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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 below 32 °C to 37 °C ables 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} - \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]_{max} = \frac{\alpha _{1}}{\mu _{1}} \end{equation}

\begin{equation} \large [C]_{max} = \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 [mV] - \mu_{4}[V] \end{equation}

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

\begin{equation} \large [V]_{max} = \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[G]}{dt} = \alpha_{6} \cdot [mG] - \mu_{6}[G] \end{equation}

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

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

Parameters and variables

Our gene circuit is made of six different variables: the concentrations of three proteins (cI, VIP and GFP) and their respective mRNA inside a cell. In table 1, the symbols for each variable are shown. Proteins are represented by a single letter and their mRNAs are represented by that same letter with a lowercase "m" before it.

Table 1.- Variables

Symbol	Definition	Gene size in bp	Source
mC, C	Transcription factor cI (mRNA and protein)	775	http://partsregistry.org/wiki/index.php?title=Part:BBa_C0051
mV, V	Insecticide protein VIP (mRNA and protein)	2412	http://www.ncbi.nlm.nih.gov/nuccore/HQ876489
mG, G	Reporter protein GFP (mRNA and protein)	876	http://parts.igem.org/wiki/index.php?title=Part:BBa_E0240

To parameterize our model, we chose to follow the approach of team Beijing 2009; they propose a relationship between the gene length in base pairs and the maximum transcription rate and, similarly, between the protein length in amino acid numbers and the maximum translation rate. Assuming that the number of polymerases and ribosomes are the average values determined for E. coli, the following equations are used:

Parameters

Symbol	Definition	Values	Formula	Source
α1	Transcription rate of cI	5.6	4200/Gene Length (nM/min)	https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters
α2	Translation rate of cI	9.6	2400RBS/Protein Length	https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters
α3	Transcription rate of VIP	1.74129353	4200/Gene Length (nM/min)	https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters
α4	Translation rate of VIP	2.985075	2400RBS/Protein Length	https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters
α5	Transcription rate of GFP	5.53359684	4200/Gene Length (nM/min)	https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters
α6	Translation rate of GFP	9.486166	2400RBS/Protein Length	https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters

For all the variables the degradation rate is expressed by the formula (ln(2)/half life)+(ln(2)/division time), with the same division time of e. coli (30 min), because all the process occurs within it. The only thing that change is the half time; for cI, VIP and GFP (mRNA) is 6.8 minutes and for cI (Selinger, GW, et al., 2003), VIP and GFP protein is more than 10 hours (Varshavsky, (1997) and Tobias et al., 1991).

Degradation

Symbol	Definition	Values	Formula	Source
μ1μ3,μ5,	Degradation rate of cI (mRNA	0.18063836	Half life = 6.8 min, Division time = 30 min	(Selinger, GW, et al., 2003)
μ2,μ4,μ6	Degradation rate of cI (protein)	0.03885825	Half life > 10 h; division time = 30 min	(Varshavsky, (1997) and Tobias et al., 1991)

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 simulations are showing our model in a temperature below 32ºC the first 200 minutes making the variables of cI protein production in their minimum value for this initial time, 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 higher of our ideal parameters of cI production (above 32ºC) allowing the transcription and translation of cI protein in a factor of 1 (represented in the equation number 5 showed above) this is the rise in the blue graph at the time 200 minutes, inhibiting the formation of the other two system parts: Vip3Ca3 and GFP proteins, simulated as the decrease of the purple and green graphics. Whose percents in the E.Coli bacterium drops to the minimum until the cI production stops again and the process restarts.

cI active-inactive simulation

The graphics that are shown represent the change of the concentration of each part in relation to time, including all the parameters that we presented and the equation 5, where we represent the repression and activation of the riborswitch making the value of its function 0 or 1. The first graph simulation is showing our model in a temperature between 32 to 37ºC in the first 200 minutes. As you can see, in the graph of cI (c0051), it is active and produced constantly, repressing the Vip3Ca3 and GFP proteins, but when the temperature changes below than the 32 to 37ºC the cI it is now turning off allowing the transcription and translation of Vip3Ca3 and GFP so changes the graphic, but We can appreciate the GFP is producing more than the Vip3Ca3, this because of the bigger base pair size of Vip3Ca3 than GFP that make it longer to be synthesis by the ribosome. This bp length of those proteins are 2412 and 876 respectively.

Probability Equation

\begin{equation} \large p i\left ( T \right )= \frac{e^{a_{i}+b_{i}T}}{1+e^{a_{i}+b_{i}T}} \end{equation}