Biochemistry/Part II: Enzymes/Enzyme Kinetics

6. Enzyme Kinetics

Reading time: ~55 minutes | Topics: Michaelis-Menten equation, catalytic efficiency, Lineweaver-Burk plots, multi-substrate reactions, computational analysis

Reaction Velocity

Enzyme kinetics is the quantitative study of how fast enzyme-catalyzed reactions proceed and how their rates respond to changes in experimental conditions. The fundamental measurement is the initial velocity ($v_0$), defined as the rate of product formation at the very beginning of the reaction (typically the first 60-120 seconds), before significant substrate depletion or product accumulation occurs.

Why Measure Initial Velocity?

At early time points, several simplifying conditions hold:

  • Negligible product: The reverse reaction $\text{E} + \text{P} \to \text{ES}$ is insignificant since $[\text{P}] \approx 0$.
  • Constant [S]: Substrate concentration has not appreciably changed from its initial value.
  • No product inhibition: Product has not accumulated enough to inhibit the enzyme.
  • Steady-state [ES]: The enzyme-substrate complex concentration has reached a steady state.

The Saturation Curve

When $v_0$ is measured at increasing substrate concentrations $[\text{S}]$ (with fixed $[\text{E}]_T$), the resulting plot is a rectangular hyperbola:

  • At low $[\text{S}]$: the rate increases nearly linearly ($v_0 \propto [\text{S}]$, first-order kinetics).
  • At intermediate $[\text{S}]$: the rate increase begins to level off (mixed-order kinetics).
  • At high $[\text{S}]$: the rate approaches a maximum value $V_{\max}$ (zero-order kinetics; enzyme is saturated).

This saturation behavior was first systematically described by Leonor Michaelis and Maud Menten in 1913 and implies the formation of a discrete enzyme-substrate complex.

The Michaelis-Menten Equation

The Michaelis-Menten model assumes a simple two-step mechanism:

$$\text{E} + \text{S} \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} \text{ES} \overset{k_{\text{cat}}}{\longrightarrow} \text{E} + \text{P}$$

Full Derivation (Steady-State Assumption)

Step 1: Write the rate of change of [ES]:

$$\frac{d[\text{ES}]}{dt} = k_1[\text{E}][\text{S}] - k_{-1}[\text{ES}] - k_{\text{cat}}[\text{ES}]$$

Step 2: Apply the steady-state assumption -- after a brief induction period, [ES] reaches a constant value ($d[\text{ES}]/dt = 0$):

$$k_1[\text{E}][\text{S}] = (k_{-1} + k_{\text{cat}})[\text{ES}]$$

Step 3: Define the Michaelis constant:

$$K_m = \frac{k_{-1} + k_{\text{cat}}}{k_1}$$

Therefore: $[\text{E}][\text{S}] = K_m[\text{ES}]$, so $[\text{ES}] = \frac{[\text{E}][\text{S}]}{K_m}$.

Step 4: Use the enzyme conservation equation $[\text{E}]_T = [\text{E}] + [\text{ES}]$ to eliminate free enzyme concentration:

$$[\text{ES}] = \frac{[\text{E}]_T [\text{S}]}{K_m + [\text{S}]}$$

Step 5: Since $v_0 = k_{\text{cat}}[\text{ES}]$ and $V_{\max} = k_{\text{cat}}[\text{E}]_T$:

$$\boxed{v_0 = \frac{V_{\max}[\text{S}]}{K_m + [\text{S}]}}$$

This is the Michaelis-Menten equation -- the most important equation in enzyme kinetics.

Physical Meaning of $K_m$ and $V_{\max}$

Setting $v_0 = V_{\max}/2$ in the Michaelis-Menten equation and solving for $[\text{S}]$:

$$\frac{V_{\max}}{2} = \frac{V_{\max}[\text{S}]}{K_m + [\text{S}]} \quad \Rightarrow \quad [\text{S}] = K_m$$

Therefore, $K_m$ is the substrate concentration at which the reaction rate is half-maximal.

  • Small $K_m$ ($\sim\!\mu$M): high substrate affinity -- enzyme reaches half-saturation at low [S]
  • Large $K_m$ (mM range): lower substrate affinity -- needs more substrate to reach $V_{\max}/2$
  • $V_{\max} = k_{\text{cat}}[\text{E}]_T$: the maximum rate when all enzyme molecules are saturated with substrate

Catalytic Efficiency

Turnover Number ($k_{\text{cat}}$)

The catalytic constant $k_{\text{cat}}$ (also called the turnover number) is the number of substrate molecules converted to product per enzyme molecule per unit time when the enzyme is fully saturated:

$$k_{\text{cat}} = \frac{V_{\max}}{[\text{E}]_T}$$

Typical values range from $10^{-1}$ to $10^7$ s$^{-1}$. Carbonic anhydrase has$k_{\text{cat}} \approx 10^6$ s$^{-1}$, meaning each enzyme molecule converts one million CO$_2$molecules per second.

The Specificity Constant ($k_{\text{cat}}/K_m$)

The ratio $k_{\text{cat}}/K_m$ is the best measure of catalytic efficiency, as it accounts for both the rate of catalysis and the enzyme's affinity for substrate. At low substrate concentrations ($[\text{S}] \ll K_m$), the Michaelis-Menten equation reduces to:

$$v_0 \approx \frac{k_{\text{cat}}}{K_m}[\text{E}]_T[\text{S}]$$

This is a second-order rate equation, with $k_{\text{cat}}/K_m$ as the apparent second-order rate constant.

The Diffusion Limit: Catalytic Perfection

The upper limit for $k_{\text{cat}}/K_m$ is set by the rate of diffusion-controlled encounter between enzyme and substrate, approximately:

$$\left(\frac{k_{\text{cat}}}{K_m}\right)_{\max} \approx 10^8 \text{ to } 10^9 \text{ M}^{-1}\text{s}^{-1}$$

Enzymes that approach this limit are termed "catalytically perfect" -- every encounter between enzyme and substrate leads to product. Examples include:

Enzyme$k_{\text{cat}}$ (s$^{-1}$)$K_m$ (M)$k_{\text{cat}}/K_m$ (M$^{-1}$s$^{-1}$)
Superoxide dismutase$10^5$$3.6 \times 10^{-4}$$3.5 \times 10^9$
Catalase$4 \times 10^7$$1.1$$4 \times 10^7$
Carbonic anhydrase$10^6$$1.2 \times 10^{-2}$$8.3 \times 10^7$
Acetylcholinesterase$1.4 \times 10^4$$9 \times 10^{-5}$$1.6 \times 10^8$
Triose phosphate isomerase$4.3 \times 10^3$$4.7 \times 10^{-4}$$2.4 \times 10^8$

Lineweaver-Burk Plot

Before the widespread availability of computer software for nonlinear regression, linear transformations of the Michaelis-Menten equation were used to determine $K_m$ and $V_{\max}$ graphically. The Lineweaver-Burk (double reciprocal) plot is the most widely known.

Taking the reciprocal of both sides of the Michaelis-Menten equation:

$$\frac{1}{v_0} = \frac{K_m}{V_{\max}} \cdot \frac{1}{[\text{S}]} + \frac{1}{V_{\max}}$$

This equation has the form $y = mx + b$, where:

Slope

$$\frac{K_m}{V_{\max}}$$

y-intercept

$$\frac{1}{V_{\max}}$$

x-intercept

$$-\frac{1}{K_m}$$

Advantages and Limitations

Advantages:

  • Easy visual determination of $K_m$ and $V_{\max}$
  • Clearly distinguishes types of enzyme inhibition
  • Linear plots are straightforward to interpret

Limitations:

  • Distorts experimental error: points at low [S] (high 1/[S]) have amplified errors
  • Unequal weighting: data points are unevenly spaced
  • Modern practice: use nonlinear regression instead

Other Linear Transformations

Two alternative linearizations distribute experimental error more evenly than the Lineweaver-Burk plot:

Eadie-Hofstee Plot

Rearranging the Michaelis-Menten equation to plot $v_0$ vs. $v_0/[\text{S}]$:

$$v_0 = -K_m \cdot \frac{v_0}{[\text{S}]} + V_{\max}$$

Slope = $-K_m$, y-intercept = $V_{\max}$, x-intercept = $V_{\max}/K_m$. This plot provides a more uniform distribution of data points, but both axes contain $v_0$, introducing correlated errors.

Hanes-Woolf Plot

Plotting $[\text{S}]/v_0$ vs. $[\text{S}]$:

$$\frac{[\text{S}]}{v_0} = \frac{1}{V_{\max}} \cdot [\text{S}] + \frac{K_m}{V_{\max}}$$

Slope = $1/V_{\max}$, y-intercept = $K_m/V_{\max}$, x-intercept = $-K_m$. This is considered the best of the three linear plots because the independent variable ($[\text{S}]$) is known with high precision.

Modern Approach: Nonlinear Regression

Today, kinetic parameters are determined by fitting the Michaelis-Menten equation directly to$v_0$ vs. $[\text{S}]$ data using nonlinear least-squares regression (e.g., Levenberg-Marquardt algorithm). This avoids the distortions inherent in linear transformations and provides accurate estimates of $K_m$, $V_{\max}$, and their confidence intervals. Software packages like GraphPad Prism, SigmaPlot, and Python's SciPy make this straightforward.

Multi-Substrate Reactions

Most enzymes in metabolism catalyze reactions involving two or more substrates (bisubstrate reactions account for approximately 60% of all enzymatic reactions). W.W. Cleland developed a systematic notation and classification for these mechanisms.

Sequential Mechanisms

Both substrates must bind to the enzyme before any product is released. There are two subtypes:

Ordered Sequential: Substrates bind in a defined order (A before B), and products are released in a defined order (P before Q). The first substrate to bind creates part of the binding site for the second substrate.
$$\text{E} \xrightarrow{\text{A}} \text{EA} \xrightarrow{\text{B}} \text{EAB} \longrightarrow \text{EPQ} \xrightarrow{-\text{P}} \text{EQ} \xrightarrow{-\text{Q}} \text{E}$$

Example: Lactate dehydrogenase (NAD$^+$ binds before lactate).

Random Sequential: Either substrate can bind first, forming a ternary complex EAB regardless of binding order.

Example: Creatine kinase (ATP and creatine bind in either order).

Ping-Pong (Double Displacement) Mechanism

The first substrate binds, transfers a group to the enzyme (forming a substituted enzyme intermediate, F), and the first product is released before the second substrate binds:

$$\text{E} \xrightarrow{\text{A}} \text{EA} \xrightarrow{-\text{P}} \text{F} \xrightarrow{\text{B}} \text{FB} \xrightarrow{-\text{Q}} \text{E}$$

Distinguishing Mechanisms: Lineweaver-Burk Patterns

The two mechanism types produce distinctive patterns on Lineweaver-Burk plots when one substrate concentration is varied at several fixed concentrations of the other:

  • Sequential: Lines intersect to the left of the y-axis (intersecting pattern).
  • Ping-Pong: Lines are parallel (same slope, different y-intercepts).

Example: Aspartate aminotransferase follows a ping-pong mechanism -- the amino group from aspartate is transferred to PLP, forming PMP, before the second substrate ($\alpha$-ketoglutarate) binds.

Python: Michaelis-Menten Curve Analysis

The interactive code below plots Michaelis-Menten curves for different $K_m$ values and generates the corresponding Lineweaver-Burk (double reciprocal) plots. You can modify the parameters, add new curves, or change the substrate range to explore how enzyme kinetic parameters affect reaction velocity.

Michaelis-Menten Kinetics Simulator

Python

Plot v vs [S] and Lineweaver-Burk transformations for different Km values

michaelis_menten.py67 lines

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Key Concepts

Michaelis-Menten Equation

  • $v_0 = V_{\max}[\text{S}]/(K_m + [\text{S}])$
  • $K_m$ = [S] at which $v_0 = V_{\max}/2$
  • $V_{\max} = k_{\text{cat}}[\text{E}]_T$
  • Derived from steady-state assumption

Catalytic Efficiency

  • $k_{\text{cat}}$: turnover number (s$^{-1}$)
  • $k_{\text{cat}}/K_m$: specificity constant
  • Diffusion limit: $\sim 10^8\text{-}10^9$ M$^{-1}$s$^{-1}$
  • Catalytically perfect enzymes approach this limit

Lineweaver-Burk Plot

  • $1/v_0$ vs. $1/[\text{S}]$: linear plot
  • Slope = $K_m/V_{\max}$
  • y-int = $1/V_{\max}$, x-int = $-1/K_m$
  • Error amplification at low [S] is a key limitation

Multi-Substrate Reactions

  • Sequential: ordered or random (ternary complex)
  • Ping-pong: substituted enzyme intermediate
  • LB patterns: intersecting vs. parallel lines
  • Cleland notation for systematic classification
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