Question

Nucleosomes in a box Repeat the derivation of nucleosome accessibility using the canonical distribution. This means that you should imagine a nucleosome in a box with \(L\) DNA-binding proteins and then work out the probability that the nucleosome will be occupied by one of those proteins as a function of their concentration and as a function of the position of the binding site on the DNA.

Short answer

The probability of a nucleosome being occupied by a protein depends exponentially on the concentration of the protein and the position of the binding site on the DNA. It was determined by applying the canonical distribution to this biological system.

Step-by-step solution

Define the Problem Variables

First, define the variables for the problem. Let \( p \) be the probability of a nucleosome being occupied by a protein, \( L \) be the total number of proteins in the system, and \( x \) be the position of the binding site on the DNA.

Apply Canonical Distribution

Next, apply the concept of the canonical distribution. According to this distribution, the relative number of particles (proteins) that have a certain level (position on the DNA) is given by an exponential factor of the energy of the state at that level. In this case, the energy is assumed to be a function of the position \( x \). The canonical distribution, hence, gives the probability that a nucleosome at position \( x \) is occupied by a protein as \( p(x) = \frac{1}{Z} e^{-E(x)} \), where \( E(x) \) is the energy at position \( x \) and \( Z \) is the partition function for normalizing the probability, given by \( Z = \int_{0}^{L} e^{-E(x)} dx \).

Calculate Protein Concentration Effect

To find the probability as a function of the protein concentration, assume that the energy required for a protein to bind to the DNA is inversely proportional to the concentration of the protein. Therefore, the energy can be written as \( E(x) = k/n \), where \( k \) is a constant and \( n \) is the concentration of the protein. Substituting into the earlier equation gives: \( p(x) = \frac{1}{Z} e^{-k/n} \).

Final Result

Thus, from the above steps, we have the probability function as a function of the concentration of proteins and position on the DNA. The main conclusion here is that the probability of a nucleosome being occupied by a protein depends on the concentration of the protein and the position of the binding site on the DNA. The higher the protein concentration, the more likely it is that a nucleosome will be occupied by a protein. Similarly, the location of the binding site on the DNA also influences this probability.

Key concepts

Canonical Distribution

The canonical distribution is a principle used to understand how particles, such as proteins, distribute themselves among various energy states. In the context of nucleosome accessibility, it helps us predict how likely a DNA nucleosome is to be occupied by a protein. Imagine each nucleosome as a box where proteins can "live."

In this scenario, each "state" or energy level depends on the position of the nucleosome, which is crucial since proteins may bind more readily to certain sites. The canonical distribution formula represents the probability of a state being occupied by a protein. This probability depends on the energy of the state and is expressed as: \[ p(x) = \frac{1}{Z} e^{-E(x)} \]Here, \(E(x)\) denotes the energy associated with a position \(x\), and \(Z\) is the partition function that ensures all probabilities across possible states sum to one.

DNA Binding Proteins

DNA binding proteins are essential molecules that interact specifically with DNA sequences. They play a critical role in regulating gene expression and maintaining cellular functions. These proteins can latch onto particular sequences of DNA known as binding sites.

Their affinity, or how strongly they bind to DNA, can depend heavily on the sequence of the binding site. Some proteins find it easier to attach to certain sequences, which can influence where and how often they bind. This behavior is integral to determining nucleosome accessibility because not every protein will bind with the same strength to every site available along the DNA. Thus, understanding which proteins are available and how selective they are provides insights into which sites on the DNA are accessible at any given time.

Protein Concentration

The concentration of proteins in a given cell environment can dramatically affect nucleosome accessibility. Protein concentration refers to how many proteins are present in a specific volume. Higher concentrations often mean more proteins are available to bind to DNA.

Imagine standing in a room full of balloons: the denser the balloon population, the likelier it is you will grab one easily. Similarly, with more DNA binding proteins floating around, the chance increases that they will find and occupy their respective binding sites on the DNA.

In our formula, as the protein concentration \(n\) increases, the effective energy \(E(x)\) necessary for a protein to bind reduces, making it easier for proteins to attach. This dynamic showcases the direct relationship between concentration and binding probability: \[ p(x) = \frac{1}{Z} e^{-k/n} \]where \(k\) is a constant indicating energy-related constraints.

Binding Site Position

The position of the binding site on the DNA plays a pivotal role in determining the likelihood of a protein binding there. DNA has a long sequence, and each segment can present different "sites" with varying affinity.

Certain positions might be energetically more favorable for protein binding, thus often being occupied. Conversely, some areas may be less accessible due to structural hindrances or lower affinity for the proteins in question.

The position of these sites dictates changes in the potential energy \(E(x)\) in our probability formula. Therefore, understanding these spatial preferences involves mapping these energetically favorable sites where proteins naturally gravitate. This concept helps explain why protein binding doesn't happen uniformly throughout the DNA, but rather, follows a distribution influenced by specific "hotspots" or preferred sites on the DNA.