Team:Freiburg/Project/modeling

Introduction

Effectors

Effector Control

Targeting

uniBAss

Manual

Modeling

Introduction

The Networks

1. dCAS-VP16

Our network includes four different components dCas-VP16, a RNA complex (tracr/cr RNA), a RNA-dCas-VP16 complex and the secreted alkaline phosphatase (SEAP). dCas-VP16 binds the RNA-complex and the whole complex binds the DNA, which leads to the production of SEAP.

Figure 1: Transcriptional Activation via dCAS-VP16: The dCAS-VP16 fusion protein is guided to the desired DNA sequence by a co-expressed crRNA and tracrRNA. The binding of the gene recognition complex leads to an expression of SEAP.

Setting up the ODE

According to the graphical reaction network the ODE can be set up.

Cas9 is constitutively expressed by the CBh promoter and degraded proportional to the current concentration. It is used to build the DNA recognition complex and produced during complex decay.

The RNA-complex is built linearly. The production constant can be seen as production constant of the lower expressed RNA, because this expression limits the complex building. It is assumed that the RNA is degraded after DNA recognition complex decay and therefore the complex decay does not lead to more RNA.

The DNA recognition complex is built, when Cas9 and RNA meets and degraded proportional to the current DNA recognition complex concentration.

There is a leaky SEAP production and one that depends on the current concentration of the Cas9/RNA Complex. This dependency is assumed to follow the Monod-kinetic. Because of the long half time (T₂ > 500 h) of SEAP we can neglect the SEAP decay [2, 3].

2. dCAS-KRAB

The dCas-KRAB sytem was modeled as a three component system. tetR as activator of gene production, dCas-KRAB as repressor and SEAP as final reporter protein.

Figure 2: Transcriptional respression via dCAS-KRAB: The dCAS-KRAB fusion protein binds to the desired target sequence at a different loci than the tetR, that binds to tetO and is assumed to repress SEAP production.

Setting up the ODE

According to the graphical reaction network the ODE can be set up.

Cas9 is constitutively expressed by the CBh promoter and degraded proportional to the current concentration, similar to the activation model. It binds to the DNA and is released afterwards. Because of the four times higher amount of transfected Cas-plasmid the basal expression rate is assuemd as four times higher than the of tetR.

tetR is assumed to have a similiar kinetic as Cas.

There is a leaky SEAP production and one that depends on the current concentration of Cas9 and tetR. This dependency is assumed to follow an allosteric inhibition process. Similar to the activation process the SEAP decay can be neglected because of the long half time (T₂ > 500 h) of SEAP [2, 3].

Finding the parameters

By setting up the ODE a n-dimensional hypothesis space (n is the number of parameters) is generated and finding the right parameter combination means finding a point in the space which fits the data best.
To find these parameters we used the maximum likelihood approach. The maximum likelihood hypothesis is the hypothesis which has the highest probability to generate the measured data. It is shown [3], that using the maximum likelihood approach and assuming gaussion noise in the data (an assumption that holds in our case) leads to a least-square error minimization problem.

A minimization problem is an optimization problem. You search for parameters (p0) for which holds, that the value of the function (f) at the point of the parameters is smaller than all other values. (f(p0)<=f(p)). In three dimensions the function can be thought as a landscape and minimization is finding the deepest valley. Depending on the method you use different problems arise. The most common problem is finding only a local minimum and not the global one (Fig. 3).

Figure 3: Example of a minimization problem. Shown is a 3D landscape. Depending on the start position (the initial parameters), the found minimum is either a local or the global one.

To avoid this and to be sure to have found a global minimum we started our minimization procedure using different start values for our parameters. To sample these parameters we used the latin hypercube sampling on a logarithmic scale (Fig. 4).

Figure 4: Illustration of the latin hypercube sampling in a two dimensional parameter space. The number of initial parameter vectors is 5. Therefore the parameter space is divided in 25 subspaces. Shown is one possible parameter combination.

The number of different initial parameter settings is set to N and thus the parameter space is divided in N*N subspaces. For the initial parameter the values are chosen so that there is only one parameter in each row and column.

The resulting errors we plotted in an increasing order to be sure to have found a global minimum.

Data generation

Cas9 is quantified by using Western blot and we used SEAP as target protein that can be quantified by a SEAP assay. For more detailed information refer our modeling notebook.

Fitting Procedure and Results

1. dCas-VP16

Assuming the given ODE and using the fminsearch-function implemented in matlab with various initial parameter vectors the fitting process results in one optimal parameter composition. Our measurement time starts with the transfection, however the first data we got at time point 6, therefore all 14 parameters including the initial concentrations were set variable. This results in unlikely high initial concentration. Unlikely, because although the transfection end point is not clearly to define after transfection there should only be a small amount of the components. A second process in which the initial concentrations were assumed as zero followed the first one. The change between fixed and variable parameters was easily to perform, because of an additional vector (qfit). This vector contains boolean values depending on wether the parameter is fixed or flexible during the fitting process. Moreover another parameter is required for adjusting the absolute values of the duplicates. The results of this second process are shown. (Table 1).

The model reflects the general construction of the network (Fig. 5). As assumed Cas9 converges asymptotically to a stable state and there is an exponential increase in the SEAP concentration, both processes are reflected in the data. The model also shows a potential behaviour of the not measured components, the free tracr/crRNA-complex and the gene recognition complex. The tracr/crRNA-complex reachs its steady state quickly. There is however no possibility to distinguish between the two different RNAs. There might be some differences in their expressions, especially because of the different promoters (crRNA expressed under U6-promoter; tracrRNA expressed under h1-promoter), however the model won't show them.
The free gene recognition complex has, because of the small degradation rate, not reached its steady state yet.

Figure 5: Modeling Result: Shown are the experimental results (purple square) in comparison to the model prediction values (cyan cross) for SEAP and Cas, as well as the model prediction for not measured components

Because of the fact that the fminsearch algorithm is not proved to converge to a minimum [1], we started at different points in the parameter space and therefore the probability of having found the global minimum is high (Fig. 6) .

Figure 6: Different error values plotted in increasing order.

2. dCas-KRAB

The same procedure was done to find the parameters for dCas-KRAB. We started setting all parameters as variable and after that we set the initial concentrations to zero. Fitting with zero initial concentrations led to a lower least square error. As assumed before the dCas-KRAB reachs its steady state within the measured time periode, approxiametly at the same time as Cas9-VP16. However the expected stop of SEAP concentration arising could not be observed. The effect of reduction is because of the high Cas9 to tetR ratio to low.

Figure 7: Modeling Result: Shown are the experimental results (purple square) in comparison to the model prediction values (cyan cross) for SEAP and Cas9, as well as the model prediction for the not measured component tetR.

This time we also fitted the different error values. (Fig.8) .

Figure 8: Different error values plotted in increasing order.

The Code Files

References