Noise drives sharpening of gene expression boundaries in the zebrafish hindbrain
2012; Springer Nature; Volume: 8; Issue: 1 Linguagem: Inglês
10.1038/msb.2012.45
ISSN1744-4292
AutoresLei Zhang, Kelly Radtke, Likun Zheng, Anna Q. Cai, Thomas F. Schilling, Qing Nie,
Tópico(s)Zebrafish Biomedical Research Applications
ResumoArticle25 September 2012Open Access Noise drives sharpening of gene expression boundaries in the zebrafish hindbrain Lei Zhang Lei Zhang Department of Mathematics, University of California, Irvine, CA, USA Center for Complex Biological Systems, University of California, Irvine, CA, USA Center for Mathematical and Computational Biology, University of California, Irvine, CA, USA Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong Search for more papers by this author Kelly Radtke Kelly Radtke Department of Development and Cell Biology, University of California, Irvine, CA, USA Search for more papers by this author Likun Zheng Likun Zheng Department of Mathematics, University of California, Irvine, CA, USA Center for Complex Biological Systems, University of California, Irvine, CA, USA Center for Mathematical and Computational Biology, University of California, Irvine, CA, USA Search for more papers by this author Anna Q Cai Anna Q Cai Department of Applied Mathematics, School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia Search for more papers by this author Thomas F Schilling Corresponding Author Thomas F Schilling Center for Complex Biological Systems, University of California, Irvine, CA, USA Department of Development and Cell Biology, University of California, Irvine, CA, USA Search for more papers by this author Qing Nie Corresponding Author Qing Nie Department of Mathematics, University of California, Irvine, CA, USA Center for Complex Biological Systems, University of California, Irvine, CA, USA Center for Mathematical and Computational Biology, University of California, Irvine, CA, USA Search for more papers by this author Lei Zhang Lei Zhang Department of Mathematics, University of California, Irvine, CA, USA Center for Complex Biological Systems, University of California, Irvine, CA, USA Center for Mathematical and Computational Biology, University of California, Irvine, CA, USA Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong Search for more papers by this author Kelly Radtke Kelly Radtke Department of Development and Cell Biology, University of California, Irvine, CA, USA Search for more papers by this author Likun Zheng Likun Zheng Department of Mathematics, University of California, Irvine, CA, USA Center for Complex Biological Systems, University of California, Irvine, CA, USA Center for Mathematical and Computational Biology, University of California, Irvine, CA, USA Search for more papers by this author Anna Q Cai Anna Q Cai Department of Applied Mathematics, School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia Search for more papers by this author Thomas F Schilling Corresponding Author Thomas F Schilling Center for Complex Biological Systems, University of California, Irvine, CA, USA Department of Development and Cell Biology, University of California, Irvine, CA, USA Search for more papers by this author Qing Nie Corresponding Author Qing Nie Department of Mathematics, University of California, Irvine, CA, USA Center for Complex Biological Systems, University of California, Irvine, CA, USA Center for Mathematical and Computational Biology, University of California, Irvine, CA, USA Search for more papers by this author Author Information Lei Zhang1,2,3,4,‡, Kelly Radtke5,‡, Likun Zheng1,2,3, Anna Q Cai6, Thomas F Schilling 2,5 and Qing Nie 1,2,3 1Department of Mathematics, University of California, Irvine, CA, USA 2Center for Complex Biological Systems, University of California, Irvine, CA, USA 3Center for Mathematical and Computational Biology, University of California, Irvine, CA, USA 4Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong 5Department of Development and Cell Biology, University of California, Irvine, CA, USA 6Department of Applied Mathematics, School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia ‡These authors contributed equally to this work. *Corresponding authors. Department of Development and Cell Biology, University of California, 4109 Natural Sciences II, Irvine, CA 92697-2300, USA. Tel.:+1 949 824 2479; Fax:+1 949 824 4709; E-mail: [email protected] of Mathematics, University of California, 540F Rowland Hall, Irvine, CA 92697-3875, USA. Tel.:+1 949 824 5530; Fax:+1 949 824 7993; E-mail: [email protected] Molecular Systems Biology (2012)8:613https://doi.org/10.1038/msb.2012.45 PDFDownload PDF of article text and main figures. Peer ReviewDownload a summary of the editorial decision process including editorial decision letters, reviewer comments and author responses to feedback. ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InMendeleyWechatReddit Figures & Info Morphogens provide positional information for spatial patterns of gene expression during development. However, stochastic effects such as local fluctuations in morphogen concentration and noise in signal transduction make it difficult for cells to respond to their positions accurately enough to generate sharp boundaries between gene expression domains. During development of rhombomeres in the zebrafish hindbrain, the morphogen retinoic acid (RA) induces expression of hoxb1a in rhombomere 4 (r4) and krox20 in r3 and r5. Fluorescent in situ hybridization reveals rough edges around these gene expression domains, in which cells co-express hoxb1a and krox20 on either side of the boundary, and these sharpen within a few hours. Computational analysis of spatial stochastic models shows, surprisingly, that noise in hoxb1a/krox20 expression actually promotes sharpening of boundaries between adjacent segments. In particular, fluctuations in RA initially induce a rough boundary that requires noise in hoxb1a/krox20 expression to sharpen. This finding suggests a novel noise attenuation mechanism that relies on intracellular noise to induce switching and coordinate cellular decisions during developmental patterning. Synopsis Fluctuations in morphogens often lead to gene expression domains with rough boundaries during development. A mechanism is described whereby intracellular noise can help coordinate cellular decisions during patterning and thereby sharpen expression boundaries. Fluorescent in situ hybridization reveals rough edges around gene expression domains during development of rhombomeres in the zebrafish hindbrain, in which cells co-express hoxb1a and krox20 on either side of a future boundary, and these domains sharpen within a few hours. Computational analysis of stochastic models demonstrates that the initial rough boundaries are caused by stochastic fluctuations in the morphogen, retinoic acid, which patterns the expression domains. Noise in gene expression induces switching between expression of one gene or the other to narrow the transition zone and enable sharpening of boundaries. Introduction A fundamental feature of developing systems is that cells sense their positions along morphogen gradients and respond collectively to form precise domains of target gene expression (Meinhardt, 2009; Wartlick et al, 2009). How do gene expression domains achieve such sharp boundaries? Cells near a future boundary experience fluctuations or 'noise' in: (1) morphogen concentration, due to varying synthesis and transport, (2) ability to respond, for example, due to differences in numbers of receptors, (3) transcription and translation rates of target genes, and (4) feedback (Kepler and Elston, 2001; Elowitz et al, 2002; Kaern et al, 2005). Various mechanisms have been proposed to attenuate these sources of noise to generate consistent gene expression domains in every individual. In spatial patterning systems, noise is generally considered as detrimental to the ultimate goal of the system. However, for systems without spatial constraints, noise can regulate biological switches between high and low gene expression states, and noise can be attenuated by an ultrasensitive signal (Hasty et al, 2000; Thattai and van Oudenaarden, 2002; To and Maheshri, 2010). Could similar switches be operating in spatial patterning systems? Bistability (distinct steady states of a regulatory gene network within a cell) can have a critical role in spatial patterning and lead to sharp borders between gene expression domains in deterministic models (Meinhardt, 1978, 1982; Ferrell, 2002; Lopes et al, 2008). Spatially constrained stochastic models, such as for segmentation of the Drosophila embryo, suggest that noise predominantly depends on transcription and translation dynamics of target gene expression (Holloway et al, 2011), but external fluctuations in signals also have an important role in these downstream responses (He et al, 2012). However, very few studies have addressed mechanisms of noise attenuation in the formation of gene expression boundaries in any system. Here, we investigate interactions between noise in a morphogen (i.e., retinoic acid—RA) and noise in its downstream, bistable regulatory gene network in boundary sharpening. RA specifies rough boundaries between segments (called rhombomeres) of the zebrafish hindbrain in a concentration-dependent manner, which subsequently become razor sharp (Giudicelli et al, 2001; Cooke and Moens, 2002; White et al, 2007; White and Schilling, 2008). Two genes downstream of RA, hoxb1a (r4) and krox20 (r3 and r5), cross-inhibit one another and auto-activate their own expression to form a bistable switch (Barrow et al, 2000; Giudicelli et al, 2001; Alexander et al, 2009). With a stochastic model that incorporates these interactions we estimate the switching probability between hoxb1a and krox20 expression at different RA concentrations based on an exponential function of Minimum Action Paths (MAPs) between stable and unstable states (Freidlin and Wentzell, 1998). Exploration of the stochastic models reveals that noise in the RA morphogen gradient can lead to rough gene expression boundaries initially, and that sharpening is driven by noise in the expression of hoxb1a and krox20, due to induced switching between expression of one gene and the other. These results reveal an unexpected positive role for noise in boundary sharpening that may be common for many patterning systems. Results hoxb1a and krox20 co-expression during rhombomere boundary sharpening To determine the temporal dynamics of hoxb1a and krox20 expression in the embryonic zebrafish hindbrain, we performed fluorescent in situ hybridization (FISH) analysis. Previous studies showed that initial boundaries of hoxb1a in r4 and krox20 expression in r3 and r5 are rough but become razor sharp between 10 and 14 h post fertilization (h.p.f.) (Figure 1A–F; Cooke and Moens, 2002; Cooke et al, 2005). Cells that find themselves on the wrong side of a boundary (i.e., surrounded by neighbors with a different pattern of gene expression) may go through a transient phase in which they express both genes and subsequently downregulate one or the other to enable sharpening (Schilling et al, 2001; Cooke and Moens, 2002). To quantify sharpness in krox20 expression, confocal stacks were collected for a minimum of 10 embryos at 6 different stages (between 10.7 and 12.7 h.p.f.) (Figure 1A–F) and the fluorescence was measured at different positions along the anterior-posterior (A-P) axis focusing on the r4/5 boundary (Figure 1G–I). This analysis demonstrated quantitatively how krox20 expression sharpens at rhombomere boundaries over time. Figure 1.Sharpening of gene expression boundaries in the zebrafish hindbrain. (A–F) Single confocal images of fluorescent in situ hybridization (FISH) for krox20 (red) mRNA, dorsal views, anterior to the left, between 10.7 and 12.7 h post fertilization (h.p.f.). (G–I) Fluorescence measurements at different positions along the anterior-posterior axis (X axis) at 11, 11.7, and 12.7 h.p.f. Lines represent four different samples. (J–L) Single confocal images of two-color FISH for hoxb1a (r4, red) and krox20 (r3 and r5, green). Insets show enlargements of cells co-expressing both (yellow). (M–O) Sample distributions of mis-expressing cells along the r4/5 boundary (black lines) between 10.7 and 12 h.p.f., anterior to the top. Cells mis-expressing krox20—green dots, hoxb1a—red dots and co-expressing cells—orange dots. Download figure Download PowerPoint To examine this more closely, we used confocal analysis and two-color FISH to colocalize hoxb1a and krox20 near the r3/4 and r4/5 boundaries at 20-min intervals between 10.6 and 12 h.p.f. (Figure 1J–L). hoxb1a expression is initiated broadly in the early gastrula (6.5 h.p.f.; Maves and Kimmel, 2005), and is preceded by its close relative hoxb1b, which is the first gene induced by RA in this system and activates hoxb1a transcription directly (McClintock et al, 2001). By 10.5 h.p.f. hoxb1a expression resolves into a strong r4 stripe 4–6 cells wide along the A-P axis while krox20 is expressed in flanking r3 and r5 stripes that overlap with hoxb1a at its edges (Figure 1J–L). Higher magnification images demonstrated that krox20 and hoxb1a mRNAs colocalize in many of these cells near future boundaries (insets) and occasional colocalization was observed as late as 12.0 h.p.f. This revealed an initial 'transition zone' containing a mixture of hoxb1a, krox20 and co-expressing cells that was ∼40 μm in length along the A-P axis and later reduced to 5–10 μm (1 cell diameter) by 12 h.p.f. Similar numbers of co-expressing cells were identified at 10.7 h.p.f. (average 7 cells, n=3) and 11.3 h.p.f. (average 7.3 cells, n=3), however, by 12 h.p.f. the number of co-expressing cells decreased (average 3, n=3). Conversely, the percentage of mis-specified cells that were expressing both genes increased from 36% and 34% at 10.7 and 11.3 h.p.f. to 56% at 12 h.p.f. The co-expressing cells were more prevalent in the r5 domain at both 10.7 and 11.3 h.p.f. (Figure 1M and N) while this bias was not observed at 12 h.p.f. (Figure 1O). Induction of stripes of hoxb1a and krox20 expression by RA requires bistability and initial expression of Hoxb1 RA activates hoxb1a expression in r4 (directly) and krox20 in r3 and r5 (indirectly through Vhnf1 and MafB) in a concentration-dependent manner (Niederreither et al, 2000; Begemann et al, 2001; Hernandez et al, 2004; Labalette et al, 2011). Our deterministic model is based on a previous continuum model of the RA signaling network that consists of diffusive extracellular and intracellular RA, and self-enhanced degradation through the enzyme Cyp26a1 (White et al, 2007), without inclusion of downstream signal responses (see Equation S1.1 in Supplementary information). In the new model, RA activates hoxb1a and krox20 expression, which in turn both positively regulate their own expression and negatively regulate each other (Barrow et al, 2000; Giudicelli et al, 2001; Alexander et al, 2009; Figure 2A). Such positive auto-regulation and mutual inhibition have been modeled and shown to result in only one gene remaining active in a particular cell (Meinhardt, 1978, 1982). Here, the dynamics of both genes are modeled using rate equations along with Hill functions for regulation, with RA as input (see Equation S1.2 in Supplementary information). Figure 2.Modeling induction of hoxb1a and krox20 expression by a gradient of retinoic acid (RA) in a noise-free system. (A) Diagram illustrating RA movement from extracellular [RA]out to intracellular [RA]in, self-enhanced degradation via Cyp26a1, and induction of hoxb1a and krox20 which undergo auto-activation and cross-inhibition. (B) In the absence of noise, a smooth RA gradient leads to sharp boundaries of gene expression—as long as there is a low initial level of hoxb1a (∼0.1). (C) Three-dimensional graph of krox20 (gk, Y axis) and hoxb1a (gh, Z axis) expression levels at different points along the RA gradient (X axis). The number of possible gene states is 5 (0<[RA]in<0.22), 3 (0.22<[RA]in 0.85) for a normalized RA concentration. (D) Phase diagram of Hoxb1 (red) and Krox20 (blue) activation illustrating effects of the initial level of Hoxb1 (Y axis) at different segmental positions (X axis). The initial level of krox20 is zero and the RA gradient used to generate the diagram is shown in (B). Download figure Download PowerPoint Exploration of the model reveals that the system robustly resolves into a striped pattern of gene expression with hoxb1a in r4 and krox20 in r3 and r5 (Figure 2B). This demonstrates that by simply including two bistable steady states and an anteriorly declining RA gradient, one can specify alternating gene expression patterns with sharp boundaries. Simulations in two dimensions show a similar striped pattern (Supplementary Figure S1). Auto-activation and mutual inhibition between Hoxb1 and Krox20 allow one to switch from the off to the on state, or vice versa, within a range of RA. In particular, at a low RA concentration (RA 0.084 in Figure 2D), it represses krox20 expression and activates its own expression in r4. Interestingly, hoxb1a is first expressed at an intermediate level of RA, not in the anterior hindbrain where RA concentrations are low. This reflects the presence of the initial low level of hoxb1a expression, which compensates for its weaker auto-activation than krox20. This is consistent with previous theoretical findings (Meinhardt, 1978, 1982) and experimental observations of extremely early onset of hoxb1a (and hoxb1b) expression during zebrafish gastrulation (McClintock et al, 2001; Maves and Kimmel, 2005). Noise in RA can generate rough boundaries of hoxb1a and krox20 expression Stochastic fluctuations in ligand-receptor binding and morphogen synthesis introduce noise into any morphogen gradient. To study the propagation of such noise and its influence on target gene expression in the hindbrain, we introduced spatial and temporal noise into the deterministic RA model: where [RA]out and [RA]in represent extracellular and intracellular RA concentrations, and , denote standard white noise in extracellular and intracellular RA concentrations, respectively. [Cyp([RA]in)] represents RA degradation by Cyp26 (see Section 1 in Supplementary information for the other parameters and boundary conditions). Because of inherent stochasticity in gene expression and other cellular components (Elowitz et al, 2002) in any gene regulatory network, we also include temporal noise in each gene equation: Here, gh and gk represent the concentrations of hoxb1a and krox20, respectively, and ω(t)ψ(t) represents white noise with amplitudes ah and ak, respectively (see Section 1 in Supplementary information for more details). To investigate the effects of noise on boundary sharpening we have varied each term in Equations (1) and (2) and modeled the outcome. First, if noise is only present in extracellular RA (i.e., ), due to fluctuations in environmental factors (e.g., availability of vitamin A) or in RA synthesis, then the boundaries between Krox20 and Hoxb1 expression domains are sharp from the outset (see Supplementary Figure S3). This is because RA induces Cyp26a1, creating 'self-enhanced degradation' feedback that makes signaling robust to fluctuations in RA (Eldar et al, 2003; White et al, 2007) resulting in a smooth gradient of intracellular RA concentration along the A-P axis. Consistent with this idea, simulations in which we have varied spatial noise demonstrate that self-enhanced degradation provides excellent noise attenuation for fluctuations in extracellular RA (see Supplementary Figures S3 and S4). In contrast, if noise is introduced into the intracellular RA concentration, [RA]in (i.e., ), for example, due to fluctuations in RA transport into cells, then boundaries between hoxb1a and krox20 never sharpen (Figure 3A). In this case, the r3 domain of krox20 expression expands as fluctuations in the RA gradient reach the threshold that induces krox20. Monte Carlo simulations indicate that there is a large variation in the distribution of gene expression around the r4/5 boundary over time when [RA]in is noisy (Figure 3B). Two-dimensional simulations also show that hoxb1a and krox20 expression domains initially form a rough r4/5 boundary, which does not sharpen (Figure 3C). An initial noisy distribution of hoxb1a expression at this boundary can also disrupt sharpening (see Supplementary Figure S5). However, if noise is only restricted to later hoxb1a and krox20 expression, and not local RA concentration (i.e., ), then boundaries tend to sharpen from the outset (Figure 3D and F) and Monte Carlo simulations confirm this prediction (Figure 3E). These results suggest that rough boundaries of gene expression between r4 and r5 arise due to noise in [RA]in or initial hoxb1a expression. Figure 3.Effects of noise either in the RA gradient or in hoxb1a/krox20 expression alone on boundary sharpening. (A–C) With noise in RA alone, boundaries are initially rough and never sharpen. (D–F) With noise in hoxb1a/krox20 expression alone boundaries start out sharp at the outset and remain sharp. (A, D) Single samples at three time points illustrating gene expression levels (Y axis) at different A-P positions in r3-5 (X axis). (B, E) Gene expression distributions (Y axis) at different positions relative to the r4/5 boundary (X axis). Solutions are at the scaled time T=50, which is typically long enough for simulations to reach steady state (1000 samples are taken to calculate the gene distributions). (C, F) 2D simulations at three time points showing the pattern of hoxb1a/krox20 gene expression around the r4/5 boundary (hoxb1a: blue; krox20: red). Download figure Download PowerPoint Noise in Hoxb1/Krox20 expression enables noise attenuation during boundary sharpening Rhombomeres form lineally-restricted compartments (Fraser et al, 1990; Jimenez-Guri et al, 2010) and single hindbrain cells can upregulate or downregulate their Hox expression according to their host environment/rhombomere (Trainor and Krumlauf, 2000; Schilling et al, 2001). This suggests that similar gene expression 'switches' occur in cells on either side of a noisy rhombomere boundary. For example, cells expressing krox20 isolated among neighbors expressing hoxb1a (Figure 1J–L) may downregulate the former and upregulate the latter, thereby attenuating the noise and sharpening the border. To study such switching from one stable gene expression state to another, we employed an MAP analysis based on the Wentzell–Freidlin theory of large deviation (Freidlin and Wentzell, 1998). This theory allows one to estimate the probability of a transition φ between two stable states X1, X2 in a stochastic dynamic system (e.g., with a form of Equation (2)). The most probable path φ* requires the least action and is called an MAP (see Section 2 in Supplementary information for more details). MAP analysis has been used primarily to model phase transitions between two states in stochastic chemical kinetics (E et al, 2004). Here, we adapt it to estimate the switching probability between two stable gene expression states. The likelihood that a system switches from X1 to X2 relies on its ability to pass through the unstable critical point Xc that lies between X1 and X2 along the path φ*. The distance |φ*(X1)−φ*(Xc)| is the minimum barrier to the stochastic transition from one state to the other. For a smooth RA gradient and a simple bistable gene expression state (Figure 2C), we calculate MAPs (E et al, 2004) at different levels of RA. At low RA concentration (e.g., at RA=0.1 μM), three MAPs connect each pair of stable states, with each MAP passing through one unstable critical point (Figure 4A). Based on MAP theory, the activation of Krox20 (Krox20-on) from a 'both-off' state requires less action (a lower barrier) than activation of Hoxb1, which helps explain why the r3 domain of Krox20 expression expands when noise increases in our models (Figure 3C). In contrast, at intermediate RA concentrations a single MAP connects the two steady states and the action to switching from Hoxb1-on to Krox20-on decreases from RA=0.5 to 0.8 μM (Figure 4A), indicating that it becomes easier to switch in this direction as RA increases. When RA levels are high, Krox20-on is the only stable state. Figure 4.Noise in hoxb1a/krox20 expression leads to boundary sharpening. (A) Minimum Action Paths (dash lines) at [RA]in=0.1, 0.5, and 0.8 (krox20-on: blue dot, hoxb1a-on: red dot, both-off: black dot, critical point: green dot). (B) Gene switching probability estimated by MAPs reveals that noise in gene expression can drive cells from co-expressing Hoxb1/Krox20 to uniform Krox20 expression when [RA]in is high, and this coincides with the results of Monte Carlo simulations. (C–E) With noise in both [RA]in and hoxb1a/krox20 expression, a transient noisy boundary becomes sharp over time: (C) single sample; (D) gene distribution at the r4/5 boundary (1000 samples are taken to calculate the gene distributions); (E) two-dimensional simulation at the r4/5 boundary (hoxb1a: blue; krox20: red). (F) Sharpness Index versus time. 'green dashed line': noise only in extracellular RA alone; 'black dashed-dotted line': noise in both extracellular and intracellular RA; 'magenta dotted line': noise in gene expression alone; 'blue solid line': noise interactions between RA and gene expression; 'red dashed line with green squares': mean value of the Sharpness Index for distributions of Krox20 obtained from the experimental data. The error bar represents the standard error of the mean. The times 11, 11.3, 11.7, 12, and 12.7 h.p.f. correspond 3, 4, 5, 6, and 8 somites, respectively, and are rescaled to 1, 9, 20, 29, and 50. Download figure Download PowerPoint To quantify such switching capability, we estimate the switching probability from X1 to X2 within a time interval [0, T] through an exponential of the minimal barrier: The switching probability from X2 to X1 is defined in a similar manner: We estimate the Hoxb1/Krox20 gene switching probabilities PH→K and PK→H using the MAP calculation of Equation 2 for a normalized RA concentration. Our models indicate that PH→K increases exponentially when [RA] is high and PK→H is low, and cells have a high probability of switching from Hoxb1 to Krox20 expression. On the other hand, Krox20 expression is more stable due to a cell's low switching (to Hoxb1 expression) probability (Figure 4B). Together, this suggests that noise in Hoxb1/Krox20 expression drives cells from occasionally co-expressing Hoxb1/Krox20 expression to a uniform Krox20 expression when RA concentrations are high, leading to a sharp boundary. In support of this analysis, direct Monte Carlo simulations of the gene system (2) of switching probability at the same time intervals provide similar MAP estimates based on Equations (3) and (4) (Figure 4B; see Section 2 in Supplementary information for more details). Thus, surprisingly, our models suggest that the combination of noise in both [RA]in and Krox20/Hoxb1 expression (i.e., ), synergize to reduce noise during boundary sharpening, at least at the r4/5 boundary (Figure 4C–E). Interestingly, the initial boundary (T=1) is established at 160±10 μm along the A–P axis and, following sharpening, the boundary is located at 144 μm (T=50) (Figure 4E). This suggests that sharpening preferentially drives cells near the initial, rough boundary to krox20 expression due to the irreversibility of gene switching. Similar directional boundary shifts in gene expression have also been observed in Drosophila (Jaeger et al, 2004). This fits well with our in vivo observation of a higher percentage of hoxb1a/krox20 co-expressing cells on the posterior side of the putative r4/5 boundary at 10.7 and 11.3 h.p.f. (Figure 1M and N), which might predict that the forming boundary shifts anteriorly. To quantify boundary sharpening more systematically, we define a 'Sharpness Index' (S), which resembles the standard deviation. To define S, we calculate the 'mean' location of the boundary between Hoxb1 and Krox20 expression domains as the intersection of their distributions at 50% of the normalized value. Using this definition, we can measure the roughness of the boundary, that is, deviation from a
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