Modeling relaxation experiments with a mechanistic model of gene expression

Abstract

Background: In the present work, we aimed at modeling a relaxation experiment which consists in selecting a subfraction of a cell population and observing the speed at which the entire initial distribution for a given marker is reconstituted.

Methods: For this we first proposed a modification of a previously published mechanistic two-state model of gene expression to which we added a state-dependent proliferation term. This results in a system of two partial differential equations. Under the assumption of a linear dependence of the proliferation rate with respect to the marker level, we could derive the asymptotic profile of the solutions of this model.

Results: In order to confront our model with experimental data, we generated a relaxation experiment of the CD34 antigen on the surface of TF1-BA cells, starting either from the highest or the lowest CD34 expression levels. We observed in both cases that after approximately 25 days the distribution of CD34 returns to its initial stationary state. Numerical simulations, based on parameter values estimated from the dataset, have shown that the model solutions closely align with the experimental data from the relaxation experiments.

Conclusion: Altogether our results strongly support the notion that cells should be seen and modeled as probabilistic dynamical systems.

Keywords: Asymptotic profile; Relaxation experiments; Two-state model.

Fig. 1

The 2-state model of gene expression. The gene opens with a $k_{\text{on}}$ rate and closes with a $k_{\text{off}}$ rate. Similarly to [11] we only consider protein (x) production (with an s rate) and degradation (with a d rate)

Fig. 2

The relaxation experiment. TF1-BA cells well labelled with an anti-CD34 antibody and FACS-sorted. The 10 percent most CD34 positive and the 10 percent most CD34 negative cells were sorted, grown in culture for the indicated period of time, where the distribution of cell-surface CD34 expression was assessed. KT: the modified Kantorovich–Rubinstein distance, defined by the Eq. (14), between the two distributions [26]

Fig. 3

Example of flow cytometry gating. Top. Example of SSC-H along FSC-H plot for raw data from the plus subpopulation on day 2. As the data contain a high proportion of debris cells, we select only those viable cells lying within the black ellipse. Bottom. Fluorescence data before gating (Ungated) and after gating (Gated). For the figures and the ellipse, we used the python package “FlowCal” [27]

Fig. 4

Estimation of the exponential growth rate $\lambda$. The red squares correspond to the number of cells (log scale) at different times for the relaxation experiments: Top. ${\text{CD34}}^{+}$, Bottom. ${\text{CD34}}^{-}$. The dotted line in black illustrates the optimal fit of the experimental data. The average of the slopes of the linear regressions minimizing the two experiments is given by the slope $\lambda \approx 0.42$

Fig. 5

Likelihood profiles for for $k_{\text{on}}$, $k_{\text{off}}$ and d in A and for $r_1$ in B. A. The blue curve represents the function $S_d$, the red curve $S_{k_{\text{off}}}$ and the green curve $S_{k_{\text{on}}}$. The function $S_d$ is introduced into Eq. (23). B. The grey area corresponds to the range of parameter values for $r_1$ such that the function r is non-positive. Compared with the other parameters, variation in the $r_1$ parameter has a small impact on the minimum Kantorovich–Rubinstein distance

Fig. 6

Comparison of model and data. On the left the fitting of the ${\text{CD34}}^{+}$ relaxation experiment (in A) and on the right in green of the ${\text{CD34}}^{-}$ (in B) experiments. Experimental data in logarithmic scale are represented by plain histograms and the numerical results of model (5) are represented by the dotted curves. We initialize the model on day 2, using the biological data. The initial condition is given by (24). Parameter values are given in Table 1. KT: the modified Kantorovich–Rubinstein distance, defined by the Eq. (14). C. Time-dependent evolution of the Kantorovich–Rubinstein distance between model and experimental data. For different days of the experiment, the modified Kantorovich–Rubinstein distance between the two relaxation experiments is depicted using black squares. The minimum distance, reached on day 26, is illustrated by a horizontal dotted line. The red crosses correspond to the modified Kantorovich–Rubinstein distance between the model for the parameter values from Table 1, and the ${\text{CD34}}^{+}$ cell relaxation experiment. Similarly, the green crosses represent the distance for the ${\text{CD34}}^{-}$ cell relaxation experiment