Team:SYSU-China/Project/Modeling

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<DIV class="chapter">
<DIV class="chapter">
<span>Project/Modeling</span>
<span>Project/Modeling</span>
 +
 +
<h1> 1.Overview </h1>
<h1> 1.Overview </h1>
-
 
<p>
<p>
Modeling is a powerful tool in synthetic biology and engineering. In the iPSCs Safeguard  
Modeling is a powerful tool in synthetic biology and engineering. In the iPSCs Safeguard  
-
 
project, modeling has provided us with an important engineering approach to characterize our  
project, modeling has provided us with an important engineering approach to characterize our  
-
 
pathways and predict their performance, thus helped us with modifying and testing our  
pathways and predict their performance, thus helped us with modifying and testing our  
-
 
designing.
designing.
</p>
</p>
<p>
<p>
-
Basically, the models built by us can be divided into two levels. On cell level, we proposed
+
Basically, the models built by us can be divided into two levels.On gene level, we hope to gain insight of the gene expression dynamics of our
-
the multi-compartment model to trace the change of the iPS cells in different time nodes,  
+
whole circuit. And also we tried to better characterize our parts, analyze our experimental data, for all of the sensor, switch, and the killer.
 +
Several tools including ODEs and interpolation are employed.
 +
</p>
 +
<p>
 +
On cell level, we proposed
 +
the multi-compartment model to trace the change of the iPS cells in different time nodes,
thus we are able to describe the growth and decay of iPSCs. The number of cells at the  
thus we are able to describe the growth and decay of iPSCs. The number of cells at the  
-
 
initial stage, growth rate and death rate of cells caused by suicide gene in our Safe-guard  
initial stage, growth rate and death rate of cells caused by suicide gene in our Safe-guard  
-
 
+
pathway were all taken into account.
-
pathway were all taken into account. Dox and miRNA concentration acted as parameters
+
-
 
+
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switching the pathway state and affected the growth dynamics of iPSCs.
+
</p>
</p>
-
<p>
 
-
On gene level, we adopt a chemical kinetics modeling approach to analyze the interaction
 
-
between transcriptional factor tTA, co-repressor Dox, target mRNA and miR122. Our analytical
 
-
framework is based on the use of ordinary differential equations, we described the
+
<h1>2.Gene expression dynamics of iPSC Safeguard</h1>
-
 
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<h2>2.1 analysis of the problem</h2>
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expression variation of target mRNA in different state of the pathway, determined by Dox and
+
-
 
+
-
miR122 level. The models at gene and cell level worked in concerts to give us a better
+
-
 
+
-
understanding of how our pathway and cell lines functions and how to improve the designing.
+
-
</p>
+
-
<h1> 2. Multi-compartment model
+
-
</h1>
+
-
<h2> 2.1 Analysis of the problem
+
-
</h2>
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<p>
<p>
-
We firstly focus on factors that regulate the performance of the whole pathway. Protein tTA  
+
Firstly we built a model to help ourselves better understand the gene expression dynamics of the whole circuit. In our device, tTA protein is
-
expressed by a EF1α promoter binds to the promoter pTRE to drive the transcription of  
+
firstly expressed by a EF1-α promoter and will then bind to TRE element, activating transcription of gene of interest, in our case suicide gene or
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target gene( in this case, eGFP or suicide gene) while Dox acts as a co-repressor
+
GFP. This process is affected by the Dox suppression effect. And the TRE element alone without tTA protein has a leaky expression. After production
-
prohibiting the transcription. MiR122 isa downstream part in the pathway after transcription
+
and functioning, the protein of interest will undergo degradation. And to simplify the model, we consider the miRNA degradation effect as part of
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of target mRNA, and mediated degradation of the mRNA, thus rescue the cell or knockdown its
+
this process.
-
 
+
-
GFP expression. However, the miR122 level in iPSC was low and insufficient to exert obvious
+
-
 
+
-
effect on the expression.
+
</p>
</p>
<p>
<p>
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Apart from Dox concentration,we also monitored other parameters, including cell number after
+
Then we wrote down the four chemical reactions in represent of each process.  
-
 
+
-
the stable infection and number of cell that survived the Suicide Gene. Moreover, we also
+
-
 
+
-
kept track of fluoresence intensity of the control group who has been transfected with GFP,
+
-
 
+
-
which can be employed to indicate the GOI expression level driven by Tet-Off system.
+
</p>
</p>
 +
<br/><img src="https://static.igem.org/mediawiki/2013/0/0a/New_model_1.png" width="200" /><br />
<p>
<p>
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In pratical, we planned to monitor the cell group scale every 5 hours and technically, we
+
The symbol declaration is:
-
 
+
-
counted the total clone area instead of cell number.
+
</p>
</p>
-
 
-
<h2> 2.2 Symbols declaration and assumption
 
-
</h2>
 
<p>
<p>
-
X1: initial number of iPS cells with Suicide Gene
+
X1:tTA protein
</p>
</p>
<p>
<p>
-
X2: number of the iPS cells whose TRE have been combined with tTA
+
D: the TRE promoter
</p>
</p>
<p>
<p>
-
X3: number of iPS cells which have died from expressing Suicide Gene
+
X1D:the tTA-TRE complex
</p>
</p>
<p>
<p>
-
k1: converting rate of the number of cells from phase X1 to phase X2
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X2: the protein of interest
</p>
</p>
<p>
<p>
-
k2: converting rate of the number of cells from phase X2 to phase X3
+
And we assigned each process with a reaction rate constant.
</p>
</p>
<p>
<p>
-
The unit of ki(i=1,2) is hour-1.We measured it by dividing the absolute value of the cell
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The first process is about binding and dissociation of molecules and is a fast reaction. Time unit for that is second. Since EF1-αis a
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number difference between former phase and latter phase, with the time period length.
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constitutive promoter, and we define the initial state as when we remove the Dox from medium, we can regard the concentration of X1, or tTA
-
</p>
+
-
<p>
+
-
Two cases are taken into account. In case (a), self-renewal and replication of cels are
+
-
ingored while in case (b), we take that into consideration. To further simplify the model,  
+
protein, as a given value, determined by intensity of EF1-α. Reaction rate constant k1,k-1 are affected by Dox. In the tolerable range descried in
-
we also assumed that every single cell in phase X1 turns into n1 state before phase X2, and
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part3, the more Dox we apply to D, the smaller value of k1 for the positive direction and the greater value of k-1 for the negative direction.
-
 
+
-
every single cell in phase X2 turns into n2 state before phase X3. We simulated the kinetic
+
-
 
+
-
process of gene expression and assumed an even distribution of cell content in the  
+
-
 
+
-
medium,after which the phase can be regarded as a compartment.
+
</p>
</p>
-
 
-
 
-
<h2> 2.3 Solution
 
-
</h2>
 
<p>
<p>
-
For each compartment, we construct unsteady state equilibrium equation, hence we obtain the
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In the second and third process, k2 and k3 are affected by what kind of TRE promoter we use. There are several available TRE promoters, with which
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ordinary equations
+
we can assemble different version of Tet-off system. These promoters differ in leaky expression and switching performance.
</p>
</p>
 +
<p>
 +
MiRNA-mediated mRNA degradation lead to a decrease in X2 production, which can be thought as identical with natural degradation of X2, the protein
-
<img src=" https://static.igem.org/mediawiki/2013/e/e3/Modeding_1.png " width="150" />
+
of interest. So K4 will be affected by the knockdown efficiency of miRNA binding site. We have constructed several binding sites to fine tune this
-
<p>
+
process.
-
For case (b), we just need to modify the scalar coefficients of the equations above, and we
+
-
 
+
-
obtain
+
</p>
</p>
-
<br /><img src=" https://static.igem.org/mediawiki/2013/5/53/Modeling_2.png " width="150" /><br />
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<h2>2.2 solutions and implication</h2>
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<br /><img src=" https://static.igem.org/mediawiki/2013/f/f8/Modeling_3.png " width="500" /><br />
+
-
<br>
+
-
Figure 1. dynamic process
+
-
<br />
+
<p>
<p>
-
We are going to solve X1(t), X2(t),X3(t), then we will plot the time course curve.
+
According the analysis, we then wrote down the ODEs:
</p>
</p>
 +
<br/><img src="https://static.igem.org/mediawiki/2013/1/10/New_model_2.png" width="400" /><br />
<p>
<p>
-
The initial conditions of the differential equations are as follows:
+
Here [D0] is the initial concentration of promoter Pcmv,c1 is the given concentration of X1 at the beginning. Next we will do some algebra to
 +
 
 +
simplify the equations and solve the differential equations.
</p>
</p>
 +
<br/><img src="https://static.igem.org/mediawiki/2013/6/60/New_model_3.png" width="400" /><br />
 +
<br/><img src="https://static.igem.org/mediawiki/2013/4/4c/New_model_4.png" width="400" /><br />
<p>
<p>
-
X1(0)= 5000 cells, X2(0)=0 cell, X3(0)=0 cell
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We use Mathematica’s Dynamic Interactive Function Manipulation to control the variation of the parameters α,β, and thus get curves with
 +
 
 +
different dynamics and steady state.
</p>
</p>
-
<p>
 
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k1=1day<sup>-1</sup>,k2=1 day<sup>-1</sup>
 
-
</p>
 
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<p>
 
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As for case b, the cell replicates every 26 hours, to simplify we consider one cell turns into 2 cells before next phase. Therefore, n1=n2=2.
 
-
</p>
 
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<p>
 
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Source code
 
-
</p>
 
-
<p>
 
-
<strong>
 
-
%igem_test1.m-Solution of the IPS cell differentiation model
 
-
</p>
 
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<p>
 
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%using MATLAB function ode45.m to integrate the differential equations
 
-
</p>
 
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<p>
 
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%that are contained in the file cell_diff_eq.m
 
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</p>
 
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<p>
 
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clc; clear all;
 
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</p>
 
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<p>
 
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%set the initial conditions, constants and time span
 
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</p>
 
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<p>
 
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xzero=[5000,0,0];tmax=4;
 
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</p>
 
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<p>
 
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k1=1; k2=1;
 
-
</p>
 
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<p>
 
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tspan=0:0.1: tmax;
 
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</p>
 
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<p>
 
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N=3;
 
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</p>
 
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<p>
 
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%Integrate the equations
 
-
</p>
 
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<p>
 
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[t X]=ode45(@cell_diff_eq,tspan,xzero,[ ],k1,k2);
 
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</p>
 
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<p>
 
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last=X(length(X),N);
 
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</p>
 
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<p>
 
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%Plot time curve
 
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</p>
 
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<p>
 
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plot(t,X(:,1),'-',t, X(:,2),'-',t, X(:,3),'-.');
 
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</p>
 
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<p>
 
-
legend('X1','X2','X3');
 
-
</p>
 
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<p>
 
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xlabel('time,days');
 
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</p>
 
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<p>
 
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ylabel('number of cells');
 
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</p>
 
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<p>
 
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function dx= cell_diff_eq(t,x,k1,k2)
 
-
</p>
 
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<p>
 
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%cell expression kinetic procedure
 
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</p>
 
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<p>
 
-
dx=[-k1*x(1);
 
-
    k1*x(1)-k2*x(2);
 
-
    k2*x(2); ];
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<div class="figure">
-
</strong>
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<img class="fig_img" height="245px" src="https://static.igem.org/mediawiki/2013/9/95/%E5%9B%BE%E7%89%875.png" />
 +
<img class="fig_img" height="245px" src="https://static.igem.org/mediawiki/2013/5/56/%E5%9B%BE%E7%89%876.png" />
 +
<img class="fig_img" height="245px" src="https://static.igem.org/mediawiki/2013/1/1e/%E5%9B%BE%E7%89%877.png" />
 +
<img class="fig_img" height="245px" src="https://static.igem.org/mediawiki/2013/9/91/%E5%9B%BE%E7%89%878.png" />
 +
<img class="fig_img" height="245px" src="https://static.igem.org/mediawiki/2013/4/4d/%E5%9B%BE%E7%89%879.png" />
 +
<img class="fig_img" height="245px" src="https://static.igem.org/mediawiki/2013/1/12/%E5%9B%BE%E7%89%8710.png" />
 +
<img class="fig_img" height="245px" src="https://static.igem.org/mediawiki/2013/7/72/%E5%9B%BE%E7%89%8711.png" />
 +
<p class="des" style="margin-top:0px;width:700px"><strong>Figure 1. Different values of α,βgenerate curves with different dynamics and steady
-
</p>
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state. </strong></p>
 +
<div class="clear"></div></div>
<p>
<p>
 +
As analyzed in 2.1, choosing different candidates with variant characteristic will affect the parameters in the chemical reactions, and change
 +
values of α and β. We’ve got 4 tet-off systems, a series of miR122 binding sites and experimental characterize them. And also we have two EF1-α
 +
 +
promoters with different enhancers(not mentioned in Result section). This implicate that we can assemble different circuits and fine-tune the
 +
 +
dynamics and steady state. We believe that this is important to practical engineering, just as basic logic is to conceptual designing.
</p>
</p>
-
<br/><img src=" https://static.igem.org/mediawiki/2013/9/95/Modeling_4.png " width="500" /><br />
 
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<br> Figure 2. The result of case (a)
 
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<p>
 
-
</p>
+
<h1>3. Dosage effect of DOX in turning off the Tet-off system</h1>
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<br/><img src=" https://static.igem.org/mediawiki/2013/a/a8/Modeling_5.png " width="500" /><br />
+
-
<br> Figure 3. The result of case (b)
+
-
</p>
+
-
<p>
+
-
<h1>
+
-
3. Dosage effect of DOX in turning off the Tet-off system
+
-
</h1>
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<p>
<p>
DOX ,as is discussed above, hinders the binding of tTA to pTRE in Tet-Off system and  
DOX ,as is discussed above, hinders the binding of tTA to pTRE in Tet-Off system and  
-
 
knockdown expression of suicide gene. In our experiment, we employ fluorescence technique to  
knockdown expression of suicide gene. In our experiment, we employ fluorescence technique to  
-
 
manifest the amount of protein product by detecting the strength of the fluorescence.
manifest the amount of protein product by detecting the strength of the fluorescence.
</p>
</p>
-
 
<br/><img src=" https://static.igem.org/mediawiki/2013/a/aa/Modeling_6.png " width="580" /><br />
<br/><img src=" https://static.igem.org/mediawiki/2013/a/aa/Modeling_6.png " width="580" /><br />
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<br>
<br>
Table 1. Stimulating data of GFP-Dox
Table 1. Stimulating data of GFP-Dox
</br>
</br>
-
 
<br/><img src=" https://static.igem.org/mediawiki/2013/1/19/Modeling_7.png " width="500" /><br />
<br/><img src=" https://static.igem.org/mediawiki/2013/1/19/Modeling_7.png " width="500" /><br />
<br>
<br>
Figure 4. GFP-Dox line chart
Figure 4. GFP-Dox line chart
</br>
</br>
-
 
<p>
<p>
Our task is to find the proper curve to fit the sample data. First of all we plot the  
Our task is to find the proper curve to fit the sample data. First of all we plot the  
-
 
scatter diagram, and according to its tendency, we use type curve to fit the relation of  
scatter diagram, and according to its tendency, we use type curve to fit the relation of  
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GFP-DOX. We use MATLAB to aid our fitting, i.e. to determine the parameter a, b and k.
GFP-DOX. We use MATLAB to aid our fitting, i.e. to determine the parameter a, b and k.
</p>
</p>
-
 
<br/><img src=" https://static.igem.org/mediawiki/2013/2/2f/Modeling_8.png " width="150" /><br />
<br/><img src=" https://static.igem.org/mediawiki/2013/2/2f/Modeling_8.png " width="150" /><br />
-
 
<strong>
<strong>
<p>
<p>
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The result :
The result :
</p>
</p>
-
 
<br/><img src=" https://static.igem.org/mediawiki/2013/a/a6/Modeling_9.png " width="500" /><br />
<br/><img src=" https://static.igem.org/mediawiki/2013/a/a6/Modeling_9.png " width="500" /><br />
<br>
<br>
Line 396: Line 280:
<p>
<p>
As is shown in the figure above, we can conclude that the amount of GFP tend to be steadily  
As is shown in the figure above, we can conclude that the amount of GFP tend to be steadily  
-
 
over 1.5 ug, the higher concentration of DOX we set, the lower GFP we expect. However, under  
over 1.5 ug, the higher concentration of DOX we set, the lower GFP we expect. However, under  
-
 
the real experimental conditions, over 2.2 ug DOX will lead to the undesired necrosis of the  
the real experimental conditions, over 2.2 ug DOX will lead to the undesired necrosis of the  
-
 
cells. This is a trial-experiment which proved that such a balance point for good turning-
cells. This is a trial-experiment which proved that such a balance point for good turning-
-
 
off effect and cell tolerance does exist in a certain interval concentration. More accurate  
off effect and cell tolerance does exist in a certain interval concentration. More accurate  
-
 
experiment should be conducted on stable-transfected iPSCs to find the best cultivating  
experiment should be conducted on stable-transfected iPSCs to find the best cultivating  
-
 
condition.
condition.
</p>
</p>
-
<h1>
+
 
-
4. Knockdown efficiency interpolation
+
<h1>4. Knockdown efficiency interpolation</h1>
-
</h1>
+
 
<p>
<p>
According to the experimental data, here we use interpolation technique to find the  
According to the experimental data, here we use interpolation technique to find the  
-
 
relationship between miRNA-122 concentration, the number of miR122 target sites and cell  
relationship between miRNA-122 concentration, the number of miR122 target sites and cell  
-
 
knockdown efficiency, which leads to a function with two variables. The knockdown efficiency  
knockdown efficiency, which leads to a function with two variables. The knockdown efficiency  
-
 
is represented by GFP expression level which is actually the ratio of the amount of GFP and  
is represented by GFP expression level which is actually the ratio of the amount of GFP and  
-
 
that of the parameter GAPDH. The knockdown efficiency then is
that of the parameter GAPDH. The knockdown efficiency then is
</p>
</p>
-
 
<br/><img src=" https://static.igem.org/mediawiki/2013/6/60/Modeling_11.png " width="500" /><br />
<br/><img src=" https://static.igem.org/mediawiki/2013/6/60/Modeling_11.png " width="500" /><br />
-
 
<br/><img src=" https://static.igem.org/mediawiki/2013/2/22/Modeling_12.png " width="400" /><br />
<br/><img src=" https://static.igem.org/mediawiki/2013/2/22/Modeling_12.png " width="400" /><br />
-
 
<br>
<br>
Figure 7. Two target sites, gradient miRNA concentration
Figure 7. Two target sites, gradient miRNA concentration
</br>
</br>
-
 
<br/><img src=" https://static.igem.org/mediawiki/2013/1/19/Modeling_13.png " width="500" /><br />
<br/><img src=" https://static.igem.org/mediawiki/2013/1/19/Modeling_13.png " width="500" /><br />
-
 
<br>
<br>
Table 2. Experimental data of 2 target sites, gradient miRNA concentration
Table 2. Experimental data of 2 target sites, gradient miRNA concentration
</br>
</br>
-
 
<br/><img src=" https://static.igem.org/mediawiki/2013/e/ec/Modeling_14.png " width="400" /><br />
<br/><img src=" https://static.igem.org/mediawiki/2013/e/ec/Modeling_14.png " width="400" /><br />
-
 
<br>
<br>
Table 3. Experimental data of 0.75ug miRNA plasmid with gradient target sites
Table 3. Experimental data of 0.75ug miRNA plasmid with gradient target sites
Line 445: Line 312:
<p>
<p>
We use the data above to do the interpolation. We use the griddata function to implement the  
We use the data above to do the interpolation. We use the griddata function to implement the  
-
 
interpolation.
interpolation.
</p>
</p>
Line 490: Line 356:
</br>
</br>
-
<h1>
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<h1> 5. The excursion of little mathematician--Data analysis of the FACS data
-
5. miRNA-mediated regulatory model
+
</h1>
</h1>
<p>
<p>
-
For any miRNA-mRNA complex, there are two architectures to describe the action of miRNA in
+
When our project was proceeding, we found out an interesting problem, that is, how to calculate the killing efficiency of each suicide gene?  And
-
the regulatory network. Correspondingly, there are two chemical kinetic models introduced to  
+
this quantity is also an important part of our modeling. Trying to solve this problem,one of our mathematician,  young Yang ZiYi, excursed a little
-
describe the performance of miRNA in the circuits.
+
bit from modeling to data analysis. And he analyzed our data of FACS of our transient transfection experiment of Hep G2.
</p>
</p>
<p>
<p>
-
The transcription-degradation architecture is based on the assumption that the miRNA takes
+
One important point in mathematical analysis about the complicated biological system is, not to draw arbitrary assumption, arbitrary assumption  
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effect at the latter stage of transcription and mediates the degradation of the mRNA that is  
+
just lead to disaster. Another important point is not to draw complicated assumption, which is hard to calculate. Base on these rules, Young  ZiYi
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undergoing formalizing. In this case the translation inhibition is speculated not to occur
+
draw some simple and reasonable assumptions:
 +
</p>
 +
<p>
 +
1. The initial condition of each parallel well is the same, that means, every well before transient transfection should have the same cell
-
simultaneously with the transcription degradation, mRNA's expression depends on upstream
+
density, and the cells' state should be approximately the same.
 +
</p>
 +
<p>
 +
2. In the GFP control group, the cells should be regarded as the same, whether they are transfected with GFP or not, since GFP do not harm the
-
factors. In our circuit, the transcriptional repressor is DOX, we denote it R, and there's
+
cells. But lipo-2000 will harm the cells, and may have some long term effect. So the GFP control group would be relatively weaker compare to
-
no activator, To model the action of downstream mRNA gene G, we implement the nonlinear
+
negative control, which without any transfection manipulation;
 +
</p>
 +
<p>
 +
3. The cells transfected with GFP should have an innate death rate r1 after 3 days cultivation. Besides, the ratio of “the initial number of
-
chemical kinetic equation. Let's denote the expression levels of mRNA, protein, miRNA and
+
cells” to “the number of cells harvested via FACS after 3 days” Should be the same, since if  you sample any part of the wells you will observe
-
the repressor respectively by g(t),p(t),m(t), R(t), then we define the individual impact of
+
the same distribution of cells, this means the cell experiment is scalable. Hence, we can define a special “state” to represent GFP cells in
-
proteinic repressor R on the change rate of g(t) by REP(t),
+
certain well, (r1,n1), n1 is the number of the cells we harvest in 30s, depends on r1 and the initial cell number.
</p>
</p>
 +
<p>
 +
4. In the well we transfect suicide genes, the cells which are transfected with suicide gene will be different from the cells which are not,
-
<br/><img src=" https://static.igem.org/mediawiki/2013/9/9e/Modeling_16.png " width="300" /><br />
+
due to the killing effect of suicide gene, and the cells without transfection will be at the same condition as GFP control and have the same r1.  
-
<p>
+
Cells transfected with suicide gene will have a different r2, we denote this part of cells by (r2,n2). If the transfection efficiency is a, then
-
Where BSR represents the number of the binding sites; u represents the affinity constants
+
 
 +
for any initial cell number N0, there will be N0×a cells transfected with suicide gene, the remaining N0×(1-a)are not. Here "a" in an unknown
-
with mRNA.
+
factor.
</p>
</p>
<p>
<p>
-
The generation proportion of mRNA committed by the DNA templates can be defined as
+
Then Young ZiYi built a model:
</p>
</p>
 +
<p>
 +
In the wells that had been tranfected with suicide genes, the final observed “state” (X, C) of the cells is an combination of two kinds of cells,
-
<br/><img src=" https://static.igem.org/mediawiki/2013/e/ea/Modeling_17.png " width="300" /><br />
+
one kind with state (r1,n1), the other kind with state  (r2,n2). The effect of these two kinds will be summed up together. Then we can write down a
 +
equation:
 +
</p>
 +
<br/><img src="https://static.igem.org/mediawiki/2013/6/6b/%E5%85%AC%E5%BC%8F.png" width="400" /><br />
<p>
<p>
-
Since there is no activator, so f(t)=REP(t). Let r1 be the degradation rate of mRNA, r2 the
+
Because we can not easily get the transfection efficiency, we select a reasonable range from experience:[30%,60%], and solve the equation with the
-
 
+
-
translation rate of mRNA, r3 the interaction rate between mRNA and miRNA. Taking an
+
-
 
+
-
arbitrary small time interval into account, namely [t,t+dt], degradation and interaction cut
+
-
down the amount of mRNA while the transcription increases the amount of mRNA, thus we
+
data from FACS, and get the following results:
-
 
+
-
obtain.
+
</p>
</p>
-
<br/><img src=" https://static.igem.org/mediawiki/2013/b/b3/Modeling_18.png " width="360" /><br />
+
<br/><img src="https://static.igem.org/mediawiki/2013/c/c8/Inate_death_rate_for_3_days.png" width="400" /><br />
 +
<p>
 +
It turns out that, the death rate of cells transfected with suicide gene will be greater than 60%, significantly higher than normal cells and GFP
 +
control cells(29.07%, according to our data).
 +
</p>
<p>
<p>
-
The second interaction architecture for miRNA-mRNA complex points out that the upstream
+
Then, We pick up one suicide gene, VP3, and take an “a” value 35% which we believe is fairly closed to the real tranfection efficiency, solve the  
-
miRNA inhibits the  
+
equation and get the result r2 and n2. And then, we substitute the calculated r2 and n2 to the left side of the equation, and raise the value of “
-
translation of the downstream mRNA gene, resulting in the repression of the expression of
+
a” to predict what the death rate and the “number per 30 seconds” would be when the transfection efficiency is raised. The result is shown in a
-
the protein generated
+
chart below
 +
</p>
 +
<br/><img src="https://static.igem.org/mediawiki/2013/f/f2/Predicted_death_rate.png" width="400" /><br />
 +
<br/><img src=" https://static.igem.org/mediawiki/2013/b/b9/Predicted_harvest_cell.png" width="400" /><br />
 +
<p>
 +
That makes our model become a real theory, because it predicts something that can be disproved. In the following day, we will try to raise our
-
by mRNA gene. Translation inhibition architecture focuses on the change of the amount of the  
+
transfection efficiency, and try to comparing the result with the prediction.
 +
</p>
 +
<p>
 +
And, let’s wait for the result.
 +
</p>
-
protein. A set of
 
-
n upstream miRNAs denoted by m1,m2,m3,…,mn. We denote the concentration at time t of  
+
<h1>  6. Multi-compartment model
-
 
+
</h1>
-
protein, mRNA, m1,m2,m3,
+
<h2>  6.1 Analysis of the problem
-
 
+
</h2>
-
…,mn by p(t), g(t), m1(t),m2(t),m3(t),,mn(t). For mi,the repressive impact on
+
<p>
 +
We firstly focus on factors that regulate the performance of the whole pathway. Protein tTA
 +
expressed by a EF1α promoter binds to the promoter pTRE to drive the transcription of
 +
target gene( in this case, eGFP or suicide gene) while Dox acts as a co-repressor
 +
prohibiting the transcription. MiR122 isa downstream part in the pathway after transcription
 +
of target mRNA, and mediated degradation of the mRNA, thus rescue the cell or knockdown its
 +
GFP expression. However, the miR122 level in iPSC was low and insufficient to exert obvious
 +
effect on the expression.
 +
</p>
 +
<p>
 +
Apart from Dox concentration,we also monitored other parameters, including cell number after
 +
the stable infection and number of cell that survived the Suicide Gene. Moreover, we also
 +
kept track of fluoresence intensity of the control group who has been transfected with GFP,  
 +
which can be employed to indicate the GOI expression level driven by Tet-Off system.
 +
</p>
 +
<p>
 +
In pratical, we planned to monitor the cell group scale every 5 hours and technically, we
 +
counted the total clone area instead of cell number.
 +
</p>
-
translation is defined as
+
<h2> 6.2 Symbols declaration and assumption
 +
</h2>
 +
<p>
 +
X1: initial number of iPS cells with Suicide Gene
 +
</p>
 +
<p>
 +
X2: number of the iPS cells whose TRE have been combined with tTA
 +
</p>
 +
<p>
 +
X3: number of iPS cells which have died from expressing Suicide Gene
 +
</p>
 +
<p>
 +
k1: converting rate of the number of cells from phase X1 to phase X2
 +
</p>
 +
<p>
 +
k2: converting rate of the number of cells from phase X2 to phase X3
 +
</p>
 +
<p>
 +
The unit of ki(i=1,2) is hour-1.We measured it by dividing the absolute value of the cell
 +
number difference between former phase and latter phase, with the time period length.
 +
</p>
 +
<p>
 +
Two cases are taken into account. In case (a), self-renewal and replication of cels are
 +
ingored while in case (b), we take that into consideration. To further simplify the model,
 +
we also assumed that every single cell in phase X1 turns into n1 state before phase X2, and
 +
every single cell in phase X2 turns into n2 state before phase X3. We simulated the kinetic
 +
process of gene expression and assumed an even distribution of cell content in the
 +
medium,after which the phase can be regarded as a compartment.
</p>
</p>
-
<br/><img src=" https://static.igem.org/mediawiki/2013/3/38/Modeling_19.png " width="300" /><br />
 
 +
<h2> 6.3 Solution
 +
</h2>
<p>
<p>
-
The comprehensive impact then is
+
For each compartment, we construct unsteady state equilibrium equation, hence we obtain the
 +
ordinary equations
</p>
</p>
-
<br/><img src=" https://static.igem.org/mediawiki/2013/4/41/Modeling_20.png " width="380" /><br />
+
<img src=" https://static.igem.org/mediawiki/2013/e/e3/Modeding_1.png " width="150" />
<p>
<p>
-
Let r1 and r2 be the protein degradation rate and translation rate respectively. Degradation
+
For case (b), we just need to modify the scalar coefficients of the equations above, and we
-
leads to the
+
obtain
-
 
+
-
reduce of protein whereas the mRNA translation contributes to the increase of protein, so
+
-
 
+
-
the ordinary
+
-
 
+
-
differential equation is as follow
+
</p>
</p>
-
<br/><img src=" https://static.igem.org/mediawiki/2013/c/c1/Modeling_21.png " width="300" /><br />
 
-
<h1> 6. The excursion of little mathematician--Data analysis of the FACS data
+
<br /><img src=" https://static.igem.org/mediawiki/2013/5/53/Modeling_2.png " width="150" /><br />
-
</h1>
+
<br /><img src=" https://static.igem.org/mediawiki/2013/f/f8/Modeling_3.png " width="500" /><br />
 +
<br>
 +
Figure 1. dynamic process
 +
<br />
<p>
<p>
-
When our project was proceeding, we found out an interesting problem, that is, how to calculate the killing efficiency of each suicide gene?  And this quantity is also an important part of our modeling. Trying to solve this problem,one of our mathematician, young Yang ZiYi, excursed a little bit from modeling to data analysis. And he analyzed our data of FACS of our transient transfection experiment of Hep G2.
+
We are going to solve X1(t), X2(t),X3(t), then we will plot the time course curve.
</p>
</p>
<p>
<p>
-
One important point in mathematical analysis about the complicated biological system is, not to draw arbitrary assumption, arbitrary assumption just lead to disaster. Another important point is not to draw complicated assumption, which is hard to calculate. Base on these rules, Young  ZiYi draw some simple and reasonable assumptions:
+
The initial conditions of the differential equations are as follows:
</p>
</p>
<p>
<p>
-
1. The initial condition of each parallel well is the same, that means, every well before transient transfection should have the same cell density, and the cells' state should be approximately the same.
+
X1(0)= 5000 cells, X2(0)=0 cell, X3(0)=0 cell
</p>
</p>
<p>
<p>
-
2. In the GFP control group, the cells should be regarded as the same, whether they are transfected with GFP or not, since GFP do not harm the cells. But lipo-2000 will harm the cells, and may have some long term effect. So the GFP control group would be relatively weaker compare to negative control, which without any transfection manipulation;
+
k1=1day<sup>-1</sup>,k2=1 day<sup>-1</sup>
</p>
</p>
<p>
<p>
-
3. The cells transfected with GFP should have an innate death rate r1 after 3 days cultivation. Besides, the ratio of “the initial number of cells” to “the number of cells harvested via FACS after 3 days” Should be the same, since if  you sample any part of the wells you will observe the same distribution of cells, this means the cell experiment is scalable. Hence, we can define a special “state” to represent GFP cells in certain well, (r1,n1), n1 is the number of the cells we harvest in 30s, depends on r1 and the initial cell number.
+
As for case b, the cell replicates every 26 hours, to simplify we consider one cell turns into 2 cells before next phase. Therefore, n1=n2=2.  
</p>
</p>
<p>
<p>
-
4. In the well we transfect suicide genes, the cells which are transfected with suicide gene will be different from the cells which are not, due to the killing effect of suicide gene, and the cells without transfection will be at the same condition as GFP control and have the same r1. Cells transfected with suicide gene will have a different r2, we denote this part of cells by (r2,n2). If the transfection efficiency is a, then for any initial cell number N0, there will be N0×a cells transfected with suicide gene, the remaining N0×(1-a)are not. Here "a" in an unknown factor.
+
Source code
</p>
</p>
<p>
<p>
-
Then Young ZiYi built a model:
+
<strong>
 +
%igem_test1.m-Solution of the IPS cell differentiation model
</p>
</p>
<p>
<p>
-
In the wells that had been tranfected with suicide genes, the final observed “state” (X, C) of the cells is an combination of two kinds of cells, one kind with state (r1,n1), the other kind with state  (r2,n2). The effect of these two kinds will be summed up together. Then we can write down a equation:
+
%using MATLAB function ode45.m to integrate the differential equations
</p>
</p>
-
<br/><img src="https://static.igem.org/mediawiki/2013/6/6b/%E5%85%AC%E5%BC%8F.png" width="400" /><br />
 
<p>
<p>
-
Because we can not easily get the transfection efficiency, we select a reasonable range from experience:[30%,60%], and solve the equation with the data from FACS, and get the following results:
+
%that are contained in the file cell_diff_eq.m
</p>
</p>
-
 
-
<br/><img src="https://static.igem.org/mediawiki/2013/c/c8/Inate_death_rate_for_3_days.png" width="400" /><br />
 
<p>
<p>
-
It turns out that, the death rate of cells transfected with suicide gene will be greater than 60%, significantly higher than normal cells and GFP control cells(29.07%, according to our data).
+
clc; clear all;
</p>
</p>
<p>
<p>
-
Then, We pick up one suicide gene, VP3, and take an “a” value 35% which we believe is fairly closed to the real tranfection efficiency, solve the equation and get the result r2 and n2. And then, we substitute the calculated r2 and n2 to the left side of the equation, and raise the value of “a” to predict what the death rate and the “number per 30 seconds” would be when the transfection efficiency is raised. The result is shown in a chart below
+
%set the initial conditions, constants and time span
</p>
</p>
-
<br/><img src="https://static.igem.org/mediawiki/2013/f/f2/Predicted_death_rate.png" width="400" /><br />
 
-
<br/><img src=" https://static.igem.org/mediawiki/2013/b/b9/Predicted_harvest_cell.png" width="400" /><br />
 
<p>
<p>
-
That makes our model become a real theory, because it predicts something that can be disproved. In the following day, we will try to raise our transfection efficiency, and try to comparing the result with the prediction.
+
xzero=[5000,0,0];tmax=4;
</p>
</p>
<p>
<p>
-
And, let’s wait for the result.
+
k1=1; k2=1;
</p>
</p>
 +
<p>
 +
tspan=0:0.1: tmax;
 +
</p>
 +
<p>
 +
N=3;
 +
</p>
 +
<p>
 +
%Integrate the equations
 +
</p>
 +
<p>
 +
[t X]=ode45(@cell_diff_eq,tspan,xzero,[ ],k1,k2);
 +
</p>
 +
<p>
 +
last=X(length(X),N);
 +
</p>
 +
<p>
 +
%Plot time curve
 +
</p>
 +
<p>
 +
plot(t,X(:,1),'-',t, X(:,2),'-',t, X(:,3),'-.');
 +
</p>
 +
<p>
 +
legend('X1','X2','X3');
 +
</p>
 +
<p>
 +
xlabel('time,days');
 +
</p>
 +
<p>
 +
ylabel('number of cells');
 +
</p>
 +
<p>
 +
function dx= cell_diff_eq(t,x,k1,k2)
 +
</p>
 +
<p>
 +
%cell expression kinetic procedure
 +
</p>
 +
<p>
 +
dx=[-k1*x(1);
 +
    k1*x(1)-k2*x(2);
 +
 +
    k2*x(2); ];
 +
</strong>
 +
</p>
 +
 +
<br/><img src=" https://static.igem.org/mediawiki/2013/9/95/Modeling_4.png " width="500" /><br />
 +
<br> Figure 2. The result of case (a)
 +
 +
<br/><img src=" https://static.igem.org/mediawiki/2013/a/a8/Modeling_5.png " width="500" /><br />
 +
<br> Figure 3. The result of case (b)
 +
</p>
 +
-
<h1>
 
7.reference
7.reference
</h1>
</h1>
Line 667: Line 644:
</p>
</p>
<p>
<p>
-
[5]On the inhibition of the mitochondrial inner membrane anion uniporter by cationic amphiphiles and other drugs. Beavis AD,J Biol Chem. 1989 Jan 25;264(3):1508-15.
+
[5]On the inhibition of the mitochondrial inner membrane anion uniporter by cationic amphiphiles and other drugs. Beavis AD,J Biol Chem. 1989 Jan  
 +
 
 +
25;264(3):1508-15.
</p>
</p>
 +
 +
 +
 +
 +
<!--正 文 部 分 结 束-->
<!--正 文 部 分 结 束-->
      
      

Latest revision as of 18:17, 28 October 2013

ipsc

Project/Modeling

1.Overview

Modeling is a powerful tool in synthetic biology and engineering. In the iPSCs Safeguard project, modeling has provided us with an important engineering approach to characterize our pathways and predict their performance, thus helped us with modifying and testing our designing.

Basically, the models built by us can be divided into two levels.On gene level, we hope to gain insight of the gene expression dynamics of our whole circuit. And also we tried to better characterize our parts, analyze our experimental data, for all of the sensor, switch, and the killer. Several tools including ODEs and interpolation are employed.

On cell level, we proposed the multi-compartment model to trace the change of the iPS cells in different time nodes, thus we are able to describe the growth and decay of iPSCs. The number of cells at the initial stage, growth rate and death rate of cells caused by suicide gene in our Safe-guard pathway were all taken into account.

2.Gene expression dynamics of iPSC Safeguard

2.1 analysis of the problem

Firstly we built a model to help ourselves better understand the gene expression dynamics of the whole circuit. In our device, tTA protein is firstly expressed by a EF1-α promoter and will then bind to TRE element, activating transcription of gene of interest, in our case suicide gene or GFP. This process is affected by the Dox suppression effect. And the TRE element alone without tTA protein has a leaky expression. After production and functioning, the protein of interest will undergo degradation. And to simplify the model, we consider the miRNA degradation effect as part of this process.

Then we wrote down the four chemical reactions in represent of each process.



The symbol declaration is:

X1:tTA protein

D: the TRE promoter

X1D:the tTA-TRE complex

X2: the protein of interest

And we assigned each process with a reaction rate constant.

The first process is about binding and dissociation of molecules and is a fast reaction. Time unit for that is second. Since EF1-αis a constitutive promoter, and we define the initial state as when we remove the Dox from medium, we can regard the concentration of X1, or tTA protein, as a given value, determined by intensity of EF1-α. Reaction rate constant k1,k-1 are affected by Dox. In the tolerable range descried in part3, the more Dox we apply to D, the smaller value of k1 for the positive direction and the greater value of k-1 for the negative direction.

In the second and third process, k2 and k3 are affected by what kind of TRE promoter we use. There are several available TRE promoters, with which we can assemble different version of Tet-off system. These promoters differ in leaky expression and switching performance.

MiRNA-mediated mRNA degradation lead to a decrease in X2 production, which can be thought as identical with natural degradation of X2, the protein of interest. So K4 will be affected by the knockdown efficiency of miRNA binding site. We have constructed several binding sites to fine tune this process.

2.2 solutions and implication

According the analysis, we then wrote down the ODEs:



Here [D0] is the initial concentration of promoter Pcmv,c1 is the given concentration of X1 at the beginning. Next we will do some algebra to simplify the equations and solve the differential equations.





We use Mathematica’s Dynamic Interactive Function Manipulation to control the variation of the parameters α,β, and thus get curves with different dynamics and steady state.

Figure 1. Different values of α,βgenerate curves with different dynamics and steady state.

As analyzed in 2.1, choosing different candidates with variant characteristic will affect the parameters in the chemical reactions, and change values of α and β. We’ve got 4 tet-off systems, a series of miR122 binding sites and experimental characterize them. And also we have two EF1-α promoters with different enhancers(not mentioned in Result section). This implicate that we can assemble different circuits and fine-tune the dynamics and steady state. We believe that this is important to practical engineering, just as basic logic is to conceptual designing.

3. Dosage effect of DOX in turning off the Tet-off system

DOX ,as is discussed above, hinders the binding of tTA to pTRE in Tet-Off system and knockdown expression of suicide gene. In our experiment, we employ fluorescence technique to manifest the amount of protein product by detecting the strength of the fluorescence.




Table 1. Stimulating data of GFP-Dox



Figure 4. GFP-Dox line chart

Our task is to find the proper curve to fit the sample data. First of all we plot the scatter diagram, and according to its tendency, we use type curve to fit the relation of GFP-DOX. We use MATLAB to aid our fitting, i.e. to determine the parameter a, b and k.



%expun.m

function y=expun(s,t) %coefficient and variable

y=s(1)+s(2)*exp(-s(3)*t)

%curvefit.m

treal=[0 0.125 0.25 0.5 1 2]; %experimental data

yreal=[25 13 10 8 6 5.7];

s0=[0.2 0.05 0.05]; %iteration initial value

sfit=lsqcurvefit('expun',s0,treal,yreal); %least square curve fit

f=expun(sfit,treal);

disp(sfit);

The result :




Figure 5. Running result

So a=6.4147,b=18.3999,k=7.3173.

Then we program the diagram file GFP-DOX.m

%GFP-DOX curve

treal=[0 0.125 0.25 0.5 1 2]; %experimental data

yreal=[25 13 10 8 6 5.7];

t=0:0.1:2.5;

a=6.4147;b=18.3999;k=7.3173;

y=a+b*exp(-k*t);

plot(treal,yreal,'rx',t,y,'g');

xlabel('Dosage of DOX');

ylabel('GFP');




Figure 6. GFP-Dox curve

As is shown in the figure above, we can conclude that the amount of GFP tend to be steadily over 1.5 ug, the higher concentration of DOX we set, the lower GFP we expect. However, under the real experimental conditions, over 2.2 ug DOX will lead to the undesired necrosis of the cells. This is a trial-experiment which proved that such a balance point for good turning- off effect and cell tolerance does exist in a certain interval concentration. More accurate experiment should be conducted on stable-transfected iPSCs to find the best cultivating condition.

4. Knockdown efficiency interpolation

According to the experimental data, here we use interpolation technique to find the relationship between miRNA-122 concentration, the number of miR122 target sites and cell knockdown efficiency, which leads to a function with two variables. The knockdown efficiency is represented by GFP expression level which is actually the ratio of the amount of GFP and that of the parameter GAPDH. The knockdown efficiency then is






Figure 7. Two target sites, gradient miRNA concentration



Table 2. Experimental data of 2 target sites, gradient miRNA concentration



Table 3. Experimental data of 0.75ug miRNA plasmid with gradient target sites

We use the data above to do the interpolation. We use the griddata function to implement the interpolation.

MATLAB codes:

clear

miRNA=[0 0.025 0.05 0.1 0.25 0.75 0.75 0.75];

site=[2 2 2 2 2 1 2 4];

KD=[0 29 43 55 64 55 39 32];

cx=0:0.01:0.75;

cy=0:0.05:4;

cz=griddata(miRNA,site,KD,cx,cy','cubic');

meshz(cx,cy,cz),rotate3d

%shading flat

xlabel('miRNA(plasmid ug)'),ylabel('Target Site'),zlabel('knockdown efficiency(%)');




Figure 8. Knockdown efficiency-mRNA-target site surface chart

5. The excursion of little mathematician--Data analysis of the FACS data

When our project was proceeding, we found out an interesting problem, that is, how to calculate the killing efficiency of each suicide gene? And this quantity is also an important part of our modeling. Trying to solve this problem,one of our mathematician, young Yang ZiYi, excursed a little bit from modeling to data analysis. And he analyzed our data of FACS of our transient transfection experiment of Hep G2.

One important point in mathematical analysis about the complicated biological system is, not to draw arbitrary assumption, arbitrary assumption just lead to disaster. Another important point is not to draw complicated assumption, which is hard to calculate. Base on these rules, Young ZiYi draw some simple and reasonable assumptions:

1. The initial condition of each parallel well is the same, that means, every well before transient transfection should have the same cell density, and the cells' state should be approximately the same.

2. In the GFP control group, the cells should be regarded as the same, whether they are transfected with GFP or not, since GFP do not harm the cells. But lipo-2000 will harm the cells, and may have some long term effect. So the GFP control group would be relatively weaker compare to negative control, which without any transfection manipulation;

3. The cells transfected with GFP should have an innate death rate r1 after 3 days cultivation. Besides, the ratio of “the initial number of cells” to “the number of cells harvested via FACS after 3 days” Should be the same, since if you sample any part of the wells you will observe the same distribution of cells, this means the cell experiment is scalable. Hence, we can define a special “state” to represent GFP cells in certain well, (r1,n1), n1 is the number of the cells we harvest in 30s, depends on r1 and the initial cell number.

4. In the well we transfect suicide genes, the cells which are transfected with suicide gene will be different from the cells which are not, due to the killing effect of suicide gene, and the cells without transfection will be at the same condition as GFP control and have the same r1. Cells transfected with suicide gene will have a different r2, we denote this part of cells by (r2,n2). If the transfection efficiency is a, then for any initial cell number N0, there will be N0×a cells transfected with suicide gene, the remaining N0×(1-a)are not. Here "a" in an unknown factor.

Then Young ZiYi built a model:

In the wells that had been tranfected with suicide genes, the final observed “state” (X, C) of the cells is an combination of two kinds of cells, one kind with state (r1,n1), the other kind with state (r2,n2). The effect of these two kinds will be summed up together. Then we can write down a equation:



Because we can not easily get the transfection efficiency, we select a reasonable range from experience:[30%,60%], and solve the equation with the data from FACS, and get the following results:



It turns out that, the death rate of cells transfected with suicide gene will be greater than 60%, significantly higher than normal cells and GFP control cells(29.07%, according to our data).

Then, We pick up one suicide gene, VP3, and take an “a” value 35% which we believe is fairly closed to the real tranfection efficiency, solve the equation and get the result r2 and n2. And then, we substitute the calculated r2 and n2 to the left side of the equation, and raise the value of “ a” to predict what the death rate and the “number per 30 seconds” would be when the transfection efficiency is raised. The result is shown in a chart below





That makes our model become a real theory, because it predicts something that can be disproved. In the following day, we will try to raise our transfection efficiency, and try to comparing the result with the prediction.

And, let’s wait for the result.

6. Multi-compartment model

6.1 Analysis of the problem

We firstly focus on factors that regulate the performance of the whole pathway. Protein tTA expressed by a EF1α promoter binds to the promoter pTRE to drive the transcription of target gene( in this case, eGFP or suicide gene) while Dox acts as a co-repressor prohibiting the transcription. MiR122 isa downstream part in the pathway after transcription of target mRNA, and mediated degradation of the mRNA, thus rescue the cell or knockdown its GFP expression. However, the miR122 level in iPSC was low and insufficient to exert obvious effect on the expression.

Apart from Dox concentration,we also monitored other parameters, including cell number after the stable infection and number of cell that survived the Suicide Gene. Moreover, we also kept track of fluoresence intensity of the control group who has been transfected with GFP, which can be employed to indicate the GOI expression level driven by Tet-Off system.

In pratical, we planned to monitor the cell group scale every 5 hours and technically, we counted the total clone area instead of cell number.

6.2 Symbols declaration and assumption

X1: initial number of iPS cells with Suicide Gene

X2: number of the iPS cells whose TRE have been combined with tTA

X3: number of iPS cells which have died from expressing Suicide Gene

k1: converting rate of the number of cells from phase X1 to phase X2

k2: converting rate of the number of cells from phase X2 to phase X3

The unit of ki(i=1,2) is hour-1.We measured it by dividing the absolute value of the cell number difference between former phase and latter phase, with the time period length.

Two cases are taken into account. In case (a), self-renewal and replication of cels are ingored while in case (b), we take that into consideration. To further simplify the model, we also assumed that every single cell in phase X1 turns into n1 state before phase X2, and every single cell in phase X2 turns into n2 state before phase X3. We simulated the kinetic process of gene expression and assumed an even distribution of cell content in the medium,after which the phase can be regarded as a compartment.

6.3 Solution

For each compartment, we construct unsteady state equilibrium equation, hence we obtain the ordinary equations

For case (b), we just need to modify the scalar coefficients of the equations above, and we obtain






Figure 1. dynamic process

We are going to solve X1(t), X2(t),X3(t), then we will plot the time course curve.

The initial conditions of the differential equations are as follows:

X1(0)= 5000 cells, X2(0)=0 cell, X3(0)=0 cell

k1=1day-1,k2=1 day-1

As for case b, the cell replicates every 26 hours, to simplify we consider one cell turns into 2 cells before next phase. Therefore, n1=n2=2.

Source code

%igem_test1.m-Solution of the IPS cell differentiation model

%using MATLAB function ode45.m to integrate the differential equations

%that are contained in the file cell_diff_eq.m

clc; clear all;

%set the initial conditions, constants and time span

xzero=[5000,0,0];tmax=4;

k1=1; k2=1;

tspan=0:0.1: tmax;

N=3;

%Integrate the equations

[t X]=ode45(@cell_diff_eq,tspan,xzero,[ ],k1,k2);

last=X(length(X),N);

%Plot time curve

plot(t,X(:,1),'-',t, X(:,2),'-',t, X(:,3),'-.');

legend('X1','X2','X3');

xlabel('time,days');

ylabel('number of cells');

function dx= cell_diff_eq(t,x,k1,k2)

%cell expression kinetic procedure

dx=[-k1*x(1); k1*x(1)-k2*x(2); k2*x(2); ];




Figure 2. The result of case (a)


Figure 3. The result of case (b)

7.reference

[1] Systems biology in practice concepts, implementation and application / (德) E. Klipp等著 ; 主译:贺福初, 杨 芃原, 朱云平 ,上海 : 复旦大学出版社, 2007

[2]Numerical methods in biomedical engineering / (美) Stanley M. Dunn, Alkis Constantinides, Prabhas V. Moghe著 ; 封洲燕译,北京 : 机械工业出版社, 2009

[3]miRNA regulatory circuits in ES cells differentiation: chemical kinetics modeling approach , Luo Z, Xu X, Gu P, Lonard D, Gunaratne PH, et al. (2011)

[4]kinetic signatures of microRNA modes of action, N Morozova, A Zinovyev, N Nonne, LL Pritchard - RNA, 2012

[5]On the inhibition of the mitochondrial inner membrane anion uniporter by cationic amphiphiles and other drugs. Beavis AD,J Biol Chem. 1989 Jan 25;264(3):1508-15.

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