Reaction Mechanism

In all three experiments, to measure the promoter function. We measure green fluorescent protein (GFP) expressed under the control of each three promoter on a plasmid, we detected elevated levels of fluorescence in response three conditions.

In sensing oxygen, we used a hypoxia-sensing promoter (pPepT) that is primarily regulated by the transcriptional activator, fumarate and nitrate reduction regulatory protein (FNR). In the absence of oxygen, FNR binds to a [4Fe–4S]2+ cluster to generate a transcriptionally active homodimer. However, the cluster is degraded in the presence of oxygen, which dissociates the FNR dimer into inactive monomers.

$$ \begin{equation} \begin{aligned} & 2FNR \underset{k_{-O_2}}{\stackrel{k_{O_2}}{\rightleftharpoons}} FNR_2\\ & pPepT+FNR_2 \stackrel{}{\longrightarrow} pPepT^+\\ \end{aligned} \end{equation} $$

As for the L-lactate, we used an L-lactate biosensor, derived from the native lldPRD operon, to detect lactic acid fermentation by host mammalian cells. This lactate-sensing system was constructed on two plasmids: a lactate-inducible reporter plasmid driving expression of a gene of interest and a repressor plasmid, which produces the repressor LldR that dimerizes to inhibit expression of the reporter gene unless bound to lactate.

$$ \begin{equation} \begin{aligned} & LldR_2 \underset{k_{-Lactate}}{\stackrel{k_{Lactate}}{\rightleftharpoons}} 2LldR\\ & pLldR+LldR_2 \stackrel{}{\longrightarrow} pLldR^-\\ \end{aligned} \end{equation} $$

Finally, we used the pH-sensitive promoter pCadC among other systems, that is regulated by a membrane-tethered activator protein (CadC), which shows increased activity in acidic medium compared with medium at a neutral pH.

$$ \begin{equation} \begin{aligned} & pCadC+CadC \stackrel{}{\longrightarrow} pCadC^+\\ \end{aligned} \end{equation} $$

To integrate three biosensors, we construct a three input AND gate, we change downstream transcription of three biosensors into specific RNA, which can interact with one another cooperatively to activate the gate RNA for AND logic. Then, the trigger RNA will bind to the switch RNA, due to the appetency variance, switch RNA stem will unwind, which exposes the ribosomal binding site (RBS) and the start codon to activate translation of the output gene.

In addition, to reduce translational leakage, we use switch designs which the trigger RNA unwinds only the lower portion of the switch RNA stem. Although the RBS remains enclosed within a stem-loop after trigger binding, the stem-loop is engineered to be sufficiently weak to allow ribosome binding and strong translation.

$$ \begin{equation} \begin{aligned} & A1+A2+A3 \stackrel{}{\longrightarrow} Trigger RNA\\ & Trigger RNA + Switch RNA \stackrel{}{\longrightarrow} Target Gene\\ \end{aligned} \end{equation} $$

Transcription Dynamics

We assume that all the promoters have active state and inhibit state

$$ \begin{equation} \begin{aligned} & pPepT^+ \stackrel{K_{a1}}{\longrightarrow} pPepT^+ + RNA_{A1}\\ & pPepT^- \stackrel{K_{a1'}}{\longrightarrow} pPepT^- + RNA_{A1}\\ & pCadC^+ \stackrel{K_{a2}}{\longrightarrow} pCadC^+ + RNA_{A2}\\ & pCadC^- \stackrel{K_{a2'}}{\longrightarrow} pCadC^- + RNA_{A2}\\ & pLIdR^+ \stackrel{K_{a3}}{\longrightarrow} pLIdR^+ + RNA_{A3}\\ & pLIdR^- \stackrel{K_{a3'}}{\longrightarrow} pLIdR^- + RNA_{A3}\\ \end{aligned} \end{equation} $$

$ K $ represent the rate of the reaction
(+) represent activate state
(-) represent inhibit state

Decay Dynamic

$$ \begin{equation} \begin{aligned} & RNA_{A1} \stackrel{K_{d}}{\longrightarrow} \varnothing\\ & RNA_{A2} \stackrel{K_{d}}{\longrightarrow} \varnothing\\ & RNA_{A3} \stackrel{K_{d}}{\longrightarrow} \varnothing\\ \end{aligned} \end{equation} $$

Combine Dynamic

$$ \begin{equation} \begin{aligned} & RNA_{A1} + RNA_{A2} + RNA_{A3} \stackrel{}{\longrightarrow} triggerRNA\\ \end{aligned} \end{equation} $$

In transcription we use Transcription rate equation base on Law of Mass Action: $$ \begin{align*} \text{Transcription Rate} &= k_{\text{on}} \cdot [\text{Transcription Factor}] \cdot [\text{Promoter}] \\ &- k_{\text{off}} \cdot [\text{Transcription Complex}] \end{align*} $$ To simplify the equation, we directly use a constant multiplate the environment factor to represent the transcription factor, for example, we use ko multiplate oxygen concentrate to represent the transcription factor. Therefore, we can get the reaction rate of transcription.

$$ \begin{equation} \begin{aligned} Ka_1 &= k_{\text{on1}} \cdot k_o \cdot O \cdot [ pPepT (+)] - k_{\text{off1}} [RNA_{A1}] \\ Ka_1' &= k_{\text{on1}} \cdot k_{s1} \cdot [ pPepT (-)] - k_{\text{off1}} [RNA_{A1}] \\ Ka_2 &= k_{\text{on2}} \cdot k_h \cdot \text{pH} \cdot [pCadC(+)] - k_{\text{off2}} [RNA_{A2}] \\ Ka_2' &= k_{\text{on2}} \cdot k_{s2} \cdot [pCadC(-)] - k_{\text{off2}} [RNA_{A2}] \\ Ka_3 &= k_{\text{on3}} \cdot k_{la} \cdot \text{lactate} \cdot [ pLIdR (+)] - k_{\text{off3}} [RNA_{A3}] \\ Ka_3' &= k_{\text{on3}} \cdot k_{s3} \cdot [pLIdR(-)] - k_{\text{off3}} [RNA_{A3}] \end{aligned} \end{equation} $$

$ k_{\text{on}} $ represents the binding rate constant
$ k_{\text{off}} $ represents the dissociation rate constant
$ k_{\text{s}} $ represents the transcription factor of inhibit state

ODE Models

In this way, we can get the ordinary differential equations (ODE) that captures the species dynamics.

$$ \begin{equation} \begin{aligned} \frac{d[RNA_{A1}]}{dt} &= k_{\text{on1}} \left[ k_o \cdot O \cdot [ pPepT^{+}] + k_{s1} \cdot [ pPepT^{-}] \right] \\ &- 2k_{\text{off}} [RNA_{A1}] - k_d [RNA_{A1}] \\ &- \left( k_{f_1} [RNA_{A2}] [RNA_{A1}] [RNA_{A3}] \right) \\ \frac{d[RNA_{A2}]}{dt} &= k_{\text{on2}} \left[ k_h \cdot \text{pH} \cdot [pCadC^{+}] + k_{s2} \cdot [pCadC^{-}] \right] \\ &- 2k_{\text{off}} [RNA_{A2}] - k_d [RNA_{A2}] \\ &- \left( k_{f_1} [RNA_{A2}] [RNA_{A1}] [RNA_{A3}] \right) \\ \frac{d[RNA_{A3}]}{dt} &= k_{\text{on3}} \left[ k_{la} \cdot \text{lactate} \cdot [ pLIdR^{+}] + k_{s3} \cdot [pLIdR^{-}] \right] \\ &- 2k_{\text{off}} [RNA_{A3}] - k_d [RNA_{A3}] \\ &- \left( k_{f_1} [RNA_{A2}] [RNA_{A1}] [RNA_{A3}] \right) \end{aligned} \end{equation} $$

$ k_{f_1} $ represents the combine rate of the three RNAs combining to trigger RNA

In our experiment we mainly focus on the trigger RNA's formation, using the original ODE function to fit the model may require huge amounts of data and time. Therefore, we simplified the data as follow.

In all three promoters, when not meeting the active environmental factors such as hypoxia, low pH, and high L-lactate, they will present a low transcription activity, but in a specific environment they will present a very high transcription activity and produce a large amount of specific RNA. This means in active form, for example, $ k_{la} \cdot \text{lactate} \cdot [ pLIdR^{+} ] \gg k_{s3} \cdot [pLIdR^{-}] $ and we can ignore the latter half. On the contrary, for example, $ k_{la} \cdot \text{lactate} \cdot [ pLIdR^{+} ] \ll k_{s3} \cdot [pLIdR^{-}] $ and we can ignore the first half, however, due to the inhibit state of the promoter, ks3 is relatively very small. Therefore, in the whole process, we can assume that the latter half is a constant multiplied promotor. So, we can simplify the ODE.

$$ \begin{equation} \begin{aligned} \frac{d[RNA_{A1}]}{dt} &= k_{\text{on1}} (k_o \cdot O + k_{s1}) [ pPepT] \\ &- 2k_{\text{off1}} [RNA_{A1}] - k_d [RNA_{A1}] \\ &- \left( k_{f_1} [RNA_{A2}] [RNA_{A1}] [RNA_{A3}] \right) \\ \frac{d[RNA_{A2}]}{dt} &= k_{\text{on2}} (k_{h2} \cdot \text{pH} + k_{s2}) [ pCadC] \\ &- 2k_{\text{off2}} [RNA_{A2}] - k_d [RNA_{A2}] \\ &- \left( k_{f_1} [RNA_{A2}] [RNA_{A1}] [RNA_{A3}] \right) \\ \frac{d[RNA_{A3}]}{dt} &= k_{\text{on3}} (k_{la} \cdot \text{lactate} + k_{s3}) [ pLIdR] \\ &- 2k_{\text{off3}} [RNA_{A3}] - k_d [RNA_{A3}] \\ &- \left( k_{f_1} [RNA_{A2}] [RNA_{A1}] [RNA_{A3}] \right) \end{aligned} \end{equation} $$

when it comes to the AND gate part, we construct three specific RNA and connect it in the downstream of three promoters respectively. When all three specific RNAs are transcribed, they will combine into trigger RNA. However, trigger RNA cannot express lysis gene CDD-iRGD on its own, we construct a switching RNA that contains our lysis gene RNA only when the switching RNA combines to trigger RNA, the stem-loop will unwind and permit the translation of our lysis gene.

Since switching RNA is background expression, we use the law of mass action to stimulate the reaction rate.

$$ \begin{equation} \begin{aligned} \frac{d[\text{switch RNA}]}{dt} &= (k_{f_5} [\text{PTet-CDD-iRGD}]) - (k_d [\text{switch RNA}]) \\ \frac{d[\text{trigger RNA}]}{dt} &= (k_{f_1} \cdot \text{RNA}_{A2} \cdot \text{RNA}_{A1} \cdot \text{RNA}_{A3}) - (k_d [\text{trigger RNA}]) \end{aligned} \end{equation} $$

Taking secretion into account, our lysis gene ODE will be.

$$ \begin{equation} \begin{aligned} \frac{d[\text{CDD-iRGD}]}{dt} = (k_{f_2} [\text{switch RNA}] [\text{trigger RNA}]) - (k_f \cdot [\text{CDD-iRGD}]) \end{aligned} \end{equation} $$

$ k_{f_1} $, $ k_{f_2} $, $ k_{f_5} $ is the rate constant of the law of mass action.
$ k_{f} $ represents the secretion rate constant.