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First published online February 22, 2008
doi: 10.1242/10.1242/dev.012062

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Precision of the Dpp gradient

Tobias Bollenbach^1,*,, Periklis Pantazis^2,*,, Anna Kicheva^2,3,*, Christian Bökel^2,, Marcos González-Gaitán^2,3,¶ and Frank Jülicher^1,¶

¹ Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany.
² Max-Planck-Institute of Molecular Cell Biology and Genetics, Pfotenhauer Strasse 108, 01307 Dresden, Germany.
³ Department of Biochemistry, Sciences II, 30 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland.

View larger version (80K): [in this window] [in a new window]   Fig. 1. Morphogen gradient formation in the presence of cell-to-cell variability. (A) Cellular individuality in the wing disk. The stochastic concentrations of intracellular and extracellular molecules and different cell shapes affect morphogen transport and degradation, independently of the transport mechanism. (B) Gradient formation in a simplified geometry. Concentration profiles for constant (left) or fluctuating (right) morphogen influx j, effective D and k. Average cell diameter, a. (C) One-dimensional lattice model of the scenarios in A. The hopping rates and describe transport between cells n and n+1. They fluctuate about their mean value p0 because of cell-to-cell variability, as indicated by arrows of different widths. The degradation rate kn also fluctuates.

View larger version (56K): [in this window] [in a new window]   Fig. 2. Cell-to-cell variability in the source or target affects the gradient fluctuations differently. (A) Idealized situation with fluctuating production rate in the source (different colors), but identical target cells. Morphogen concentration c along a slice (dashed line) in the y-direction close to the source (bottom left), and at a larger distance from the source (bottom right). (B) Theoretical (x) for the situation in A based on the model in Fig. 1C in two dimensions. Here, j fluctuates, while D and k are constant. (x) decreases with increasing distance x from the source, as in A. (C) The opposite situation to A: different target cells, identical source. (D) Calculation corresponding to C based on the model in Fig. 1C in two dimensions. D and k fluctuate, j is constant. Following an abrupt decrease very close to the source, (x) increases with increasing x. (E) (x) when cell-to-cell variability affects both the source and the target, i.e. with fluctuating j, D and k. The effects in A and C are superimposed. A pronounced minimum of (x) at a finite distance from the source occurs. Whereas its location and the magnitude of (x) depend on parameter choice, the qualitative behavior of the curve is independent of these parameters for different noise intensities of the same order of magnitude. Parameters in B,D,E are D/a=7, D/D0=1, k/k0=1, j/j0=0.37 (for details, see continuum limit and Fig. S2 in the supplementary material).

View larger version (19K): [in this window] [in a new window]   Fig. 3. Role of tissue dimensionality for morphogen gradient precision. (A) In a one-dimensional chain of cells, morphogens cannot get beyond the defective cell (red), resulting in a large impact on the gradient. (B) In two-dimensional tissues, the effect of one defective cell is smaller because it can be bypassed (arrows) if morphogen transport is non-directional. (C) In three-dimensional tissues, the effect is even smaller because there are more paths to bypass the defective cell. (D) Logarithmic plot of theoretical (x) calculated for different dimensions, but otherwise identical parameters. Only the target cells are subject to cell-to-cell variability (solid lines), or both the source and the receiving cells (dashed lines). D/a=7, D/D0=0, k/k0=1, j/j0=0.37 (for details, see Fig. S3 in the supplementary material).

View larger version (58K): [in this window] [in a new window]   Fig. 4. Fluctuations of the GFP-Dpp gradient in the absence of endogenous Dpp for 30 gradients. Genotype: dppd8/dppd12; dppGal4/UAS-GFP-Dpp. (A,B) GFP-Dpp gradient (A, green) and PMad immunostaining (B, red) to identify the source boundary (dotted line), located at x=0 in C-F. Disk size L (yellow line). The white box indicates the square which is scanned to perform the measurement. Scale bars: 10 µm. Posterior, right. (C) GFP-fluorescence intensity (FI) cDpp(x) in the white squares shown in A as a function of the distance to the source. Two profiles (in the dorsal and ventral compartments) were recorded per disk. (D) Mean FI c-Dpp(x) at each position (yellow line) with error bars (black region surrounding the yellow line) showing the standard deviation. Red line, fit of the function C0exp(-x/Dpp) to the mean profile. The uncertainty of the FI leads to an uncertainty of the position where this FI is present. (E) Standard deviation of the FI. (F) Relative uncertainty of the FI for the ensemble of gradients. Note the qualitative similarity to Fig. 2E. Insets in C-F show corresponding results for the normalized Dpp profiles. Green horizontal bars in C-F, 5 cell diameters (13 µm).

View larger version (38K): [in this window] [in a new window]   Fig. 5. Precision of the PMad gradient. (A) Immunostaining for PMad (red). Same set of disks, orientation, source boundary and scale bar as in Fig. 4A. (B) PMad FI in individual nuclei as a function of the distance x from the Dpp source, normalized to the total FI. The average PMad profile (red line) is approximated by an exponential function c-PMad(x)=P0exp(-x/PMad) with PMad=25.0 µm=9.6 cells. Inset, uncertainty of the FI. (C) Normalized PMad correlated to non-normalized GFP-Dpp level in the same disk and for the same position. Inset, the same but normalized GFP-Dpp data. (D) Relative concentration uncertainty of PMad. Note the similarity to Fig. 4F. (E) Positional uncertainty of the PMad gradient as a function of the positional uncertainty of the non-normalized GFP-Dpp gradient, . Inset, same but with normalized GFP-Dpp data. and are proportional with a proportionality constant (slope of the red line) close to , non-normalized GFP-Dpp data, and , normalized. Green horizontal bars in B,D, 5 cell diameters (13 µm).

View larger version (34K): [in this window] [in a new window]   Fig. 6. Precision of the expression domain of the Dpp target gene sal. (A) Sal immunostaining. Same set of disks, orientation, source boundary and scale bar as in Fig. 4A. (B) The sal range x*=d+w was determined by fitting the sigmoidal function s(x)=s0(tanh((d-x)/w)+1)/2 (red line) to the Sal profile. Quantification and normalization as for GFP-Dpp (Fig. 4). (C) Sal profiles from the dorsal (black) and ventral compartments (blue). (D) Non-normalized Dpp concentration cDpp at x* from the same wing disks in which x* was determined. Dorsal profiles, black; ventral, blue. Green horizontal bars in B-D, 5 cell diameters (13 µm).

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