Extended Data Fig. 4: Mean transcription rate explains dynamic pattern establishment.

(a) A simple model to estimate protein accumulation from measured mean transcription rates. The mean transcription rate (left column), across space and time, is computed by normalizing the measured activity by the elongation time and applying a minor correction for the loop-insertion delay (<1 min). Horizontal white dashed lines mark the mitotic transition from NC13 to NC14. Protein accumulation (middle column) is modeled as the convolution of the transcription rate with a kernel that captures protein decay, diffusion, and a combined time delay due to mRNA export, translation, and nuclear import. This model involves three free parameters: protein lifetime, diffusion constant, and time delay (see panel b), which were optimized to minimize the mean squared error relative to gap gene protein patterns measured via antibody staining of carefully staged embryos (right column; Dubuis et al., 2013). Minor residual differences between model and experiment can be attributed to staging inaccuracies or, in the case of hb, to unmodeled maternal mRNA contributions (see c). (b) Estimated model parameters for protein accumulation, as described in (a). Parameters were either fitted individually for each gene (colored bars) or globally across all genes (dashed lines, used for middle column in a). The inferred values are broadly consistent with previous estimates⁶⁷. (c) Quantitative comparison between modeled and measured protein patterns shown in (a). Absolute errors were computed across the patterns in space and time. The generally low errors confirm a strong match between model and experiment. For hb, larger discrepancies are observed in the anterior, likely due to maternal mRNA contributions not captured by the model. (d) Overall root mean square errors (RMSE) between the measured protein pattern (from antibody staining), the modeled protein pattern (with either individual or global parameters), and the transcription rate data. Protein models (individual: colored, global: light gray) yield low RMSE values (mean 0.09), while using the transcription rate directly leads to higher error (mean 0.23), mainly due to the time lag between transcription and protein accumulation. For reference, the RMSE between live and fixed (smFISH) mean profiles (see Fig. 1c) is shown as a dashed line at 0.08. (e) Structural comparison of modeled and measured protein patterns, assessing pattern shape and features (for example, peaks, boundaries). We computed the local spatiotemporal correlation (akin to a structural similarity index without luminance or contrast terms) using a 4-AP-bin (6% egg length) and 5-min window (black rectangle). Local correlations are generally high (near 1), indicating strong structural agreement. Discrepancies mainly occur during the first 20 minutes, when few stained embryos were available for accurate staging (see Dubuis et al., 2013). As in panel c, hb displays residual differences at later stages due to unmodeled maternal input. (f) Mean correlation across space and time between measured transcription rate (dark gray), modeled protein concentration (individual: color; global: light gray), and measured protein pattern. Correlation values were averaged over the full spatiotemporal pattern using the local measure from (e). The modeled protein patterns show high agreement with experimental data (mean correlation ≈ 0.9), while transcription rates alone correlate less well (≈0.6), consistent with expected temporal lag between transcription and protein levels.