A compartmental model for the bicoid gradient

Abstract

The anterior region of the Drosophila embryo is patterned by the concentration gradient of the homeodomain transcription factor bicoid (Bcd). The Bcd gradient was the first identified morphogen gradient and continues to be a subject of intense research at multiple levels, from the mechanisms of RNA localization in the oocyte to the evolution of the Bcd-mediated patterning events in multiple Drosophila species. Critical assessment of the mechanisms of the Bcd gradient formation requires biophysical models of the syncytial embryo. Most of the proposed models rely on reaction-diffusion equations, but their formulation and applicability at high nuclear densities is a nontrivial task. We propose a straightforward alternative in which the syncytial blastoderm is approximated by a periodic arrangement of well-mixed compartments: a single nucleus and an associated cytoplasmic region. We formulate a compartmental model, constrain its parameters by experimental data, and demonstrate that it provides an adequate description of the Bcd gradient dynamics.

Figure 1

(A) Model geometry: red spheres illustrate the nucleus in each compartment, and light blue denotes the surrounding cytoplasmic island. (B) Variation of the width of each compartment between two successive cycles, i and i+1. If the number of nuclei doubles then the total volume is conserved when w_i+1=2^−1/2 w_i. (C) Expanded view of three adjacent compartments. Mass is transferred within each compartment from the cytoplasm to the nucleus at a rate proportional to the coefficient k_in, and from the nuclear phase to the cytoplasmic at a rate proportional to the coefficient k_out. Cytoplasmic Bcd is transferred across compartments at a rate proportional to the coefficient Γ.

Figure 2

(A) Spatiotemporal evolution of nuclear Bcd predicted for D=3μm²/s, Γ=0.1 μm/s, k_in=108μm/s and k_out=20 μm/s (k_in/k_out=5). These parameter values produce Bcd dynamics that are consistent with the Bcd shape criterion (λ = 0.16) and the nuclear Bcd stability criterion (~9% accuracy). (B) Snapshots of the nuclear Bcd gradient used for the computation of the stability measure. The Bcd gradients are ~9% accurate over cycles 10–14. (C) Temporal evolution of nuclear Bcd concentration at dimensionless distance z=0.1, showing a rapid increase at the beginning of each cycle; nuclear Bcd then starts decreasing progressively as the volume occupied by the nuclear phase grows.

Figure 3

(A) Distribution of the acceptable values of the equilibrium constant for nucleocytoplasmic shuttling, K. (B) Distribution of the sampled acceptable values of Bcd diffusivity, D, during cycles 1–9.

Figure 4

Distribution of the acceptable values of the diffusivity (A), equilibrium constant for nucleocytoplasmic shuttling (B) and intercompartmental transport (C) for the model with the Bicoid production sources with different levels of localization (see text for details).