Comprehensive Lecture Notes on Enzyme Kinetics and Michaelis-Menten Models

Fundamentals of Enzyme Kinetics and Initial Velocity

• Definition of Enzyme Kinetics: The quantitative study of chemical reaction rates catalyzed by purified enzymes.

• Significance:

  • Reveals the specific molecular mechanisms by which an enzyme converts substrate into product.

  • Elucidates the mode of action of regulatory ligands, including allosteric activators and inhibitors.

• Reaction Progress and Equilibrium:

  • In a standard enzymatic reaction where substrate ($S$) is converted to product ($P$):     $S \rightarrow P$

  • As time progresses, product accumulates and substrate is depleted, causing the reaction rate to continuously decrease until chemical equilibrium is reached:     $S \rightleftharpoons P$

  • At equilibrium, the rate of forward product formation equals the rate of reverse substrate formation.

• Initial Velocity ($V_0$):

  • Defined as the number of moles of product formed per second at the very start of the reaction ($t \rightarrow 0$).

  • Measured when only a tiny fraction of substrate has reacted (typically less than $5\%$).

  • Standardizing measurements to initial velocity eliminates complications caused by substrate depletion, product inhibition, or the reverse reaction ($P \rightarrow S$).

• Determinants of Initial Velocity:

  • Total enzyme concentration ($[E_T]$).

  • Substrate concentration ($[S]$).

  • For a fixed enzyme concentration, increasing $[S]$ increases $V_0$ up to a theoretical limit known as maximum velocity ($V_{\max}$), where all enzyme active sites are occupied.

Experimental Measurement: Stopped-Flow Kinetics

• Purpose: Used to measure rapid kinetic events and initial reaction rates during the first few milliseconds ($\text{ms}$) after mixing enzyme and substrate.

• Mechanism of a Stopped-Flow Apparatus:

  • Purified enzyme and substrate solutions are loaded into separate precision syringes.

  • A drive mechanism rapidly forces both solutions into a high-efficiency mixing chamber.

  • The liquids meet and mix thoroughly in a mixing tube at flow rates reaching up to $1000\,cm/s$.

  • The reaction mixture flows past an optical detector positioned within $1\,cm$ of the mixing junction.

  • This arrangement allows real-time measurement of product formation or intermediate species when reactions are only a few milliseconds old.

The Michaelis-Menten Model and Rate Constants

• Reaction Scheme:   $E + S \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} ES \xrightarrow{k_{\text{cat}}} E + P$

• Reaction Steps:

  • Binding Step: Reversible association of free enzyme ($E$) and substrate ($S$) to form the enzyme-substrate complex ($ES$).

  • Catalytic Step: Breakdown of the $ES$ complex to regenerate free enzyme ($E$) and release product ($P$).

• Individual Rate Constants:

  • $k_1$: Second-order rate constant for the formation of the $ES$ complex ($M^{-1}\,s^{-1}$).

  • $k_{-1}$: First-order rate constant for the dissociation of the $ES$ complex back to $E$ and $S$ ($s^{-1}$).

  • $k_2$ or $k_{\text{cat}}$: First-order rate constant for the conversion of $ES$ to $E + P$ ($s^{-1}$).

  • $k_{-2}$: Rate constant for the reverse reaction forming $ES$ from free $E$ and $P$. Under initial velocity ($V_0$) conditions where $[P] \approx 0$, this reverse step is negligible.

• Dissociation Constant ($K_D$):   $K_D = \frac{k_{-1}}{k_1} = \frac{[E][S]}{[ES]}$

  • Reflects the thermodynamic binding affinity of the enzyme for its substrate.

  • A lower $K_D$ value represents a tighter binding affinity between $E$ and $S$.

Derivation of the Michaelis-Menten Equation

• Steady-State Assumption:

  • During the primary reaction window, the concentration of the $ES$ complex remains nearly constant over time ($\frac{d[ES]}{dt} = 0$).

  • This occurs because the rate of $ES$ formation equals its total rate of breakdown.

• Step-by-Step Mathematical Derivation:

  1. Rate of ES Formation:      $\text{Rate of formation} = k_1 [E][S]$

  2. Rate of ES Breakdown:      $\text{Rate of breakdown} = k_{-1} [ES] + k_{\text{cat}} [ES] = (k_{-1} + k_{\text{cat}})[ES]$

  3. Equating Formation and Breakdown Rates:      $k_1 [E][S] = (k_{-1} + k_{\text{cat}})[ES]$

  4. Rearranging and Defining the Michaelis Constant ($K_m$):      $[E][S] = \left(\frac{k_{-1} + k_{\text{cat}}}{k_1}\right)[ES]$      $K_m = \frac{k_{-1} + k_{\text{cat}}}{k_1}$      $[E][S] = K_m [ES]$

  5. Applying Conservation of Enzyme Mass:      Total enzyme ($[E_T]$) is divided between free enzyme ($[E]$) and complexed enzyme ($[ES]$):      $[E_T] = [E] + [ES] \implies [E] = [E_T] - [ES]$

  6. Substituting Free Enzyme Concentration:      $([E_T] - [ES])[S] = K_m [ES]$      $[E_T][S] - [ES][S] = K_m [ES]$      $[E_T][S] = K_m [ES] + [S][ES]$      $[E_T][S] = (K_m + [S])[ES]$

  7. Solving for [ES]:      $[ES] = \frac{[E_T][S]}{K_m + [S]}$

  8. Formulating Initial Velocity ($V_0$):      $V_0 = k_{\text{cat}} [ES] = \frac{k_{\text{cat}} [E_T] [S]}{K_m + [S]}$      Because maximum velocity ($V_{\max}$) occurs when the enzyme is completely saturated with substrate ($[ES] = [E_T]$):      $V_{\max} = k_{\text{cat}} [E_T]$

  9. Final Michaelis-Menten Equation:      $V_0 = \frac{V_{\max} [S]}{K_m + [S]}$

Key Simplifying Assumptions

• Substrate Excess ($[S] \gg [E_T]$): The total concentration of substrate is vastly greater than the total concentration of enzyme. Thus, the fraction of substrate bound to the enzyme at any moment is negligible, keeping free $[S]$ essentially constant during initial velocity measurements.

• Zero Product Approximation ($[P] \approx 0$): Initial velocity ($V_0$) is measured before significant product accumulates, ensuring the reverse reaction ($E + P \rightarrow ES$) does not take place.

• Immediate Steady State: The $ES$ complex achieves dynamic steady state almost instantaneously after reactant mixing.

• Two-State Enzyme Distribution: Enzyme molecules exist exclusively in either free form ($E$) or substrate-bound form ($ES$), allowing the total enzyme concentration to be expressed strictly as $[E_T] = [E] + [ES]$.

Kinetic Parameters: Vmax, Km, and kcat

• Maximum Velocity ($V_{\max}$):

  • Represents the highest theoretical turnover rate obtainable when all active sites are fully occupied by substrate ($[S] \rightarrow \infty$).

  • Depends directly on total enzyme concentration: $V_{\max} = k_{\text{cat}} [E_T]$.

• Michaelis Constant ($K_m$):

  • Algebraically defined as $K_m = \frac{k_{-1} + k_{\text{cat}}}{k_1}$.

  • Equal to the specific substrate concentration ($[S]$) at which initial reaction velocity reaches half of maximum velocity ($V_0 = \frac{1}{2}V_{\max}$).

  • Relationship to Binding Affinity: When $k_{-1} \gg k_{\text{cat}}$, the value of $K_m$ simplifies to $K_m \approx K_D = \frac{k_{-1}}{k_1}$. Under these conditions, $K_m$ serves as an estimate of enzyme-substrate affinity; a lower $K_m$ indicates tighter binding.

• Turnover Number ($k_{\text{cat}}$):

  • The number of substrate molecules converted into product per enzyme active site per unit time when the enzyme is fully saturated with substrate.

  • Calculated as $k_{\text{cat}} = \frac{V_{\max}}{[E_T]}$ with standard units of $s^{-1}$.

• Turnover Numbers ($k_{\text{cat}}$) of Representative Enzymes:

Kinetic Behavior across Substrate Regimes

• Substrate-Limiting Regime ($[S] \ll K_m$):

  • $K_m + [S] \approx K_m$

  • $V_0 \approx \left(\frac{V_{\max}}{K_m}\right) [S]$

  • Displays first-order kinetics; initial velocity is directly proportional to substrate concentration. Active sites are mostly unoccupied.

• Half-Saturation Threshold ($[S] = K_m$):

  • $V_0 = \frac{V_{\max} K_m}{K_m + K_m} = \frac{1}{2} V_{\max}$

  • Exactly half of all enzyme active sites are occupied by substrate molecules.

• Enzyme-Saturating Regime ($[S] \gg K_m$):

  • $K_m + [S] \approx [S]$

  • $V_0 \approx \frac{V_{\max} [S]}{[S]} = V_{\max}$

  • Displays zero-order kinetics; reaction rate becomes independent of substrate concentration because all active sites are filled.

Double-Reciprocal Analysis: Lineweaver-Burk Plot

• Algebraic Derivation:   Taking the reciprocal of the Michaelis-Menten equation:   $\frac{1}{V_0} = \frac{K_m + [S]}{V_{\max} [S]} = \left(\frac{K_m}{V_{\max}}\right)\left(\frac{1}{[S]}\right) + \frac{1}{V_{\max}}$

• Linear Equation Parameters ($y = mx + b$):

  • Y-variable ($y$): $\frac{1}{V_0}$

  • X-variable ($x$): $\frac{1}{[S]}$

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

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

  • X-intercept: $-\frac{1}{K_m}\n\n\n\n- **Determining Kinetic Values from Intercepts:**\n - V_{\max} = \frac{1}{\text{Y-intercept}}\n - K_m = -\frac{1}{\text{X-intercept}}\n\n# Quantitative Practice and Worked Example\n\n- **Sample Kinetics Dataset 1:**\n\n| [S]\,(mM)$|$V_0$|$\frac{1}{[S]}\,(mM^{-1})$|$\frac{1}{V_0} |\n| :--- | :--- | :--- | :--- |\n| 2.5$|$32.3$|$0.40$|$0.031 |\n| 3.5$|$40.0$|$0.29$|$0.025 |\n| 5.0$|$50.8$|$0.20$|$0.020 |\n| 10.0$|$72.0$|$0.10$|$0.014 |\n| 20.0$|$87.7$|$0.05$|$0.011 |\n| 50.0$|$115.4$|$0.02$|$0.009 |\n\n- **Worked Linear Regression Problem:**\n\n| [S]\,(\mu M)$|$V_0\,(nmol/min)$|$\frac{1}{[S]}\,(\mu M^{-1})$|$\frac{1}{V_0}\,(nmol^{-1}\,min) |\n| :--- | :--- | :--- | :--- |\n| 0.20$|$1.43$|$5.00$|$0.70 |\n| 0.26$|$1.67$|$3.85$|$0.60 |\n| 0.33$|$2.08$|$3.03$|$0.48 |\n| 1.00$|$3.33$|$1.00$|$0.30 |\n\n- **Regression Line Calculation:**\n - Linear equation fitted to reciprocal data:\n    y = 0.1016 x + 0.1928\n - **Calculating V_{\max}:**\n    \text{Y-intercept} = 0.1928\n    V_{\max} = \frac{1}{0.1928} = 5.19\,nmol/min\n - **Calculating K_m:**\n    \text{X-intercept} = -1.9\n    K_m = -\frac{1}{-1.9} = 0.53\,\mu M\n\n\n\n# Biological Significance of Km Values\n\n- **Experimental K_m Values across Enzymes:**\n\n| Enzyme | Substrate | K_m\,(\mu M) |\n| :--- | :--- | :--- |\n| Chymotrypsin | Acetyl-L-tryptophanamide | 5000 |\n| Lysozyme | Hexa-N-acetylglucosamine | 6 |\n| \beta$-Galactosidase | Lactose |$4000 |\n| Threonine deaminase | Threonine | 5000 |\n| Carbonic anhydrase | \text{CO}2$|$8000 |\n| Penicillinase | Benzylpenicillin | 50 |\n| Pyruvate carboxylase | Pyruvate | 400 |\n| Pyruvate carboxylase | \text{HCO}_3^-$|$1000 |\n| Pyruvate carboxylase | ATP | 60 |\n| Arginine-tRNA synthetase | Arginine | 3 |\n| Arginine-tRNA synthetase | tRNA | 0.4 |\n| Arginine-tRNA synthetase | ATP | 300 |\n\n- **Physiological Adaptation Strategies:**\n - **K_m \approx Normal Cellular Concentration:**\n - Reaction rate fluctuates dynamically in proportion to changes in substrate levels.\n - Enables metabolic pathways to automatically adjust flux in response to substrate availability.\n - **K_m \ll Normal Cellular Concentration:**\n - Enzyme is perpetually saturated with substrate and operates continuously at maximum rate (V{\max}).\n - Protects critical cellular reactions from being affected by minor fluctuations in substrate supply.\n - **K_m \gg Normal Cellular Concentration:**\n - Enzyme remains mostly idle under basal conditions.\n - Used when multiple enzymes compete for a shared substrate pool, ensuring this specific enzyme acts only when substrate levels become exceptionally high.\n\n# Isozymes and Metabolic Regulation: Hexokinase vs. Glucokinase\n\n- **Reaction Catalyzed:** Both enzymes catalyze the phosphorylation of glucose:\n  \text{Glucose} + \text{ATP} \rightarrow \text{Glucose-6-phosphate} + \text{ADP}\n- **Hexokinase:**\n - Found in most extrahepatic tissues.\n - Possesses a very low K_m$for glucose ($< 1\,mmol/L).\n - Operates near maximum velocity at normal blood glucose levels (\approx 5\,mmol/L), prioritizing continuous glucose utilization for energy production via glycolysis.\n- **Glucokinase:**\n - Found in liver parenchymal cells.\n - Possesses a high K_m$for glucose ($\approx 10\,mmol/L).\n - Inactive at basal blood glucose levels (5\,mmol/L); activates significantly when blood glucose rises following carbohydrate consumption, trapping excess glucose for liver glycogen synthesis.\n\n# Physiological Implications: Alcohol Metabolism and Flushing\n\n- **Two-Step Enzymatic Oxidation of Ethanol:**\n 1. \text{Ethanol} + \text{NAD}^+ \xrightarrow{\text{Alcohol dehydrogenase}} \text{acetaldehyde} + \text{H}^+ + \text{NADH}\n 2. \text{Acetaldehyde} + \text{NAD}^+ \xrightarrow{\text{Aldehyde dehydrogenase}} \text{acetate} + 2\text{H}^+ + \text{NADH}\n- **Aldehyde Dehydrogenase Isozymes:**\n - **Mitochondrial Aldehyde Dehydrogenase:** Low K_m for acetaldehyde; rapidly metabolizes trace amounts of acetaldehyde into harmless acetate.\n - **Cytoplasmic Aldehyde Dehydrogenase:** High K_m for acetaldehyde; operates effectively only at elevated acetaldehyde concentrations.\n- **Genetic Mutation (Facial Flushing Phenotype):**\n - A single point mutation causing a \text{Glu487} \rightarrow \text{Lys} substitution inactivates mitochondrial aldehyde dehydrogenase.\n - Individuals with this mutation rely solely on the high-K_m cytoplasmic enzyme, resulting in acetaldehyde accumulation after ethanol consumption.\n - Accumulation of acetaldehyde causes facial flushing, vasodilation, elevated heart rate, and nausea.\n\n# Specificity Constant and Catalytic Perfection\n\n- **Definition of Specificity Constant (\frac{k_{\text{cat}}}{K_m}):**\n - Combines information about catalytic conversion speed (k_{\text{cat}}$) and substrate binding affinity ($K_m).\n - Measures catalytic efficiency under sub-saturating conditions ([S] \ll K_m):\n    V_0 = \left(\frac{k_{\text{cat}}}{K_m}\right) [E][S]\n- **Upper Limit of Efficiency (Diffusion Control):**\n - Expanding K_m:\n    \frac{k_{\text{cat}}}{K_m} = \frac{k_{\text{cat}}}{\frac{k_{-1} + k_{\text{cat}}}{k_1}} = \frac{k_1 k_{\text{cat}}}{k_{-1} + k_{\text{cat}}}\n - When catalytic rate exceeds dissociation rate (k_{\text{cat}} \gg k_{-1}$),$\frac{k_{\text{cat}}}{K_m} \approx k_1\n - The physical rate limit for enzyme-substrate collisions in aqueous solution (diffusion-controlled rate) is 10^8 \text{ to } 10^9\,M^{-1}\,s^{-1}.\n- **Catalytic Perfection:**\n - An enzyme is defined as catalytically perfect when its \frac{k_{\text{cat}}}{K_m}$ratio approaches the diffusion limit ($10^8 \text{ to } 10^9\,M^{-1}\,s^{-1}).\n - Virtually every collision between enzyme and substrate results in product formation.\n - Does not require an exceptionally low K_m$or high$k_{\text{cat}}, but rather that their ratio is maximized.\n\n- **Catalytic Efficiency Comparison Table:**\n\n| Enzyme | Substrate | k_{\text{cat}}\,(s^{-1})$|$K_m\,(M)$|$\frac{k_{\text{cat}}}{K_m}\,(s^{-1}\,M^{-1}) |\n| :--- | :--- | :--- | :--- | :--- |\n| Acetylcholinesterase | Acetylcholine | 1.4 \times 10^4$|$9 \times 10^{-5}$|$1.6 \times 10^8 |\n| Carbonic anhydrase | \text{CO}_2$|$6.0 \times 10^5$|$8.0 \times 10^{-3}$|$8.3 \times 10^7 |\n| Catalase | \text{H}_2\text{O}_2$|$4.0 \times 10^7$|$1.0$|$4.0 \times 10^7 |\n| Crotonase | Crotonyl-CoA | — | — | 2.8 \times 10^8 |\n| Fumarase | Fumarate | 8.0 \times 10^2$|$5.0 \times 10^{-6}$|$1.6 \times 10^8 |\n| Triose phosphate isomerase | Glyceraldehyde 3-phosphate | — | — | 2.4 \times 10^8 |\n| \beta$-Lactamase | Penicillin | — | — |$1.0 \times 10^8 |\n| Superoxide dismutase | Superoxide | — | — | 7.0 \times 10^9$|</p></li></ul></li></ul><p></p><h4 id="fe3f783e-a189-4ed6-8c55-528f5bf09b34" data-toc-id="fe3f783e-a189-4ed6-8c55-528f5bf09b34" collapsed="false" seolevelmigrated="true">Fundamentals of Enzyme Kinetics and Initial Velocity</h4><ul><li><p><strong>Definition of Enzyme Kinetics:</strong> The quantitative study of chemical reaction rates catalyzed by purified enzymes under controlled experimental conditions.</p></li><li><p><strong>Significance:</strong></p><ul><li><p>Reveals the specific step-by-step molecular mechanisms by which an enzyme binds substrate, lowers activation energy, and converts substrate into product.</p></li><li><p>Elucidates the mode of action of regulatory ligands, including allosteric activators, competitive inhibitors, and non-competitive inhibitors.</p></li></ul></li><li><p><strong>Reaction Progress and Equilibrium:</strong></p><ul><li><p>In a standard enzymatic reaction converting substrate ($S$) to product ($P$):$S \rightarrow P$</p></li><li><p>Over time, product accumulates while substrate is depleted, causing the forward reaction rate to continuously decrease until dynamic chemical equilibrium is established:$S \rightleftharpoons P$</p></li><li><p>At equilibrium, the rate of forward product formation ($S \rightarrow P$) precisely equals the rate of reverse substrate formation ($P \rightarrow S$), resulting in zero net change in product concentration.</p></li></ul></li><li><p><strong>Initial Velocity ($V_0$):</strong></p><ul><li><p>Defined as the rate of product formation per unit time at the exact onset of the reaction ($t \rightarrow 0$).</p></li><li><p>Standardized measurements require determining$V_0$when only a tiny fraction of substrate has reacted (typically less than$5\%$).</p></li><li><p>Measuring initial velocity eliminates experimental artifacts such as substrate depletion, product inhibition, or reverse reaction turnover ($P \rightarrow S$).</p></li></ul></li><li><p><strong>Determinants of Initial Velocity:</strong></p><ul><li><p>Total enzyme concentration ($[E_T]$).</p></li><li><p>Substrate concentration ($[S]$).</p></li><li><p>For a fixed concentration of enzyme, increasing$[S]$increases$V_0$asymptotically up to a maximum velocity ($V_{\max}$), where all active sites are saturated.</p></li></ul></li></ul><h4 id="d50be50f-12ee-4c53-8ada-08106a1ea3e6" data-toc-id="d50be50f-12ee-4c53-8ada-08106a1ea3e6" collapsed="false" seolevelmigrated="true">Experimental Measurement: Stopped-Flow Kinetics</h4><ul><li><p><strong>Purpose:</strong> Enables observation and quantitative measurement of rapid kinetic events and initial rates occurring within the pre-steady state phase (the first few milliseconds,$\text{ms}$).</p></li><li><p><strong>Mechanism of a Stopped-Flow Apparatus:</strong></p><ul><li><p>Purified enzyme and substrate solutions are loaded into separate precision drive syringes.</p></li><li><p>A pneumatic drive mechanism forces both solutions rapidly into a high-efficiency micro-mixing chamber.</p></li><li><p>Reactants meet and mix completely in a mixing tube at flow velocities up to$1000\,cm/s$</p></li><li><p>The reaction mixture flows past an optical observation channel positioned within$1\,cm$of the mixing junction.</p></li><li><p>This arrangement allows optical detectors (absorbance or fluorescence) to monitor the appearance of products or short-lived enzyme-substrate intermediates when the reaction is only a few milliseconds old.</p></li></ul></li></ul><p></p><img src="https://assets.knowt.com/pdf-flow-prod/759ae92d-e662-4bbe-9556-b86bd1c52ef5-figures/1.jpg" data-width="50$E + S \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} ES \xrightarrow{k_{\text{cat}}} E + P$</p></li><li><p><strong>Reaction Steps:</strong></p><ul><li><p><strong>Binding Step:</strong> Reversible association between free enzyme ($E$) and substrate ($S$) to form the non-covalent enzyme-substrate complex ($ES$).</p></li><li><p><strong>Catalytic Step:</strong> Breakdown of the$ES$complex to regenerate free enzyme ($E$) and release final product ($P$).</p></li></ul></li><li><p><strong>Individual Rate Constants:</strong></p><ul><li><p>$k_1$: Second-order rate constant for the association of$E$and$S$to form$ES$($M^{-1}\,s^{-1}$).</p></li><li><p>$k_{-1}$: First-order rate constant for the dissociation of$ES$back to free$E$and$S$($s^{-1}$).</p></li><li><p>$k_2$or$k_{\text{cat}}$: First-order rate constant for the conversion of$ES$into free$E$and product$P$($s^{-1}$).</p></li><li><p>$k_{-2}$: Rate constant for the reverse reaction ($E + P \rightarrow ES$). Under initial velocity conditions where$[P] \approx 0$, this reverse step is virtually zero and ignored.</p></li></ul></li><li><p><strong>Dissociation Constant ($K_D$):</strong></p><ul><li><p>$K_D = \frac{k_{-1}}{k_1} = \frac{[E][S]}{[ES]}$</p></li><li><p>Measures the thermodynamic binding affinity of the enzyme for its substrate.</p></li><li><p>A lower$K_D$value signifies higher thermodynamic binding affinity.</p></li></ul></li></ul><h4 id="845916cb-2e90-4026-b846-cbb1b4a81870" data-toc-id="845916cb-2e90-4026-b846-cbb1b4a81870" collapsed="false" seolevelmigrated="true">Derivation of the Michaelis-Menten Equation</h4><ul><li><p><strong>Steady-State Assumption:</strong></p><ul><li><p>Shortly after mixing, the concentration of the$ES$complex reaches a dynamic steady state where its rate of synthesis equals its rate of breakdown ($\frac{d[ES]}{dt} = 0$).</p></li><li><p>This keeps$[ES]$constant throughout the initial velocity measurement window.</p></li></ul></li></ul><p></p><img src="https://assets.knowt.com/pdf-flow-prod/759ae92d-e662-4bbe-9556-b86bd1c52ef5-figures/5.png" data-width="50$\text{Rate of formation} = k_1 [E][S]$</p></li><li><p><strong>Rate of ES Breakdown:</strong>$\text{Rate of breakdown} = k_{-1} [ES] + k_{\text{cat}} [ES] = (k_{-1} + k_{\text{cat}})[ES]$</p></li><li><p><strong>Equating Rates at Steady State:</strong>$k_1 [E][S] = (k_{-1} + k_{\text{cat}})[ES]$</p></li><li><p><strong>Rearranging and Defining Michaelis Constant ($K_m$):</strong>$[E][S] = \left(\frac{k_{-1} + k_{\text{cat}}}{k_1}\right)[ES] \implies K_m = \frac{k_{-1} + k_{\text{cat}}}{k_1} \implies [E][S] = K_m [ES]$</p></li><li><p><strong>Mass Balance for Enzyme:</strong> Total enzyme ($[E_T]$) is split between free enzyme ($[E]$) and complexed enzyme ($[ES]$):$[E_T] = [E] + [ES] \implies [E] = [E_T] - [ES]$</p></li><li><p><strong>Substitution:</strong>$([E_T] - [ES])[S] = K_m [ES] \implies [E_T][S] = (K_m + [S])[ES]$</p></li><li><p><strong>Solving for [ES]:</strong>$[ES] = \frac{[E_T][S]}{K_m + [S]}$</p></li><li><p><strong>Formulating Initial Velocity ($V_0$):</strong>$V_0 = k_{\text{cat}} [ES] = \frac{k_{\text{cat}} [E_T] [S]}{K_m + [S]}$</p></li><li><p><strong>Defining Maximum Velocity ($V_{\max}$):</strong> At full saturation ($[ES] = [E_T]$),$V_{\max} = k_{\text{cat}} [E_T]$.</p></li><li><p><strong>Final Michaelis-Menten Equation:</strong>$V_0 = \frac{V_{\max} [S]}{K_m + [S]}$</p></li></ol></li></ul><h4 id="373d6fed-16f5-4e67-aa49-cd7f47f5706d" data-toc-id="373d6fed-16f5-4e67-aa49-cd7f47f5706d" collapsed="false" seolevelmigrated="true">Key Simplifying Assumptions</h4><ul><li><p><strong>Substrate Excess ($[S] \gg [E_T]$):</strong> Total substrate concentration is vastly higher than total enzyme concentration, ensuring free$[S]$remains essentially unchanged during initial velocity measurements.</p></li><li><p><strong>Zero Product Approximation ($[P] \approx 0$):</strong> Measurements are taken immediately after initiation so product concentration is close to zero, preventing reverse catalysis.</p></li><li><p><strong>Immediate Steady State:</strong> Dynamic equilibrium of$[ES]$is achieved almost instantaneously post-mixing.</p></li><li><p><strong>Two-State Enzyme Distribution:</strong> Enzyme molecules exist strictly as free enzyme ($[E]$) or bound complex ($[ES]$), validating$[E_T] = [E] + [ES]$.</p></li></ul><h4 id="eb5fcad2-4bcb-42bb-bcb6-6d69719581e5" data-toc-id="eb5fcad2-4bcb-42bb-bcb6-6d69719581e5" collapsed="false" seolevelmigrated="true">Kinetic Parameters: Vmax, Km, and kcat</h4><ul><li><p><strong>Maximum Velocity ($V_{\max}$):</strong></p><ul><li><p>The maximum rate attainable when enzyme active sites are completely saturated with substrate ($[S] \rightarrow \infty$).</p></li><li><p>Proportional to enzyme concentration:$V_{\max} = k_{\text{cat}} [E_T]$.</p></li></ul></li><li><p><strong>Michaelis Constant ($K_m$):</strong></p><ul><li><p>Defined algebraically as$K_m = \frac{k_{-1} + k_{\text{cat}}}{k_1}$.</p></li><li><p>Numerically equal to the substrate concentration ($[S]$) at which reaction velocity reaches half its maximum ($V_0 = \frac{1}{2} V_{\max}$).</p></li><li><p><strong>Relationship to Binding Affinity:</strong> When$k_{-1} \gg k_{\text{cat}}$,$K_m \approx K_D = \frac{k_{-1}}{k_1}$. In this scenario,$K_m$serves as an estimate of binding affinity, where a smaller$K_m$reflects stronger binding.</p></li></ul></li><li><p><strong>Turnover Number ($k_{\text{cat}}$):</strong></p><ul><li><p>The number of substrate molecules converted into product per active site per second when saturated.</p></li><li><p>Calculated as$k_{\text{cat}} = \frac{V_{\max}}{[E_T]}$with units of$s^{-1}$.</p></li></ul></li><li><p><strong>Turnover Numbers ($k_{\text{cat}}) of Representative Enzymes:   | Enzyme | Turnover Number (s^{-1}) |   | :--- | :--- |   | Carbonic anhydrase | 600,000 |   | 3-Ketosteroid isomerase | 280,000 |   | Acetylcholinesterase | 25,000 |   | Penicillinase | 2,000 |   | Lactate dehydrogenase | 1,000 |   | Chymotrypsin | 100 |   | DNA polymerase I | 15 |   | Tryptophan synthetase | 2 |   | Lysozyme | 0.5$|</p></li></ul><h4 id="f447a2b2-80ff-48d1-a580-690cde7f9d42" data-toc-id="f447a2b2-80ff-48d1-a580-690cde7f9d42" collapsed="false" seolevelmigrated="true">Kinetic Behavior across Substrate Regimes</h4><p></p><img src="https://assets.knowt.com/pdf-flow-prod/759ae92d-e662-4bbe-9556-b86bd1c52ef5-figures/29.jpg" data-width="50$[S] \ll K_m$):</strong></p><ul><li><p>$K_m + [S] \approx K_m$</p></li><li><p>$V_0 \approx \left(\frac{V_{\max}}{K_m}\right) [S]$</p></li><li><p>Demonstrates first-order kinetics; rate scales directly and linearly with substrate concentration.</p></li></ul></li><li><p><strong>Half-Saturation Threshold ($[S] = K_m$):</strong></p><ul><li><p>$V_0 = \frac{V_{\max} K_m}{K_m + K_m} = \frac{1}{2} V_{\max}$</p></li><li><p>Exactly$50\%$of enzyme active sites are bound to substrate.</p></li></ul></li><li><p><strong>Enzyme-Saturating Regime ($[S] \gg K_m$):</strong></p><ul><li><p>$K_m + [S] \approx [S]$</p></li><li><p>$V_0 \approx \frac{V_{\max} [S]}{[S]} = V_{\max}$</p></li><li><p>Demonstrates zero-order kinetics; rate is maximum and completely independent of changes in$[S].