Team:Sheffield/Chimeric protein system

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Model of the chimeric protein system

Explanation of the system

The diagram below explains the process of our cholera detection biosensor with the chimeric protein system.

Signalling molecule CAI-1 binds to CqsS protein and this causes a change in the fusion receptor. This change triggers autophosphorylation of the histadine kinase BarA and it starts the phosphorylation pathway which ends in the response regulator UvrY. Phosphorylated UvrY acts as a transcription factor activating the promoter which causes the expression of the reporter gene.

Model for signal transduction

The phosphorylation pathway in e-coli is modelled as a two component systems starting with a histadine kinase and ending in a response regulator. Research into the literature [1,2] has shown that these two component systems contain intermediate layers of proteins which accept and transfer the phosphates during phosphorylation

The BarA/UvrY two component system can be modelled as having 4 layers [1,2] which are Histadine Kinase(HK - BarA ) , proteins with receiver domains (REC), His-containing phosphotransfer proteins (Hpt) and Response Regulator (RR - UvrY) in the given order as shown in Figure 2. When BarA starts phosphorylation a phosphate is passed through all these layers resulting in a phosphorylated response regulator UvrY. It has been shown through previous literature[2] that the phosphorylated response regulator can undergo dephosphorylation due to several reasons such as hydrolysis activity intrinsic to RRs or due to a specific phosphatise or a bifunctional HK.

This process can be represented by the following chemical equations. ( the equations were obtained from reference [2] )

where:

Li represents ith layer

ki represent reaction rate constants

Equation (1) describes the autophosphorylation of the first layer that occurs when CAi – 1 is detected.

Equations (2), (3) and (4) represent the phosphate passing from the upper layer to the layer below.

Equation (5) shows the dephosphorylation of the last layer.

Using these equations we create the following mathematical model for the system with the assumptions that

1. The input CAI-1 will control the rate of autophosphorylation of BarA and therefore is included in the rate constant for the reaction.

2. ATP is in abundance in e-coli and therefore will not be a parameter in the system signalling process.

3. The total concentration of all the proteins will remain constant throughout the signalling process.

4. The reverse reactions occurring in system (if any) are negligible.

According to these assumptions the mathematical model is developed as shown below.

Model for the gene expression

Our chimeric system works on the principle that; phosphorylated UvrY which is the transcription factor produced at the end of the phosphorylation kinase binds to our chosen reporter DNA gene which causes transcription of the reporter DNA gene to mRNA to begin. Interactions between the phosphorylated UvrYs and the RNA polymerase transcriptional machinery however plays an important role in determining the order of transcriptional activation which as a result affect the frequency at which mRNA transcripts are produced. Also, the effects of phosphorylated UvrYs at different promoters could vary due to changes in the sequence of the DNA binding site and its distance from the orientation with respect to the promoter. These discrepancies therefore tend to influence the order in which the promoter respond to increasing concentration of the phosphorylated UvrY and seemingly could have important functional consequences. Taking into consideration the above factors stated we constructed a mathematical model for transcription; the first procedure in gene expression, using these following assumptions:

1. The RNA polymerase in the E. Coli is in abundant, hence all the phosphorylated UvrYs produced, bind to the reporter DNA gene resulting to high level of transcription activation.

2. That the PgaAB promoter used is always active and tight; meaning there is high expression of gene, hence all the DNAs are transcribed to mRNA.

3. The degradation rate of mRNA is very negligible, so most of the mRNAs are translated to proteins.

With these assumptions we create following differential equation for the transcription.

where:

t1 = the rate of transcription

UvrYp = phosphorylated UvrY

d1 = transcription degradation

After the reporter DNA genes have been transcribed to mRNAs, the next step in the gene expression is the mRNAs being translated to proteins. In the presence of tRNAs, the mRNAs get translated into proteins and therefore to construct a mathematical model for translation for our chimeric system, the following assumptions were also made;

1. The reporter DNA gene expression is limited at the beginning of the translation process. [3]

2. Also, tRNAs are abundant in the E. Coli and that there are over – expressing rare tRNAs and so all the mRNAs are translated to proteins. [4]

3. The degradation rate of the protein is very negligible and therefore the protein produced have longer half – life.

With these assumptions we create following differential equation for the translation.

where :

t2 = the rate of translation

d2 = translation degradation

Simulation of the model

We first analysed the behaviour of the different layers of proteins in the phosphorylation pathway by simulating the model for the signal transduction using ode solver in matlab to solve the Equations (6) – (9).

For the simulations we set the total amount of each protein layer to 10 and all the rate constants to 1 and the steady state responses of each of the layers were investigated for different levels of input. Graph 1 below shows the results from this simulation.

Graph 1

From this graph it could be seen that the amount of phosphorylated proteins in each layer increases as the input increase and they saturate after a certain input level with the 4th layer being the fastest. The saturation rate of each protein depends on the saturation of the succeeding layers and these levels could be changed by altering the rate constants for each of the reactions [2].

In the complete system these would not always tend to a steady state as the response regulator at the final layer binds to the reporter DNA to facilitate transcription of the reporter gene.

In simulating the mathematical model for the complete system consisting of the six differential equations (Equations (6) – (11)) we again input the equations into matlab and used an ode solver to solve the equations. (The code can be found in the Appendix)

Again we set the total amount of each protein layer to 10 and all the rate constants to 1 except for transcription/translation degradation which are at 0.01.

We first investigated the expression of the reporter gene over 1000 seconds to determine the production of the colour.

Graph 2

It can be seen that as in Graph 1 the colour increases and tend to a steady state at around 900 seconds. From this we could estimate that it will take about 15 minutes for the colour to be produced. The input was kept constant at 1 over this period.

The curve shows that the initial rate of translation was low for sometime before becoming high as the translation curves were both sigmoidal. From this, it therefore could be concluded that fluorescence light produced from the reporter protein in the system may not be visible to the naked eye initially when the E.coli detects the CAI-1.

As the real system parameters will not be 1, we also created a stochastic model where the system was simulated with parameters selected at random. The result for this simulation is given in Graph 3 below.

Graph 3

This graph shows that the system still follows the same shape as before with only a little amount of noise. Therefore it can be deduced that the chimeric protein system is highly robust for variations in system parameters.

We then investigated how the steady state response for the reporter protein would change with different levels of inputs. This is shown by Graph 2 below.

Graph 4

It can be seen that from 0 to 1.2 moles of CAI-1 (estimated values with systems parameters set to 1) the steady state levels increase and at these levels there would be a difference in colour intensity seen. After 1.2 the concentration of the protein tends to a steady state and any more increase in the input would not change the intensity of the colour. Therefore we can deduce that we can use the biosensor to measure the amount of cholera present only up to a certain level of autoinducer.

Different parameter and input values could change the output values of the system this can be analysed using the interface we have created in matlab. More information can be found at Graphical User Interface.

References

[1] Kim JR, Cho KH ,2006 ,The multi-step phosphorelay mechanism of unorthodox two-component systems in E-coli realizes ultrasensitivity to stimuli while maintaining robustness to noises, Computational Biology and Chemistry , [online] Available at: <http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B73G2-4PNJ1K7-7&_user=10&_coverDate=12%2F31%2F2006&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=4ed1aecda98446af7276385b09dff8b2&searchtype=a> [Accessed 24 October 2010] pp. 438 - 439

[2] Csikasz-Nagy A. ,Cardelli, L. Soyer OS.(2010) ,Response Dynamics of phosphorelays suggest their potential utility in cell signalling, [online] Available at: http://rsif.royalsocietypublishing.org/content/early/2010/08/06/rsif.2010.0336.short?rss=1 [Accessed 24 October 2010] pp. 1 – 3 ,7-8 [3]Kudla G, Murray AW, Tollervery D Plotkin JB (2009); “Coding sequence determinants of gene expression in Escherichia coli”

[4] Kane JF (1995); Effects of rare codon clusters on high - levels expression of the heterologous proteins in Escherichia coli. Curr Opin Biotechnol 6: 494 - 500