Question

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).

Short answer

The binding polynomial is \(Q = 1 + K_{\text{olim}} x^{10} + K_{\text{mpel}} x^{100}\). The average number of protein molecules per object is \(v = \frac{x + 10 K_{\text{olim}} x^{10} + 100 K_{\text{mpel}} x^{100}}{1 + K_{\text{olim}} x^{10} + K_{\text{mpel}} x^{100}}\).

Step-by-step solution

Understand the equilibrium reactions

Identify the given equilibrium reactions: \[10 A_{1} \stackrel{K_{\text {olim}}}{\longrightarrow} A_{10}\] \[100 A_{1} \stackrel{K_{\text {mpel}}}{\longrightarrow} A_{100}\]

Define the binding polynomial Q

The binding polynomial \(Q\) is a sum of the contributions of each state, weighted by their equilibrium constants and the concentration of the monomers: $$Q = 1 + K_{\text{olim}} x^{10} + K_{\text{mpel}} x^{100}$$

Express the average number of molecules per object (v)

The average number \(v\) is calculated as follows: $$v = \frac{\text{Total number of molecules in all states}}{\text{Total number of objects}}$$ We set up the expression:$$ v = \frac{x + 10 K_{\text{olim}} x^{10} + 100 K_{\text{mpel}} x^{100}}{1 + K_{\text{olim}} x^{10} + K_{\text{mpel}} x^{100}}$$

Key concepts

Aggregation States

Let's start by understanding the aggregation states of the protein molecules in this fibrillization model.
The model describes three states:

• State \(A_1\): monomer, the protein is a single molecule in solution.
  
• State \(A_{10}\): oligomer, where 10 protein molecules are hydrophobically clustered together.
  
• State \(A_{100}\): fibril, where 100 protein molecules aggregate into long fibrils.
  

These states represent different ways proteins in Alzheimer's disease can aggregate, going from individual molecules to larger and more complex structures.
Understanding these states is crucial as they are fundamental in explaining how the protein's behavior changes in different conditions.

Binding Polynomial

The binding polynomial \(Q\) is an equation that helps express the relationship between the concentration of monomers and equilibrium constants.
In our case, it sums contributions from the different protein states: the monomers, oligomers, and fibrils.
Using the provided equilibrium constants and monomer concentration \(x\), the binding polynomial is:
\[Q = 1 + K_{\text{olim}} x^{10} + K_{\text{mpel}} x^{100}\]
Here:

• \(K_{\text{olim}}\) and \(K_{\text{mpel}}\) are the equilibrium constants for the oligomer and fibril formations respectively.
  
• \(x\) represents the concentration of the monomer state.
  

The polynomial helps in quantifying the distribution of proteins across different states.
It gives insight into the predominance of each state under varied conditions of monomer concentration.

Equilibrium Constants

Equilibrium constants are essential for understanding the balance between different states in a reaction.
In the Alzheimer's fibrillization model, they tell us how readily proteins form oligomers and fibrils:

• \(K_{\text{olim}}\) describes the equilibrium between monomers and oligomers:
  \[10 A_1 \rightleftharpoons A_{10} \]
  
• \(K_{\text{mpel}}\) describes the equilibrium between monomers and fibrils:
  \[100 A_1 \rightleftharpoons A_{100} \]
  

These constants stem from a balance of forward and reverse reaction rates.
A larger \(K\) value implies a stronger tendency to form oligomers or fibrils.
In practical applications, knowing these constants aids in predicting which state will dominate under specific conditions.

Average Number of Molecules

The average number of molecules per object, \(v\), is another critical metric in this model.
It tells us on average how many protein molecules are found in each object (monomer, oligomer, or fibril).
The formula is:
\[v = \frac{x + 10 K_{\text{olim}} x^{10} + 100 K_{\text{mpel}} x^{100}}{1 + K_{\text{olim}} x^{10} + K_{\text{mpel}} x^{100}}\]
Here:

• The numerator represents the total number of molecules across all states.
  
• The denominator is the total number of objects (monomers, oligomers, and fibrils).
  

This ratio helps in determining the average cluster size.
It underscores the extent of aggregation and is pivotal for examining protein behavior in Alzheimer's disease.