Difference between revisions of "Team:UFSCar-Brasil/part3.html"
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<div class="row"> | <div class="row"> | ||
<h3 class="ui header" id="overview">Simulations and Results</h3> | <h3 class="ui header" id="overview">Simulations and Results</h3> | ||
− | <p> | + | <p> Our main goal was to determine a model capable to explain Zn(II) concentration in cellular environment as a function of time. We have found one function that describes production of smtA and zur, proteins that associate with zinc, versus time. Thus we will consider that the smtA production would be equal to GFP produciton by uspA promoter, meaning that if we modeled GFP production versus time, will be possible to use the same curve to describe smtA.</p> |
− | <p> As we have not obtained specific experimental points to adjust an empiric curve, we will consider that the GFP molecules number in function of time is rule by a logistic function, because is the behavior | + | <p> As we have not obtained specific experimental points to adjust an empiric curve, we will consider that the GFP molecules number in function of time is rule by a logistic function, because is the behavior appearing in majority of promoter studies.</p> |
<p>Logistic function: | <p>Logistic function: | ||
− | $$f(x)=L/(1+e^{-k(x-x_0)} | + | (Eq. 9)$$f(x)=L/(1+e^{-k(x-x_0)})$$ |
− | <p>Where \(x_0\) is the \(x\) midpoint of the sigmoid curve, \(L\) the maximum value and \(k\) the slope of the curve in middle.</p> | + | <p>Where: \(x_0\) is the \(x\) midpoint of the sigmoid curve, \(L\) the maximum value and \(k\) the slope of the curve in middle.</p> |
− | <p>Thus the values funded for molecules number in equation (4), that is referent to a stabilization of growth after four hours of experiment, was used as the saturation value \(L\) and the half-life \(x_0\) was half of the experiment time.</p> | + | <p>Thus the values funded for molecules number in equation (Eq. 4), that is referent to a stabilization of growth after four hours of experiment, was used as the saturation value \(L\) and the half-life \(x_0\) was half of the experiment time.</p> |
− | <p> | + | <p><b>Table 6:</b> of constants of equation (Eq. 9) used to model the Figure 4.</p> |
<img class="ui image" src=" https://static.igem.org/mediawiki/2015/0/08/UFSCar-Brasil_grafico4-ks.jpg"> | <img class="ui image" src=" https://static.igem.org/mediawiki/2015/0/08/UFSCar-Brasil_grafico4-ks.jpg"> | ||
− | <h5 class="ui center aligned header"><b> Graph 4 </b>: GFP molecules curve as a function of time for different concentrations of PEG.</h5> | + | <h5 class="ui center aligned header"><b>Graph 4</b>: GFP molecules curve as a function of time for different concentrations of PEG.</h5> |
− | <p>The code lines used in MatLab for making all the graphs in this report can be founded | + | <p>The code lines used in MatLab for making all the graphs in this report can be founded in appendix.</p> |
− | <p> Observing | + | <p> Observing Figure 4, we can realize that as lower PEG concentration is, a higher function growth in the number of GFP molecules will be. It is still possible to see a fast function growth in the early hours and reach a maximum after approximately four or five hours.</p> |
− | <p> | + | <p>Possessing this equation that describes the smtA production, we will model another equations using the same logistic function for describe the zur protein production, but with the values obtained from the equation (Eq. 5), since it was promoted by J23101.</p> |
− | <p> | + | <p><b>Table 7:</b> of constants of equation (Eq. 9) used to model the Figure 5.</p> |
<img class="ui image" src=" https://static.igem.org/mediawiki/2015/5/57/UFSCar-Brasil_grafico5-ks.jpg"> | <img class="ui image" src=" https://static.igem.org/mediawiki/2015/5/57/UFSCar-Brasil_grafico5-ks.jpg"> | ||
− | <h5 class="ui center aligned header"><b> | + | <h5 class="ui center aligned header"><b>Figure 5</b>: Zur molecules number versus time for different concentrations of PEG.</h5> |
<p>Similarly to what occurs with the smtA, the zur curve is growing rapidly in the early hours and reaches a plateau after approximately four hours and the lower the concentration of PEG is, the greater is this growth. Is worth emphasizing that even with a very similar behavior between the two curves, the zur molecules values is about ten times lower than the numbers of molecules referent to smtA graph.</p> | <p>Similarly to what occurs with the smtA, the zur curve is growing rapidly in the early hours and reaches a plateau after approximately four hours and the lower the concentration of PEG is, the greater is this growth. Is worth emphasizing that even with a very similar behavior between the two curves, the zur molecules values is about ten times lower than the numbers of molecules referent to smtA graph.</p> |
Revision as of 20:59, 16 September 2015
Kill Switch Modelling
How to ensure the Bug Shoo biosafety?
Curve fitting
Based on laboratory experiments was possible to plot a standard curve of fluorescence as a function of Green Fluorescent Protein (GFP) concentration. This adjustment was essential to facilitate the process of obtaining the GFP molecules production rate just using the fluorescence values obtained in lab.
The equation that better fitted the fluorescence standard curve points was linear: (Eq. 1) $$ F_{luo}=a[GFP]+b$$
Where: parameters a and b are constants, [GFP] is GFP concentrations given in nM and \(F_{luo}\) is fluorescence in arbitrary units.
![](https://static.igem.org/mediawiki/2015/8/81/UFSCar-Brasil_grafico1-ks.jpg)
Figure 1: Linear fitting of GFP concentrations versus fluorescence.
The Figure 1 fitting constants results and their respective uncertainties is described in Table 1 in appendix. The result was: (Eq. 2) $$ F_{luo}=113[GFP]+94$$
For further analyses, we will transform this equation to GFP molecules number per hour, resulting in the following equation: (Eq. 3) $$ F_{luo}=1.85*10^{⁻13}[GFP]+117$$
Using the lab results of promoters characterization, respectivelly, pUspA and pJ23101, we fitted the GFP molecules production rate in function of PEG 6000 concentration with a exponential function: \(y=y_0+Ae^{R_0x}\) and made the Figure 2.
![](https://static.igem.org/mediawiki/2015/6/61/UFSCar-Brasil_grafico2-ks.jpg)
Figure 2: Exponential fitting of GFP molecules number per hour as a function of PEG 6000 concentration to promoters uspA and J23101.
Both curves have a coefficient of determination value of \(r^2\) = 0.97, meaning almost 97% of the experimental dots described by this fit. Moreover, the constants uncertainties are relatively low for both adjustment as can be seen in the Tables 2 and 3 in appendix.
The exponential fitting curve for uspA promoter: (Eq. 4) $$ GFP_{ih}= 1,51* 10^{16}e^{-0,106[PEG]}$$
And the exponential fitting curve for J23101 promoter: (Eq. 5) $$GFP_{ih}= 5,4*10^{15}e^{-0,082[PEG]}$$
However, we need to associate these values with osmotic pressure values, an universal measurement. So, it was used a set of equations described by Braccini et al. (1996). Through them, PEG 6000 concentration and temperature in the moment of the experiment were correlated to osmotic pressure. Where \(\psi_osm \) is osmotic pressure (bar), \(C\) represents PEG 6000 concentration (g/L) and \(T\) is environmental temperature given in Celsius degrees. (Eq. 6) $$ \psi_{osm}= -(1,18*10^{-2})C- (1,18*10^{-4})C^2+ (2,67*10^{-4})CT+(8,39*10^{-7})C²T$$
This possibilitates the creation of a new plot associating GFP and PEG values of Figure 2 with their respective values in osmotic pressure at 37ºC calculated by equation (Eq. 6).
![](https://static.igem.org/mediawiki/2015/9/9f/UFSCar-Brasil_grafico3-ks.jpg)
Figure 3: Exponential fitting of GFP molecules number per hour as a function of osmotic pressure for uspA and J23101 promoters at 37ºC.
Using the same exponential fitting applied in the PEG data, we reached to a coefficient of determination values of 0.92 for uspA and about 0.96 for J23101, according to Tables 4 and 5 in appendix. The new fitting equation for uspA promoter is: (Eq. 7) $$ GFP_{ih}=2,3*10^{15}+(1,2*10^{16})e^{1,5\psi_{osm}}$$
And the for J23101 promoter is: (Eq. 8) $$ GFP_{ih}=9,6*10^{14}+(4,1*10^{15})e^{0,1\psi_{osm}}$$
Simulations and Results
Our main goal was to determine a model capable to explain Zn(II) concentration in cellular environment as a function of time. We have found one function that describes production of smtA and zur, proteins that associate with zinc, versus time. Thus we will consider that the smtA production would be equal to GFP produciton by uspA promoter, meaning that if we modeled GFP production versus time, will be possible to use the same curve to describe smtA.
As we have not obtained specific experimental points to adjust an empiric curve, we will consider that the GFP molecules number in function of time is rule by a logistic function, because is the behavior appearing in majority of promoter studies.
Logistic function: (Eq. 9)$$f(x)=L/(1+e^{-k(x-x_0)})$$
Where: \(x_0\) is the \(x\) midpoint of the sigmoid curve, \(L\) the maximum value and \(k\) the slope of the curve in middle.
Thus the values funded for molecules number in equation (Eq. 4), that is referent to a stabilization of growth after four hours of experiment, was used as the saturation value \(L\) and the half-life \(x_0\) was half of the experiment time.
Table 6: of constants of equation (Eq. 9) used to model the Figure 4.
![]( https://static.igem.org/mediawiki/2015/0/08/UFSCar-Brasil_grafico4-ks.jpg)
Graph 4: GFP molecules curve as a function of time for different concentrations of PEG.
The code lines used in MatLab for making all the graphs in this report can be founded in appendix.
Observing Figure 4, we can realize that as lower PEG concentration is, a higher function growth in the number of GFP molecules will be. It is still possible to see a fast function growth in the early hours and reach a maximum after approximately four or five hours.
Possessing this equation that describes the smtA production, we will model another equations using the same logistic function for describe the zur protein production, but with the values obtained from the equation (Eq. 5), since it was promoted by J23101.
Table 7: of constants of equation (Eq. 9) used to model the Figure 5.
![]( https://static.igem.org/mediawiki/2015/5/57/UFSCar-Brasil_grafico5-ks.jpg)
Figure 5: Zur molecules number versus time for different concentrations of PEG.
Similarly to what occurs with the smtA, the zur curve is growing rapidly in the early hours and reaches a plateau after approximately four hours and the lower the concentration of PEG is, the greater is this growth. Is worth emphasizing that even with a very similar behavior between the two curves, the zur molecules values is about ten times lower than the numbers of molecules referent to smtA graph.
Now that we have the functions that describe the smtA and zur as functions of time, we can create an equation that controls the number of zinc ions in the external environment. We want to model the concentration of \(Zn^2+\) because intracellularly (after imported by the cell) he associate with zur and keep ufscarA promoter blocked, and the absence of that block starts the process of cell death, what is a result of the action of our killswitch. When the number of free zinc ions in the intracellular environment begins to decrease, zur molecules begin to disengage from the promoter region allowing the transcription of death genes. This decrease of free ions in the cellular environment is due to association of this metal with the smtA proteins. This information is important to determine which is the lifetime of our bacteria (after how long the killswitch is active) and if that time is enough to produce good amount of limonene.
According to the literature, each SMTA molecule binds to four ions zinc and each zur binds with up to 6 ions zinc, and when the number of produced zur molecules exceeds the initial amount of zur in the cell that connection 6 ions start decrease. To model this behavior we will use the following function for zur: $$f(t)=(6-zur_0/zur(t)) (10)$$
Where \(zur_0\) is the initial amount of zur and \(zur(t)\) quantity produced over time. Thus, the number of zinc decays according to the following equation: $$[Zn]_{total}=[Zn]_0-4[smtA(t)-K_{smtA}smtA(t)]-f(t)(zur_0+zur_p(t)-K_{zur}(zur_0+zur_p(t))) (11)$$
Where \([Zn]_{total}\) is the total amount of zinc over time, \([Zn]_0\) the amount of zinc initially,\(K_{smtA}\) is the smtA degradation rate, \(K_{zur}\) is the zur degradation rate, \(smtA(t)\) the function smtA over time given by graph 4 and \(zur_p(t)\) function zur given by graph 5.
Where as the degradation rate of the smtA and zur are equal and they vary from 2.5% to 6% per hour of the total amount according to Nath K. e Koch A. L. et al. (1971). For this modeling will consider the highest value of protein degradation.
To find out what the initial values of zinc and zur must use for the amount of zinc fall to zero, we apply a limit for time going to infinity and make that equal to zero.
$$\lim_{t\rightarrow \infty}[Zn]_{total}(t)=0 \rightarrow Zn_0= 0,94[4L_{smt}+(6-L_{zur}/zur_0)(zur_0+L_{zur})] (12)$$Using Matlab, we simulate this curve for different PEG concentrations.
![](https://static.igem.org/mediawiki/2015/b/bc/UFSCar-Brasil_grafico6-ks.jpg)
Graph 6 : Initial zinc values versus zur molecules for three different PEG concentrations.
For practical purposes we can discard negative values of number of zinc. With this relation, we took two values of each curve, one at the beginning and other at the end, for applying the equation (11) and simulate the curves of zinc ion versus time for three different concentrations of PEG.
![](https://static.igem.org/mediawiki/2015/c/ce/UFSCar-Brasil_grafico7-ks.jpg)
Graph 7 : Zinc ion versus time simulations for three different concentrations of PEG with initials values of zinc and zur distinct according to graph 6.
On the graph it is evident that the best PEG concentration values for the bacteria survive longer and thus produce more limonene is between 5% and 10% of PEG.
It is also possible to conclude that the bacteria will die after about 4 hours, depending on the initial values. This time can be extended a little up to about 5 hours, but will not going to far beyond that for the case studied. If it is desired to further increase the time before the activation of UfscarA is necessary to increase the half-life value (\(x_0\)) of production of smtA and zur.
Conclusions
We noticed that the UspA promoter has higher activity than J23101 as is evident from graph 2 and 3. Thus, through adjustments in these points we were able to simulate the decay of zinc ions in the cellular environment in function of time for three different PEG concentrations and also different initial conditions of zur and \(Zn^2+\). We conclude that with 5% PEG will have a longer bacteria’s life time of about 4.5 hours to initial conditions \(2,297*10^{17}\) \(Zn^{2+0}\) and \(9,168*10^{15}\) zur molecules. Is worth emphasizing that for 10% PEG concentration also has acceptable values of about 4 hours, and this time may be extended a little further increasing the initial conditions of zinc ions to values greater than \(210^{17}\), according to the curve of graph 6.
References
BRACCINI, A. L.; RUIZ, H. A.; BRACCINI, M. C. L.; REIS, M. S. Germinação e vigor de sementes de soja sob estresse hídrico induzido por soluções de cloreto de sódio, manitol e polietileno glicol. Revista Brasileira de Sementes, Brasília. v. 18, n. 1, p. 10-16. 1996.
NATH, K., and KOCH, A. L. ; Protein degradation in Escherichia coli. J. Biol. Chem., 6963 (1971).
Appendix
List of Tables
Table 1 – Reta da fluorescência (TABELA 1 - APP)
Tabela 2 – Ajuste exp. Do UspA (TABELA 2 - APP)
Tabela 3 – Ajuste exp. Do J23101 (TABELA 3 - APP)
Tabela 4 – Ajuste exp. Do UspA osmótica (TABELA 4 - APP)
Tabela 5 – Ajuste exp. Do J23101 osmotica (TABELA 5 - APP)
Code Lines
Código referente ao gráfico 4 de gfp em função do tempo
clc
clear all
%-------------------- Constants Values -----------------
L5=4*8.7670*10^15; %max value for 5% PEG
L10=4*5.2420*10^15; %max value for 10% PEG
L15=4*3.035*10^15; %max value for 15% PEG
k=1.5; %inclination
X=2; %half-life
%-------------------------------------------------------
x5=0:0.1:7
x10=x5;
x15=x10;
y5=L5./(1+exp(-k*(x5-X)));
y10=L10./(1+exp(-k*(x10-X)));
y15=L15./(1+exp(-k*(x15-X)));
plot(x5,y5,x10,y10,x15,y15);
xlabel('Time in hours');
ylabel('GFP molecules');
legend('5% PEG','10% PEG','15% PEG');
Código referente ao gráfico 5 de Zur em função do tempo
clc
clear all
%-------------------- Constants Values -----------------
L5=4*3.554703*10^15; %max value for 5% PEG
L10=4*2.393924*10^15; %max value for 10% PEG
L15=4*1.572847*10^15; %max value for 15% PEG
k=1.5; %inclination
X=2; %half-life
%-------------------------------------------------------
x5=0:0.1:6
x10=x5;
x15=x10;
y5=L5./(1+exp(-k*(x5-X)));
y10=L10./(1+exp(-k*(x10-X)));
y15=L15./(1+exp(-k*(x15-X)));
plot(x5,y5,x10,y10,x15,y15);
xlabel('Time in hours');
ylabel('Zur molecules');
legend('5% PEG','10% PEG','15% PEG');
Codigo do gráfico 6 Variando Zur
clc
clear all
k=1.5; %quanto menor o valor mais demora p/ atingir o plato
t0=2; % tempo de meia vida
degra= 6/100;
Zur0=4*10.^14:10.^10:1*10.^16;
%------------------ Valores 15% PEG ---------------
Lsmt15=4*3.035*10^15;
Lzur15=4*1.572847*10^15;
%--------------------------------------------------
%------------------ Valores 10% PEG ---------------
Lsmt10=4*5.2420*10^15;
Lzur10=4*2.393924*10^15;
%--------------------------------------------------
%------------------ Valores 5% PEG ---------------
Lsmt5=4*8.7670*10^15;
Lzur5=4*3.554703*10^15;
%--------------------------------------------------
Zn01=0.94.*(4*Lsmt15+(6-Lzur15./Zur0).*(Zur0+Lzur15));
Zn02=0.94.*(4*Lsmt10+(6-Lzur10./Zur0).*(Zur0+Lzur10));
Zn03=0.94.*(4*Lsmt5+(6-Lzur5./Zur0).*(Zur0+Lzur5));
plot(Zur0,Zn01,Zur0,Zn02,Zur0,Zn03);
xlabel('Zur Molecules');
ylabel('Number of Zinc ions');
legend('15 % PEG','10 % PEG','5 % PEG');
Codigo do gráfico 7 para ions de zinco em função do tempo
clc
clear all
t=0:0.1:6;
k=1.5; %quanto menor o valor mais demora p/ atingir o plato
t0=2; % tempo de meia vida
degra= 6/100; % taxa de degradação
%--------------Valores iniciais de Zur e Zn -- 15% --------
Z0115=6.7*10.^16;
zur0115=1.94*10.^15;
Z0215=1.23*10.^17;
zur0215=9.228*10.^15;
%--------------------------------------------------------
%--------------Valores iniciais de Zur e Zn -- 10% --------
Z0110=1.66*10.^17;
zur0110=9.19*10.^15;
Z0210=9.116*10.^16;
zur0210=1.96*10.^15;
%--------------------------------------------------------
%--------------Valores iniciais de Zur e Zn -- 5% ------
Z0105=2.297*10.^17;
zur0105=9.168*10.^15;
Z0205=1.138*10.^17;
zur0205=1.97*10.^15;
%--------------------------------------------------------
%------------------ Valores 15% PEG ---------------
Lsmt15=4*3.035*10^15;
Lzur15=4*1.572847*10^15;
yzur15=Lzur15./(1+exp(-k*(t-t0))); % curva zur
ysmt15=Lsmt15./(1+exp(-k*(t-t0))); %curva smt
%--------------------------------------------------
%------------------ Valores 10% PEG ---------------
Lsmt10=4*5.2420*10^15;
Lzur10=4*2.393924*10^15;
yzur10=Lzur10./(1+exp(-k*(t-t0))); % curva zur
ysmt10=Lsmt10./(1+exp(-k*(t-t0))); %curva smt
%--------------------------------------------------
%------------------ Valores 5% PEG ---------------
Lsmt5=4*8.7670*10^15;
Lzur5=4*3.554703*10^15;
yzur5=Lzur5./(1+exp(-k*(t-t0))); % curva zur
ysmt5=Lsmt5./(1+exp(-k*(t-t0))); %curva smt
%--------------------------------------------------
Znt1=Z0115-4*(1-degra)*(ysmt15)-(6-yzur15./zur0115)*(1-degra).*(zur0115+yzur15);
Znt2=Z0110-4*(1-degra)*(ysmt10)-(6-yzur10./zur0110)*(1-degra).*(zur0110+yzur10);
Znt3=Z0105-4*(1-degra)*(ysmt5)-(6-yzur5./zur0105)*(1-degra).*(zur0105+yzur5);
Znt4=Z0215-4*(1-degra)*(ysmt15)-(6-yzur15./zur0215)*(1-degra).*(zur0215+yzur15);
Znt5=Z0210-4*(1-degra)*(ysmt10)-(6-yzur10./zur0210)*(1-degra).*(zur0210+yzur10);
Znt6=Z0205-4*(1-degra)*(ysmt5)-(6-yzur5./zur0205)*(1-degra).*(zur0205+yzur5);
plot(t,Znt1,t,Znt2,t,Znt3,t,Znt4,t,Znt5,t,Znt6);
xlabel('Time in hours');
ylabel('Number of Zinc ions');
legend('Zn_0=6.7 10^{16}; Zur_0=1.94 10^{15}; PEG 15% ','Zn_0=1.66 10^{17}; Zur_0=9.19 10^{15}; PEG 10%',...
'Zn_0=2.297 10^{17}; Zur_0=9.168 10^{15}; PEG 05%','Zn_0=1.23 10^{17}; Zur_0=9.228 10^{15}; PEG 15%',...
'Zn_0=9.116 10^{16}; Zur_0=1.96 10^{15}; PEG 10%','Zn_0=1.138 10^{17}; Zur_0=1.97 10^{15}; PEG 05%')
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