Week 5: Enzyme Kinetics, Inhibition Mechanisms, and Allosteric Regulation

Fundamentals of Enzymology and Catalysis

• Definition of Catalysts:

  • Catalysts are chemical agents that accelerate the rates of chemical reactions without themselves undergoing any permanent chemical modification or being consumed by the end of the reaction.

  • Most biological catalysts are proteins (enzymes), though catalytic ribonucleic acids (ribozymes) also exist.

• Thermodynamic Principles:

  • Enzymes do not alter the chemical equilibrium of a reaction.

  • The position of the final equilibrium point remains identical whether an enzyme is present or absent.

  • The acceleration achieved by an enzyme ensures that thermodynamic equilibrium is attained significantly faster than in the uncatalyzed process.

• Magnitude of Rate Enhancements:

  • Biological reactions in the absence of catalysts possess extremely long half-lives, ranging from days to millions of years.

  • Enzymes accelerate reaction rates by factors typically between $10^6$ and $10^{17}$.

  • Experimental rate enhancement data compiled by A. Radzicka and R. Wolenden (Science 267:90–93, 1995):

  • OMP (Orotidine monophosphate) decarboxylase:

    • Nonenzymatic half-life: $78,000,000\,\text{years}$

    • Uncatalyzed rate ($k_{\text{un}}$!): $2.8 \times 10^{-16}\,\text{s}^{-1}$

    • Catalyzed rate ($k_{\text{cat}}$!): $39\,\text{s}^{-1}$

    • Rate enhancement ($k_{\text{cat}} / k_{\text{un}}$!): $1.4 \times 10^{17}$

  • Staphylococcal nuclease:

    • Nonenzymatic half-life: $130,000\,\text{years}$

    • Uncatalyzed rate ($k_{\text{un}}$!): $1.7 \times 10^{-13}\,\text{s}^{-1}$

    • Catalyzed rate ($k_{\text{cat}}$!): $95\,\text{s}^{-1}$

    • Rate enhancement ($k_{\text{cat}} / k_{\text{un}}$!): $5.6 \times 10^{14}$

  • AMP (Adenosine monophosphate) nucleosidase:

    • Nonenzymatic half-life: $69,000\,\text{years}$

    • Uncatalyzed rate ($k_{\text{un}}$!): $1.0 \times 10^{-11}\,\text{s}^{-1}$

    • Catalyzed rate ($k_{\text{cat}}$!): $60\,\text{s}^{-1}$

    • Rate enhancement ($k_{\text{cat}} / k_{\text{un}}$!): $6.0 \times 10^{12}$

  • Carboxypeptidase A:

    • Nonenzymatic half-life: $7.3\,\text{years}$

    • Uncatalyzed rate ($k_{\text{un}}$!): $3.0 \times 10^{-9}\,\text{s}^{-1}$

    • Catalyzed rate ($k_{\text{cat}}$!): $578\,\text{s}^{-1}$

    • Rate enhancement ($k_{\text{cat}} / k_{\text{un}}$!): $1.9 \times 10^{11}$

  • Ketosteroid isomerase:

    • Nonenzymatic half-life: $7\,\text{weeks}$

    • Uncatalyzed rate ($k_{\text{un}}$!): $1.7 \times 10^{-7}\,\text{s}^{-1}$

    • Catalyzed rate ($k_{\text{cat}}$!): $66,000\,\text{s}^{-1}$

    • Rate enhancement ($k_{\text{cat}} / k_{\text{un}}$!): $3.9 \times 10^{11}$

  • Triose phosphate isomerase:

    • Nonenzymatic half-life: $1.9\,\text{days}$

    • Uncatalyzed rate ($k_{\text{un}}$!): $4.3 \times 10^{-6}\,\text{s}^{-1}$

    • Catalyzed rate ($k_{\text{cat}}$!): $4,300\,\text{s}^{-1}$

    • Rate enhancement ($k_{\text{cat}} / k_{\text{un}}$!): $1.0 \times 10^{9}$

  • Chorismate mutase:

    • Nonenzymatic half-life: $7.4\,\text{hours}$

    • Uncatalyzed rate ($k_{\text{un}}$!): $2.6 \times 10^{-5}\,\text{s}^{-1}$

    • Catalyzed rate ($k_{\text{cat}}$!): $50\,\text{s}^{-1}$

    • Rate enhancement ($k_{\text{cat}} / k_{\text{un}}$!): $1.9 \times 10^{6}$

  • Carbonic anhydrase:

    • Nonenzymatic half-life: $5\,\text{seconds}$

    • Uncatalyzed rate ($k_{\text{un}}$!): $1.3 \times 10^{-1}\,\text{s}^{-1}$

    • Catalyzed rate ($k_{\text{cat}}$!): $1 \times 10^{6}\,\text{s}^{-1}$

    • Rate enhancement ($k_{\text{cat}} / k_{\text{un}}$!): $7.7 \times 10^{6}$

Enzyme Specificity and Binding Models

• Active Site Organization:

  • The active site is a specialized three-dimensional crevice or pocket formed by specific amino acid residues distributed throughout the primary sequence of the enzyme protein.

  • Substrates bind within the active site via noncovalent interactions, placing functional catalytic groups in precise stereochemical proximity to target chemical bonds.

• Enzyme Specificity:

  • Specificity represents the capacity of an enzyme to selectively distinguish its cognate substrate from closely related molecular analogues.

  • Proteolytic Cleavage Examples:

  • Trypsin: A digestive protease that hydrolyzes peptide bonds specifically on the carboxyl side of Lysine ($\text{Lys}$) or Arginine ($\text{Arg}$) residues; cleavage is strictly prevented if the subsequent residue is Proline ($\text{Pro}$).

  • Thrombin: A critical blood-clotting endopeptidase possessing stringent specificity; it cleaves Arginine-Glycine ($\text{Arg-Gly}$) bonds exclusively within the extended sequence: $\text{Lys-Val-Pro-Arg-Gly-Ser}$.

• Models of Enzyme-Substrate Binding:

  • Lock and Key Model (Emil Fischer, 1890):

  • Proposes that the unbound active site possesses a static, rigid geometry complementary to the conformation of the substrate prior to binding.

  • The substrate fits into the preformed active site in a complementary manner analogous to a key fitting into a lock.

  • Induced Fit Model (Daniel E. Koshland Jr, 1958):

  • Posits that the active site is structurally flexible rather than rigid.

  • Initial noncovalent contact with the substrate induces a dynamic conformational change in the enzyme, aligning key catalytic amino acid residues precisely around the substrate to form the functional enzyme-substrate ($\text{ES}$) complex.

Chemical Kinetics Principles

• Reaction Velocity Definitions:

  • For a fundamental reaction converting substrate $\text{S}$ to product $\text{P}$ ($\text{S} \rightarrow \text{P}$):

  • Velocity ($V$) is mathematically defined as the rate of disappearance of substrate or appearance of product over time ($t$):   $V = -\frac{\Delta[\text{S}]}{\Delta t} = \frac{\Delta[\text{P}]}{\Delta t}$

• Reaction Order:

  • First-Order Reactions:

  • The reaction velocity is directly proportional to the concentration of a single reactant:     $V = k[\text{S}]$

  • The rate constant $k$ has units of $\text{s}^{-1}$.

  • Second-Order Reactions:

  • Reactions requiring two colliding reactant molecules:     $2\text{S} \rightarrow \text{P} \quad \implies \quad V = k[\text{S}]^2$     $\text{S}_1 + \text{S}_2 \rightarrow \text{P} \quad \implies \quad V = k[\text{S}_1][\text{S}_2]$

  • The rate constant $k$ has units of $\text{M}^{-1}\,\text{s}^{-1}$.

  • Pseudo-First-Order Reactions:

  • Occurs when a second-order reaction involves two substrates, but one substrate is maintained at a vast stoichiometric excess relative to the other.

  • The concentration of the excess reactant changes negligibly during the course of the reaction, causing the reaction velocity to appear directly proportional strictly to the concentration of the limiting reactant.

  • Most in vivo enzymatic reactions follow pseudo-first-order kinetics (e.g., hydrolytic reactions where water is a substrate present in massive excess).

Michaelis-Menten Kinetic Theory

• Reaction Scheme and History:

  • Formulated in 1913 by Leonor Michaelis and Maud Menten to describe single-substrate enzyme kinetics:   $\text{E} + \text{S} \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \text{ES} \underset{k_{-2}}{\overset{k_2}{\rightleftharpoons}} \text{EP} \rightleftharpoons \text{E} + \text{P}$

  • Simplified two-step progression:   $\text{E} + \text{S} \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \text{ES} \overset{k_2}{\longrightarrow} \text{E} + \text{P}$

• Initial Velocity Assumption ($V_0$):

  • Measurements are conducted during the very initial phase of the reaction, when substrate has barely been depleted and product concentration ($[\text{P}]$) is approximately zero.

  • Under these initial conditions, the reverse reaction converting product back to intermediate ($\text{E} + \text{P} \rightarrow \text{ES}$) is negligible:   $k_{-2}[\text{E}][\text{P}] = 0$

• Rate-Limiting Step and Kinetic Parameters:

  • Product synthesis from the $\text{ES}$ complex is slower than the reversible association/dissociation of $\text{E}$ and $\text{S}$.

  • The second chemical step ($\text{ES} \overset{k_2}{\rightarrow} \text{E} + \text{P}$) represents the rate-limiting step; thus, overall velocity depends directly on the concentration of the intermediate complex ($[\text{ES}]$):   $V_0 = k_2[\text{ES}]$

  • Catalytic Constant ($k_{\text{cat}}$!):

  • $k_2$ corresponds to the catalytic constant ($k_{\text{cat}}$), also termed the turnover number.

  • Equivalent to the maximal number of substrate molecules converted to product per unit of time by a single enzyme molecule when the enzyme is completely saturated with substrate.

  • Maximal Velocity ($V_{\text{max}}$!):

  • The maximum initial velocity reached when all active sites on the total enzyme population are occupied by substrate ($[\text{ES}] = [\text{E}]_{\text{total}}$).

• Michaelis Constant ($K_M$):

  • Mathematically defined in terms of individual elementary rate constants:   $K_M = \frac{k_{-1} + k_2}{k_1}$

  • Operational definition: the specific substrate concentration ($[\text{S}]$) at which the initial reaction velocity ($V_0$) is exactly half of maximal velocity ($V_0 = \frac{1}{2}V_{\text{max}}$).

  • Represents the substrate concentration at which exactly half ($50\%$) of the available enzyme binding sites are filled.

  • Provides a physiological index of the substrate concentration necessary for substantial catalysis to proceed.

  • Quantitative interpretation:

  • Low $K_M$ indicates high affinity of the enzyme for its substrate (strong binding; small concentration needed to half-saturate active sites).

  • High $K_M$ indicates low affinity (weak binding; large concentration required to half-saturate active sites).

  • Dependence: value depends on the identity of the substrate, temperature, $\text{pH}$, and solution ionic strength.

• Representative $K_M$ Values (Lehninger Table 6-6):

  • Hexokinase (brain):

  • Substrate ATP: $K_M = 0.4\,\text{mM}$

  • Substrate D-Glucose: $K_M = 0.05\,\text{mM}$

  • Substrate D-Fructose: $K_M = 1.5\,\text{mM}$

  • Carbonic anhydrase:

  • Substrate $\text{HCO}_3^-$: $K_M = 26\,\text{mM}$

  • Chymotrypsin:

  • Substrate Glycyltyrosinylglycine: $K_M = 108\,\text{mM}$

  • Substrate N-Benzoyltyrosinamide: $K_M = 2.5\,\text{mM}$

  • $\beta$-Galactosidase:

  • Substrate D-Lactose: $K_M = 4.0\,\text{mM}$

  • Threonine dehydratase:

  • Substrate L-Threonine: $K_M = 5.0\,\text{mM}$

The Lineweaver-Burk Double Reciprocal Transformation

• Equation Derivation:

  • Taking the algebraic reciprocal of the hyperbolic Michaelis-Menten equation yields the linear Lineweaver-Burk expression:   $\frac{1}{V_0} = \left(\frac{K_M}{V_{\text{max}}}\right)\frac{1}{[\text{S}]} + \frac{1}{V_{\text{max}}}$

  • Conforms to the standard equation for a straight line: $y = mx + b$.

• Key Graphical Coordinates:

  • Independent variable ($x$-axis): $\frac{1}{[\text{S}]}$

  • Dependent variable ($y$-axis): $\frac{1}{V_0}$

  • Slope ($m$): $\frac{K_M}{V_{\text{max}}}$

  • $y$-intercept ($b$): $\frac{1}{V_{\text{max}}}$

  • $x$-intercept (where $\frac{1}{V_0} = 0$): $-\frac{1}{K_M}$

Enzyme Inhibition Principles

• Taxonomy of Inhibitors:

  • Enzyme inhibitors decrease the catalytic rate of an enzymatic reaction.

  • Classified broadly into irreversible and reversible modalities:

  • Irreversible Inhibitors

  • Reversible Inhibitors:

    • Competitive Inhibitors

    • Uncompetitive Inhibitors

    • Noncompetitive Inhibitors

      

• Irreversible Inhibition:

  • Dissociates extremely slowly from the target enzyme because it becomes tightly bound via covalent modifications or extremely strong noncovalent associations.

  • Causes permanent inactivation of the specific enzyme molecules bound.

  • Classical Examples:

  • Penicillin: Irreversibly inactivates the bacterial enzyme transpeptidase via covalent acylation of an active-site serine, blocking peptidoglycan cross-linking in the bacterial cell wall, resulting in cell lysis and death.

  • Aspirin (Acetylsalicylic acid): Covalently modifies cyclooxygenase (COX) through transesterification/acetylation, suppressing the enzymatic biosynthesis of pro-inflammatory prostaglandins and thromboxanes.

  • Suicide Inhibitors (Mechanism-Based Inactivators):

  • A specialized class of irreversible inhibitors structurally resembling the normal substrate.

  • Binds reversibly to the enzyme active site and initiates the standard initial catalytic steps.

  • Instead of being converted into normal product, the catalytic machinery transforms the inhibitor molecule into an exceptionally reactive chemical intermediate.

  • This reactive intermediate immediately forms an irreversible covalent bond with active site amino acid residues, permanently disabling the enzyme.

  • Example: Fluoro-uracil (5-Fluorouracil, 5-FU), a suicide inhibitor used extensively as an anticancer chemotherapy agent.

• Reversible Inhibition:

  • Characterized by rapid, noncovalent equilibrium binding and dissociation between the enzyme and inhibitor.

  • Activity of the enzyme is restored upon dilution or physical removal of the inhibitor.

Reversible Inhibition Mechanisms and Kinetics

• Competitive Inhibition:

  • Mechanism of Action:

  • The competitive inhibitor ($\text{I}$) possesses structural similarity to the natural substrate ($\text{S}$).

  • Competes directly with substrate for occupancy of the active site.

  • Prevents substrate binding by generating a dead-end enzyme-inhibitor ($\text{EI}$) complex:     $\text{E} + \text{I} \underset{K_I}{\rightleftharpoons} \text{EI}$

  • The equilibrium dissociation constant for the inhibitor is given by:     $K_I = \frac{[\text{E}][\text{I}]}{[\text{EI}]}$

  • Overcoming Competitive Inhibition:

  • Can be completely outcompeted by increasing substrate concentration ($[\text{S}]$); at saturating levels of substrate, all active sites remain occupied by $\text{S}$.

  • Kinetic Consequences:

  • $V_{\text{max}}$ remains unchanged because infinitely high substrate concentrations outcompete the inhibitor.

  • Apparent $K_M$ increases ($K_{M,\text{apparent}} > K_M$); more substrate is required to achieve half-maximal velocity.

  • Lineweaver-Burk Diagnostic Profile:

  • Lines for uninhibited and competitively inhibited enzyme intersect directly on the $y$-axis at identical $\frac{1}{V_{\text{max}}}$ values.

  • The slope ($\frac{K_M}{V_{\text{max}}}$) increases in the presence of inhibitor.

  • The $x$-intercept ($-\frac{1}{K_{M,\text{apparent}}}$) moves closer to the origin (becomes less negative).

• Uncompetitive Inhibition:

  • Mechanism of Action:

  • The inhibitor binds exclusively to an allosteric pocket that is created or exposed only after substrate is bound to the active site.

  • Does not bind to the free enzyme ($\text{E}$); binds strictly to the enzyme-substrate ($\text{ES}$) complex to produce an inactive enzyme-substrate-inhibitor ($\text{ESI}$) complex:     $\text{ES} + \text{I} \underset{K_I'}{\rightleftharpoons} \text{ESI}$

  • The ternary $\text{ESI}$ complex is catalytically non-productive and cannot form product.

  • Kinetic Consequences:

  • Cannot be overcome by increasing substrate concentration.

  • $V_{\text{max}}$ decreases to an apparent lower value ($V_{\text{max,apparent}} < V_{\text{max}}$) because a fraction of the enzyme is trapped in the non-functional $\text{ESI}$ complex.

  • Apparent $K_M$ decreases by the exact same mathematical proportion as $V_{\text{max}}$ ($K_{M,\text{apparent}} < K_M$) due to Le Chatelier depletion of the $\text{ES}$ pool, which pulls the $\text{E} + \text{S} \rightleftharpoons \text{ES}$ equilibrium forward (reflecting an apparent increase in substrate binding affinity).

  • Lineweaver-Burk Diagnostic Profile:

  • Generates a series of parallel lines when comparing inhibited and uninhibited states.

  • The slope ($\frac{K_M}{V_{\text{max}}}$) remains constant because $K_M$ and $V_{\text{max}}$ decrease by the identical ratio.

  • The $y$-intercept ($\frac{1}{V_{\text{max,apparent}}}$) shifts higher up the axis.

  • The $x$-intercept ($-\frac{1}{K_{M,\text{apparent}}}$) shifts further to the left (becomes more negative).

• Noncompetitive Inhibition:

  • Mechanism of Action:

  • The inhibitor binds to a regulatory/allosteric site distinct from the active catalytic pocket.

  • Exhibits equal affinity for both the free enzyme ($\text{E}$) and the enzyme-substrate complex ($\text{ES}$):     $\text{E} + \text{I} \rightleftharpoons \text{EI} \quad \text{and} \quad \text{ES} + \text{I} \rightleftharpoons \text{ESI}$

  • Substrate binding is not prevented; the enzyme can simultaneously bind both substrate and inhibitor.

  • However, the resulting $\text{ESI}$ complex is catalytically inactive and cannot progress to product.

  • Kinetic Consequences:

  • Cannot be overcome by increasing substrate concentration.

  • $V_{\text{max}}$ decreases to a new, lower apparent value ($V_{\text{max,apparent}} < V_{\text{max}}$) as active enzyme is converted to non-productive forms.

  • $K_M$ remains unchanged because inhibitor binding does not disrupt substrate affinity for the remaining free catalytic sites.

  • Lineweaver-Burk Diagnostic Profile:

  • Plots intersect precisely on the negative horizontal axis at $-\frac{1}{K_M}$.

  • The slope ($\frac{K_M}{V_{\text{max}}}$) increases in the presence of inhibitor.

  • The $y$-intercept ($\frac{1}{V_{\text{max,apparent}}}$) shifts higher on the vertical axis.

• Side-by-Side Lineweaver-Burk Comparison:

  • Competitive: Lines intersect on the $y$-axis; $V_{\text{max}}$ unchanged; $K_M$ increases.

  • Uncompetitive: Parallel lines; slope identical; both $V_{\text{max}}$ and $K_M$ decrease.

  • Noncompetitive: Lines intersect on the $x$-axis; $K_M$ unchanged; $V_{\text{max}}$ decreases.

Pharmacological Assessment: Half Maximal Inhibitory Concentration

• Potency Parameters in Biological Assays:

  • In drug discovery, pharmacological potency is quantified using functional inhibition assays.

  • $\text{IC}_{50}$ (Half maximal inhibitory concentration): The precise concentration of an inhibitor compound required to reduce an enzymatic or biological response to $50\%$ of its baseline value in the absence of inhibitor.

  • $\text{IC}_{90}$: The concentration of an inhibitor compound required to reduce an enzymatic or biological response to $10\%$ of its uninhibited baseline (i.e., achieving $90\%$ inhibition).

• The Cheng–Prusoff Relationship:

  • Formulated by biochemists Yung-Chi Cheng and William Prusoff to relate the functional metric $\text{IC}_{50}$ to the thermodynamic dissociation constant ($K_i$) of the enzyme-inhibitor complex.

  • Under competitive inhibition conditions, the relationship is given by the Cheng–Prusoff equation:   $\text{IC}_{50} = K_i \left(1 + \frac{[\text{S}]}{K_M}\right)$

  • Highlights that experimental $\text{IC}_{50}$ values depend directly upon the ambient substrate concentration ($[\text{S}]$) and substrate affinity ($K_M$), whereas $K_i$ is an absolute thermodynamic equilibrium parameter.

Allosteric Regulation and Non-Michaelis-Menten Kinetics

• Characteristics of Allosteric Enzymes:

  • Allosteric enzymes do not obey standard hyperbolic Michaelis-Menten kinetics.

  • They function via the reversible, noncovalent binding of regulatory molecules known as allosteric modulators or allosteric effectors (frequently small cellular metabolites or cofactors).

  • Effectors alter catalytic efficiency by inducing concerted conformational changes across multiple protein subunits.

  • Modulator Classes:

  • Stimulatory (Positive) Modulators: Enhance enzyme activity or substrate affinity.

  • Inhibitory (Negative) Modulators: Decrease enzyme activity or substrate affinity.

  • Homotropic Regulation: The substrate and the regulatory modulator are chemically identical molecules (cooperativity across identical active sites).

  • Heterotropic Regulation: The regulatory modulator is a distinct chemical entity different from the enzyme's substrate.

• Distinction from Uncompetitive and Noncompetitive Inhibitors:

  • While uncompetitive, noncompetitive, and allosteric inhibitors all bind to sites distinct from the catalytic active pocket, uncompetitive and noncompetitive inhibitors do not necessarily trigger cooperative quaternary conformational changes.

  • Their underlying kinetic profiles and saturation geometries are consequently distinct.

• Subunit Structural Dynamics:

  • Allosteric enzymes generally consist of multi-subunit complexes comprising discrete Catalytic ($\text{C}$) and Regulatory ($\text{R}$) subunits.

  • Sequential Activation Mechanism:

  1. In the absence of positive effector, the enzyme exists in a low-activity (low-affinity) state.

  2. A positive modulator ($\text{M}$) binds specifically to the regulatory subunit ($\text{R}$).

  3. Binding triggers an allosteric conformational shift within the $\text{R}$ subunit.

  4. This structural reorganization propagates across subunit interfaces to induce a high-affinity conformational state within the adjacent catalytic ($\text{C}$) subunit.

  5. The catalytic subunit binds substrate ($\text{S}$) with significantly elevated affinity, forming the active enzyme-substrate complex.

• Sigmoidal Kinetic Profiles:

  • Unlike the classic hyperbolic curves seen in Michaelis-Menten enzymes, homotropic allosteric enzymes display a distinctive sigmoidal (S-shaped) velocity versus substrate concentration ($V_0$ vs. $[\text{S}]$) curve due to positive cooperativity.

  • Half-Saturation Parameter ($K_{0.5}$!):

  • For sigmoidal kinetics, the substrate concentration required to produce half of the maximal velocity ($\frac{1}{2}V_{\text{max}}$) is designated as $K_{0.5}$ (or $[\text{S}]_{0.5}$) rather than $K_M$.

• Modulator Effects on Saturation Curves:

  • Positive Modulator ($\oplus$!):

  • Shifts the sigmoidal curve toward the left.

  • Lowers the apparent $K_{0.5}$ ($K_{0.5}^\oplus < K_{0.5}$), increasing substrate affinity.

  • Transitions the kinetic profile toward a hyperbolic shape.

  • Negative Modulator ($\ominus$!):

  • Shifts the sigmoidal curve toward the right.

  • Increases the apparent $K_{0.5}$ ($K_{0.5}^\ominus > K_{0.5}$), reducing substrate affinity.

  • Exaggerates the sigmoidal character of the velocity curve.

  • Maximal velocity ($V_{\text{max}}$) remains essentially constant across these transitions in typical $K$-system allosteric enzymes.

Comparative Review of Kinetic and Regulatory Features

• Enzymatic Catalysis Core Summary:

  • Enzymes dramatically increase reaction rates ($10^6$ to $10^{17}$ fold) without modifying final reaction equilibrium.

  • Specificity is achieved via stereochemical and chemical complementarity, characterized historically by the Lock and Key model and dynamically by the Induced Fit model.

• Michaelis-Menten Kinetic Parameters:

  • $V_0 = \frac{V_{\text{max}}[\text{S}]}{K_M + [\text{S}]}$

  • $V_{\text{max}}$: Maximal initial velocity under full active-site enzyme saturation.

  • $K_M = \frac{k_{-1} + k_2}{k_1}$; substrate concentration producing half-maximal velocity ($\frac{1}{2}V_{\text{max}}$).

  • $k_{\text{cat}} = \frac{V_{\text{max}}}{[\text{E}]_{\text{total}}}$: Catalytic turnover number per unit time.

• Reversible Inhibition Profiles:

  • Competitive: Inhibitor binds to free $\text{E}$ at the active site; $V_{\text{max}}$ unchanged; $K_M$ increases.

  • Uncompetitive: Inhibitor binds strictly to the $\text{ES}$ complex; both $V_{\text{max}}$ and $K_M$ decrease proportionally; Lineweaver-Burk lines are parallel.

  • Noncompetitive: Inhibitor binds equally to free $\text{E}$ and $\text{ES}$ complexes; $V_{\text{max}}$ decreases; $K_M$ unchanged; Lineweaver-Burk lines intersect at the $x$-axis.

• Irreversible and Suicide Inhibition:

  • Irreversible agents bind covalently or with very slow dissociation kinetics (e.g., penicillin targeting transpeptidase; aspirin targeting cyclooxygenase).

  • Suicide inhibitors (e.g., 5-fluorouracil) are mechanism-based chemical trojans processed by catalytic steps into reactive inhibitors that destroy the enzyme.

• Allosteric Enzymes:

  • Multi-subunit regulatory proteins that do not conform to Michaelis-Menten kinetics.

  • Exhibit sigmoidal substrate saturation curves governed by positive cooperativity.

  • Characterized by the half-saturation constant $K_{0.5}$.

  • Positive modulators shift curves to the left (decreasing $K_{0.5}$, increasing affinity); negative modulators shift curves to the right (increasing $K_{0.5}$, decreasing affinity).