Biochem lec 16-22

Kinetics of Bisubstrate Enzyme Reactions

• Prevalence and General Characteristics:

  • Greater than $60\%$ of all biochemical reactions involve two or more substrates (bisubstrate reactions).

  • Overall representation of a bisubstrate reaction: $A + B \rightleftharpoons P + Q$, catalyzed by enzyme $E$.

  • Substrate interactions occur either through ternary complex formation (single displacement) or through modified enzyme intermediates (double displacement).

• Single Displacement (Sequential) Reactions:

  • In a single displacement mechanism, both substrates ($A$ and $B$) must bind to the enzyme active site simultaneously before any product can be formed or released.

  • The rate-limiting chemical transformation step is the interconversion of the ternary enzyme-substrates complex to the ternary enzyme-products complex: $AEB \rightleftharpoons PEQ$.

  • Random Single Displacement:

  • Substrates $A$ and $B$ bind in no obligatory sequence; either $A$ can bind to $E$ first ($E + A \rightleftharpoons AE$, followed by $AE + B \rightleftharpoons AEB$) or $B$ can bind to $E$ first ($E + B \rightleftharpoons EB$, followed by $EB + A \rightleftharpoons AEB$).

  • Product release from the ternary product complex $PEQ$ can likewise occur in any order ($P$ leaves first or $Q$ leaves first).

  • Both substrates are present simultaneously in the active site.

  • Representative Example: Creatine Kinase catalyzes the reversible reaction: $\text{Creatine (Cr)} + \text{ATP} \rightleftharpoons \text{Phosphocreatine (CrP)} + \text{ADP}$. The overall direction is determined by the relative physiological concentrations of $\text{ATP}$, $\text{ADP}$, $\text{Cr}$, $\text{CrP}$, and the equilibrium constant $K_{\text{eq}}$. Most kinases operate through this random sequential pathway.

• Ordered Single Displacement:

  • A mandatory leading substrate ($A$) must bind to free enzyme $E$ before the second substrate ($B$) can bind ($E + A \rightleftharpoons AE$, then $AE + B \rightleftharpoons AEB$).

  • Substrate $B$ cannot bind to enzyme $E$ in the absence of substrate $A$.

  • The interconversion $AEB \rightleftharpoons PEQ$ is rate-limiting.

  • Product release occurs in an obligatory, ordered manner ($PEQ \rightleftharpoons PE + Q$, followed by $PE \rightleftharpoons E + P$).

• Steady-State Kinetic Profiling of Single Displacement:

  • Both random and ordered sequential mechanisms generate identical Lineweaver-Burk patterns.

  • Plots of $1/v$ versus $1/[A]$ at multiple fixed concentrations of the second substrate ($[B]$, $2[B]$, $3[B]$) yield a family of intersecting lines.

  • The lines intersect to the left of the $y$-axis (above or on the $x$-axis), resembling the graphical profile of noncompetitive inhibition.

  • The intersection point coordinates are given by:

  • $x\text{-coordinate} = -\frac{1}{K_S^A}$

  • $y\text{-coordinate} = \frac{1}{V_{\max}}\left(1 - \frac{K_m^A}{K_S^A}\right)$

  • The slope of each line at a given concentration of $B$ is determined by:     $\text{Slope} = \frac{1}{V_{\max}}\left(K_m^A + \frac{K_S^A K_m^B}{[B]}\right)$

  • Saturation is achieved when further increasing $[B]$ produces no additional shift in the Lineweaver-Burk profile.

• Double Displacement (Ping-Pong) Reactions:

  • The two substrates bind and react sequentially in a distinct ping-pong mechanism, precluding the formation of a ternary complex.

  • Substrate $A$ binds to free enzyme $E$ to form complex $AE$.

  • Substrate $A$ transfers a chemical group to the enzyme, producing the first product $P$ and generating a covalently modified enzyme intermediate designated $E'$ ($AE \rightleftharpoons PE' \rightleftharpoons E' + P$).

  • The first product $P$ dissociates from $E'$ before the second substrate ($B$) can bind.

  • The second substrate $B$ binds to the modified intermediate $E'$ to form $E'B$.

  • The transferred group on $E'$ is passed to substrate $B$, generating the second product $Q$ and regenerating native enzyme $E$ ($E'B \rightleftharpoons EQ \rightleftharpoons E + Q$).

  • Substrates $A$ and $B$ are never bound to the enzyme active site simultaneously.

  • Reactions conforming to this mechanism exhibit "burst kinetics," in which rapid formation of the first product $P$ is observed prior to the release of $Q$.

• Steady-State Kinetic Profiling of Double Displacement:

  • Double displacement reactions yield a family of strictly parallel lines on a Lineweaver-Burk reciprocal plot ($1/v$ versus $1/[A]$) as the concentration of $B$ is varied.

  • The slope remains invariant across different concentrations of $B$: $\text{Slope} = \frac{K_m^A}{V_{\max}}$.

  • The $y$-intercepts and $x$-intercepts are given by:

  • $y\text{-intercept} = \frac{1}{V_{\max}}\left(1 + \frac{K_m^B}{[B]}\right)$

  • $x\text{-intercept} = -\frac{1}{K_m^A}\left(1 + \frac{K_m^B}{[B]}\right)$

  • The parallel line pattern graphically resembles uncompetitive inhibition because the second substrate ($B$) interacts only with an enzyme form ($E'$) produced after the interaction with the first substrate ($A$).

• Representative Biochemical Example: Glutamate:Aspartate Aminotransferase:

  • Uses a pyridoxal phosphate (PLP) cofactor covalently bound to the resting enzyme ($E$ form).

  • Step 1: Glutamate binds to the enzyme-PLP complex and donates its amino group to the cofactor.

  • Step 2: The deaminated carbon skeleton departs as $\alpha$-ketoglutarate ($P$), leaving the enzyme in a covalently modified pyridoxamine phosphate form ($E'$ form).

  • Step 3: Oxaloacetate ($B$) binds to the $E'$ form and accepts the amino group from the modified cofactor.

  • Step 4: Aspartate ($Q$) is released, restoring the enzyme to the resting enzyme-PLP complex ($E$).

Diagnosing Bisubstrate Reaction Mechanisms

• Diagnostic Experimental Approaches:

  • 1. Kinetic Analysis (Lineweaver-Burk Plots):

  • Distinguishes single displacement from double displacement.

  • Intersecting lines indicate a single displacement mechanism (ternary complex formed).

  • Parallel lines indicate a double displacement (ping-pong) mechanism.

  • 2. Substrate Binding Assays:

  • Differentiates random from ordered single displacement pathways.

  • Assesses whether individual substrates can bind independently to free enzyme $E$.

  • If both $A$ and $B$ can independently bind to $E$, the mechanism is random.

  • If $B$ binds only after $A$ is pre-bound, the mechanism is ordered.

  • 3. Isotope Exchange Reactions:

  • Tests the capacity of an enzyme to catalyze partial reverse isotope exchange between a product and substrate in the complete absence of the co-substrate.

  • A double displacement enzyme facilitates isotopic exchange because it forms a covalent intermediate with half of the reaction components.

  • A single displacement enzyme cannot catalyze partial isotopic exchange because all substrates must be present simultaneously to proceed through the ternary complex.

• Case Study: Sucrose Phosphorylase versus Maltose Phosphorylase:

  • Both enzymes cleave disaccharides and attach inorganic phosphate ($\text{P}_i$) to yield glucose-1-phosphate ($\text{G-1-P}$):

  • Sucrose Phosphorylase: $\text{Sucrose} + \text{P}_i \rightleftharpoons \text{Glucose-1-P} + \text{Fructose}$

  • Maltose Phosphorylase: $\text{Maltose} + \text{P}_i \rightleftharpoons \text{Glucose-1-P} + \text{Glucose}$

  • Exchange Assay: Unlabeled $\text{G-1-P}$ is mixed with radiolabeled inorganic phosphate (${}^{32}\text{P}_i$) in the absence of the corresponding monosaccharide (fructose or glucose):     $\text{G-1-P} + {}^{32}\text{P}_i \rightleftharpoons \text{P}_i + \text{G-1-}^{32}\text{P}$

  • Results & Mechanistic Deduction:

  • Sucrose Phosphorylase catalyzes the exchange reaction efficiently in the absence of fructose.

    • Mechanism: Double displacement (Ping-Pong).

    • $\text{Sucrose} + E \rightleftharpoons E\text{-glucose} + \text{Fructose}$

    • $E\text{-glucose} + \text{P}_i \rightleftharpoons E + \text{Glucose-1-P}$

    • The glucosyl-enzyme covalent intermediate ($E'$) readily transfers glucose reversibly between phosphate and enzyme without needing fructose.

    • Structural data: Bifidobacterium, $113\,\text{kDa}$ homodimer, PDB 1R7A resolved at $1.77\,\text{Å}$, EC classification 2.4.1.7.

  • Maltose Phosphorylase fails to catalyze isotopic exchange in the absence of glucose.

    • Mechanism: Single displacement.

    • $\text{Maltose} + E + \text{P}_i \rightleftharpoons \text{Maltose}:E:\text{P}_i \rightleftharpoons \text{Glucose-1-P} + \text{Glucose}$

    • No covalent intermediate exists; all components of the ternary complex are required for bond rearrangement.

Energetics of Catalysis and Transition-State Stabilization

• The Transition State ($X^\ddagger$):

  • All chemical reactions proceed through an activated, high-energy transition state ($X^\ddagger$) positioned at the peak of the free energy barrier along the reaction coordinate.

  • $X^\ddagger$ represents a short-lived ($\approx 10^{-14}$ to $10^{-13}\,\text{s}$), unstable atomic arrangement where chemical bonds are in the process of forming or breaking (possessing partial bond character).

  • The transition state is an energetic maximum, not a chemical intermediate. It cannot be isolated or trapped.

  • The rate of a reaction is directly proportional to the concentration of reactant molecules that reach the transition state configuration.

• Thermodynamic Principles of Enzyme Action:

  • Enzymes accelerate reaction rates typically by factors of $10^7$ to $10^{14}$ (and up to $10^{21}$) compared to uncatalyzed reactions.

  • Enzymes accelerate the attainment of equilibrium ($K_{\text{eq}}$), but do not alter $K_{\text{eq}}$.

  • Enzymes do not alter the overall standard free energy of reaction ($\Delta G_{\text{rx}}$ or $\Delta G^\circ$).

  • Enzymes achieve acceleration by selectively lowering the activation energy barrier ($\Delta G^\ddagger$).

  • The free energy barrier between the enzyme-substrate complex and the enzyme-transition state complex ($ES \rightarrow EX^\ddagger$, designated $\Delta G_{\text{e}}^\ddagger$) is significantly lower than the barrier between free substrate and transition state ($S \rightarrow X^\ddagger$, designated $\Delta G_{\text{u}}^\ddagger$).

• Selective Binding Affinity ($K_T$ vs $K_S$):

  • Enzymes exhibit much higher binding affinity for the transition state ($X^\ddagger$) than for the ground-state substrate ($S$).

  • The dissociation constant of the enzyme for the transition state ($K_T$) is extraordinarily low, ranging from $10^{-20}\,\text{M}$ to $10^{-26}\,\text{M}$, whereas $K_S$ is typically $10^{-3}\,\text{M}$ to $10^{-6}\,\text{M}$.

  • Thermodynamic Pit Warning: If an enzyme binds the ground-state substrate too tightly ($K_S$ excessively small), the $ES$ complex falls into a deep thermodynamic energy well. The activation energy required to reach $EX^\ddagger$ would then equal or exceed the uncatalyzed barrier, abolishing catalysis.

• Mechanisms of $ES$ Complex Destabilization:

  • Formation of the $ES$ complex involves a loss of translational and rotational entropy ($\Delta S < 0$), which increases the intrinsic binding energy of the complex. To compensate, enzymes deliberately incorporate destabilizing features:

  • Structural Strain and Distortion (Induced Fit):

    • The "Lock and Key" model assumes static, complementary matching, which favors ground-state stabilization rather than catalysis.

    • The "Induced Fit" model posits flexible active sites that undergo conformational shifts upon binding, geometrically straining the substrate toward the transition state geometry.

  • Desolvation:

    • Solvation shells stabilize charged and polar groups on substrates in aqueous solution.

    • Transfer of substrate into the hydrophobic active site sheds hydrating water molecules, increasing electrostatic reactivity and ground-state free energy.

  • Electrostatic Destabilization:

    • Positioning like charges in close proximity within the active site creates unfavorable electrostatic repulsion in the $ES$ complex, which is relieved upon conversion to the transition state.

• Quantitative Magnitudes of Rate Accelerations:

Transition-State Analogs in Enzymology and Medicine

• Principles of Transition-State Analogs (TSAs):

  • While true transition states are unstable with half-lives on the femtosecond timescale, transition-state analogs (TSAs) are stable chemical compounds designed to mimic the planar geometry, charge distribution, and stereochemistry of $X^\ddagger$.

  • Because enzymes have evolved active sites complementary to the transition state, TSAs bind with much higher affinities than substrate molecules ($K_i$ values often $10^{-9}$ to $10^{-12}\,\text{M}$ or tighter).

  • TSAs act as potent, highly selective competitive inhibitors.

• Biochemical Examples:

  • Proline Racemase:

  • Converts L-proline to D-proline via a planar carbanion transition state.

  • The planar heterocyclic carboxylates pyrrole-2-carboxylate and $\Delta$-1-pyrroline-2-carboxylate mimic this planar intermediate and inhibit the enzyme.

• Yeast Aldolase:

  • Converts dihydroxyacetone phosphate (DHAP) and glyceraldehyde-3-phosphate into fructose-1,6-bisphosphate via an enediolate transition state coordinated to an active site $\text{Zn}^{2+}$ ion.

  • Phosphoglycolohydroxamate mimics the enediolate intermediate.

  • Kinetic metrics:

  • Substrate $K_m = 4 \times 10^{-4}\,\text{M}$

  • Inhibitor $K_i = 1 \times 10^{-8}\,\text{M}$

  • Affinity ratio: $\frac{K_m}{K_i} = 4 \times 10^4$ (binds $40{,}000$-fold more tightly than substrate)

• Pharmacological and Medicinal Applications:

  • Statins (e.g., Lipitor / atorvastatin calcium):

  • Potent cholesterol-lowering drugs that inhibit HMG-CoA reductase, the rate-limiting enzyme in cholesterol biosynthesis, by functioning as transition-state analogs of the mevalonate intermediate.

  • Antihypertensive Agents (Enalapril and Aliskiren):

  • Enalapril inhibits angiotensin-converting enzyme (ACE); Aliskiren inhibits renin.

  • Antiviral Neuraminidase Inhibitors (Tamiflu / oseltamivir):

  • Influenza infects $5\%$ to $15\%$ of the global population annually, causing up to $500{,}000$ deaths.

  • Viral neuraminidase is a membrane envelope glycoprotein essential for cleaving terminal sialic acid residues to release progeny virions from host cells.

  • Tamiflu mimics the oxocarbonium ion transition state of the neuraminidase reaction, preventing viral shedding and propagation.

  • HIV Protease Inhibitors (Invirase / saquinavir and Crixivan / indinavir):

  • Structure-based peptidomimetic drugs targeting viral polyprotein cleavage.

  • Agricultural Applications:

  • Inhibitors targeting insect juvenile hormone esterase serve as selective pesticides.

Chemical and Physical Mechanisms of Catalysis

• Electron Pushing Conventions:

  • Curved arrows illustrate the movement of electrons from filled orbitals (electron-rich nucleophiles) to electron-deficient orbitals (electrophiles).

  • A double-barbed (full) arrowhead indicates the concerted movement of an electron pair.

  • A single-barbed (fishhook) arrowhead indicates the movement of a single electron.

  • In bond cleavage, the arrow originates at the center of the covalent bond and points toward the electronegative atom taking the electron pair.

• Nucleophiles and Electrophiles in Biological Systems:

  • Nucleophiles: Electron-rich entities donating unshared electron pairs. Examples include halides ($\text{Cl}^-$, $\text{Br}^-$, $\text{I}^-$), oxygen species ($\text{H}_2\text{O}$, $\text{HO}^-$, $\text{ROH}$, $\text{RO}^-$, $\text{RCOO}^-$), sulfur species ($\text{H}_2\text{S}$, $\text{HS}^-$, $\text{RSH}$, $\text{RS}^-$), nitrogen species ($\text{NH}_3$, $\text{N}_3^-$), and carbanions/acetylides ($^-:\text{C}\equiv\text{N}$, $\text{R-C}\equiv\text{C}^-$).

  • Electrophiles: Electron-poor atoms accepting electron pairs. Major biological electrophiles include carbonyl carbons ($\text{C}=\text{O}$), phosphorus in phosphoryl groups ($\text{P}=\text{O}$), and protonated basic groups.

  • Proton Transfers: Protonation transforms poor leaving groups into neutral, stable leaving groups; deprotonation converts weak nucleophiles into strong nucleophiles.

• Primary Mechanisms of Catalysis:

  • 1. Catalysis by Proximity and Orientation (Near-Attack Complexes, NACs):

  • Enzymes immobilize substrates in precise geometric orientations within the active site, increasing effective local concentration and productive collision frequencies.

  • In solution, random thermal collisions align reacting atoms productively only $0.0001\%$ of the time; in an enzyme active site, this frequency rises to $1\%$ to $70\%$.

  • Pre-organized active sites form Near-Attack Complexes (NACs), defined by two criteria:

    • Reacting atoms are situated within van der Waals contact at distances $\le 3.2\,\text{Å}$.

    • Approach angle is within $\pm 15^\circ$ of the covalent bonding angle formed in the transition state.

  • Example: Alcohol dehydrogenase aligns the zinc-coordinated benzyl alcohol substrate and the C4 atom of the $NAD^+$ nicotinamide ring within strict NAC geometry alongside Ser48.

  • Protein Dynamics: Enzymes are non-rigid structures; continuous femtosecond-to-millisecond motions (bond stretching, side-chain rotations, loop swinging, and collective domain breathing) drive substrate binding, NAC formation, transition state stabilization, and product expulsion.

• 2. Covalent Catalysis:

  • An active-site nucleophile attacks an electrophilic center on the substrate, transiently forming a covalent enzyme-substrate intermediate. Subsequent hydrolysis or group transfer cleaves the bond, regenerating free enzyme.

  • Essential participant in all ping-pong (double displacement) mechanisms.

• 3. General Acid-Base Catalysis:

  • Specific acid-base catalysis relies on hydronium ($\text{H}_3\text{O}^+$) or hydroxide ($\text{OH}^-$) ions diffusing from bulk solvent.

  • General acid-base catalysis involves direct, active-site-mediated proton transfers occurring within the transition state via amino acid side chains or bound structural waters.

  • Proton transfers activate nucleophiles, stabilize developing negative or positive charges on transition state intermediates, and facilitate leaving group departure.

  • Catalytic residues commonly serving this role: Glu, Asp, His, Cys, Lys, and Tyr.

  • Histidine's Central Role: Because the imidazole ring has an intrinsic $pK_a \approx 6.0$, near physiological pH, Histidine readily cycles between protonated (acid) and unprotonated (base) states.

  • Microenvironmental Shifts in $pK_a$: Active-site local environments (dielectric constant, neighboring hydrophobic regions, proximity to like or opposite charges) induce dramatic shifts in amino acid $pK_a$ values relative to standard aqueous values:

• 4. Metal Ion Catalysis:

  • Metal cofactors ($\text{Zn}^{2+}$, $\text{Mg}^{2+}$, $\text{Fe}^{2+}$, $\text{Mn}^{2+}$) function as electrophilic catalysts because they carry high positive charge densities.

  • They orient substrates via coordination bonds, shield/polarize negative charges, and stabilize transient negative charges on intermediates.

  • Nucleophile Generation: In carbonic anhydrase, $\text{Zn}^{2+}$ is tetrahedrally coordinated to three active-site Histidine residues and one water molecule. Coordination lowers the $pK_a$ of bound water from $15.7$ to $7.0$, generating an active zinc-bound hydroxide ion ($\text{Zn}^{2+}\text{-OH}^-$) at physiological pH to attack carbon dioxide ($\text{CO}_2$).

• 5. Low-Barrier Hydrogen Bonds (LBHBs):

  • Conventional hydrogen bonds exhibit lengths of $\approx 0.28\,\text{nm}$ ($2.8\,\text{Å}$), have strengths of $10$ to $30\,\text{kJ/mol}$, and possess a double-well potential where the proton remains localized to one heteroatom.

  • When heteroatom separation decreases to $< 0.25\,\text{nm}$ ($< 2.5\,\text{Å}$) and their $pK_a$ values match closely, a Low-Barrier Hydrogen Bond (LBHB) forms.

  • In an LBHB, the energy barrier separating the two potential wells drops below the zero-point vibrational energy of hydrogen, allowing the proton to be shared equally (bond order $\approx 0.5$ for each interaction).

  • LBHB stabilization energies approach $60\,\text{kJ/mol}$, providing significant driving force for transition-state stabilization.

Serine Proteases: Catalytic Mechanism and Specificity

• Superfamily Members and Biological Functions:

  • Trypsin, Chymotrypsin, Elastase: Pancreatic digestive endopeptidases secreted as inactive zymogens.

  • Thrombin: Regulated endopeptidase operating within the blood coagulation cascade.

  • Subtilisin: Bacterial serine protease (example of convergent evolution sharing the catalytic triad).

  • Plasmin: Endopeptidase that cleaves fibrin polymers to dissolve blood clots.

  • Tissue Plasminogen Activator (tPA): Cleaves plasminogen to active plasmin; administered clinically to prevent or treat myocardial infarction and ischemic stroke.

  • Acetylcholinesterase: Hydrolyzes acetylcholine at neuromuscular junctions; shares an identical Ser-His-Asp triad mechanism despite not acting as a peptide hydrolase.

• Substrate Specificity and Binding Pockets:

  • Serine proteases cleave peptide bonds on the carboxyl side of specific residues (scissile bond: $\text{R}_1\text{-C(O)-NH-R}_2$), unless the following residue ($\text{R}_2$) is Proline (the rigid cyclic structure of proline blocks productive binding).

  • Trypsin:

  • Specificity: Cleaves on the C-terminal side of positively charged, basic amino acids (Arg, Lys).

  • Active Site Pocket: Deep, narrow, negatively charged pocket containing an Asp189 residue at the base, which forms an electrostatic salt bridge with the basic side chain of substrate.

  • Chymotrypsin:

  • Specificity: Cleaves on the C-terminal side of bulky aromatic amino acids (Phe, Trp, Tyr).

  • Active Site Pocket: Deep, wide, hydrophobic pocket lined by uncharged residues, with a neutral Ser189 located at the bottom (not to be confused with the catalytic Ser195).

  • Elastase:

  • Specificity: Cleaves on the C-terminal side of small, neutral residues (Ala, Gly, Val).

  • Active Site Pocket: Shallow pocket blocked by bulky, hydrophobic Val216 and Thr226 residues located near the entrance, excluding large or charged side chains.

• Pre-Steady-State Kinetics: Burst Kinetics:

  • Demonstrated using the artificial chromogenic ester substrate $p$-nitrophenylacetate ($p\text{-NPA}$).

  • The reaction produces $p$-nitrophenolate (yellow, absorbs strongly at $400\,\text{nm}$) and acetate.

  • Experimental mixing reveals biphasic kinetics:

  • Burst Phase: Immediate, rapid stoichiometric release of $p$-nitrophenolate equal to the molar concentration of active enzyme sites.

  • Steady-State Phase: Slower, linear rate of product formation corresponding to rate-limiting deacylation.

  • This confirms a two-step mechanism: fast acylation of the enzyme yielding an acyl-enzyme covalent intermediate, followed by rate-limiting hydrolytic deacylation by water.

• Detailed Catalytic Mechanism of Chymotrypsin:

  • The Catalytic Triad: Formed by Asp102, His57, and Ser195, which are widely separated in primary sequence but brought together in the active site.

  • Asp102: Anchors His57 in proper orientation via hydrogen bonding; forms a Low-Barrier Hydrogen Bond (LBHB) with protonated His57 to stabilize its positive charge.

  • His57: Functions as a general base to extract a proton from Ser195, and later as a general acid.

  • Ser195: Acts as the primary nucleophile attacking the carbonyl carbon of the scissile peptide bond.

  • Step-by-Step Reaction Sequence:

  • 1. Substrate Binding: Substrate aromatic ring inserts into the hydrophobic pocket, positioning the scissile carbonyl carbon adjacent to Ser195.

  • 2. Nucleophilic Attack (Acylation): His57 acts as a general base, deprotonating the $\text{-OH}$ of Ser195. The resulting alkoxide attacks the carbonyl carbon, transforming trigonal planar ($sp^2$) geometry into a tetrahedral oxyanion intermediate (Intermediate 1). Asp102 forms an LBHB with protonated His57.

  • 3. Oxyanion Stabilization: The negative charge on the tetrahedral oxygen is stabilized by hydrogen bonding from backbone amide $\text{-NH-}$ groups of Gly193 and Ser195 within the oxyanion hole.

  • 4. Cleavage of Peptide Bond: The tetrahedral intermediate collapses; the carbonyl double bond reforms. His57 acts as a general acid, donating a proton to the departing amine leaving group (the C-terminal fragment of the substrate, $P_1$), which diffuses out of the active site. The N-terminal fragment remains covalently attached as an acyl-enzyme intermediate.

  • 5. Hydrolytic Attack (Deacylation): A water molecule enters the active site. His57 acts as a general base, deprotonating water to generate a hydroxide nucleophile, which attacks the acyl carbonyl carbon, creating a second tetrahedral oxyanion intermediate (Intermediate 2) stabilized by the oxyanion hole.

  • 6. Intermediate Collapse and Regeneration: The second tetrahedral intermediate collapses. His57 acts as a general acid, transferring a proton back to the Ser195 oxygen. The carboxylate product ($P_2$) dissociates, and the active site catalytic triad is restored.

Aspartic Proteases: Catalysis and Drug Targeting

• General Characteristics and Representatives:

  • Aspartic proteases are endopeptidases that utilize two catalytic aspartate residues in the active site.

  • They prefer to cleave peptide bonds positioned between two bulky hydrophobic or aromatic residues.

  • Unlike serine proteases, no covalent enzyme intermediate is formed during catalysis.

• Structural Architecture:

  • Monomeric Aspartic Proteases (e.g., Pepsin): Consist of a single polypeptide chain folded into two structural lobes (N-terminal and C-terminal lobes) resulting from ancient gene duplication. Each lobe contributes one catalytic aspartate (Asp32 and Asp215 in pepsin) to form the active site cleft.

  • Dimeric Aspartic Proteases (e.g., HIV-1 Protease): Consist of a homodimer composed of two identical 99-amino acid subunits, exhibiting two-fold rotational ($C_2$) symmetry. Each monomer contributes a single catalytic aspartate (Asp25 and Asp25'). Flexible glycine-rich "flaps" sit atop the cleft and close over the substrate during catalysis. Homodimeric organization conserves viral genetic coding capacity.

• pH Dependence and Ionization State:

  • Aspartic proteases function at acidic pH (pepsin optimal activity occurs at pH $1.5$ to $2.5$).

  • The bell-shaped pH activity profile reflects the ionization states of the two catalytic aspartates:

  • One aspartate acts as a general base and must be unprotonated/ionized ($\text{Asp-COO}^-$, observed $pK_a \approx 1.5$\text{--}$2.0$ due to active-site microenvironment).

  • The other aspartate acts as a general acid and must be protonated/neutral ($\text{Asp-COOH}$, observed $pK_a \approx 4.5$\text{--}$5.0$).

• Catalytic Mechanism:

  • Catalysis occurs via general acid-base mechanisms assisted by Low-Barrier Hydrogen Bonds (LBHBs) that facilitate hydrogen tunneling between aspartyl carboxyl groups.

  • Step 1: Substrate binds between the catalytic aspartates.

  • Step 2: The unprotonated aspartate ($\text{Asp}^-$) acts as a general base to activate a catalytic water molecule, which directly attacks the scissile carbonyl carbon. Simultaneously, the protonated aspartate ($\text{Asp-H}$) acts as a general acid, polarizing the carbonyl oxygen.

  • Step 3: A non-covalent tetrahedral dihydroxy intermediate is generated without the formation of an acyl-enzyme bond.

  • Step 4: Proton rearrangement drives the collapse of the tetrahedral intermediate, cleaving the peptide bond and releasing the two peptide fragments while re-establishing the protonation states of the two aspartates.

• Retroviral Processing and Antiretroviral Drug Design:

  • HIV-1 synthesizes its structural proteins and replication enzymes as giant polyproteins from three primary reading frames: gag, pol, and env.

  • Translation produces the gag-pol polyprotein, which must be cleaved by HIV-1 protease into individual functional proteins: p17 (matrix), p24 (capsid), p15 (further cleaved to p7 nucleocapsid and p6), p11 (active protease), p66/p51 (reverse transcriptase), and p32 (integrase).

  • Inhibition of HIV-1 protease prevents viral maturation, yielding non-infectious, immature viral particles.

  • Protease Inhibitors (AIDS Drugs): High-affinity peptidomimetic transition-state analogs were designed to bind the symmetric active site under the flaps, including Crixivan (indinavir) and Invirase (saquinavir).

Regulatory Mechanisms of Enzyme Activity

• Hierarchical Levels of Regulation:

  • 1. Enzyme Abundance: Controlled at the genetic level by altering rates of transcription, mRNA processing, translation, and protein degradation.

  • 2. Enzyme Catalytic Activity: Regulates existing enzyme molecules directly to match immediate physiological and metabolic demands.

• Mass Action and Product Inhibition:

  • Reaction velocity is governed by substrate and cofactor availability ($E + S \rightleftharpoons ES \rightarrow E + P$).

  • As product accumulates, the velocity of the forward reaction decreases: $\frac{d[P]}{dt}$.

  • When the ratio of $[P]/[S]$ approaches the thermodynamic equilibrium constant ($K_{\text{eq}} = [P]/[S]$), the net rate becomes zero due to an equal rate of the reverse reaction.

• Five Primary Mechanisms Regulating Enzyme Activity:

  • 1. Reversible Covalent Modification:

  • Chemical modification of amino acid side chains alternates enzymes between active and inactive states.

  • Protein Phosphorylation: Most prevalent regulatory covalent modification in eukaryotes. Protein kinases transfer the $\gamma$-phosphoryl group of $\text{ATP}$ to specific hydroxyl side chains (Ser, Thr, Tyr) or carboxylates (Asp). Phosphoprotein phosphatases reverse this process by hydrolyzing phosphoryl groups.

• Lysine Acetylation: Acetylation of the $\epsilon\text{-NH}_3^+$ group on Lys neutralizes positive charge to an uncharged amide. Catalyzed by acetyl-CoA-dependent lysine acetyltransferases (KATs; $>30$ known mammalian isoforms); reversed by lysine deacetylases (KDACs). Regulates primary metabolic flux (e.g., malate dehydrogenase acetylation stimulates relative MDH activity to $\approx 200\%$).

  • 2. Zymogen (Proenzyme) Activation:

• Synthesis of enzymes as catalytically dormant precursors prevents premature degradation of producing cells or tissues.

• Irreversible proteolytic cleavage of one or more specific peptide bonds establishes catalytic competency.

• Proinsulin Processing: Synthesized as an 86-amino acid inactive polypeptide. Endopeptidic excision of the connecting peptide (C-peptide, residues 31–65) yields active mature insulin consisting of two chains linked by three disulfide bonds.

• Digestive Zymogens of Pancreas and Stomach: Enteropeptidase and trypsin initiate a proteolytic activation cascade in the duodenum:

• Chymotrypsinogen Activation Cascade:

  • Chymotrypsinogen ($245$ amino acids, single chain, inactive) is cleaved between Arg15 and Ile16 by trypsin, generating active $\pi$-chymotrypsin.

  • Two $\pi$-chymotrypsin molecules autolytically cleave one another, excising two dipeptides: Ser14-Arg15 and Thr147-Asn148.

  • The final stable product, active $\alpha$-chymotrypsin, consists of three polypeptide chains (A chain: residues 1–13; B chain: residues 16–146; C chain: residues 149–245) interconnected by two interchain disulfide bonds.

• 3. Isozymes (Isoenzymes):

  • Enzymes catalyzing identical reactions that differ in amino acid composition, kinetic parameters ($K_m$, $V_{\max}$), and regulatory properties.

  • Lactate Dehydrogenase (LDH): Tetrameric enzyme assembled from two distinct subunit types: $A$ (or M, muscle-predominant) and $B$ (or H, heart-predominant). Five distinct tetrameric combinations assemble with tissue-specific distribution:

  • $A_4$ (LDH-5): Dominates in skeletal muscle and liver (adapted to function in anaerobic conditions).

  • $A_3B$ (LDH-4): Prevalent in white blood cells.

  • $A_2B_2$ (LDH-3): Prevalent in brain and kidney.

  • $AB_3$ (LDH-2): Prevalent in red blood cells.

  • $B_4$ (LDH-1): Dominates in cardiac muscle (adapted to aerobic conditions, strongly inhibited by excess pyruvate).

• 4. Control by Modulatory Regulatory Proteins:

  • Enzyme activation or inhibition mediated by reversible physical binding of specialized protein subunits.

  • cAMP-Dependent Protein Kinase (Protein Kinase A / PKA): In resting cells, PKA exists as an inactive heterotetramer: $R_2C_2$, consisting of two regulatory ($R$) subunits and two catalytic ($C$) subunits. Elevated cyclic AMP ($\text{cAMP}$) binds allosterically to the regulatory dimer (two $\text{cAMP}$ per $R$ chain, totaling four $\text{cAMP}$). This induces dissociation of the complex:     $R_2C_2\text{ (inactive)} + 4\,\text{cAMP} \rightleftharpoons R_2(\text{cAMP})_4 + 2C\text{ (catalytically active)}$

• 5. Allosteric Regulation:

  • Non-covalent, reversible binding of low-molecular-weight metabolites (allosteric effectors) at regulatory sites spatially distinct from the catalytic active site.

  • Provides rapid response times to maintain metabolic homeostasis via positive (activation) or negative (inhibition) control.

  • Feedback Inhibition: The end product of a multi-step biosynthetic pathway ($F$) acts as an allosteric inhibitor of the first committed enzyme ($E_1$) in the sequence ($A \xrightarrow{E_1} B \xrightarrow{E_2} C \xrightarrow{E_3} D \xrightarrow{E_4} E \xrightarrow{E_5} F$).

  • Allosteric enzymes deviate from hyperbolic Michaelis-Menten kinetics, displaying sigmoidal (S-shaped) velocity versus $[S]$ plots indicative of positive cooperativity.

  • Allosteric enzymes are multi-subunit oligomers with multiple interacting active sites.

Allosteric Regulation: Theoretical Models (MWC vs KNF)

• The Monod-Wyman-Changeux (MWC) Model (Symmetry / Concerted Model):

  • Core Assumptions:

  • Allosteric proteins are symmetric oligomers consisting of identical protomers/subunits.

  • The oligomer exists in an equilibrium between two distinct conformational states: the Tense ($T$) state (low substrate affinity) and the Relaxed ($R$) state (high substrate affinity).

  • Concerted Symmetry Conservation: All subunits in an oligomer must transition simultaneously; hybrid states (e.g., $R_1T_{n-1}$) are forbidden. All subunits are either all $T$ or all $R$.

  • In the absence of ligand, the allosteric equilibrium constant $L$ heavily favors the unliganded $T_0$ state:       $L = \frac{[T_0]}{[R_0]} \gg 1$

  • Substrate binding affinities are described by dissociation constants $K_R$ and $K_T$, with substrate binding preferentially to the $R$ state ($K_R \ll K_T$; ratio $c = K_R / K_T \ll 1$).

  • Mechanism of Cooperativity:

  • Substrate ($S$) binds preferentially to the low-abundance $R_0$ form, trapping it as $R_1$.

  • By Le Chatelier's principle, removal of free $R_0$ pulls the pre-existing $T_0 \rightleftharpoons R_0$ equilibrium toward the $R$ conformation, increasing the number of high-affinity sites available for subsequent substrate molecules (positive homotropic cooperativity).

  • Allosteric Effectors:

  • Positive Heterotropic Effector (Allosteric Activator, $A$): Binds selectively to the $R$ state, shifting the equilibrium toward $R$. This decreases apparent $L$, shifts the sigmoidal velocity curve to the left, decreases $K_{0.5}$, and reduces cooperativity (making the curve more hyperbolic).

  • Negative Heterotropic Effector (Allosteric Inhibitor, $I$): Binds selectively to the $T$ state, shifting the equilibrium toward $T$. This increases apparent $L$, shifts the sigmoidal velocity curve to the right, increases $K_{0.5}$, and enhances sigmoidal character/cooperativity.

  • K-Systems versus V-Systems:

  • K-Systems: The allosteric effectors modify the substrate concentration required to reach half-maximal velocity ($K_{0.5}$); $V_{\max}$ remains unchanged. This occurs when $R$ and $T$ states have different affinities for substrate.

  • V-Systems: The allosteric effectors modify maximum velocity ($V_{\max}$), while $K_{0.5}$ remains unchanged. This occurs when $R$ and $T$ states have identical affinity for substrate ($K_R = K_T$) but differ in their catalytic rate constants ($k_{\text{cat}}$) and effector affinities.

  • Theoretical Limitation: Because the MWC model requires concerted transitions and $c \le 1$, the MWC model cannot explain negative cooperativity.

• The Koshland-Némethy-Filmer (KNF) Model (Sequential / Induced-Fit Model):

  • Core Assumptions:

  • In the absence of ligand, the protein exists in a single stable conformation. There is no pre-existing equilibrium between unliganded $R$ and $T$ states.

  • Substrate binding induces a conformational change in that specific subunit via an induced-fit mechanism.

  • Subunits change conformation sequentially as ligands bind.

  • The conformational change in the bound subunit alters the packing interactions across inter-subunit interfaces, transmitting mechanical strain to adjacent unliganded subunits.

  • This transmitted strain can either increase the neighboring subunits' affinity for substrate (positive cooperativity) or decrease their affinity for substrate (negative cooperativity).

  • Hybrid oligomeric intermediates containing mixed subunit conformations (e.g., one square and one diamond in a dimer) exist along the binding pathway.

• Comparison of MWC and KNF Models:

Allosteric Regulation and Cooperativity in Hemoglobin and Myoglobin

• Structural Properties of Hemoglobin and Myoglobin:

  • Myoglobin (Mb): Monomeric oxygen-storage hemeprotein found in muscle tissue ($153$ amino acids, molecular weight $17{,}200\,\text{Da}$). Composed of $8$ $\alpha$-helices designated A through H, cradling a single heme prosthetic group.

  • Hemoglobin (Hb): Tetrameric oxygen-transport protein found in red blood cells. Heterotetramer ($\alpha_2\beta_2$) containing two $\alpha$-chains ($141$ amino acids each) and two $\beta$-chains ($146$ amino acids each). Each of the four subunits possesses an independent heme prosthetic group and an 8-helix fold homologous to Mb.

  • Heme Architecture: Protoporphyrin IX ring coordinating a central ferrous ion ($\text{Fe}^{2+}$).

  • $\text{Fe}^{2+}$ has six coordination positions: four coordinated to pyrrole nitrogen atoms, the fifth coordinated to the imidazole ring of the proximal histidine (His F8), and the sixth available for reversible $O_2$ binding.

  • Oxidation of $\text{Fe}^{2+}$ (ferrous) to $\text{Fe}^{3+}$ (ferric) produces metmyoglobin or methemoglobin, which cannot bind oxygen.

• Oxygen Binding Curves and Cooperativity:

  • Myoglobin: Displays a hyperbolic binding curve, exhibiting high oxygen affinity ($P_{50} \approx 1\text{--}2\,\text{torr}$). It remains saturated at typical tissue oxygen pressures, serving as an oxygen reservoir.

  • Hemoglobin: Displays a sigmoidal binding curve, reflecting positive cooperativity among its four heme groups. The Hill coefficient ($n$) for adult Hb is approximately $2.8$.

  • Physiological Transport Efficiency: Hemoglobin achieves near-complete saturation ($\approx 98\%$) in the high oxygen environment of the lungs ($p\text{O}_2 \approx 100\,\text{torr}$), yet readily unloads oxygen in systemic capillaries ($p\text{O}_2 \approx 40\,\text{torr}$ in resting tissue, $p\text{O}_2 \approx 20\,\text{torr}$ in exercising muscle).

• Structural Mechanism of Cooperative Transition:

  • In deoxygenated hemoglobin (T state), the high-spin $\text{Fe}^{2+}$ ion has an atomic radius too large to fit within the porphyrin ring and sits approximately $0.055\text{--}0.06\,\text{nm}$ ($0.6\,\text{Å}$) out of the heme plane toward His F8.

  • Oxygen binding to the sixth coordination site draws electrons into a low-spin configuration, shrinking the iron radius and pulling $\text{Fe}^{2+}$ by $0.039\,\text{nm}$ into the porphyrin plane (leaving it only $0.026\,\text{nm}$ displaced).

  • The inward motion of $\text{Fe}^{2+}$ pulls proximal His F8, translating the entire F helix, EF corner, and FG corner.

  • This localized tertiary movement propagates to the subunit interfaces:

  • The $\alpha_1\beta_1$ and $\alpha_2\beta_2$ dimer interfaces remain rigid (packing contacts involving helices B, G, H, and the GH corner).

  • The sliding contacts at the $\alpha_1\beta_2$ and $\alpha_2\beta_1$ interfaces undergo a concerted $15^\circ$ rotation of one $\alpha\beta$ dimer relative to the other.

  • The quaternary transition from the T state to the R state ruptures 8 inter-subunit salt bridges that stabilize the deoxy conformation, including bridges involving C-terminal residues:

  • Carboxylate of $\beta_1\text{His146}$ with $\epsilon$-amino of $\alpha_2\text{Lys40}$.

  • Imidazole of $\beta_1\text{His146}$ with carboxylate of $\beta_1\text{Asp94}$.

  • Salt bridges between N-terminal and C-terminal residues of the $\alpha$-chains.

• Hybrid Allosteric Behavior of Hemoglobin:

  • Cooperativity in Hb incorporates features of both classical models:

  • KNF Characteristics: In deoxy-Hb, the heme groups of the $\beta$-chains are sterically inaccessible. Oxygen binds first to an $\alpha$-subunit, inducing localized tertiary changes that enhance the oxygen affinity of the neighboring $\alpha$-subunit by 3-fold.

  • MWC Characteristics: Once two oxygen molecules are bound, the energetic barrier is overcome, and the entire tetramer undergoes a concerted quaternary transition to the R state, shifting all four subunits into the high-affinity conformation.

• Allosteric Effectors of Hemoglobin:

  • 1. The Bohr Effect (Protons, $\text{H}^+$):

  • Hydrogen ions act as negative heterotropic effectors: elevated proton concentration (decreased pH) lowers oxygen affinity, shifting the binding curve to the right.

  • Structural Mechanism: Lower pH protonates specific residues, particularly $\beta\text{His146}$. Protonated $\beta\text{His146}$ forms a salt bridge with $\beta\text{Asp94}$, stabilizing the low-affinity T state.

  • Physiological Benefit: Actively metabolizing tissues produce lactic acid and carbonic acid, generating $\text{H}^+$ that triggers oxygen release. In pulmonary capillaries, oxygen binding drives proton release, shifting bicarbonate toward $\text{CO}_2$ for exhalation.

• 2. Carbon Dioxide ($\text{CO}_2$):

  • Decreases oxygen affinity through two mechanisms:

  • Indirect: Hydration by carbonic anhydrase produces carbonic acid, which dissociates into bicarbonate and protons, driving the Bohr effect:       $\text{CO}_2 + \text{H}_2\text{O} \rightleftharpoons \text{H}_2\text{CO}_3 \rightleftharpoons \text{H}^+ + \text{HCO}_3^-$

  • Direct (Carbamate Formation): $\text{CO}_2$ reacts directly with uncharged $\alpha$-amino groups at the N-termini of globin chains to form carbamate adducts, releasing a proton:       $\text{R-NH}_2 + \text{CO}_2 \rightleftharpoons \text{R-NH-COO}^- + \text{H}^+$

  • The negatively charged carbamate groups form additional salt bridges that stabilize the deoxy (T) state.

• 3. 2,3-Bisphosphoglycerate (2,3-BPG):

  • Synthesized in red blood cells as an allosteric regulator. In the absence of 2,3-BPG ("stripped hemoglobin"), hemoglobin exhibits a hyperbolic oxygen binding curve with high affinity.

  • 2,3-BPG acts as a negative heterotropic effector; its presence establishes the characteristic sigmoidal binding curve.

  • Binding Cavity: A single molecule of 2,3-BPG binds in the central cavity between the two $\beta$-subunits, existing only in the T state.

  • Electrostatic Contacts: The negative charges of 2,3-BPG form salt bridges with 8 basic amino acid groups lining the cavity:

  • Two $\alpha$-amino groups of the $\beta$-chain N-termini (Val1)

  • Two Lys82 residues

  • Four His residues (His2 and His143 from each $\beta$-chain)

  • Transition to the R state narrows the central cavity, expelling 2,3-BPG.

• 4. Fetal Hemoglobin (Hb F):

  • Consists of an $\alpha_2\gamma_2$ subunit composition, in which $\gamma$-chains replace adult $\beta$-chains.

  • At position 143, the $\gamma$-chain possesses an uncharged Serine residue in place of the basic Histidine (His143) found in $\beta$-chains.

  • This substitution removes two positive charges from the central cavity, significantly reducing its binding affinity for 2,3-BPG.

  • Consequently, fetal hemoglobin binds 2,3-BPG less tightly and exhibits higher oxygen affinity than adult hemoglobin (Hb A), allowing efficient oxygen transfer from maternal erythrocytes to fetal blood across the placenta.

Questions & Discussion

• Question: In the presence of a positive effector, how will the fractional saturation curve ($Y_S$ versus $[S]$) shift?

  • Response: The curve will shift to the left. A positive allosteric effector stabilizes the high-affinity Relaxed ($R$) state, lowering the concentration of substrate required to achieve half-maximal saturation ($K_{0.5}$) and reducing cooperativity.

• Question: What effect does the binding of a negative allosteric effector have on an allosteric enzyme (K-system) that shows positive cooperativity in substrate binding?

  • Response: It causes a shift to the right in the sigmoidal curve of velocity ($v$) versus $[S]$. Negative effectors stabilize the Tense ($T$) state, increasing $L$ ($T_0/R_0$ ratio), which decreases substrate affinity and elevates $K_{0.5}$.

• Question: How does the addition of a negative allosteric effector affect an enzyme operating as a V-system in terms of half-maximum velocity ($K_{0.5}$) and maximum velocity ($V_{\max}$)?

  • Response: $K_{0.5}$ remains constant while $V_{\max}$ decreases. In V-systems, both $R$ and $T$ states have equivalent affinities for substrate ($K_R = K_T$), but the effector modulates the catalytic rate constant ($k_{\text{cat}}$), thereby altering $V_{\max}$ without shifting $K_{0.5}$.

• Question: Under conditions of systemic alkalosis (serum pH higher than normal, $> 7.45$), how does hemoglobin's affinity for oxygen change, and in which direction does the binding curve shift?

  • Response: Hemoglobin's affinity for oxygen increases, and the binding curve shifts to the left. Reduced proton concentration favors the deprotonation of critical salt-bridge-forming residues (such as $\beta\text{His146}$), destabilizing the T state and favoring the oxygen-bound R state via the Bohr effect.