UPDATE 09/18/2013

1.Overview

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

Basically the models built by us can be divided into two levels. In 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 iPSC. 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. Dox and miRNA concentration acted as parameters switching the pathway state and affected the growth dynamics of iPSC.

In 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 expression variation of target mRNA in different state of the pathway, determined by Dox and miR122 level. The models in 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.

2. Multi-compartment model

2.1 Analysis of the problem

We first outlined factors that regulate the performance of the whole pathway. Protein tTA expressed by a EF1a 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 hindering the transcription.MiR122 functions as downstream 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.

Two core factors in our modeling are the concentration of Dox，and the number of sites of miR122 targets on targeted mRNA. In our modeling, the growth process of iPS cells which have been transfected with our iPSC Safeguard pathway is tracked. Initial cell number after the transfection process was counted and after that, Dox would be added differently.

The differentiation process of iPS cells were first tracked, after which the initial number of iPS cells which had been transfected by suicide genes and survived the selection was counted. With the concentration of Dox and number of miR122 targets manually altered as gradient, the number of iPS cells expressing reporting gene(GFP),as well as those which remained alive after expressing suicide genes，were counted.

Under our real experimental conditions, we kept note of the cell number every 5 hour. Using cell florescence technology , we counted the cell number referring to the florescent strength of observatory area.

2.2 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: the 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, the method of measurement is using the number of cells in latter phase from the number of cells in former phases, keeping the absolute value, dividing it by the time lasting.

Two cases are taken into account. In case (a), self-renewal and replication of cell are assumed not to occurred. In case (b), we take the replication phenomenon into consideration, to simplify the model, we assume that every single cell in phase X1 turns into n1 cells before phase X2, and every single cell in phase X2 turns into n2 cells before phase X3. We simulate the kinetic process of gene expression. And we assume that in every phase the distribution of the cell content solution is even, the phase can be regarded as a compartment.

2.3 Solution

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

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

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 replicate every 26 hours, to simplify we consider one cell turns into 2 cells before next phase. Therefore, n1=n2=2. We use MATLAB ode45 function to solve the problem.

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); ];

The result of case (a):

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

DOX ,as we have discussed above, is used to hinder the combination of tTA and TRE in tet-off system, thus the synthesis of the mRNA is blocked and the expression of the suicide gene is reduced, the produce of the suicide protein product will correspondingly decline. In our experiment, we use fluorescence technique to manifest the amount of protein product. The magnitude of fluorescence GFP indicates the protein amount in cells of unit amount.

TABLE 1. Experimental data

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 :

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');

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 can 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 is to prove 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 iPSC 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 4. 2 target sites, gradient miRNA concentration

TABLE2. Experimental data of 2 target sites, gradient miRNA concentration

TABLE3. 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(%)');

5. miRNA-mediated regulatory model

For any miRNA-mRNA pair, there’re 2 architectures to describe the action of miRNA in the regulatory network. Correspondingly, there’re 2 chemical kinetic models introduced to describe the performance of miRNA in the circuits.

The transcription-degradation architecture is based on the assumption that the miRNA takes effect at the latter stage of transcription and mediates the degradation of the mRNA that is undergoing formalizing. In this case the translation inhibition is speculated not to occur simultaneously with the transcription degradation, mRNA’s expression depends on upstream factors. In our circuit, the transcriptional repressor is DOX, we denote it R, and there’s no activator, To model the action of downstream mRNA gene G, we implement the nonlinear chemical kinetic equation. Let’s denote the expression levels of mRNA , protein, miRNA and the repressor respectively by g(t),p(t),m(t), R(t), then we define the individual impact of proteinic repressor R on the change rate of g(t) by REP(t),

Where BSR represents the number of the binding sites; u represents the affinity constants with mRNA.

The generation proportion of mRNA committed by the DNA templates can be defined as

Since there is no activator, so f(t)=REP(t). Let r1 be the degradation rate of mRNA, r2 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 obtain.

The second interaction architecture for miRNA-mRNA pair points out that the upstream miRNA inhibits the translation of the downstream mRNA gene, resulting in the repression of the expression of the protein generated by mRNA gene. Translation inhibition architecture focuses on the change of the amount of the protein. A set of n upstream miRNAs denoted by m1,m2,m3,…,mn. We denote the concentration at time t of protein, mRNA, m1,m2,m3,…,mn by p(t), g(t), m1(t),m2(t),m3(t),…,mn(t). For mi，the repressive impact on translation is defined as

The comprehensive impact then is

Let r1 and r2 be the protein degradation rate and translation rate respectively. Degradation leads to the reduce of protein whereas the mRNA translation contributes to the increase of protein, so the ordinary differential equation is as follow

6.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