Question

Nucleosome formation and assembly (a) Repeat the derivation given in the chapter for the nucleosome formation energy, but now assuming that there is a discrete number of contacts \((N=14)\) between the DNA and the histone octamer. (b) Use the discrete model to calculate the equilibrium accessibility of binding sites wrapped around nucleosomes. Apply this model to the data by Polach and Widom (1995) and fit the adhesive energy per contact \(\gamma\) discrete (c) Reproduce Figure 10.25 and compare your results for the equilibrium accessibility versus burial depth from (a) and (b) with the continuum model. (d) Look at some of the binding affinities of different DNA sequences to histones reported by Lowary and Widom \((1998) .\) Once again, assume that the electrostatic interaction between the histone and the different DNA molecules does not vary, that is, it is not sequence-dependent. This is equivalent to saying that the difference between each sequence lies in its flexibility, in its bending energy. What would one expect the difference in their persistence lengths to be? (e) Model the case of having two binding sites for the same DNA-binding protein on a DNA molecule that is wrapped around a histone octamer. How does the equilibrium accessibility depend on the protein concentration and the relative position of the binding site? How does the problem change if the two binding sites correspond to two different DNA-binding proteins? Relevant data for this problem is provided on the book's website.

Short answer

The exercise analyzed various aspects of DNA-histone interaction in nucleosomes using the discrete model. The equilibrium accessibility of the binding sites was calculated for different scenarios, compared with the continuum model, and factors such as protein concentration, relative position of the binding site, and the sequence dependence of DNA bending energy were considered.

Step-by-step solution

Derive the Energy for Nucleosome Formation

Knowing that there are 14 discrete points of contact, derive the energy for nucleosome formation using the following formula - \( \Delta G = -N \cdot k \cdot T \cdot ln(p) \) where N is the number of contacts, k is the Boltzmann constant, T is the absolute temperature, and p is the protein concentration.

Calculate the Equilibrium Accessibility

Using the discrete model, calculate the equilibrium accessibility of the binding sites wrapped around the nucleosomes. The equilibrium accessibility is calculated as \( \text{Accessibility} = \frac{1}{1 + e^{\Delta G/kT}} \) where \(\Delta G\) is the energy calculated in step 1.

Compare with the Continuum Model

Reproduce Figure 10.25 with the data calculated from the discrete model. Compare the results obtained from the discrete model with the continuum model to understand the differences in the equilibrium accessibility of binding sites.

Analyse the Binding Affinities of DNA Sequences

Look at the binding affinities of different DNA sequences to histones. The difference between each sequence lies in its flexibility, that is, in its bending energy. Consider the electrostatic interaction between the histone and the different DNA molecules to be constant, which would mean that the bending energy is sequence-dependent.

Case of Two Binding Sites

Model the case where there are two binding sites for the same DNA-binding protein on a DNA molecule wrapped around a histone octamer. Compute the equilibrium accessibility as a function of the protein concentration and relative position of the binding site. Further, study how the problem changes if the two binding sites correspond to two different DNA binding proteins.

Key concepts

Nucleosome Formation Energy

The energy required to form a nucleosome, which consists of DNA wrapped around a histone octamer, is fundamental to understanding chromatin structure. In the provided exercise, students are tasked with deriving the nucleosome formation energy given a discrete number of contacts (\(N=14\) points where DNA interacts with the histones).

The derivation uses the formula \( \Delta G = -N \cdot k \cdot T \cdot ln(p) \), where \( \Delta G \) represents the change in Gibbs free energy of nucleosome formation, \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin, and \( p \) signifies protein concentration. A negative \( \Delta G \) implies that the nucleosome formation is energetically favorable.

This concept is paramount as it provides insight into the stability of nucleosomes and their role in gene regulation. The energy landscape of nucleosome formation impacts how tightly DNA is packaged and consequently, how accessible it is for transcriptional machinery. Understanding this energy opens the door to grasping more complex genetic regulation mechanisms.

Equilibrium Accessibility of Binding Sites

Equilibrium accessibility refers to how readily transcription factors and other DNA-binding proteins can access the DNA sequences they need to bind, even when these sequences are wrapped around histones in nucleosomes. The exercise challenges students to compute this accessibility using a discrete model and compare it to a continuum model.

The calculated accessibility is represented by the equation \( \text{Accessibility} = \frac{1}{1 + e^{\Delta G/kT}} \), highlighting the dependence on the Gibbs free energy derived earlier. When comparing this discrete model to the continuum model, students visualize the nuances of protein-DNA interactions. In the grand scheme, equilibrium accessibility is critical as it influences the frequency and efficiency with which genes are expressed, thereby controlling cellular function and organism development.

DNA-Histone Binding Affinity

DNA-histone binding affinity can be understood as the strength of the interaction between DNA and histone proteins. In the exercise, variation in binding affinity is attributed to the flexibility of different DNA sequences—essentially their bending energy—while keeping electrostatic interactions constant, suggesting a lack of sequence dependence in this parameter.

The analysis calls for comparison between different DNA sequences' affinities to histones. This assumption simplifies the model, focusing the investigation on the physical properties of the DNA. DNA sequences that are more flexible, or have lower bending energy, are expected to have higher affinity for histones as they can more easily wrap around the histone core. Such insights are crucial in biotechnology and medicine where understanding protein-DNA interactions can lead to advancements in gene therapy and epigenetics.

Protein-DNA Interactions

The final part of the exercise delves into quantitative modeling of protein-DNA interactions. Modeling scenarios where DNA is complexed with histones, and there are binding sites for DNA-binding proteins, students must consider both the concentration of the binding proteins and their spatial arrangement on the DNA.

In case of a single type of binding protein, the equilibrium accessibility must be calculated considering both the protein concentration and the competition between protein binding and DNA-histone interaction. When different proteins are involved, the complexity increases as each protein may have a different affinity and kinetic properties towards the binding sites. These interactions play a pivotal role in cellular processes such as replication, transcription, and repair. Understanding these dynamics aids in the design of targeted drugs and in the study of genetic diseases where these interactions are often disrupted.