Model

Overview

As we began to develop our project, a question emerged: How to set up a production system that guarantees maximum production of the molecules of interest? We seek to provide an answer through the development of mathematical models. On this page you can read how our model:

Optimize the production process of the molecules of interest, by knowing the key input control parameters.
Evaluate the scalability and profitability of the production process.

iRNAldo welcoming you to our model page.

BACKGROUND

The molecules we are interested in producing are antimicrobial peptides. Our plan was to design a system that would allow us to control the amount of peptides obtained. Thus we chose to make an IPTG-inducible system. We chose to use a mathematical model to decide what the best concentrations of IPTG in our system are. Through a system of ordinary differential equations (ODEs), we predicted the production of our interest molecules.

Additionally, we used our system to select the promoter to use in our expression vectors.

THEORY BEHIND THE MODEL

In our project we plan to produce antimicrobial peptides. We designed genetic constructs for the expression of these molecules under an IPTG-inducible promoter. So, our system has two states: turned off (when there is no IPTG in the medium) and turned on (when there is IPTG).

System turned off

When we do not add IPTG to our system, expression is being repressed due to the presence of LacI dimer. E coli BL21 (DE3) has a gene that naturally encodes for LacI protein (Kwon et al., 2015). The gene will then express the mRNA coding for the LacI protein for its further translation. When two LacI proteins interact, a dimer can be formed. The LacI dimer finally binds to the operator and prevents the translation of the gene of our molecule of interest from happening.

System turned on

When IPTG is added to the medium, it enters the bacterium by diffusion. Once inside, it binds to the LacI dimer, inhibiting it from repressing LacI expression. This allows the expression of our molecules of interest.

Schematic representation of the IPTG-inducible and LacI-repressible expression system. — **Figure 1.** Production process of one antimicrobial peptide in a single cell. It starts with the uptake of IPTG, then the formation of IPTG-LacI complex, and finally the expression of the peptide gene.

MODEL

The model is based on the production of antimicrobial peptides that are shown in Figure 1. The model is a qualitative analysis that allows us to understand the expression system by looking at the change in concentration of each of the components/molecules over time.

ASSUMPTIONS OF THIS MODEL

The assumptions made in this model are shown below.

The concentration of antimicrobial peptides in the cell is assumed to have no impact on the functionality of the cell.
The LacI gene present in the E. coli BL21 (DE3) genome is consistently expressed.
The system is initially repressed (turned off).
The operation of the expression system is the same for both peptides (PcOSM and (CBD)2-DrsB1).
The amount of species transformed by the reactions only depends of the current amount of species, the rates which these reaction proceed, and the stoichiometry of the reaction.
The number of copies of the gene LacI keeps constant over time (because it is a chromosomal gene).
The plasmids that contain the gene of our interest molecule are first duplicated, and then the number of this plasmids will keep constant over time. Our system only takes into account the moment when the number of plasmids is constant.
It models what happens in a single cell.

PARAMETERS

Table 1 lists the parameters that influence our protein expression model. Most of the values we used for modeling were taken from the literature. Table 2 shows the initial conditions of the model.

Table 1. Parameters used on our protein expression model.

Name	Notation	Value	Unit	Reference
Constitutive expression of LacI
Transcription rate of LacI gene	$$ K_{Li} $$	0.23	$$ nM/min $$	(Stamatakis & Mantzaris, 2009)
Translation rate of LacI	$$ K_{pLi} $$	15	1/min	(Stamatakis & Mantzaris, 2009)
Degradation rate of mRNA of LacI	$$ K_{dLi} $$	0.693	1/min	(Wong et al., 1997)
LacI protein degradation rate	$$ K_{dpLi} $$	0.1	1/min	(Wong et al., 1997)
LacI dimers formation
LacI dimer formation constant	$$ K_{fLi} $$	50	1/(nM*min)	(Stamatakis & Mantzaris, 2009)
LacI dimer dissociation constant	$$ K_{sLi} $$	0.001	1/min	(Stamatakis & Mantzaris, 2009)
Repression mechanism
Union between dimer and operator constant	$$ K_{LiO} $$	960	1/(nM*min)	(Stamatakis & Mantzaris, 2009)
Dissociation between dimer and operator constant	$$ K_{sLiO} $$	2.4	1/min	(Stamatakis & Mantzaris, 2009)
Intracellular IPTG: Diffusion process
Permeability	$$ \varphi $$	0.65	1/min	(Yildrim et al., 2003)

1st derepression mechanism
IPTG and Li2 association rate constant	$$ K_{uLi2I} $$	3*10^-7	1/(min*nM^2)	(Stamatakis & Mantzaris, 2009; Zygourakis, 2011)
IPTG and Li2 dissociation rate constant	$$ K_{sLi2I} $$	12	1/min	(Stamatakis & Mantzaris, 2009; Zygourakis, 2011)
2nd derepression mechanism
Represor-operator and inducer association rate constant	$$ \psi_{f} $$	3*10^-7	1/(min*nM^2)	(Stamatakis & Mantzaris, 2009; Zygourakis, 2011)
Represor-inducer complex and operator dissociation rate constant	$$ \psi_{s} $$	4.8*10^3	1/(min*nM^2)	(Stamatakis & Mantzaris, 2009; Zygourakis, 2011)
Protein of interest expression
Transcription rate of interest peptide	$$ K_{peptide} $$	1000	nM/min	(Basu et al., 2005)
Translation rate of interest peptide	$$ \alpha_{peptide} $$	15	1/min	(Stamatakis & Mantzaris, 2009)
Degradation rate of mRNA of interest peptide	$$ \lambda{peptide} $$	20.2	1/min	(Marshall & Noireaux, 2019)
Interest peptide degradation rate	$$ \beta{peptide} $$	0.017	nM/min-1	(Wong et al., 1997)

Operator total	$$ O $$	34	nM	estimated

Representation of LacI mRNA. — **Table 2.** Initial conditions for the model.

Irnaldo waving — **Table 2.** Initial conditions for the model.

Variable	Description	Value	Unit
mLi	LacI mRNA	5	nM
Li	LacI protein	0	nM
Li2	LacI dimer	0	nM
Li2O	LacI dimer attached to the operator	$$ O_{total} $$	nM
O	Free operator	0	nM
Li2I2	LacI dimer attached to two molecules of IPTG	0	nM
mProt	Protein of interest mRNA	0	nM
PProt	Protein of interest	0	nM
I	IPTG intracellular	0	nM
$$ I_{out} $$	IPTG intracellular	1000000	nM

RESULTS

Based on the initial conditions and parameters shown in Tables 1 and 2, the model was run. It is assumed that the IPTG is added at t= 100 min. This is to simulate an estimated time in which our cell is left to grow before the induction.

Due to the constitutive expression of the LacI gene in the E. coli BL21 (DE3) genome and the absence of IPTG, during the first 100 minutes the expression system is repressed. The results of the model in the first 100 minutes are shown in Figure 2. During this time, we can observe interesting events, such as the constant accumulation of the LacI dimer. This is due to to the abscence of IPTG in the medium to occupy the dimer. Thus, it tends to grow non-stop during this period. Another event worth discussing is the behavior of the LacI-Operator complex. Due to the absence of IPTG in the medium, it is natural for the system to remain repressed. Although some fluctuations exist during the first minutes, the change that occurs is minimal. Hence, it is safe to say that the behavior of the complex remains constant.

**Figure 2.** Changes in the concentration of each variable over time. The graph shows an accumulation of the LacI dimer (Li2), because it is being produced and can only bind to the operator, which is fully occupied (Li2O).

After 100 minutes, we are assuming that our cells have achieved an adequate growth (OD600 0.4-0.6). Thus, IPTG is now added into the medium. The changes of this phenomenom can be observed in Figure 3. First, the concentration of extracellular IPTG starts diminishing, while the intracellular concentration naturally begins to increase. The presence of IPTG inside the cell now induces the start of the derepression mechanisms. As the free LacI dimer and the Dimer-Operator complex begin to reduce, the LacI-IPTG complex begins to form. The derepression of the system allows our peptide to be transcripted and and produced.

Behavior of our system after induction with IPTG at t=100 mins. — **Figure 3.** Behavior of our system after induction with IPTG at t=100 mins

Three different scenarios

Below we show three different scenarios in which our model can be applied. Click on each of the buttons to read each of the situations.

Using a T7 promoter

An agricultural company wants to create a recombinant protein without antimicrobial activity. For this, they decided to use a construct with a T7 promoter that has a higher rate of production. To prove it they use the mathematical model by modifying the variable $ K_{peptide} $ (This can be taken from the literature, or adjusted from experimental data). The model can predict the new protein concentrations (which can be observed to be higher in this scenario compared to the previous one). In this case, the behavior only changes in the second scenario (when we add IPTG, i.e. when the system is on) since the variable being changed only affects when there is transcription of the protein of interest.

Analysis of a change of the promoter — **Figure 4.** Changes in the behavior of the different elements over time when switching to a stronger promoter. It is observed that the concentrations of the peptide mRNA (mPeptide) and of the peptide as such (pPeptide) are higher than those used in the first results of our model.

It is important to clarify that the T7 promoter has slight differences in performance with the promoters that are modeled by our system. For example, it is necessary to consider that IPTG will not only release the operator, but will also promote the production of T7 RNA polymerase (enzyme necessary to initiate transcription under a T7 promoter). Therefore, parameters and variables would have to be added to obtain a behavior more in line with the real one. However, the model as proposed here can give a first approach.

Using a different concentration of IPTG

Suppose you want to obtain an approximate amount of protein, since you want to find what is the highest amount of protein the cell can produce without affecting its growth. To do this, you set up an experiment where you measure the cell growth of different samples after inducing protein expression.

Suppose that after analyzing the data, it is obtained that the highest amount of protein that the cell can produce without affecting its growth rate was when IPTG was applied at a concentration of 0.4 mM. The model can be used to give an approximation of the amount of protein the cells will produce.

In this case, the initial value $ I_{out} $ is changed to 400000 nM. We note the change in the second scenario since the IPTG only takes effect when the system is on.

Using a different cell as chasis

We want to test the expression system on another strain. Therefore, E. coli TOP10 is chosen to express the protein with the constructs. This strain does not encode for the LacI protein, so there will be no presence of the repressor protein.

For this case we will compare two cases: with IPTG and without IPTG.

To model this situation we will change the LacI transcription rate $ K_{Li} $ = 0 (since this strain does not produce LacI). Likewise, the initial amount of operator occupied $ Li_{2} O $ will be 0, since there is no LacI repressing expression. And the amount of free operator will be equal to the amount of total operator.

Without IPTG

With IPTG

**Figure 7.** Changes in the behavior of the different elements over time when there is no gene coding for LacI in the strain where it is expressed. LacI ( $ Li $ ) is not present, as shown in the graph.

BUILDING AND RUNNING THE MODEL

The equations used in this model were constructed from mass balances. For this purpose, the biochemical reactions present in our system were analyzed, then they were rewritten as differential equations and finally the mass balances were performed to construct the system of Ordinary Differential Equations.

Biochemical reaction and differential equations

Intracellular IPTG: Difussion process

To induce protein expression under a LacI-repressible promoter, it's necessary to add IPTG to the medium. This way, it enters the cell and avoids repression of expression. The following chemical system describes the diffusion process of IPTG.

$$ \begin{align} \ce{I <=>[\varphi][\varphi cf]I_{total}} \tag{1} \label{Eq1} \\ \\ \end{align} $$

First reaction: IPTG is found extracellular and can enter thorugh the membrane but it can also get out of the cell.

We defined the extracellular IPTG as $ I_{out} $, at the initial time this value equals $ I_{total} $. While defining intracellular IPTG 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.

$$ \begin{align} cf= \frac{E. Coli Vol}{Medium Vol.} \tag{2} \label{Eq2} \\ \\ \end{align} $$

Using the law of mass action, this biochemical reaction can be expressed as an (ODE).

$$ \begin{align} I_{out} &= \varphi (I) - \varphi (cf) (I_{out}) \tag{3} \label{Eq3} \\ \\ I &= \varphi (cf) (I_{out}) - \varphi (I) - 2 \psi_{f} (Li_{2}O) (I^{2}) + 2 \psi_{s} (Li_{2} I_{2}) (O) \tag{4} \label{Eq4} \\ \\ \end{align} $$

Constitutive expression of the LacI gene

The antimicrobial peptide expression system is repressible by LacI. Thus our first step for repression is the constitutive expression of the LacI gene. Constitutive expression includes the transcription of the gen LacI to mRNA (Eq.\ref{Eq5}), and the translation of mRNA to LacI protein (Eq.\ref{Eq7}). The degradation of mRNA (Eq.\ref{Eq6}) and protein (Eq.\ref{Eq8}) is also considered.

$$ \begin{align} \ce{->[\ce{KLi}] m_{Li}} \tag{5} \label{Eq5} \\ \\ \ce{m_{Li} ->[KdLi] 0} \tag{6} \label{Eq6} \\ \\ \end{align} $$

Second reaction: messenger LacI can get translated an ready for the next reaction or translated and degradated.

$$ \begin{align} \ce{m_{Li} ->[\ce{KpLi}] Li} \tag{7} \label{Eq7} \\ \\ \ce{Li ->[\ce{KdpLi}] 0} \tag{8} \label{Eq8} \\ \\ \end{align} $$

Once there is LacI protein it can get dimerized for the next reaction or can be degradated.

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

$$ \begin{align} m_{Li} &= K_{Li} - K_{dLi} (m_{Li}) \tag{9} \label{Eq9} \\ \\ Li &= K_{pLi} (m_{Li}) - K_{dpLi} (Li) \tag{10} \label{Eq10} \\ \\ \end{align} $$

Dimerization of LacI

To repress protein expression, we assumed that a dimer formed by two LacI molecules must bind to the operator. The dimerization of LacI can be described by the following biochemical reaction:

$$ \begin{align} \ce{2Li <=>[K_{fLi}][K_{sLi}] Li_{2}} \tag{11} \label{Eq11} \\ \\ \end{align} $$

When aplying the law of mass action, the set of ODEs are:

$$ \begin{align} Li_{2} &= K_{fLi} (Li^2) - K_{SLi} (Li_{2}) \tag{12} \label{Eq12} \\ \\ Li &= 2 K_{sLi} (Li_{2}) - 2 K_{fLi} (Li^2) \tag{13} \label{Eq13} \\ \\ \end{align} $$

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 following biochemical reaction describes the repression action:

$$ \begin{align} \ce{Li_{2} + O <=>[K_{LiO}][K_{sLiO}] Li_{2}O} \tag{14} \label{Eq14} \\ \\ \end{align} $$

LacI dimer is reponsable for the repression mechanism so it joins the operator when there is a lack of IPTG.

The differentials equations are:

$$ \begin{align} O &= K_{sLio} (Li_{2}O) - K_{LiO} (Li_{2}) (O) \tag{15} \label{Eq15} \\ \\ Li_{2} &= K_{sLio} (Li_{2}O) - K_{LiO} (Li_{2}) (O) \tag{16} \label{Eq16} \\ \\ Li_{2}O &= K_{Lio} (Li_{2}) (O) - K_{sLiO} (Li_{2}O) \tag{17} \label{Eq17} \\ \\ \end{align} $$

1st derepression mechanism

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

The first derepression mechanism is when IPTG binds to a free LacI dimer. The following biochemical reaction model this process:

$$ \begin{align} \ce{Li_{2} + 2 I <=>[K_{uLi_{2}I}][K_{sLi_{2}I}] Li_{2}I_{2}} \tag{18} \label{Eq18} \\ \\ \end{align} $$

When IPTG is foun intracellularly it bind to a LacI dimer creating a complex (This is the first repression memchanism wich by the way, is reversible).

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

$$ \begin{align} Li_{2}I_{2} &= K_{uLi_{2}I} (Li_{2}) (I^2) - K_{sLi_{2}I} (Li_{2}I{2}) \tag{19} \label{Eq19} \\ \\ Li_{2} &= K_{sLi_{2}I} (Li_{2}I_{2}) - K_{uLi_{2}I} (Li_{2}) (I^2) \tag{20} \label{Eq20} \\ \\ \end{align} $$

2nd derepression mechanism

The second derepression mechanism takes into account IPTG binding to LacI-Operator complex. Thus, it leaves the operator free. The following biochemical reaction models this process:

$$ \begin{align} \ce{2I+Li_{2}O <=>[\psi_{f}][\psi_{s}] Li_{2}I_{2} + O} \tag{21} \label{Eq21} \\ \\ \end{align} $$

IPTG can join to a dimerized LacI on the Lac operator (This is the second repression mechanism).

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

$$ \begin{align} Li_{2}I_{2} &= \psi_{f} (Li_{2}O) (I^2) - \psi_{s} (Li_{2}I_{2}) (O) \tag{22} \label{Eq22} \\ \\ Li_{2}O &= \psi_{s} (Li_{2}I{2}) (O) - \psi{f} (I^2) (Li_{2}O) \tag{23} \label{Eq23} \\ \\ O &= \psi_{f} (Li_{2}O) (I^2) - \psi_{s} (Li_{2}I_{2}) (O) \tag{24} \label{Eq24} \\ \\ \end{align} $$

Deterministic model: mass balance

We have modeled both of the scenarios of our expression system. Considering the total operator is equal to the free operator $ [O] $ plus the dimer-operator complex $ [Li_{2}O] $, the mass balances for both scenarios are:

$$ \begin{align} O_{total} &= Li_{2}O + O \tag{25} \label{Eq25} \\ \\ \end{align} $$

When the system is turned off (no IPTG)

$$ \begin{align} \frac{dm_{Li}}{dt} &= K_{mLi} - K_{dLi} (m_{Li}) \tag{26} \label{Eq26} \\ \\ \frac{dLi}{dt} &= K_{pLi} (m_{Li}) - K_{dpLi} (Li) - 2 K_{fLi} (Li^2) + 2 k_{sLi} (Li_{2}) \tag{27} \label{Eq27} \\ \\ \frac{d{Li}_{2}}{dt} &= K_{fLi} (Li^2) - K_{sLi} (Li_{2}) - K_{LiO} (Li_{2}) (O_{total} - Li_{2}O) + K_{sLio} (Li_{2}O) \tag{28} \label{Eq28} \\ \\ \frac{d{Li}_{2}O}{dt} &= K_{LiO} (O_{total} - Li_{2}O) (Li_{2}) - K_{sLiO} (Li_{2}O) \tag{29} \label{Eq29} \\ \\ \end{align} $$

When the system is turned on (with IPTG)

$$ \begin{align} I_{out} &= \varphi (I) - \varphi (cf) (I_{out}) \tag{30} \label{Eq30} \\ \\ I &= \varphi (cf) (I_{out}) - \varphi (I) - 2 \psi_{f} (Li_{2}O) (I^{2}) + 2 \psi_{s} (Li_{2} I_{2}) (O) \tag{31} \label{Eq31} \\ \\ \frac{dm_{Li}}{dt} &= K_{mLi} - K_{dLi} (m_{Li}) \tag{32} \label{Eq300} \\ \\ \frac{dLi}{dt} &= K_{pLi} (m_{Li}) - K_{dpLi} (Li) - 2 K_{fLi} (Li^2) + 2 k_{sLi} (Li_{2}) \tag{33} \label{Eq301} \\ \\ \frac{d{Li}_{2}}{dt} &= K_{fLi} (Li^2) - K_{sLi} (Li_{2}) - K_{LiO} (Li_{2}) (O) + K_{sLio} (Li_{2}O) + K_{sLi2I} (Li_{2}I_{2}) - K_{uLi2i} Li_{2} (I^2) \tag{34} \label{Eq35502} \\ \\ \frac{d{Li}_{2}O}{dt} &= K_{LiO} (O) (Li_{2}) - K_{sLiO} (Li_{2}O) + \psi_{s} (Li_{2}I{2}) (O) - \psi{f} (I^2) (Li_{2}O) \tag{35} \label{Eq303} \\ \\ \frac{O}{dt} &= K_{LiO} (O) (Li_{2}) - K_{sLiO} (Li_{2}O) - \psi_{s} (Li_{2}I{2}) (O) + \psi{f} (I^2) (Li_{2}O) \tag{36} \label{Eq36} \\ \\ \frac{dm_{Prot}}{dt} &= K_{Prot} (O) - \gamma_{Prot} (m_{Prot}) \tag{37} \label{Eq37} \\ \\ \frac{dP_{Prot}}{dt} &= \alpha_{Prot} (m_{Prot}) - Dm_{Prot} (P_{Prot}) \tag{38} \label{Eq38} \\ \\ \end{align} $$

We solved our ODEs to study the behavior of our expression system. Then we found the steady state concentrations of each molecule. Below are the MATLAB scripts we wrote to solve our system of equations for the expression systems of an antimicrobial peptide.


  % This programs solves the ODE systems that represents the expresion of the
% osmotin and dermaseptin proteins under LacI promoter
% iGEM team Tec-Chihuahua
% October 10th 2022  

clear
clc

global time
%PARAMETERS
p.KmLi=0.23; %transcription rate constant [nmol*min-1]
p.KdLi=0.693; % mRNA degradation rate constant [min^-1]
p.KpLi=15; % translation rate constant [min^-1] (15)
p.KdpLi=.1; % protein LacI degradation rate constant [min^-1]

p.KfLi=50; % dimer formation constant[nmol-1*min-1]
p.KsLi=0.001; % dimer dissociation constant[min-1]


p.KLiO=960; % union constant between LacI dimer and operator [nmol-1*min-1]
p.KsLiO=2.4; % separation constant between Lac I dimer and operator [min-1
%p.KsLiO=1E-8; 


p.Ototal=17*2.08; %Total Operon [nM]


% Initial conditions
mRNALacI_0 = 5;    % x(1) mRNA_LacI [nM]
Li_0    = 0;        % x(2) Li LacI protein [nM]
Li2_0   = 0;        % x(3) Li2 Dimers LacI [nM]
Li2O_0 = p.Ototal;        % x(4) Li2O LacI and operon [nM]

%Init=[mRNALacI_0, Li_0, Li2_0, Li2I2_0, Op_0]; %Initial values

Init=[mRNALacI_0, Li_0, Li2_0, Li2O_0];

tspan=0:0.01:100;
opti=odeset('AbsTol', 1e-10, 'RelTol', 1e-10); 
[t,x]=ode15s(@(t,x) const_model_28SEP2022a(t,x,p),tspan,Init); 

%PLOTS

%Graph mRNA LacI
f1=figure(1);
set(gcf,'color','w');
title('Concentration vs time')
nexttile

plot(t,x(:,1),'.-r', 'LineWidth',3)
legend("mRNA LacI")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph Protein LacI
plot(t,x(:,2),'.-g', 'LineWidth',3)
legend("Li")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph IPTG
plot(t,x(:,3),'.-b', 'LineWidth',3)
legend("Li2")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph LacI dimer
plot(t,x(:,4),'.-k', 'LineWidth',3)
legend("Li2O")
xlabel('Time [min]');
ylabel('Concentration [nM]')

%%

% Last value from the above simulation



%PARAMETERS
p.ctePerm=0.65; % permeability cell constant [min-1]
p.cf=1/10; % volume of E coli/Volume of medium (mL/mL)
p.It=(10)*1e6; % quantity of IPTG in the medium (nmol)

p.Yf=3e-5; %IRepresor-operator and inducer association rate constant [nmol-2*min-1]
p.Ys=4.8e3; % represor-inducer complex and operator dissociation rate constant [nmol-2*min-1]

p.KuLi2I=3e-7; % union constant between the dimer and IPTG [nmol-2*min-1]
p.KsLi2I=12000; % dissociation constant between the dimer and IPTG [min-1]*

p.apeptide=15; % translation rate constant [min^-1]
p.Kpeptide=1; % transcription rate constant [nM*min-1]
p.Lpeptide=20.2;% constant of degradation of RNA* [min^-1]
p.Bpeptide=0.17;% degradation rate of protein* [min^-1]


iptge_0 = p.It;
iptgi_0 = 0;
Li2O_0 = x(end,4);
Li2_0 = x(end,3);
Li2I2_0 = 0;
Op_0    = 0;
mprod_0 = 0;
pprod_0 = 0;
Li_0 = x(end,2);
mLi_0 = x(end, 1);

Init=[p.It, iptgi_0, Li2O_0, Li2_0, Li2I2_0, Op_0,mprod_0,pprod_0, Li_0, mLi_0];

tspan=100:0.1:115;

opti=odeset('AbsTol', 1e-10, 'RelTol', 1e-10); 
[ty,y]=ode15s(@(t,x) const_model_28SEP2022b(t,x,p),tspan,Init); 


%Graph mRNA LacI
f2=figure(2);
set(gcf,'color','w');
title('Concentration vs time')
nexttile

plot(ty,y(:,1),'-.y','LineWidth',3)
legend("iptge")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph Protein LacI
plot(ty,y(:,2),'-.m', 'LineWidth',3)
legend("iptgi")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph LacI dimer
plot(ty,y(:,10),'-.r', 'LineWidth',3)
legend("mRNA LacI ")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph LacI dimer
plot(ty,y(:,9),'-.g', 'LineWidth',3)
legend("Li protein")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph LacI dimer
plot(ty,y(:,4),'-.b', 'LineWidth',3)
legend("Li2")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph IPTG
plot(ty,y(:,3),'-.k', 'LineWidth',3)
legend("Li2O")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile



%Graph LacI dimer
plot(ty,y(:,5),'-.m', 'LineWidth',3)
legend("Li2I2")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph LacI dimer
plot(ty,y(:,6),	'-.b', 'LineWidth',3)
legend("Op")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile


%Graph LacI dimer
plot(ty,y(:,7),'-.c', 'LineWidth',3)
legend("mPeptides")
xlabel('Time [min]');
ylabel('Concentration [nM]')
nexttile

%Graph LacI dimer
plot(ty,y(:,8),'-.r', 'LineWidth',3)
legend("pPeptide")
xlabel('Time [min]');
ylabel('Concentration [nM]')


%Funtion to solve Ordinary Differential Equations (ODEs)

function [dxdt,Op]=const_model_28SEP2022(t,x,p)
global time
% x(1) mRNA LacI [nM]
% x(2) Li LacI protein [nM]
% x(3) Li2 Dimers LacI [nM]
% x(4) Li2O LacI and operon [nM]
% x(5) Free operon (Ready to begin transcription) [nM]


mLacI = x(1);
Li    = x(2);
Li2   = x(3);
Li2O  = x(4);

%ODEs
dxdt(1,1)  =p.KmLi-p.KdLi*mLacI;
dxdt(2,1)  =p.KpLi*mLacI - p.KdpLi*Li - p.KfLi*Li^2 + p.KsLi*Li2;
dxdt(3,1)  =p.KfLi*Li^2 - p.KsLi*Li2 - p.KLiO *Li2*(p.Ototal-Li2O)+ p.KsLiO*Li2O; 
dxdt(4,1)  =p.KLiO*(p.Ototal-Li2O)*Li2- p.KsLiO*Li2O;

end

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%Funtion to solve Ordinary Differential Equations (ODEs)

function [dxdt,Op]=const_model_28SEP2022b(t,x,p)


iptge = x(1);
iptgi = x(2);
Li2O  = x(3);
Li2 = x(4);
Li2I2 = x(5);
Op    = x(6);
mprod = x(7);
pprod = x(8);
Li=x(9);
mLi=x(10);

%ODEs
%dxdt(6,1)=(p.Ototal-Li2O);
dxdt(1,1) = p.ctePerm*iptgi - p.ctePerm*p.cf*(p.It-iptgi);
dxdt(2,1)  =p.ctePerm * p.cf*iptge - p.ctePerm *iptgi - 2*p.Yf*Li2O*iptgi^2+ 2*p.Ys*Li2I2*Op + 2*p.KsLi2I*Li2I2 - 2*p.KuLi2I*(iptgi)^2*Li2;
dxdt(3,1)  = -p.Yf*Li2O*iptgi^2 + p.Ys*Li2I2*Op + p.KLiO*Op*Li2- p.KsLiO*Li2O;
dxdt(4,1) = p.KfLi*Li^2 - p.KsLi*Li2 - p.KLiO*Li2*Op + p.KsLiO*Li2O - p.KuLi2I*Li2*iptgi^2 + p.KsLi2I*Li2I2;
dxdt(5,1)  = p.Yf*Li2O*iptgi^2 - p.Ys*Li2I2*Op + p.KuLi2I*Li2*iptgi^2 - p.KsLi2I*Li2I2 ;
dxdt(6,1)  =p.Yf*Li2O*iptgi^2 - p.Ys*Li2I2*Op + p.KsLiO*Li2O - p.KLiO*Op*Li2 ;
dxdt(7,1)  =p.Kpeptide*Op - p.Lpeptide*mprod;
dxdt(8,1)  =p.apeptide*mprod- p.Bpeptide*pprod;
dxdt(9,1) = p.KpLi*mLi - p.KdpLi*Li - 2*p.KfLi*(Li)^2 + 2*p.KsLi*Li;
dxdt(10,1) = p.KmLi - p.KdLi*mLi;
end

SENSITIVITY ANALYSIS

Once the model was built, we wanted to see how parameter perturbation affected our model. We did this for two purposes:

Theoretically, many of our parameters cannot be exactly accurate. Then, checking that small variations do not affect the behavior of our model is important.

Also, knowing which parameters have the greatest influence on the final production of the molecule of interest is important to see how we can use them to our advantage to increase expression.

To evaluate the effect of parameter variations on our model we have performed a sensitivity analysis. We performed a global sensitivity analysis using the Morris method. For this, we used the ODEsensitivity package in R. We set a time of 10 minutes in a step of 0.1 minutes. We analyzed the impact of all parameters on the variable pPeptide, which is the concentration of the antimicrobial peptide. Figure 10 shows a graph reflecting the impact that each of the parameters of our model impact on the production of our peptide over time.

Sensitivity analysis plot of the parameters involved in our model. The graph on the right shows the different values of µ* (The value of µ* indicates how important each input variable is in relation to the assessed output—a higher value for µ* indicates a greater impact—given the variation in each input). The graph on the left shows the values of σ (is the standard deviation of the variation and gives insight into how much each parameter interacts with other parameters) for each parameter with respect to time. — **Figure 10.** Sensitivity analysis plot of the parameters involved in our model. The graph on the right shows the different values of µ* (The value of "µ*" indicates how important each input variable is in relation to the assessed output—a higher value for "µ*" indicates a greater impact—given the variation in each input). The graph on the left shows the values of σ (is the standard deviation of the variation and gives insight into how much each parameter interacts with other parameters) for each parameter with respect to time.

It can be seen that the variable that has the most impact on the production of our peptide is the amount of IPTG we add. This is positive since it is an easy parameter to control and ensure that we have optimal values. Also, the variable of transcription strength of the promoter (Kpeptide). This parameter can be modified by switching to stronger or weaker promoters. In addition, parameters such as the rate of repressor cleavage and repressor binding to the operator have almost no impact. This is positive, since obtaining values for these parameters can be inaccurate.

In addition, we can see that variables such as translation and transcription rates of the antimicrobial peptide have a high interaction with other parameters relative to each other.

To have a more than visual interpretation of our sensitivity analysis, we performed a Tukey test to see which variables' effects are statistically different from each other. The last 10 plotted times of µ* were analyzed for each of the parameters using a Tukey's test in Minitab.

Impact of IPTG concentration is greater than the impact of the other parameters. It also confirms that the parameters KuLi2I, KsLiO, KLiO, KsLi, KLi are in the same group, which shows a null impact on peptide production. — **Table 3** Table that shows the grouping information of the impact of each parameter on the peptide production using the Tukey Method. Differences between means that share a letter are not statistically significant.

Table 3 shows that the impact of IPTG concentration is greater than the impact of the other parameters. It also confirms that the parameters KuLi2I, KsLiO, KLiO, KsLi, KLi are in the same group, which shows a null impact on peptide production.

USING MEASUREMENTS TO IMPROVE OUR MODEL

This model is a qualitative analysis of the behavior of our expression system. And it can provide an approximation of the variables that most impact the production of our molecules of interest, as well as an approximation of the final concentration of our peptides. This gives us the possibility to start experiments in the Lab with ideas on how to optimize our system. However, it is possible to improve the system through experiments with measurable signals. The diagram below shows a visualization of the experiment that can be performed to improve the model.

Process carried out: 1.Transform the cells with a plasmid that contains the expression system and mCherry gene 2.Grows the cells until it reach an OD600 = 0.4 - 0.6. 3.Add different concentrations of IPTG. 4.Measure de OD600 and flourescence at 4h.

To analyze the data from the proposed experiment it is first necessary to convert the flourescence to concentration using the calibration curve (see measurement page). The OD600 is also converted to E. coli cell number using the equivalence of $ \frac{2 e 7 cell}{mL * ODE_{600}} $ (Sezonov et al., 2007).

1.Measuring flourescence. 2.Converting flourescence to concentration using calibration curves 3.Calculating the number of estimated cells in the well. 4.Calculating the concentration per cell

Once the data is analyzed. It is possible to adjust some parameters of the model so that the data better reflect the real behavior of our system. And thus, the model works better as a predictive tool for peptide concentration.

We started with experiments to improve our model. However, due to time constraints in the lab we were not able to finish them.

CONCLUSION

In our project, we are going to produce antimicrobial peptides in E. coli. To this end, our model provides insight into the expression processes of it under the control of an IPTG-inducible promoter. Our model is designed to predict the optimal concentrations of IPTG to produce more peptides. Also, our model can be easily adapted to other expression systems using inducible promoters.

REFERENCES

Basu, S., Gerchman, Y., Collins, C. H., Arnold, F. H., & Weiss, R. (2005). A synthetic multicellular system for programmed pattern formation. Nature, 434(7037), 1130–1134. doi:10.1038/nature03461

Bell, C., & Lewis, M. (200). A closer view of the conformation of the Lac repressor bound to operator. Nature Structural Biology volume 7, pages209–214 (2000), 1130–1134. doi: 10.1038/73317

Kwon, S. K., Kim, S. K., Lee, D. H., & Kim, J. F. (2015). Comparative genomics and experimental evolution of Escherichia coli BL21 (DE3) strains reveal the landscape of toxicity escape from membrane protein overproduction. Scientific reports, 5, 16076. https://doi.org/10.1038/srep16076

Marshall, R., & Noireaux, V. (2019). Quantitative modeling of transcription and translation of an all-E. coli cell-free system. Nature, Article number: 11980 (2019). https://www.nature.com/articles/s41598-019-48468-8

Sezonov, G., Joseleau-Petit, D., & D'Ari, R. (2007). Escherichia coli physiology in Luria-Bertani broth. Journal of bacteriology, 189(23), 8746–8749. https://doi.org/10.1128/JB.01368-07

Stamatakis, M., & Mantzaris, N. V. (2009). Comparison of deterministic and stochastic models of the lac operon genetic network. Biophysical journal, 96(3), 887–906. https://doi.org/10.1016/j.bpj.2008.10.028

Stamatakis, M., & Zygourakis, K. (2011). Deterministic and stochastic population-level simulations of an artificial lac operon genetic network. BMC bioinformatics, 12(1), 1-17.

Wong, P., Gladney, S., & Keasling, J. D. (1997). Mathematical Model of the lac Operon: Inducer Exclusion, Catabolite Repression, and Diauxic Growth on Glucose and Lactose. Biotechnology Progress, 13(2), 132–143. doi:10.1021/bp970003o

Yildirim, N., & Mackey, M. C. (2003). Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data. Biophysical journal, 84(5), 2841-2851.

Overview

BACKGROUND

THEORY BEHIND THE MODEL

System turned off

System turned on

MODEL

ASSUMPTIONS OF THIS MODEL

PARAMETERS

RESULTS

Three different scenarios

Using a T7 promoter

Using a different concentration of IPTG

Using a different cell as chasis

BUILDING AND RUNNING THE MODEL

Biochemical reaction and differential equations

Intracellular IPTG: Difussion process

Constitutive expression of the LacI gene

Dimerization of LacI

Repression mechanism

1st derepression mechanism

2nd derepression mechanism

Deterministic model: mass balance

Script that graphs the solutions of our system

Funtion to solve ODEs of the first state of the model (off)

Funtion to solve ODEs of the second state of the model (on)

SENSITIVITY ANALYSIS

USING MEASUREMENTS TO IMPROVE OUR MODEL

CONCLUSION

REFERENCES