Fundamentals of Biochemistry - A. A. Anisimov 1986

Enzymes
Kinetics of Enzymatic Reactions

3.7.1. Units of enzymatic activity. Kinetics is The Study of reaction rates and their dependence on various factors. The rate of an enzymatic reaction, like that of any chemical reaction, is determined by The amount of substances reacting per unit time under given conditions. The rate of an enzymatic reaction depends on enzyme activity, which can be expressed in various units. Until recently, the most widely accepted unit of activity was the so-called standard unit (U). It is defined as the amount of enzyme that catalyzes The conversion of 1 μM of substrate per 1 min under optimal conditions for the given enzyme (t°, pH, [S]). According to the latest international agreement, 1 unit of any enzyme is the amount of enzyme that, under specified conditions, catalyzes substrate conversion at a rate of 1 mol/s. This unit is called the katal (abbreviated as kat; 1 kat = 6·10⁷ standard units). Nano- and picokatals are commonly used to express enzyme activity.

There is also METABOLISM/2.html">THE CONCEPT OF specific activity, which is expressed as the number of units of enzyme activity per 1 mg of protein in an enzymatic preparation (U/mg). Currently, it is recommended to express specific activity in kat/kg. If the molecular mass of the enzyme is known, one can calculate the molecular activity (turnover number), which is characterized by the number of moles of substrate converted by 1 mole of enzyme per 1 min. When an enzyme has multiple active sites, the catalytic center activity is determined as the number of moles of substrate converted by one mole of catalytic centers (the molar concentration of the enzyme multiplied by the number of active sites in the enzyme molecule).

When determining enzyme activity, initial reaction rates (V) are measured in the stationary phase (Fig. 3.10). The reaction section preceding the ESTABLISHMENT OF THE steady state is called the transition phase. After the stationary phase, the reaction rate decreases.

This may be due to a decrease in the initial Substrate Concentration, accumulation of reaction products and their inhibition of the enzyme, Enzyme inactivation during incubation, and A number of other reasons.

3.7.2. Effect of enzyme and substrate concentration on the initial reaction rate. The dependence of the reaction rate on the Enzyme Concentration is linear (Fig. 3.11). Non-linear dependence at high enzyme concentrations is observed due to a shortage of substrate or activator, or aggregation of the enzyme protein molecules leading to the masking of active sites. The absence of a directly proportional relationship at low enzyme concentrations may result from the presence of toxic impurities in the incubation medium that bind to the enzyme and inactivate it.

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Fig. 3.10. Dependence of the enzymatic reaction rate on time:

I — transition phase. II — initial rate phase, III — main reaction phase

Fig. 3.11. Graph of the initial reaction rate (V) versus enzyme concentration [F]:

1 — normal, 2 — presence of trace amounts of impurities toxic to the enzyme in the system

Fig. 3.12. Dependence of the reaction rate (V) on substrate concentration [S]

In almost all cases, the graph of the initial reaction rate versus substrate concentration is a hyperbola (Fig. 3.12).

As the substrate concentration increases, the curve reaches a plateau, and the rate attains its maximum value (V_max). This indicates that all enzyme molecules (their active sites) are fully saturated with the substrate.

The hyperbolic plot of V versus [S] demonstrates that the reaction order changes during Enzymatic Catalysis. At low substrate concentrations, a first-order reaction takes place (V is proportional to [S]). At saturating substrate concentrations, the rate is independent of it, representing a zero-order reaction. At intermediate substrate concentrations, a mixed-order reaction occurs.

To describe the dependence of the initial reaction rate on substrate concentration, L. Michaelis and M. Menten derived an equation for a single-substrate reaction:

This equation was named the Michaelis–Menten Equation (K_S is the dissociation constant of the enzyme-substrate complex, equal to k_-1/k₁, and V_max is the maximum reaction rate). In deriving the equation, L. Michaelis and M. Menten based their work on the following General mechanism of a single-substrate reaction. The enzyme interacts with the substrate to form an enzyme-substrate complex (ES), which then dissociates into the free enzyme and the reaction product:

where k₁, k_-1, k₂, and k_-2 are the rate constants of the respective reaction steps.

The reverse reaction step — The formation of ES from E and P — is usually disregarded because initial reaction rates are considered, under which only a very small fraction of S is converted into P. In this case, virtually no ES is formed from P.

Later, J. Briggs and J. Haldane suggested writing the Michaelis–Menten equation in the following form:

where K_m is the Michaelis constant. K_m = (k_-1 + k₂)/k₁ = K_S + k₂/k₁.

This transformation of the equation stems from the fact that Michaelis and Menten initially assumed that The First stage of the reaction (E + S = ES) is in equilibrium, meaning that equilibrium is reached much faster than the ES complex can dissociate to form the reaction product. However, this is far from always valid. Although this stage proceeds quite rapidly and reversibly, it is not an equilibrium state. Because Enzymes exhibit high catalytic activity, under physiological conditions the system maintains a steady state where the concentration of the intermediate complex (ES) remains constant—that is, the rates of its formation and breakdown (including into the final product) are equal. This is accounted for by introducing K_m instead of K_s. K_m ≂ K_s only when k₂ ≪ k_-1.

The Michaelis constant (K_m) is an essential enzyme characteristic used to evaluate the enzyme's affinity for the substrate and its capacity to form reaction products. However, the true affinity of an enzyme for a substrate can only be determined from the value of K_s, which is experimentally much harder to measure than K_m. K_m is expressed in mol/L. A high K_m indicates that the E_S complex readily dissociates into its starting components, resulting in a slow reaction. Conversely, low K_m values (high k₊₁) correspond to fast reactions. For most single-substrate enzymatic reactions, K_m ranges from 1 ⋅ 10^-2 to 1 ⋅ 10^-5 mol/L. For a given enzyme and substrate under specific environmental parameters (pH, Temperature, etc.), K_m and V_max are constants. For enzymes with relative Specificity, each substrate is characterized by its own K_m value.

K_m is determined graphically from measurements of V at various substrate concentrations. In the specific case when

hence K_m = [S], meaning that the Michaelis constant equals the substrate concentration at which the reaction velocity is equal to half of the maximum velocity (see Fig. 3.12). Determining K_m accurately from the hyperbolic dependence of V on [S] is difficult because V_max is an asymptotic value and cannot be measured with sufficient precision. Consequently, several transformations of the Michaelis-Menten Equation have been proposed to yield a linear relationship between V and [S]. By inverting both sides of the Michaelis-Menten equation, H. Lineweaver and D. Burk derived the following relationship:

where K_m and V_max are constants under given experimental conditions, while V and [S] are variables. This is a linear equation (y = kx+b). The plot of 1/V versus 1/[S] is a straight line with a slope of K_m/V_max, intercepting the ordinate axis at 1/V_max and the abscissa axis at -1/K_m (Fig. 3.13). The value of K_m is determined by measuring the intercept on the abscissa axis.

Fig. 3.13. Lineweaver-Burk plot

Multiplying both sides of the Lineweaver-Burk equation by [S] yields a new linear equation, which is also used for the graphical Determination of k_m (Fig. 3.14):

Fig. 3.14. Plot of [S]/V versus [S] used to determine K_m and V_max

Fig. 3.15. Plot of V versus V/[S] (Eadie-Hofstee plot)

Multiplying both sides of the Lineweaver-Burk equation by V_max provides another form of the equation whose graphical representation is a straight line (Fig. 3.15):

A. Cornish-Bowden proposed the following linear plot for determining K_m and V_max:

Constructing this plot requires no calculations, and each line on the graph corresponds to a single observation (Fig. 3.16).

In deriving the equation for the dependence of reaction velocity on substrate concentration, L. Michaelis and M. Menten relied on a simplified scheme of enzymatic reactions. However, most enzymatic reactions are not single-substrate processes; furthermore, they involve the formation of multiple enzyme-substrate complexes: E + S ⇄ ES ⇄ ESP' ⇄ EP ⇄ E + P. The appearance of additional complexes does not alter the general form of the Michaelis-Menten equation, but K_m and V_max become more complex constants that incorporate a larger number of rate constants compared to single-step reactions.

Fig. 3.16. Linear plot of V_max versus K_m (Cornish-Bowden plot)

The rate equation for two-substrate reactions is also similar to the Michaelis-Menten equation: these reactions proceed via the formation of enzyme-substrate complexes, with each substrate characterized by its own K_m. As with single-substrate reactions, the Michaelis constant is determined graphically from the dependence of V on the concentration of one substrate at a constant, saturating concentration of the other. Alternatively, K_m can be determined with respect to an enzyme activator, provided that the substrate concentration is held fixed.

Plotting kinetic curves of V versus [S] is useful not only for determining K_m values but also for elucidating the mechanisms of two-substrate reactions and the type of inhibition (see Section 3.7.5).

3.7.3. Effect of temperature on the Rate of Enzymatic reaction. The rate of an enzymatic reaction strongly depends on temperature. The dependence of the reaction rate constant (k) on temperature (T) is expressed by the Arrhenius equation: 2.3 lgk = B—E_a/RT, where E_a is the activation energy (J/mol), B characterizes the collision frequency of molecules and their mutual orientation, T is the absolute temperature, and R is the gas constant. In the equation, E_a, B, and R are constant values under given conditions. Consequently, the plot of lgk versus 1/T should be linear (Fig. 3.17).

The slope of the line equals E_a/2.3R; therefore, the activation energy of the reaction can be determined from the temperature dependence of the reaction rate. More precisely, the graph yields only E_a of the rate-limiting step of the reaction, which is the slowest step, i.e., the step with the highest activation energy (Fig. 3.18).

Fig. 3.17. Effect of temperature on the rate constant (k) of an enzymatic reaction (dependence of lg k on the reciprocal of absolute temperature)

Fig. 3.18. Energy profile of an enzymatic reaction:

ES* — Transition State (activated complex—a short-lived association of E and S formed upon their approach, which then converts into the conventional ES complex—intermediate state); E_a — activation energy, P — reaction product; I, II — reaction steps (step II is the rate-limiting step)

The plot of lgk versus 1/T is linear only within certain limits. This is because, as the temperature rises, the rate of enzymatic reactions increases up to a certain maximum value; further temperature increases lead to a decline in the reaction rate due to the onset of thermal Denaturation of the enzyme (Fig. 3.19). When heated above 80°C, the vast majority of enzymes undergo complete denaturation.

The temperature optimum for most enzymes lies within the range of 40–60°C. Enzymes from thermophilic microorganisms exhibit high thermostability; some of them are even capable of withstanding brief boiling without a significant loss of activity. The thermostability of enzymes increases in the presence of substrate(s). In crystalline form, enzymes are more heat-resistant. Most enzymes preserve their activity well at low and sub-zero temperatures. Only a few cold-labile enzymes are inactivated upon cooling from 30 to 0°C.

3.7.4. Effect of pH on the rate of enzymatic reaction. The graphical dependence of the rate of most enzymatic reactions on pH has a bell-shaped curve (Fig. 3.20). The pH value at which the reaction rate is maximal is termed the pH optimum; any deviation of pH from this value in either direction decreases the reaction rate. Enzymatic reactions are sensitive to pH changes because enzymes contain A large number of ionizable groups. The ionization state of the groups most critical for enzyme function, which depends on pH, affects the catalytic activity of the enzyme, since the enzyme can exhibit maximal activity only in a specific state of these groups: ionized or unionized. In many cases, certain groups of the substrate are also capable of ionization, which likewise affects the pH optimum for enzyme action, as the presence of the substrate in a specific form is essential for the reaction.

Fig. 3.19. Dependence of the enzymatic reaction rate on temperature

Fig. 3.20. Effect of the active reaction of the medium on the rate of the enzymatic reaction

In addition, pH can exert an indirect effect on enzyme activity by destabilizing their Structure and disrupting the strength of the bond between the apoenzyme and the non-protein component. Changes in pH can also affect the state of enzyme activators and inhibitors. The pH value corresponding to the optimum does not always coincide with the pH characteristic of the intracellular environment. Therefore, pH can be one of the factors responsible for regulating enzymatic activity within The Cell.

3.7.5. Enzyme Inhibitors. The rate of an enzymatic reaction can be decreased, and in some cases completely halted, by the action of inhibitors—substances that, for one reason or another, partially or completely prevent the formation of a productive enzyme-substrate complex. In particular, the toxicity of many poisons to living organisms and the therapeutic effect of a number of drugs are due to their inhibitory action on enzymes. Inhibitors include both synthetic

substances and natural metabolites. Inhibitors are characterized by specificity of action. Irreversible (and non-specific) denaturation of enzymes under The Influence of heat, strong acids, or any other factors is generally not regarded as inhibition. Non-natural inhibitors include, for example, sodium fluoride—an inhibitor of Phosphatases; phenanthroline—of metal-containing enzymes; p-chloromercuribenzoate—of SH-enzymes; and diisopropyl fluorophosphate (DFP)—of Serine proteinases and esterases. The action of such inhibitors is usually irreversible.

Examples of reversible inhibitors include cellular metabolites and their structural analogues that reduce enzyme activity. There are Three types of reversible Enzyme Inhibition: competitive, non-competitive, and uncompetitive. Competitive Inhibitors are substances capable of binding to the Active Site of the enzyme and, consequently, competing with the substrate. The interaction of the enzyme with a competitive inhibitor is written as the equation: E + I ⇄ EI, where I is the inhibitor; EI is the enzyme-inhibitor complex that does not convert into a product; K_i is the inhibition constant, representing the dissociation constant of EI, K_i = [E][I]/[EI]. K_i reflects the affinity of the inhibitor for the enzyme; the smaller the K_i, the stronger the inhibitor.

As K_i → 0, inhibition becomes irreversible. The reaction rate in the presence of a competitive inhibitor depends on The ratio of [S] to [I]. An example of competitive inhibition is the action of malonic acid and a number of other dicarboxylic acids on succinate dehydrogenase (SDH). Substances that are structurally similar to the substrate act as competitive inhibitors. In the case of SDH, these are substances of the following structure:

Among them, malonic acid is the most potent inhibitor of SDH, reducing the enzymatic reaction rate by 50% at [I]/[E] = 50. An increase in substrate concentration leads to a decrease in the degree of inhibition, as the substrate displaces the inhibitor from the active site.

A non-competitive inhibitor binds to the enzyme outside the active site; therefore, no competitive relationship exists between the substrate and the inhibitor, and the degree of inhibition depends solely on the concentration of the latter. Upon binding to the enzyme, non-competitive inhibitors induce significant Changes in the enzyme conformation and the Spatial Structure of the active site, which disrupts normal substrate binding.

Uncompetitive inhibition is observed when the inhibitor reversibly interacts with the enzyme only after the formation of ES: ES + I ⇄ ESI. Uncompetitive inhibition is most characteristic of enzymes whose catalytic action proceeds via a "ping-pong" mechanism.

In addition to these types of reversible inhibition, a mixed type of inhibition is also distinguished.

The type of inhibition can be identified through kinetic analysis by studying the dependence of V on [S] at various inhibitor concentrations (Fig. 3.21). The Effect of a competitive inhibitor on the reaction rate is that it increases K_m while leaving V_max unchanged. Uncompetitive inhibition is characterized by a decrease in both V_max and K_m to the same extent, resulting in a constant V_max/K_m ratio.

Non-competitive inhibitors affect enzyme activity by decreasing the V_mах value while leaving К_m unchanged.

When plotting V_mах against К_m, the type of inhibition is characterized by a shift in the intersection point: to the right for competitive inhibition, toward the origin for uncompetitive inhibition, and an intermediate shift for mixed inhibition.

Many pharmacologically active compounds alter metabolic pathways by acting on enzymes. Some of these compounds act as competitive inhibitors due to structural similarities with natural metabolites, and are therefore referred to as antimetabolites. For instance, synthetic antibacterial agents known as sulfonamides compete with p-aminobenzoic acid (PABA). Bacteria utilize PABA to synthesize Folic acid, an essential growth factor. Sulfonamides inhibit the enzymatic step in folic acid synthesis by competing with PABA for the active site of the enzyme. This inhibition of folic acid synthesis by sulfonamides results in a bacteriostatic effect. Because humans do not synthesize folic acid from PABA and other metabolites (see Section 10.3), therapeutic doses of sulfonamides do not significantly affect The Human Body. However, they do suppress beneficial gut microflora, which serves as a source of various Vitamins for humans; consequently, Treatment with sulfonamides (such as sulfadimezine, ethazole, sulfamonomethoxine, etc.) is typically accompanied by multivitamin intake.

Fig. 3.21. Eadie-Hofstee (left) and Lineweaver-Burk (right) plots for various types of inhibition: a — competitive, b — uncompetitive, c — non-competitive