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From 2013hs.igem.org

Mathematical modeling

System topology

Our system is composed by two parts, the first consists of the transcription factor cI (c0051) under the regulation of a promoter with a riboswitch (J23100 and k110517) that can be activated or repressed depending on the temperature; we call this our thermosensor. The second part consist of two genes -the insecticide protein VIP and the green fluorescent protein, GFP- that are under the regulation of cI.

The thermosensor is repressed when the temperature is below 32 °C; when the temperatures reaches a range between 32 and 37 °C, the thermosensor is activated.

cI (c0051) is a gene that constitutively represses the promoter r0051. Once our promoter is activated starts the production of Vip3Ac3 and GFP. This is means that a range of 32 °C to 37 °C enables the production of the Vip and GFP.

Deterministic model

Our deterministic model represents the change in time of the concentrations of mRNAs and their corresponding proteins. It assumes that the variables behave continuously and obey kinetic rules that can be represented by constants. We are aware that, in practice, the components and variables in the model may not fall under the assumptions of a deterministic model and that there is always the chance that noise effects are being grossly underestimated; however, we propose this model as a general framework for thermoregulator modeling, upon which further work can be done.

Equations

\begin{equation} \large \frac{d[mC]}{dt} = \alpha_{1} \cdot \frac{K_{D}^h}{K_{D}^h + [cI]^h} - \mu_{1}[mC] \end{equation}

\begin{equation} \large \frac{d[C]}{dt} = \alpha_{2} \cdot f_{RBS} \cdot [mC] - \mu_{2}[C] \end{equation}

\begin{equation} \large [mC] = \frac{\alpha _{1}}{\mu _{1}} \end{equation}

\begin{equation} \large [C] = \frac{\alpha _{1} \cdot \alpha _{T}} {\mu _{1} \cdot \mu _{T}} \end{equation}

\begin{equation} \large f_{RBS} = \left\{ \begin{array}{rcl} 0 & \mbox{if} & t \geq ON \\ 1 & \mbox{if} & t < ON \end{array} \right. \end{equation}

\begin{equation} \large \frac{d[mV]}{dt} = \alpha_{3} \cdot \frac{K_{D}^h}{K_{D}^h + [C]^h} - \mu_{1}[mV] \end{equation}

\begin{equation} \large \frac{d[V]}{dt} = \alpha_{4} \cdot f_{RBS} \cdot [mV] - \mu_{4}[V] \end{equation}

\begin{equation} \large [mV] = \frac{\alpha _{3}}{\mu _{3}} \end{equation}

\begin{equation} \large [V] = \frac{\alpha _{3} \cdot \alpha _{4}} {\mu _{3} \cdot \mu _{4}} \end{equation}

\begin{equation} \large \frac{d[mG]}{dt} = \alpha_{5} \cdot \frac{K_{D}^h}{K_{D}^h + [C]^h} - \mu_{5}[mG] \end{equation}

\begin{equation} \large \frac{d[C]}{dt} = \alpha_{6} \cdot f_{RBS} \cdot [mG] - \mu_{6}[G] \end{equation}

\begin{equation} \large [mG] = \frac{\alpha _{5}}{\mu _{5}} \end{equation}

\begin{equation} \large [G] = \frac{\alpha _{5} \cdot \alpha _{6}} {\mu _{5} \cdot \mu _{6}} \end{equation}

Parameters

Symbol

Definition

Values

Formula

Source

Simulations

The saturation values for mRNAs and proteins were calculated analytically; but since there are several variables, it becomes complicated to integrate by analytical methods, so we use methods of numerical integration in a computer program by called Simulink. The values of the parameters (rate of transcription, translation, degradation and dissociation) are the ones we have found so far, but we continue researching in order to improve and expand our model.

cI inactive-active simulation

This simulation are showing our model in a temperature between 20 to 37ºC making the variables of cI protein production in their minimum value for the first 30 minutes, creating the possibility of Vip3Ca3 and GFP proteins to be processed in the bacteria at maximum capacity (maximum capacities shown in the parameter table). Past the half hour the temperature is below or higher of our ideal parameters of cI production (20 to 37ºC) allowing the transcription and translation of cI protein in a factor of 0 (represented in the equation number 5 showed above), inhibiting the formation of the other two system parts: Vip3Ca3 and GFP proteins. Whose percents in the E.Coli bacterium drops to the minimum until the cI production stops again and the process restarts.