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 eukaryotes 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 complex 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 simple repression using transcriptional machinery such as that proposed in (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 (a) and (b) for a case where \(Y\) binds to a site on the DNA that is near the X-binding site, and there is an interaction energy between \(X\) and \(Y\)

Short answer

In the eukaryotic case scenario, gene regulation can be reduced to an one-molecule effective problem similar to bacterial case. The fold-change in gene expression can be determined in both scenarios using the association constants and the weak promoter assumption. When an activator can contact \(Y\), this will affect the fold change. In scenario where \(Y\) binds near \(X\)-binding site, interaction energy should be incorporated in calculations.

Step-by-step solution

Calculation of Probability of Binding

Assuming that the subunits \(X\) and \(Y\) bind sequentially, the concentration of the complex \(X+Y\) can be written as \( [X+Y] = [X][Y]K_xK_y \) where \( K_x \) and \( K_y \) are the association constants for \(X\) and \(Y\) binding. Thus, the probability \(P\) that the complex \(X+Y\) is bound to the promoter is \(P = [X+Y]K_p / (1 + [X+Y]K_p) \) where \( K_p \) is the promoter association constant. This can be rearranged to form an effective one-molecule problem (like a bacterial case) with an effective association constant \( K' = K_xK_yK_p \) and \( P = [X][Y]K' / (1 + [X][Y]K') \).

Calculate Fold Change in Expression

The fold-change in gene expression for simple repression can be defined as the ratio of gene expression with a repressor present to the gene expression without it. With the weak promoter assumption (which allows us to neglect the terms involving the promoter), the fold change can be reduced to that corresponding to the bacterial case. By calculating the expression levels in both scenarios, folding-change can be found. In case where an activator can contact \(Y\), the scenario will slightly change but similar calculations apply.

Analyzing a Different Binding Scenario

In the scenario where \(Y\) binds to a site on the DNA near the \(X\)-binding site, and there is an interaction energy \(E_{int}\) between \(X\) and \(Y\), the probability of \(X+Y\) binding and fold-change in expression should be recalculated. Here, the binding would be a competitive binding between \(X\) and all other possible interacting partners including \(Y\). In addition, interaction energy \(E_{int}\) between \(X\) and \(Y\) should be factored in the equations.

Key concepts

Thermodynamic Models

Thermodynamic models provide a way to predict how biological molecules such as DNA, RNA, and proteins interact with each other within the cell. In the context of gene regulation, these models help us to understand how the binding of molecules to DNA can affect the rate at which genes are turned on or off.

Through these models, we apply the principles of energy landscapes and binding affinities to determine the probability that certain configurations—like a transcriptional machinery binding to a promoter—will occur. For instance, transcription factors can have different binding strengths to a DNA sequence, which can be quantified with association constants (\( K \)) in our equations.

This approach assumes a dynamic equilibrium, where binding and unbinding processes are continually occurring, and the probability of any given state can be illustrated by calculating the ratio of concentrations of bound and unbounded molecules. In the exercise given, a simplified model is applied to eukaryotic transcription, highlighting the complexity of multi-subunit assembly on promoter regions in contrast to the single molecular species assumption used for prokaryotic systems.

Transcriptional Machinery

The transcriptional machinery of a eukaryotic cell is substantially more complex than that of a prokaryotic cell. It includes a multitude of proteins and subunits that must come together to initiate transcription. This assembly is often referred to as the pre-initiation complex and facilitates the accurate and timely transcription of genes.

In prokaryotes, the RNA polymerase may directly bind to DNA and start the transcription process, whereas in eukaryotes, various transcription factors (subunits) must assemble in a coordinated fashion. For example, general transcription factors facilitate the binding of RNA polymerase to the DNA, and other coactivators might alter the structure of chromatin to enable transcription. As explained in the exercise, considering just two subunits (\( X \) and \( Y \)) is already a step towards capturing the complexity of eukaryotic gene regulation, by assuming a sequential binding to DNA.

Probability of Gene Expression

The probability of gene expression can be conceptualized as the likelihood that a gene will be transcribed to produce messenger RNA (mRNA) and then translated into a functional protein. This probability is influenced by various factors such as the presence of transcription factors, the strength of promoter sequences, and DNA accessibility.

To quantify this probability, we calculate the likelihood that transcriptional machinery will be properly assembled on the gene's promoter. In simple models, this might be represented as the fraction of time a promoter is occupied by RNA polymerase. In more complex scenarios, such as the one we face with eukaryotic transcription involving multiple subunits, calculations must account for the concentration of subunits and their interactions. The provided exercise uses thermodynamic principles to calculate the likelihood that the transcriptional complex \( X+Y \) forms and binds to the promoter region, which directly affects the gene's expression level.

Simple Repression

Simple repression is a form of gene regulation where a repressor protein binds to specific DNA sequences known as operator sites, which are generally found in the vicinity of the promoter region. By binding to these sites, repressors physically block RNA polymerase from transcribing the gene, thus reducing or completely inhibiting gene expression.

In the exercise, 'simple repression' is analyzed in the context of a two-subunit transcriptional machinery. The 'fold-change' in gene expression—calculated as the ratio of gene expression levels with and without a repressor—provides a measure of how effective the repression is. When applying the assumption of a weak promoter, one that has a low probability of RNA polymerase binding, the calculations can be simplified. This approximation allows us to reformulate complex eukaryotic systems to mirror simpler bacterial situations, thus extending the concept of simple repression to more elaborate models of transcriptional regulation.