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Parameters

As our circuit is relatively complex, we have many factors and thus many parameters with which to describe the function of these factors mathematically.

These parameters are as follows:

1. δ is a function designed to represent the presence/absence of light in a Boolean fashion. δ=0 without light and 1 with light.
2. c_1 represents the rate of conformational change from LovTAP to LovTAP* when irradiated with 470nm light.
3. c_2 represents the rate of SupD + T7ptag --> T7 polymerase
4. α represents the maximum protein synthesis rate in nanomoles/sec.
5. β represents the basal synthesis rate. We assume ∀x β_x= .01α_x
6. μ represents the protein degradation rate.
7. n represents a ligand’s hill coefficient.
8. K_m represents the ligand’s dissociation constant

Deterministic Modeling of a Bacterial Light Recognition Circuit

Players

The following are the players that are used in the modeling of our circuit:

Transcription factors

1. LovTAP

2. LovTAP* (After light-induced conformational change)

3. tetR

4. Mnt

5. AraC

6. LacI

7. CI

8. CI434

9. GAL4

Reporters

1. S1

2. S2

3. S3

4. S4

Constants

The following are the constants for which we need to find online or determine experimentally:

1. $\delta = \left\{ \begin{array}{ll} 0 & \mbox{without light}\\ 1 & \mbox{with light} \end{array} \right.$

2. $c$ represents the rate of conformation change from LovTAP to LovTAP* under 470nm light.

Parameters

1. $\alpha_x$ represents the $x$th synthesis rate $\left(\dfrac{\text{nanomoles}}{\text{min}}\right)$.
   $\alpha=\left(\text{rate of transcription}\right)\times\left(\text{rate of translation}\right)$
   For promoters for which reliable data is not available, we assume an average E. coli transcription speed to be 70 nucleotides/second = 4,200 nucleotides/min, and an average translation speed of 40 amino acids/second = 2400, which is then further regulated by the appropriate ribosome binding site, represented as a normalized constant. Thus, we use the following equation to calculate unknown synthesis rates[0]: $$\alpha = \left(\frac{4200}{\text{gene length}}\right)\times\left(\frac{2400\times\text{RBS Strength}}{\text{protein length}}\right) $$

   $\alpha_1$ LovTAP synthesis rate constant $= 1.06432\times10^{-13}$ nanomoles. $\alpha_2$ tetR synthesis rate constant $= 1.15318\times10^{-13}$ nanomoles. $\alpha_3$ Mnt synthesis rate constant $= 6.0562\times*10^{-13}$ nanomoles. $\alpha_4$ AraC synthesis rate constant $= 5.99989\times10^{-14}$ nanomoles. $\alpha_5$ LacI synthesis rate constant $= 3.94791\times10^{-14}$ nanomoles. $\alpha_6$ CI synthesis rate constant $= 8.93023\times10^{-14}$ nanomoles. $\alpha_7$ CI434 synthesis rate constant $= 1.2236\times10^{-13}$ nanomoles. $\alpha_8$ [check]SupD synthesis rate constant $= 1.2236\times10^{-13}$ nanomoles. $\alpha_9$ [check]T7ptag synthesis rate constant $= 1.2236\times10^{-13}$ nanomoles. $\alpha_{10}$ [check]GAL4 synthesis rate constant $= 1.06432\times10^{-13}$ nanomoles.

   $\alpha_{11}$ S1 synthesis rate constant $= 9.68992\times10^{-14}$ nanomoles. $\alpha_{12}$ S2 synthesis rate constant $= 9.68992\times10^{-14}$ nanomoles. $\alpha_{13}$ S3 synthesis rate constant $= 9.68992\times10^{-14}$ nanomoles. $\alpha_{14}$ S4 synthesis rate constant $= 9.68992\times10^{-14}$ nanomoles.

2. $\beta_x $ represents $x$th basal (un-induced or un-repressed) synthesis rate of a given promoter. We assume that, for all promoters, this is equal to 1% of the synthesis rate constant. $$\forall x \ \beta_x = 0.1\alpha_x$$

3. $\mu_x$ represents the degradation rate of a given protein. When the degredation rate is unknown, we assume a decay of 0.012 proteins/min (half-life of one hour) $\mu_1= 0.0453$ Rate of LovTAP degredation $(1/sec)$$\mu_2= 0.0453$ Rate of LovTAP* degredation $(1/sec)$$\mu_3 = 0.0453$ tetR degredation $(1/sec)$$\mu_4= 0.0453$ Rate of Mnt degredation $(1/sec)$$\mu_5= 0.0453$ Rate of AraC degredation $(1/sec)$$\mu_6= 0.0453$ Rate of LacI degredation $(1/sec)$$\mu_7= 0.0453$ Rate of CI degredation $(1/sec)$$\mu_8= 0.0453$ Rate of CI434 degredation $(1/sec)$$\mu_9= 00453$ Rate of SupD degredation $(1/sec)$$\mu_{10}= 0.0453$ Rate of T7ptag degredation $(1/sec)$$\mu_{11}= 0.012$ Rate of T7 polymerase degredation $(1/sec)$$\mu_{12}= 0.012$ Rate of GAL4 degredation $(1/sec)$$\mu_{13}= 0.012$ Rate of S1 degredation $(1/sec)$$\mu_{14}= 0.012$ Rate of S2 degredation $(1/sec)$$\mu_{15}= 0.012$ Rate of S3 degredation $(1/sec)$$\mu_{16}= 0.012$ Rate of S4 degredation $(1/sec)$

4. $n_x$ represents the hill coefficient for a given protein. $n_1 = 1$ for LovTAP*

   $n_2 = 3$ for tetR)

   $n_3 = 2$ for CI

   $n_4 = 2$ for CI434

   $n_5 = 2$ Hill coefficient of GAL4

   $n_6 = 2$ Hill coefficient of AraC

   $n_7 = 2$ Hill coefficient of T7

   $n_8 = 2$ Hill coefficient of LacI

   $n_9 = 1$ Hill coefficient of Mnt

5. $K_x$ represents, for a given protein, the ligand concentration producing half occupation (ligand concentration occupying half of the binding sites) in nanomoles. This is also the infamous "microscopic dissociation constant". $K_{d1} = 142$ for LovTAP*

   $K_{d2} = 0.179$ for tetR

   $K_{d3} = 50$ for CI

   $K_{d4} = 40$ for CI434

   $K_{d5} = 0.5$ for GAL4

   $K_{d6} = 14$ for AraC

   $K_{d7} = 2$ for T7

   $K_{d8} = 800$for LacI

   $K_{d9} = 50$ for Mnt

6. $c$ $c_1 = 20.20$, which represents the rate of light-induced LovTAP to LovTAP* conformation change $(1/sec)$ $c_2 = 20.20$ which represents the rate of the reaction SupD + T7ptag $\rightarrow$ T7

7. $\delta = \left\{ \begin{array}{ll} 0 & \mbox{without light}\\ 1 & \mbox{with light} \end{array}\right.$

For a complete list of values and sensitivities, please see our spreadsheet of parameters and sensitivities.