Stochastic model for gene transcription on Drosophila melanogaster embryos

Guilherme N. Prata, José Eduardo M. Hornos, and Alexandre F. Ramos

Phys. Rev. E 93, 022403 – Published 2 February 2016

Abstract

We examine immunostaining experimental data for the formation of stripe 2 of even-skipped (eve) transcripts on D. melanogaster embryos. An estimate of the factor converting immunofluorescence intensity units into molecular numbers is given. The analysis of the eve dynamics at the region of stripe 2 suggests that the promoter site of the gene has two distinct regimes: an earlier phase when it is predominantly activated until a critical time when it becomes mainly repressed. That suggests proposing a stochastic binary model for gene transcription on D. melanogaster embryos. Our model has two random variables: the transcripts number and the state of the source of mRNAs given as active or repressed. We are able to reproduce available experimental data for the average number of transcripts. An analysis of the random fluctuations on the number of eves and their consequences on the spatial precision of stripe 2 is presented. We show that the position of the anterior or posterior borders fluctuate around their average position by $\sim 1 %$ of the embryo length, which is similar to what is found experimentally. The fitting of data by such a simple model suggests that it can be useful to understand the functions of randomness during developmental processes.

Received 10 November 2014
Revised 19 November 2015

DOI:https://doi.org/10.1103/PhysRevE.93.022403

Physics Subject Headings (PhySH)

Nucleic acid replication & information processes Stochastic processes

Physics of Living Systems

Authors & Affiliations

Guilherme N. Prata^1,*, José Eduardo M. Hornos^2,†, and Alexandre F. Ramos^1,3,4,‡

¹Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, Avenida Arlindo Béttio, 1000, Ermelino Matarazzo, São Paulo, SP CEP 03828-000, Brazil
²Instituto de Física de São Carlos, Universidade de São Paulo, Av. Trabalhador São-Carlense, 400, São Carlos, SP CEP 13566-590, Brazil
³Departamento de Radiologia - Faculdade de Medicina, Universidade de São Paulo, São Carlos, SP CEP 13566-590, Brazil
⁴Núcleo de Estudos Interdisciplinares em Sistemas Complexos, Universidade de São Paulo, São Carlos, SP CEP 13566-590, Brazil

^*gnprata@cdcc.usp.br
^†Deceased.
^‡alex.ramos@usp.br

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Vol. 93, Iss. 2 — February 2016

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Images

Figure 1
The mRNA profiles of the $e v e$ 's stripe 2 is presented for C13 and for the eight time classes of C14, ending by the onset of gastrulation. The red lines indicate the experimental data while average mRNA numbers predicted by the stochastic binary model for gene transcription in fruit flies is given by the solid green circles. The standard deviation on the mRNA numbers at each position along the AP axis is represented by the vertical black bars.
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Figure 2
The two-phase character of the dynamics of the eve's gene transcription at the position 41.5% of EL is shown here. The experimental data is given by the solid red circles while the green line indicates the average mRNA number as predicted by our model.
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Figure 3
A comparison between the Gaussian shape for critical time as obtained with Eq. (23) and experimental data on transcriptional activity around the peak of expression of eve's stripe 2, using experimental data presented in Ref. [40]. The experimental data are shown as the solid red circles and the green lines are indicating the model's results.
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Figure 4
The implications of the fluctuations on the mRNA numbers on the position of the anterior and posterior borders of eve's stripe 2 domain at time class T5. We consider three particular curves, one (green color) for the average number of mRNA synthesized by the eve's gene and the remaining two (red color) given by the average plus or minus the standard deviation on the number of transcripts.
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Figure 5
Heaviside-like time dependence in the rates $h_{i}$ and $f_{i}$ . Each phase of the dynamics of the mRNA expression at 41.5% $A P$ position is associated with an evolution toward seemingly asymptotic level. This, in turn, is associated with a pair of transition rates: the regulatory effect is represented by ${\bar{h}}_{i}$ and ${\bar{f}}_{i}$ in the first phase and by ${\tilde{h}}_{i}$ and ${\tilde{f}}_{i}$ in the second phase. Since the first and second phases are respectively characterized by the formation and disappearance of stripe 2, this means that whereas the gene performs the transition off $\to$ on faster than on $\to$ off during the first phase, the inverse occurs during the second phase. Mathematically, this means that ${\bar{f}}_{i} > {\tilde{f}}_{i}$ and ${\tilde{h}}_{i} < {\bar{h}}_{i}$ .
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Figure 6
Heaviside-like time dependence in the rates $q_{i}$ . Heaviside-like time dependence in rates $h$ and $f$ also makes the asymptotic probabilities ${\bar{a}}_{i}$ and ${\tilde{a}}_{i}$ vary suddenly over time. It is convenient to work with the $q_{i}$ , given by $q_{i} = {(ε ρ)}^{- 1} f_{i}$ , because it has units of probability, that is, $0 \leq q_{i} \leq 1$ , which is useful to investigate the regulatory information and to run optimizations for parameters of the model.
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