BIOCHEM2027 Specialised Subject · 4.4
2027 · High School Specialised Subject · Biochemistry

6.3 Enzyme Kinetics as a Route into Mechanism

Enzyme kinetics

Initial rate, the Michaelis-Menten equation, Km, kcat and kcat/Km, double-reciprocal plots, multi-substrate mechanisms, pH effects, pre-steady state and inhibition.

8 SECTIONS17 FIGURESEnglish edition
01

Substrate concentration affects the rate

  • Initial rate v0 is the product formation rate over the first short interval, when substrate depletion, product inhibition, reverse reaction and inactivation are small
  • For a 1:1 reaction S to P, v0 also equals -d[S]/dt
  • Measurement conditions hold temperature, pH, ionic strength, enzyme amount, cofactors and time window constant
  • Run an enzyme-free blank and replicates, and take the slope of the interval where product increases linearly
  • When A = epsilon l c holds, dc/dt = (1/epsilon l)(dA/dt) gives the concentration rate
Figure 6.3-41Time course in which the initial product rate rises with substrate concentration; distinguish the average slope from the initial slope.
Figure 6.3-43The slope of the product curve changes as the reaction proceeds, so the whole trace must not be treated as an initial rate.
02

A quantitative relation links substrate concentration and rate

  • Minimal mechanism E + S to ES to E + P, with k1 for binding, k-1 for dissociation and k2 for product formation
  • Assumptions: negligible reverse reaction, [E]T much less than [S], and quasi-steady state for ES
  • d[ES]/dt = k1[E][S] - (k-1 + k2)[ES] is about zero, giving [ES] = [E]T[S]/(Km + [S]) with Km = (k-1 + k2)/k1
  • v0 = Vmax[S]/(Km + [S]) with Vmax = kcat[E]T, and v0 = Vmax/2 when [S] = Km
  • Km is generally not the true Kd, and Km approaches Kd only when k2 is much smaller than k-1
Figure 6.3-44Minimal reaction model for a single-substrate enzyme; at quasi-steady state ES formation and consumption are nearly equal.
Figure 6.3-45Michaelis-Menten saturation curve using the teaching values Vmax = 120 micromolar per minute and Km = 2.00 mM.
Figure 6.3-46Changing total enzyme concentration scales Vmax proportionally while Km of the simple model stays the same, given identical conditions.
03

Kinetic parameters compare enzyme activity

  • Vmax depends on enzyme concentration, so activity is normalised as kcat = Vmax/[E]T
  • kcat is the turnover per active site per second and kcat/Km the efficiency index at low [S]
  • When [S] is much less than Km, v0 approximates (kcat/Km)[E]T[S], a second-order form
  • Comparing purified enzymes requires active-site concentration, degree of inactivation, temperature, pH and cofactors
  • Comparing kcat between enzymes is only meaningful when substrate, conditions and rate-limiting step are comparable
표 6.3-1Meaning of the kinetic parameters
04

[Separate note 6-1] Double-reciprocal transformation

  • 1/v0 = (Km/Vmax)(1/[S]) + 1/Vmax, with slope Km/Vmax, y intercept 1/Vmax and x intercept -1/Km
  • The Lineweaver-Burk plot is useful for visualising theoretical differences between inhibition types
  • Its weakness is that small rate errors at low [S] are amplified by the reciprocal transformation
  • Modern practice fits the original [S] and v0 data by non-linear regression and checks weighting, residuals and uncertainty
Figure 6.3-48Slope and two intercepts of an ideal double-reciprocal plot; the x intercept is read by extrapolation into negative values.
Figure 6.3-49Reciprocal transformation of the same data magnifies small rate errors at low substrate concentration; the data are simulated.
05

Most enzyme reactions involve two or more substrates

  • Phosphate transfer by transferases and electron transfer by oxidoreductases both involve two substrates
  • A sequential mechanism forms a ternary EAB complex, either ordered with a fixed binding order or random
  • In a ping-pong mechanism the first product leaves, the enzyme persists as a modified form F, and the second substrate restores E
  • The simple ping-pong model does not require a ternary complex
Figure 6.3-50Sequential and ping-pong pathways compared; note the order of substrate binding and the modified enzyme form.
Figure 6.3-51Model lines at several concentrations of B against reciprocal concentration of one substrate; ideal sequential gives crossing lines and simple ping-pong tends to parallel lines.
06

Enzyme activity depends on pH

  • Protonation of active-site acid and base groups, substrate ionisation and structural stability all create pH dependence
  • Bell-shaped activity-pH curves are common but not universal
  • The optimum pH can shift with measurement conditions and substrate
  • Equal total substrate at each pH can still mean different reactive ionised fractions, so buffer, ionic strength and pre-treatment must be controlled
Figure 6.3-52Relative activity against pH assuming two ionisable groups; structural denaturation and substrate ionisation are simplified.
07

Pre-steady-state kinetics gives evidence on individual steps

  • Pre-steady state is the fast phase right after mixing, before ES reaches quasi-steady state
  • Stopped-flow and rapid-quench methods reach timescales inaccessible to manual sampling
  • A burst is a fast initial product phase followed by a slower linear phase
  • The burst suggests a fast first step and a slower later step such as product release, but reading its amplitude as active-site concentration requires a verified pathway
Figure 6.3-53Pre-steady-state burst model: a fast initial product phase followed by a slower steady state.
08

Enzymes are inhibited reversibly or irreversibly

  • Reversible inhibitors act through non-covalent binding and activity can recover when the inhibitor is removed
  • v0 = Vmax[S]/(αKm + α′[S]) with α = 1 + [I]/Ki and α′ = 1 + [I]/Ki′, so Vmax,app = Vmax/α′ and Km,app = αKm/α′
  • Competitive raises apparent Km with unchanged Vmax, uncompetitive lowers both, pure non-competitive lowers Vmax with unchanged Km, and mixed lowers Vmax with variable Km
  • Irreversible or time-dependent inhibition reduces the amount of active enzyme, often through covalent modification
  • Deepening inhibition over time alone does not prove irreversibility, so recovery after dilution or dialysis and mass data are needed
표 6.3-2Four models of reversible inhibition
Figure 6.3-55Enzyme species available for inhibitor binding in the four idealised models of reversible inhibition.
Figure 6.3-56Ideal saturation curves for competitive, uncompetitive, pure non-competitive and mixed inhibition; all values are simulated.
Figure 6.3-57The same four inhibition types on double-reciprocal plots; the pattern is a clue, not a proof.
Figure 6.3-58Concept comparison of reversible inhibition that recovers on dilution with inhibition that deepens over time; not real data.
Figure 6.3-59Recommended workflow for inhibition studies: initial rates at many concentrations, non-linear fitting, residual analysis, then independent mechanistic tests.
Figure 6.3-61Concept diagram of ornithine metabolism and the target action of eflornithine (DFMO); not a dosing diagram.
2027 High School Specialised Subject · Biochemistry · Sangil Park (Asan High School) · Reference material is printed as an appendix.
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