Question

In the thermodynamic models of gene regulation discussed in the chapter the RNA polymerase is treated as a single molecular species. While this might be a reasonable assumption for transcription in prokaryotes, in eukary. otes tens of different molecules need to come together in order to form the transcriptional machinery. The objective of this problem is to develop intuition about the requirements for our simple model to apply in such a com plex case by assuming that the transcriptional machinery is made out of two different subunits. \(X\) and \(Y\), that come together at the promoter. (a) Calculate the probability of finding the complex \(x+Y\) bound to the promoter in the case where unit \(x\) binds to DNA and unit \(Y\) binds to \(X\). Can you reduce this to an effective one molecule problem such as in the bacterial case? (b) Calculate the fold change in gene expression for \(\operatorname{sim}\) ple repression using transcriptional machinery such as that proposed in part (a). Explore the weak promoter assumption in order to reduce the expression to that corresponding to the bacterial case. Repeat this for the case where an activator can contact \(Y\) (c) Repeat parts \((a)\) and \((b)\) for a case when \(Y\) binds to a site on the DNA which is near the \(X\) binding site, and there is an interaction energy between \(X\) and \(Y\)

Short answer

In this complex scenario of gene regulation, the probability calculations and gene expression analyses depend on where and how the different subunits bind and interact. The calculations could be reduced to a more straightforward one molecule model in some cases, but in others, like an activator interacting with one of the subunits, it becomes considerably more complex. The interaction between subunits near the promoter also changes dynamics and requires extra considerations during calculations.

Step-by-step solution

Calculate the Probability for Subunits \(X\) and \(Y\) Binding to the Promoter

First, start by defining the rate of assembly of the subunits, represented by \(k_{on}\) for association and \(k_{off}\) for dissociation. Then define the concentration for the two subunits \(X\) and \(Y\) as [\(X\)] and [\(Y\)]. Calculate the probability (\(P\)) of finding the complex \(X+Y\) bound to the promoter as the assembly divided by the sum of the assembly and disassembly rates: \[P = \frac{k_{on}[X][Y]}{k_{on}[X][Y] + k_{off}}\]. This can be transformed to a form similar to the one molecule problem by treating the effective concentration [\(XY\)] as [\(X\)][\(Y\)].

Calculate the Fold Change in Gene Expression

Next, calculate the fold change in gene expression by comparing the levels of gene expression in the presence and absence of the repressor. For this, it's necessary to define the promoter activity as the rate of transcription events per unit time in a cell with repressor protein bound and its absence respectively. Use the weak promoter assumption that assumes the rate of transcription initiation is much slower than the rate of promoter binding and unbinding. The fold-change is expressed as: \[FC = \frac{1}{1 + R/(N_{NS}\cdot P)}\], where \(R\) is the number of repressors per cell, \(N_{NS}\) is the number of non-specific binding sites in the genome and \(P = \frac{1}{1+e^{-\beta\Delta\epsilon}}\) is the probability that the promoter is in active state. This can be reduced to the bacterial case, unless an activator molecule contacts \(Y\) and changes the transcription dynamics.

Repeat the Calculations for a Different Binding Scenario

Repeat Steps 1 and 2 for the case where \(Y\) binds to a different site near the \(X\) binding site. The presence of an interaction energy between \(X\) and \(Y\) will influence the probability calculations. This interaction changes the binding dynamics and will require recalculating \(P\), the fold change in gene expression, with this interaction energy included in the calculations.

Key concepts

Transcriptional Machinery in Eukaryotes

Understanding the transcriptional machinery in eukaryotes is crucial for grasping the complexities of gene regulation. Contrary to prokaryotes, eukaryotic transcription involves a multitude of proteins and regulatory sequences that come together to initiate the process.

In eukaryotes, transcription is carried out by three different RNA polymerases (Pol I, Pol II, Pol III), and each is responsible for synthesizing different types of RNA. For instance, RNA Pol II is primarily responsible for transcribing mRNA, which translates into proteins. The actual initiation of transcription, where these polymerases can begin their work, requires the assembly of several transcription factors and other proteins at the promoter region of a gene.

The transcription factors and other regulatory proteins function as subunits, such as the hypothetical units X and Y in our exercise, that must bind together at the promoter to form a functional pre-initiation complex (PIC). This complex assembly process is tightly regulated and influenced by various mechanisms including protein-protein interactions, DNA-protein interactions, and the accessibility of the DNA wrapped in chromatin.

To understand the efficiency of this complex formation, we use models to calculate the probabilities of these subunits coming together. This is where thermodynamics plays a key role as it helps in predicting how likely is the formation of such a complex under certain conditions of concentration and affinity between the interacting subunits.

Probability of Complex Binding to Promoter

The probability of a transcriptional machinery complex binding to a promoter is fundamental in determining the overall rate of transcription initiation. As transcription initiation is a critical step in gene expression, understanding this probability helps us predict how genes will be expressed under different conditions.

In the example of a complex made up of units X and Y, we calculate the probability ((P)) of finding this complex at the promoter. This calculation involves considering the rates at which these units come together (association) and fall apart (dissociation), as well as their concentrations.

The rate of assembly, represented by ((k_{on})), reflects how rapidly the units can form a stable complex, while the rate of dissociation ((k_{off})) indicates how quickly the complex can break apart. The respective concentrations of each subunit ([(X)] and [(Y)]) are also critical. By combining these variables, one can calculate the likelihood of a stable complex present at the promoter site.

Moreover, sometimes it's possible to simplify these interactions to an effective one molecule problem if the units interact with high specificity and affinity, making the situation somewhat analogous to prokaryotic transcription. However, this simplification may not always be valid due to the more intricate interactions and regulatory mechanisms present in eukaryotic cells, such as chromatin remodeling and the involvement of enhancers and silencers that can influence the transcription machinery from a distance.

Fold Change in Gene Expression

The 'fold change' in gene expression is a metric used to quantify the difference in gene expression levels when comparing two different states—often with and without a regulatory molecule like a repressor or activator. In the context of the exercise, it symbolizes the ratio of gene expression levels when the repressor is bound to the DNA versus when it is not.

The fold change helps reveal the strength and effect of regulatory elements on gene expression. By comparing the activity levels of a promoter in the presence and absence of repressions or activation, we can derive a numerical value that indicates the magnitude of regulation. This is especially helpful in eukaryotic systems where the presence of activators and complex machinery might complicate direct analysis.

The weak promoter assumption simplifies this calculation by assuming that the transcription initiation is much slower compared to the binding and unbinding of the transcriptional machinery to the DNA. This helps reduce the complexity of the equation to analyze fold change, allowing it to align more closely to models used for bacterial systems, where fewer varied components are involved in transcription initiation.

By calculating fold changes across different conditions — for instance, with varying amounts of regulatory molecules or changes in their affinity — researchers can predict and compare the impacts of different regulatory scenarios on gene expression, thereby furthering our understanding of the dynamics at play within eukaryotic gene regulation.