Calculation of steady-state rate equations and the fluxes between substrates and products in enzyme reactions.

1. Two methods are described for deriving the steady-state velocity of an enzyme reaction from a consideration of fluxes between enzyme intermediates. The equivalent-reaction technique, in which enzyme intermediates are systematically eliminated and replaced by equivalent reactions, appears the most generally useful. The methods are applicable to all enzyme mechanisms, including three-substrate and random Bi Bi Ping Pong mechanisms. Solutions are obtained in algebraic form and these are presented for the common random Bi Bi mechanisms. The steady-state quantities of the enzyme intermediates may also be calculated. Additional steps may be introduced into enzyme mechanisms for which the steady-state velocity equation is already known. 2. The calculation of fluxes between substrates and products in three-substrate and random Bi Bi Ping Pong mechanisms is described. 3. It is concluded that the new methods may offer advantages in ease of calculation and in the analysis of the effects of individual steps on the overall reaction. The methods are used to show that an ordered addition of two substrates to an enzyme which is activated by another ligand will not necessarily give hyperbolic steady-state-velocity kinetics or the flux ratios characteristic of an ordered addition, if the dissociation of the ligand from the enzyme is random.     

Use of alternate substrates to probe the order of substrate addition to dopamine beta-hydroxylase

In order to determine the order of substrate binding to dopamine beta-hydroxylase during catalysis, the effect of alternate substrates upon kinetic parameters was examined. The V/K value for ascorbate was unchanged when tyramine, phenylpropylamine, p-Cl-phenethylamine, p-CH3O-phenethylamine, or phenethylamine was the hydroxylated substrate. The V/K values for tyramine and oxygen were similarly unchanged when ferrocyanide was used as the reductant in place of ascorbate. In order to use ferrocyanide as reductant it was necessary to include copper to alleviate the substrate inhibition seen with this substrate. The pattern of substrate inhibition observed with ferrocyanide was consistent with a small amount of free cyanide present in the ferrocyanide. With ferrocyanide as reductant and [2,2-2H2]tyramine as substrate, there was a measurable isotope effect on the V/K value for oxygen, but none on the values of Vmax or V/K for tyramine. These results are consistent with a ping-pong mechanism in which tyramine binds to the enzyme after the release of oxidized ascorbate. Subsequently, oxygen binds to form a ternary complex.     

Use of an enzyme thermistor in continuous measurements and enzyme reactor control.

The enzyme thermistor measures the heat produced by the action of an immobilized enzyme on a substrate present in the sample. Its application in analysis of discrete samples, e.g., in clinical chemistry, is well documented, but it has not been used so far for continuous measurements. We decribe here the application of the enzyme thermistor for continuous monitoring and control of enzyme reactors. An enzyme thermistor filled with coimmobilized glucose oxidase and catalase was used to measure the amount of glucose in the outflow from a column reactor containing immobilized lactase acting on a lactose solution pumped through the reactor. The lactose conversion was kept on a constant level, irrespective of the actual enzymatic activity in the reactor, by regulating the flow through the reactor. The experiments were carried out with aqueous solutions of lactose as well as with whey from cow's milk.     

Metabolic fluxes, pools, and enzyme measurements suggest a tighter coupling of energetics and biosynthetic reactions associated with reduced pyruvate kinase flux.

In this study, it is found that, for Bacillus subtilis, citrate-glucose cometabolism leads to zero acid production over a wide range of growth rates and nearly theoretical carbon yield. Experimental results are presented that point to pyruvate kinase (PYK) as a site of citrate-mediated glycolytic flux attenuation. First, the measured fluxes show that, compared with cultures grown on glucose, the PYK flux drops by more than tenfold when citrate is added. Second, relative to cultures metabolizing glucose, the phosphoenolpyruvate (PEP) pool elevates substantially, whereas the pyruvate pool drops, when citrate is present. Finally, our modeling results indicate that maximizing carbon yield corresponds to nearly eliminating pyruvate kinase (PYK) flux and that the pyruvate supplied by the PEP-consuming glucose transport system can supply the biosynthetic requirements. A literature review suggests some mechanisms for how PYK attenuation by citrate addition can occur. At this juncture, we hypothesize that direct PYK inhibition occurs which, in turn, also leads to phosphofructokinase inhibition via the elevated PEP pool. These two inhibition events combine to throttle glycolytic flux; minimize acid formation; and substantially increase cellular, product, and energetic yields.     

Enhancement of product selectivity via enzyme immobilization in sequential degradation reactions of polymeric substrates

The balance equations pertaining to the modelling of a slap-shaped bead containing immobilized enzyme uniformly distributed which catalyzes the sequential reactions of degradation of a polymeric substrate were written and analytically solved in dimensionless form. The effect of the Thiele modulus on the selectivity of consumption of each multimeric product was studied for a simple case. Whereas plain diffusional regime leads to lower selectivities than plain kinetic regime, improvements in selectivity of species A _i relative to species A_i+1 may be obtained at the expense of higher Thiele moduli within a limited range when the diffusivity of A _i is larger than that of A _i +1, or when the pseudo first order kinetic constant describing the rate of consumption of A _i is lower than that of A_i+1.List of Symbols A _i polymeric substrate containing i monomeric subunits - C _i mol·m^–3 normalized counterpart of C _i - C _i mol·m^–3 concentration of substrate A _i - C _i,0 mol·m^–3 initial concentration of substrate A _i - C _i,0 normalized counterpart of C _i,0 - D _ap,i m²·s^–1 apparent diffusivity of substrate A _i - k _i s^–1 pseudo-first order rate constant - K _m,i mol·m^–3 Michaelis-Menten constant associated with substrate A _i - L m half-thickness of the catalyst slab - N number of monomeric subunits of the largest substrate molecule - Th Thiele modulus - V _i mol·m^–3·s^–1 rate of rection of substrate A _i - V_max,i mol·m^–3·s^–1 maximum rate of reaction under saturating conditions of substrate A _i - x m longitudinal coordinate - S _i,i+1 selectivity of enzyme with respect to substrates with consecutive numbers of monomeric subunits Greek Symbols _i ratio of maximum rates of reaction - _i ratio of apparent diffusivities     

Deviations from the flux ratio equation for chloride ions in ouabain- and acetazolamide-treated frog skin.

In order to establish whether or not chloride ions behave as freely moving particles in "passive", i.e. ouabain- and acetazolamide-treated, frog skin, tracer fluxes of 36Cl-have been measured while a voltage (generally +40 mV, serosal side positive) across the skin was applied. Ussing's flux ratio equation has been used as a criterion for this type of transport. One group of skin samples exhibited significant exchange diffusion phenomena. Most samples in a second group either behaved according to the flux ratio equation of showed significant and extreme exchange diffusion. From flux ratios obtained at two different voltages across various skin samples, showing extreme exchange diffusion, it appeared that the simple form of Kedem and Essig's law derived from irreversible thermodynamics, which is valid for homogeneous systems, does not apply to the type of exchange diffusion found. The system can, however, be described by a 1:1 exchange mechanism working in parallel with a diffusional pathway. The ratio exchange flux/observed efflux must then have a constant value (0.83) at the voltages appled, which implies that the exchange flux is voltage dependent. By comparison with iodide flux experiments as carried out by Ussing, it is shown that iodide exhibits the same type of exchange diffusion. A carrier, possibly responsible for the observed behaviour, is described.