A biophysical model of supercoiling dependent transcription predicts a structural aspect to gene regulation

Background Transcription in Escherichia coli generates positive supercoiling in the DNA, which is relieved by the enzymatic activity of gyrase. Recently published experimental evidence suggests that transcription initiation and elongation are inhibited by the buildup of positive supercoiling. It has therefore been proposed that intermittent binding of gyrase plays a role in transcriptional bursting. Considering that transcription is one of the most fundamental cellular processes, it is desirable to be able to account for the buildup and release of positive supercoiling in models of transcription. Results Here we present a detailed biophysical model of gene expression that incorporates the effects of supercoiling due to transcription. By directly linking the amount of positive supercoiling to the rate of transcription, the model predicts that highly transcribed genes’ mRNA distributions should substantially deviate from Poisson distributions, with enhanced density at low mRNA copy numbers. Additionally, the model predicts a high degree of correlation between expression levels of genes inside the same supercoiling domain. Conclusions Our model, incorporating the supercoiling state of the gene, makes specific predictions that differ from previous models of gene expression. Genes in the same supercoiling domain influence the expression level of neighboring genes. Such structurally dependent regulation predicts correlations between genes in the same supercoiling domain. The topology of the chromosome therefore creates a higher level of gene regulation, which has broad implications for understanding the evolution and organization of bacterial genomes. Electronic supplementary material The online version of this article (doi:10.1186/s13628-016-0027-0) contains supplementary material, which is available to authorized users.

This derivation is mostly taken from Sen et al. where an even more in depth explanation of the supercoiling free energy can be found. This derivation treats the DNA molecule as a "homopolynucleotide" in the shape of a circle where there is no base specificity. This DNA molecule has a total of N base pairs and a linking number equal to Lk, where a linking number is the number of base pairs per turn of the B form DNA. When a DNA molecule is supercoiled the linking number will change, ∆Lk, which is just the difference between the relaxed linking number and the linking number with supercoiling. The number of base pairs per twist is equal to 10.4 and will be assigned to the letter "A". The twist rate is defined as 2π/A and the change in the twist rate in the helical regions will be assigned τ h . Likewise, the twist rate in the melted regions will be τ c . In order for the molecule to be stable the following equation must be obeyed: Where c h and c c are defined as the "torsional stiffness constants." If we then examine the linking number of a partially melted DNA molecule, Lk', with n melted base pairs Lk' can be represented by the following equation: Where the first term is the linking number due to supercoiling and the melted region and the second region is due to the regular melted regions. It is fundamental that the linking number has to remain constant throughout, Lk'=Lk, considering the molecule is a closed loop. Setting these two equations equal to each other and then solving the previous equations for τ h and τ c the following is obtained: The supercoiling energy was derived in the studies by Benham and is beyond the scope of this SI for an in depth derivation. [2,3,4].
Taking the previous equations for τ h and τ c and plugging them into the equation and using σ = ∆Lk·A/N , the supercoiling density, gives the free energy of supercoiling for the partially melted DNA molecule.
The total free energy of the DNA molecule can then be written as the following: This equation represents having all of the n melted base pairs being distributed into the n j junctions. But here we are only interested in DNA molecules at room temperature, which corresponds to a small fraction of the molecule being melted. If we look at the average number of n j for a certain amount of n melted base pairs we see that for small n the number of base pairs in each junction is very small [1]. This allows us to approximate the probability of having a certain number of melted base pairs inside of the promoter to be independent. This is where the binomial distribution results from in the text.

Derivation With Supercoiling Influencing the Rate k cat
In the main text we have the change in free energy caused by supercoiling influencing the off rate in the kinetic model. Due to the fact that this assumption is a result from studies conducted with T7 RNAP, it is very possible that the free energy of the DNA could influence the transition to the open complex with the RNAP from E. coli. Here we show that we obtain the same results by modifying the rate k cat .

Starting at Eq. 3 in the main text
If instead of associating the exponential with k off we attached the exponential to the rate k cat , this would result in the following equation: Where there is a minus sign in the exponential because RNAP must melt the DNA to get over the free energy barrier. Interestingly, this is the exact same result as that found in the main text. If we simply simplify this equation it results in the following: which is the exact same as Eq. 3 in the main text.

Sensitivity to Supercoiling is Dependent Upon the Biophysical Properties of the DNA
Here we discuss how Eq. 2 in the main text could be used with future research to predict which regions of the chromosome would have supercoiling sensitive gene expression. The main two parameters that could differ along the different regions of the chromosome are α and C. These parameters are related to the bending and torsional stiffness constants of the DNA [5]. C is only dependent upon the properties of the helical region, while α is dependent upon both the helical and coiled regions of the DNA. If future research investigating specific DNA has been done to determine the two parameters of various sections of the chromosome, Eq. 2 would be able to show how the free energy of the transition state varies with these different parameters.
Hypothetically, if we assume we are looking at different DNA loops with similar C values and varying values of α we would obtain the data shown in Fig. S1. This shows the smaller α is the more sensitive to supercoiling the system will be. In the future one could potentially computationally determine these two parameters for different sections of DNA and predict which regions of the chromosome would be most affected by supercoiling. This also brings up the point that a transcription event may generate more positive supercoiling in different regions of the chromosome due to the biophysical properties of the DNA. If the specific biophysical parameters for different loops could be obtained experimentally and data describing how the initiation rate decreases with each transcription event could be obtained, one could use Eq. 2 to approximate the amount of supercoiling density generated with each transcription event. This could provide interesting information regarding the chromosome structure/dynamics.