Enzyme Reaction Kinetics

Author: Polar Glacier

Reviewer: Weiming

  Note: This is a long article, and some readers may find parts of it difficult. It offers a substantial treatment of the subject and is recommended for students, enthusiasts, and readers in related fields.

I. Common equations in chemical kinetics

  Before discussing enzyme kinetics, let us review several common cases in chemical reaction kinetics.

1. First-order reactions

  A first-order reaction is one whose rate is proportional to the first power of the reactant concentration.

  The reaction is shown below:

  Its rate equation can therefore be written as:

  From the definition of a first-order reaction:

  Therefore:

  Rearranging:

  Solving gives:

2. Second-order reactions

  In a second-order reaction, the reaction rate is proportional to the square of the reactant concentration. A typical form is:

  For the reaction:

  At any point during the reaction:

  The rate equation is:

  Combining the equations gives:

  Solving gives:

  For the reaction:

  There are two cases. If reactants A and B have the same initial concentration, the derivation is essentially the same as above and gives:

  If A and B have different initial concentrations:

  Therefore:

  For reactions of integer order three or greater, the rate equation is derived in much the same way as for a simple second-order reaction.

3. Zero-order reactions

  A zero-order reaction has a constant rate that is independent of reactant concentration. Such reactions are uncommon, although some reactions on solid surfaces are zero order. One example is the decomposition of ammonia on catalysts such as tungsten or iron.

  When reactant molecules adsorb onto a solid catalyst and their concentration is high enough to cover the surface completely, the rate depends only on the catalyst’s surface area, not on the reactant concentration.

  The rate equation can be written as:

  With those basic equations reviewed, we can turn to enzyme kinetics.

II. The intermediate-complex hypothesis and derivation of the Michaelis–Menten equation

  It has long been known from practical chemical production that reaction rates rise as reactant concentrations increase. In 1903, Victor Henri of the Sorbonne in Paris studied the hydrolysis of sucrose by invertase. Holding the invertase concentration constant while varying the sucrose concentration, he observed a rectangular-hyperbolic relationship—the familiar graph found in secondary-school textbooks:

  The graph shows that at very low substrate concentration, the reaction rate rises sharply and in direct proportion to substrate concentration, so the reaction is first order. As substrate concentration increases, the rate no longer rises proportionally and each further increase becomes smaller. At still higher concentrations, the rate stops increasing and the reaction behaves as zero order. The enzyme is then saturated with substrate. All enzymes show saturation, although the required substrate concentration differs.

  Henri used these results to propose that an enzyme and its substrate combine to form an unstable intermediate during conversion of substrate to product. This intermediate forms readily and also breaks down readily, releasing product and regenerating the enzyme. This is the intermediate-complex hypothesis. Henri also proposed a preliminary equation for the invertase reaction:

  Here K, Φ, m, and n are constants, v is the reaction rate, [S] is the substrate concentration, and [P] is the product concentration.

  From a modern perspective, this equation describes substrate inhibition, a typical form of enzyme inhibition.

  In 1913, L. Michaelis and M. Menten reexamined Henri’s results. Starting from the intermediate-complex hypothesis, they considered a rapid-equilibrium process, derived a new mathematical expression, and published the results in Biochemische Zeitschrift (the Journal of Biochemistry, a predecessor of the FEBS Journal).

  They proposed a rapid-equilibrium model for a single-substrate enzyme reaction. The process can be written as:

  Here E is free enzyme, S is substrate, ES is the unstable intermediate, and P is product.

Michaelis (left) and Menten (right)

  From this model they obtained the rate equation for an irreversible, single-substrate enzyme reaction: the well-known Michaelis–Menten equation.

  Here V₀ is the initial rate, V_max is the maximum rate, [S] is the substrate concentration, and K_s is the dissociation constant, equal to K₂/K₁.

  The derivation of the Michaelis–Menten equation rests on three assumptions:

• (1) At the beginning of the reaction, very little product has formed, so the reverse reaction can be ignored.

• (2) The substrate concentration is much greater than the concentration of free enzyme and therefore remains approximately constant during the reaction.

• (3) The second step is rate-limiting, with K₂ ≫ K₃, so the conversion of ES to P is too slow to disrupt the equilibrium between E and ES.

  The derivation proceeds as follows:

  Assume that E + S → ES rapidly reaches equilibrium and that the second step is much slower than the first. The overall rate is then determined by the second step:

  Because [ES] is small, unstable, and difficult to measure experimentally, it must be expressed in terms of other measurable quantities. For the dissociation of ES into E and S during the rapid first-step equilibrium, the rate constant is:

  Thus:

  The total amount of enzyme remains constant during the reaction, giving the enzyme-conservation equation:

  Therefore:

  Substitute the preceding expression into

  to obtain:

  Rearranging gives:

  Substitute

  into the preceding equation to obtain:

  The maximum rate occurs when all enzyme molecules are bound to substrate and participating in the reaction. Therefore:

  Substitution gives:

  Although this model was a major advance in explaining enzyme kinetics, its assumptions are not universally valid. There is no reason to expect K₂ ≫ K₃ in every enzyme reaction, and the model does not allow for reversibility in the second step. In 1925, G. E. Briggs and J. B. S. Haldane therefore revised the Michaelis–Menten model and proposed the steady-state theory of enzyme kinetics.

  In the steady-state theory, Briggs made the following assumptions:

• (1) At the beginning of the reaction, product concentration is extremely low. The rate of E + S → ES in the second step is therefore negligible, and this step can be ignored.

• (2) Initially, substrate concentration is much greater than enzyme concentration and may be treated as constant during the early reaction.

• (3) After a very short interval of a few milliseconds, [ES] becomes approximately constant. Although the concentrations of S and P continue to change, the formation rate v_f and decomposition rate v_d of ES remain nearly equal for a period of time, so the net rate of ES formation is approximately zero.

  The mathematical derivation is as follows:

  The formation rate v_f of the enzyme–substrate complex ES is:

  The decomposition rate v_d of ES is:

  At steady state, [ES] remains constant and v_f = v_d:

  From the enzyme identity

  we obtain:

  Substitution gives:

  Rearranging gives:

  Let:

  Then:

  Solving for [ES] gives:

  Substitute this into

  to obtain:

  The steady-state and rapid-equilibrium models produce equations with the same mathematical form, but the steady-state model is more general. In honor of Michaelis and Menten, both expressions are called the Michaelis–Menten equation, and K_m is called the Michaelis constant.

  Now return to Henri’s graph.

  A simple calculation shows that:

  The rate is proportional to the first power of substrate concentration, so the reaction is first order.

  The rate is constant, so the reaction is zero order.

  When substrate concentration lies between 0.01K_m and 100K_m, the rate follows the Michaelis–Menten equation and the reaction has mixed order.

  This agrees with the experimental results.

  When [S] = K_m,

  the Michaelis constant is therefore the substrate concentration at which the reaction reaches half its maximum rate.

III. The Michaelis constant

  The Michaelis constant also has several other implications:

• (1) It is an important quantity in enzymology. From the derivation of K_m:

  Every term in this expression is constant for a given enzyme under specified conditions, so the Michaelis constant has a definite value under those conditions. It is one of an enzyme’s characteristic constants: it depends on the enzyme and the type of substrate, but not on enzyme concentration. Different enzymes have different values of K_m.

• (2) An enzyme with broad substrate specificity has a different K_m value for each substrate. The substrate with the smallest value is the enzyme’s optimal substrate. For chymotrypsin, for example, benzoyltyrosinamide is the optimal substrate.

  Why? K_m is the substrate concentration that produces half the maximum rate. A smaller K_m means the substrate saturates the enzyme more readily and the rate rises faster for a unit increase in substrate concentration; that is, v is more sensitive to Δs.

  A close comparison of the two models used to derive the Michaelis–Menten equation reveals another point.

  Consider the reaction:

  In the rapid-equilibrium model, the equilibrium constant K_s is:

  In the steady-state model, the Michaelis constant K_m is:

  Comparing the two expressions:

  When k₁ ≫* k₂*, so that ES → P + E is an extremely slow step in the overall reaction:

  Thus, the rapid-equilibrium model is a special case of the steady-state model. When ES → P + E is extremely slow and k₂/k₁ is very small, steady state becomes effectively equivalent to rapid equilibrium. Viewed another way, steady state adds a slow equilibrium to a rapid one: steady state = rapid equilibrium + slow equilibrium.

IV. Linearizing the Michaelis–Menten equation

  The Michaelis–Menten equation is the classic biochemical expression relating the rate of an enzyme-catalyzed reaction to its initial substrate concentration. Its hyperbolic curve can introduce substantial error during fitting. A convenient response is to rearrange the equation into a linear form. Common approaches include the Lineweaver–Burk double-reciprocal plot, Eadie–Hofstee plot, Hanes–Woolf plot, and Eisenthal plot. The Lineweaver–Burk plot is the most widely used.

  The Lineweaver–Burk double-reciprocal plot was proposed by Hans Lineweaver and Dean Burk in 1934. It is obtained by taking the reciprocal of both sides of the Michaelis–Menten equation and rearranging:

  This plot gives the two important Michaelis–Menten parameters directly and can also be applied to inhibited enzyme reactions. Its weakness is that experimental points cluster excessively in the lower-left part of the line, while points at low concentration acquire large errors after reciprocation and often lie far from the line. This reduces the accuracy of both parameter estimates. The Eadie–Hofstee or Hanes–Woolf method can reduce the errors of the Lineweaver–Burk method at very low or very high substrate concentrations.

V. The Michaelis–Menten equation under reversible inhibition

  Inhibition occurs when an inhibitor acts on groups essential to an enzyme and reduces or abolishes its activity without denaturing it. The effect may resemble denaturation, but the mechanisms differ. Denaturation disrupts the secondary interactions that maintain the enzyme’s conformation—hydrogen bonds, hydrophobic interactions, van der Waals forces, and sometimes disulfide bonds—causing loss of three-dimensional structure and biological activity. Inhibition instead targets particular reactive groups. Diisopropyl fluorophosphate, for example, forms a phosphate ester bond at an active-site hydroxyl group in proteases and esterases, blocking normal activity; organomercury compounds and halogenated alkylating agents can react with sulfhydryl groups. The Chinese source identifies that hydroxyl as belonging to tryptophan; biochemically, the classic target is the active-site serine residue.

  During inhibition, the enzyme’s overall conformation and secondary interactions remain intact. Only particular functional groups are modified, reducing enzyme activity.

  According to how tightly an inhibitor binds to functional groups in the enzyme, inhibition may be irreversible or reversible. Reversible inhibition is divided by mechanism into competitive, noncompetitive, and uncompetitive inhibition.

(1) Competitive inhibition

  As the name suggests, competitive inhibition involves competition. The inhibitor competes with the substrate for the enzyme’s active site and binds reversibly to free enzyme, forming an enzyme–inhibitor complex. By occupying the active site, it prevents substrate binding and lowers the reaction rate.

  A competitive inhibitor resembles the substrate in structure. Malonate, for example, competes with succinate for the active site of succinate dehydrogenase. The enzyme may therefore bind the inhibitor instead, or steric hindrance may block normal substrate binding. Inhibitor-bound enzyme can neither bind the correct substrate nor catalyze the normal reaction. The effective enzyme concentration falls, which appears macroscopically as reduced enzyme activity.

  Succinic acid, also called butanedioic acid or ethane-1,2-dicarboxylic acid, has the molecular formula C₄H₆O₄. Succinate dehydrogenase converts it to fumaric acid.

  Competitive inhibition can be viewed as two simultaneous reactions involving enzyme E:

  These are parallel reactions. From chemical-equilibrium principles, increasing the concentration of substrate S shifts the equilibrium of

  to the right, while the reaction

  is suppressed. Increasing [S] can therefore overcome inhibition by inhibitor I.

  The rate equation is derived as follows:

  Substitute

  into the expression for K_i to obtain:

  From the Michaelis hypothesis:

  Substitute into the enzyme-conservation equation

  to obtain:

  Substitute into

  to obtain:

  Rearranging gives:

  Mathematically, the degree of competitive inhibition is directly proportional to inhibitor concentration [I] and inversely proportional to substrate concentration [S]. At fixed [I] and [S], inhibition is stronger when K_i is smaller and K_m is larger.

  The Lineweaver–Burk plot shows the characteristic kinetics of competitive inhibition: the maximum rate v_max is unchanged, while the apparent Michaelis constant K_m increases.

(2) Noncompetitive inhibition

  A noncompetitive inhibitor binds to a group outside the active site and changes the enzyme’s conformation so it can no longer catalyze formation of product. It can bind either free enzyme or the enzyme–substrate complex, forming the ternary EIS complex. Binding does not prevent substrate binding, but it stops the enzyme–substrate complex from proceeding to product, so enzyme activity falls. Inhibition by some heavy-metal ions, including Cu²⁺, Pb²⁺, and Hg²⁺, belongs to this category.

  From

  we obtain:

  Similarly, from

  we obtain:

  From

  solving gives:

  Substitute the preceding results into:

  Rearranging gives:

  The double-reciprocal equation is:

  The Michaelis–Menten equation for noncompetitive inhibition shows that the degree of inhibition depends only on inhibitor concentration [I] and K_i, not on substrate concentration [S] or the substrate’s Michaelis constant K_m. The graph shows that under noncompetitive inhibition, K_m remains unchanged,

(3) Uncompetitive inhibition

  In contrast to competitive inhibition, an uncompetitive inhibitor binds only to the enzyme–substrate complex, forming a ternary complex that cannot release product. The effective enzyme concentration and enzyme activity both fall. This mechanism is common in multisubstrate enzyme reactions.

  The rate equation is:

  The double-reciprocal equation is:

  The derivation is similar to that for competitive inhibition and is omitted here.

  The kinetic signature of uncompetitive inhibition is a proportional decrease in v_max and K_m, producing a family of parallel lines. The degree of inhibition depends on K_i, K_m, [S], and [I]. It is directly proportional to both [S] and [I]. At fixed [I] and [S], inhibition decreases as either K_i or K_m increases.

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