Exact reductions of Markovian dynamics for ion channels with a single permissive state

We show that the cooperative model for the kinetics of a tetrameric potassium ion channel derived in Nekouzadeh et al. (Biophys J 95(7):3510–3520, 2008) is an invariant manifold reduction of the full master equation for the channel kinetics. We further establish the validity of this reduction for ion channel models consisting of multiple independent subunits with cooperative transitions from a single permissive state to a conducting state. Finally, we conclude that solutions of the reduced model are globally asymptotically stable solutions of the full master equation system.     

Cooperative reactions of poly-L-lysine-heme complex with molecular oxygen, carbon monoxide, or cyanide ion.

The interaction of the alpha-helical poly-L-lysine-heme complex with molecular oxygen, carbon monoxide, or cyanide ion was studied. Binding equilibrium curve and activation parameters for the reactions were determined. Sigmoid responses were observed for the absorption of molecular oxygen or carbon monoxide by the complex and the cooperative parameter was found to be 2.1. This indicated a cooperative interaction between hemes situated on a cylindrical alpha-helix of poly-L-lysine. But those of other polymer-ligand-heme complexes were 1.0. The cooperative reaction mechanism, in which an alpha-helical poly-L-lysine plays an important role, was suggested.     

Cooperation effects in binding of large ligands to DNA. II. Contact interactions between adsorbed ligands

Cooperative effects arising upon binding of biologically active ligands to DNA are considered. Equations are derived which enable one to describe the binding of two different ligands to DNA. We also consider the case when ligand can form two type of DNA complexes. The cooperative binding of the ligand in the vicinity of saturation level of binding can be described with a good accuracy by equation derived for the non-cooperative adsorption of the same ligand with some effective binding constant Keff. It is shown that cooperative effects arising upon binding of proteins and other ligands to DNA can be divided into two groups depending on the symmetry of interactions between the bound ligand molecules. In particular, if such interactions favor the formation of dimeric ligand species on the DNA, Keff approximately a1/2, where a is the ligand-ligand interaction constant. If cooperative interactions favor the formation of aggregates of unrestricted size, then Keff approximately aL+Y, where L is the size of the binding site for the ligand on DNA.     

Mathematical model of the interaction of cations and anions in a channel

A mathematical model of the ionic channel permeable both to anions and cations is considered. The model takes into account the electrostatic interaction between oppositely charged ions and does not suppose single-file movement. An equation for zero-current potential is derived, which leads to the Goldman equation in the limit of low ion concentrations. The model is used to describe concentration relationships of zero-current potentials on a lipid bilayer with amphotericin B channels which cannot be described on the basis of the independence principle.     

Dependence of reactivity and cooperativity in normal human erythrocyte glucose-6-phosphate dehydrogenase on ionic strength,pH, and temperature

The steady-state kinetics of the binding reaction of NADP⁺ to normal human erythrocyte glucose-6-phosphate dehydrogenase were studied as a function of pH, ionic strength, and temperature. The interaction coefficient obtained according to the Hill equation increases with increase in pH, ionic strength, and temperature. The observed variation of cooperative interaction is interpreted in terms of an increase in the percentage of the dimer as these environmental parameters increase. Activation energy decreases with increase in pH, the activation energy at the lowest pH being almost halved at the most alkaline pH. This behavior is explained in terms of a differential reactivity between the dimeric and tetrameric forms of the enzyme. The observation of cooperative and more reactive dimer is postulated to be a regulatory mechanism by the cell for shifting the equilibrium from one quaternary structure to the other depending on the need for NADPH.     

Single file ion diffusion and the Schmoluchowski equation

The Schmoluchowski equation is introduced into the problem of single file ion diffusion in a channel. The ions mutually interact due to coulomb repulsion and are also subject to a single ion potential due to the channel. The positions of the ions are represented by a continuous co-ordinate. The problem is reduced to the solution of a pair of transfer integral equations. The resistivity of finite and infinite channels is calculated for various dielectric constants and mean ionic separations. The ionic density for finite channels is also calculated. The results clearly demonstrate that strong coulomb interaction leads to a co-operative motion of the ions across channels.     

Thermodynamic model of cooperativity in a dimeric protein: unique and independent parameters formulation.

A model of the cooperative interaction of ligand binding to a dimeric protein is presented based upon the unique and independent parameters (UIP) thermodynamic formulation (Gutheil and McKenna, Biophys. Chem. 45 (1992) 171-179). The analysis is developed from an initial model which includes coupled conformational and ligand binding equilibria. This completely general model is then restricted to focus on conformationally mediated cooperative interactions between the ligands and the expressions for the apparent ligand binding constant and the apparent ligand-ligand interaction constant are derived. The conditions under which there is no cooperative interaction between the ligands are found as roots to a polynomial equation. Consideration of the distribution of species among the various conformational states in this general model leads to a set of inequalities which can be represented as a two dimensional plot of boundaries. By superimposing a contour plot of the value of the apparent ligand-ligand interaction constant over the plot of boundaries a complete graphical representation of this system is achieved similar to a phase diagram. It is found that the parameter space homologous to Koshland-Nemethy-Filmer type of model is most consistent with both positive and negative cooperativity in this model. The maximal amount of positive and negative cooperativity are found to be simple functions of Kc, the equilibrium constant associated with the change of a subunit and ligand from the unligated to ligated conformation. It is shown that under certain limiting conditions the apparent allosteric interaction between ligands is equal to the conformational interaction between subunits. The methods presented are generally applicable to the theoretical analysis of thermodynamic interactions in complex systems.     

Protein-free models for the proton-coupled reduction of haemoproteins

Reduction of the bis-pilocarpate-haemin complex at pH greater than or equal to 10 involves the simultaneous uptake of an electron by the Fe(III) ion and a proton by the pendant alkoxide group of an axial ligand. This provides a protein-free model for reactions such as the proton-coupled reduction of cytochromes which involve cooperative Coulombic interaction between two non-bonded sites.     

Cooperative specific adsorption of ions at charged sites in an electric field

The adsorption of two cations at the anionic sites of a polymer (e.g., such as a protein) in an electric fields is discussed, taking into account cooperative interaction of the cations mediated through the backbone of the polymer. The calculation of the grand partition function explicitly considers the vacant negative sites of the polymer. As in the case without cooperative interaction, the problem reduces to the determination of the largest eigenvalue of asymmetric matrices. The weights of the different neighbor configurations are determined. Approximate formulae for the grand partition function and for those weights are derived. The formal analogy of these cooperative phenomena and those occurring in quantum (bio)chemistry is pointed out exemplifying an earlier suggestion about the basis of quantum biology.     

Kinetic theory of a gas-discharge positive column and wall sheath

The positive column and wall sheath in a gas discharge are studied with allowance for ion collisions in a plasma and ion reflection from a solid surface under conditions of incomplete ion neutralization. The kinetic equation for ions in a positive column is reduced to a Fredholm equation of the second kind. This makes it possible to solve the kinetic equation using a resolvent and thereby derive a single integrodifferential equation for the potential, which is referred to as a generalized plasma-sheath equation. Specific versions of the plasma-sheath equation are obtained that take into account charge exchange of the ions in a plasma and the thermal spread in velocities of the ionization-produced ions.     

Description of interacting channel gating using a stochastic Markovian model

Single-channel recordings from membrane patches frequently exhibit multiple conductance levels. In some preparations, the steady-state probabilities of observing these levels do not follow a binomial distribution. This behavior has been reported in sodium channels, potassium channels, acetylcholine receptor channels and gap junction channels. A non-binomial distribution suggests interaction of the channels or the presence of channels with different open probabilities. However, the current trace sometimes exhibits single transitions spanning several levels. Since the probability of simultaneous transitions of independent channels is infinitesimally small, such observations strongly suggest a cooperative gating behavior. We present a Markov model to describe the cooperative gating of channels using only the all-points current amplitude histograms for the probability of observing the various conductance levels. We investigate the steady-state (or equilibrium) properties of a system ofN channels and provide a scheme to express all the probabilities in terms of just two parameters. The main feature of our model is that lateral interaction of channels gives rise to cooperative gating. Another useful feature is the introduction of the language of graph theory which can potentially provide a different avenue to study ion channel kinetics. We write down explicit expressions for systems of two, three and four channels and provide a procedure to describe the system ofN channels.     

Regulatory properties of lysine-sensitive aspartokinase under equilibrium conditions

The regulatory properties of the lysine-sensitive aspartokinase (ATP : L-aspartate 4-phosphotransferase, EC 2.7.2.4) have been studied under equilibrium conditions by determining the effects of modifiers on the rate of equilibrium isotope exchange between ADP and ATP. The extent of inhibition by lysine, leucine or phenylalanine is almost independent of substrate concentration but is influenced by the substrate/product ratio. Inhibition by a given concentration of inhibitor is increased when the ADP/ATP ratio is increased indicating a regulatory interaction between end products and cellular energy metabolism. Lysine inhibition is cooperative under equilibrium conditions and the parameters of the Hill equation are nearly identical to those obtained in initial velocity studies. A cooperative heterotropic interaction between lysine and leucine is also observed by the ATP-ADP exchange assay just as it is in initial velocity assays. Thus, the regulatory features of aspartokinase that are observed in initial velocity studies are also manifest under equilibrium conditions as revealed by equilibrium isotope exchange rates.     

Site—site interaction in the lipid activation of β-hydroxybutyrate dehydrogenase

Kinetic data for the activation of the β-hydroxybutyrate dehydrogenase by long-chain lecithins [(1979) Biochemistry 18, 2420–2429; (1983) J. Biol. Chem. 258, 208–214] are analyzed. A previous kinetic model [(1982) Biochemistry 21, 3899–3908] is shown not to apply. Instead, the use of a two-site Adair equation points to a strongly cooperative interaction between the lecithin binding sites (ΔG, ?2.8 kcal/mol).     

Hill coefficient for estimating the magnitude of cooperativity in gating transitions of voltage-dependent ion channels

A frequently used measure for the extent of cooperativity in ligand binding by an allosteric protein is the Hill coefficient, obtained by fitting data of initial reaction velocity (or fractional binding saturation) as a function of substrate concentration to the Hill equation. Here, it is demonstrated that the simple two-state Boltzmann equation that is widely used to fit voltage-activation data of voltage-dependent ion channels is analogous to the Hill equation. A general empiric definition for a Hill coefficient (n(H)) for channel gating transitions that is analogous to the logarithmic potential sensitivity function of Almers is derived. This definition provides a novel framework for interpreting the meaning of the Hill coefficient. In considering three particular and simple gating schemes for a voltage-activated cation channel, the relation of the Hill coefficient to the magnitude and nature of cooperative interactions along the reaction coordinate of channel gating is demonstrated. A possible functional explanation for the low value of the Hill coefficient for gating transitions of the Shaker voltage-activated K(+) channel is suggested. The analogy between the Hill coefficients for ligand binding and for channel gating transitions further points to a unified conceptual framework in analyzing enzymes and channels behavior.     

Experimental evidence for allosteric modifier saturation as predicted by the bi-substrate Hill equation

The cooperative enzyme reaction rates predicted by the bi-substrate Hill equation and the bi-substrate Monod-Wyman-Changeux (MWC) equation when allosterically inhibited are compared in silico. Theoretically, the Hill equation predicts that when the maximum inhibitory effect at a certain substrate condition has been reached, an increase in allosteric inhibitor concentration will have no effect on reaction rate, that is the Hill equation shows allosteric inhibitor saturation. This saturating inhibitory effect is not present in the MWC equation. Experimental in vitro data for pyruvate kinase, a bi-substrate cooperative enzyme that is allosterically inhibited, are presented. This enzyme also shows inhibitor saturation, and therefore serves as experimental evidence that the bi-substrate Hill equation predicts more realistic allosteric inhibitor behaviour than the bi-substrate MWC equation.     

Binding of the environmental pollutant naphthol to bovine serum albumin

The interactions between naphthol and bovine serum albumin (BSA) were investigated by spectroscopy. Our results prove the formation of complex between naphthol and BSA. Hydrophobic interaction dominates in the association reaction. The isomers stack with the aromatic residues in their binding sites with different geometries. Effects of BSA on the excited-state proton transfer and fluorescence spectra of the isomers indicate the different characters of their binding sites. 1-Naphthol inserts deeply into a hydrophobic cavity whereas 2-naphthol is in a basic environment on the surface of BSA. Naphthol statically quenches the fluorescence of BSA in a concentration-dependent manner positively deviating from the linear Stern-Volmer equation. Naphthol binds near Trp-134 in the subdomain IA of the native BSA and is accessible to Trp-212 when BSA is unfolded by naphthol. The folding pattern of the main chain is altered at high naphthol concentration as revealed by the change in the secondary structure. The binding of 1-naphthol is more cooperative than that of 2-naphthol. The extent of cooperativity was estimated by the Hill equation.     

The folding of the hairpin ribozyme: dependence on the loops and the junction

In its natural context, the hairpin ribozyme is constructed around a four-way helical junction. This presents the two loops that interact to form the active site on adjacent arms, requiring rotation into an antiparallel structure to bring them into proximity. In the present study we have compared the folding of this form of the ribozyme and subspecies lacking either the loops or the helical junction using fluorescence resonance energy transfer. The complete ribozyme as a four-way junction folds into an antiparallel structure by the cooperative binding of magnesium ions, requiring 20-40 microM for half-maximal extent of folding ([Mg2+]1/2) and a Hill coefficient n = 2. The isolated junction (lacking the loops) also folds into a corresponding antiparallel structure, but does so noncooperatively (n = 1) at a higher magnesium ion concentration ([Mg2+]1/2 = 3 mM). Introduction of a G + 1A mutation into loop A of the ribozyme results in a species with very similar folding to the simple junction, and complete loss of ribozyme activity. Removal of the junction from the ribozyme, replacing it either with a strand break (serving as a hinge) or a GC5 bulge, results in greatly impaired folding, with [Mg2+]1/2 > 20 mM. The results indicate that the natural form of the ribozyme undergoes ion-induced folding by the cooperative formation of an antiparallel junction and loop-loop interaction to generate the active form of the ribozyme. The four-way junction thus provides a scaffold in the natural RNA that facilitates the folding of the ribozyme into the active form.     

Collective Equilibrium Behaviour of Ion Channel Gating in Cell Membranes: An Ising Model Formulation

A statistical mechanical model for voltage-gated ion channels in cell membranes is proposed using the transfer matrix method. Equilibrium behavior of the system is studied. Representing the distribution of channels over the cellular membrane on a one-dimensional array with each channel having two states (open and closed) and incorporating channel–channel cooperative interactions, we calculate the fraction of channels in the open state at equilibrium. Experimental data obtained from batrachotoxin-modified sodium channels in the squid giant axon, using the cut-open axon technique, is best fit by the model when there is no interaction between the channels.     

Generalized transition state method and continuous diffusion in multi-dimensional systems with relation to ion transport in channels of biological membranes

An exact expression for the escape rate of a particle in a multi-dimensional system, with respect to an arbitrary reaction coordinate, is derived from first principles according to the transition state method, using a simple geometrical argument. It is shown how the mutual coupling of all degrees of freedom due to the interaction forces leads to the appearance of an effective mass and the potential of the mean force. The same relevant quantities dominate the effective one-particle Fokker-Planck equation, which is derived by a similar projection procedure from the multi-dimensional transport equation. In the limit of a large, position-dependent friction the respective effective Smoluchowski equation is obtained. It allows for the discussion of a diffusing particle which is subject to a temperature bath only through the coupled motion with the constituent lattice particles, or ligands in the case of a molecular ion channel. This treatment is of particular importance for the analysis of ion transport in membrane pores in which the ionic motion is mediated by internal ligand motion.