Latest revision as of 22:53, 19 November 2018

This is an IPTG-inducible model for the expression of the fusion protein CBD cipA-BMP2 under the control of Plac promoter and cloned in psb1c3. The vector does not contain the ORF of LacI and thus the LacI repressor protein concentration is that in E. coli strains (10nM per cell).1
The model consists of systems of differential equations that were solved in Python using Spyder 3.6 (Anaconda).

LIST OF VARIABLES AND CONSTANTS

Variable/Constant	Definition	Unit/Value
[IPTG] TOTAL	Total IPTG concentration in the medium	nM
[IPTG]ext	IPTG concentration outside E. coli	nM
[IPTG]int	IPTG concentration inside E. coli	nM
[IPTG]TOTAL	Total LacI concentration in E. coli	nM
[LacI]act	Active LacI concentration in E. coli	nM
[CB]mRNA	CBD-BMP2 mRNA concentration in E. coli	nM
[CB]	CBD-BMP2 protein concentration in E. coli	nM
δ[CB]	Degradation rate of CBD-BMP2 mRNA	min-1
β[CB]	Translational rate of CBD-BMP2 protein	min-1
σ[CB]	Degradation rate of CBD-BMP2 protein	min-1
t	time	min
α[CB]	Transcriptional leakage of Plac promoter	%
kPlac	Maximum transcription rate of Plac promoter	0.5 nM min-1
kIPTGupt	Rate constant of IPTG uptake	0.92 min-1
kIPTGout	Rate constant of IPTG output	0.05 min-1
KIPTG	Michaelis constant of IPTG-LacI binding	600 nM
hIPTG	Hill coefficient of IPTG-LacI binding	2
HLacI	Hill coefficient of LacI-Plac promoter binding	2

IPTG-INDUCIBLE MODEL FOR CBD CIPA-BMP2 PROTEIN EXPRESSION IN E. coli

Entry of IPTG into E. coli
Mimicking lactose, IPTG is a chemical widely used in scientific research because it cannot be metabolized. IPGT induces the expression of genes under the control of Plac promoter.

IPTG enters into E. coli cells in a concentration dependent manner. This uptake is mediated by lactose permease and passive diffusion at low and high concentrations respectively.2 The IPTG uptake through passive diffusion can be described with the law of mass action expressing the balance between IPTG concentration outside (IPTGext) and inside (IPTGint) of E. coli :

Eq 1. The first term on the right side of Eq 1 represents the forward rate where kIPTGupt is the rate constant of IPTG uptake (0.92 min-1)3 while the second term has negative sign because it represents the backward rate where kIPTGout is the rate constant of IPTG output (0.05 min-1).3 Because of [IPTG]ext = [IPTG]TOTAL - [IPTG]int, it can be replaced in equation 1 in order to have IPTGint as the unique independent variable:

Eq 2. The LacI-IPTG interaction is assumed to be so fast and when the IPTG concentration largely exceeds that of lacI. The LacI active ([LacI]act), i.e that unbounded to IPTG, can be described with the Hill repression function.4,2

Eq 3. Equation 3 (Eq 3) describes the amount of unbounded LacI that actively repress the expression of a protein driven by the Plac promoter; the expression of CBD cipA-BMP2 in the present study. [LacI]TOTAL is the concentration of LacI inside E. coli, KIPTG is the Michaelis constant of IPTG-LacI binding (6000 M) 5 and hIPTG is the associated Hill coefficient (2).3,6

CBD cipA-BMP2 EXPRESSION

CBD cipA-BMP2 mRNA

The CBD cipA-BMP2 mRNA concentration ([CB]mRNA) at time t can be described as follows:

Eq 4.

The first term on the right side of the equation 2.6 represents the synthesis of mRNA. kPlac is the maximum transcriptional rate corresponding to the Plac promoter (0.5 nM min-1).3 αCEB+(1-αCEB) is the leakiness factor where αCEB is the transcriptional leakage of Plac promoter. is the repressing Hill function of Plac promoter where KLacI is the Michaelis constant of LacI-Plac promoter binding (6000 M), 3 and hLacI is the associated Hill coefficient(2).3 The second term represents the degradation of the [CB]mRNA where δ[CB] is the degradation rate.3,6,7
Because there is not constitutive expression of LacI protein from the plasmid cloned in E. coli carrying the CBD cipA-BMP2 construct and based on our preliminary measurements of total protein concentration using BCA method, the transcriptional leakage can be taken to be 0.8 and thus the balance for [CB]mRNA becomes:

Eq 5.

CBD cipA-BMP2 protein

The concentration of CBD cipA-BMP2 protein ([CB]) can be described as follows:

Eq 6. The first term on the right side of the equation 6 represents the synthesis of [CB] where β[CB] is the translational rate of [CB]. The second term represents the degradation of [CB] where σ[CB] is the degradation rate.

KINETIC CONSTANTS
The time cell division of E. coli is function of factors such as pressure, temperature and culture medium and ranging from 20-30 min under optimal conditions. Culturing in universal common medium such as Nutrient Broth and Luria-Bertani at 35 ℃, the time cell division can be assumed to be 30 min.8,9,10
The half life of mRNAs in E. coli has been well reported by Bernstein and coworkers who determined that it depends of culture medium. In, LB 99% of mRNAs had a half-life time between 1-15 min with a mean of 5.2 min.11 Therefore, 5.2 min can be assumed as the half-life time of CBD cipA -BMP2 mRNA. Consequently, δ[CB]= 0.226 min-1.
In E. coli , the translation takes place at 12-21 amino acids per second (aa s-1).12 We choose 19 aa s-1 because the codon optimization. Because CBD cipA-BMP2 contains 292 aa, β[CB] can be taken to be 3.904 min-1.The half life of CBD cipA-BMP2 protein can be assumed to be 60 min-1, 13 consequently, σ[CB]= 0.05min-1:

RESULTS OF THE IPTG-INDUCIBLE MODEL FOR CBD cipA-BMP2 PROTEIN EXPRESSION IN E. coli

Calculation of entry of IPTG into E. coli
Figure 1 is a rate balance plot derived from Equation 1 whith both forward and backward rates as functions of the normalized [IPTG]int concentration whose value in the equilibrium (steady state) can be obtained where the two functions intersect. Analitically, this point is [IPTG]int =0.9485[IPTG]TOTAL

rate

Figure 1. Rate balance of IPTG diffusion through E. coli membrane. Forward and backward rates are plotted versus normalized [IPTG]int

IPTG-LacI repressor interaction
Because of IPTG is the inducer, it is important to evaluate its interaction with LacI repressor at different initial concentrations added to the system ([IPTG]TOTAL). For these values the [LacI]act can be calculated with Equation 3 (Table 1). From table 1 and figure 4, it can be deduced that a concentration of IPTG = 0.1 mM is saturating because the solutions derived from higher IPTG values converge.

Table 1. [LacI]act for different initial IPTG concentrations.

[IPTG]TOTAL (mM)	[LacI]act (nM)
0,001	9.7562
0,02	0.9095
0,03	0.4257
0.1	0.0399
1.0	0.0004

Calculation and validation of CBD cipA-BMP2 expression

The mathematical model for the expression of CBD cipA-BMP2 is:

Eq 7.

The stability of the system can be analyzed with the Jacobian matrix associated to the system of differential equations:

JACOBIAN

Because the two igenvalues (λ1= -0.226 y λ2 = -0.05) are negative, the equilibrium point is a stable node.

Time step selection and phase plane
A reasonable error less than 0.01 for both mRNA and protein concentrations was obtained with a time step Dt=0.1/8. Therefore, this Dt was considered to evaluate a phase plane plot which indicates the equilibrium points of the mathematical model. To determine the maximum concentrations of CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein, a saturating concentration of IPTG (IPTG=0.1 mM) was evaluated. This is a monostable system where the equilibrium point at the steady state (Figure 2) is 2.21 nM and 172.69nM for CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein respectively.

Plano

Figure 2. Phase plane plot of the molecular species of CBD cipA-BMP2 with IPTG=0.1 mM as saturating concentration. The stable equilibrium point is at 2.21 nM and 172.69 nM for [CB]mRNA and [CB], respectively.

Expression at different IPTG inducer concentrations
Different values of [LacI]act cause a different effect in the mathematical model for expression of [CB]mRNA and [CB]. Figure 3 and Figure 4 show the concentrations of [CB]mRNA and [CB], respectively, as functions of time with a time step size Dt=0.1/8. It is worth to note that the higher IPTG dosage the higher [CB]mRNA and [CB] until the saturating level of IPTG = 0.1 mM.

RNA

Figure 3. CBD cipA-BMP2 mRNA concentration as function of time. The concentration is dependent of levels of IPTG inducer until the saturating concentration (IPTG=0.1 mM).

CB protein

Figure 4. CBD cipA-BMP2 protein concentration as function of time. The concentration is dependent of the IPTG inducer until the saturating level (IPTG=0.1 mM).

CONCLUSIONS

The present IPTG-inducible mathematical model, where a fusion protein CBD cipA-BMP2 was cloned in psb1c3, can be analyzed in two stages. First, the concentration of the IPTG inducer added to the culture medium interacts with the repressor protein LacI following the Hill equation. Then, the expression of CBD cipA-BMP2 becomes relieved, and at saturating concentrations of IPTG ≥ 0.1 mM it reaches the highest value (CBD cipA-BMP2 protein = 172.69nM, CBD cipA-BMP2 mRNA=2.21 nM).

The IPTG concentration inside E. coli is 94,845% of the total IPTG added to the culture medium. Moreover, the phase line analysis of the ODE representing the IPTG concentration inside E. coli as function of time showed that the equilibrium point is stable.

The linearization of the mathematical model revealed an stable equilibrium point for the expression of CBD cipA-BMP2 because two negative igenvalues,

REFERENCES

1. Oehler, S. (2009). Feedback Regulation of Lac Repressor Expression in Escherichia coli. Journal of Bacteriology, 191(16), 5301–5303. http://doi.org/10.1128/JB.00427-09
2. Ukkonen, K. Vasala, A. (2015). Protein expression by IPTG autoinduction in EnPresso B:Protocol with minimal manual work and superior yields compared to other media.BioSilta. Retrieved on june 24th, 2018 from https://bioscience.co.uk/userles/pdf/Application%20Note%20-%20Protein%20expression%20by%20IPTG%20autoinduction%20in%20EnPresso%20B.pdf.
3. Team TUDelft. (2009).Bacterial Relay Race. International Genetically Enginereed Machines. Retrieved on june 25th, 2018 from https://2009.igem.org/Team:TUDelft.

4. Semsey, S., Jaured, L., Csiszovszki, Z., Erdssy, J., Stger, V., Hansen, S. Krishna, S. (2014).The effect of LacI autoregulation on the performanceof the lactose utilization system in Escherichia coli. Nucleic Acids Research,41 (13),6381?6390.https://academic.oup.com/nar/article/41/13/6381/1120421.

5. University of California Berkeley.(2006).Berkeley. International Genetically Enginereed Machines.Retrieved on july 14th, 2018 from https://2006.igem.org/wiki/index.php/Berkeley
6. Ozbudak, E., Thattai, M., Lim, H., Shraiman, B. Van Oudenaarden, A. (2004).Multistability in the lactose utilization network of Escherichia coli. Nature, 427 (1476-4687), 737?740.https://www.nature.com/articles/nature02298.

7. Michel, D. (2013). Kinetic approaches to lactose operon induction and bimodality. Journal of Theoretical Biology 325, 6275.https://www.sciencedirect.com/science/article/pii/S0022519313000659?via

8. Moulton, G. (2014). Fed-Batch Fermentation A Practical Guide to Scalable Recombinant Protein Production in Escherichia coli. Retrieved from Google Books.

9. Galli, E., Midonet, C., Paly, E. Barre, F. (2017).Pressure and Temperature Dependence of Growth and Morphology of Escherichia coli:Experiments and Stochastic Model. Plos Genetics,13(3):e1006702.https://doi.org/10.1371/journal.pgen.1006702. ://jour-nals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1006702

10. Hovgaard, L., Frokjaer, S. van de Weert, M. (2013).Pharmaceutical Formulation Development of Peptides and Proteins. Second Edition. Retrieved from Google Books.

11. Bernstein, J., Khodursky, A., Lin, P., Chao, S. Cohen, S. (2002). Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two color fluorescent DNA microarrays. Proceedings of the National Academy of Sciences of the United States of America. 99 (15) 9697-9702. ://www.pnas.org/content/99/15/9697

12. Bremer, H., Dennis, P. P. (1996) Modulation of chemical composition and other parameters of the cell by growth rate. Neidhardt, et al. eds. Escherichia coli and Salmonella typhimurium: Cellular and Molecular Biology, 2nd ed. chapter 97, pp. 1559, Table 3

13. Team PKU Beijing.(2009).PKU IGEM . International Genetically Enginereed Machines. Retrieved on july 16th, 2018 from://2009.igem.org/Team:PKU Beijing/Modeling/Parameters

The main goal of our work is to develop a functionalized cellulose-based biomaterial for bone and cartilage regeneration in order to allow a faster injuries recovery time and overcome remaining challenges in repairing these connective tissues. In biomedicine, emergent strategies have focused the use biomaterials loaded with drugs.1-3 We aimed to use bacterial cellulose, a biocompatible biomaterial, as scaffold for the delivery of the human bone morphogenetic protein 2 (BMP2). It is worth to note that the diffusion of a drug far of the action site can become ectopic tissue formation, representing a serious problem. Therefore, we focused to attach the BMP2 by the use of a cellulose binding domain (CBD). We choose the CBD of the scaffolding protein of C. thermocellum (CBD cipA) because the higher affinity compared to other CBDs, reported by the Team Imperial_2014. Moreover, in an effort for a longer broad application of a functionalized bacterial cellulose, such as wound healing, we also fused the CBD cipA with an elastin like polypeptide (ELP_C5) derived from the ELP- [V-150].

@@ Line 1: / Line 1: @@
 {{Ecuador}}
 <html>
 <head>
 <meta charset="utf-8">
-<title>C-lastin, Modeling</title>
+<title>C-lastin, Interlab</title>
 <style>
-@charset "utf-8";
-/* CSS Document */
-@import url('https://fonts.googleapis.com/css?family=Raleway');
-    *{
-        margin: 0px;
-        font-family: 'Raleway', sans-serif;
-}
-.ecuador_menu {
-    font-size: 14px;
-    width: 100%;
-    height: 80px;
-    position: fixed;
-    left: 0px;
-    top: 15px;
-    background-color: #2471A3;
-    animation-name: fade-in;
-    animation-duration: 2s;
-    -webkit-box-shadow: 0px 3px 11px 0px rgba(0,0,0,0.75);
-    -moz-box-shadow: 0px 3px 11px 0px rgba(0,0,0,0.75);
-    box-shadow: 0px 3px 11px 0px rgba(0,0,0,0.75);
-    display: flex;
-    z-index: 5;
-  }
-    .ecuador_home {
-      width: 16%;
-      height: 100%;
-      display: flex;
-      justify-content: center;
-      background-color: #1F618D;
-      position: absolute;
-      align-items: center;
-      margin: auto auto;
-      left: 1.5%;
-      color: #FFF;
-      transition: 0.5s;
-      -webkit-box-shadow: inset -50px -42px 114px -30px rgba(0,0,0,0.75);
-      -moz-box-shadow: inset -50px -42px 114px -30px rgba(0,0,0,0.75);
-      box-shadow: inset -50px -42px 114px -30px rgba(0,0,0,0.75);
-      z-index: 10;
-    }
-    .ecuador_home:hover {
-      cursor: pointer;
-      font-size: 16px;
-      background-color: #1A5276;
-      box-shadow: none;
-    }
-    .menu_options_container {
-    width: 79%;
-    height: 50px;
-    position: absolute;
-    left: 19%;
-    top: 15px;
-    display: flex;
-    justify-content: center;
-  }
-  .menu_option {
-    width: 14.28%;
-    height: 100%;
-    position: relative;
-    justify-content: center;
-    align-items: center;
-    display: flex;
-    color: #FFFFFF;
-    transition: 0.2s;
-    text-align: center;
-  }
-    #project {
-    animation-name: slide-in-up-oposite;
-    animation-duration: 0.5s;
-    animation-delay: 0.5s;
-    animation-fill-mode: backwards;
-    position: absolute;
-    left: 0%;
-    z-index: 9;
-  }
-  #parts {
-    animation-name: slide-in-up-oposite;
-    animation-duration: 0.5s;
-    animation-delay: 0.6s;
-    animation-fill-mode: backwards;
-    position: absolute;
-    left: 12.5%;
-    z-index: 9;
-  }
-  #human_practice {
-    animation-name: slide-in-up-oposite;
-    animation-duration: 0.5s;
-    animation-delay: 0.7s;
-    animation-fill-mode: backwards;
-    position: absolute;
-    left: 25%;
-    z-index: 9;
-  }
-  #safety {
-    animation-name: slide-in-up-oposite;
-    animation-duration: 0.5s;
-    animation-delay: 0.8s;
-    animation-fill-mode: backwards;
-    position: absolute;
-    left: 37.5%;
-    z-index: 9;
-  }
-  #interlab {
-    animation-name: slide-in-up-oposite;
-    animation-duration: 0.5s;
-    animation-delay: 0.9s;
-    animation-fill-mode: backwards;
-    position: absolute;
-    left: 50%;
-    z-index: 9;
-  }
-  #judging_forms {
-    animation-name: slide-in-up-oposite;
-    animation-duration: 0.5s;
-    animation-delay: 1s;
-    animation-fill-mode: backwards;
-    position: absolute;
-    left: 62.5%;
-    z-index: 9;
-  }
-  #team {
-    animation-name: slide-in-up-oposite;
-    animation-duration: 0.5s;
-    animation-delay: 1.1s;
-    animation-fill-mode: backwards;
-    position: absolute;
-    left: 75%;
-    z-index: 9;
-  }
-	#atributions {
-    animation-name: slide-in-up-oposite;
-    animation-duration: 0.5s;
-    animation-delay: 1.2s;
-    animation-fill-mode: backwards;
-    position: absolute;
-    left: 87.5%;
-    z-index: 9;
-  }
-	.ecuador_href_text {
-    width: 14.28% !important;
-    height: 100% !important;
-    position: relative !important;
-    justify-content: center !important;
-    align-items: center !important;
-    display: flex !important;
-    color: #FFFFFF !important;
-    transition: 0.2s !important;
-    text-align: center !important;
-    text-decoration: none !important;
-    font-weight: bold !important;
-  }
-  .ecuador_href_sub_text {
-    width: 100% !important;
-    height: 40px !important;
-    position: relative !important;
-    text-align: left !important;
-    text-indent: 20px !important;
-    display: flex !important;
-    background-color: #FFF !important;
-    margin: auto auto !important;
-    align-items: center !important;
-    transition: 0.2s !important;
-    color: #D5DBDB !important;
-    text-decoration: none !important;
-  }
-    .menu_option:hover {
-    margin-top: 3px;
-    cursor: pointer;
-  }
-  .menu_option:hover div.menu_sub_options_container{
-    display: block;
-  }
-  .menu_sub_options_container {
-    width: 100%;
-    height: auto;
-    position: absolute;
-    top: 50px;
-    display: none;
-  }
-    .sub_menu_option {
-    width: 100%;
-    height: 40px;
-    position: relative;
-    text-align: left;
-    text-indent: 20px;
-    display: flex;
-    background-color: #000;
-    margin: auto auto;
-    align-items: center;
-    transition: 0.2s;
-    color: #D5DBDB;
-  }
-  .sub_menu_option:hover {
-	text-indent: 71px;
-	background-color: #2E86C1;
-  }
-	.ecuador_footer {
-      width: 100%;
-      height: 250px;
-      position: relative;
-      left: 0px;
-      background-color: #212F3C;
-    }
-    .ecuador_footer_igem_logo {
-      width: 180px;
-      height: 180px;
-      position: absolute;
-      right: 80px;
-      top: 35px;
-      background-image: url(https://static.igem.org/mediawiki/2018/5/55/T--Ecuador--iGEM--Logo--Black.png);
-      background-size: 100%;
-      background-repeat: no-repeat;
-      background-position: center;
-    }
-    .ecuador_footer_information_layout {
-      width: 75%;
-      height: 80%;
-      position: absolute;
-      left: 60px;
-      top: 15%;
-      display: flex;
-    }
-    .ecuador_footer_information_container {
-      width: 33.33%;
-      height: 100%;
-      position: relative;
-    }
-    .ecuador_footer_information_title_container {
-      width: 100%;
-      height: 25px;
-      position: relative;
-      top: 0px;
-      text-align: center;
-      font-size: 20px;
-      font-weight: bold;
-      color: #B2BABB;
-    }
-    .ecuador_follow_us_icons_container {
-	width: 50%;
-	height: 60px;
-	position: relative;
-	top: 40px;
-	margin: auto;
-	display: flex;
-    }
-	.ecuador_information_data_container{
-		width: 80%;
-      height: 60px;
-      position: relative;
-      top: 25px;
-      margin: auto;
-      display: flex;
-		font-size: 17px;
-	}
-    .ecuador_follow_us_icon {
-      width: 70px;
-      height: 60px;
-      position: relative;
-      margin: auto;
-    }
-    #facebook {
-      background-image: url(https://static.igem.org/mediawiki/2018/4/42/T--Ecuador--Facebook.png);
-      background-repeat: no-repeat;
-      background-position: center;
-      background-size: 80%;
-      transition: 0.4s;
-      border-radius: 5px;
-    }
-    #facebook:hover {
-      background-image: url(https://static.igem.org/mediawiki/2018/4/40/T--Ecuador--Facebook--White.png);
-      cursor: pointer;
-    }
-    #instagram {
-      background-image: url(https://static.igem.org/mediawiki/2018/9/90/T--Ecuador--Instagram.png);
-      background-repeat: no-repeat;
-      background-position: center;
-      background-size: 80%;
-      transition: 0.4s;
-      border-radius: 5px;
-    }
-    #instagram:hover {
-      background-image: url(https://static.igem.org/mediawiki/2018/4/49/T--Ecuador--Instagram--White.png);
-      cursor: pointer;
-    }
-	.ecuador_sponsors_icons_container {
-	width: 100%;
-	height: 60px;
-	position: relative;
-	top: 40px;
-	margin-top: auto;
-	margin-right: auto;
-	margin-left: auto;
-	margin-bottom: auto;
-	display: flex;
-    }
-	.ecuador_sponsors_icon {
-	width: 60px;
-	height: 60px;
-	position: relative;
-	margin-right: 4px;
-	margin-left: 4px;
-    }
-	#ing_bio{
-      background-image: url(https://static.igem.org/mediawiki/2018/thumb/1/17/T--Ecuador--Logo--BIO.png/902px-T--Ecuador--Logo--BIO.png);
-      background-repeat: no-repeat;
-      background-position: center;
-      background-size: 90%;
-      transition: 0.4s;
-      border-radius: 5px;
-    }
-	#ing_bio:hover {
-      background-size: 93%;
-      cursor: pointer;
-    }
-	#espe{
-      background-image: url(https://static.igem.org/mediawiki/2018/thumb/d/d2/T--Ecuador--Logo--ESPE.png/1066px-T--Ecuador--Logo--ESPE.png);
-      background-repeat: no-repeat;
-      background-position: center;
-      background-size: 120%;
-      transition: 0.4s;
-      border-radius: 5px;
-    }
-	#espe:hover {
-      background-size: 125%;
-      cursor: pointer;
-    }
-	#idgen{
-      background-image: url(https://static.igem.org/mediawiki/2018/e/e2/T--Ecuador--Logo--Idgen.png);
-      background-repeat: no-repeat;
-      background-position: center;
-      background-size: 90%;
-      transition: 0.4s;
-      border-radius: 5px;
-    }
-	#idgen:hover {
-      background-size: 93%;
-      cursor: pointer;
-    }
-	#tukii{
-      background-image: url(https://static.igem.org/mediawiki/2018/thumb/2/26/T--Ecuador--Logo--Tukii.png/1024px-T--Ecuador--Logo--Tukii.png);
-      background-repeat: no-repeat;
-      background-position: center;
-      background-size: 90%;
-      transition: 0.4s;
-      border-radius: 5px;
-    }
-	#tukii:hover {
-      background-size: 93%;
-      cursor: pointer;
-    }
 @charset "utf-8";
 /* CSS Document */
 .ec--modeling--container{
 	padding-top: 100px;
-	padding-right: 300px;
+	padding-right: 14%;
 	padding-bottom: 100px;
-	padding-left: 300px;
+	padding-left: 38%;
 }
 .ec--main--title{
@@ Line 411: / Line 29: @@
 }
 .ec--modeling--items{
-	display: flex;
+	width: 270px;
+	display: block;
 	position: fixed;
-	float: right;
+	float: left;
-	right: 64px;
+	top: 250px;	/*position: -webkit-sticky;
-	top: 200px;
+	position: sticky;
+	padding-bottom: 0px;
+	margin-top: 11.4em;*/
+	left: 14%;
+	line-height: 1.3;
+	text-indent: inherit;
+}
+ul.circle {
+    list-style-type: circle;
 }
 .ec--biob--cont{
@@ Line 460: / Line 87: @@
 }
 #leftBox {
-  float:left;
+	float: left;
-  height:300px;
+	height: 300px;
-  width: 390px;
+	width: 390px;
-  background: pink;
+	margin: 0;
-  margin:0;
 }
 #rightBox {
@@ Line 470: / Line 96: @@
    height:300px;
    width: 390px;
-  background: pink;
 }
 .ec--biobr--location--container{
@@ Line 478: / Line 103: @@
 	padding-left: 25px;
 }
+.ec--h2{
+	font-size: 25px;
+	font-weight: bolder;
+	display: block;
+	margin-top: 15px;
+	margin-bottom: 15px;
+	clear: both;
+	padding-top: 20px;
+	padding-bottom: 20px;
+	color: black;
+	line-height: 40px;
+}
+.ec--p{
+	display: block;
+	line-height: 2;
+	text-align: justify;
+font-size: 16px;
+}
+.ec--coll--video{
+	display: block;
+	clear: both;
+	text-align: center;
+	padding-top: 30px;
+	padding-bottom: 30px;
+}
+.ec--coll--letter{
+	text-align: justify;
+	display: block;
+	margin-left: 20px;
+	margin-right: 20px;
+	margin-top: 20px;
+	margin-bottom: 20px;
+	width: 900px;
+}
+.ec--img--surv{
+	top: 0;
+	bottom: 0;
+	left: 0;
+	right: 0;
+	margin: auto;
+	padding-top: 20px;
+	padding-bottom: 20px;
+}
+.anchorOffset{
+display: block !important;
+position: relative !important;
+top: -135px !important;
+visibility: hidden !important;
+}
+a.lateral{
+  color: black;
+  text-decoration: inherit;
+  opacity: 1;
+  color: gray;
+  font-weight: 500;
+  font-size: 16px;
+}
+a.inner-link--active {
+  opacity: 1;
+  animation: bulge .5s ease;
+  -webkit-animation: bulge .5s ease;
+}
+.ec--h3{
+	font-size: 22px;
+	font-weight: bolder;
+	display: block;
+	margin-top: 15px;
+	margin-bottom: 15px;
+	clear: both;
+	padding-top: 20px;
+}
 </style>
 </head>
 <body>
-	<div class="ecuador_menu">
+		<div class="ec--modeling--container">
-		<a href="https://2018.igem.org/Team:Ecuador">
-	    <div class="ecuador_home">
-			iGEM ECUADOR
-		  </div>
-		</a>
-		<div class="menu_options_container">
-		  <div id="project" class="menu_option">
-			  <a class="ecuador_href_text" href="https://2018.igem.org/Team:Ecuador/Description">
-			PROJECT
-			  </a>
-			<div class="menu_sub_options_container">
-			  <a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Description">
-				<div class="sub_menu_option">
-				  DESCRIPTION
-				</div>
-			  </a>
-			  <a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Background">
-				<div class="sub_menu_option">
-				  BACKGROUND
-				</div>
-			  </a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Model">
-			  <div class="sub_menu_option">
-					MODELING
-			  </div>
-			  </a>
-			  <a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Design">
-				<div class="sub_menu_option">
-				  DESIGN
-				</div>
-			  </a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Experiments">
-				<div class="sub_menu_option">
-				  EXPERIMENTS
-				</div>
-			  </a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Notebook">
-				<div class="sub_menu_option">
-				  NOTEBOOK
-				</div>
-			  </a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Model">
-				<div class="sub_menu_option">
-				  MODELLING
-				</div>
-			  </a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Results">
-				<div class="sub_menu_option">
-				  RESULTS
-				</div>
-			  </a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Demonstrate">
-				<div class="sub_menu_option">
-				  DEMOSTRATE
-				</div>
-			  </a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Improve">
-				<div class="sub_menu_option">
-				  IMPROVE
-				</div>
-			  </a>
-			</div>
-		  </div>
-			  <div id="parts" class="menu_option">
-			  <a class="ecuador_href_text" href="https://2018.igem.org/Team:Ecuador/Parts">
-				PARTS
-			  </a>
-			<div class="menu_sub_options_container">
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Parts">
-				  <div class="sub_menu_option">
-						  PARTS OVERVIEW
-				  </div>
-				</a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Basic">
-				  <div class="sub_menu_option">
-						  BASIC PARTS
-				  </div>
-				</a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Composite">
-				  <div class="sub_menu_option">
-						  COMPOSITE PARTS
-				  </div>
-				</a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/List">
-				  <div class="sub_menu_option">
-						  PARTS COLLECTION
-				  </div>
-				</a>
-			</div>
-		  </div>
-			<div id="human_practice" class="menu_option">
-			  <a class="ecuador_href_text" href="https://2018.igem.org/Team:Ecuador/Human_Practices">
-				HUMAN PRACTICES
-			  </a>
-			<div class="menu_sub_options_container">
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Human_Practices">
-				  <div class="sub_menu_option">
-						HUMAN PRACTICES
-				  </div>
-				</a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Public_Engagement">
-				  <div class="sub_menu_option">
-						PUBLIC ENGAGEMENT
-				  </div>
-				</a>
-			</div>
-		  </div>
-		<div id="safety" class="menu_option">
-				<a class="ecuador_href_text" href="https://2018.igem.org/Team:Ecuador/Safety">
-				SAFETY
-				</a>
-		  </div>
-			<div id="interlab" class="menu_option">
-			<a class="ecuador_href_text" href="http://2018.iGEM.org/Team:Ecuador/Interlab">
-			  INTERLAB
-			</a>
-		  </div>
-			<div id="judging_forms" class="menu_option">
-			<a class="ecuador_href_text" href="https://igem.org/2018_Judging_Form?id=2804">
-			  JUDGING FORMS
-			</a>
-		  	</div>
-		  <div id="team" class="menu_option">
-			  <a class="ecuador_href_text" href="https://2018.igem.org/Team:Ecuador/Team">
-			TEAM
-			  </a>
-			<div class="menu_sub_options_container">
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Team">
-					<div class="sub_menu_option">
-						TEAM MEMBERS
-					</div>
-				</a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Collaborations">
-				  <div class="sub_menu_option">
-					COLLABORATIONS
-				  </div>
-				</a>
-			</div>
-		  </div>
-			<div id="atributions" class="menu_option">
-			<a class="ecuador_href_text" href="https://2018.igem.org/Team:Ecuador/Attributions">
-				  ATTRIBUTIONS
-			  </a>
-				<div class="menu_sub_options_container">
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Attributions">
-				  <div class="sub_menu_option">
-					ATRIBUTTIONS
-				  </div>
-				</a>
-				<a class="ecuador_href_sub_text" href="https://2018.igem.org/Team:Ecuador/Sponsoring">
-				  <div class="sub_menu_option">
-					SPONSORING
-				  </div>
-				</a>
-				</div>
-		  	</div>
-		</div>
-</div>
-	<div class="ec--modeling--container">
 	<div class="ec--first--section">
 		<div class="ec--main--title">
-			<h1><a name="MOD">Modeling</a></h1>
+			<a name="MOD"  style="text-decoration: none;"><a name="INT" style="text-decoration:none;"></a>
+				<h1>
+					MATHEMATICAL MODELING
+				</h1>
+			</a>
 		</div>
 		<div class="ec--modeling--items">
-			<ul>
+			<h3><a href="#INT" style="color:black; text-decoration: none">Index</a></h3>
-				<li><a href="#MOD">Modeling</a></li>
+			<ul class="circle">
-				<li><a href="#WWD">WHAT DID WE DO?</a></li>
+				<li><a href="#LOV" style="color:black; text-decoration: none">List of variables and  constants</a></li>
-				<li><a href="#HWD">HOW DID WE DO IT?</a></li>
+                <li>I<a href="#IND3" style="color:black; text-decoration: none">PTG-inducible model</a></li>
-				<li><a href="#RES">RESULTS</a></li>
+                <ul>
-				<li><a href="#EXP">EXPERIENCE</a></li>
+                  <li><a href="#IND4" style="color:black; text-decoration: none">Entry of IPTG into the  cell</a></li>
-				<li><a href="#ACK">ACKNOWLEDGEMENT</a></li>
+                  <li><a href="#IND5" style="color:black; text-decoration: none">CBD cipA-BMP2  expression</a></li>
+                  <li><a href="#IND6" style="color:black; text-decoration: none">Kinetic constants</a></li>
+                </ul>
+                <li><a href="#IND7" style="color:black; text-decoration: none">Results of the  IPTG-inducible model</a></li>
+                <ul>
+                  <li><a href="#IND8" style="color:black; text-decoration: none">Calculation of entry of  IPTG into the cell</a></li>
+                  <li>I<a href="#IND9" style="color:black; text-decoration: none">PTG-LacI repressor  interaction</a></li>
+                  <li><a href="#IND10" style="color:black; text-decoration: none">Calculation and  validation of CBD cipA-BMP2 expression</a></li>
+                </ul>
+                <li><a href="#IND11" style="color:black; text-decoration: none">Conclusions</a></li>
 			</ul>
 		</div>
-		<div class="ec--img--home--slider">Reference image</div>
 	</div>
 		<div class="ec--nau--content">
 			<div class="ec--sect--title">
-				<h2><a name="WWD">WHAT DID WE DO?</a></h2>
-			  <p>Parrafo<br></p>
-				  <div class="ec--img--wwd--cont">
-				<p><img width="700" height="500" src="recursos extras/modeling.jpeg" alt="Transfected with GFP (Green Fluorescent Protein) Recorded using a 63x objective 3 days in vitro mouse Made by Jurjen Broeke"></p>
-			  </div>
-	          <h3>Converting between  absorbance of cells to absorbance of a known concentration of beads.</h3>
-              <p>...</p>
-              <h3>Counting  colony-forming units (CFUs) from the sample.</h3>
-				<p>…</p>
-			  <h2><a name="HWD">HOW DID WE DO IT?</a></h2>
-				<p>In order to avoid distortions in the results due to  differences in the experimentation processes, we follow the procedures  described in the iGEM protocol (<a href="https://static.igem.org/mediawiki/2018/0/09/2018_InterLab_Plate_Reader_Protocol.pdf">Plate reader protocol</a>, <a href="http://parts.igem.org/Help:Protocols/Competent_Cells">component_cells</a>,  <a href="http://parts.igem.org/Help:Protocols/Transformation">Transformation</a>)
-				<br>
-				</p>
-				<br>
-              <h3>BIOBRICKS</h3>
-              <br>
-			  <div class="ec--biob--cont">
+				<div class="ec--h2">
-				 <div id="bigBox">
+				INTRODUCTION
-					<div id="leftBox">
-			    	<table border="0" cellspacing="8px" cellpadding="0">
-		          <tr>
-		            <td width="148" valign="top"><p ><strong>Device</strong><strong> </strong></p></td>
-		            <td width="132" valign="top"><p ><strong>Part Number</strong><strong> </strong></p></td>
-		            <td width="104" valign="top"><p ><strong>Location</strong><strong> </strong></p></td>
-	              </tr>
-		          <tr>
-		            <td width="148" valign="top"><p id="label-neg-c">Negative control</p></td>
-		            <td width="132" valign="top"><p>BBa_R0040 </p></td>
-		            <td width="104" valign="top"><p>Well 2D </p></td>
-	              </tr>
-		          <tr>
-		            <td width="148" valign="top"><p id="type">Positive control </p></td>
-		            <td width="132" valign="top"><p>BBa_I20270 </p></td>
-		            <td width="104" valign="top"><p>Well 2B</p></td>
-	              </tr>
-		          <tr>
-		            <td width="148" valign="top"><p id="label-td1">Test Device 1 </p></td>
-		            <td width="132" valign="top"><p>BBa_J364000 </p></td>
-		            <td width="104" valign="top"><p>Well 2F    &nbsp; </p></td>
-	              </tr>
-		          <tr>
-		            <td width="148" valign="top"><p id="label-td2">Test Device 2 </p></td>
-		            <td width="132" valign="top"><p>BBa_J364001 </p></td>
-		            <td width="104" valign="top"><p>Well 2H    &nbsp; </p></td>
-	              </tr>
-		          <tr>
-		            <td width="148" valign="top"><p id="label-td3">Test Device 3 </p></td>
-		            <td width="132" valign="top"><p>BBa_J364002 </p></td>
-		            <td width="104" valign="top"><p>Well 2J </p></td>
-	              </tr>
-		          <tr>
-		            <td width="148" valign="top"><p id="label-td4">Test Device 4 </p></td>
-		            <td width="132" valign="top"><p>BBa_J364007 </p></td>
-		            <td width="104" valign="top"><p>Well 2L    &nbsp; </p></td>
-	              </tr>
-		          <tr>
-		            <td width="148" valign="top"><p id="label-td5">Test Device 5 </p></td>
-		            <td width="132" valign="top"><p>BBa_J364008 </p></td>
-		            <td width="104" valign="top"><p>Well 2N </p></td>
-	              </tr>
-		          <tr>
-		            <td width="148" valign="top"><p id="label-td6">Test Device 6 </p></td>
-		            <td width="132" valign="top"><p>BBa_J364009 </p></td>
-		            <td width="104" valign="top"><p>Well 2P    &nbsp; </p></td>
-	              </tr>
-	            </table>
-			  		</div>
-					<div id="rightBox">
-					 	<div class="ec--biobr--location--container">
-						 	<div class="loc-neg-c">
-								<p><strong>Location</strong></p><br>
-								<p><img width="359" height="75" src="modeling_6_oct_clip_image002.png"> </p>
-								<br>
-								<p>length: 54 bp </p>
-							</div>
-							<div class="textdis" id="pos_c">
-								<p><strong>Location</strong></p><br>
-								<p><img width="195" height="70" src="modeling_6_oct_clip_image003.png"></p>
-								<br>
-								<p>length: 919 bp</p>
-							</div>
-							<div class="loc-td1">
-								<p><strong>Location</strong></p><br>
-								<p><img width="195" height="51" src="modeling_6_oct_clip_image004.png" > </p>
-								<br>
-								<p>length: 918 bp </p>
-							</div>
-							<div class="loc-td2">
-								<p><strong>Location</strong></p><br>
-								<p><img width="189" height="54" src="modeling_6_oct_clip_image005.png" ></p>
-								<br>
-								<p>length: 918 bp</p>
-							</div>
-							<div class="loc-td3">
-								<p><strong>Location</strong></p><br>
-								<p><img width="197" height="63" src="modeling_6_oct_clip_image006.png" > </p>
-								<br>
-								<p>length: 918 bp </p>
-							</div>
-							<div class="loc-td4">
-								<p><strong>Location</strong></p><br>
-								<p><img width="196" height="52" src="modeling_6_oct_clip_image007.png"></p>
-								<br>
-								<p>length: 918 bp</p>
-							</div>
-							<div class="loc-td5">
-								<p><strong>Location</strong></p><br>
-								<p><img width="188" height="64" src="modeling_6_oct_clip_image008.png"  > </p>
-								<br>
-								<p>length: 918 bp </p>
-							</div>
-							<div class="loc-td6">
-								<p><strong>Location</strong></p><br>
-								<p><img width="194" height="55" src="modeling_6_oct_clip_image009.png"></p>
-								<br>
-								<p>length: 918 bp</p>
-							</div>
-					  	</div>
-					</div>
-				  </div>
 				</div>
-			<h2><a name="RES">RESULTS</a></h2>
-              <strong>Particle standard curves</strong>
+                <div class="ec--p">
-			  <div class="ec--img--wwd--cont">
-				<p><img src="recursos extras/A-PSC.1.png" width="775" height="445" alt=""/></p>
+                    This  is an IPTG-inducible model for the expression of the fusion protein CBD  cipA-BMP2 under the control of Plac promoter and cloned in psb1c3. The vector  does not contain the ORF of LacI and thus the LacI repressor protein concentration  is that in <em>E</em>. <em>coli</em> strains (10nM per cell).1<br>
+                    <a name="LOV" style="text-decoration:none;"></a>The  model consists of systems of differential equations that were solved in Python  using Spyder 3.6 (Anaconda).
 				</div>
-				<div class="ec--img--wwd--cont">
-				  <p><img src="recursos extras/A-PSC.2.png" width="779" height="445" alt=""/></p>
+				<a name="LOV2" style="text-decoration:none;">
-			  </div>
+				<div class="ec--h2">
-				<br>
+				LIST OF VARIABLES AND CONSTANTS
-              <p><strong>Fluorescein standard curves</strong>				</p>
+				</div>
-              <div class="ec--img--wwd--cont">
+				</a>
-			    <p><img src="recursos extras/A-FSC.1.png" width="779" height="445" alt=""/><br></p>
+                  <table border="1" cellspacing="0" cellpadding="0" width="571">
-			  </div>
+                    <tr>
-              <div class="ec--img--wwd--cont">
+                      <td width="142" nowrap valign="top"><div class="ec--p"><strong>Variable/Constant</strong></div></td>
-				  <img src="recursos extras/A-FSC.2.png" width="779" height="445" alt=""/><br>
+                      <td width="328" nowrap valign="top"><div class="ec--p"><strong>Definition</strong></div></td>
-			  </div>
+                      <td width="101" nowrap valign="top"><div class="ec--p"><strong>Unit/Value</strong></div></td>
-                <strong>Net  Absorbance 600 as number of particles/100 uL</strong></p>
+                    </tr>
-			<div class="ec--img--wwd--cont">
+                    <tr>
-              <p><img src="recursos extras/image019.png" width="752" height="452" alt=""/></p>
+                      <td width="142" nowrap valign="top"><div class="ec--p">[IPTG] TOTAL</div></td>
-			</div>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Total IPTG    concentration in the medium</div></td>
-			<div class="ec--img--wwd--cont">
+                      <td width="101" nowrap valign="top"><div class="ec--p">nM</div></td>
-              <p><img src="recursos extras/image021.png" width="752" height="452" alt=""/></p>
+                    </tr>
-			</div>
+                    <tr>
-		<h2><a name="EXP">EXPERIENCE</a></h2>
+                      <td width="142" nowrap valign="top"><div class="ec--p">[IPTG]ext</div></td>
-            <div class="ec--img--wwd--cont">
+                      <td width="328" nowrap valign="top"><div class="ec--p">IPTG    concentration outside E. coli</div></td>
-              <p><img src="https://svgsilh.com/svg/2324013-2196f3.svg" width="752" height="452" alt=""/></p>
+                      <td width="101" nowrap valign="top"><div class="ec--p">nM</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">[IPTG]int</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">IPTG    concentration inside E. coli</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">nM</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">[IPTG]TOTAL</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Total LacI    concentration in E. coli</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">nM</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">[LacI]act</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Active LacI    concentration in E. coli</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">nM</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">[CB]mRNA</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">CBD-BMP2 mRNA    concentration in E. coli</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">nM</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">[CB]</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">CBD-BMP2    protein concentration in E. coli</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">nM</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">δ[CB]</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Degradation    rate of CBD-BMP2 mRNA</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">min-1</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">β[CB]</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Translational    rate of CBD-BMP2 protein</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">min-1</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">σ[CB]</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Degradation    rate of CBD-BMP2 protein</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">min-1</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">t</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">time</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">min</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">α[CB]</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Transcriptional    leakage of Plac promoter</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">%</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">kPlac</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Maximum    transcription rate of Plac promoter</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">0.5 nM min-1</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">kIPTGupt</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Rate constant    of IPTG uptake</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">0.92 min-1</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">kIPTGout</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Rate constant    of IPTG output</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">0.05 min-1</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">KIPTG</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Michaelis    constant of IPTG-LacI binding</div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">600 nM</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p"><a name="IND3"></a>hIPTG</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Hill coefficient of IPTG-LacI binding </div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">2</div></td>
+                    </tr>
+                    <tr>
+                      <td width="142" nowrap valign="top"><div class="ec--p">HLacI</div></td>
+                      <td width="328" nowrap valign="top"><div class="ec--p">Hill coefficient of LacI-Plac promoter binding </div></td>
+                      <td width="101" nowrap valign="top"><div class="ec--p">2</div></td>
+                    </tr>
+                  </table>
+                  <div class="ec--h2"><strong><a name="IND4"></a>IPTG-INDUCIBLE MODEL FOR CBD CIPA-BMP2 PROTEIN  EXPRESSION IN </strong><strong><em>E</em></strong><strong><em>. </em></strong><strong><em>coli</em></strong><strong> </strong></div>
+                  <div class="ec--p"><strong>Entry of IPTG into <em>E. coli</em></strong><br>
+                    Mimicking lactose, IPTG is a chemical widely used  in scientific research because it cannot be metabolized. IPGT induces the  expression of genes under the control of Plac promoter. </div>
+                  <div class="ec--p">IPTG enters into <em>E</em>. <em>coli</em> cells in a concentration dependent manner. This  uptake is mediated by lactose permease and passive diffusion at low and high  concentrations respectively.2 The IPTG uptake through passive  diffusion can be described with the law of mass action expressing the balance  between IPTG concentration outside (IPTGext) and inside (IPTGint) of <em>E</em>. <em>coli</em> :
+					  <br><center>
+<img src="https://static.igem.org/mediawiki/2018/1/1a/T--Ecuador--Ecuacion1.jpg"/>
+					  <br>Eq 1.
+</center>
+  The first term on the right side of Eq 1  represents the forward rate where kIPTGupt is the rate constant of IPTG uptake (0.92 min-1)3 while the second term has negative  sign because it represents the backward rate where kIPTGout is the rate constant of IPTG output (0.05 min-1).3 Because of [IPTG]ext = [IPTG]TOTAL - [IPTG]int, it can be replaced in equation 1 in order to have IPTGint as the unique independent variable:
+					  <br><center>
+<img src="https://static.igem.org/mediawiki/2018/d/de/T--Ecuador--Ecuacion2.jpg"/>
+					  <br>Eq 2.
+</center>
+  The LacI-IPTG interaction is assumed to be so  fast and when the IPTG concentration largely exceeds that of lacI. The LacI  active ([LacI]act), i.e that unbounded to IPTG, can be described  with the Hill repression function.4,2<br>
+					   <br><center>
+<img src="https://static.igem.org/mediawiki/2018/8/80/T--Ecuador--Ecuacion3.jpg"/>
+					  <br>Eq 3.
+</center>
+  Equation 3 (Eq 3) describes the amount of  unbounded LacI that actively repress the expression of a protein driven by the  Plac promoter; the expression of CBD cipA-BMP2 in the  present study. <a name="IND5"></a>[LacI]TOTAL is the concentration of LacI inside <em>E</em>. <em>coli</em>, KIPTG is the Michaelis constant of IPTG-LacI binding (6000 M) 5 and hIPTG is the associated Hill coefficient (2).3,6</div>
+<div class="ec--p">&#160;</div>
+<div class="ec--p"><strong>CBD cipA-BMP2 EXPRESSION</strong></div>
+<div class="ec--p"><strong>            </strong><strong><u>CBD cipA-BMP2 mRNA</u></strong></div>
+<div class="ec--p">The CBD  cipA-BMP2 mRNA concentration ([CB]mRNA) at time t can be described as follows:</div>
+				<br><center>
+<img src="https://static.igem.org/mediawiki/2018/1/1d/T--Ecuador--Ecuacion4.jpg"/>
+					  <br>Eq 4.
+</center>
+<div class="ec--p">The first term on the right side of the equation  2.6 represents the synthesis of mRNA. kPlac is the maximum transcriptional rate corresponding to the Plac promoter (0.5 nM min-1).3 αCEB+(1-αCEB) is the leakiness factor where αCEB is the  transcriptional leakage of Plac promoter.  is the repressing Hill function of Plac promoter where KLacI is the Michaelis constant of LacI-Plac promoter binding (6000 M), 3 and hLacI is the associated Hill coefficient(2).3 The second term  represents the degradation of the [CB]mRNA where δ[CB]<strong> </strong>is the degradation rate.3,6,7<br>
+  Because there is not constitutive expression of  LacI protein from the plasmid cloned in <em>E</em>. <em>coli</em> carrying the CBD cipA-BMP2 construct and based on our preliminary measurements of total  protein concentration using BCA method, the transcriptional leakage can be  taken to be 0.8 and thus the balance for [CB]mRNA becomes:
+	<br><center>
+<img src="https://static.igem.org/mediawiki/2018/b/b2/T--Ecuador--Ecuacion5.jpg"/>
+					  <br>Eq 5.
+</center>
+<div class="ec--p"><strong>&#160;</strong></div>
+<div class="ec--p"><strong><u>CBD cipA-BMP2 protein</u></strong><br>
+  <br>
+  The concentration of CBD cipA-BMP2 protein  ([CB]) can be described as follows:
+	<br><center>
+<img src="https://static.igem.org/mediawiki/2018/d/d8/T--Ecuador--Ecuacion6.jpg"/>
+					  <br>Eq 6.
+</center>
+  <a name="IND6"></a>The first term on the right side of the equation  6 represents the synthesis of [CB] where β[CB] is the translational rate of [CB]. The second term represents the degradation of [CB] where σ[CB] is the degradation rate.</div>
+<div class="ec--p"><strong>KINETIC CONSTANTS</strong><br>
+  The time cell division of E. coli is function of factors such as pressure, temperature and culture medium  and ranging from 20-30 min under optimal conditions. Culturing in universal  common medium such as Nutrient Broth and Luria-Bertani at 35 ℃, the time cell division can be assumed to be 30  min.8,9,10<br>
+  The half life of mRNAs in E. coli has been well reported by Bernstein and coworkers who determined that it  depends of culture medium. In, LB 99% of mRNAs had a half-life time between  1-15 min with a mean of 5.2 min.11 Therefore, 5.2 min can be assumed as the  half-life time of CBD cipA -BMP2 mRNA. Consequently, δ[CB<strong>]</strong>= 0.226 min-1. <br>
+  In <em>E</em>. <em>coli</em> , the translation takes place at 12-21 amino  acids per second (aa s-1).12 We choose 19 aa s-1 because  the codon optimization. <a name="IND7"></a><strong><a name="IND8"></a></strong>Because CBD cipA-BMP2 contains 292 aa, β[CB]<strong> </strong>can be taken to be 3.904 min-1.The  half life of CBD cipA-BMP2 protein can be assumed to be 60 min-1, 13 consequently,  σ[CB]= 0.05min-1:</div>
+<div class="ec--h2">RESULTS OF THE IPTG-INDUCIBLE MODEL FOR CBD cipA-BMP2  PROTEIN EXPRESSION IN E. coli</div>
+<div class="ec--p"><strong>Calculation of entry of IPTG into </strong><strong><em>E</em></strong><strong><em>. </em></strong><strong><em>coli</em></strong><br>
+  Figure 1 is a rate balance plot derived from  Equation 1 whith both forward and backward rates as functions of the normalized  [IPTG]int concentration whose value in the equilibrium (steady  state) can be obtained where the two functions intersect. Analitically, this  point is [IPTG]int =0.9485[IPTG]TOTAL <br>
+<div>
+  <div class="ec--p">rate</div>
+	<br><center>
+<img src="https://static.igem.org/mediawiki/2018/1/1f/T--Ecuador--rate.jpg"/>
+	</center>
+</div>
+<p><a name="IND9"></a>
+</p>
+<p>&nbsp;</p>
+</div>
+  <strong>Figure 1.</strong> Rate balance of IPTG diffusion through E. coli membrane. Forward and  backward rates are plotted versus normalized [IPTG]int </div>
+				<div class="ec--p"><strong>IPTG-LacI repressor interaction</strong><br>
+  Because of IPTG is the inducer, it is important  to evaluate its interaction with LacI repressor at different initial  concentrations added to the system ([IPTG]TOTAL). For these  values the [LacI]act can be calculated with Equation 3 (Table 1). From table 1 and figure 4, it  can be deduced that a concentration of IPTG = 0.1 mM is saturating because the  solutions derived from higher IPTG values converge. </div>
+<div class="ec--p">Table 1. [LacI]act for different initial IPTG concentrations. </div>
+<div align="center">
+  <table border="1" cellspacing="0" cellpadding="0" width="437">
+    <tr>
+      <td width="212" nowrap valign="top">
+		 <div class="ec--p"><strong>[IPTG]TOTAL    (mM)</strong></div></td>
+      <td width="225" nowrap valign="top"><div class="ec--p"><strong>[LacI]act (nM)</strong></div></td>
+    </tr>
+    <tr>
+      <td width="212" nowrap valign="top"><div class="ec--p">0,001</div></td>
+      <td width="225" nowrap valign="top"><div class="ec--p">9.7562</div></td>
+    </tr>
+    <tr>
+      <td width="212" nowrap valign="top"><div class="ec--p">0,02</div></td>
+      <td width="225" nowrap valign="top"><div class="ec--p">0.9095</div></td>
+    </tr>
+    <tr>
+      <td width="212" nowrap valign="top"><div class="ec--p">0,03</div></td>
+      <td width="225" nowrap valign="top"><div class="ec--p">0.4257</div></td>
+    </tr>
+    <tr>
+      <td width="212" nowrap valign="top"><div class="ec--p">0.1</div></td>
+      <td width="225" nowrap valign="top"><div class="ec--p">0.0399</div></td>
+    </tr>
+    <tr>
+      <td width="212" nowrap valign="top"><div class="ec--p">1.0</div></td>
+      <td width="225" nowrap valign="top"><div class="ec--p">0.0004</div></td>
+    </tr>
+  </table>
+  <a name="IND10"></a></div>
+				<div class="ec--p"></div>
+				<div class="ec--p"><br></div><div class="ec--p"><br></div>
+<div class="ec--p"><strong>&#160;</strong></div>
+<div class="ec--p"><strong>Calculation and validation of CBD cipA-BMP2  expression</strong></div>
+<div class="ec--p">The mathematical model for the expression of CBD  cipA-BMP2 is:
+	<br><center>
+<img src="https://static.igem.org/mediawiki/2018/4/4b/T--Ecuador--Ecuacion7.jpg"/>
+					  <br>Eq 7.
+</center>
+<div class="ec--p">The stability of the system can be analyzed with the  Jacobian matrix associated to the system of differential equations:<br>
+<div>
+  <div class="ec--p">JACOBIAN</div>
+	<br><center>
+<img src="https://static.igem.org/mediawiki/2018/2/22/T--Ecuador--Jacobian.jpg"/>
+</center>
+</div>
+  Because the two igenvalues (λ1= -0.226 y λ2 = -0.05) are negative, the equilibrium point is a  stable node.</div>
+<div class="ec--p">&#160;</div>
+<div class="ec--p"><strong>          <u>Time step selection and phase plane</u></strong><br>
+  A reasonable error less than 0.01 for both mRNA and  protein concentrations was obtained with a time step Dt=0.1/8. Therefore, this Dt was considered to evaluate a phase plane plot  which indicates the equilibrium points of the mathematical model. To determine  the maximum concentrations of CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein, a  saturating concentration of IPTG (IPTG=0.1 mM) was evaluated. This is a  monostable system where the equilibrium point at the steady state (Figure 2) is  2.21 nM and 172.69nM for CBD cipA-BMP2 mRNA and CBD cipA-BMP2 protein  respectively.</div>
+<div class="ec--p">
+<div>
+  <div class="ec--p">Plano </div>
+	<br><center>
+<img src="https://static.igem.org/mediawiki/2018/5/55/T--Ecuador--Plano.jpg"/>
+</center>
+</div>
+</div>
+<div class="ec--p"><strong>Figure 2.</strong> Phase plane plot of the molecular species of  CBD cipA-BMP2 with IPTG=0.1 mM as saturating concentration. The stable  equilibrium point is at 2.21 nM and 172.69 nM for [CB]mRNA and [CB], respectively. </div>
+<div class="ec--p"><strong>          </strong><strong><u>Expression at different IPTG inducer concentrations</u></strong><strong><u> </u></strong><br>
+  Different values of [LacI]act cause a different effect in the mathematical  model for expression of [CB]mRNA and [CB]. Figure 3 and Figure 4 show the  concentrations of [CB]mRNA and [CB], respectively, as functions of time with a time step size Dt=0.1/8. It is worth to note that the higher IPTG dosage  the higher [CB]mRNA and [CB] until the saturating level of IPTG = 0.1 mM.</div>
+<div class="ec--p">
+<div>
+  <div class="ec--p">RNA</div>
+	<br><center>
+<img src="https://static.igem.org/mediawiki/2018/0/09/T--Ecuador--RNA.jpg"/>
+</center>
+</div>
+</div>
+<div class="ec--p"><strong>Figure 3.</strong> CBD cipA-BMP2 mRNA concentration  as function of time. The concentration is dependent of levels of  IPTG inducer until the saturating  concentration (IPTG=0.1 mM). <br>
+<div>
+  <div class="ec--p">CB protein</div>
+	<br><center>
+<img src="https://static.igem.org/mediawiki/2018/9/9e/T--Ecuador--CB_protein.jpg"/>
+</center>
+</div>
+&#160;<a name="IND11"></a>
+</div>
+<div class="ec--p"><strong>Figure 4.</strong> CBD cipA-BMP2 protein concentration  as function of time. The concentration is dependent of the IPTG  inducer until the saturating level (IPTG=0.1 mM).</div>
+<div class="ec--p">&#160;</div>
+<div class="ec--h2">CONCLUSIONS</div>
+<div class="ec--p">The present IPTG-inducible mathematical model,  where a fusion protein CBD cipA-BMP2 was cloned in psb1c3, can be analyzed in  two stages. First, the concentration of the IPTG inducer added to the culture  medium interacts with the repressor protein LacI following the Hill equation.  Then, the expression of CBD cipA-BMP2 becomes relieved, and at saturating  concentrations of IPTG ≥ 0.1 mM it reaches the highest value (CBD  cipA-BMP2 protein = 172.69nM, CBD cipA-BMP2 mRNA=2.21 nM).</div><br>
+	<div class="ec--p">
+		The IPTG concentration inside E. coli is 94,845% of the total IPTG added to the culture  medium. Moreover, the phase line analysis of the ODE representing the IPTG  concentration inside <em>E</em><em>. </em><em>coli</em> as function of time showed that the equilibrium point  is stable.</div>
+  The linearization of the mathematical model revealed  an stable equilibrium point for the expression of CBD cipA-BMP2 because two  negative igenvalues,</div>
+<div class="ec--p">&#160;</div>
+<div class="ec--p">&#160;</div>
+	<div class="ec--h2">REFERENCES</div>
+	<div class="ec--p">
+. Oehler, S.  (2009). Feedback Regulation of Lac Repressor Expression in <em>Escherichia coli</em>. Journal of Bacteriology, 191(16), 5301–5303.  http://doi.org/10.1128/JB.00427-09<br>
+. Ukkonen, K.  Vasala, A. (2015). Protein expression by IPTG autoinduction in EnPresso  B:Protocol with minimal manual work and superior yields compared to other  media.BioSilta. Retrieved on june 24th, 2018 from  https://bioscience.co.uk/userles/pdf/Application%20Note%20-%20Protein%20expression%20by%20IPTG%20autoinduction%20in%20EnPresso%20B.pdf.<br>
+. Team TUDelft. (2009).Bacterial Relay Race.  International Genetically Enginereed Machines. Retrieved on june 25th, 2018  from https://2009.igem.org/Team:TUDelft.</div>
+<div class="ec--p">4. Semsey, S., Jaured, L., Csiszovszki, Z.,  Erdssy, J., Stger, V., Hansen, S. Krishna, S. (2014).The effect of LacI  autoregulation on the performanceof the lactose utilization system in <em>Escherichia coli</em>. <em>Nucleic Acids Research</em>,41 (13),6381?6390.https://academic.oup.com/nar/article/41/13/6381/1120421.</div>
+<div class="ec--p">5.   University of California Berkeley.(2006).Berkeley. International  Genetically Enginereed Machines.Retrieved on july 14th, 2018 from  https://2006.igem.org/wiki/index.php/Berkeley<br>
+. Ozbudak, E., Thattai, M., Lim, H., Shraiman,  B. Van Oudenaarden, A. (2004).Multistability in the lactose utilization network  of <em>Escherichia coli</em>. Nature, 427  (1476-4687), 737?740.https://www.nature.com/articles/nature02298.</div>
+<div class="ec--p">7. Michel, D. (2013). Kinetic approaches to  lactose operon induction and bimodality. Journal of Theoretical Biology 325,  6275.https://www.sciencedirect.com/science/article/pii/S0022519313000659?via</div>
+<div class="ec--p">8. Moulton, G. (2014). Fed-Batch Fermentation A  Practical Guide to Scalable Recombinant Protein Production in Escherichia coli. Retrieved from <a href="https://books.google.com.ec/books?id=MTpYc3fAyICpg=20lpg=PA20dq=doubling+time+recombinant+e+coli+30+minutessource=ots=97TxFmAXwsig=W3OJJYLqLEF5Z48fXRfVTPqPP4hl=essa=Xved=ahUKEwi505aivqTcAhVSuVkKHVaDTgQ6AEwCHoECAgQAQv=onepageq=20-30">Google Books</a>.</div>
+<div class="ec--p">9. Galli, E., Midonet, C., Paly, E. Barre, F.  (2017).Pressure and Temperature Dependence of Growth and Morphology of <em>Escherichia coli</em>:Experiments and Stochastic Model. Plos Genetics,13(3):e1006702.https://doi.org/10.1371/journal.pgen.1006702.  ://jour-nals.plos.org/plosgenetics/article?id=10.1371/journal.pgen.1006702</div>
+<div class="ec--p">10. Hovgaard, L., Frokjaer, S. van de Weert, M.  (2013).Pharmaceutical Formulation Development of Peptides and Proteins. Second  Edition. Retrieved from <a href="https://books.google.com.ec/books?id=MTpYc3fAyICpg=PA20lpg=20dq=doubling+time+recombinant+e+coli+30+minutessource_blots=p97TxFmAXwsig=W3OJJYLqLEF5Z48fXRfVTPqPP4hl=essa=Xved=ahUKEwi505aivqTcAhVSuVkKHVaDTgQ6AEwCHoECAgQAQv=onepageq=20-30">Google Books</a>. </div>
+<div class="ec--p">11. Bernstein, J., Khodursky, A., Lin, P., Chao,  S. Cohen, S. (2002). Global analysis of mRNA decay and abundance in Escherichia coli at single-gene resolution using two color  fluorescent DNA microarrays. Proceedings of the National Academy of Sciences of the United States  of America. 99 (15) 9697-9702. ://www.pnas.org/content/99/15/9697</div>
+<div class="ec--p">12. Bremer, H.,  Dennis, P. P. (1996) Modulation of chemical composition and other parameters of  the cell by growth rate. Neidhardt, et al. eds. Escherichia coli and Salmonella  typhimurium: Cellular and Molecular Biology, 2nd ed. chapter 97, pp. 1559,  Table 3</div>
+<div class="ec--p">13. Team PKU  Beijing.(2009).PKU IGEM . International Genetically Enginereed Machines.  Retrieved on july 16th, 2018 from://2009.igem.org/Team:PKU  Beijing/Modeling/Parameters</div>
+<div class="ec--p">&#160;</div>
+                  <div class="ec--p">The main goal of our work is to develop a functionalized cellulose-based biomaterial for bone and cartilage regeneration in order to allow a faster injuries recovery time and overcome remaining challenges in repairing these connective tissues. In biomedicine, emergent strategies have focused the use biomaterials loaded with drugs.1-3 We aimed to use bacterial cellulose, a biocompatible biomaterial, as scaffold for the delivery of the human bone morphogenetic protein 2 (BMP2). It is worth to note that the diffusion of a drug far of the action site can become ectopic tissue formation, representing a serious problem. Therefore, we focused to attach the BMP2 by the use of a cellulose binding domain (CBD). We choose the CBD of the scaffolding protein of C. thermocellum (CBD cipA) because the higher affinity compared to other CBDs, reported by the Team Imperial_2014. Moreover, in an effort for a longer broad application of a functionalized bacterial cellulose, such as wound healing, we also fused the CBD cipA with an elastin like polypeptide (ELP_C5) derived from the ELP- [V-150].</div>
+                </div>
 			</div>
-              <p><em>As a team the interlab  challenge was useful&nbsp;because we learn about the protocols and the handling  of the plate reader and how it can be used in later processes of our project</em></p>
-				  <br>
-		<h2><a name="ACK">ACKNOWLEDGEMENT</a></h2>
-				  <br>
-                As a team, we would like to thank the  Biomedicine team of Universidad Tecnica Equinoccial and especially Linda  Guaman, member of their research team, for allowing us to use their measurement  equipment, which were essential for the completion of the Interlab.</div>
-			<div class="ec--sect--text"></div>
 		</div>
 	</div>
-	<div class="ec--hump--section">
-	  <div class="ec--hump--content">
-		</div>
-	</div>
-	</div>
-	</div>
-	 <div class="ecuador_footer">
-      <div class="ecuador_footer_igem_logo">
-      </div>
-      <div class="ecuador_footer_information_layout">
-        <div class="ecuador_footer_information_container">
-          <div class="ecuador_footer_information_title_container">
-            SPONSORS
-		<div class="ecuador_sponsors_icons_container">
-			<a href="http://www.espe.edu.ec/">
-				<div id="espe" class="ecuador_sponsors_icon">
-				</div>
-			</a>
-			<a href="https://biotecnologia.espe.edu.ec/">
-			<div id="ing_bio" class="ecuador_sponsors_icon">
-			</div>
-			</a>
-              		<a href="https://www.facebook.com/IDgenlaboratorio/">
-              		<div id="idgen" class="ecuador_sponsors_icon">
-              		</div>
-              		</a>
-              		<a href="https://www.facebook.com/wawatukii/">
-              		<div id="tukii" class="ecuador_sponsors_icon">
-              		</div>
-              		</a>
-            </div>
-          </div>
-        </div>
-        <div class="ecuador_footer_information_container">
-          <div class="ecuador_footer_information_title_container">
-            FOLLOW US
-            <div class="ecuador_follow_us_icons_container">
-				<a href="https://www.facebook.com/BiosinEcuador/">
-				<div id="facebook" class="ecuador_follow_us_icon">
- 				</div>
-				</a>
-				<a href="https://www.instagram.com/igem.espe.ecuador/">
-				<div id="instagram" class="ecuador_follow_us_icon">
-				</div>
-				</a>
-            </div>
-          </div>
-        </div>
-        <div class="ecuador_footer_information_container">
-          <div class="ecuador_footer_information_title_container">
-            CONTACT US
-			<div class="ecuador_information_data_container">
-				biosin.espe@outlook.es<br>
-				<br>
-			UNIVERSIDAD DE LAS FUERZAS ARMADAS ESPE.<br>
-			Ecuador,Sangolquí, Av. General Rumiñahui s/n y Ambato
-			</div>
-          </div>
-        </div>
-      </div>
-    </div>
-</body>
 <script src="https://ajax.googleapis.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
 <script>

Difference between revisions of "Team:Ecuador/Model"

Latest revision as of 22:53, 19 November 2018

MATHEMATICAL MODELING

Index