Team:Alma/Model

Mathematical Modeling

For our biosensor, we each designed five different genetic circuits with the general goal of detecting DDT (Dichlorodiphenyltrichloroethane). It is not efficient for our team to construct each circuit in the wet-lab, and then determine whether it works well. We decided to utilize math modelling to predict how each circuit will function. This was done by the dry-lab team. Each model shows what the circuit is predicted to produce at different given levels of DDT. They also each have specific goals for detection.

Below, you can find some general notes on our math model. We have also made an expanded version of this available as a PDF, which may be a more convenient viewing format.

Parameters Key for All Circuits:

trxn: Transcription
trnsl: Translation
deg: Degradation
d: Disassociation
K: Constant
lac: Lac Promoter
Ktrnsl (general): 2360.0 time
Ktrxn (general): 0.88 substance
Kdeg (general): 2000000.0 time
KtrnslhERa: 2360.0 time
KtrnslTet: 2360.0 time
KtrxnRFP: 0.88 time
Kd (general): 1.0 E13 second
KtrnslRFP: 0.009 time
KdegRFPmRNA: 2000000.0 time
Ktrxnlac: 2470000.0 time

Circuit #1

Idea – In this circuit, the rainbow trout estrogen receptor (rtER) will be able to bind and prevent expression of any gene that is controlled by the Lac promoter with the appropriate operator (rtERE, or Estrogen Response Element). The human estrogen receptor (hERa) will likewise prevent transcription of pTet since there is a hERE element there. Both rtER and hERa can bind both DDT and Estradiol but do so with different affinities (in other words, the kd for rtER binding DDT is lower than that for hERa binding DDT).

Goal – We strived to maximize the RFP signal when and only when DDT is present – the presence of Estradiol should inhibit the expression.

Math Model:

Figure 2. Model of Circuit 1

m = was used sometimes to denote mRNA Species Key: s12= hERa and TetR gene, s18= hERa and TetR mRNA, s29= activated rtER protein, s19= TetR protein, s17= inactivated hERa protein, s25= RFP gene, s31= activated hERa protein, s24= RFP mRNA, s27= RFP protein, s49= Estradiol, s39= rtER gene, s38= rtER mRNA, s47= rtER protein, s50= DDT

Reactions (Equations):

Transcription of TetR and hERa mRNA repressed by activated rtER:
s12 * Ktrxn * 1 / 1 + pow(s29 / Kd, 2) - Kdeg * s1
Translation of TetR:
s18 * KtrnslTet - Kdeg * s19
Translation of hERa:
s18 * KtrnslhERa - Kdeg * s17
Transcription of RFP repressed by TetR and activated hERa:
s25 * KtrxnRFP * 1 / 1 + pow(s19 / Kd, 2) * 1 / 1 + pow(s31 / Kd, 2) - KdegRFPmRNA * s24
Translation of RFP:
s24 * KtrnslRFP - Kdeg * s27
Activation of hERa:
1 + pow(Kd * (s17 * s49 / Kd), 2) - Kdeg * s31
Transcription of rtER:
s39 * Ktrxn - Kdeg * s38
Translation of rtER:
s38 * Ktrnsl - Kdeg * s47
Activation of rtER:
1 + pow(Kd * (s47 * s50 / Kd), 2) - Kdeg * s29

Circuit #2

Idea – Here, the goal is to create/model a circuit that is able to produce one color (green) at intermediate levels of DDT, and another color (red) at dangerously high levels. In this circuit, the Estrogen receptor blocks expression of both cI and TetR repressor. Estrogen receptor, when activated, blocks production of both cI and TetR. GFP responds to only TetR, and so is somewhat less sensitive (and can be tuned with aTC), while RFP is repressed by both, and levels of TetR and cI must decrease to a sufficient level to allow expression of this color.

Goal – We wanted to be able to distinguish between a low (1uM) and high (40uM) level of DDT.

Math Model:

Figure 4. Model of Circuit 2

Species Key: s1= cI and TetR gene, s18= activated hERa protein, s2= cI and TetR mRNA, s3= cI protein, s7= GFP gene, s6= TetR protein, s8= GFP mRNA, s9= GFP protein, s10= RFP gene, s11= RFP mRNA, s12= RFP protein, s13= hERa gene, s14= hERa mRNA, s16= DDT, s17= inactivated hERa protein

Reactions (Equations):

Transcription of TetR and cI repressed by activated hERa:
s1 * Ktrxnlac * 1 / 1 + pow(s18 / Kd, 2) - Kdeg * s2
Translation of cI:
s2 * Ktrnsl - Kdeg * s3
Transcription of GFP repressed by TetR:
s7 * Ktrxn * 1 / 1 + pow(s6 / Kd, 2) - Kdeg * s8
Translation of GFP:
s8 * Ktrnsl - Kdeg * s9
Transcription of RFP repressed by TetR and cI:
s10 * KtrxnRFP * 1 / 1 + pow(s3 / Kd, 2) * 1 / 1 + pow(s6 / Kd, 2) - KdegRFPmRNA * s11
Translation of RFP:
s11 * KtrnslRFP - Kdeg * s12
Transcription of hERa:
s13 * Ktrxn - Kdeg * s14
Translation of TetR:
s2 * KtrnslTet - Kdeg * s6
Translation of hERa:
s14 * KtrnslhERa - Kdeg * s17
Activation of hERa:
1 + pow(Kd * (s17 * s16 / Kd), 2) - Kdeg * s18

Circuit #3

Idea – Here, the goal is to create/model a circuit that is able to produce one color (green) at intermediate levels of DDT, and another color (red) at dangerously high levels – this is the idea, although in this circuit the colors might have been swapped! In this circuit, TetR is under the control of at least two tandem hERE elements – so repression of this gene is easier to achieve by the estrogen receptor, and less DDT is necessary to repress it. You might reflect this by changing the hill coefficient or otherwise multiplying the repressor binding term in this model.

Goal – We wanted to be able to distinguish between a low (1uM) and high (40uM) level of DDT.

Math Model:

Figure 6. Model of Circuit 3

Species Key: s13= TetR gene, s14= TetR mRNA, s15= TetR protein, s16= hERa gene, s17= hERa mRNA, s18= inactivated hERa, s19= cI gene, s20= cI mRNA, s21= cI protein, s22= GFP gene, s23= GFP mRNA, s24= GFP protein, s25= RFP gene, s26= RFP mRNA, s27= RFP protein, s28= activated hERa, s29= DDT

Reactions (Equations):

Transcription of TetR repressed by activated hERa:
s13 * Ktrxnlac * 1 / 1 + pow(s28 / Kd, 2) - Kdeg * s14
Translation of TetR:
s14 * KtrnslTet - Kdeg * s15
Transcription of hERa:
s16 * Ktrxn - Kdeg * s17
Translation of hERa:
s17 * KtrnslhERa - Kdeg * s18
Transcription of cI repressed by activated hERa:
s19 * Ktrxnlac * 1 / 1 + pow(s28 / Kd, 2) - Kdeg * s20
Translation of cI:
s20 * Ktrnsl - Kdeg * s21
Transcription of GFP repressed by cI:
s22 * Ktrxn * 1 / 1 + pow(s21 / Kd, 2) - Kdeg * s23
Translation of GFP:
s23 * Ktrnsl - Kdeg * s24
Transcription of RFP repressed by TetR:
s25 * KtrxnRFP * 1 / 1 + pow(s15 / Kd, 2) - KdegRFPmRNA * s26
Translation of RFP:
s26 * KtrnslRFP - Kdeg * s27
Activation of hERa:
1 + pow(Kd * (s18 * s29 / Kd), 2) - Kdeg * s28

Circuit #4

Idea – This circuit is designed as an IFFL, which should detect DDT within a particular range. In this circuit, estrogen receptor activated by DDT represses expression of TetR, which in turns represses both cI and RFP. RFP is also repressed by cI, thanks to a hybrid promoter.

Goal – We be able to detect a specific range of DDT (i.e., between 1 and 40uM).

Math Model:

Figure 8. Model of Circuit 4

Species Key: s1= hERa gene, s2= hERa mRNA, s3= inactivated hERa, s4= TetR gene, s13= activated hERa, s5= TetR mRNA, s6= TetR protein, s7= cI gene, s8= cI mRNA, s9= cI protein, s10= RFP gene, s11= RFP mRNA, s12= RFP protein, s14= DDT

Reactions (Equations):

Transcription of hERa: s1 * Ktrxn - Kdeg * s2
Translation of hERa: s2 * KtrnslhERa - Kdeg * s3
Transcription of TetR repressed by activated hERa: s4 * Ktrxnlac * 1 / 1 + pow(s13 / Kd, 2) - Kdeg * s5
Translation of TetR: s5 * KtrnslTet - Kdeg * s6
Transcription of cI repressed by TetR: s7 * Ktrxn * 1 / 1 + pow(s6 / Kd, 2) - Kdeg * s8
Translation of cI: s8 * Ktrnsl - Kdeg * s9
Transcription of RFP repressed by TetR and cI: s10 * KtrxnRFP * 1 / 1 + pow(s6 / Kd, 2) * 1 / 1 + pow(s9 / Kd, 2) - KdegRFPmRNA * s11
Translation of RFP: s11 * KtrnslRFP - Kdeg * s12
Activation of hERa: 1 + pow(Kd * (s3 * s14 / Kd), 2) - Kdeg * s13

Circuit #5

Idea-This circuit is meant to detect DDT at varying degrees with red fluorescence, depending on the concentration of DDT the microbes are exposed to. In this circuit, the estrogen receptor activated by DDT represses the expression of TetR. Which stops it from repressing the expression of RFP. Red fluorescence would then appear in the presence of high DDT.

Goal- We strived to maximize the RFP signal when and only when DDT is present – the presence of Estradiol should inhibit the expression.

Math Model:

Figure 10. Model of Circuit 5

Species Key: s1= RFP gene, s6= TetR protein, s2= RFP mRNA, s3= RFP protein, s4= TetR gene, s10= activated hERa, s5= TetR mRNA, s7= hERa gene, s8= hERa mRNA, s9= inactivated hERa, s11= DDT

Reactions (Equations):

Transcription of RFP repressed by TetR: s1 * KtrxnRFP * 1 / 1 + pow(s6 / Kd, 2) - KdegRFPmRNA * s2
Translation of RFP: s2 * KtrnslRFP - Kdeg * s3
Transcription of TetR repressed by activated hERa: s4 * Ktrxnlac * 1 / 1 + pow(s10 / Kd, 2) - Kdeg * s5
Translation of TetR: s5 * KtrnslTet - Kdeg * s6
Transcription of hERa: s7 * Ktrxn - Kdeg * s8
Translation of hERa: s8 * KtrnslhERa - Kdeg * s9
Activation of hERa: 1 + pow(Kd * (s9 * s11 / Kd), 2) - Kdeg * s10

Parts:

These are some of the biobricks or parts necessary for these circuits.

Plasmid Backbone: BBa_K3445000
Estrogen response element-controlled inverter: BBa_K3445001
RFP: BBa_K3445002
BBa_I135221
Estrogen receptor and TetR: BBa_K123002
Promoter: BBa_K123003
RFP coding device: BBa_J04450