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The Signalling Cascade

Protein FrmR acts as a regulator, it can form DNA-protein cross-links with a special promotor P_frmAB to prevent activating downstream transcription of Mecherry RFP gene. Formaldehyde is able to covalently modify protein FrmR, inhibiting its function, so when formaldehyde exists, promotor P_frmAB is capable of activating RFP by the destruction of Promoter_frmAB-- FrmR protein cross-link. Therefore Promoter_frmAB is activated and promote the transcription of Mecherry rfp gene.

Kinetics of Signalling

In cell signaling, a ligand formaldehyde binds to its receptor site FrmR, which triggers a conformational change in the receptor. This leads to a change in functionality. The formaldehyde bind their receptors at different rates and affinities according to the molecules involved.
These binding rates are critical because they determine the rate at which the biosensor responds to the Formaldehyde it detects. Low rates will lead to low expression, high rates will lead to high expression. We use the Law of Mass Action kinetics model to model these rates. The principle determines that the reaction rate
(d / dt) is proportional to the concentration of the substrate.

For specific reaction, for example:

The rate can be mathematically converted into:

But most of the biological reactions are reversible, so when we want to model the reaction speed, we must take this into account by adding an additional term.

Applying these principles to the binding of FrmR and Promoter frmAB yields:

Transcription

Transcription rates may be affected by various factors. However, it has been determined that transcription initiation is a major factor in determining the overall rate of mRNA production. Transcription initiation is a process that involves several key steps, including: promoter escape (Kugel & Goodrich, 1998),[2] RNA polymerase binding to the promoter, isomerization and formation of closed RNA Pol complexes from open complexes (Hakkinen et al. 2013).[3] In this model, we assume that the binding of RNA polymerase is the rate-determining step of transcription initiation and is therefore the rate determination step for the entire transcription.

Thus, our key assumptions for transcription are:
1. The total transcription rate is mainly determined by the amount of substrate – FrmR dissociates with promoter_frmAB and the intensity of the promoter. When all binding sites are destroyed, the maximum transcription rate occurs.
2.The promoter intensity (in polymerase per second) is a constant that represents the maximum expression rate of transcription.
3. Thus, the total transcription rate is the function of the maximum transcription rate and the fraction of the signal molecule bound to its binding site.
4. For constitutive expression of genes - FrmR, we can assume that it is the maximum transcript of all t.

①Finding parameters for FrmR (Constitutive Transcription)
In our final construct, FrmR is placed under the control of promoterfrmR, and Mecherry rfp is placed under the control of promoter_frmAB.

②Finding parameters for the Mecherryrfp reporter protein (Induced Transcription)
For our reporter genes, the model for expression is complicated by the binding of signalling molecules. In contrast to the constitutively expressed genes, we must assume that the transcription rate is a function of the amount of signalling molecules that are bound.

This rate can be modelled as below:

Promoter _frmAB − FrmR + Formaldehyde → FrmR − Formaldehyde + Promoter_frmAB

Signalling Cascade Model Parameters	Value	Units [1]
Kf2(M −1s−1 ) b	Formaldehyde binding constant	5660
Kr2(s−1) a	Formaldehyde dissociation	0.003

Translation

After the formation of Mecherry rfp mRNA transcripts, it must be translated into functional chromogenic proteins. The translation involves three major phases: the initiation, extension and termination of the polypeptide. The key molecular mechanism driving the process is ribosomes, whose binding and movement determine the overall transcription rate.

Translation rate can be affected by 3 major processes:

1. The binding of the ribosome to the RBS
2. Formation of initiation complexes
3. Rate of the ribosome across the mRNA transcript (Proshkin et al. 2010).[4]

In this model, we will assume that the movement of ribosomes (in amino acids formed per second) is the rate determination step in protein translation.

Thus, our major assumptions for translation are:

1. The main factor influencing the translation rate is the rate of ribosomes across the mRNA, which can be approximated by constant rates.
2. The translation rate is limited by the amount of available mRNA transcripts that bind with ribosome.
3. Thus, the translation rate is a function of the ribosome velocity and the mRNA transcript concentration.

Finding parameters for translation

The literature search showed an average translation rate of 17.1 amino acids per second (Young and Bremer, 1976[5] and Proshkin et al., 2010[4]), so the transcription factor would depend on the length of the mRNA transcript divided by 17.1 in the table below.

 

Protein	Length (amino acids)	Translation rate (1/s)
FrmR	101	0.168
Mecherry rfp	236	0.072

 

These translation rate coefficients are transmitted by a single mRNA molecule in the ribosome. In fact, our cells will contain multiple copies of the same mRNA molecule. We multiply these translation rate coefficients by the total number of mRNA molecules to get the total translation rate per second of each DNA construct.

Degradation and Dilution

The concentration of each species in the model is affected by its degradation (molecular inherent) and dilution (due to cell growth). Although these processes are completely unrelated, they have the same net effect: to reduce the concentration of species. Thus, we represent these ratios as the combined degradation / dilution parameters in our model.
The life of the mRNA is usually very limited. mRNA is degraded prior to any significant dilution compared to cell growth rate or cell dilution.
The opposite is true for proteins. While proteins are victims of degradation, it is not the major factor in the reduction of a protein’s concentration. Proteins tend to far outlive the mRNA molecules that produced them. It is often the changes in the protein’s environment that lead to it’s decrease in concentration. As a cell grows in size, the amount of Mecherry rfp inside it remains relatively constant, but the concentration of Mecherry rfp decreases.
The opposite is true for proteins. Although the protein is the victim of degradation, it is not a major factor in reducing protein concentration. Proteins tend to go far beyond the mRNA molecules that produce them. Changes in the protein environment often result in a decrease in concentration. When the cell growth size, the internal Mecherry rfp volume remained relatively constant, but the concentration of Mecherry rfp decreased.
In our model, we only consider the degradation / dilution of Mecherry rfp mRNA and Mecherry rfp . In any case, the following sensitivity analysis shows that degradation and dilution have minimal impact on overall biosensor performance compared to other more important parameters.

Name	Value	Units
Mecherry rfp mRNA degradation/dilution	0.0022	molecules/sec
Mecherry rfp degradation/dilution	0.0012	Molecules/sec

Cellular Equations

Using the above parameters, we can construct a series of ordinary differential equations representing our biosensor network.

The system described by these equations can be represented diagramatically by SimBiology:

Result

Our model shows the process of formaldehyde induction after the promter_frmAB- FrmR binding has been constructed. We assume the initial FrmR concentration was 0.1 mM, and the initial concentration of promter_frmAB- FrmR binding complex was 0.1mM.The initial formaldehyde concentration was 20 mM for a 10 hour period of time.
The following figure shows the time course concentration of each species, where similar concentrations of material are plotted on separate axes.

 

Reference:

1. Denby, K. J. et al: The mechanism of a formaldehyde-sensing transcriptional regulator. Sci. Rep. 10.1038 (2016). web. 09 Dec. 2016.
2. Kugel J. F. and J. A. Goodrich: Promoter Escape Limits The Rate Of RNA Polymerase II Transcription And Is Enhanced By TFIIE, TFIIH, And ATP On Negatively Supercoiled DNA. Proceedings of the National Academy of Sciences 95.16 (1998): 9232-9237. Web.
3. Häkkinen, Antti et al: Effects Of Rate-Limiting Steps In Transcription Initiation On Genetic Filter Motifs". PLoS ONE 8.8 (2013): e70439. Web.
4. Proshkin, S. et al: Cooperation Between Translating Ribosomes And RNA Polymerase In Transcription Elongation. Science 328.5977 (2010): 504-508. Web.
5. Young, R and H Bremer: Polypeptide-Chain-Elongation Rate In Escherichia Coli B/R As A Function Of Growth Rate. Biochem. J. 160.2 (1976): 185-194. Web.

 

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