Enzymes

Introduction to Biological Catalysis:

Enzymes are biological molecules that function as catalysts, significantly increasing the rate of chemical reactions without being consumed in the process.

• Molecular Composition: While the vast majority of enzymes are proteins, certain catalytic RNA molecules exist, known as ribozymes.

• Fundamental Role: The primary objective of an enzyme is to facilitate reactions that would otherwise occur too slowly to support life. They achieve this by providing an alternative reaction pathway with a lower energy requirement.

Chemical Kinetics and Reaction Rates:

To understand enzyme action, one must first understand the simplest chemical reaction where a reactant (A) is converted into a product (B).

• Rate Equations: The reaction rate is defined by the change in concentration of reactants or products over time. In a first-order reaction, the rate is dependent on the concentration of the reactants: Reaction rate = k[A].

• The Rate Constant ($k$): This is a constant of proportionality that describes the proportion of reactant that will react per unit of time, typically measured in s^-1.

• Exponential Decay: As the reaction proceeds, the decrease in substrate concentration follows an exponential decay model, expressed as:

                [A] = [A]_0 \exp(-k_1t) , where [A]₀ is the initial concentration and t is time.

Thermodynamics vs. Reaction Kinetics:

A critical distinction must be made between whether a reaction can happen (thermodynamics) and how fast it happens (kinetics).

Thermodynamic Favourability:

Reactions are depicted using free energy diagrams where the progress is shown along a reaction coordinate.

• Exothermic Reactions: When the change in free energy (△G_R-P) is less than zero (△G < 0), the reaction is thermodynamically favourable and spontaneous.

• The Spontaneity Paradox: Spontaneous does not mean fast. For example, the hydrolysis of ATP to ADP and P_i has a highly favourable △G⁰ = -32kJmol^-1, yet ATP remains stable in solution for days at room temperature.

Rate Enhancement by Enzymes:

Without enzymes, biological reactions are incredibly sluggish. For instance, the half-life for glycine decarboxylation is 1.1 billion years. Enzymes provide massive rate enhancements, typically ranging from 10⁵ to 10¹⁷ fold.

• Chorismate mutase: Increases rate by 10⁶.

• Triosephosphate isomerase: Increases rate by 10⁹.

• Carboxypeptidase: Increases rate by 10¹³.

Transition State Theory:

Transition State Theory provides the framework for understanding how enzymes accelerate reactions by focusing on the high-energy barrier between reactants and products.

The Transition State (TS):

The transition state is a high-energy, very unstable chemical species that exists briefly 10^-13 to 10^-14s during the reaction. It is a point where original bonds are partially broken and new bonds are partially formed.

Activation Energy (△G^‡):

The difference in free energy between the reactants in their ground state and the transition state is the activation energy (△G^‡).

• Rate Relationship: The rate constant (k) is inversely and exponentially related to the activation energy. This is described by the Eyring Equation:

                                                                                                           k = \frac{\kappa T}{h} \exp\left(-\frac{\Delta G^{\ddagger}}{RT}\right)

• Where κ is the Boltzmann constant, h is Planck’s constant, R is the gas constant, and T is temperature.

  • Enzyme Action: Enzymes speed up reactions by reducing △G^‡ through the stabilisation of the transition state. Crucially, the overall △G of the reaction remains unchanged.

Catalytic Mechanisms:

Enzymes reduce the activation energy by addressing the enthalpic and entropic components of the reaction.

Enthalpic Stabilisation (△H^‡):

This involves the direct stabilisation of charges and the facilitation of bond making/breaking.

• General Acid-Base Catalysis: This mechanism prevents the development of unfavourable charges.

  • Acid Catalysis: Stabilises developing negative charges by transferring a proton (H⁺) from an acidic group on the enzyme.

  • Base Catalysis: Stabilises developing positive charges by accepting a proton.

• Electrostatic Catalysis: The enzyme active site uses charged side chains or metal ions (like Zn²⁺ in carbonic anhydrase) to create an electric field that stabilises transition state charges.

Entropic Stabilisation (△S^‡):

Entropy is a measure of disorder; more disorder is generally energetically favourable.

• Proximity and Orientation: Enzymes bring substrates together in the correct alignment for a reaction to occur.

  • This effectively converts a bimolecular reaction (high entropy loss) into a unimolecular-like reaction (low entropy loss), significantly increasing the effective concentration of reactants.

• Covalent Catalysis: In some cases, the enzyme forms a temporary covalent bond with the substrate, providing a highly reactive intermediate that further lowers the energy barrier.

  • Chymotrypsin is a classic example of an enzyme utilising this mechanism.

Introduction to Enzyme Kinetics:

Enzyme kinetics involves the characterisation of biological phenomena with respect to time, specifically measuring the change in concentration of chemical species as a reaction proceeds.

• Role of Enzymes: It is fundamental to note that enzymes affect only the kinetics (rate) of a reaction and do not alter the thermodynamic equilibrium.

• Experimental Approach: To analyse kinetics, data is typically collected at a constant enzyme concentration ([E]) using several different substrate concentrations ($[S]$). The initial rate or velocity (V₀) is then calculated for each.

• Reaction Velocity: A plot of V₀ against [S] typically yields a hyperbolic curve.

  • Region I (Low [S]): The velocity is approximately linear with respect to the substrate concentration.

  • Region II (High [S]): The velocity becomes approximately independent of the substrate concentration as the enzyme reaches saturation.

The Michaelis-Menten Model:

To quantify kinetic data, the Michaelis-Menten equation is employed: V_0 = \frac{V_{max}[S]}{[S] + K_m} .

Kinetic Parameters:

• V_max (Maximum Velocity): This represents the maximum reaction rate catalysed by a given concentration of enzyme when it is fully saturated with substrate.

• K_m (Michaelis Constant): This is defined as the substrate concentration at which the reaction rate is exactly half of the maximal velocity (V_max / 2).

Mathematical Derivation and Assumptions:

• Initial Velocity Assumption: By measuring only the initial rate (V₀), the back reaction of product (P) to substrate (S) (governed by k_-2) is assumed to be insignificant and can be ignored.

• Steady-State Approximation: When [E] \ll [S], it is assumed that the concentration of the enzyme-substrate complex ([ES]) remains constant over time. Therefore, the rate of ES formation equals the rate of ES breakdown:   k_1[E][S]=(k_{-1}+k_2)[ES]

• The Michaelis Constant: K_m is mathematically derived from these rate constants as

                                               K_m = \frac{k_{-1} + k_2}{k_1}

Biological Significance of K_m:

K_m provides vital information regarding the affinity of an enzyme for its substrate.

• Affinity Levels: A low K_m indicates strong substrate binding (often in the micromolar range), whereas a high K_m suggests relatively weak binding (often in the millimolar range).

• Case Study: Alcohol Metabolism: The importance of K_m is illustrated by aldehyde dehydrogenase (ALDH).

  • Most individuals possess a low K_m (high affinity) form of ALDH that efficiently clears toxic acetaldehyde.

  • Some individuals carry a point mutation (E487K) resulting in a high K_m (low affinity) enzyme. This leads to acetaldehyde build-up, causing symptoms such as facial flushing and tachycardia

Turnover Number and Catalytic Efficiency:

Beyond V_max and K_m, other parameters are used to assess how "good" an enzyme is.

• Turnover Number (k_cat): Also known as the molecular activity, k_cat represents the number of substrate molecules converted to product per unit of time (usually s^-1) per active site when the enzyme is saturated. It is calculated as: k_{cat} = \frac{V_{max}}{[E]_{total}} .

• The Specificity Constant (k_cat/K_m): This parameter allows for the comparison of different enzymes or the same enzyme acting on different substrates. It is considered a better indicator of overall catalytic efficiency than k_cat alone.

• Upper Limits of Catalysis: The value of k_cat/K_m has a physical upper limit determined by the rate of diffusion of the substrate to the enzyme's active site.

  • Diffusion-Limited Reactions: In these highly efficient enzymes (e.g. carbonic anhydrase), every collision between enzyme and substrate results in catalysis.

    • The limit for such reactions is approximately 10⁸ to 10⁹M^-1s^-1.

Analysing Kinetic Data:

While modern techniques use non-linear fitting methods to determine V_max and K_m with confidence intervals, historical methods relied on linear transformations.