'Steady/equilibrium approximation' in relaxation and fluctuation. I. Procedure to simplify first-order reaction

The quasi-equilibrium assumption for Bi-Bi ordered bisubstrate enzymatic reaction. How to discriminate the mechanism correctly

Application of the quasi-equilibrium assumption for the steady-state kinetics of bisubstrate irreversible enzymatic reactions in the case of ordered binding of substrates (Bi-Bi ordered mechanism) is considered. The necessary and sufficient conditions for application of the quasi-equilibrium assumption have been found and accuracy of this assumption has been numerically evaluated. The limitations on application of the quasi-equilibrium assumption have been shown and errors of its application have been analyzed. It is shown that possible discrimination of substrate binding order using asymmetrical expressions grounded on the quasi-equilibrium assumption is inconsistent because such asymmetrical expressions arise from incorrect application of the quasi-equilibrium assumption. Moreover, it has been proved in the general case that mechanisms generating such substrate-asymmetrical expressions for the steady-state rate of enzymatic reaction do not exist. The error source when using graphical interpretation for discrimination of mechanisms of bisubstrate enzymatic reactions has been determined. The strategy to avoid such errors is pointed out.

A multi-time-scale analysis of chemical reaction networks: I. Deterministic systems

We consider deterministic descriptions of reaction networks in which different reactions occur on at least two distinct time scales. We show that when a certain Jacobian is nonsingular there is a coordinate system in which the evolution equations for slow and fast variables are separated, and we obtain the appropriate initial conditions for the transformed system. We also discuss topological properties which guarantee that the nonsingularity condition is satisfied, and show that in the new coordinate frame the evolution of the slow variables on the slow time scale is independent of the fast variables to lowest order in a small parameter. Several examples that illustrate the numerical accuracy of the reduction are presented, and an extension of the reduction method to three or more time scale networks is discussed.

Quasi-equilibrium assumption in enzyme kinetics. Necessary and sufficient conditions and accuracy of its application for single-substrate reactions

Steady-state kinetics of compulsory-ordered single-substrate irreversible and reversible enzyme reactions with two, three, and arbitrary number of intermediates were observed. Necessary and sufficient conditions for application of the quasi-equilibrium assumption and restrictions of this assumption were found in cases of two and three intermediates in the equilibrium segment. For all cases, accuracy of the quasi-equilibrium assumption was evaluated.

'Steady/equilibrium approximation' in relaxation and fluctuation. II. Mathematical theory of approximations in first-order reaction

A mathematical theory of the steady/equilibrium approximation for first-order reactions is presented. This gives the theoretical basis for the methods of simplifying the complex first-order reactions described in the preceding work The steady/equilibrium relation holds on every fast component after a proper inducation period T degrees T degrees is either of O(1) or less, or nearly of O(1/epsilon) depending on the reaction scheme and on the initial condition but is always less than O(1/epsilon) (as in the preceding paper [1], we use the symbol O(1) to denote a positive number of the order of unity). In the open group, the determinant of the submatrix M(p), representing the interconversion between the fast components in the group and their dissipation, is of O(1). The concentration of the fast components in the open group can thus be expressed as a linear combination of those components neighboring the group after the establishment of a steady/equilibrium relation, and can be eliminated from the reaction scheme leaving the pathway through them. On the other hand, in the closed group the determinant of Mp is of O(epsilon) or less and the components in the group are in quasi equilibrium with each other after T degrees . They are eliminated from the reaction scheme leaving the sum of the components in the closed group as a slow component.

A Ca2+-induced Ca2+ release mechanism involved in asynchronous exocytosis at frog motor nerve terminals

The extent to which Ca2+-induced Ca2+ release (CICR) affects transmitter release is unknown. Continuous nerve stimulation (20-50 Hz) caused slow transient increases in miniature end-plate potential (MEPP) frequency (MEPP-hump) and intracellular free Ca2+ ([Ca2+]i) in presynaptic terminals (Ca2+-hump) in frog skeletal muscles over a period of minutes in a low Ca2+, high Mg2+ solution. Mn2+ quenched Indo-1 and Fura-2 fluorescence, thus indicating that stimulation was accompanied by opening of voltage-dependent Ca2+ channels. MEPP-hump depended on extracellular Ca2+ (0.05-0.2 mM) and stimulation frequency. Both the Ca2+- and MEPP-humps were blocked by 8-(N, N-diethylamino)octyl3,4,5-trimethoxybenzoate hydrochloride (TMB-8), ryanodine, and thapsigargin, but enhanced by CN-. Thus, Ca2+-hump is generated by the activation of CICR via ryanodine receptors by Ca2+ entry, producing MEPP-hump. A short interruption of tetanus (<1 min) during MEPP-hump quickly reduced MEPP frequency to a level attained under the effect of TMB-8 or thapsigargin, while resuming tetanus swiftly raised MEPP frequency to the previous or higher level. Thus, the steady/equilibrium condition balancing CICR and Ca2+ clearance occurs in nerve terminals with slow changes toward a greater activation of CICR (priming) during the rising phase of MEPP-hump and toward a smaller activation during the decay phase. A short pause applied after the end of MEPP- or Ca2+-hump affected little MEPP frequency or [Ca2+]i, but caused a quick increase (faster than MEPP- or Ca2+-hump) after the pause, whose magnitude increased with an increase in pause duration (<1 min), suggesting that Ca2+ entry-dependent inactivation, but not depriming process, explains the decay of the humps. The depriming process was seen by giving a much longer pause (>1 min). Thus, ryanodine receptors in frog motor nerve terminals are endowed with Ca2+ entry-dependent slow priming and fast inactivation mechanisms, as well as Ca2+ entry-dependent activation, and involved in asynchronous exocytosis. Physiological significance of CICR in presynaptic terminals was discussed.

Kinetic and pharmacological properties of the GABA-induced chloride current in Aplysia neurones: a 'concentration clamp' study

1. gamma-Aminobutyric acid (GABA) was applied by the 'concentration clamp' technique to isolated neurones of Aplysia. GABA induced a chloride current (ICl) due to activation of a single class of chloride-channel. 2. The concentration-response curve for the peak ICl gave an apparent dissociation constant of 6.4 X 10(-5) M and a Hill coefficient of 0.88. The current-voltage relationship was linear in the voltage range examined (-40 to +10 mV). 3. The activation phase of the ICl could be fitted to a single exponential function and desensitization followed the sum of two exponential functions. The time constants of activation and desensitization decreased with increasing concentrations of GABA but were voltage-independent. The recovery process from desensitization also followed the sum of two exponential functions. 4. As for the rate-limiting step of the channel activation, the hyperbolic relationship between the activation rate and GABA concentration showed that the rapid binding assumption holds, suggesting that the isomerization step is rate-limiting. The apparent channel closing rate constant was estimated to be 10 s-1 from the ordinate intercept of the linear part of the above relationship at lower concentrations. 5. Muscimol and beta-alanine induced a ICl, which cross-desensitized with that evoked by GABA. The GABA-ICl was not enhanced by diazepam (10(-6) M) or alpha-chloralose (10(-3) M), in fact depressant effects were evident. 6. Pentobarbitone decreased the GABA-ICl non-competitively without altering activation or desensitization kinetics. The concentration-inhibition curve gave a KD value of 8.9 x 10(-5) M and a Hill coefficient of 1.0. 7. These results suggest that GABA activates a single class of Cl channel in Aplysia neurones, which have one binding site for the agonist. The GABA receptor-Cl channel complex in Aplysia is pharmacologically and perhaps structurally different from that in vertebrates.

Kinetic analysis of acetylcholine-induced chloride current in isolated Aplysia neurones

(1) Kinetics of activation and desensitization phases of the ACh-induced chloride current (ICl) were studied in isolated single neurones of Aplysia kurodai, using the 'concentration clamp' technique which combines internal perfusion and rapid exchange of the external solution within a few milliseconds (2) The dose-response curve for the peak ICl gave a dissociation constant of 6.7 X 10(-6) M and a Hill coefficient of 1.7. (3) The current-voltage relationship was linear in the voltage range examined (-70 to +30 mV). The reversal potential (EACh) was -7.1 +/- 1.8 mV (n = 14). The value was close to the calculated equilibrium potential for chloride ions (ECl). (4) The activation phase of the ICl was single exponential and the desensitization proceeded double exponentially to a steady state level. The time constants of both phases decreased with increasing concentrations of ACh but showed no potential dependency. The desensitizing component of the ICl was generated by activation of a single population of the receptor-channel complex. (5) The recovery from desensitization of the ICl induced by 6 X 10(-6) M ACh proceeded double exponentially, with time constants of 6.5 and 43 s at a holding potential of -30 mV. (6) Noise analysis performed on the steady state of ICl induced by low concentrations of ACh (3 X 10(-7) M to 3 X 10(-6) M) showed that the steady ICl was due to activation of a single population of the receptor-channel complex with a single channel conductance of 23.3 +/- 4.3 pS (n = 9).(ABSTRACT TRUNCATED AT 250 WORDS)

Kinetic analysis of glutamate-induced chloride current in Aplysia neurones: a 'concentration clamp' study

L-Glutamate (Glu) applied by the 'concentration clamp' technique to isolated neurones of Aplysia induced a chloride current (ICl) by activating a single population of the channel. The concentration-response curve for the peak ICl gave a dissociation constant of 1.3 x 10(-4) M and a Hill coefficient of 1.8. The current-voltage relationship was linear in the voltage range examined (-60 to +10 mV). The activation phase of the ICl followed a single-exponential time course and desensitization was complete with a double-exponential time course. The time constants for activation and desensitization decreased with increasing concentrations of Glu but were voltage-independent. The process of recovery from desensitization was also double-exponential. The single-channel conductance estimated by ensemble noise analysis was 50 +/- 4.7 pS (n = 4). These results suggest that the Glu receptor-Cl channel complex in Aplysia neurones consists of a single population with two binding sites for the agonist.

Receptor current fluctuation analysis in the labellar sugar receptor of the fleshfly

Fluctuations in the receptor current of the labellar sugar receptor of the fleshfly were analyzed. The receptor current was recorded extracellularly as a drop in potential between the tip and the base of the taste sensillum. After treatment with tetrodotoxin, the taste cells completely lost their impulses but retained their receptor currents, thus facilitating analysis of the receptor current without disturbance by impulses. The current fluctuation increased markedly when the sensillum was stimulated with effective sugars: maltose, sucrose, and fructose. The fluctuation increased in parallel with development of the receptor current, which indicates that it occurs as soon as the sugar reaches the apex of the sensory process. Analysis of fluctuations by computation of autocorrelation functions (ACFs) or power spectra (PS) revealed that: (a) the variance (mean square) of fluctuation vs. sugar concentration curve reached a maximum, in contrast to the monotonic increase shown by the receptor current; (b) the ACF was approximated by an exponential term, and its time constant differed according to the sugars used and their concentrations. The time constants for fructose and maltose decreased with increases in sugar concentration. At the concentrations of sugars evoking the same magnitude of receptor current, the time constant for fructose was the largest and that for maltose was the smallest. It was strongly suggested that transduction ion channels are present at the tip region of the sensory process of the sugar receptor cell and are operated directly by sugars.

Kinetic properties of the pentobarbitone-gated chloride current in frog sensory neurones

1. The kinetic properties of the activation and inactivation (desensitization) phases of pentobarbitone (PB)-induced inward Cl- current (ICl) were studied in isolated frog sensory neurones, following suppression of Na+, K+ and Ca2+ currents, using the concentration jump technique which combines the internal perfusion and the rapid exchange of the external solutions surrounding a neurone with time constants of 2-3 ms. The results were compared with those of the gamma-aminobutyric acid (GABA)-gated ICl. 2. The PB dose-response curve was bell-shaped and the maximum peak value was less than the current induced by 1.7 X 1.5(-5) M-GABA, the concentration at which GABA evoked a half-maximum response. 3. The activation and inactivation phases of PB-induced ICl consisted of double-exponential, fast and slow components, respectively. The time constant of the fast component (tau af) of the activation was relatively stable in a concentration range between 3 X 10(-4) and 6 X 10(-3) M. The time constant of the slow component (tau as) of the activation decreased with increasing PB concentrations. Both the fast and slow components (tau if and tau is) of the inactivation decreased with increasing PB concentrations. 4. Over a wide range of concentrations the tau af and tau as values of the PB-induced ICl were 10-30 times greater than the respective values of GABA-induced ICl. 5. At concentrations below 10(-3) M the PB-induced ICl was voltage dependent at more negative potentials than -20 mV. 6. The PB-induced ICl was blocked by bicuculline and by picrotoxin, but in a different manner. Bicuculline increased the time constants of the activation and inactivation. Picrotoxin had little effect on the activation phase but markedly facilitated the inactivation phase. 7. High concentrations of PB (over 10(-3) M) led to a decline in both the peak and plateau currents of the PB-induced ICl. A transient 'hump' current appeared with wash-out of the external solutions containing high concentrations of PB. This hump current was blocked by bicuculline in a dose-dependent manner. 8. The results suggest the possibilities that the PB receptor-ionophore complexes consist of at least two different components having different affinities and kinetics and that the PB and GABA binding sites are closely located.

Statistical analysis of channel current from a membrane patch. II. A stochastic theory of a multi-channel system in the steady-state

A general stochastic theory is presented for analysis of current records of a patch containing an arbitrary number (N) of independent homologous channels in the steady-state. We give the "basic theorem" that at the instant of any open (or shut) transition of a channel, the other N-1 channels are located in each state with a probability equal to those in the steady-state, if enough transitions are observed. Using the "basic theorem", we derived: (a) the time-dependent open and shut frequencies after a definite type of transition, and (b) the probability density functions (pdf) of the duration of any period between two successive transitions. Briefly, the main results obtained were: (1) The time-dependent open (or shut) transition frequency after every shut (or open) transition at t = 0 in an N-channel patch, fJSh,Op(t)(N) (or fJOp,Sh(t)(N)), is the same as that of a one-channel patch except for the value of the constant. (2) In the all-shut (or all-open) period of a patch, the average duration of the period is 1/N, and the slowest exponential decay constant contained in the pdf is N times those of a single channel patch, respectively. (3) An example calculation for small N showed that the stochastic properties of a single channel can be obtained even when N is uncertain, if the channel open probability is small and exponential decay constants are separated. (4) When the channels are in equilibrium, the pdf of duration of every type of period in the patch is described by a sum of exponential terms with positive coefficients. This also holds for fJSh,Op(t)(N) and fJOp,Sh(t)(N).

Simplifying principles for chemical and enzyme reaction kinetics

Tihonov's Theorems for systems of first-order ordinary differential equations containing small parameters in the derivatives, which form the mathematical foundation of the steady-state approximation, are restated. A general procedure for simplifying chemical and enzyme reaction kinetics, based on the difference of characteristic time scales, Is presented. Korzuhin's Theorem, which makes it possible to approximate any kinetic system by a closed chemical system, is also reported. The notions and theorems are illustrated with examples of Michaelis-Menten enzyme kinetics and of a simple autocatalytic system. Another example illustrates how the differences in the rate constants of different elementary reactions may be exploited to simplify reaction kinetics by using Tihonov's Theorem. All necessary mathematical notions are explained in the appendices. The most simple formulation of Tihonov's 1st Theorem 'for beginners' is also given.