Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 28: Problem 6

Relating stoichiometric to site constants. Derive the relationship between stoichiometric-model binding constants \(K_{1}\) and \(K_{2}\) and site-model binding constants \(K_{a}\) and \(K_{b}\). Suppose the sites are independent, \(K_{c}=K_{a} K_{b}\).

Short Answer

Expert verified
The relationship is \( K_a = K_1 \) and \( K_b = K_2 \). Since \( K_c = K_a K_b \), we get \( K_c = K_1 K_2 \).

Step by step solution

01

- Understand the Binding Constants

Identify the given constants: stoichiometric-model binding constants are denoted as \( K_1 \) and \( K_2 \), while site-model binding constants are \( K_a \) and \( K_b \). We are to derive a relationship connecting these constants.
02

- Write the Expression for the Total Binding Affinity

Given the sites are independent, we have \( K_c = K_a K_b \). This means the overall binding affinity for the stoichiometric model is a product of the individual site binding constants.
03

- Relate \( K_1 \) and \( K_c \)

In the stoichiometric model, for the first binding event, the binding constant \( K_1 \) can be related directly to \( K_a \) since it represents the binding at one site: \( K_1 = K_a \).
04

- Relate \( K_2 \) and \( K_c \)

Similarly, the second binding constant \( K_2 \) in the stoichiometric model relates to the second site: \( K_2 = K_b \).
05

- Combine the Relationships

Combine the two expressions: \( K_c = K_1 K_2 \) after substituting \( K_a \) and \( K_b \) with \( K_1 \) and \( K_2 \). Thus, \( K_a = K_1 \) and \( K_b = K_2 \) and \( K_c = K_1 K_2 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binding Affinity
Binding affinity measures how strongly two molecules interact with each other. It's crucial in understanding molecular interactions.
In the context of the problem, we're examining the binding affinity between two sites and their respective molecules. When we talk about binding constants, we're talking about these affinities. Each constant tells us how likely a molecule is to bind to a specific site.
For example, if a site-model binding constant is very high, it indicates a strong binding affinity between the molecule and the site. Conversely, a low constant means weak affinity. In mathematical terms:
  • High binding constant <=> High affinity

  • Low binding constant <=> Low affinity
Understanding these interactions helps us predict how molecules behave in different environments, which is essential for many applications in chemistry and biology.
Independent Sites
Sites being independent means that the binding of a molecule to one site does not affect the binding of a molecule to another site. This is a critical assumption in this exercise.
When sites are independent, the overall binding constant can be derived from the individual site constants. This simplifies the calculations and allows us to understand each site separately before combining them.
For example:
  • If site 1 has a binding constant of \( K_a \) and site 2 has a binding constant of \( K_b \), then the overall binding constant, \( K_c \), can be calculated as:
    \[ K_c = K_a K_b \]
Thus, independence allows for straightforward multiplication to find the overall binding interaction, without worrying about interference between the sites.
Site-Model Binding Constants
Site-model binding constants, denoted as \( K_a \) and \( K_b \), represent the binding affinities for individual sites in a given model.
The site-model approach is beneficial as it breaks down complex interactions into simpler, more manageable segments. Each constant represents one site's interaction with a molecule:
  • \( K_a \): Binding constant for the first site

  • \( K_b \): Binding constant for the second site
When calculating total binding affinity for these independent sites, you combine these individual constants.
This approach is pivotal for understanding complex binding scenarios, as it allows easy visualization and computation of how each site contributes individually to the overall binding affinity.
Stoichiometric Model
The stoichiometric model focuses on the overall binding constants, denoted as \( K_1 \) and \( K_2 \), representing the binding events sequentially.
In this model, the constants correspond directly to the binding events:
  • \( K_1 \): First binding event

  • \( K_2 \): Second binding event
If the binding sites are independent, the overall binding constant \( K_c \) can be derived as a product of these individual event constants.
This is why we can relate the site-model binding constants to the stoichiometric-model constants, combining them as:
\[ K_c = K_a K_b \ K_1 = K_a \ K_2 = K_b \]
Thus, simplifying the process of understanding and calculating complex molecular interactions.
Thermodynamics
Thermodynamics helps us understand the energy changes during molecular interactions. Binding constants are directly linked to these energy changes.
The relationship between binding constants and free energy changes (\( \triangle G \)) is given by:
\[ \triangle G = -RT \text{ ln } K \]
where:

  • \( R \): Universal gas constant
  • \( T \): Temperature in Kelvin
  • \( K \): Binding constant

Understanding this equation allows us to see that a higher binding constant (stronger binding affinity) corresponds to a more negative \( \triangle G \), indicating more favorable binding from an energy perspective.
Key takeaways:
  • High \( K \) => Strong binding and more negative \( \triangle G \)

  • Low \( K \) => Weak binding and less negative \( \triangle G \)

Thus, thermodynamics provides the energetic context to the binding constants, helping us understand the favorability of binding interactions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Three-site binding. A ligand \(X\) can bind to a macromolecule \(P\) at three different binding sites with binding constants \(K_{1}, K_{2}\), and \(K_{3}\) : $$ \begin{aligned} &X+P \stackrel{K_{1}}{\longrightarrow} P X, \quad X+P X \stackrel{K_{2}}{\longrightarrow} P X_{2}, \\ &\text { and } X+P X_{2} \stackrel{K_{3}}{\longrightarrow} P X_{3} . \end{aligned} $$ (a) Write the binding polynomial, \(Q\). (b) Write an expression for the number of ligands \(v\) bound per \(P\) molecule. (c) Compute \(v\) for \(x=[X]=0.05\), assuming \(K_{1}=1\), \(K_{2}=1\), and \(K_{3}=1000\). (d) Assume the same \(K\) values as in (c). Below ligand concentration \(x=x_{0}\), most of the macromolecular \(P\) molecules have zero ligands bound. Above \(x=x_{0}\), most of the \(P\) molecules have three ligands bound. Compute \(x_{0}\). (e) For \(x=x_{0}\) in (d), show the relative populations of the ligation states with zero, one, two, and three ligands bound.

Cooperative assembly of membrane proteins. You have constructed an alpha- helical peptide. It partitions into lipid bilayer membranes and assembles into oligomeric complexes within the membrane. You measure the concentration \(x\) of the peptide in the membrane and the fraction \(\theta\) of helical peptide molecules that form oligomeric complexes in the membrane. At temperature \(T=300 \mathrm{~K}\), the peptide concentrations \(x\) and the ratios of oligomeric complexes \(\theta /(1-\theta)\) are as given in Table 28.1. Table 28.1 \begin{tabular}{lr} \hline\(x(\mu \mathrm{M})\) & \(\theta /(1-\theta)\) \\ \hline 1 & 10 \\ 2 & 160 \\ 3 & 810 \\ 4 & 2560 \\ 5 & 6250 \\ \hline \end{tabular} (a) What are the association constant \(K\) and the Hill coefficient \(n\) ? (b) Make a Hill plot. (c) Plot \(\theta\) versus \(x\). (d) At temperature \(T=330 \mathrm{~K}\), you find that all the values of \(\theta /(1-\theta)\) in the table are doubled. Compute the \(\Delta H\) and \(\Delta S\) of binding.

Drug binding to a protein. A drug \(D\) binds to a protein \(P\) with two different equilibrium binding constants, \(K_{1}\) and \(K_{2}\) : $$ D+P \stackrel{K_{1}}{\longrightarrow} P D_{1}, \quad \text { and } \quad D+P D_{1} \stackrel{K_{2}}{\longrightarrow} P D_{2}, $$ where \(K_{1}=1000 \mathrm{M}^{-1}\), and \(K_{2}=2000 \mathrm{M}^{-1}\) at \(T=300 \mathrm{~K}\). (a) Write an algebraic expression for the fraction of protein molecules that have two drug molecules bound as a function of drug concentration \(x=[D]\). (b) If the drug concentration is \(x=10^{-3} \mathrm{M}\), what is the fraction of protein molecules that have one drug molecule bound? (c) If the drug concentration is \(x=10^{-3} \mathrm{M}\), what is the fraction of protein molecules that have zero drug molecules bound? (d) If the drug concentration is \(x=10^{-3} \mathrm{M}\), what is the fraction of protein molecules that have two drug molecules bound?

The entropy in the grand ensemble. Write an expres sion for the entropy in the grand canonical ensemble \((T, V, \mu)\).

A model for Alzheimer's fibrillization. Consider a solution of protein molecules in three possible states of aggregation: in state \(A_{1}\), the protein is a monomer in solution; in state \(A_{10}\), the protein is hydrophobically clustered into oligomers having 10 molecules each; and in state \(A_{100}, 100\) protein molecules are all aggregated into long fibrils. The equilibrium is given by $$ \begin{gathered} 10 A_{1} \stackrel{K_{\text {olim }}}{\longrightarrow} A_{10} \\ 100 A_{1} \stackrel{K_{\text {mpel }}}{\longrightarrow} A_{100} \end{gathered} $$ (a) Express the binding polynomial \(Q\) for this association equilibrium in terms of \(K_{\text {allgo, }} K_{\text {tharil }}\), and \(x\), where \(x\) is the concentration of protein \(A\). (b) Write an expression for the average number \(v\) of protein molecules per object (monomer, oligomer, or fibril).
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free