A two-dimensional simulation model of the bicoid gradient in Drosophila

Abstract

Background: Bicoid (Bcd) is a Drosophila morphogenetic protein responsible for patterning the anterior structures in embryos. Recent experimental studies have revealed important insights into the behavior of this morphogen gradient, making it necessary to develop a model that can recapitulate the biological features of the system, including its dynamic and scaling properties.

Methodology/principal findings: We present a biologically realistic 2-D model of the dynamics of the Bcd gradient in Drosophila embryos. This model is based on equilibrium binding of Bcd molecules to non-specific, low affinity DNA sites throughout the Drosophila genome. It considers both the diffusion media within which the Bcd gradient is formed and the dynamic and other relevant properties of bcd mRNA from which Bcd protein is produced. Our model recapitulates key features of the Bcd protein gradient observed experimentally, including its scaling properties and the stability of its nuclear concentrations during development. Our simulation model also allows us to evaluate the effects of other biological activities on Bcd gradient formation, including the dynamic redistribution of bcd mRNA in early embryos. Our simulation results suggest that, in our model, Bcd protein diffusion is important for the formation of an exponential gradient in embryos.

Conclusions/significance: The 2-D model described in this report is a simple and versatile simulation procedure, providing a quantitative evaluation of the Bcd gradient system. Our results suggest an important role of Bcd binding to non-specific, low-affinity DNA sites in proper formation of the Bcd gradient in our model. They demonstrate that highly complex biological systems can be effectively modeled with relatively few parameters.

Figure 1. Simulated Bcd distributions in a Drosophila embryo.

A. A simulated embryo at nuclear cycle 14 showing the local total Bcd concentration [B_tot] (arbitrary units). The A–P position is shown as absolute distance x (in µm) from the anterior. The ratio of total Bcd molecules in the cortical layer to those in the inner part of the embryo is 1.88 at nuclear cycle 14 (see text for further details). B. A plot of local DNA-bound Bcd concentration [B_bound] (arbitrary units) within the cortical layer as a function of fractional embryo length x/L. In this and other figures presented in this report, [B_bound] at each A–P position represents the mean [B_bound] value of all cubes within the cortical layer of the embryo at that A–P position. C. Same as B except [B_bound] is on ln scale. Linearity of ln[B_bound] indicates an exponential Bcd protein gradient; see text and Fig. 2 legend for more information about Adjusted R ² values to further evaluate the quality of exponential fitting of the simulated data.

Figure 2. Stability of nuclear Bcd concentrations.

A. A plot of [B_bound] within the cortical layer as a function of x/L, at nuclear cycles 10–14. B. Same as in A, except now showing nuclear Bcd concentrations ([B_n][B_bound]/nuclear number; all in arbitrary units) within the cortical layer at nuclear cycles 10–14. While [B_bound] increases after each nuclear division (as seen in panel A), [B_n] remains stable in the simulated embryo (as seen in panel B). The length constant λ of the simulated [B_bound] within the cortical layer at nuclear cycles 10–14 is: 72, 79, 85, 89 and 92 µm, respectively. The Adjusted R ² values of exponential fitting of simulated [B_bound] during nuclear cycles 10–14 are, respectively: 0.9985, 0.9988, 0.9992, 0.9996 and 0.9998 for the fitting region of x/L = 0.2 to 0.7; and 0.9454, 0.9470, 0.9456, 0.9402, and 0.9296 for the fitting region of x/L = 0 to 0.7.

Figure 3. Scaling properties of the Bcd gradient.

A. A plot of [B_bound] (arbitrary units) within the cortical layer of two simulated embryos with distinct lengths (550 µm and 600 µm) as a function of absolute distance x (in µm) from the anterior. In this simulation, the amount of bcd mRNA is correlated with the volume of the embryo, i.e., the aggregate J value is correlated with embryo volume. B. Same as in A, except with the use of normalized A–P position x/L. Note the convergence of the two profiles. C. Noise (standard deviation divided by the mean) of [B_bound] within the cortical layer in 50 simulated embryos, plotted as a function of either absolute distance x (in µm) from the anterior (red, upper scale) or normalized A–P position x/L (blue, lower scale). Error bars are from bootstrapping. The lengths of simulated embryos are variable and follow normal distribution with a mean of 550 µm and standard deviation of 20 µm. In this plot (as in panels A and B), the total amount of bcd mRNA is correlated with embryo volume. D. Same as C, except that there is no correlation between the total amount of bcd mRNA and embryo volume. Color codes are the same as in C.

Figure 4. Systematic evaluation of parameter values on system behavior.

3-D presentation of parameter space satisfying each of the three criteria: [B_n] stability as measured by g (panel A), gradient shape as measured by length constant λ at nuclear cycle 14 (panel B), and cortical enrichment as measured by the ratio of total Bcd molecules in the cortical layer to those in the inner part of the embryo at nuclear cycle 14 (panel C). In our systematic sampling, we tested all possible parameter value combinations (within the tested ranges at the tested increments) and the simulated results are evaluated according to the three criteria. Also see Fig. S2 for presentations showing the effects of changes in individual parameters while holding the other two at set values used in the main model simulation.

Figure 5. The effects of K_A on simulated Bcd gradient properties.

A. A plot of [B_n] at nuclear cycles 10–14 in a simulation in which K_A = 10⁷ M⁻¹. This figure is the same as Fig. 2B, except the difference in K_A values used in simulations (K_A = 5×10⁵ M⁻¹ for Fig. 2B). Note the instability of [B_n] profiles between different nuclear cycles (g = −0.3557) and a reduction in length constant (λ/L = 0.12 at nuclear cycle 14) in this simulation. B. A simulated embryo showing [B_tot] at nuclear cycle 14. This figure is the same as Fig. 1A except that different K_A values are used in simulations (K_A = 10⁴ M⁻¹ for this figure and K_A = 5×10⁵ M⁻¹ for Fig. 1A). Note a lack of cortical layer enrichment of Bcd molecules in this simulation, with a ratio of total Bcd molecules in the cortical layer to those in the inner part of the embryo, Ratio = 0.6328.

Figure 6. Simulating the effects of the dynamics of bcd mRNA distributions.

A. A simulated embryo at nuclear cycle 14 showing the [B_tot] (arbitrary units). Although the profiles of the redistributed bcd mRNA within the cortical layer have been reported , their precise functions have not been well characterized. Here we approximate the redistributed bcd mRNA profile during nuclear cycles 10–14 using two phases of decreases along the projected A–P axis (from the anterior, the relative bcd mRNA concentration within the cortical layer decreases linearly to 0.15 at x/L = 0.3 and further drops linearly to 0 at x/L = 1). Parameter values for simulations described in this figure are: D = 2 µm²s⁻¹, ω = 0.0005 s⁻¹, K_A = 2×10⁵ M⁻¹, and [DNA_sites]₁₀ = 3×10⁻⁷ M. B. Same simulation as in A, showing the stability of [B_n] (arbitrary units) within the cortical layer during nuclear cycles 10–14. C. Same as in B, except with a progressive reduction (at selected intervals shown for clarity) of diffusion constant D (2, 1, 0.2, and 0.1 µm²s⁻¹) to determine the role of Bcd protein diffusion in gradient formation. [B_bound] within the cortical layer is normalized to 1 at x/L = 0 to facilitate the comparison between results obtained under different simulation conditions. D. Same as in C, except the normalized [B_bound] is on ln scale. Linearity of ln[B_bound] indicates an exponential Bcd protein gradient. See Fig. S4A for a plot of Adjusted R ² values as a function of D.