LOS JÓVENES VINCULADOS AL CENTRO DE FORMACIÓN JUVENIL ACJ-YMCA

Figures 5.4 A and 5.5 B).

The fit parameter distributions obtained by fitting traces of 283 mRNA transfected cells are shown in Fig. 5.5 C-F. In Fig. 5.5 C the distribution of ton for pDNA transfected cells

(in red) is given for comparison. The distributions of the degradation rates δ and β were fitted with a Gaussian1. This yieldsδ = 0.056±0.021h−1 and β = 0.051±0.022h−1. This is slightly bigger than a value of 0.028h−1 for δ reported previously [105]. The result is smaller than the degradation rate of endogenous mRNA (0.1h−1_{) which was to be expected} due to the stabilizing cap that the in vitro mRNA carries. Note that in the model kT L

and m0 are linked and therefore only the product of both appears as a fit parameter (Fig. 5.5 D). However, the number of mRNA molecules that are successfully transfected per cell would be of great interest. Since transfection is an inherently stochastic process it can be assumed that the greater part of the observed variance ofkT L·m0 is due to an underlying distribution of m0 and not due to great variability of kT L. Therefore, in the following

section a stochastic mRNA delivery model will be developed to account for this variance.

5.5

Stochastic mRNA delivery model

The assumed mRNA delivery process to transfected cells is depicted in Fig. 5.6. mRNA and the lipid mixture lipofectamine form complexes with on average m mRNA molecules per complex. These complexes sediment on to the cells and are taken up via the endocytotic pathway with on averageN endosomes per cell. The complexes then escape the endosomes with a lysis probabilitykand are unpacked with probabilityq. Afterwards the freed mRNA molecules are available for translation.

The number of endosomes as well as the number of complexes in an endosome are assumed to be small and to arise from stochastic and independent processes. Therefore, Poisson distributions are adopted for both. The probability to have N endosomes in a given cell is then described by:

PN(k) = Nk

k! e

−N _(5.7)

Endosomes lyse with probabilityk. The probability distribution of N0 lysed endosomes is a convolution of eq. (5.7) with a binomial with probability k since lysis can happen for

1_{The Gaussian or normal distribution is given byf(x) =} 1

σ√2πe

−(x₂−_σµ2)2 _{whereµis the mean andσthe}

64 5. Modeling of exogenous gene expression in eukaryotic cells

Figure 5.5: A) single cell traces of mRNA transfected cells; B) traces from A) shifted by

ton and normalized; C) distribution of ton for pDNA (red) and mRNA (blue) transfected

cells; D)-F) distributions of independent fit parameters of 283 analyzed A549 single cell traces (mRNA transfection). Refer to the text for details.

5.5 Stochastic mRNA delivery model 65

Figure 5.6: It is assumed that lipoplexes are taken up by the cell via the endocytotic path- way. First, lipoplexes sediment on to the cell membrane. Then the membrane invaginates around these lipoplexes and is chocked off creating an endosome with the lipoplexes inside. Subsequently the endosome lyses and the mRNA molecules are released into the cytosol. In the context of the stochastic delivery model described here it is assumed that the average number of endosomes per cellN as well as the average number of lipoplexes per endosome

care small. Therefore, N and cfollow Poisson distributions.

allN ≥N0: P(k|N0, N) = N N0 kN0(1−k)N−N0 ·PN(k) (5.8)

Equation (5.8) can simply be reduced to a Poissonian with the parameter N k. As motivated above the probability distribution of complexes per endosome is also a Pois- sonian with parameter c. The probability to have c0 complexes from lysed endosomes in a cell is then a convolution of both processes. Using the fact that a convolution of N0

Poisson distribution with parameter c is itself a Poisson distribution with parameter N0c

and accounting for all possible values of N0 yields:

P(c0) = ∞ X N0₌₀ (N k)N0 N0_! e −N k(N 0_c)c0 c0_! e −N0c _(5.9)

In the above equation it is implied that all complexes are identical which is an unrealistic assumption. One would rather expect that the complex size and therefore mRNA payload to follow a normal distribution. This means a convolution of equation (5.9) with a normal

66 5. Modeling of exogenous gene expression in eukaryotic cells

Figure 5.7: A) Probability distribution of mRNA transfected A549 cells at various mRNA concentrations generated from flow cytometry data. At higher concentrations the fraction of transfected cells increases but not the average fluorescence per cell (the position of the peak does not shift to the right). B) Fraction of transfected cells generated by integrating the data from (A). A stochastic mRNA delivery model with only one Poisson process does not satisfactorily fit the data (dashed red line). However, the model with two independent consecutive Poisson processes (equation (5.10)) presented in the text does (blue line). Here the fit parameters are N0 = 0.9±0.2 andc0 = 1.1±0.5.

distribution.

A further quantity of interest is the transfection ratio, that is the percentage of trans- fected cells at a given concentration of lipoplexes (that is mRNA) in the cell culture. This is the probability that in a cell at least one lipoplex is released from the endosomes (note that here the unpacking probability of complexes q in Fig. 5.6 is assumed to equal one for simplicities sake). The transfection ratio is obtained by summing over eq. (5.9) from

c0 = 1 to infinity:

P(c0 ≥1) = 1−exp[N0(e−c0 −1)] (5.10) In Fig. 5.7 A the probability distributions of A549 cells transfected with various con- centrations of mRNA are depicted. The data were recorded using flow cytometry. Higher concentrations of mRNA increase the number of transfected cells. However, interestingly the average brightness of the cells does not increase (that is, the peak in Fig. 5.7 A is not shifted to the right at higher mRNA concentrations). The dose response relationship between mRNA concentration and fraction of transfected cells is depicted in Fig. 5.7 B. It was obtained by integrating the probability distributions in Fig. 5.7 A. The fact that only

5.5 Stochastic mRNA delivery model 67

Figure 5.8: Distribution of mRNA molecules per transfected cell m0 obtained by dividing the distribution ofkT Lm0 in Fig. 5.5 D by the literature value kT L= 180h−1. The fit (blue

line) was obtained by inserting N0 = 0.9 and c0 = 1.1 into equation (5.9) and convolving with a normal distribution. This is necessary to account for a distribution in complex size and therefore mRNA payload. The fit parameters are the standard deviation σ = 2 and the average number of mRNA molecules per complexNmRN A = 300.

up to 60% of all cells are transfected indicates that some bottleneck in the transfection process limits this number. The simplest model of such a bottleneck would be a single Bernoulli process2 _{with a low probability of success. The resulting probability distribution} would then be a Poissonian and the dose response relationship would therefore feature a simple exponential increase. This case is depicted with the doted red line in Fig. 5.7 B. Ob- viously, this model does not yield an acceptable fit to the data. An acceptable fit however can be obtained by inwoking equation (5.10) of the double Poissonian model discussed above (blue line in Fig. 5.7 B). The fit returns the average number of lysed endosomes

N0 = 0.9±0.2 as well as the average number of unpacked complexesc0 = 1.1±0.5. Using equation (5.9) the model can furthermore reproduce the probability distribution of successfully transfected mRNA molecules per cell m0. In Fig. 5.8 the distribution of m0 of 283 A549 cells obtained with quantitative fluorescence microscopy and image analysis is depicted. Note that these are the same data as shown previously in Fig. 5.5 D albeit divided by kT L = 180h−1 (see Tab. 5.1). When comparing these data with the

predicted distribution of equation (5.9) with N0 = 0.9 and c0 = 1.1 it was found that the predicted distribution is narrower than the one obtained from single cell data. However, if one assumes that the complex size follows a normal distribution one obtains the standard

2_{In probability theory a Bernoulli process is a discrete stochastic process which has only two possible}

68 5. Modeling of exogenous gene expression in eukaryotic cells

deviationσ as a fit parameter that broadens the distribution. The second fit parameter is the average number of mRNA molecules per complex NmRN A. The fit depicted in Fig. 5.8

was obtained with σ = 2 andNmRN A= 300.