Curve fitting

Based on laboratory experiments was possible to plotting fluorescence standard curve as a function of GFP concentration (Green Fluorescent Protein). This adjustment was essential to facilitated the process of obtaining the GFP molecules number just using the fluorescence values obtained in lab.

The curve that better fitted the fluorescence standard curve points was the straight: $$ F_{luo}=a[GFP]+b (1)$$

Where the parameters a and b are constants, [GFP] is GFP concentrations given by nM and \(F_{luo}\) is fluorescence in u.a. (IMAGEM 1)

The graph 1 fitting constants results and their respective uncertainties is described in table 1 on appendix (link). The result was: $$ F_{luo}=113[GFP]+94 (2)$$

For further analyses we will transform this equation for GFP molecules number, resulting in the following equation: $$ F_{luo}=1.85 *10^{⁻13}[GFP]+117 (3)$$

Using the lab results (link) of uspA and J23101, we fitted the GFP molecules number in function of PEG (polyethylene glycol) concentration with the exponential function: \(y=y_0+Ae^{R_0x}\) and made the graph 2. (IMAGEM 2)

This curves have both the coefficient of determination value \(r^2\) = 0.97,it is the number that indicates how well data fit a statistical model , in other words, almost 97% of the experimental dots are described by this fit. Moreover, the constants uncertainties are relatively low for both adjustment as can be seen in the tables 2 and 3 on appendix (link).

The exponential fitting curve for uspA promoter: $$ GFP_{ih}= 1,51* 10^{16}e^{-0,106[PEG]} (4)$$

And the exponential fitting curve for J23101 promoter: $$GFP_{ih}= 5,4*10^{15}e^{-0,082[PEG]} (5)$$

However, we need to associate this values not with PEG concentrations, but with the osmotic pressure. So was used the equations described in the works of Braccini et al. (1996) to correlate osmotic pressure as a function of PEG concentration and temperature in the moment of the experiment. Where \(\psi_osm \) is osmotic pressure (bar), \(C\) represent PEG 6000 concentration (g/L) and \(T\) is environmental temperature given in degrees Celsius. $$ \psi_{osm}= -(1,18*10^{-2})C- (1,18*10^{-4})C^2+ (2,67*10^{-4})CT+(8,39*10^{-7})C²T (6)$$

This way was possible to create a new graph that associate GFP and PEG values of graph 2 with their respective values in osmotic pressure at 37ºC calculated by equation (6). (IMAGEM 3)

Using the same exponential fitting applied in the PEG dates we reached coefficient of determination values of 0.92 for uspA and about 0.96 for J23101, according to the tables 4 and 5 on appendix(link). The new fitting equation for uspA promoter: $$ GFP_{ih}=2,3*10^{15}+(1,2*10^{16})e^{1,5\psi_{osm}} (7)$$

And the for J23101 promoter: $$ GFP_{ih}=9,6*10^{14}+(4,1*10^{15})e^{0,1\psi_{osm}}(8)$$

Simulations and Results

For reach one of our main goals, that is model Zn concentration in cellular environment as a function of time, we have to find one function that describes production of smtA and zur, proteins that associate with zinc, versus time. Thus we will consider that the smtA production equals of GFP produciton by uspA promoter, what means 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 that appears in majority of studies.

Logistic function: $$f(x)=L/(1+e^{-k(x-x_0)}) (9)$$

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 (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.

Tables of constants of equation (9) used to model the graph 4. (TABELA 1) (GRAFICO 4)

The code lines used in MatLab for making all the graphs in this report can be founded on appendix.

Observing graph 4 we can realize that as lower PEG concentration is, the greater growth in the number of GFP molecules will be. Also is possible to see a fast growth in the early hours and reach a maximum after approximately four or five hours.

Now that we have the equation that describe 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 (5). For that will consider that he is produced in same rate as GFP productions induced by J23101 promoter.

Tables of constants of equation (9) used to model the graph 5.(TABELA 2)(GRAFICO 5)

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. (GRAFICO 6)

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.(GRAFICO 7)

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%')