# 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:
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:
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:
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:
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:
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