Question

Protein aggregation. You have a solution with mole fraction \(x\) of proteins. The proteins can aggregate and thus change their local concentration. The entropy of mixing is $$ \Delta S=-k[x \ln x+(1-x) \ln (1-x)] $$ and the internal energy is \(\Delta U=\chi x(1-x)\). Draw curves for the free energy of the system as a function of \(x\) (between 0 and 1 ), for \(X=0,1,2\), and \(4 k T\). Identify the stable states of the system. For what values of \(x\) does the system prefer to phase separate?

Short answer

Plot \( \Delta F\) as a function of \(x\) for \(\chi = 0, 1, 2, 4 kT\). Identify \(x\) where \(\Delta F\) is minimized and indicate phase separation when multiple minima exist.

Step-by-step solution

Understand the given equations

Given the entropy of mixing \( \Delta S=-k[x \ln x+(1-x) \ln (1-x)] \) and the internal energy \( \Delta U=\chi x(1-x) \), where \(x\) is the mole fraction of proteins, \(k\) is the Boltzmann constant, and \(\chi \) is the interaction parameter that can take values 0, 1, 2, or 4.

Define the free energy

The free energy, \(\Delta F\), combines the entropy and internal energy. It is given by: \[\Delta F = \Delta U - T \Delta S = \chi x(1-x) + kT [x \ln x + (1-x) \ln (1-x)]\]

Plot the free energy

To understand the behavior of the system, plot \(\Delta F\) as a function of \(x\) for different values of \(\chi \). Use \(\chi = 0, 1, 2, 4kT\) and plot for \(0 \leq x \leq 1\).

Analyze stability

Stable states correspond to local minima in the free energy curve. For each curve, identify the values of \(x\) where the free energy is minimized.

Determine phase separation

Phase separation occurs when the free energy curve has multiple minima separated by a maximum, indicating that the system can exist in more than one phase. Determine the values of \(\chi \) for which this occurs and the corresponding values of \(x\).

Key concepts

Entropy of Mixing

In the context of protein aggregation, the entropy of mixing helps us understand how the mixture of proteins and solvent molecules affects the disorder and randomness of the system. The entropy of mixing is given by the formula:

\[ \text{ S}=-k[x \text{ ln } x + (1-x) \text{ ln } (1-x)] \]

where:

• x = mole fraction of proteins
• k = Boltzmann constant
• The terms inside the logarithms represent the contributions of proteins and solvent molecules, respectively.

Entropy of mixing is always negative because the mixing process always leads to an increase in disorder (positive entropy change) in the system. This is due to the fact that the mixture is more disordered than the pure components alone.

In simpler terms, when proteins mix with other molecules, the system becomes more chaotic, and entropy (a measure of disorder) increases. This change in entropy plays a crucial role in determining the overall free energy and stability of the system.

Free Energy

Free energy, denoted as \( \text{{Delta F}} \), is critical in determining the stability of a system involving protein aggregation. The free energy combines both internal energy \( \text{{Delta U}} \) and entropy \( \text{{Delta S}} \) into a single value, defined by:

\[ \text{ \text{{Delta F}}= \text{{Delta U}} - T \text{{Delta S}} } \]
Substituting the given expressions for \( \text{{Delta U}} \) and \( \text{{Delta S}} \), we get:

\[ \text{{Delta F}} = \text{chi x}(1-x) + kT[x \text{ln } x + (1-x) \text{ ln } (1-x)] \]

This equation can be broken down as:

• The first term \( \text{{chi x}}(1-x) \) represents the internal energy, which depends on the interaction parameter \( \text{{chi}} \) and the mole fraction \( x \).
• The second term \( kT[x \text { ln } x + (1-x) \text{ ln } (1-x)] \) represents the entropy change, scaled by temperature \( T \).

By plotting \( \text{{Delta F}} \) versus \( x \), one can observe the stability of the system for different values of \( \text{{chi}} \). The stable states correspond to the local minima in the \( \text{{Delta F}} \) curve. A deeper local minimum means a more stable state for that specific mole fraction of proteins.

Phase Separation

Phase separation in a protein aggregation system occurs when the free energy curve \( \text{{Delta F}} \) shows multiple minima separated by a maximum. In other words, the system can favor multiple distinct phases instead of a single mixed phase.

To identify phase separation:

• Draw the free energy \( \text{{Delta F}} \) as a function of mole fraction \( x \) for various values of the interaction parameter \( \text{{chi}} \).
• Check for curves with more than one minimum. Such curves indicate more than one stable state, which suggests phase separation.

For example, when \( \text{{chi}} \) is high (e.g., 4 \( kT \)), there could be multiple local minima on the free energy curve, indicating regions where the system is stable in different configurations. At a certain \( x \), the system may prefer to split into two phases rather than remain as a mixed phase.

In simpler terms, phase separation means that the proteins prefer to clump together separately rather than evenly mixing with the solvent. This is often observed when the interaction parameter \( \text{{chi}} \) is high, causing distinct regions of protein aggregates and solvent to form within the solution.

By analyzing these free energy curves, one can determine the specific values of \( x \) where phase separation is favored and predict the behavior of protein solutions under different conditions.