Chapter 8: Kinetics and Regulation

Fundamentals of Reaction Kinetics

• Kinetics is defined as the study of the rates of chemical reactions. When applied specifically to biological catalysts, the study of the rates of enzyme-catalyzed reactions is termed enzyme kinetics.

• The primary function of enzymes in biological systems is to accelerate reaction velocities.

• Reaction velocity ($V$) for a simple conversion of reactant $A$ to product $P$ ($A \rightarrow P$) represents the quantity of reactant that disappears per unit time ($t$), which is equivalent to the quantity of product that appears per unit time:   $V = -\frac{d[A]}{dt} = \frac{d[P]}{dt}$

• Velocity is directly proportional to reactant concentration through a proportionality constant known as the rate constant ($k$):   $V = k[A]$

• Kinetic orders of chemical reactions:

  • First-order reactions: The velocity is directly proportional to a single reactant concentration. First-order rate constants possess the unit of $s^{-1}$.

  • Second-order reactions: Biomolecular reactions involving two reactants. Second-order rate constants possess the unit of $M^{-1}s^{-1}$.

• Quantitative kinetic parameters provide critical physiological insight into:

  • The upper limit of an enzyme's catalytic speed.

  • Reaction rates at substrate concentrations encountered in vivo within cellular environments.

  • Substrate preference and catalytic specificity.

• Cellular metabolic context:

  • Cellular metabolism consists of dozens of metabolic pathways involving thousands of distinct reactions, each catalyzed by specific enzymes.

  • Michaelis–Menten enzymes generally lack sophisticated regulatory mechanisms and operate via mass action: catalysis occurs whenever substrate is present.

  • Allosteric enzymes function as information sensors and control points, regulating metabolic flux to avoid metabolic chaos and coordinating interconnected pathways.

The Michaelis–Menten Model of Enzyme Kinetics

• The standard catalytic model involves reversible substrate binding followed by catalytic conversion and product dissociation:   $E + S \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} ES \underset{k_{-2}}{\overset{k_2}{\rightleftharpoons}} E + P$

  • $k_1$ is the rate constant for the formation of the enzyme–substrate complex ($ES$).

  • $k_{-1}$ is the rate constant for the dissociation of the $ES$ complex back to free enzyme ($E$) and substrate ($S$).

  • $k_2$ is the rate constant for the conversion of $ES$ into product ($P$) and free enzyme ($E$).

  • $k_{-2}$ is the rate constant for the reverse reaction reforming $ES$ from $E + P$.

• Initial velocity ($V_0$) measures the rate of catalysis (moles of product formed per second) at the very onset of the reaction ($t \approx 0$).

  • Because product concentration is negligible early in the reaction, the reverse reaction governed by $k_{-2}$ can be ignored.

  • Reaction rate diminishes over time as substrate is depleted and equilibrium is approached.

• At constant enzyme concentration ($[E]_T$), plotting initial reaction velocity ($V_0$) against initial substrate concentration ($[S]$) yields a rectangular hyperbola.

  • At low substrate concentrations, $V_0$ increases linearly with $[S]$.

  • At high substrate concentrations, the rate asymptotes toward a maximal velocity ($V_{\max}$).

• The Michaelis–Menten Equation quantitatively relates initial velocity to substrate concentration:   $V_0 = \frac{V_{\max}[S]}{[S] + K_M}$

• The Michaelis constant ($K_M$) is defined as:   $K_M = \frac{k_{-1} + k_2}{k_1}$

  • $K_M$ is unique to each enzyme-substrate pair and is completely independent of enzyme concentration.

  • $K_M$ characterizes the affinity and properties of the enzyme-substrate interaction.

  • When $V_0 = \frac{V_{\max}}{2}$, substitution into the equation yields $[S] = K_M$. Thus, $K_M$ is equal to the substrate concentration at which the reaction velocity is half-maximal.

• Maximal velocity ($V_{\max}$):   $V_{\max} = k_2[E]_T$

  • $V_{\max}$ occurs exclusively when all active sites are occupied by substrate ($[ES] = [E]_T$).

  • Unlike $K_M$, $V_{\max}$ is directly dependent on enzyme concentration.

• Kinetic regimes of the Michaelis–Menten equation:

  • When $[S] \ll K_M$: $V_0 \approx \left(\frac{V_{\max}}{K_M}\right)[S]$. The reaction displays first-order kinetics with respect to substrate.

  • When $[S] \gg K_M$: $V_0 \approx V_{\max}$. The enzyme is fully saturated, displaying zero-order kinetics independent of $[S]$.

• Most enzymes are simple, monomeric, and unregulated catalysts whose rates are governed by mass action.

• Historical contribution of Maud Menten:

  • Maud Menten was among the first Canadian women to earn an MD (1911) and later a PhD (1916).

  • Co-developed the Michaelis–Menten model as a graduate researcher with Leonor Michaelis.

  • Conducted research at the University of Pittsburgh and the British Columbia Medical Research Institute, publishing over 70 scientific papers.

  • Invented an alkaline phosphatase histochemical assay, characterized bacterial toxins and early vitamin C deficiency, and pioneered protein electrophoresis.

Kinetic Parameters and Catalytic Efficiency

• Derivation of $K_M$ and $V_{\max}$ is achieved computationally through non-linear curve fitting or algebraically via the Lineweaver–Burk (double-reciprocal) transformation:   $\frac{1}{V_0} = \left(\frac{K_M}{V_{\max}}\right)\frac{1}{[S]} + \frac{1}{V_{\max}}$

  • Plotting $\frac{1}{V_0}$ ($y$-axis) versus $\frac{1}{[S]}$ ($x$-axis) yields a straight line.

  • $y$-intercept = $\frac{1}{V_{\max}}$

  • Slope = $\frac{K_M}{V_{\max}}$

  • $x$-intercept = $-\frac{1}{K_M}$

• Magnitude and physiological meaning of $K_M$:

  • Physiological $K_M$ values typically range between $10^{-1}\,M$ and $10^{-7}\,M$.

  • The value of $K_M$ varies depending on substrate identity, pH, temperature, and ionic strength.

  • Table of representative $K_M$ values for enzymes and specific substrates:

  • Carbonic anhydrase ($CO_2$): $8000\,\mu M$

  • Chymotrypsin (Acetyl-L-tryptophanamide): $5000\,\mu M$

  • $\beta$-Galactosidase (Lactose): $4000\,\mu M$

  • Penicillinase (Benzylpenicillin): $50\,\mu M$

  • Lysozyme (Hexa-N-acetylglucosamine): $6\,\mu M$

• Elasticity of enzymes:

  • Elasticity describes the degree to which an enzyme's rate responds to variations in substrate concentration.

  • Natural enzymes commonly evolve to have $K_M$ values that approximate physiological in vivo substrate concentrations ($[S] \approx K_M$).

  • At $[S] \ll K_M$: Activity is low, but sensitivity (elasticity) to substrate shifts is high.

  • At $[S] \gg K_M$: Catalytic activity approaches $V_{\max}$, but the enzyme is insensitive (inelastic) to substrate concentration fluctuations.

  • At $[S] \approx K_M$: The enzyme operates at approximately half-maximal velocity ($\frac{1}{2}V_{\max}$) while remaining responsive to regulatory changes in substrate levels.

• Turnover Number ($k_{\text{cat}}$):

  • Defined as the number of substrate molecules converted into product per active site per unit time when saturated:     $k_{\text{cat}} = \frac{V_{\max}}{[E]_T}$

  • Table of representative turnover numbers ($s^{-1}$):

  • Carbonic anhydrase: $600{,}000\,s^{-1}$

  • 3-Ketosteroid isomerase: $280{,}000\,s^{-1}$

  • Acetylcholinesterase: $25{,}000\,s^{-1}$

  • Penicillinase: $2000\,s^{-1}$

  • Lactate dehydrogenase: $1000\,s^{-1}$

  • Chymotrypsin: $100\,s^{-1}$

  • DNA polymerase I: $15\,s^{-1}$

  • Tryptophan synthetase: $2\,s^{-1}$

  • Lysozyme: $0.5\,s^{-1}$

• Catalytic efficiency and the specificity constant ($k_{\text{cat}}/K_M$):

  • When $[S] \ll K_M$, the rate equation becomes:     $V_0 = \left(\frac{k_{\text{cat}}}{K_M}\right)[S][E]_T$

  • The ratio $k_{\text{cat}}/K_M$ reflects both substrate binding affinity and catalytic rate.

  • The upper limit of catalytic efficiency is determined by the diffusion-controlled limit of molecular encounter in solution, which is between $10^8\,M^{-1}s^{-1}$ and $10^9\,M^{-1}s^{-1}$.

  • Enzymes operating near the diffusion-controlled limit exhibit kinetic perfection:

  • Acetylcholinesterase: $1.6 \times 10^8\,M^{-1}s^{-1}$

  • Carbonic anhydrase: $8.3 \times 10^7\,M^{-1}s^{-1}$

  • Catalase: $4 \times 10^7\,M^{-1}s^{-1}$

  • Crotonase: $2.8 \times 10^8\,M^{-1}s^{-1}$

  • Fumarase: $1.6 \times 10^8\,M^{-1}s^{-1}$

  • Triose phosphate isomerase: $2.4 \times 10^8\,M^{-1}s^{-1}$

  • $\beta$-Lactamase: $1 \times 10^8\,M^{-1}s^{-1}$

  • Superoxide dismutase: $7 \times 10^9\,M^{-1}s^{-1}$

Multi-Substrate Reactions

• Most physiological reactions are bisubstrate reactions that involve two substrates and yield two products:   $A + B \rightarrow P + Q$

• Many bisubstrate processes involve the transfer of a functional group from one substrate to another.

• Bisubstrate reaction classes:

  • Sequential reactions: All substrates must bind the enzyme before any product is released. A ternary complex ($EAB$) forms.

  • Ordered sequential: Substrates bind in a specific, mandatory sequence (e.g., Lactate Dehydrogenase binds $NADH$ first, then pyruvate; it releases lactate first, then $NAD^+$).

  • Random sequential: Substrates bind and products release in an arbitrary sequence.

• Double-displacement (ping-pong) reactions: One or more products dissociate from the enzyme before all substrates have bound.

  • Defined by the formation of a temporarily modified substituted enzyme intermediate.

  • Example: Aspartate Aminotransferase transfers an amino group from aspartate onto the enzyme, releasing oxaloacetate; $\alpha$-ketoglutarate subsequently binds the aminated enzyme intermediate ($E\text{-}NH_3$) to yield glutamate.

Allosteric Enzymes and Metabolic Regulation

• Allosteric enzymes regulate the rate of biochemical flux through metabolic pathways, typically catalyzing the committed step.

• Structural and kinetic hallmarks of allosteric enzymes:

  • Quaternary structure comprising multiple active and regulatory sites across distinct subunits.

  • Cooperative substrate binding, which generates a sigmoidal velocity-versus-substrate curve rather than a hyperbolic Michaelis–Menten curve.

  • Direct responsiveness to environmental signals and allosteric effectors.

• Feedback inhibition:

  • Pathway end-products inhibit the enzyme catalyzing the committed step of their synthesis.

  • Feedback inhibitors bind to distinct regulatory sites rather than the catalytic active site and frequently display no structural resemblance to the pathway's substrates or initial products.

• Cross-pathway regulation in branched pathways (e.g., synthesis of product $K$ from intermediate precursors $F$ and $I$):

• Product $K$ feedback-inhibits the initial committed enzymes $e_1$ and $e_{10}$.

• To maintain stoichiometric balance between intermediates $F$ and $I$: excess compound $F$ stimulates $e_{10}$ while inhibiting $e_1$. Conversely, excess compound $I$ stimulates $e_1$ while inhibiting $e_{10}$.

  • The Concerted Model for Allostery (Monod–Wyman–Changeux / MWC Model):

• The enzyme exists in two distinct conformations: the relaxed ($R$) state (active conformation) and the tense ($T$) state (significantly less active conformation).

• In the absence of ligand, $T$ and $R$ states are in equilibrium, with the $T$ state being thermodynamically favored ($L_0 = T/R$, often in the hundreds).

• Symmetry rule: All subunits within the multimer must occupy the same conformational state simultaneously (all $T$ or all $R$; no hybrid states are permitted).

• Substrate binds preferentially to the $R$ state. Binding of substrate traps the enzyme in the $R$ conformation, shifting the global equilibrium toward $R$.

  • The Sequential Model (Koshland–Némethy–Filmer / KNF Model):

• Substrate binding to one subunit induces a local conformational transition that influences neighboring subunits.

• Unlike the concerted model, the sequential model accommodates negative cooperativity, in which binding of substrate to one site decreases the affinity of adjacent sites.

  • The Threshold Effect:

• Cooperativity produces a switch-like response over a narrow concentration range of substrate.

• Comparison of the substrate fold-change required to accelerate velocity from $10\%$ to $80\%$ of $V_{\max}$:

  • Michaelis–Menten enzyme: $\frac{[S]_{0.8V_{\max}}}{[S]_{0.1V_{\max}}} = 27$

  • Allosteric enzyme: $\frac{[S]_{0.8V_{\max}}}{[S]_{0.1V_{\max}}} = 4$

• Heterotropic versus Homotropic Effects:

  • Homotropic effects: Cooperative interactions generated by identical molecules (substrates binding to catalytic sites).

  • Heterotropic effects: Regulatory interactions driven by distinct molecules (allosteric effectors binding to non-catalytic regulatory sites).

  • Positive effectors (activators) bind and stabilize the $R$ state, shifting the sigmoidal curve to the left and lowering the substrate threshold.

  • Negative effectors (inhibitors) bind and stabilize the $T$ state, shifting the sigmoidal curve to the right and raising the substrate threshold.

  • Aspartate transcarbamoylase (ATCase) demonstrates both effects: activated by $ATP$ ($2\,mM$ shifts curve left) and feedback-inhibited by $CTP$ ($0.4\,mM$ shifts curve right).

Clinical Pathologies of Kinetic and Allosteric Dysfunction

• Ethanol Sensitivity and Aldehyde Dehydrogenase Isozymes:

  • Hepatic alcohol dehydrogenase converts ethanol into acetaldehyde:     $\text{CH}_3\text{CH}_2\text{OH} + \text{NAD}^+ \rightleftharpoons \text{CH}_3\text{CHO} + \text{NADH} + \text{H}^+$

  • Acetaldehyde is subsequently converted to acetate by aldehyde dehydrogenase.

  • Humans express two principal aldehyde dehydrogenase isozymes:

  • A low-$K_M$ mitochondrial form.

  • A high-$K_M$ cytoplasmic form.

  • Individuals sensitive to alcohol carry a point mutation causing a single amino acid substitution that inactivates the mitochondrial enzyme.

  • In these individuals, acetaldehyde can only be cleared by the high-$K_M$ cytoplasmic isozyme, which requires elevated concentrations of acetaldehyde to function at an appreciable rate.

  • Acetaldehyde escapes into the circulation, causing facial flushing, tachycardia, and nausea.

• Gout and Loss of Allosteric Feedback Control:

  • Gout results from the crystallization of excess sodium urate within joint linings and synovial fluid, triggering painful inflammation when engulfed by immune cells.

• Phosphoribosylpyrophosphate synthetase (PRS) catalyzes a critical committed step in nucleotide synthesis:     $\text{Ribose 5-phosphate} + \text{ATP} \xrightarrow{\text{Phosphoribosylpyrophosphate synthetase}} \text{5-phosphoribosyl-1-pyrophosphate} + \text{AMP}$

• Under normal conditions, PRS is feedback-inhibited by purine nucleotides.

• Mutations altering the allosteric regulatory site render PRS insensitive to purine feedback inhibition while leaving catalytic activity intact.

• The resulting uncontrolled production of 5-phosphoribosyl-1-pyrophosphate (PRPP) leads to an overproduction of purine nucleotides, which are degraded into excess urate.

Single-Molecule Enzymology Versus Ensemble Studies

• Ensemble Studies:

  • Conventional kinetic assays analyze solutions containing billions of enzyme molecules simultaneously.

  • Assumes all enzyme molecules in the population behave uniformly.

  • The measured parameters (such as $K_M$ and $V_{\max}$) represent an average of the entire heterogeneous assembly.

• Single-Molecule Studies:

  • Experimental methods track individual enzyme molecules in isolation.

  • Enables detection of rare or transient conformational intermediates, fleeting reaction steps, and mechanical forces.

  • Pitfall: Risk of misinterpreting statistical outliers as representative behaviors.

• Molecular heterogeneity example:

  • An enzyme population exists in three interconverting conformational states:

  • Form 1 ($45\%$ of population): Activity = $1$

  • Form 2 ($20\%$ of population): Activity = $2$

  • Form 3 ($35\%$ of population): Activity = $3$

  • Ensemble analysis yields a single average activity value:     $(0.45 \times 1) + (0.20 \times 2) + (0.35 \times 3) = 0.45 + 0.40 + 1.05 = 1.9$

  • Single-molecule analysis reveals three distinct peaks corresponding to each conformational state.

Quantitative Problem Solving and Worked Examples

• Problem 1: Penicillinase Kinetics and Turnover Number:

  • Penicillinase (molecular mass = $29.6\,\text{kDa} = 29{,}600\,\text{g/mol}$) from Staphylococcus aureus was assayed in a $10\,\text{ml}$ solution containing $10^{-9}\,\text{g}$ of purified enzyme. Penicillin hydrolysis was tracked over $1\,\text{minute}$.

  • Experimental data and reciprocal transformations:

  • $[\text{Penicillin}] = 1\,\mu M$, $v_0 = 0.11\,\text{nmol/min}$ $\rightarrow$ $1/[\text{Penicillin}] = 1\,\mu M^{-1}$, $1/v_0 = 9.09\,\text{min/nmol}$

  • $[\text{Penicillin}] = 3\,\mu M$, $v_0 = 0.25\,\text{nmol/min}$ $\rightarrow$ $1/[\text{Penicillin}] = 0.33\,\mu M^{-1}$, $1/v_0 = 4.00\,\text{min/nmol}$

  • $[\text{Penicillin}] = 5\,\mu M$, $v_0 = 0.34\,\text{nmol/min}$ $\rightarrow$ $1/[\text{Penicillin}] = 0.20\,\mu M^{-1}$, $1/v_0 = 3.00\,\text{min/nmol}$

  • $[\text{Penicillin}] = 10\,\mu M$, $v_0 = 0.45\,\text{nmol/min}$ $\rightarrow$ $1/[\text{Penicillin}] = 0.10\,\mu M^{-1}$, $1/v_0 = 2.22\,\text{min/nmol}$

  • $[\text{Penicillin}] = 30\,\mu M$, $v_0 = 0.58\,\text{nmol/min}$ $\rightarrow$ $1/[\text{Penicillin}] = 0.033\,\mu M^{-1}$, $1/v_0 = 1.72\,\text{min/nmol}$

  • $[\text{Penicillin}] = 50\,\mu M$, $v_0 = 0.61\,\text{nmol/min}$ $\rightarrow$ $1/[\text{Penicillin}] = 0.020\,\mu M^{-1}$, $1/v_0 = 1.63\,\text{min/nmol}$

  • Lineweaver–Burk regression parameters:

  • $y$-intercept = $1.4602\,\text{min/nmol}$

  • Maximal velocity: $V_{\max} = \frac{1}{1.4602} = 0.68\,\text{nmol/min}$

  • Slope = $7.6225\,\text{min}\cdot\mu M\,\text{nmol}^{-1}$

  • Michaelis constant: $K_M = \frac{\text{slope}}{y\text{-intercept}} = \frac{7.6225}{1.4602} = 5.22\,\mu M$

  • Turnover number calculation ($k_{\text{cat}}$, or $k_2$):

  • $V_{\max} = 0.68\,\text{nmol/min} = \frac{0.68 \times 10^{-9}\,\text{mol}}{60\,\text{s}} = 1.13 \times 10^{-11}\,\text{mol/s}$

  • Total enzyme in moles: $\frac{10^{-9}\,\text{g}}{29{,}600\,\text{g/mol}} = 3.38 \times 10^{-14}\,\text{mol}$

  • Turnover number:       $k_{\text{cat}} = \frac{0.68}{60} \times 10^{-9} \times \frac{29{,}600}{10^{-9}} \approx 340\,s^{-1}$

• Problem 2: Initial Velocity at Defined Substrate Concentration:

  • Given: An enzyme with $V_{\max} = 100\,\text{moles/sec}$ and $K_M = 5000\,M$ operating at $[S] = 100\,M$.

  • Michaelis–Menten calculation:     $V_0 = V_{\max} \times \frac{[S]}{[S] + K_M}$     $V_0 = \frac{100\,\text{moles}}{\text{sec}} \times \frac{100\,M}{100\,M + 5000\,M}$     $V_0 = \frac{100\,\text{moles}}{\text{sec}} \times \frac{100\,M}{5100\,M} = \frac{100\,\text{moles}}{\text{sec}} \times \frac{1}{51}$     $V_0 = \frac{100\,\text{moles}}{51\,\text{sec}} \approx 1.96\,\text{moles/sec}$

Questions & Discussion

• Rate Constant Units:

  • Prompt: Which statement is correct?

  • First-order rate constants have the unit of $s$.

  • First-order rate constants have the unit of $M^{-1}$.

  • Second-order rate constants have the unit of $s^{-2}$.

  • Second-order rate constants have the unit of $M^{-1}s^{-1}$.

  • Answer: Second-order rate constants have the unit of $M^{-1}s^{-1}$. (First-order rate constants have units of $s^{-1}$).

• Calculation of Initial Reaction Velocity:

  • Prompt: What is the initial velocity of catalysis for an enzyme with a $V_{\max} = 100\,\text{moles/sec}$ and a $K_M = 5000\,M$ at a substrate concentration of $100\,M$?

  • $2500\,\text{moles/sec}$

  • $50\,\text{moles/sec}$

  • $1.96\,\text{moles/sec}$

  • $0.02\,\text{moles/sec}$

  • Answer: $1.96\,\text{moles/sec}$.

• Cooperativity in Allosteric Models:

  • Prompt: True or False: The sequential model of allosteric enzymes more readily accommodates positive cooperativity.

  • Answer: False. The sequential model more readily accommodates negative cooperativity.

• Characteristics of Ensemble Studies:

  • Prompt: Which statement(s) describes ensemble studies?

  • Experiments allowing one individual enzyme molecule to be examined at a time.

  • Experiments on the scale of a few microliters of such a solution, containing millions of enzyme molecules.

  • Generates a single value representing the average enzyme activity for a heterogeneous assembly.

  • Generates multiple values for enzyme activity which represent each of the observed forms of a heterogeneous assembly.

  • Answer:

  • Experiments on the scale of a few microliters of such a solution, containing millions of enzyme molecules.

  • Generates a single value representing the average enzyme activity for a heterogeneous assembly.

• Characteristics of Single-Molecule Studies:

  • Prompt: Which statement(s) describes single-molecule studies?

  • Experiments allowing one individual enzyme molecule to be examined at a time.

  • Experiments on the scale of a few microliters of such a solution, containing millions of enzyme molecules.

  • Generates a single value representing the average enzyme activity for a heterogeneous assembly.

  • Generates multiple values for enzyme activity which represent each of the observed forms of a heterogeneous assembly.

  • Answer:

  • Experiments allowing one individual enzyme molecule to be examined at a time.

  • Generates multiple values for enzyme activity which represent each of the observed forms of a heterogeneous assembly.