and bound ligand, on degradation and internalization of ligand and on degradation and internalization of receptor is shown. Equilibrium constants derived from the Scatchard plots are generally incorrect but the derived receptor concentration is often correct. The simulations lead to possibilities for distinction among the several side reactions in ligand-receptor binding systems.     

The exponential model for a regulatory enzyme: Its extension to describe the binding of two ligands

The exponential model for a regulatory enzyme (Ainsworth, 1977a) is extended to deal explicitly with the presence in solution of a second ligand. This is achieved by introducing exponential interaction coefficients which respectively describe how the affinity of the free and bound forms of the protein for the ligand depend on its fractional saturation by the second ligand. The basic equations, so derived, are applied to binding experiments where the ligands bind independently or competitively and to rate experiments where the ligands represent two substrates or one substrate and a modifier which may be either competitive or non-competitive in type. The conditions required to display linkage between the binding of the two ligands are established and it is also shown that rate data may display a maximum as one ligand concentration is varied at a fixed concentration of the other. The equations that are derived are tested by application to experimental data and the conditions that have to be met to justify such an application are discussed.     

Negatively co-operative ligand binding

Simple systems are considered in which the binding of a ligand at a single site exhibits a doubly sigmoid curve when saturation is plotted against the logarithm of ligand concentration, i.e. where a fraction of the site exhibits one dissociation constant and the rest exhibits another. The condition for this to occur is that the ligand should also combine at another site on the binding molecule with comparable affinity and that the binding at one site should markedly lower affinity at the other. The protonation of simple compounds such as cysteine and 3-hydroxypyridine is taken as an example of this process and the equations derived are also applied to the binding of substrates to enzymes.     

Kinetics of large-ligand binding to one-dimensional lattices: theory of irreversible binding

Exact solutions are obtained for the time dependence of the extent of irreversible binding of ligands that cover more than one lattice site to a homogeneous one-dimensional lattice. The binding may be cooperative or noncooperative and the lattice either finite or infinite. Although the form of the solution is most convenient when the ligand concentration is buffered, exact numerical or approximate analytical solutions, including upper and lower bounds, can be derived for the case of variable ligand concentration as well. The physical reason behind the relative simplicity of the kinetics of irreversible as opposed to reversible binding in such systems is discussed.     

Time-dependent ligand current into a single cell performing chemoreception

We determine the ligand current into a single spherical cell which carries a large number of receptors on its surface. Initially, this cell is placed into a medium which contains ligands at uniform concentration. The time-dependent ligand distribution is calculated, from which the time-dependent ligand current into the cell is derived. If the ligand concentration is kept constant at distances comparable to the radius of the cell the stationary state sets in at times comparable to the T1 necessary for ligands to travel a distance of the order of the radius of the cell. If the ligand concentration is kept constant at infinity the stationary state sets in at a time which is about 1000T1 for typical values of the parameters.     

Platymonas subcordiformis Channelrhodopsin-2 (PsChR2) Function: II. RELATIONSHIP OF THE PHOTOCHEMICAL REACTION CYCLE TO CHANNEL CURRENTS*

Channelrhodopsins, such as the algal phototaxis receptor Platymonas subcordiformis channelrhodopsin-2 (PsChR2), are light-gated cation channels used as optogenetic tools for photocontrol of membrane potential in living cells. Channelrhodopsin (ChR)-mediated photocurrent responses are complex and poorly understood, exhibiting alterations in peak current amplitude, extents and kinetics of inactivation, and kinetics of the recovery of the prestimulus dark current that are sensitive to duration and frequency of photostimuli. From the analysis of time-resolved optical absorption data, presented in the accompanying article, we derived a two-cycle model that describes the photocycles of PsChR2. Here, we applied the model to evaluate the transient currents produced by PsChR2 expressed in HEK293 cells under both fast laser excitation and step-like continuous illumination. Interpretation of the photocurrents in terms of the photocycle kinetics indicates that the O states in both cycles are responsible for the channel current and fit the current transients under the different illumination regimes. The peak and plateau currents in response to a single light step, a train of light pulses, and a light step superimposed on a continuous light background observed for ChR2 proteins are explained in terms of contributions from the two parallel photocycles. The analysis shows that the peak current desensitization and recovery phenomena are inherent properties of the photocycles. The light dependence of desensitization is reproduced and explained by the time evolution of the concentration transients in response to step-like illumination. Our data show that photocycle kinetic parameters are sufficient to explain the complex dependence of photocurrent responses to photostimuli.     

Theoretical characterization of ion channel blockade: ligand binding to periodically accessible receptors

With repetitive stimulation, the time course of use-dependent blockade as assessed by peak membrane ion currents can be described by a sequence of blocking relationships that have the form of recurrence equations. The equations of the sequence describe blockade acquired during each interval of a stimulus where the possibly different binding and unbinding rates are assumed constant during each interval. The solution predicts that use-dependent uptake follows an exponential time course. Furthermore, the exponential uptake rate is a linear function of uptake rates associated with the stimulus time intervals. Similarly, the fraction of blocked channels at steady state is a linear function of the interval dependent blockade equilibria. Several novel tests of consistency between the model and observations are derived from these theoretical results. It is also shown that as the stimulus interval increases to infinity, steady state dissociation constants measured by peak membrane currents are theoretically equivalent to those measured with true equilibrium methods such as radioligand binding studies.     

Time course and Ca(2+) dependence of sensitivity modulation in cyclic GMP-gated currents of intact cone photoreceptors

We determined the Ca(2+) dependence and time course of the modulation of ligand sensitivity in cGMP-gated currents of intact cone photoreceptors. In electro-permeabilized single cones isolated from striped bass, we measured outer segment current amplitude as a function of cGMP or 8Br-cGMP concentrations in the presence of various Ca(2+) levels. The dependence of current amplitude on nucleotide concentration is well described by the Hill function with values of K(1/2), the ligand concentration that half-saturates current, that, in turn, depend on Ca(2+). K(1/2) increases as Ca(2+) rises, and this dependence is well described by a modified Michaelis-Menten function, indicating that modulation arises from the interaction of Ca(2+) with a single site without apparent cooperativity. (Ca)K(m), the Michaelis-Menten constant for Ca(2+) concentration is 857 +/- 68 nM for cGMP and 863 +/- 51 for 8Br-cGMP. In single cones under whole-cell voltage clamp, we simultaneously measured changes in membrane current and outer segment free Ca(2+) caused by sudden Ca(2+) sequestration attained by uncaging diazo-2. In the presence of constant 8Br-cGMP, 15 micro, Ca(2+) concentration decrease was complete within 50 ms and membrane conductance was enhanced 2.33 +/- 0.95-fold with a mean time to peak of 1.25 +/- 0.23 s. We developed a model that assumes channel modulation is a pseudo-first-order process kinetically limited by free Ca(2+). Based on the experimentally measured changes in Ca(2+) concentration, model simulations match experimental data well by assigning the pseudo-first-order time constant a mean value of 0.40 +/- 0.14 s. Thus, Ca(2+)-dependent ligand modulation occurs over the concentration range of the normal, dark-adapted cone. Its time course suggests that its functional effects are important in the recovery of the cone photoresponse to a flash of light and during the response to steps of light, when cones adapt.     

The concentration dependence of active K+ transport in the turkey erythrocyte. Hill analysis and evidence for positive cooperativity between ion binding sites

A mathematical model is presented which describes the theoretical relationship between ligand concentration and physiological response for systems in which the response is dependent upon simultaneous occupancy of two receptor ligand-binding sites. The treatment considers both the possibility of intrinsic differences between the binding sites with regard to ligand affinity, as well as the possibility of mutually induced changes in affinity resulting from allosteric interactions. Unlike the Monod-Wyman-Changeux formulation for allosteric enzymes, the general model put forward here makes double occupancy an absolute requirement for enzymatic function. It is shown that such a model leads to the prediction of a curvilinear Hill plot from which one can obtain an explicit estimate of the degree of allosteric interaction between the two ligand binding sites as well as the Gibbs standard free energy change for the overall binding reaction. It is then shown that, in the specific instance of Na, K-ATPase-mediated K+ transport by the turkey erythrocyte, the configuration of the Hill curve describing the rate of ouabain-sensitive K+ transport as a function of external K+ concentration conforms closely to that predicted by the model described above. The results are of particular interest because they indicate a strongly cooperative interaction between the two K+ binding sites on the transport protein such that occupancy of one site results in an enhancement of the affinity of the other site for K+ by a minimum of 15- to 20-fold. Finally, we consider in detail a model of the Monod-Wyman- Changeux type in which, by contrast, both singly and doubly occupied forms of the enzyme are assumed to be catalytically active, and which we analogously extend to allow for the possibility of asymmetry between the two ligand binding sites. Although it is shown that the two models can not be differentiated from each other in the present experimental system, they yield virtually identical estimates for the degree of positive cooperativity between the two K+ binding sites.     

A dynamic dialysis method for studying protein-ligand binding using chromatographic theory

A new dynamic dialysis method has been developed for studying protein-ligand binding phenomena. The method depends on analysis of the elution pattern of ligand in a single dialyzing process where the ligand concentration in the sample compartment changes greatly with time. The dialyzer is composed of a long, narrow chamber (the sample compartment) between two sheets of semipermeable membrane and two outside chambers (the sink compartment) connected as a single path. Eluting buffer flows in the sink compartment to exchange the ligand with the solution in the sample compartment. Therefore, the ligand concentration gradient in the sink compartment is in the longitudinal direction. The mathematical expressions to analyze the experimental data were derived from a modified theory of chromatography. Examination of the binding of sulfanilamide to bovine serum albumin using this method shows that these equations are valid for use in studying protein-ligand binding.     

Ligand accumulation in autocrine cell cultures

Cell-culture assays are routinely used to analyze autocrine signaling systems, but quantitative experiments are rarely possible. To enable the quantitative design and analysis of experiments with autocrine cells, we develop a biophysical theory of ligand accumulation in cell-culture assays. Our theory predicts the ligand concentration as a function of time and measurable parameters of autocrine cells and cell-culture experiments. The key step of our analysis is the derivation of the survival probability of a single ligand released from the surface of an autocrine cell. An expression for this probability is derived using the boundary homogenization approach and tested by stochastic simulations. We use this expression in the integral balance equations, from which we find the Laplace transform of the ligand concentration. We demonstrate how the theory works by analyzing the autocrine epidermal growth factor receptor system and discuss the extension of our methods to other experiments with cultured autocrine cells.     

Stimulation of Kv1.3 potassium channels by death receptors during apoptosis in Jurkat T lymphocytes

The loss of intracellular potassium is a pivotal step in the induction of apoptosis but the mechanisms underlying this response are poorly understood. Here we report caspase-dependent stimulation of potassium channels by the Fas receptor in a human Jurkat T cell line. Receptor activation with Fas ligand for 30 min increased the amplitude of voltage-activated potassium currents 2-fold on average. This produces a sustained outward current, approximately 10 pA, at physiological membrane potentials during Fas ligand-induced apoptosis. Both basal and Fas ligand-induced currents were blocked completely by toxins that selectively inhibit Kv1.3 potassium channels. Kv1.3 stimulation required the expression of Fas-associated death domain protein and activation of caspase 8, but did not require activation of caspase 3 or protein synthesis. Furthermore, Kv1.3 stimulation by Fas ligand was prevented by chronic stimulation of protein kinase C with 20 nm phorbol 12-myristate 13-acetate during Fas ligand treatment, which also blocks apoptosis. Thus, Fas ligand increases Kv1.3 channel activity through the same canonical apoptotic signaling cascade that is required for potassium efflux, cell shrinkage, and apoptosis.     

Gonadotropin-releasing hormone receptor expression in Xenopus oocytes

The rodent GnRH receptor was characterized in Xenopus oocytes injected with RNA isolated from rat pituitary and from a gonadotrope cell line, alpha T3, derived from a transgenic mouse. Three to 4 days after 150-200 ng RNA injection, 93% of the oocytes, which were recorded by voltage clamp, responded to 10(-7) M GnRH. The mean inward currents obtained after RNA injection were 620 +/- 88 nA (n = 22) with pituitary RNA and 1415 +/- 598 (n = 4) with alpha T3 RNA. The threshold GnRH concentration able to evoke the dose dependent current after pituitary RNA injection was 3 x 10(-9) M GnRH. The GnRH receptor response of the oocyte was antagonized by [D-Phe2,6,Pro3] GnRH and [N-Ac-D-Na](2)1, D-alpha D-Me, pCl-Phe2, D-Arg6, D-Ala10-NH2]GnRH and could be elicited by D-Ser(But)6,Pro9-N-ethylamide GnRH (buserelin). The reversal potential of the GnRH generated current as determined by voltage-ramp was -22.5 +/- 1.0 mV (n = 7) and -25.6 +/- 3.3 mV (n = 3) in pituitary and cell line RNA-injected oocytes respectively, consistent with the chloride reversal potential. The GnRH receptor response was virtually eliminated by intracellular EGTA injection but was unaffected by ligand application in calcium-free perfusate. The GnRH-evoked response is mimicked by intracellular injection of inositol 1,4,5-trisphosphate. To determine the size of the GnRH receptor mRNA, alpha T3 RNA was size fractionated through a sucrose gradient. The maximal GnRH response was induced by a fraction larger than the 28S ribosomal peak. Thus we find that oocytes injected with RNA from an appropriate source develop an electrophysiological response to GnRH which is dependent on intracellular calcium mobilization, is independent of extracellular calcium, and may be mediated by inositol 1,4,5-trisphosphate.     

Equilibrium competition binding assay: inhibition mechanism from a single dose response

Inhibition of a receptor by a small-molecule compound in many cases is achieved via a competitive, uncompetitive or non-competitive mechanism. The receptor-inhibitor interaction is often probed through the displacement of a ligand in an equilibrium competition binding experiment. The previous solutions to receptor inhibition mechanisms were borrowed from steady-state enzyme inhibition mechanisms. The inhibition mechanism is determined by a visual inspection or a global fit of ligand dose response curves at a series of inhibitor concentrations. However these solutions only apply to situations when both the ligand and the inhibitor are not significantly depleted by the receptor. In most published equilibrium receptor binding studies, only the relative potency of the inhibitor is calculated. Ranking inhibitors tested under differing experimental conditions is often not possible. In the current paper, we offer exact mathematical solutions to uncompetitive and non-competitive inhibition, and demonstrate that in most cases both the inhibition mechanism and absolute potency of an inhibitor can be simultaneously determined from a single dose response of the inhibitor at a fixed concentration of the ligand. Therefore, an equilibrium competition assay provides a quick and facile method to determine the inhibition mechanism of a large number of inhibitors. The theory herein described is applicable to equilibrium competition binding experiments such as radioligand assays and fluorescence polarization assays.     

DNA bis-intercalation: Theoretical calculation of binding curves

A statistical mechanical calculation of the binding properties of DNA bis-intercalators is presented, based on the sequence-generating function method of Lifson. The effects of binding by intercalation of one or both chromophores of a bifunctional intercalating agent are examined. The secular equation for a general model that includes the effects of neighbor (nearest and non-nearest) exclusion and/or cooperativity in the binding of both singly and doubly intercalated ligands is derived. Numerical results for binding curves are presented for a more restricted model in which each type of bound ligand rigorously excludes its nearest neighbor and the total number of sites covered by a doubly intercalated ligand is variable. At low values of free ligand concentration bis-intercalation dominates the binding process, while at high value of free ligand concentration, intercalation of only one chromophore per ligand becomes significant due to the unavailability of contiguous free sites required for bis-intercalation. Also, depending on the binding parameters, the free energy of the system can be lowered by a loss of doubly intercalated ligands in favor of singly intercalated ones. Corresponding to this transition in binding mode, the average number of sites occupied by a bound ligand decreases from that characteristic of bis-intercalation to that characteristic of mono-intercalation as free ligand concentration increases. An analysis of Scatchard plots describing bis-intercalation is presented.     

On the stochastic properties of single ion channels

It is desirable to be able to predict, from a specified mechanism, the appearance of currents that flow through single ion channels (a) to enable interpretation of experiments in which single channel currents are observed, and (b) to allow physical meaning to be attached to the results observed in kinetic (noise and relaxation) experiments in which the aggregate of many single channel currents is observed. With this object, distributions (and the means) are derived for the length of the sojourn in any specified subset of states (e.g. all shut states). In general these are found to depend not only on the state in which the sojourn starts, but also on the state that immediately follows the sojourn. The methods described allow derivation of the distribution of, for example, (a) the number of openings, and total length of the burst of openings, that may occur during a single occupancy, and (b) the apparent gap between such bursts. The methods are illustrated by their application to two simple theories of agonist action. The Castillo-Katz (non-cooperative) mechanism predicts, for example, that the number of openings per occupancy, and the apparent burst length, are independent of agonist concentration whereas a simple cooperative mechanism predicts that both will increase with agonist concentration.     

Requirement of the proteasome for the trimming of signal peptide-derived epitopes presented by the nonclassical major histocompatibility complex class I molecule HLA-E

The nonclassical major histocompatibility complex class I molecule HLA-E acts as a ligand for CD94/NKG2 receptors on the surface of natural killer cells and a subset of T cells. HLA-E presents closely related nonameric peptide epitopes derived from the highly conserved signal sequences of classical major histocompatibility complex class I molecules as well as HLA-G. Their generation requires cleavage of the signal sequence by signal peptidase followed by the intramembrane-cleaving aspartic protease, signal peptide peptidase. In this study, we have assessed the subsequent proteolytic requirements leading to generation of the nonameric HLA-E peptide epitopes. We show that proteasome activity is required for further processing of the peptide generated by signal peptide peptidase. This constitutes the first example of capture of a naturally derived short peptide by the proteasome, producing a class I peptide ligand.