Gradient formation is a fundamental patterning mechanism during embryo development, commonly related to secreted proteins that move along an existing field of cells. Here, we mathematically address the feasibility of gradients of mRNAs and non‐secreted proteins. We show that these gradients can arise in growing tissues whereby cells dilute and transport their molecular content as they divide and grow, a mechanism we termed ‘cell lineage transport.’ We provide an experimental test by unveiling a distal‐to‐proximal gradient of Hoxd13 in the vertebrate developing limb bud driven by cell lineage transport, corroborating our model. Our study indicates that gradients of non‐secreted molecules exhibit a power‐law profile and can arise for a wide range of biologically relevant parameter values. Dilution and nonlinear growth confer robustness to the spatial gradient under changes in the cell cycle period, but at the expense of sensitivity in the timing of gradient formation. We expect that gradient formation driven by cell lineage transport will provide future insights into understanding the coordination between growth and patterning during embryonic development.
Gradients of secreted proteins have been repeatedly observed during embryonic development in a variety of multicellular organisms. By eliciting different responses according to concentration thresholds, gradients of secreted molecules (morphogens) have been proposed as a morphogenetic mechanism that can pattern the early embryo (Wolpert, 1969; Driever and Nusslein‐Volhard, 1988a; Gurdon and Bourillot, 2001; Lawrence, 2001). Commonly, morphogen gradient formation dynamics involves molecular transport mechanisms such as passive diffusion or active transport through vesicles, among others. Moreover, several data point towards the importance of cell‐driven molecular transport mechanisms for gradient formation during morphogenesis (Zhu and Scott, 2004). Specifically, it has been shown that an mRNA gradient can be formed through the mRNA decay in tissues with polarized growth (Dubrulle and Pourquié, 2004). However, a full characterization, involving mathematical analyses and further experimental evidence of the dynamics of gradient formation based on cellular transport processes, is missing. Specifically, the implications of cell division and cell growth dynamics in the formation and shaping of gradients of non‐secreted proteins and mRNAs have not been studied yet.
In order to address these issues, we have used an in silico approach that characterizes a basic scenario for the formation of gradients of mRNAs and non‐secreted proteins in growing tissues. In addition, we have exemplified such gradient formation through novel data on a graded expression of the homeobox transcription factor Hoxd13 in the developing vertebrate limb.
We have proposed a mathematical model in which cells divide and grow, diluting and transporting their molecular content, a mechanism we termed ‘cell lineage transport.’ In the model, only some of the dividing cells, located at one end of the growing tissue, transcribe the mRNA, whereas protein translation occurs in all cells containing mRNA. In addition, cells divide and grow forming an elongated distribution of cell descendants along the same spatial axis.
Our computational and analytical results indicate that mRNA gradients driven by cell lineage transport can be formed, which in turn always create gradients of non‐secreted proteins (Figure 2A–C). In addition, we find that gradients of non‐secreted proteins can be formed in the absence of graded distributions of mRNA (Figure 2A). Our results show that longer cell cycles elicit steeper gradients (Figure 2C), whereas longer molecular half‐lives make gradients shallower (Figure 2A and B).
All these gradients can be characterized by two overlapping, but distinct in size, spatial domains (Figure 2D). The large domain is the spatial region over which the gradient is formed. The small domain is the region where cells transcribe mRNA. Importantly, these two domains can be detected experimentally, through the exonic/protein and intronic spatial expressions, respectively (Dubrulle and Pourquié, 2004). Our model indicates that the differences in size between the exonic and intronic domains of expression are controlled by the ratio between the mRNA half‐life and the duration of cell cycle.
Our results support that the molecular turnover becomes essential in forming a gradient when cell proliferation occurs only in a pool of mRNA transcribing cells, as previously reported (Gaunt et al, 2003; Dubrulle and Pourquié, 2004). On the other hand, when all cells proliferate, as in cell lineage gradients, clonal dilution alone can create gradients of very stable molecules. Through our theoretical analysis, we have uncovered that these diverse tissue dynamics elicit gradients with distinct profiles and thus different properties. Cell lineage gradients are characterized by power‐law decays, whereas gradients formed by molecular degradation in non‐proliferating cells exhibit an exponential profile. Nonlinear growth and clonal dilution in cell lineage gradients make their spatial extension less sensitive to alterations in the duration of the cell cycle, but at the expense of more variable timing in gradient formation.
To find an example of an mRNA gradient formed by clonal dilution, we focused on the early stages of development of the chick limb, as it is characterized by the outgrowth and elongation of the initial limb bud through evenly proliferating cells (Hornbruch and Wolpert, 1970; Sun et al, 2002). Moreover, cell tracing experiments have shown that the spatial distribution of descendants of cells located distally at early stages form an elongated shape along the proximo‐distal axis and correlates with the dynamical expression of Hoxd13 (Vargesson et al, 1997). We have found that Hoxd13 exonic expression spans more proximally than its intronic expression, which is located at the most distal end (Figure 6A). Quantitative PCR and fluorescent in situ hybridization confirm the graded expression of Hoxd13 along the proximo‐distal axis (Figure 6B and C). Altogether, our results indicate that Hoxd13 expression in the chick limb bud is graded, suggesting, at the same time, that cell lineage transport is the mechanism driving such gradient formation.
One of the interesting features of the cell lineage transport mechanism resides in its ability for generating gradients of non‐secreted molecules, such as receptors, transcription factors and second messengers (mRNA, microRNA, etc.), indicating that not only secreted proteins, but also non‐secreted proteins, should be the focus of studying biologically relevant graded molecular distributions. In addition, cell lineage transport could also participate in the formation of gradients of secreted proteins. As gradients driven by cell lineage transport are formed as tissue grows, linking the timing of mitotic divisions with the steepness of the gradient, cell lineage transport gradients might represent a mechanism that coordinates the dynamics of patterning and growth.
Gradients of mRNAs and non‐secreted proteins can arise in growing tissues whereby cells dilute and transport their molecular content as they divide and grow, a mechanism we termed ‘cell lineage transport.’
The dilution and nonlinear growth involved in the cell lineage transport mechanism elicit power‐law gradients and can confer spatial robustness under changes in the cell cycle period, but at the expense of sensitivity in the timing of gradient formation.
Hoxd13 forms a distal‐to‐proximal gradient in the vertebrate developing limb driven by cell lineage transport.
Molecules distributed along gradient profiles have been repeatedly observed during embryonic development in a variety of multicellular organisms. Known examples include, in Drosophila, bicoid, decapentaplegic (dpp), hedgehog and wingless (Driever and Nusslein‐Volhard, 1988a, 1988b; Teleman and Cohen, 2000; Tabata, 2001; Martinez‐Arias, 2003; Ephrussi and St Johnston, 2004); in Xenopus, Activin (Green, 2002); in mouse and chick, fibroblast growth factor 8 (fgf8) (Dubrulle and Pourquié, 2004); and Auxin in Arabidopsis (Bhalerao and Bennett, 2003). Gradients of secreted molecules (morphogens) have been proposed as a morphogenetic mechanism that can pattern the early embryo by conferring positional information to the cells, eliciting different responses according to concentration thresholds (Turing, 1952; Wolpert, 1969; Driever and Nusslein‐Volhard, 1988b; Lawrence and Struhl, 1996; Gurdon and Bourillot, 2001; Lawrence, 2001). Indeed, gradients of the transcription factor bicoid in the Drosophila syncytium and the secreted protein dpp have been reported to act in such a morphogen‐like manner (Driever and Nusslein‐Volhard, 1988b; Teleman and Cohen, 2000; Ephrussi and St Johnston, 2004).
The most widely recognized mechanism for protein gradient formation is the secretion of a protein that passively diffuses away from its source of production (Crick, 1970). However, several concerns have been raised about the ability of a protein to effectively form a gradient through passive diffusion alone because diffusion may be hampered by the insolubility of ligands and their interaction with receptors (Kerszberg and Wolpert, 1998; Teleman et al, 2001; Lander et al, 2002; Vincent and Dubois, 2002; Kruse et al, 2004; Mann and Culi, 2005). As a result, and according to several observations (e.g., the intracellular accumulation of ligands in vesicles), active protein transport mechanisms have been proposed such as transcytosis (Entchev et al, 2000; Strigini and Cohen, 2000; Pfeiffer et al, 2002), cytoneme‐mediated (Ramirez‐Weber and Kornberg, 1999), and vesicle or lipoprotein transport (Greco et al, 2001; Panakova et al, 2005; Tanaka et al, 2005). These transport mechanisms have been also analyzed computationally and mathematically (Kerszberg and Wolpert, 1998; Eldar et al, 2002, 2003; Lander et al, 2002; Kruse et al, 2004; Bollenbach et al, 2005; Shimmi et al, 2005), generating new perspectives on the role of diffusion and on the nature and formation of molecular gradients.
The common denominator of all the above‐mentioned gradient formation dynamics is the spatial transport of the protein by means of a specific molecular mechanism (e.g., passive diffusion, active transport through vesicles, etc.) over the underlying field of cells. Other transport mechanisms could involve cells acting as active molecular carriers through cell proliferation, cell growth, or cell migration (Lecuit and Cohen, 1998; Tabata, 2001; Teleman et al, 2001). For example, it has been found that cells dividing and leaving the wingless expression domain in Drosophila embryos retain the inherited wingless protein in secretory vesicles and carry them over distances of several cell diameters (Pfeiffer et al, 2000). Furthermore, it has been proposed that molecular decay coupled to these transport mechanisms could elicit molecular gradients (Gaunt et al, 2003; Dubrulle and Pourquié, 2004; Gaunt, 2004). Indeed, it has been shown that mRNAs can be distributed in a gradient profile formed by the mRNA decay in tissues with polarized growth (Dubrulle and Pourquié, 2004).
All these data point towards the importance of cell‐driven molecular transport mechanisms for gradient formation during morphogenesis (Zhu and Scott, 2004). However, a full characterization, involving mathematical analyses and further experimental evidence, of the dynamics of gradient formation based on such transport processes is missing. Specifically, the implications of cell division and cell growth dynamics in the formation of gradients of non‐secreted proteins and mRNAs have not been studied yet. Herein we focus on the dilution and transport of mRNAs and non‐secreted proteins driven by cell division and cell growth dynamics, a mechanism we call ‘cell lineage transport.’ Gradients driven by ‘cell lineage transport’ emerge under several specific conditions, which we will explore in this paper, involving particular dynamics of tissue growth coupled with a local source of mRNA transcription. To characterize this mechanism for gradient formation, we introduce a computational model, supporting it with novel in vivo data on mRNA gradients. In addition, we extend our model to gradients formed by molecular decay in growing tissues (Dubrulle and Pourquié, 2004), which do not involve clonal dilution, comparing them with gradients driven by cell lineage transport. Our mathematical analysis shows that both mRNAs and non‐secreted proteins can form a gradient through cell lineage transport and unveils the dynamics and features of their gradient profiles.
A framework for mRNA and protein gradient formation through cell lineage transport
We characterized a basic scenario for the formation of gradients of mRNA and non‐secreted proteins through cell lineage transport (Figure 1). In this scenario, when a cell divides, the daughter cells inherit, on average, half the amount of mRNA and protein of its progenitor, diluting its molecular content. Cell descendants rapidly grow after dividing to reach the diameter l of the progenitor cell, carrying their molecular content along the same spatial axis.
In our model, we also assumed that only a group of N cells situated in one of the tips of the growing tissue (hereafter called ‘domain of transcription (DoT)’) transcribes mRNA, such that a constant amount of mRNA is maintained in the transcribing cells. In addition, protein is translated in all cells containing mRNA proportionally to their amount of mRNA. The mRNA and protein turnovers were also included, as indicated by their half‐lives (Hm and Hp for the mRNA and protein, respectively).
In order to characterize this gradient formation dynamics, we elaborated on a novel computational model, using a time‐discrete modeling approach, which could include cells dividing in synchrony (Supplementary information).
According to the dynamics described, the molecular content in a cell becomes spatially transported when the cell is displaced and it evolves over time due to dilution, molecular decay, protein translation and mRNA transcription. Therefore, in the presence of a localized source of mRNA with proliferating and growing cells (the DoT), we expected the formation of mRNA and non‐secreted protein gradients.
mRNA and protein gradients formed by cell lineage transport
Our computational analysis showed that, following the cell lineage transport dynamics, mRNA gradients with a maximum at the DoT and a decreasing minimum at the opposite end of the growing tissue are formed (Figure 2A–C). Note that these mRNA cell lineage gradients are characterized by two overlapping, but distinct in size, spatial domains (Figure 2D). The large domain is the spatial region over which the gradient is formed. The small domain is the DoT. Importantly, these two domains can be detected experimentally (Dubrulle and Pourquié, 2004). Transcription synthesizes from DNA premature mRNA, which contains both exonic and intronic sequences. These transcripts are then spliced such that mature mRNA, which contains only exonic sequences, is formed. Therefore, probes directed against intronic sequences detect the nascent transcripts and can be used to mark the DoT. In contrast, the exonic expression domain spans over those cells containing mature mRNA above a threshold, detecting the spatial region where the gradient is formed (Figure 2D). Therefore, in vivo, cell lineage gradients will exhibit graded exonic expression domains, which span more broadly than the intronic domain.
Our mathematical analysis indicated that mRNA cell lineage gradients have an average spatial profile, which is steepest at times just before the next cell division occurs (Supplementary information). Specifically, mRNAs have a power‐law profile, with no characteristic length, due to the nonlinear growth of the tissue (Materials and methods). Neither the level of synchrony of the cell cycle between cells, nor fluctuations in the duration of the cell cycle and molecular amount inherited by each cell descendant alters this profile significantly (Supplementary information).
We obtained similar dynamics for non‐secreted proteins, which also become distributed along gradients (Figure 2A–C). Experimentally, these gradients will show a protein expression domain, spanning over the whole gradient, larger than the intronic domain of expression. Our analysis indicated that the average gradient profile for proteins is characterized by a combination of power‐like decays modulated by the mRNA and protein half‐lives (Materials and methods). In the absence of an mRNA distribution (i.e. the mRNA is located only at the DoT), non‐secreted proteins can form a gradient (Figure 2A; Materials and methods). In addition, when the mRNA forms a gradient, the protein is always distributed in a graded profile (Figure 2B).
In our model, cells reach a specific position outside the DoT after an average constant number of divisions, independently of how fast the cell cycle is. Thus, at this spatial position, the molecular content has decayed through dilution in the same proportion for fast and slow cell cycles. However, cells with faster cell cycles reach this position earlier and, as molecules degrade over time, contain more molecular content. Accordingly, our results showed that mRNA and protein gradients become steeper as the mRNA half‐life is reduced or the cell cycle lasts longer (Figure 2A and C). In addition, longer protein half‐lives make protein gradients shallower (Figure 2A and B; Materials and methods). Thus, extremely short mRNA half‐lives compared to the cell cycle do not give rise to mRNA gradients, but protein gradients can still be formed if the protein is stable enough (Figure 2A).
Spatial properties of gradients driven by cell lineage transport
Molecular gradients in developing embryos should vary significantly in order to elicit different responses and thus be biologically relevant. Specifically, gradients with 10‐fold variations along the overlying field of cells have been measured in different scenarios (Entchev et al, 2000; Houchmandzadeh et al, 2002; Dubrulle and Pourquié, 2004). Accordingly, we focused on the features of cell lineage gradients with 10‐fold decays and defined the spatial extent of the gradient (L10m,p) as the space occupied by all those cells with an amount of mRNA/protein equal to or higher than 10% of the amount at the DoT (Figure 2D; Materials and methods). Note that, as indicated, the exonic/protein expression domain will span over this domain, whereas the intronic expression will extend over the DoT. Thus, by mathematically computing the ratio between the spatial extent of the mRNA gradient and the size of the DoT (L10m/LI; hereafter called differential expression), we could characterize the differential exonic and intronic domains of expression that are expected for cell lineage gradients (Table I). When only the cells at the DoT contain mRNA and, hence, no mRNA gradient is formed, there is no differential expression (L10m/LI=1) and the exonic and intronic domains of expression will span over the same spatial region.
As shown in Table I, the differential expression depends on the parameter γm≡Hm/Cc alone, which we defined as the ‘relative half‐life’ of the mRNA, as it measures the mRNA half‐life compared to the duration of the cell cycle (i.e. large/small relative half‐lives correspond to molecular half‐lives much longer/shorter than the cell cycle). Our results indicated that the differential expression increases with the relative half‐life until it is saturated (Figure 3A). Even if the mRNA half‐life is rather shorter than the cell cycle length, a differential expression can arise (Figure 3A–C). For molecular decays that are not significant, mRNA gradients are still formed owing to the process of dilution inherent in cell proliferation. In this case, the gradients have a maximum extension proportional to the size of the DoT (Figure 3A). In addition, long‐range mRNA gradients can arise for a wide range of values of the relative molecular half‐lives (Figure 3A–C).
Our results indicated that protein gradients, arising from either mRNA gradients or non‐graded mRNAs, behave in a similar manner qualitatively and thus can be characterized equivalently (Materials and methods).
Temporal properties of gradients driven by cell lineage transport
As gradients created by cell lineage transport are formed as more and more cycles of cell division occur, and thus as a function of time, it was also important to know how many cell divisions and how long these gradients need to be formed (Materials and methods).
We found that the average number of cell divisions required to form a gradient with a 10‐fold decay is rather small and saturated for large relative half‐lives (Table I and Supplementary Figure S2A). Our results also indicated that the time gradients take to be formed ranges from minutes to hours depending on the specific values of the molecular half‐lives and the duration of the cell cycle (Table I and Figure 4A). This time depends only on the duration of the cell cycle and thus on how fast the tissue grows for large relative half‐lives, whereas the molecular half‐life controls the time of gradient formation for small relative half‐lives (Figure 4A).
Our numerical results showed that gradients formed by cell lineage transport are maintained as long as the tissue is growing and the cell lineage transport is active: once the tissue stops growing, no transport or dilution takes place. Thus, these gradients disappear in a time period controlled by the molecular dynamics alone.
Cell lineage transport gradients can be formed within large parameter ranges that are biologically relevant
We computed for which parameter values mRNA and protein gradients with a 10‐fold decay over a spatial region at least 50% larger than the DoT and not requiring many cell divisions can be produced. We found that mRNA and protein gradients satisfying these conditions can arise in a very large region of the parameter space, which extends to smaller relative half‐lives if the molecular content varies largely (100‐fold; Figure 4B). In the case of proteins, the ranges of relative half‐lives that can elicit protein gradients are much larger, because, as indicated, protein gradients are created even if either the mRNA or the protein relative half‐life is very small.
In eukaryotic cells, the half‐life of mRNAs varies between seconds and hours, whereas the time between divisions varies between minutes and several hours (Peltz et al, 1991; Sachs, 1993; Ross, 1995; Varshavsky, 1996; Figueroa et al, 2003). As shown in 3, Figure 4B, our results indicated that these biological timescales allow for the formation of mRNA and protein gradients by cell lineage transport in several developmental contexts, both for very short and long molecular half‐lives and cell cycles.
Molecular dilution can confer robustness to the spatial gradient
We analyzed a particular case of the model in which only the mRNA transcribing cells could divide (Supplementary information). For this tissue dynamics, cells outside the DoT just transport their molecular content when being displaced by proliferating cells at the DoT. As before, we considered that the mRNA and the protein are degraded, and the protein can be translated in cells outside the DoT. However, in contrast with the dynamics described so far, this case scenario did not involve clonal dilution, as cells outside the DoT cannot divide.
Our analysis revealed that the molecular turnover becomes essential for the formation of gradients in this tissue dynamics, as previously proposed (Dubrulle and Pourquié, 2004). As this scenario applies to fgf8 gradients formed by molecular decay in the presomitic mesoderm of chick and mouse embryos (Dubrulle and Pourquié, 2004), we confirmed that our model for this tissue dynamics could appropriately reproduce these experimental data (Supplementary information).
An analysis of the gradients arising when only the cells at the DoT divided revealed that the absence of proliferating cells outside the DoT, and thus the absence of clonal dilution, changes many of the properties of the molecular gradients. Owing to the linear growth involved in this tissue dynamics, the gradient profile becomes exponential with a characteristic length proportional to the relative molecular half‐lives and to the size of the DoT (Supplementary information). Hence, in this case, the length of the gradient and the number of divisions can increase indefinitely as the relative half‐life augments (Figure 3D and Supplementary Figure S2B). Compared to gradients driven by cell lineage transport, the differential expression is larger and a higher average number of cell divisions per cell and longer times are required (Figures 3D and 4C, and Supplementary Figure S2B). Accordingly, when taking into account that cells can divide only a limited number of times (Conlon and Raff, 1999), we obtained that above a maximum relative half‐life the molecular content spans too homogeneously and does not form a gradient (Figure 3E and F, and Supplementary Figure S2B). Therefore, in the absence of successive dilutions, the molecular half‐lives cannot be arbitrarily long compared to the cell cycle duration in order to form a gradient with a relevant decay, and thus gradient formation occurs in a smaller parameter region (Figure 4D).
Robustness of gradients can be analyzed through its sensitivity to parameter changes (Eldar et al, 2003; Bollenbach et al, 2005; Materials and methods). Specifically, we focused on the sensitivity of the differential expression and the timing of gradient formation to variations in the duration of the cell cycle, as changes in the temperature at which an embryo is developing alter the duration of the cell cycle but do not necessarily induce patterning defects. Our results showed that each mechanism exhibits very different behaviors. First, gradients formed by cell lineage transport, involving repeated dilution owing to successive cell divisions, have a less sensitive differential expression when the cell cycle is altered (Figure 5A and B). Specifically, the differential expression and thus the average number of cell divisions involved in the gradient become robust for large relative half‐lives (Figure 5A; data not shown). Second, for gradients formed when only the cells at the DoT divide, the time of formation is independent of the duration of the cell cycle (Table I, Figure 4C and Supplementary information), in contrast with the timing of cell lineage gradients, which is very sensitive to variations in the cell cycle (Figure 5C). Note that at large relative half‐lives, whereas the differential expression of gradients driven by cell lineage transport becomes robust, their timing of formation becomes extremely sensitive to variations in the cell cycle (Figure 5A and C).
Therefore, for gradients involving nonlinear growth with repeated divisions and dilutions, our analysis revealed that the differential spatial domains of expression become less sensitive to alterations in the duration of the cell cycle, but at the expense of more variable timing in gradient formation. In contrast, gradients not involving repeated dilution of the molecular content have a fixed timing of gradient formation but a non‐robust differential expression for changes in the duration of the cell cycle.
A Hoxd13 gradient in the vertebrate limb created by cell lineage transport
Until now, no mRNA gradient involving clonal dilution and transport of the molecular content driven by evenly proliferating cells has been reported in a developing embryo. Therefore, we searched for an example of a developmental scenario where cell lineage transport could elicit a gradient.
The early stages of the chick limb are characterized by the outgrowth and elongation of the initial limb bud. During these early stages, it has been readily demonstrated that cells proliferate evenly all over the limb bud (Hornbruch and Wolpert, 1970; Vargesson et al, 1997; Sun et al, 2002). Moreover, cell tracing experiments have shown that the spatial distribution of descendants of cells located distally at early stages form an elongated shape along the proximo‐distal axis (Supplementary Figure S4A; Vargesson et al, 1997; Dudley et al, 2002). Therefore, the vertebrate limb bud has appropriate morphogenetic dynamics to analyze if mRNA gradients are formed during early development according to cell lineage transport dynamics involving clonal dilution.
Hox genes are known to play critical roles during vertebrate limb patterning and outgrowth (Dolle et al, 1993; Davis and Capecchi, 1994; Morgan and Tabin, 1994; Fromental‐Ramain et al, 1996; Goff and Tabin, 1997; Heanue et al, 1997; Shubin et al, 1997; Zakany et al, 2004; Kmita et al, 2005). Hoxd13, in particular, is the most 5′‐Hoxd gene and is shown to be necessary for proper development of the autopod in vertebrates. Moreover, it has been shown that Hoxd13 can trigger the appearance of digits (Kmita et al, 2005). Hoxd13 shows a dynamic and restricted expression pattern in the developing limb (Nelson et al, 1996). Initially, Hoxd13 expression is restricted to the distal posterior region (Hamburger and Hamilton stages 19–22) and, later on, it extends anteriorly within the distal domain (Hamburger and Hamilton, 1951; Nelson et al, 1996). Importantly, cell fate analyses have revealed that the dynamic expression pattern of Hoxd13 can be understood in terms of the expansion of former Hoxd13‐expressing cells (Vargesson et al, 1997). For the purpose of this paper, we focused only on characterizing whether Hoxd13 expression is graded and driven by cell lineage transport. Owing to the correlation of its expression with autopod development, we expect that our characterization of Hoxd13 expression formation will be able to give some hints to future work on the specific function of this gene, which still requires further elucidation.
As shown in our previous analyses, cell lineage gradients require a DoT. We reasoned that in the limb bud the DoT should be located preferentially on the distal end (see Discussion). According to previous cell fate analyses on distally located cells (Supplementary Figure S4B; Vargesson et al, 1997), we inferred that cell descendants of the DoT would be located along the proximo‐distal axis, with older cells found more proximally, able to potentially elicit a distal‐to‐proximal cell lineage mRNA gradient. As previously described, if such a gradient is formed, the intronic domain of expression would be located at the distal end, whereas the exonic signal will be graded and will span over a domain much larger and extending more proximally than the intronic expression.
By reverse transcription–PCR (RT–PCR) analysis, we confirmed intronic expression of Hoxd13 in the distal region of the limb bud in both forelimbs and hindlimbs during stages 21–26 (Figure 6A and Supplementary Figures S3 and S4B; data not shown). This expression did not exhibit any significant graded profile (data not shown), discarding the potential emergence of an mRNA gradient through graded transcription. The width (along the proximo‐distal axis) of the intronic expression increased at stage 25, especially in the hindlimb (Figure 6A and Supplementary Figure S4B; data not shown).
Whole‐mount in situ hybridization depicted an exonic domain of expression much larger and spanning more proximally than the intronic domain of expression (Figure 6A and Supplementary Figure S4B), pinpointing where a gradient of Hoxd13 formed by cell lineage transport could be present. Specifically, our data revealed that exonic expression extends over the spatial region potentially occupied by former transcribing cells at the intronic domain (Materials and methods).
A graded expression of Hoxd13 was first confirmed qualitatively by in situ hybridization and semiquantitative RT–PCR, showing a higher level of exonic expression at the distal end, as expected (Supplementary Figure S4C; data not shown). Quantitative PCR (qPCR) on three (proximal, medial and distal) regions of the chick limb bud, normalized to the housekeeping gene Gapdh (Materials and methods), clearly evidenced such a gradient and indicated a 10‐fold decay between the most distal and medial regions (Figure 6B). Fluorescent in situ hybridization signal (Materials and methods), despite not being linearly proportional to the real mRNA gradient owing to enzymatic amplification, further confirmed the existence of a distal‐to‐proximal Hoxd13 gradient. We note that the graded fluorescent signal in anterior and posterior regions of distal and middle regions was rather similar (Figure 6C). From the quantitative data on the graded expression of Hoxd13 at stage 25 obtained by qPCR (Figure 6B) and using our mathematical model, we could estimate the molecular half‐life of Hoxd13. Our results indicate a range of relative half‐lives γm≈0.3–0.8 (Materials and methods), which for cell cycles between 8 and 12 h (Vargesson et al, 1997) corresponds to half‐lives in the range of 2.4–9.6 h. As expected, the differential size between exonic and intronic domains of expression at stage 25 also supported a relative half‐life within this range (Materials and methods).
Altogether, our results indicate that Hoxd13 expression in the chick limb bud is graded and pinpoints cell lineage transport as the mechanism driving such gradient formation. We expect future work to elucidate whether other Hox genes exhibit similar graded expressions and how limb development depends on them.
Gradients of secreted proteins transported through different molecular mechanisms have been extensively analyzed in developmental biology. Generally, these mechanisms of gradient formation do not take into account the spatiotemporal dynamics of cells or the mRNA dynamics (Zhu and Scott, 2004). Here we have shown a full mathematical framework, which for the first time takes into account the dynamics of cells, mRNAs and proteins, and we have provided new experimental evidence for a less understood mechanism of gradient formation that might prove to play an important role during embryo development. In this mechanism, cells act as the carriers of molecules such as mRNAs and non‐secreted proteins, and, through cell division, dilute this molecular content over their progeny. Our model shows how this cell lineage transport mechanism can elicit mRNA gradients, which in turn create gradients of non‐secreted proteins. In addition, our results indicate that cell lineage transport can account as well for the formation of non‐secreted protein gradients from local sources of protein translation not involving graded distributions of mRNA.
One of the interesting features of the cell lineage transport mechanism resides in its ability to generate gradients of non‐secreted molecules, such as receptors, transcription factors and second messengers (mRNA, microRNA, etc.), indicating that not only secreted proteins, but also non‐secreted proteins, should be the focus of studying biologically relevant graded molecular distributions. In addition, cell lineage transport could also participate in the formation of gradients of secreted proteins. Accordingly, the mathematical framework presented herein can be extended to secreted proteins by introducing the appropriate molecular transport dynamics (diffusion, vesicle trafficking, etc.).
Our results exemplify how cell proliferation and cell growth can elicit molecular gradients solely by diluting the molecular content, without requiring any molecular degradation. In contrast, our results showed that the molecular turnover becomes essential in forming a gradient when cell proliferation occurs only in a pool of mRNA transcribing cells, as previously reported (Gaunt et al, 2003; Dubrulle and Pourquié, 2004; Gaunt, 2004). Through our theoretical analysis, we have uncovered that these diverse tissue dynamics elicit gradients exhibiting very different profiles that lead to specific spatial and temporal sensitivities under changes of the duration of the cell cycle.
The present study has focused on a proliferating population of cells whose descendants become located one next to another creating an elongated structure. However, our mathematical framework could be extended to growing morphologies that involve cell rearrangements, intermingling or migration. In this case, the critical parameter characterizing the cell dynamics would be the timing of the spatial displacements of cells, which would involve the dynamics of the cell cycle. In addition, our mathematical framework could be extended to growing populations of cells that do not distribute along a specific spatial axis but which have a fixed DoT. In all these cases, we can expect the formation of gradients driven by cell lineage transport, as long as younger cells become located on average closer to the DoT than older cells. These gradients are expected to exhibit spatial fluctuations, like gradients emerging from non‐constant cell cycles. In all these scenarios, and in order to buffer such fluctuations, pattern formation could occur through the integration of the graded signal in neighboring cells.
The mechanism underlying cell lineage transport implies that gradients are formed only over the progeny of a pool of cells, that is, those initially at the DoT. In most invertebrate species, embryonic structures emerge from cells genealogically related, producing lineage‐based cellular patterning. In these cases, the formation of mRNA and protein gradients through cell lineage transport might be an important mechanism of pattern formation. In contrast, in early development of vertebrate embryos, cell dispersal is present in many instances, as revealed by clonal analyses in mouse, Xenopus, fish and chick embryos, leading to embryonic structures that arise from cells that do not descend from a common progenitor (Mathis and Nicolas, 2002). This implies that the formation of relevant cell lineage mRNA and protein gradients in vertebrate embryos might be restricted to very local structures, such as the limbs, for instance, and, possibly, not as common as in invertebrates.
An essential element for the formation of gradients driven by cell lineage transport is the presence of a local domain of mRNA transcription (the DoT) that contains proliferating cells. This domain might be sustained by signals coming from surrounding tissues, like those coming from the apical ectodermal ridge for mesenchymal cells in the vertebrate limb bud. Note that the DoT could be thought of as acting as a kind of niche of stem cells: cells are actively proliferating and self‐renewing such that some of the progeny remains at the DoT and the rest of the progeny is displaced outside the DoT through oriented cell division, for instance (Yamashita et al, 2003).
A major question in developmental biology is the coordination between patterning and growth. Gradient formation of secreted proteins has been proposed to pattern developing embryos and control the growth of a tissue through its steepness (Lawrence, 2001). As gradients driven by cell lineage transport are formed as tissue grows, linking the timing of mitotic divisions with the steepness of the gradient, cell lineage transport gradients might represent a mechanism that coordinates the dynamics of patterning and growth.
Materials and methods
Analytical gradient profiles of gradients driven by cell lineage transport
Our simulations start with N cells located at the DoT (x∈[0, Nl−l]; Supplementary information). Thus, when all cells divide in synchrony, the spatial dynamics of the cell furthest away from the DoT is given by , where τ indicates the total number of cell divisions, with cell cycle Cc. From the equations in Figure 1, the amount of mRNA and protein inside this cell as a function of the amounts of mRNA and protein its progenitor had at the DoT just before dividing and letting its descendant outside this domain (m0 and p0, respectively) is given by
where U=Cc has been introduced. By introducing the spatial position of this cell in equations (1a), (1b) and (1c), we obtained a continuous expression for the gradient profile at times just before another cell divided:
for x>Nl−l and where α=0.5 and γm,p≡Hm,p/Cc has been set.
As shown in Figure 2 and Supplementary Figure S1, these analytical profiles correctly fit the numerical gradients, obtained either for synchronous or non‐synchronous cell divisions, and fluctuating cell cycles and/or inherited fractions of the molecular content.
For very small relative half‐lives (short mRNA half‐lives compared to the cell cycle, γm≪1), no mRNA gradient is created () but still a protein gradient can be formed. In this case, only the cells in the DoT translate the protein and thus the DoT becomes the unique source of protein translation. This protein gradient is characterized by the single power‐law decay of equation (2a) ().
The analytical results derived from these profiles were computed with Mathematica software (Wolfram Research Inc.).
Length, number of divisions and time
Herein we derive the spatial and temporal properties of mRNA cell lineage gradients. As, as indicated, protein gradients formed by non‐graded mRNAs exhibit a single power‐law profile, equivalent mathematical expressions to those for mRNAs apply for protein gradients (Table I).
The spatial extent of the mRNA gradient was defined as L10m=x10m+l where x10m is the spatial position where the gradient has decayed 10‐fold (m(x10m)= m0/10, with m0 being the constant amount of mRNA at the DoT, and m(x) is given by equation (2a)). For other choices of decay of the gradient (q‐fold decays), we have m(xqm)=m0/q, which leads to . Note that the extent of mRNA gradients increases proportionally to the number N of progenitor cells at the DoT and the cell diameter l, as expected, and saturates to . The spatial extent of protein gradients can be analogously computed, although no closed analytical expression can be obtained when the mRNA is distributed in a graded profile (Supplementary information).
The average number of divisions per cell at the DoT required for creating an mRNA or protein gradient through cell lineage transport with a q‐fold decay satisfies , where the nonlinear growth of the tissue has been taken into account (Table I and Supplementary information). Once cells at the DoT have started dividing, the time required for the formation of a gradient with a q‐fold decay is given by (Table I). Note for 10‐fold decay gradients.
The derivation of these properties for gradients that do not involve clonal dilution is detailed in Supplementary information.
A measure for robustness and sensitivity
We quantified the robustness R of a property or function F under changes in the variable X as , where Δ is the magnitude of the change of X, ΔrX=Δ/X is the relative change and ΔrF=(F(X+Δ)−F(X))/F(X) characterizes the sensitivity of F. We set h=0.1. Note that R ranges between 0 and 1, R=1 (R=0) corresponding to robust (non‐robust) dynamics. Therefore, robustness emerges for low sensitivities of F (h−1ΔrF/ΔrX≪1). This measure of robustness was applied to the differential expression (F=L10m/LI) and to the time needed to form a gradient (F=t) under changes in the duration of the cell cycle (X=Cc). For F=L10m/LI, we measured how the sensitivity changed when clonal dilution was participating. Accordingly, we took the ratio between the sensitivity Sd for gradients driven by cell lineage transport involving clonal dilution owing to cell proliferation and the sensitivity Snd for gradients that had proliferating cells only at the DoT and thus did not involve massive dilution (Sd/Snd=ΔrF(dilution)/ΔrF(no dilution)). Sd/Snd<1 indicates that dilution decreases the sensitivity to changes in the duration of the cell cycle of the differential domains of expression.
Hoxd13 expression in the chick limb bud
Whole‐mount in situ hybridization was carried out according to standard procedures (Wilkinson, 1993). The chick Hoxd13 exon probe contains 1.2 kb 3′‐UTR. To clone an intron region of the chick Hoxd13 gene, we found the Hoxd cluster on chromosome 7 in the NCBI database. The second intron (590 bp) was amplified by PCR from chick genomic DNA, subcloned into pCR‐II (Invitrogen) and confirmed by nucleotide sequencing.
Fluorescent in situ hybridization
Chick forelimbs at stage 25/26 were sectioned at 10 μm after paraffin embedding. In situ hybridization was performed using standard procedures (Izpisua‐Belmonte et al, 1991) coupled to the TSA fluorescence system according to the manufacturer's instructions (Perkin Elmer), and Hoxd13 mRNA was detected as deposited fluorescent signal by the TSA fluorescence system.
Fluorescent images were analyzed by Leica SP5 tandem‐AOBS_MP multiphoton laser microscope with software AFS Leica. The intensity was quantified along the proximo‐distal axis on four different limb sections. The total average gradient profile was obtained from the data of nine gradients. The signal over longer distances along the proximo‐distal axis was averaged over four gradients. The average was computed by setting the distal end of all gradients at the same spatial position.
Chick forelimb buds from stage 25/26 embryos were dissected into distal, medial and proximal regions (Figure 6B). One sample consisted of 5–6 forelimbs, and five samples for the distal, middle and proximal were used for the analysis. Total RNA was isolated by using Trizol (Invitrogen) according to the manufacturer's instructions. cDNA was synthesized with 3 μg of total RNA with the Superscript II and oligo(dT) primer (Invitrogen), and used after 10‐fold dilution in order to monitor PCR products in the range of a relative standard curve. qPCR analysis was performed with the SYBR green system according to the manufacturer's instructions (Applied Biosystems) with a two‐step PCR amplification program (10 min at 95°C followed by 40 cycles of 15 s at 95°C, 45 s at 53°C). A relative standard curve was constructed using 10‐fold serial dilutions of cDNA derived from distal limb buds at stage 24–26. Data were calculated based on the threshold cycle, where the SYBR green signal becomes higher than that of the background. For quantitative comparison of gene expression, the relative expression level of Hoxd13 against GAPDH was calculated. The following primers for chick Hoxd13 were used: sense 5′‐AGACATGGTCTCAACGTTTGGC and antisense 5′‐GCACATCTCGGGCTGGTTTAG. For GAPDH, we used the following primers: sense 5′‐GGACACTTCAAGGGCACTGT and antisense 5′‐GGTGAAGACACCAGTGGACT. The amplification product sizes are 193 and 150 bp for Hoxd13 and GAPDH, respectively.
Hoxd13 expression in terms of the cell lineage transport model
Our data support that Hoxd13 exonic expression spans within the region potentially occupied by cell descendants of the intronic domain. Hoxd13 expression starts at stages 18/19 (Nelson et al, 1996), and the limb takes 12–19 h to develop to stage 22 (Hamburger and Hamilton, 1951). As the duration of the cell cycle during limb development has been reported as 8–12 h (Vargesson et al, 1997), cell descendants of the intronic domain (DoT) are expected to extend up to a region at most five times larger than the DoT. Therefore, at stages 21/22, we expected an exonic domain of expression at most five times larger than the intronic one, as we found indeed (n=4 limbs (intron), n=12 limbs (exon); Supplementary Figure S4B). Similar conclusions can be obtained from the expression patterns at later stages. In addition, these results are in agreement with previously reported correlations between exonic Hoxd13 expression and cell lineage analysis in the chick limb bud (Vargesson et al, 1997).
qPCR measured the amount of mature mRNA over a spatial region. In terms of the cell lineage transport model, the total amount at the distal region, which extends over a region 1+ε times the intronic domain's length, is , whereas the total amounts at the medial and proximal regions are respectively and , assuming all regions have the same size. Thus, and, using the value indicated by the qPCR, we obtained γm≈0.3. From MM/MD and setting ε⩽1 (Figure 6A and B), we obtained γm⩽0.8. Note that this value is lowered if we take into account that in proximal regions, the exonic expression does not extend over the whole antero‐posterior axis.
qPCR could not be performed at earlier stages owing to the small size of limbs. Assuming a 10‐fold decay over the exonic region, from the ratio between exonic and intronic expression domains (Figure 6A and Supplementary Figure S4B) and identifying this ratio as L10m/LI, we could estimate a plausible range of relative half‐lives, as well, through . At stage 25/26, we had (n=6 limbs (intron), n=6 limbs (exon)), which lead to , which is in agreement with the previous analysis. At stage 21/22, we had (n=4 limbs (intron), n=12 limbs (exon)), which lead to . A decrease in the relative half‐lives during limb development could result from an increase in the cell cycle, which is supported by previously reported data on the mitotic index (Hornbruch and Wolpert, 1970).
We thank Concepción Rodríguez‐Esteban, Mercè Martí, Joaquin Rodríguez‐León, Hiroko Kawakami and Carolina Soler for their help with experiments and embryo pictures. We also thank the late Francis Crick, Marcos González‐Gaitán and Javier Capdevila for their helpful insights at the early stages of this work. MI has been partially supported by the Fulbright Program and Generalitat of Catalunya and by the Ramon y Cajal Program from the Spanish Ministerio de Educación y Ciencia. This work was supported by grants from the Human Frontiers Science Program, the NIH, BBVA, Fundacion Cellex and the G Harold and Leila Y Mathers Charitable Foundation.
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