Global sensitivity analysis of a dynamic model for gene expression in Drosophila embryos

Abstract

It is well known that gene regulation is a tightly controlled process in early organismal development. However, the roles of key processes involved in this regulation, such as transcription and translation, are less well understood, and mathematical modeling approaches in this field are still in their infancy. In recent studies, biologists have taken precise measurements of protein and mRNA abundance to determine the relative contributions of key factors involved in regulating protein levels in mammalian cells. We now approach this question from a mathematical modeling perspective. In this study, we use a simple dynamic mathematical model that incorporates terms representing transcription, translation, mRNA and protein decay, and diffusion in an early Drosophila embryo. We perform global sensitivity analyses on this model using various different initial conditions and spatial and temporal outputs. Our results indicate that transcription and translation are often the key parameters to determine protein abundance. This observation is in close agreement with the experimental results from mammalian cells for various initial conditions at particular time points, suggesting that a simple dynamic model can capture the qualitative behavior of a gene. Additionally, we find that parameter sensitivites are temporally dynamic, illustrating the importance of conducting a thorough global sensitivity analysis across multiple time points when analyzing mathematical models of gene regulation.

Keywords: Gene expression; Parameter sensitivity; Reaction–diffusion; Transcription; Translation.

Figure 1. Schematic of the biological procceses represented in the ODE model.

In (A) the Reaction terms of the model are illustrated. These include the synthesis of new mRNA through transcription, the synthesis of new protein through translation of mRNA, mRNA decay, and protein decay. In B, the Diffusion terms of the model are illustrated. These include both mRNA diffusion and protein diffusion to/from neighboring nuclei in an early Drosophila embryo.

Figure 2. Qualitative similarities between parameter sensitivities and experimental measurements.

(A) Ubiquitous gene with initial concentrations of 1.0; First and second-order sensitivities at the middle nucleus at t = 4 min. (B) Anterior maternally deposited gene; First and second-order sensitivities at the middle nucleus at t = 2 min. In both, along the x-axis are the parameters corresponding to: 1. Transcription, 2. mRNA Diffusion, 3. mRNA Decay, 4. Translation, 5. Protein Diffusion, and 6. Protein Decay.

Figure 3. Comparison of parameter sensitivities to experimental measurements at a later time point.

(A) Ubiquitous gene with initial concentrations of 1.0; First and second-order sensitivities at the middle nucleus at t = 10 min. (B) Anterior maternally deposited gene; First and second-order sensitivities at the middle nucleus at t = 10 min. In both, along the x-axis are the parameters corresponding to: 1. Transcription, 2. mRNA Diffusion, 3. mRNA Decay, 4. Translation, 5. Protein Diffusion, and 6. Protein Decay.

Figure 4. Temporal dynamics of parameter sensitivities.

(A) First-order parameter sensitivities at the middle nucleus over time for a ubiquitous gene with initial concentrations of 1.0. (B) First-order parameter sensitivities at the middle nucleus over time for an anterior maternally deposited gene.