## Tag Archives: limiting rate

## On true and apparent michaelis constants in enzymology. I. Differences

**S. O. Karakhim**

*Palladin Institute of Biochemistry, National Academy of Sciences of Ukraine, Kyiv;*

* e-mail: laserlab@biochem.kiev.ua*

Differences between both true and apparent rate constants and Michaelis constants have been examined. Rate constants of elementary stages of real mechanisms are true ones. True Michaelis constant Km is expressed by equation *K*_{m} = (*k*_{-1} + *k*_{2})/*k*_{1}. True constants may be determined for reliable mechanism only for which the equation of initial rate was obtained which displays physical sense of these constants and permits to find the method of their calculation. The true constant values are independent of concentration of reactants, activators, inhibitors, extraneous agents and pH.

The apparent rate constants are such constants of the composite reaction which are observed when this reaction is described by the equation of simple reaction. Michaelis constant calculated by a half of the ultimate constant is an apparent constant. The apparent constants may be functions of several true rate constants and/or concentrations of reacting substances. The evident physical sense of apparent constants being absent, only formal relation between the reaction rate and reactant concentration independent of the investigated mechanism is provided.

## On true and apparent Michaelis constants in enzymology. III. Is it linear dependence between apparent Michaelis constant and limiting rate and is it possible to determine the substrate constant value using this dependence?

**S. O. Karakhim**

*Palladin Institute of Biochemistry, National Academy of Sciences of Ukraine, Kyiv;*

* e-mail: laserlab@biochem.kiev.ua*

The Slater-Bonner method which is used for graphic determination of substrate constant (*K*_{s}) by linear dependence of apparent Michaelis constant (*K*_{m}^{app}) on the limiting rate (*V*^{app}) of enzyme-catalysed reactions with activator participation has been critically analysed. It has been shown that although it is possible to record the mechanisms of such reactions as a scheme similar to Michaelis-Menten model which allow to find correlation *K*_{m}^{app} and *V*^{app} as equation *K*_{m}^{app} = *K*_{s} + *V*^{app}/*k*_{1}[*E*]_{0} ([*E*]_{0} is a total enzyme concentration, *k*_{1} is a rate constant of enzyme-substrate complex formation from free enzyme and substrate) in order to calculate *K*_{s} and individual rate constants (*k*_{1}, *k*_{-1}), but this approach for investigation of all reactions with activator participation ought not to be used. The above equation is not obeyed in general, it may be true for some mechanisms only or under certain ratios of kinetic parameters of enzyme-catalysed reactions.

## On true and apparent Michaelis constants in enzymology. II. Is the equation K(m)(app) = K(s) + k(cat)/k(1) true for enzyme-catalysed reactions with activator participation?

**S. O. Karakhim**

*Palladin Institute of Biochemistry, National Academy of Sciences of Ukraine, Kyiv;*

* e-mail: laserlab@biochem.kiev.ua*

The article is dedicated to analysis of equation which expresses apparent Michaelis constant *K*_{m}^{app} of enzyme-catalysed reactions with activator participation by means of the substrate constant *K*_{s} and rate constant of enzyme-substrate complex decomposition kcat. It has been shown that although it is possible to record the mechanisms of such reactions as a scheme similar to Michaelis-Menten model and to derive equation of apparent Michaelis constant as *K*_{m}^{app} = *K*_{s} +* k _{cat}*/

*k*

_{1}, but this approach cannot be used for investigation of all reactions with activator participation. The equation mentioned above is not obeyed in the general case, it may be true for some mechanisms only or under certain ratio of kinetic parameters of enzyme-catalysed reactions.

## Mathematical modeling of calcium homeostasis in smooth muscle cells while activity of plasma membrane calcium pump is modulated

**S. O. Karakhim, V. F. Gorchev, P. F. Zhuk, S. O. Kosterin**

*Palladin Institute of Biochemistry, National Academy of Sciences of Ukraine, Kyiv;*

* e-mail: laserlab@biochem.kiev.ua; kinet@biochem.kiev.ua*

A mathematical model of intracellular calcium homeostasis in smooth muscle cells has been investigated by computer modelling method. The results of calculations showed that for the plasma membrane calcium pump (PMCA) the limiting rate (*V _{mPM}*) increasing or the Michaelis constant (

*K*) decreasing result in a lowering of the Ca

_{mPM}^{2+}concentration in cytosol and sarcoplasmic reticulum (SR); the slight

*V*decreasing or

_{mPM}*K*increasing result in fluent cytosolic Ca

_{mPM}^{2+}strengthening due to slow basal influx (SBI) since a massive release of Ca

^{2+}from SR does not occur. The further

*V*decreasing or

_{mPM}*K*increasing stimulate the Ca

_{mPM}^{2+}-induced Ca

^{2+}release from SR and the system passes into oscillation mode; when the certain low

*V*or high

_{mPM}*K*level is reached the oscillations of Ca

_{mPM}^{2+}concentration in cytosol are stopped, there is only first oscillation after which a new level of cytosolic Ca

^{2+}concentration is formed fluently: this level is higher than in the initial basal condition (IBC). Sensitivity of myocytes with the lowering of

*V*or increasing

_{mPM}*K*to agonist action is rising but sensitivity of myocytes with increasing

_{mPM}*V*or decreasing

_{mPM}*K*to agonist action is reducing. If the PMCA parameters (

_{mPM}*V*or

_{mPM}*K*) are changed then passive influx of Ca

_{mPM}^{2+}in cytosol from extracellular space remains virtually invariable and it is equal to SBI value during the whole process. Initial rate of PMCA in a new equilibrium condition (NEC) is equal virtually to initial rate in IBC: it allows to calculate a new value

*V*or

_{mPM}*K*from cytosolic Ca

_{mPM}^{2+}concentration in NEC.