8.2: Introduction

Kinetics

Chemical kinetics is the study of rates of chemical processes. Kinetic studies include investigation of how different experimental conditions influence the rate of a chemical reaction. The data derived from a kinetic study can give insight into a reaction's mechanism, the nature of its transition states, and can be used to construct of mathematical models that describe the reaction.

Biochemical reactions take place in aqueous media where they are catalyzed by protein molecules, called enzymes. Enzymes, like all proteins, are polypeptide polymers made up of successive \(\alpha\)-amino acid residues. Although twenty different types of amino acids are used, the resultant polypeptide chain may contain hundreds or even thousands of amino acid residues. Each type of enzyme (or protein) is a unique chemical species having a fixed number of specific amino acids arranged in a particular sequence along the chain(s).

The catalytic activity of an enzyme depends upon the fact that in aqueous solution each enzyme assumes a particular three-dimensional structure (the native tertiary or quaternary structure) containing an "active site". The reactant molecules (usually small organic molecule) is bound in the active site, where a specific chemical reaction is catalyzed to form products.

An enzyme loses its catalytic ability if it is denatured (unfolded from its native state). Enzymes can be denatured by a change in pH, by the addition of denaturing agents such as detergents, urea, or guanidine hydrochloride, or, in some cases, by raising the temperature. Some catalytic activity may also be lost if a competing substrate, or inhibitor, is added to the solution along with the desired substrate. Many enzyme-catalyzed reactions proceed via a mechanism that can be kinetically analyzed using a formalism developed by Michaelis and Menton.

The Michaelis-Menton Mechanism for Enzyme-Catalyzed Reactions

The Michaelis-Menton treatment assumes that the enzyme (\(\mathrm{E}\)) first reacts with the substrate (\(\mathrm{S}\)) to form the enzyme-substrate complex (\(\mathrm{ES}\)) which breaks down in a second step to give free enzyme and products (\(\mathrm{P}\)).

\[ \ce{E + S <=>[{k_1}][{k_{-1}}] ES} \label{1} \]

\[ \ce{ES ->[k_2] P} \label{2} \]

Steady state hypothesis

The rate of formation of \(\mathrm{ES}\) from \(\mathrm{E}\) + \(\mathrm{P}\) is very small and may be neglected. If the concentration of the complex \(\mathrm{ES}\) is assumed to achieve a steady state

\[ \frac{d[\mathrm{ES}]}{dt}=k_1[\mathrm{E}][\mathrm{S}]-k_{-1}[\mathrm{ES}]-k_2[\mathrm{ES}]=0 \nonumber\]

then

\[ [\mathrm{ES}]=\frac{k_1[\mathrm{E}][\mathrm{S}]}{k_{-1}+k_2} \label{3} \]

In general, we would know the initial concentration, \([\mathrm{E}_o]\), of enzyme before the enzyme-substrate complex is formed, but not \( [ \mathrm{ES} ] \) or \( [ \mathrm{E} ] \) after the reaction is underway. But we can use mass balance to know that the total enzyme is equal to free enzyme plus enzyme in the \(\mathrm{ES}\) complex. And, since the total enzyme is the same as the initial amount of enzyme:

\[ \left[\mathrm{E}_{0}\right]=[\mathrm{ES}]+[\mathrm{E}] \label{4} \]

We can substitute \([ \mathrm{E}]=[ \mathrm{E}_0]-[ \mathrm{ES}]\) from \ref{4} into \ref{3} to get the equation in terms of initial concentration of enzyme:

\[ [E S]=\frac{k_1}{k_{-1}+k_2}[\mathrm{~S}]\left(\left[\mathrm{E}_{0}\right]-[\mathrm{ES}]\right) \label{5} \]

Now \ref{5} is simplified to yield an expression that to gives \([\mathrm{ES}]\) in in terms of \([\mathrm{E}_0]\):

\[ [\mathrm{ES}]=\frac{k_1[\mathrm{~S}]\left[\mathrm{E}_{0}\right]}{\left(k_{-1}+k_2\right)+k_1[\mathrm{~S}]} \label{6} \]

Finding the initial rate, \(V_0\)

The rate (\(V\)) of the reaction can be be expressed as the rate of formation of product with respect to time (see Equation \ref{2}). The formation of product is related to the disappearance of the \(\mathrm{ES}\) complex and the rate constant \(k_2\). We can express the rate of formation of products using Equation \ref{7a}:

\[V_0 = \left( \frac{d[\mathrm{P}]}{dt} \right)=k_2[\mathrm{ES}] \label{7a}\]

We can apply the steady state hypothesis by substituting Equation \ref{6} for \([ES]\) in Equation \ref{7a}. This allows us to express the initial rate, in terms of \(\mathrm{E}_0\):

\[ V_0=\frac{d[\mathrm{P}]}{d t}=k_2[E S]=\frac{k_1 k_2\left[E_o\right][\mathrm{S}]}{\left(k_{-1}+k_2\right)+k_1[\mathrm{S}]} \label{7} \]

This relation is usually rewritten in a modified form by dividing numerator and denominator by \(k_1\) and introducing a new constant, known as the Michaelis constant (\(K_M\)):

\[K_M = \dfrac{(k_{–1} + k_2)}{k_1} \label{KM}\]

Thus, the Michaelis-Menton rate law is given by Equation \ref{8}.

\[ V=\frac{k_2\left[E_o\right][\mathrm{S}]}{K_M+[\mathrm{S}]} \label{8} \]

Notice that if we determine the initial rate of reaction, both \( [\mathrm{E}_{0} ] \) and \([ \mathrm{S}] \) are the known initial concentrations, so that \ref{8} expresses the rate in terms of known concentrations.

Finding maximum velocity, \(V_{max}\)

Next, we examine some of the implications of the Michaelis-Menton rate law.

Recall Equation \ref{3}. That equation can be rearranged to show that \(K_M\) is an equilibrium constant:

\[ K_M=\frac{k_{-1}+k_2}{k_1}=\frac{[\mathrm{E}][\mathrm{S}]}{[\mathrm{ES}]} \label{9} \]

We see that \(K_M\) is an apparent dissociation constant of the complex (\(\ce{ES<=>[{k_{-1}}][{k_1}]E + S}\)). However, from Equation \ref{1}, we can see that the true dissociation constant is \(K_D = \dfrac{k_{–1}}{k_1}\). The \(K_M = K_D\) only if \(k_2 << k_{–1}\). This equality will be valid when most of the complex dissociates to \(\mathrm{E+S}\) rather than reacting to form product; that is to say, the initial reactants \(\mathrm{E}\) and \(\mathrm{S}\) are basically in equilibrium with the complex \(\mathrm{ES}\).

When the concentration of substrate (reactant) is so large that it pushes the equilibrium of Equation \ref{1} (\(\ce{E + S <=>[{k_{1}}][{k_{-1}}] ES}\)) far to the right, essentially all of the enzyme is in the form of the complex. Under these conditions, the reaction proceeds at its maximum rate:

\[ \text { Max. Rate }=\mathrm{V}_{\max }=k_2\left[\mathrm{E}_0\right] \; \; \; \; \text{when } [\mathrm{S}] >> K_M \label{10} \]

Under these conditions, increasing \([\mathrm{S}]\) does not affect the reaction rate.

On the other hand, when the substrate concentration is small enough that \([\mathrm{S}] < K_M\), then the rate varies linearly with substrate concentration and the rate can be expressed as:

\[\mathrm{V}=\frac{k_2\left[\mathrm{E}_{0}\right][\mathrm{S}]}{k_{\mathrm{M}}} \quad \text{when }[\mathrm{S}]<K_M \label{11} \]

Hence, the reaction rate depends upon substrate concentration in the manner shown in Figure \(\PageIndex{1}\).

Figure \(\PageIndex{1}\): Variation of Reaction Velocity with Substrate Concentration According to the Michaelis-Menton Model

If we substitute \(V_{max}=k_2 [\mathrm{E}_{0} ] \) in \ref{8} we obtain the following form of the Michaelis-Menten equation.

\[ \mathrm{V}=\frac{\mathrm{V}_{\max }[\mathrm{S}]}{K_M +[\mathrm{S}]} \label {12} \]

This equation is important because it accurately reproduces the velocity as a function of substrate concentration behavior observed for enzyme systems. Additionally, if one measures V as a function of substrate concentration it is possible to obtain values for \(K_M\) and \)(V_{max}\).

Extensions of Michaelis-Menton:

Lineweaver-Burk

We might also take the reciprocal of \ref{12};

\[ \frac{1}{V}=\frac{K_M}{V_{\max }}\left(\frac{1}{\mathrm{S}}\right)+\frac{1}{V_{\max }} \label{13} \]

This equation has the form of a straight line and is the basis for the Lineweaver-Burk plot, an example of which appears in Figure \(\PageIndex{2}\). If the enzyme concentration, pH, and temperature are held constant, then a plot of \(\dfrac{1}{[V]}\) vs. \(\dfrac{1}{[\mathrm{S}]}\) for different substrate concentrations should yield a line having an intercept at \(\dfrac{1}{V_{max}}\) and a slope of \(\dfrac{K_M}{V_{max}}\). Knowing \(V_{max}\) and \(\mathrm{E}_0\) we can calculate \(k_2\) from \ref{10}.

Figure \(\PageIndex{2}\): CLineweaver-Burk plot.

Related methods for treatment of data include the Hanes-Wolf plot and the Eadie-Hoftstee plot.

Hanes-Wolf

The equation of the Hanes-Wolf plot is obtained by multiplication of \ref{13} by \([\mathrm{S}]\)) and giving:

\[ \frac{[\mathrm{S}]}{\mathrm{V}}=\frac{k_{\mathrm{M}}}{\mathrm{V}_{\max }}+\frac{[\mathrm{S}]}{\mathrm{V}_{\max }} \label{14} \]

Eadie-Hofstee

The equation for the Eadie-Hofstee plot is found by multiplication of \ref{13} by \(V_{max}\):

\[ \mathrm{V}=\mathrm{V}_{\max }-k_{\mathrm{M}} \frac{\mathrm{V}}{[\mathrm{S}]} \label {15}\]

Tyrosinase Enzyme

The copper-containing enzyme tyrosinase (found in plant and animal tissues) consists of four identical subunits, each of molecular weight 32,000. This enzyme catalyzes the oxidation of the \(\alpha\)-amino acid, L-tyrosine, via a number of intermediates to a dark colored polyphenolic compound, melanin:

The first step is the oxidation of tyrosine to form L-DOPA (3,4-dihydroxyphenylalanine). This step occurs slowly in the presence of tyrosinase, and is autocatalyzed by the product, L-DOPA.

The second step is the enzyme-mediated oxidation of L-DOPA. This step proceeds in a fashion that is more typical of enzymes and in a manner consistent with the kinetic models described above.¹ Quinone spontaneously reacts to form Dopachrome.

The reaction of L-DOPA (second step above) can be measured by monitoring either the concentration the reactant (L-DOPA) or the formation products. We can monitor concentrations of these species spectrophotometrically using UV-visible light because tthe absorbance of light is related to concentration by the Lambert-Beer law :

\[ \text { Transmittance }=\mathrm{I} / \mathrm{I}_0=10^{-\varepsilon c \ell} \label{18} \]

Where

\(\mathrm{I}\) = intensity of light transmitted through the sample

\(\mathrm{I}_0\) = intensity of the incident light

\(\varepsilon\)= molar absorptivity of the absorbing species

\(\ell\) = path length of light through the sample

\(c\) = concentration of adsorbing species

The Beer-Lambert law may also be written as

\[ A=\varepsilon c \ell \label{19} \]

Where

\(A = \log(\mathrm{I}_0/\mathrm{I}) = -log (\text { Transmittance })\)

\(A\) = absorbance of the solution at a specific wavelength.

If multiple species (eg species 1, 2, and 3) all absorb at the specific wavelength measured, then the absorbance is equal to the sum of absorbance from each species (eg: A=A1+A2+A3).