Ribonucleic acid polymerase (RNAP) is an enzyme ubiquitous in biological systems. It catalyzes the synthesis of RNA biomolecules from DNA in the process of transcription. RNAP binds to a receptor in a probabilistic manner. We calculated the probability of RNAP binding from the Boltzmann equation and the law of mass action in an E. coli cell under biological conditions and derived a quantitative model which predicts the probability of RNAP binding in varying concentrations of repressor and nucleoid proteins.

1. A Theoretical Model of Repression of Transcription by Nucleoid Proteins
Alec Hoyland, Sumana Shashidhar, Tim Harden, and Jané Kondev
Dept. of Quantitative Biology, Brandeis University, Waltham, MA 02454
Introduction
Ribonucleic acid polymerase (RNAP) is an enzyme ubiquitous
in biological systems. It catalyzes the synthesis of RNA
biomolecules from DNA in the process of transcription. RNAP
binds to a receptor in a probabilistic manner. We calculated
the probability of RNAP binding from the Boltzmann
equation and the law of mass action in an E. coli cell under
biological conditions and derived a quantitative model which
predicts the probability of RNAP binding in varying
concentrations of repressor and nucleoid proteins.
Nucleoid-associated proteins (NAP) bind tightly to DNA in
order to compact DNA to fit within the bacterial cell. Using
the previously derived equation, we developed a model for
NAP repressing transcription under biological conditions.
Our theory allows us to predict biological processes which
would be difficult to observe in vivo. Theoretical models give
quantitative understanding to qualitative experiments by
verifying empirical data. Our research supports the use of
equilibrium models in biological physics.
Promoters have been characterized as a function
of the number of repressors (Oehler et al. 1994;
Rosenfeld et al. 2005; Garcia & Kondev 2011).
As repressor concentration increases, the probability
of RNAP binding decreases. As RNAP concentration
increases, the probability of RNAP binding increases.
Figure 4. Proof of Principle: A Prototypical Model
Figure 1. Empirical Support for Theoretical Models
There is a very high probability that the RNA polymerase
will be bound when the dissociation constant (𝐾 𝑑) is equal
to 3 nM. The next model will therefore look only at
patterns present at this 𝐾 𝑑value.
𝐾 𝑑 =
𝑃 𝐿
𝑃𝐿
Figure 3. Two models of non specific binding of RNA
Polymerase based on NAP Protein HU concentration
Figure 2. Control model with no repressor in system
Model 1 takes into consideration the
possible arrangements for the NAP
proteins binding to the promoter
region, as seen by the equation picture
in Figure 4, while Model 2 does not
consider arrangement. There is a
significant difference between the two
models, indicating that the
arrangement of the NAP proteins does
influence RNAP binding.
Theory & Discussion
The cytoplasm within an E. coli cell was modeled as a fluid
lattice. Using the Boltzmann equation, we were able to
solve for the kinetic weights of the energies as exponential
terms. The number of thermodynamic arrangements were
found using the binomial theorem.
Stirling’s approximation was used to simplify the factorial
expression. Michaelis-Menten kinetics were also assumed,
for thermodynamic equilibrium in biological conditions.
The prototypical model of simple repression describes the
probability of RNAP binding in vivo with high accuracy
when there exists one binding site for both repressor and
RNAP. Model 1 indicates conclusively that thermodynamic
arrangement of NAP proteins significantly affects binding
probability.
Special Thanks
We would like to thank Tim Harden for his ardent guidance.
The probability is a function of ligand concentration,
repressor concentration, and the dissociation
constant.
Results
Figure 4. Model of promoter states and
NAP arrangement and equation used to
calculate arrangements for Model 1