Comprehensive Biochemistry Notes: Ligand Binding and Enzyme Kinetics

THEME E: BINDING OF LIGANDS TO PROTEINS LECTURE 1: BINDING SITES OF PROTEINS

Most dynamic proteins ($P$) perform their biological functions by binding to a Ligand ($L$). A ligand is defined as a smaller molecule that binds reversibly to a protein. This reversible interaction is the critical first step in various biological processes. Key examples of protein-ligand functions include enzymes binding to substrates, inhibitors, or activators; hormone receptors interacting with hormones; immunoglobulins (antibodies) binding to antigens; and hemoglobin/myoglobin transporting $O_2$ and $CO_2$.

The binding site is a relatively small part of the overall protein structure. It is highly specific, designed to recognize and accommodate particular ligands. While the binding site itself is small, the remainder of the polypeptide chain is essential as it contributes to the formation and stabilization of the protein's three-dimensional (3D) structure. This specific folding allows for the creation of the binding site at a precise location on the protein.

Physically, a binding site structure is often a 3D groove on the surface of the protein or an embedded pocket. These sites typically have a non-polar or hydrophobic nature. Because water competes for hydrogen-bonding interactions, it is excluded from the site to facilitate ligand binding. Non-polar amino acid residues make up the majority of the binding groove and are primarily involved in the binding interaction. In contrast, polar residues within the site lend specificity to the interaction and are often involved in the catalysis of reactions (turning substrate into product), in which case the site is termed a catalytic site.

LYSOZYME AND BINDING SPECIFICITY

Lysozyme serves as a primary example of a protein with a binding site groove/cavity. In its primary structure, specific residues like $Asp52$ and $Glu35$ are far apart ($52^{nd}$ and $35^{th}$ amino acids respectively). However, the folding of the redundant residues into a stable 3D structure brings these polar residues close together to form the catalytic site. Lysozyme functions by hydrolyzing peptidoglycans in bacterial cell walls, thereby inhibiting bacterial growth.

Binding is characterized by high specificity, which arises from the precise arrangement of amino acid residues in the binding site. A ligand will only bind if the binding site is complementary to the ligand's structure. It is important to note that while ligands bind at specific sites, inhibitors and activators can bind to other parts of the protein to regulate function, rather than directly at the primary ligand-binding site.

HYPOTHESES OF LIGAND BINDING AND FORCES INVOLVED

There are two primary models for protein-ligand interaction. Fisher's Lock-and-key hypothesis suggests a rigid, pre-existing complementarity between the protein and ligand. Conversely, Koshland's induced fit model suggests that the ligand binds specifically only after a conformational change occurs in the protein to make the site complementary. In this model, the binding site forms around the ligand. While induced fit is more common in nature, there must still be a high degree of initial complementarity to maintain specificity.

The forces involved in ligand binding are primarily weak non-covalent forces. Hydrogen bonds (H-bonds) are critical, with amino acids like $Trp$, $Arg$, and $Lys$ acting as donors, while $Ser$, $Thy$, $Asn$, $Gln$, and $Cys$ act as either donors or acceptors. $Asp$ and $Glu$ act as donors if protonated or acceptors if charged. Electrostatic forces involve positively charged groups (N-terminal, $Arg$, $Lys$, and sometimes $His$) and negatively charged groups (C-terminal, $Asp$, $Glu$, $Cys$, and sometimes $Tyr$). Dipoles involved in binding are found in residues containing $-OH$, $-SH$, and $C=O$ groups. Finally, Van der Waals and hydrophobic interactions occur between non-polar residues.

THE BINDING EQUATION AND FRACTIONAL SATURATION

The binding reaction for one mole of protein with one binding site is expressed as $P + L \rightleftharpoons PL$. The forward rate is $V_f = k_1 [P][L]$ and the reverse rate is $V_r = k_{-1} [PL]$. The binding constant ($K_b$) at equilibrium is defined as:

$K_b = \frac{[PL]}{[P][L]} = \frac{k_1}{k_{-1}}$

Conversely, the dissociation equilibrium ($PL \rightleftharpoons P + L$) uses the dissociation constant ($K_d$):

$K_d = \frac{[P][L]}{[PL]} = \frac{k_{-1}}{k_1}$

The units for $K_d$ are $mol\,dm^{-3}$ or $M$. A smaller $K_d$ value indicates a higher affinity of the protein for the ligand and stronger bonding. Typical $K_d$ values range from $10^{-2}$ to $10^{-8}\,M$.

Fractional saturation ($Y$) is a dimensionless measure of how much protein is bound in the PL-complex, ranging from 0 to 1. It is defined as:

$Y = \frac{[PL]}{[P]_{tot}} = \frac{[PL]}{[P] + [PL]}$

Substituting the relationships from the dissociation constant leads to the hyperbolic binding equation:

$Y = \frac{[L]}{K_d + [L]}$

If $[L] = 0$, then $Y = 0$. If $[L] = K_d$, then $Y = 0.5$, meaning half the protein is bound to the ligand. As $[L]$ becomes much greater than $K_d$ ($[L] \gg K_d$), $Y$ approaches 1, indicating the protein is saturated.

TRANSFORMATIONS AND MULTIPLE BINDING SITES

To analyze binding data, the hyperbolic equation can be transformed into a linear form using the Hughes-Klotz (double-reciprocal) plot:

$\frac{1}{Y} = \frac{K_d}{[L]} + 1$

In this equation, a plot of $\frac{1}{Y}$ versus $\frac{1}{[L]}$ yields a straight line where the slope is $K_d$ and the y-intercept is 1. When a protein has $n$ independent binding sites, the equation becomes:

$\frac{1}{Y} = \frac{K_d}{n} \times \frac{1}{[L]} + \frac{1}{n}$

In this case, the y-intercept is $\frac{1}{n}$ and the x-intercept is $-\frac{1}{K_d}$.

In most experimental or cellular conditions, the total ligand concentration is much higher than the total protein concentration ($[L]_{tot} \gg [P]_{tot}$). Because $[PL]$ cannot exceed $[P]_{tot}$, the formation of the complex does not significantly deplete the free ligand, allowing the approximation $[L]_{free} \approx [L]_{tot}$.

CO-OPERATIVE BINDING AND OLIGOMERIC PROTEINS

Co-operative binding occurs in oligomeric proteins with more than one dependent binding site. In these cases, the binding curves are not hyperbolic. Positive co-operativity occurs when the binding of the first ligand induces a conformational change that enhances the binding of subsequent ligands, resulting in a sigmoid (S-shaped) binding curve. Negative co-operativity occurs when subsequent binding is made more difficult, leading to a flattened curve.

Myoglobin (Mb) is a monomer with only one subunit and one binding site, resulting in a hyperbolic binding curve. In contrast, Hemoglobin (Hb) is a tetramer (four subunits) that exhibits positive co-operative binding, resulting in a sigmoidal curve. Transformations such as double-reciprocal graphs for these co-operative proteins are not linear.

THEME F: ENZYMES - CLASSES AND CHARACTERISTICS

Enzymes are biological catalysts, usually specialized proteins, that speed up reactions without being consumed or changed. A catalyst is defined as a compound that increases the rate of reaction. The substances enzymes act upon are called substrates, and the end results are products. Enzymes are characterized by the acronym CASPER: Catalytic ability, Specificity, and Regulatability.

Enzymes are classified into six main classes, designated by an EC number. The first number represents the class, and the following three numbers describe the reaction, bonds, and substrates.

1. Oxidoreductases: Catalyze oxidation-reduction reactions (e.g., Lactate: NAD oxidoreductase, EC 1.1.1.27).

2. Transferases: Catalyze group transfer reactions, essentially single replacements (e.g., L-alanine: 2-oxoglutarate aminotransferase, EC 2.6.1).

3. Hydrolases: Catalyze hydrolysis reactions (hydrolytic cleavage) using $H_2O$ (e.g., Diphosphate phosphohydrolase, EC 3.6.1.1).

4. Lyases: Catalyze elimination reactions to form double bonds or the addition of groups to double bonds (e.g., 2-oxo-acid carboxy-lyase, EC 4.1.1.1).

5. Isomerases: Catalyze intramolecular rearrangements (e.g., Alanine racemase, EC 5.1.1.1).

6. Ligases (Synthetases): Link two substrates using chemical potential energy, such as ATP (e.g., L-glutamate: ammonia ligase, EC 6.3.1.2).

CHEMICAL KINETICS AND REACTION ORDERS

In chemical kinetics, the reaction rate ($V$) is the amount of product formed per unit time. For a simple reaction $S \rightarrow P$, the rate is $V = k[S]$, where $k$ is the rate constant (measured in $s^{-1}$). This is a first-order reaction because it depends on the concentration of one reactant. For bi-reactant reactions ($S_1 + S_2 \rightarrow P_1 + P_2$), the rate is $V = k[S_1][S_2]$, which is second-order (overall order = 2).

Reversible reactions are expressed as $S \rightleftharpoons P$, where the net velocity is $V = k_1 [S] - k_{-1} [P]$. Progression curve analysis measures the concentration of product over time. The initial reaction rate ($V_0$) is determined by drawing a tangent to the start of the curve where $[P] \approx 0$. The slope of the plot of $V_0$ versus $[S]$ represents the rate constant $k$.

ENZYMES AS BIOLOGICAL CATALYSTS

Reaction rates are not explained solely by the free energy of substrates and products; reactions must proceed via a high-energy transition state ($\ddagger$), which is an intermediate between the substrate and product. Enzymes speed up reactions by a factor of $10^8$ to $10^{13}$ by decreasing the activation energy ($E_a$) of the transition state for both forward and reverse reactions. Catalysts do not change the equilibrium of the reaction.

The rate enhancement is calculated as $\frac{k_{cat}}{k_{non}}$. Key factors in enzyme catalysis include:

1. Weak binding of substrates: Binding must not be too strong, or it creates a "thermodynamic pit" where the enzyme-substrate (ES) complex is too stable to reach the transition state. $K_m$ must be optimized to allow proximity without excessive stability.

2. Proximity effect: Enzymes bring substrates together in the correct orientation, lowering degrees of freedom and loss of entropy, which speeds up the reaction.

3. Induced fit: Distortion of the enzyme and substrate structure toward the transition state.

4. Stabilization of the Transition State (TS): This is the most important factor. Enzymes bind to the transition state more strongly than to the substrate itself. As per Fisher's redefined model, the transition state is the "key," not the substrate.

MECHANISMS OF ENZYMATIC CATALYSIS

Mechanistic descriptions of enzyme action involve molecular, atomic, and sub-atomic events. Common types include nucleophilic substitution, where a nucleophile (electron-rich) attacks an electrophile (electron-poor), and cleavage reactions. Heterolytic cleavage involves an atom or proton keeping both electrons, while homolytic cleavage involves an equal split. Redox reactions involve the loss of electrons (oxidation) or gain of electrons (reduction).

Chemical catalysis types include:

1. Acid-base catalysis: Increases the rate through proton transfer. Specific acid-base catalysis involves $H^+$ or $OH^-$ from the solvent. General acid-base catalysis involves a molecule or functional group (general acid or base) with a $pK_a$ near the solution pH. Histidine, with an imidazole group and a $pK_a$ of 6-7, is an ideal general acid-base catalyst.

2. Covalent catalysis: The substrate binds covalently to the enzyme, forming a temporary intermediate before release.

MECHANISM OF CHYMOTRYPSIN

Chymotrypsin is a serine protease that hydrolyzes peptide bonds at amino acids with large side chains. It utilizes a catalytic triad consisting of $Asp102$, $His57$, and $Ser195$. The mechanism involves several steps:

1. The enzyme binds the substrate (E-S).

2. $His57$ acts as a general base to deprotonate $Ser195$, which then acts as a nucleophile to attack the substrate carbonyl carbon, forming the first tetrahedral intermediate ($TI_1$). This is stabilized by H-bonds in the oxyanion hole provided by residues like $Gly193$.

3. $TI_1$ collapses; $His57$ acts as a general acid to donate a proton to the leaving amine group. The first product ($P_1$, an amine) is released, leaving an Acyl-Enzyme intermediate.

4. Water enters the site. $His57$ acts as a general base to pull a proton from water, and the resulting $OH^-$ attacks the acyl-enzyme carbonyl.

5. This forms the second tetrahedral intermediate ($TI_2$), again stabilized by the oxyanion hole.

6. $TI_2$ collapses; $His57$ acts as a general acid to donate a proton back to $Ser195$, releasing the second product ($P_2$, a carboxylate). The enzyme is restored to its original state.

MICHAELIS-MENTEN KINETICS

The Michaelis-Menten (M-M) model describes the kinetics of enzymes that form an ES complex: $E + S \rightleftharpoons ES \rightarrow E + P$. The chemical conversion of ES to product ($k_2$ or $k_{cat}$) is often the rate-determining step. The M-M equation is:

$V_0 = \frac{V_{max}[S]}{K_m + [S]}$

Where $V_{max}$ is the maximum velocity when the enzyme is saturated, and $K_m$ is the Michaelis constant. $K_m = \frac{k_{-1} + k_2}{k_1}$. It is the substrate concentration at which $V_0 = 0.5 V_{max}$ and is a measure of the enzyme's affinity for the substrate.

If $[S] \ll K_m$, the rate is linear with respect to $[S]$: $V_0 = \frac{V_{max}[S]}{K_m}$. If $[S] \gg K_m$, the enzyme is saturated and $V_0 = V_{max}$. Double-reciprocal analysis (Lineweaver-Burk plot) uses the linear form:

$\frac{1}{V_0} = \left( \frac{K_m}{V_{max}} \right) \frac{1}{[S]} + \frac{1}{V_{max}}$

In this plot, the slope is $\frac{K_m}{V_{max}}$, the y-intercept is $\frac{1}{V_{max}}$, and the x-intercept is $-\frac{1}{K_m}$.

ENZYME EFFICIENCY AND INHIBITION

$k_{cat}$ is the catalytic constant or turnover number, indicating the number of substrate molecules converted to product per catalytic site per second ($s^{-1}$) under saturating conditions. Enzyme efficiency is measured by the ratio $\frac{k_{cat}}{K_m}$. A larger ratio indicates a more efficient enzyme.

Reversible enzyme inhibition occurs when a small molecule binds reversibly to reduce activity. The types include:

1. Competitive: Inhibitor binds to the active site.

2. Uncompetitive: Inhibitor binds only to the ES complex.

3. Mixed: Inhibitor binds to both E and ES.

4. Non-competitive: A sub-type of mixed inhibition.

REGULATION OF ENZYME ACTIVITY

Enzyme regulation prevents the accumulation of intermediates and wasteful usage of substrates. Methods include:

1. Substrate level control: Rate increases with substrate and decreases with product.

2. Feedback control: The final product of a pathway inhibits the first step.

3. Non-covalent allosteric modulation: Multimeric enzymes that undergo allosteric transitions between a T (tense/inactive) state and an R (relaxed/active) state. Modulators (inhibitors or activators) bind to change $K_m$ or $V_{max}$. Binding curves are sigmoidal.     - Symmetry Model (MWC): All subunits are concurrently in either T or R state.     - Sequential Model (KNF): T and R states can coexist in the same molecule.

4. Covalent modification: Attachment or removal of functional groups (e.g., phosphorylation and dephosphorylation) by other enzymes to alter activity. This is slower than allosteric regulation but reversible.

QUESTIONS & DISCUSSION

Question: How does the double-reciprocal plot help in determining multiple binding sites?

Answer: When a protein has $n$ independent binding sites, the Hughes-Klotz plot of $\frac{1}{Y}$ versus $\frac{1}{[L]}$ results in a y-intercept of $\frac{1}{n}$. By extrapolating the line to the y-axis, researchers can calculate the total number of binding sites on the protein. Similarly, the x-intercept provides $-\frac{1}{K_d}$, allowing the determination of the dissociation constant.