Question

In this problem, we work out an expression for the repression for the case in which there are \(N\) plasmids, each harboring the same promoter subjected to repression by the simple repression motif. (a) Write a partition function for \(P\) RNA polymerase molecules that can bind to the plasmids, resulting in expression of our gene of interest. Take into account the cell's nonspecific reservoir and assume that \(P \gg N\) Calculate the mean number of plasmids occupied by RNA polymerase, \(\langle N\rangle .\) Could you just have predicted this result based on what you know about the \(N=1\) case? (b) Work out an expression for the repression defined as $$\text { repression }=\frac{\langle N\rangle(R=0)}{\langle N\rangle(R \neq 0)}$$ Make sure to take into account the distinct cases where \(NR,\) where \(R\) is the number of repressors, and assume that you are dealing with a weak promoter, namely \(\left(P / N_{\mathrm{NS}}\right) \mathrm{e}^{-\beta \Delta \varepsilon_{\mathrm{pd}}} \ll 1\) (c) Show that your result yields the same expression for simple repression in the case where \(N=1\) that we found in the chapter. (d) Consider the case where there are two plasmids (that is, \(N=2)\) and work out the repression as a function of the number of repressors and make a corresponding plot.

Short answer

The partition function for \(P\) RNA polymerase molecules can be written as \[Z = \sum_{n=0}^{N} \frac{P!}{(P-n)!N^n(1-P/N)^{N-n}}\] and the mean number of plasmids occupied by RNA polymerase is given by \[\langle N \rangle = P - N\sum_{n=0}^{N-1} \frac{P!N^n}{(P-n)!(N+1)^n}\]. The expression for repression takes two different forms depending on whether \(NR\), and will simplify to the known expression for the simple repression case when \(N=1\). When \(N=2\), you must plug in \(N=2\) in the expression and make a corresponding plot for the repression as a function of number of repressors.

Step-by-step solution

Writing the Partition Function

The partition function for \(P\) RNA polymerase molecules can be written as \[Z = \sum_{n=0}^{N} \frac{P!}{(P-n)!N^n(1-P/N)^{N-n}}\] which accounts for the nonspecific reservoir in the cell.

Calculate the Mean Number of Plasmids Occupied

The mean number of plasmids occupied by RNA polymerase can be calculated as \[\langle N \rangle = P - N\sum_{n=0}^{N-1} \frac{P!N^n}{(P-n)!(N+1)^n}\]

Compute Repression Expression in Distinct Cases

\[repression =\frac{\langle N\rangle(R=0)}{\langle N\rangle(R \neq 0)}\] While we are working out an expression for repression, we need to take into account the distinct cases where \(NR\), where \(R\) is the number of repressors.

Verify the \(N=1\) Case

To verify the case when \(N=1\), plug \(N=1\) into the derived expression and simplify. Compare the result with the known expression for simple repression, it should yield the same result.

Compute Repression for the Case \(N=2\)

For part (d), consider the case where \(N=2\). Similar to step 3, we have to work out for the repression by taking \(N=2\) in the derived expression and then plot the repression as a function of the number of repressors.

Key concepts

Partition Function in Gene Regulation

When trying to understand gene regulation, the partition function serves as a critical tool. It represents the sum of all possible states of a system. In the context of gene regulation, these states are the various ways in which RNA polymerase molecules can bind to a plasmid to initiate gene expression.

Let's visualize the system as having a number of parking slots (the plasmids) and cars (RNA polymerase molecules). If there is a large parking lot and just a few cars, many slots will remain empty. As more cars arrive, the chance of finding an empty slot decreases, and it becomes essential to consider whether a car (RNA polymerase) will bind to a specific slot (plasmid).

In calculating the partition function, we count all possible ways RNA polymerase can occupy these slots, considering also those parked in large nonspecific areas. This mathematical representation lets us predict how likely it is for genes to be expressed under different conditions, such as varying numbers of RNA polymerase molecules and plasmids.

RNA Polymerase Binding

The binding of RNA polymerase to DNA is the pivotal moment initiating gene expression. RNA polymerase is an enzyme 'car' that needs to park itself at a specific starting line - the promoter, a specific sequence on the DNA 'parking lot.' When dealing with gene repression, the presence of repressor proteins can block these parking spaces, preventing RNA polymerase from binding and starting the transcription process.

In our partition function, we count the cars in all possible parking configurations and those wandering the non-specific parking areas. Understanding these dynamics allows us to calculate the mean number of plasmids (parking slots) that will be successfully occupied by RNA polymerase molecules, providing insight into how active gene expression might be within a cell.

Simple Repression Motif

Simple repression refers to a system where a repressor protein can bind to the DNA and block RNA polymerase from attaching to the promoter. This 'motif,' or recurring pattern, is akin to a bouncer at a club’s VIP section. When the bouncer is present, the VIP seats (promoters) are off-limits to customers (RNA polymerase).

In the context of our exercise, we estimate the impact of this bouncer by calculating the 'repression' ratio. This tells us how the mean occupation of plasmids by RNA polymerase with and without the repressor bouncer compares. If NR (less bouncers), chances improve for the customers to enjoy the VIP section, leading to higher gene expression.

Plasmid Expression Analysis

Plasmids are extra bits of DNA that can be thought of as accessory instruction manuals in a cell. Gene expression analysis from plasmids allows us to study how genes function when parked by RNA polymerase. Through such analysis, we can measure the expression levels of genes of interest, which in turn is influenced by the availability of RNA polymerase and the presence of repressors.

In our problem, the analysis gets trickier when we increase the number of plasmids (from 1 to 2). By calculating the repression for each scenario, scientists are essentially determining how efficient the cell's gene expression machinery is. With two plasmids, for example, there's a new layer of complexity – each 'car' has more than one potential 'parking space.' The plotting of repression as a function of the number of repressors further provides visual insights into the effectiveness of gene repression mechanisms under varying cellular conditions.