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Overview

Here will be a summary of everything that was done during the project, so that the reader can get an idea by looking at this and the menu on the left.

Gold Medal Criterion #3

Use modeling to gain insight into how your project works or should be implemented. Explain your model's assumptions, data, parameters, and results in a way that anyone could understand.


Please see the 2022 Medals Page for more information.

Best Model Special Prize

Models and computer simulations provide a great way to describe the functioning and operation of BioBrick Parts and Devices. Synthetic biology is an engineering discipline and part of engineering is simulation and modeling to determine system behavior before building your design. Designing and simulating can be iterated many times in a computer before moving to the lab. This award is for teams who build a model of their system and use it to inform system design or simulate expected behavior before or in conjunction with experiments in the wetlab.

To compete for the Best Model prize, please describe your work on this page and also fill out the description on the judging form.


Please see the 2022 Awards Page for more information.

Overview


Mathematical models and computer simulations provide a great way to describe the function and operation of Parts and Devices. Synthetic Biology is an engineering discipline, and part of engineering is simulation and modeling to determine the behavior of your design before you build it. Designing and simulating can be iterated many times in a computer before moving to the lab.

Please note you can compete for both the Gold Medal criterion #3 and the Best Model prize with this page.

Biochemical reactions and differential equations

Intracellular IPTG: Diffusion process

To induce protein expression under a LacI-repressible promote, it is necessary to add IPTG to the medium so that it enters the cell and from there avoids repression of expression. The diffusion process of IPTG can be modelled as the following chemical system:

\begin{equation} \label{Eq1} \ce{I <=>[\varphi][\varphi cf]I_{out}} \end{equation}

The extracellular IPTG is denoted as $I_{out}$ while, intracellular IPTG is denoted as $I$. $\varphi$ is the permeability of the cell and $cf$ is the ratio between the volume of the cell and the volume of the medium (Eq. \ref{Eq2}) , this variable represents the probability of interaction between IPTG and the cell.

\begin{equation} \label{Eq2} cf= \frac{Vol. E. Coli}{Vol. Medio} \end{equation}

Using the Law of mass this biochemical reaction can be expressed as an ordinary differential equation (ODE).

\begin{equation} \label{Eq3} \mathring{I}=\varphi(cf*(I_{Total}-I)-I) \end{equation}

Constitutive expression of the gene LacI

The antimicrobial peptide expression system is repressible by LacI, so we consider as a first step for repression the constitutive expression of the LacI gene. Constitutive expression of the gene LacI includes the transcription of the gen LacI to mRNA (Eq.\ref{Eq4}), the translation of mRNA to LacI protein (Eq.\ref{Eq6}).The degradation of mRNA (Eq.\ref{Eq5}) and protein (Eq.\ref{Eq7}) is also considered.

\begin{equation} \label{Eq4} \ce{->[\ce{KLi}] m_{Li}} \end{equation} \begin{equation} \label{Eq5} \ce{m_{Li} ->[KdLi] 0} \end{equation} \begin{equation} \label{Eq6} \ce{m_{Li} ->[\ce{KpLi}] Li} \end{equation} \begin{equation} \label{Eq7} \ce{Li ->[\ce{KdpLi}] 0} \end{equation}

These equations can also be written as ODEs based on the Law of mass action.

\begin{equation} \label{Eq8} m_{Li}=K_{Li} \end{equation} \begin{equation} m_{Li}=-K_{dLi}(m_{Li}) \end{equation} \begin{equation} \label{Eq9} Li=K_{pLi}(m_{Li}) \end{equation} \begin{equation} Li=-K_{dpLi}(Li) \end{equation}

Dimerization of LacI

For protein expression to be repressed, a dimer formed by two LacI molecules must bind to the operator.The dimerization of LacI can be described by the following biochemical reaction:

\begin{equation} \label{Eq10} \ce{2Li <=>[K_{fLi}][K_{sLi}]Li_{2}} \end{equation}

Repression mechanism

We assume that one LacI dimer binds to one operator. And that the LacI monomer is unable to bind to the operator and exert any repressive effect.Under these assumptions the repression action of LacI is described by the following biochemical reaction:

\begin{equation} \ce{Li_{2}+O <=>[K_{LiO}][K_{sLiO}]Li_{2}O} \end{equation}

Also, we write the corresponding differentials equations for the LacI dimer and for the free operator:

\begin{equation} O=K_{sLio}(Li_{2}O)-K_{LiO}(Li_{2})(O) \end{equation} \begin{equation} Li_{2}=K_{sLio}(Li_{2}O)-K_{LiO}(Li_{2})(O) \end{equation}

1st derepression mechanism

The induction of the protein expression of our system is assumed to take place with IPTG as inducer Two molecules of IPTG binds strongly to the LacI dimer, this interaction describe the mechanism of the allosteric regulation (Bell et. al.,} 2000). We assumed that will be a cooperative binding, and this reaction will be reversible.

The first derepression mechanism is when IPTG binds to free LacI dimer. This process can be modeled as the following biochemical reaction:

\begin{equation} \ce{Li_{2}+2I <=>[K_{uLi_{2}I}][K_{sLi_{2}I}]Li_{2}I_{2}} \end{equation}

The ODEs for concentration of LacI dimer and IPTG-LacI complex are:

\begin{equation} \ce{Li_{2}+2I <=>[K_{uLi_{2}I}][K_{sLi_{2}I}]Li_{2}I_{2}} \end{equation}

2nd derepression mechanism

The second derepresion mechanism is taken account IPTG binding to LacI-Operator complex, thus leaving the operator free This process can be modeled as the following biochemical reaction:

\begin{equation} \ce{2I+Li_{2}O <=>[\psi_{f}][\psi_{s}]Li_{2}I_{2}+O} \end{equation}

he ODEs for concentration of LacI dimer and IPTG-LacI complex are:

\begin{equation} Li_{2}I_{2}=\psi_{f}(Li_{2}O)(I)^{2}-\psi_{s}(Li_{2}I_{2})(O) \end{equation}

Deterministic model: Mass balance

Taking into account that the total operator is equal to the free operator $[O]$ plus the operator bound to the LacI dimer $[Li_{2}O]$), the mass balances are written as:

\begin{equation} m_{Li}=K_{m_{Li}}-K_{dLi}(m_{Li}) \end{equation} \begin{equation} Li=K_{pLi}(m_{Li})-K_{dpLi}(Li)-k_{fLi}(Li)^{2}+k_{sLi}(Li_{2}) \end{equation} \begin{equation} \mathring{I}=\varphi(cf*(I_{Total}-I)-I) \end{equation} \begin{equation} \mathring{L}i_{2}=k_{fLi}(Li)^{2}-k_{sLi}(Li_{2})-K_{LiO}(Li_{2})(O)+K_{sLio}(O_{total}-O)-K_{uLi_{2}I}(Li_{2})(I)^{2}+K_{sLi_{2}I}(Li_{2}I_{2}) \end{equation} \begin{equation} \mathring{L}i_{2}I_{2}=K_{ULi_{2}I}(Li_{2})(I)^{2}-K_{SLi_{2}I_{2}}(Li_{2}I_{2}) \end{equation} \begin{equation} \mathring{O}=\psi_{f}(O_{total}-O)(I)^{2}-\psi_{s}(Li_{2}I_{2})(O)-K_{LiO}(Li_{2})(O)+K_{sLio}(O_{total}-O) \end{equation} \begin{equation} m_{PcOS}=K_{PcOS}(O)-\gamma_{PcOS}(m_{PcOS}) \end{equation} \begin{equation} P_{PcOS}=\alpha_{PcOS}(m_{PcOS})-Dm_{PcOS}(P_{PcOS}) \end{equation}

To study the behavior of our expression system as predicted by the deterministic model, we first solved the ODEs, and find the steady state concentrations of each molecule.

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