Enzyme kinetics Study Guide

📖 Core Concepts Enzyme kinetics – study of how fast enzyme‑catalyzed reactions proceed and how conditions affect rate. Enzyme–substrate complex (ES) – transient binding of substrate at the active site; the source of the Michaelis complex. Rate‑determining step – the slowest elementary step that sets the overall reaction speed. \(V{\max}\) – maximal velocity when every enzyme molecule is saturated with substrate. \(KM\) – substrate concentration at which the reaction rate is half of \(V{\max}\); a measure of apparent affinity. \(k{cat}\) – turnover number; substrates converted per enzyme per second at saturating substrate. Catalytic efficiency – expressed as \(k{cat}/KM\); the slope of the rate vs. \([S]\) line at low \([S]\). Reversible vs. irreversible inhibition – reversible inhibitors bind non‑covalently (competitive, non‑competitive, uncompetitive); irreversible inhibitors form covalent bonds (affinity labeling, mechanism‑based). Multi‑substrate mechanisms – ternary‑complex (both substrates bind before chemistry) vs. ping‑pong (product released before second substrate binds). Cooperativity – binding of one substrate alters affinity at other sites; described by the Hill equation and coefficient \(n\). --- 📌 Must Remember Michaelis–Menten equation: \[ v0 = \frac{V{\max}[S]}{KM + [S]} \] \(k{cat} = V{\max}/[E]0\) (units s\(^{-1}\)). \(k{cat}/KM\) = catalytic efficiency; diffusion limit ≈ \(10^8\)–\(10^{10}\ \text{M}^{-1}\text{s}^{-1}\). Competitive inhibition: increases apparent \(KM\) (slope ↑), \(V{\max}\) unchanged. Non‑competitive inhibition: decreases \(V{\max}\) (y‑intercept ↑), \(KM\) unchanged. Uncompetitive inhibition: both \(KM\) and \(V{\max}\) decrease (parallel shift on Lineweaver–Burk). Ping‑pong plot: parallel secondary Lineweaver–Burk lines when varying the second substrate. Ternary‑complex plot: intersecting secondary lines. Hill equation: \[ v = V{\max}\frac{[S]^n}{K{0.5}^n + [S]^n} \] – \(n>1\) = positive cooperativity, \(n<1\) = negative. Burst phase in pre‑steady‑state kinetics indicates a rapid single turnover, revealing active‑enzyme concentration. --- 🔄 Key Processes Determining \(V{\max}\) & \(KM\) (steady‑state) Measure initial rates \(v0\) at several \([S]\). Fit data to the Michaelis–Menten equation (prefer non‑linear regression). Lineweaver–Burk linearization (if needed) Plot \(1/v0\) vs. \(1/[S]\); intercepts give \(-1/KM\) (x) and \(1/V{\max}\) (y). Identifying inhibition type Perform assay at multiple \([S]\) with/without inhibitor. Compare changes in intercepts on Lineweaver–Burk or use secondary plots. Multi‑substrate kinetic analysis Fix substrate A, vary B; generate secondary Lineweaver–Burk plots. Intersecting lines → ternary complex; parallel lines → ping‑pong. Pre‑steady‑state burst analysis Rapid‑mix enzyme & substrate, monitor product formation within milliseconds. Fit to a biphasic model: rapid burst (k\(\text{burst}\)) + slower steady‑state phase. --- 🔍 Key Comparisons Competitive vs. Non‑competitive Competitive: inhibitor ↔ active site; ↑\(KM\), \(V{\max}\) unchanged. Non‑competitive: inhibitor ↔ allosteric site; ↓\(V{\max}\), \(KM\) unchanged. Ternary‑complex vs. Ping‑pong Ternary: both substrates present in one enzyme complex; intersecting secondary LB lines. Ping‑pong: enzyme forms a modified intermediate (E); parallel secondary LB lines. Positive vs. Negative Cooperativity Positive: first binding ↑ affinity of remaining sites; Hill \(n>1\). Negative: first binding ↓ affinity of remaining sites; Hill \(n<1\). Reversible vs. Irreversible Inhibition Reversible: inhibitor dissociates; kinetic parameters change but enzyme is intact. Irreversible: covalent modification; activity loss follows first‑order kinetics until saturation. --- ⚠️ Common Misunderstandings “\(KM\) = affinity” – Only true when the chemical step is much slower than substrate dissociation; otherwise \(KM\) ≈ \( (k{-1}+k{\text{cat}})/k1\). Linear transformations are always accurate – Double‑reciprocal plots overweight low‑\([S]\) points; modern non‑linear regression is preferred. All inhibitors that lower \(V{\max}\) are non‑competitive – Mixed inhibition can also reduce \(V{\max}\) while affecting \(KM\). A sigmoidal curve always means cooperativity – Substrate inhibition or allosteric regulation can also produce curvature. --- 🧠 Mental Models / Intuition “Traffic jam” analogy: \(V{\max}\) is the speed limit when the highway (enzyme) is full of cars (substrate). Adding more cars (higher \([S]\)) does nothing once the road is saturated. “Key‑and‑lock vs. glove‑hand”: Competitive inhibitors are keys that fit the lock (active site); non‑competitive inhibitors are gloves that jam the hand (allosteric site) regardless of the key. “Two‑step relay race”: In ping‑pong, the baton (product) is passed to a new runner (modified enzyme) before the second substrate joins. In ternary‑complex, both runners line up together before the exchange. --- 🚩 Exceptions & Edge Cases Diffusion‑limited enzymes – When \(k{cat}/KM\) approaches the diffusion limit, further improvements in chemistry are impossible; rate is set by substrate arrival. Substrate inhibition – At very high \([S]\), rate can decrease; not covered by simple Michaelis–Menten. Mixed inhibition – Inhibitor binds both free enzyme and ES with different affinities; produces both \(KM\) and \(V{\max}\) changes. Enzymes with multiple conformational states – The simple two‑step model may not capture induced‑fit dynamics; kinetic traces can show pre‑steady‑state bursts. --- 📍 When to Use Which | Situation | Preferred Method / Plot | Reason | |-----------|--------------------------|--------| | Quick estimate of \(KM\) & \(V{\max}\) | Linear Lineweaver–Burk (if data are limited) | Easy visual intercepts | | Accurate parameter extraction | Non‑linear regression of Michaelis–Menten | Minimizes weighting bias | | Identify inhibition type | Compare Lineweaver–Burk with/without inhibitor | Distinct intercept shifts | | Distinguish ternary vs. ping‑pong | Secondary LB plots varying second substrate | Intersecting vs. parallel lines | | Detect covalent intermediates | Pre‑steady‑state burst analysis | Burst magnitude ≈ active‑enzyme concentration | | Assess cooperativity | Hill plot (log\(v/(V{\max}-v)\) vs. log\([S]\)) | Slope = Hill coefficient \(n\) | | Measure very fast rates | Rapid‑mixing, stopped‑flow spectroscopy | Captures sub‑second kinetics | --- 👀 Patterns to Recognize Linear increase → plateau → classic Michaelis–Menten saturation. Double‑reciprocal plot with unchanged y‑intercept → competitive inhibition. Parallel secondary LB lines → ping‑pong mechanism. Sigmoidal velocity curve → cooperative binding; check Hill coefficient. Burst phase followed by slower steady‑state → covalent intermediate or rapid chemistry followed by rate‑determining step. Large \(k{cat}/KM\) (≥ 10⁸ M⁻¹ s⁻¹) → diffusion‑limited (kinetically perfect) enzyme. --- 🗂️ Exam Traps Mistaking a change in slope for a change in intercept – remember which inhibition alters which axis on a Lineweaver–Burk plot. Assuming any sigmoidal curve = positive cooperativity – verify Hill \(n>1\); substrate inhibition can mimic a sigmoid. Using \(KM\) as a direct measure of binding affinity for all enzymes – only true when \(k{\text{cat}} \ll k{-1}\). Choosing linear transformations over non‑linear fitting – linear plots distort error; exam may penalize inaccurate \(KM\) or \(V{\max}\). Overlooking the burst amplitude – the burst size equals active enzyme concentration; forgetting this leads to mis‑interpreting enzyme purity. Confusing mixed inhibition with non‑competitive – mixed inhibition changes both \(KM\) and \(V{\max\)}\) but not in the simple patterns of pure non‑competitive. ---