IC.Y3.AND.Gate.with.RiboJ:Dry Labs

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Differential equations

Simple Chain Reaction [math]\displaystyle{ A \xrightarrow{k_{1}} B \xrightarrow{k_{2}} C }[/math] [math]\displaystyle{ \begin{alignat}{2} \frac{d[A]}{dt} & = - k_{1}*[A] \\ \frac{d[B]}{dt} & = k_{1}*[A] -k_{2}*[B] \\ \frac{d[C]}{dt} & = k_{2}*[B] \end{alignat} }[/math]	Enzymatic Reaction [math]\displaystyle{ E + S \begin{matrix} k_1 \\ \longrightarrow \\ \longleftarrow \\ k_{2} \end{matrix} ES \xrightarrow{k_{3}} E + P }[/math] [math]\displaystyle{ \begin{alignat}{2} \frac{d[E]}{dt} & = k_{2}[ES] - k_{1}[E][S] + k_{3}[ES] \\ \frac{d[S]}{dt} & = k_{2}[ES] - k_{1}[E][S] \\ \frac{d[ES]}{dt} & = k_{1}[E][S] - k_{2}[ES] - k_{3}[ES] \\ \frac{d[P]}{dt} & = k_{3}[ES] \end{alignat} }[/math]
Constitutive Gene Expression [math]\displaystyle{ Gene \xrightarrow{k_{1}} mRNA \xrightarrow{k_{2}} Protein }[/math] [math]\displaystyle{ \begin{alignat}{1} \frac{d[mRNA]}{dt} & = k_{1} - d_{1}[mRNA] \\ \frac{d[Protein]}{dt} & = k_{2}[mRNA] - d_{2}[Protein] \\ \end{alignat} }[/math]	Simplified Constitutive Gene Expression [math]\displaystyle{ Gene \xrightarrow{k_{1}} Protein }[/math] [math]\displaystyle{ \begin{alignat}{2} \frac{d[Protein]}{dt} = s - d[Protein] \\ \end{alignat} }[/math]
Repressed Gene Expression [math]\displaystyle{ \begin{align} & Repressor \\ & \bot \\ Gene &\rightarrow mRNA \rightarrow Protein \end{align} }[/math] Hill function for transcriptional repression: [math]\displaystyle{ \begin{align} \\ transcriptionRate=\frac{k_1.{K_m}^n}{{K_m}^n+R^n} \end{align} }[/math] [math]\displaystyle{ k_1 }[/math]: maximal transcription rate [math]\displaystyle{ K_m }[/math]: repression coefficient [math]\displaystyle{ n }[/math]: Hill coefficient R = [repressor] [math]\displaystyle{ \begin{alignat}{1} \frac{d[mRNA]}{dt} & = \frac{k_{1}.{K_m}^n}{{K_m}^n+R^n} - d_{1}[mRNA] \\ \frac{d[Protein]}{dt} & = k_{2}[mRNA] - d_{2}[Protein] \\ \end{alignat} }[/math]	Activated Gene Expression [math]\displaystyle{ \begin{align} & Activator \\ & \downarrow \\ Gene & \rightarrow mRNA \rightarrow Protein \end{align} }[/math] Hill function for transcriptional activation: [math]\displaystyle{ \begin{align} \\ transcriptionRate=\frac{k_1.{A}^n}{{K_m}^n+A^n} \end{align} }[/math] [math]\displaystyle{ k_1 }[/math]: maximal transcription rate [math]\displaystyle{ K_m }[/math]: activation coefficient [math]\displaystyle{ n }[/math]: Hill coefficient A=[activator] [math]\displaystyle{ \begin{alignat}{1} \frac{d[mRNA]}{dt} & = \frac{k_{1}.A^n}{{K_m}^n+A^n} - d_{1}[mRNA] \\ \frac{d[Protein]}{dt} & = k_{2}[mRNA] - d_{2}[Protein] \\ \end{alignat} }[/math]
Repressilator [math]\displaystyle{ \begin{alignat}{1} \frac{d[mRNA]_{i}}{dt} & = \frac{a}{1+{[Protein]_{j}}^n} - [mRNA]_{i} \\ \frac{d[Protein]_{i}}{dt} & = b[mRNA]_{i} - b[Protein]_{i} \\\\ \ i=1,2,3; \\ \ j=3,1,2; \\ \end{alignat} }[/math]	Toggle Switch [math]\displaystyle{ \begin{alignat}{1} \frac{d[A]}{dt} & = \frac{\alpha}{1+[B]^4} - \gamma*[A] \\ \frac{d[B]}{dt} & = \frac{\beta}{1+[A]^4} - \gamma*[B] \end{alignat} }[/math]
Responses to various promoters [math]\displaystyle{ \begin{alignat}{1} f([I]) & = \frac{\alpha+[I]^{n_i}}{[K_1]^{n_i}+[I]^{n_i}}*k \end{alignat} }[/math] [math]\displaystyle{ [I] }[/math]: concentration of the inducer [math]\displaystyle{ K_1 }[/math]: Hill constant [math]\displaystyle{ n_i }[/math]: Hill coefficient [math]\displaystyle{ k }[/math]: the maximum expression level due to induction [math]\displaystyle{ \alpha }[/math]: constant relating to the basal level of the promoter due to leakage	AND gate [math]\displaystyle{ \begin{alignat}{1} f([R], [S]) & = \frac{[G]}{[G]_{max}} & = \frac{{(\frac{[R]}{K_R})^{n_R}}*{(\frac{[S]}{K_S})^{n_S}}}{{(1+\frac{[R]}{K_R})^{n_R}}*{(1+\frac{[S]}{K_S})^{n_S}}} \end{alignat} }[/math] [math]\displaystyle{ [R],[S] }[/math]: concentration of the activator proteins [math]\displaystyle{ [G] }[/math]: output [math]\displaystyle{ K_R,K_S }[/math]: Hill constants [math]\displaystyle{ n_R, n_S }[/math]: Hill coefficients

Schedule

Week 1

Monday	Tuesday	Wednesday	Thursday	Friday
	11:00am: Group Meeting		11:30am: Lab Induction	02:00pm: Tutorial 1 Derived and simulated mass action and enzymatic reactions from first principles

Week 2

Monday	Tuesday	Wednesday	Thursday	Friday
	11:00am: Group Meeting			02:00pm: Tutorial 2 Derived and created standard models for constitutive gene expression Derived and created standard models for activated and repressed gene expression

Week 3

Monday	Tuesday	Wednesday	Thursday	Friday
	11:00am: Group Meeting			02:00pm: Tutorial 3 Derived and simulated a repressilator from first principles

Week 3

Monday	Tuesday	Wednesday	Thursday	Friday
	11:00am: Group Meeting			02:00pm: Computational Modelling Modelled Hill functions using parameters describing the best fits of the characterised promoter responses using different RBS found by Wang et al. 2011

Week 4

Monday	Tuesday	Wednesday	Thursday	Friday
	11:00am: Group Meeting			02:00pm: Computational Modelling Modelled transfer function describing biological AND gate for differing inducer concentrations

Computational Modelling

We replicated the best fits of the characterised promoter responses, using different RBSs, from the parameters described by Wang et al. in Engineering modular and orthogonal genetic logic gates for robust digital-like synthetic biology, 2011.

Dose responses of Plac, Pbad and Plux using different RBSs

Using parameters KR =206.1±32.5, KS=3135±374, nR =2.381±0.475, and nS=1.835±0.286 we then replicated the output response of the AND gate for different inducer concentrations.

Output of engineered AND gate for different inducer concentrations