Question

Suppose that the dissociation constant of an \(\mathrm{F}_{\mathrm{ab}}-\) hapten complex is \(3 \times 10^{-7} \mathrm{M}\) at \(25^{\circ} \mathrm{C}\). (a) What is the standard free energy of binding? (b) Immunologists often speak of affinity \(\left(K_{\mathrm{a}}\right),\) the reciprocal of the dissociation constant, in comparing antibodies. What is the affinity of this \(\mathrm{F}_{\mathrm{ab}}\) ? (c) The rate constant for the release of hapten from the complex is \(120 \mathrm{s}^{-1}\). What is the rate constant for association? What does the magnitude of this value imply about the extent of structural change in the antibody on binding hapten?

Short answer

(a) \(37.45\,\mathrm{kJ/mol}\); (b) \(3.33\times10^6\,\mathrm{M}^{-1}\); (c) \(3.996\times10^8\,\mathrm{M}^{-1}\mathrm{s}^{-1}\); indicates minimal structural change.

Step-by-step solution

Calculate Standard Free Energy of Binding

The standard free energy change for a process can be calculated using the Gibbs free energy formula: \[\Delta G^\circ = -RT \ln K_d\]where \(R\) is the universal gas constant \(8.314 \text{ J/mol K}\), \(T\) is the temperature in Kelvin \(\text{(298 K for 25}^{\circ}\text{C)}\), and \(K_d\) is the dissociation constant. Substituting the given values:\[\Delta G^\circ = - (8.314 \text{ J/mol K})(298 \text{ K}) \ln(3 \times 10^{-7} \text{ M})\]\[\Delta G^\circ \approx - (8.314)(298) \times (-15.03) \approx 37.45 \text{ kJ/mol}\]

Calculate Affinity (Ka)

Affinity \(K_a\) is the reciprocal of the dissociation constant \(K_d\). Thus:\[K_a = \frac{1}{K_d}\]Substituting the given dissociation constant:\[K_a = \frac{1}{3 \times 10^{-7}} = 3.33 \times 10^6 \text{ M}^{-1}\]

Calculate the Rate Constant for Association (ka)

The association constant \(k_a\) can be found using the relationship between the rate constants and equilibrium constant, which is:\[K_a = \frac{k_a}{k_d}\]Here \(k_d = 120 \text{ s}^{-1}\) (the rate constant for dissociation), and we have already calculated \(K_a\). Solving for \(k_a\):\[k_a = K_a \times k_d = (3.33 \times 10^6 \text{ M}^{-1})(120 \text{ s}^{-1})\]\[k_a = 3.996 \times 10^8 \text{ M}^{-1}\text{s}^{-1}\]A high \(k_a\) value indicates a fast association rate, implying minimal structural changes are required for binding.

Key concepts

Dissociation Constant

The dissociation constant, often represented by the symbol \(K_d\), is a crucial concept in biochemistry used to describe the affinity between two molecules, such as an antibody and a hapten. It quantifies how easily a complex dissociates into its components. A smaller \(K_d\) indicates higher affinity, meaning the molecules prefer to stay bound rather than separate. The value of \(K_d\) is expressed in molarity (M), showing the concentration at which half of the molecules are bound in a complex. In our example, the dissociation constant was given as \(3 \times 10^{-7} \text{ M}\), which signifies a fairly strong binding between the \(\mathrm{F}_{\mathrm{ab}}\) region of the antibody and the hapten.

Standard Free Energy

Standard free energy of binding is a thermodynamic metric calculated through the Gibbs free energy equation: \[\Delta G^\circ = -RT \ln K_d\] It offers insights into the spontaneity of the binding process. Here, \(R\) is the universal gas constant (8.314 J/mol K), \(T\) is the absolute temperature in Kelvin, and \(K_d\) is the dissociation constant. Using these values, we can determine how favorable a binding interaction is. In our scenario, the standard free energy was approximately 37.45 kJ/mol, signifying that formation of the complex is a spontaneous and energetically favorable process.

Antibody Affinity

Affinity in immunology refers to the strength with which an antibody binds to its specific antigen, or hapten in this case. It is the inverse of the dissociation constant, represented as \(K_a\). This can be calculated using the formula: \[K_a = \frac{1}{K_d}\] By understanding affinity, researchers can compare different antibodies and determine which binds more effectively. In this example, the affinity was found to be \(3.33 \times 10^6 \text{ M}^{-1}\), indicating a high affinity and therefore a strong interaction between the antibody and hapten.

Rate Constant

The rate constant for association, \(k_a\), describes the speed at which two molecules come together to form a complex. It can be related to the dissociation rate constant \(k_d\) and the equilibrium constant \(K_a\) using the equation: \[K_a = \frac{k_a}{k_d}\] A high association rate constant implies that molecules rapidly find and bind to one another. In our case, the calculated \(k_a\) was \(3.996 \times 10^8 \text{ M}^{-1}\text{s}^{-1}\), indicating a very fast association rate.

Structural Change in Antibodies

Structural changes in antibodies occur upon binding to their haptens or antigens, often influencing the rate at which these molecules associate or dissociate. Minimal structural changes imply that the antibody can quickly and easily accommodate the hapten without the need to rearrange its structure significantly. This is inferred from a high association rate constant, as seen in our example. A swift binding process suggests the optimized design of an antibody's binding site, efficiently recognizing and attaching to the hapten with little reconfiguration needed.