On the importance of protein diffusion in biological systems: The example of the Bicoid morphogen gradient

BBA - Proteins and Proteomics 1865 (2017) 1676–1686

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Review

On the importance of protein diffusion in biological systems: The example of the Bicoid morphogen gradient☆

MARK

Cécile Fradin Dept. of Physics and Astronomy, McMaster University, 1280 Main St W., Hamilton, ON L8S 4M1, Canada

A R T I C L E I N F O

A B S T R A C T

2000 MSC: 30 50.020 50.040 60.230 70.020 90.060

Morphogens are proteins that form concentration gradients in embryos and developing tissues, where they act as postal codes, providing cells with positional information and allowing them to behave accordingly. Bicoid was the first discovered morphogen, and remains one of the most studied. It regulates segmentation in flies, forming a striking exponential gradient along the anterior-posterior axis of early Drosophila embryos, and activating the transcription of multiple target genes in a concentration-dependent manner. In this review, the work done by us and by others to characterize the mobility of Bicoid in D. melanogaster embryos is presented. The central role played by the diffusion of Bicoid in both the establishment of the gradient and the activation of target genes is discussed, and placed in the context of the need for these processes to be all at once rapid, precise and robust. The Bicoid system, and morphogen gradients in general, remain amongst the most amazing examples of the coexistence, often observed in living systems, of small-scale disorder and large-scale spatial order. This article is part of a Special Issue entitled: Biophysics in Canada, edited by Lewis Kay, John Baenziger, Albert Berghuis and Peter Tieleman.

Keywords: Morphogen gradients Bicoid Hunchback Diffusion Fluorescence correlation spectroscopy Fluorescence recovery after photobleaching

1. Introduction Studies of diffusion have often been used to answer questions at the intersection between biology and physics. Famously, the observation, by the botanist Robert Brown, of the motion of particles contained in pollen [1] motivated Einstein's work on diffusion [2]. Einstein demonstrated that if Brownian motion was, as many believed, the manifestation of the thermal motions of the fluid molecules, then they should follow the laws of diffusion, namely a mean-squared displacement linear in time, ⟨r2⟩ = 6Dt, and a diffusion coefficient D = kT/6πηR depending on absolute temperature (T), particle radius (R) and fluid viscosity (η). The quantitative study of colloidal particles motion done by Jean Perrin shortly afterwards proved these predictions to be correct, providing a strong support for the existence of molecules [3]. Around the same time, the term “random walk” was coined by the mathematician Karl Pearson, when his help was solicited by a colleague who was trying to find a model for the spread of malaria by mosquitoes [4]. Above 1 μm, particles are too large to be significantly impacted by thermal collisions with solvent molecules, while below 1 Å, interactions involving valence electrons become predominant. But at the scale of biopolymers or small biomolecular assemblies, i.e. between 1 nm (e.g. the size of a lipid molecule) and 100 nm (e.g. the size of the pollen

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particles observed by Brown), the value of the thermal energy stored in a particle, kT, is comparable to that of other types of energies (electrostatic, mechanical, visible photon absorption, breaking of a chemical bound) [5]. Thus thermal motions play an important role for nm-size particles. The exchanges between thermal energy and the other different types of energies that can uniquely happen at this scale give soft matter and biological systems their defining responsiveness [6], and is what permits life as we know it. Since their size makes them so susceptible to Brownian motions, all proteins, unless they are bound to larger structures, are diffusing. Thus all cellular processes need to work either with the help, or in spite, of diffusion. For processes relying on protein diffusion, the relationship between the size of the relevant cell or compartment, L, and the time necessary for the proteins to explore this space, t = L2/2D (assuming an elongated and almost one-dimensional geometry as for example that of an E. coli cell), sets the timescale of the process. This relationship is illustrated in Fig. 1 for proteins with different mobilities. Soluble proteins can diffuse across bacterial cells in a fraction of a second, allowing fast signalling processes based on diffusion (e.g. in E. coli the control of the flagellar motor rotation by the diffusing signalling protein CheY [7]). On the other hand, membrane proteins might take almost a minute to equilibrate across a bacterial cell, allowing for example the sustenance of the slow ≃ 2 Hz MinD concentration oscillations that

This article is part of a Special Issue entitled: Biophysics in Canada, edited by Lewis Kay, John Baenziger, Albert Berghuis and Peter Tieleman.

http://dx.doi.org/10.1016/j.bbapap.2017.09.002 Received 29 April 2017; Received in revised form 16 August 2017; Accepted 5 September 2017 Available online 13 September 2017 1570-9639/ © 2017 Elsevier B.V. All rights reserved.

BBA - Proteins and Proteomics 1865 (2017) 1676–1686

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example in work we have done on the pro-apoptotic protein Bid, which showed that this protein has a loosely-bound membrane state with D ≃ 20 μm2/s, invisible in SPT experiments [23]. The limitations of SPT have also been discussed in the context of the diffusion of lipids, which usually have D = 1–5 μm2/s, approaching the limits of what can be detected with this technique [24,25]. Fluorescence recovery after photobleaching (FRAP) has even more stringent limitations in terms of its range. FRAP measurements are limited by the “halo effect”, i.e. the diffusion of molecules away from the photobleached area during the photobleaching step, which can effectively make the photobleached area larger than intended [26,27]. This problem is exacerbated by the fact that, with the popularization of commercial confocal microscopes with FRAP modules, most current FRAP experiments are performed in the raster-scan mode using low power lasers. Such experiments require long photobleaching steps, in which case, unless the appropriate analysis tools are used, the range of accessible diffusion coefficient is limited to D < ∼ 1 μm2/s. This very serious limitation is not always properly recognized. In the following, we discuss the role of diffusion for morphogens, with an emphasis on Bicoid. In particular, we review measurements of the diffusion coefficients of Bicoid in the fly embryo, and highlight the importance of accurate measurements in order to validate or invalidate models for either the establishment or the interpretation of morphogen concentration gradients.

Fig. 1. Approximate time required for a protein to explore a space of a certain size. The green shaded area corresponds to the expected range of values for soluble proteins (D = 2 − 10 μm2/s), while the blue shaded area corresponds to the expected range for mobile membrane proteins (D = 0.1 − 1 μm2/s). It takes less than 1 s for a soluble protein to equilibrate in a bacterial cell, and more than 1 h in a large embryo like the fly embryo.

provides positional information during cell division in E. coli [8]. In contrast, in large cells such as embryos, the timescale associated with protein diffusion is much larger. It makes signalling through protein diffusion a very slow and inefficient process, but allows the long-term persistence of protein gradients across the cell, which establish cell polarity and drive morphogenesis. Characterizing the diffusion of proteins in cells and tissues, and obtaining an accurate measurement of their diffusion coefficient, is essential in order to understand the dynamics of cellular processes. However, it is a difficult endeavour, for at least two reasons. The first is the complexity of the intra- and extracellular media. The viscosity of the cellular medium is roughly 5 times that of water, but varies with cell type and with location within the cell, and is often found to increase both with the size of the diffusing particles and with the scale at which diffusion is probed [9–11]. This scale-dependence highlights the inhomogeneous nature of the cellular interior, crowded with macromolecules, membranes and filaments, which tend to restrict the longrange motions of larger molecules. Hence diffusion in cells is often labeled as “anomalous”, a term which includes a number of diffusive processes with a dependence on scale [12]. In addition, proteins of interest are often subject to specific or non-specific interactions with other cellular components, which alters their mobility and often makes the interpretation of diffusion measurements difficult [13,14]. Finally, due to their inherently out-of-equilibrium nature, cells produce nonthermal noise (for example through the action of molecular motors), which can both enhance and complicate cellular diffusion [15–18]. Although a lot of work has been done, in particular in our lab [19,20], in order to understand the origin of the complex diffusive behaviour observed for proteins in crowded media and in cells, it often remains difficult to interpret the result of cellular diffusion measurements. The second issue encountered when trying to measure diffusion in cells is that the methods available to probe protein diffusion (primarily fluorescence methods) can each only measure diffusion coefficients within a certain range. Fluorescence correlation spectroscopy (FCS), for example, a technique based on the analysis of fluorescence fluctuations due to the passage of molecules through a diffraction-limited detection volume, is best suited to the characterization of fast diffusive processes. Because slow molecules photobleach while traversing the detection volume, FCS is not usually appropriate for measuring diffusion processes for which D < 1 μm2/s, as discussed in several of our publications [21–24]. Single particle tracking (SPT), for its part, is limited by the time required to record enough photons for accurate particle localization. As a result, standard implementations of SPT cannot detect particles with D > 5 μm2/s. This particular point came through for

2. The Bicoid morphogen gradient 2.1. Morphogens Morphogenesis, that is the emergence of shape in embryos and tissues, is controlled by a set of molecules called morphogens. These molecules combine two very specific properties: 1) they form concentration gradients across fields of cells, and 2) they affect cells within this field in a concentration-dependant manner [28,29]. Morphogens thus act in a way similar to the characters in a postal code: each cells, by “reading” the concentration of the different morphogens present, obtains information on its precise location within the organism, and reacts accordingly. In animals, morphogens are mostly proteins, and the first demonstration of their existence was obtained through a vast genetic screen performed in Drosophila [30]. Some morphogens discovered in this way and shaping the early fly embryo turned out to be transcription factors, which modify the behaviour of cells by directly affecting the expression of a particular subset of genes, their “target genes”. Other morphogens, including some morphogens acting in the fly embryos at later stages, instead act on cells by binding to extracellular receptors. In both cases, the signalling cascade initiated by the morphogen affects cell fate and identity, leading to changes in gene expression and, in turn, to cell differentiation. Morphogen action also often leads to modification in cell mobility and mechanics, inducing tissue modelling during development [31]. In the simplest model for morphogen gradient readout, the so-called “french flag model”, target genes are expressed only above (if the morphogen acts as an activator) or below (if it acts as a repressor) a certain threshold concentration of the morphogen [28]. However, it has become increasingly evident that many factors, in particular the presence of other morphogens, modulate the response of target genes to a morphogen, maybe affecting its precision and robustness [32]. The term “morphogen” was coined by Alan Turing, before their actual discovery, as he presented the idea that morphogenesis must be controlled by “a system of chemical substances (…) reacting together and diffusing through a tissue” [33]. Diffusion has therefore been believed to play a role in morphogenesis for a long time. Remarkably, diffusion in fact plays a major role in both aspects of the morphogen function. It drives the formation of morphogen concentration gradients, influencing both their range and the speed at which they can be 1677

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Fig. 2. (a) Schematic Bcd sequence showing the different domains present in the protein [39–44]. (b) Structure of the homeodomain of Bicoid specifically bound to DNA (PDB ID: 1ZQ3 [45]). Positively charged residues presumably involved in non-specific DNA recognition are indicated in blue. The inset in the lower left corner is a helical wheel representation of the helix interacting with the DNA major groove, showing a cluster of positively charged residues on one side of the helix. (c) Structure of hunchback proximal enhancer region, showing known binding sites for three different transcription factors (Bicoid [46,47], Hunchback [48] and Krüppel [48]). The different shades of green used for the Bcd binding sites roughly reflect different binding affinities, with the darker sites (X1, X2 and X3) having a higher affinity than the lighter ones (A1, A2, A3, B1 and B2). Binding sites for Caudal [49] and Zelda [50,51] have also been reported (not shown), and additional binding sites are present in the distal enhancer regions of hb (not shown), which are involved in later stage expression of hb.

is a small (55 kDa) homeobox protein, whose sequence contains a number of functional domains, roughly organized into DNA binding domains at the N-terminal half, and activating domains at the C-terminal half (Fig. 2a, b). Bicoid's homeodomain is unique in that it allows Bicoid to bind both DNA and RNA [45,54,59], probably adopting slightly different conformations in order to do so [60]. Accordingly, Bicoid acts both as a transcription factor activating the expression of many different target genes (a systematic search for Bicoid targets has uncovered no less than 66 of them [61]), and as a repressor of the protein Caudal, blocking its translation by binding to caudal mRNA [59,62]. The binding of Bicoid to DNA has shown to be slightly cooperative, with Bicoid-Bicoid interactions mediated by regions on either sides of the homeodomain [41,47,63,64]. Bicoid also contains a PEST sequence, a region usually implicated in protein degradation, but which in this case has also been shown to mediate Bicoid's interaction with mRNA [42]). The readout of the Bicoid's gradient has been most studied in the context of the regulation of the expression of the protein Hunchback (Hb) [46,56,65-69]. Hunchback is itself a transcription factor and a

established [34–36]. It also plays a part in the signalling function of morphogens, by setting limits on the speed and precision at which morphogen gradients can be interpreted [37,38]. 2.2. The Bicoid gradient Bicoid (Bcd) is a master regulator of patterning along the anteriorposterior (AP) axis of Drosophila embryos (Figs. 2 and 3) [52,53]. bicoid mRNA is deposited by maternal nurse cells at the anterior pole of the oocyte before egg laying, and this localization is maintained during the early development of the embryo [54]. As a consequence, the Bicoid protein adopts a very distinct spatial distribution, forming within an hour an exponential gradient between the anterior and posterior poles of the embryo [52]. In D. melanogaster this gradient has a characteristic length between λ = 80 μm and λ = 120 μm (as measured by immunostaining [55,56], or by imaging embryos expressing a fluorescent Bcd fusion, as in Fig. 3 [57,58]). This value corresponds to ∼ 20% of the total D. melanogaster egg length (λ ≃ 0.2 L where L ≃ 500 μm is the egg length, which remains constant during early development). Bicoid 1678

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Fig. 3. (a) Confocal image of a D. melanogaster embryo expressing Bcd-EGFP (nuclear cycle 12), showing the characteristic Bicoid exponential concentration profile. Image reproduced from [58]. (b) Transcriptional activity of the hb promoter, visualized by confocal microscopy using the MS2 system (each red dot represents an active hb promoter region, grey circles show nuclear membranes), as explained in [69,78]. The channel showing hb transcriptional activity has been filtered and thresholded for clarity, and pseudo-colors are used for both channels. (c) Schematic profile showing how the smooth distribution of the nuclear concentration of the Bcd protein is turned into a sharp hb transcriptional response. The length of the Bcd gradient and precision of the hb response shown here corresponds to what is measured for this system at nuclear cycles 11–13 (length of the Bicoid gradient λ = 100 μm =0.2 L, position of the hb border at x = 0.5 L, and readout precision of dx = 0.02 L corresponding to δc/c = 0.1) [58,77].

a

b

c

binding cooperativity observed in vitro [41,47,63,64]. This response is also astonishingly precise: By looking directly at the production of nascent mRNA at the hb locus (Fig. 3b), our collaborator, Dr. Dostatni, has shown that, by nuclear cycle 11, nuclei can reliably make opposite decisions about hb expression if they are placed more than 10 μm, or 0.02L apart, in other words they are able to “measure” a difference in Bicoid concentration of only 10% [69,77]. This means that two neighbouring nuclei in the fly embryo may robustly adopt very different fates, one expressing hb, and the other not, on the basis of a very small difference in Bcd concentration (Fig. 3c). What is more, this readout is reproducible from embryo to embryo, where the average position of the hb border is found between 0.43 and 0.5% of egg length between nuclear cycle 10 and 13 (where position is taken from the anterior pole), with a reproducibility between different embryos of 2 − 3% starting at cycle 11 [56,77]. As the first morphogen shown to actually form a concentration gradient [52,53], Bicoid remains one of the most studied. It is an ideal model system for several reasons. It can be addressed genetically, allowing in particular the generation of fly stocks expressing fluorescent protein fusions, and it is amenable to biophysical studies and quantification, in particular due to the position of nuclei in the cortical region

morphogen [70]. Its proximal enhancer region, which controls hb anterior expression in the early embryo, contains at least 8 Bicoid binding sites (consistent with the fact that Bicoid directly activates hb’s transcription in a synergistic manner), 6 of them present in a region very close to the beginning of the hb gene (Fig. 2c) [46,47,63]. The proximal enhancer region of hb also contains at least 3 Hb binding sites, suggesting that Hb can regulate its own expression [48,71]. Other binding sites are present, notably for Krüppel [48], Caudal [49] and Zelda (a protein providing temporal information in the early fly embryo) [50,51]. Two additional enhancer regions have been identified for the hb gene, a distal shadow enhancer that also responds to Bicoid activation and seems to have a role in refining the readout of the Bicoid gradient performed by the proximal enhancer [72,73], and a posterior stripe enhancer that responds to transcription factors other than Bicoid and drives the establishment of a Hb stripe observed in the posterior part of the embryo at later developmental stages [73,74]. Hb gene transcription responds to the Bicoid gradient in an almost all-or-nothing manner (“on” in the anterior part of the embryo, and “off” in the posterior part) [75]. The remarkable sharpness of this response (characterized by Hill coefficients between 6 and 10, depending on developmental stage [76]) can only be partially explained by the weak DNA 1679

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Fig. 4. (a) Average autocorrelation functions obtained from multiple FCS measurements performed in the anterior cortical cytoplasm of D. melanogaster embryos (nuclear cycles 12 to 14), expressing either Bcd-EGFP (black line) or NLS-EGFP (grey line) [58]. Error bars are the standard error of the mean. (b) Residuals and corresponding normalized χ2 values show the results of fits of three different diffusion models to the Bcd-EGFP data. Two-species diffusion refers to the coexistence of two different populations of Bicoid proteins, each with a different diffusion coefficient, presumably due to binding to other molecules. Anomalous diffusion refers to a diffusion process with a meansquared displacement proportional to a power-law in time, ⟨r2⟩ = tα, and further assumed to have a Gaussian propagator, leading to an analytical form of the autocorrelation function [12]. Anomalous diffusion of Bicoid could be due to obstruction caused by other molecules or molecular structures in the embryo. As a reminder, the normalized chi-squared χ2/ν is calculated by taking the average value of the square of the residuals divided by the error on the data at each point, then dividing this average by N − r, where N is the number of points in the autocorrelation function and r is the number of variables in the model used to fit the data. A good fit of the data is characterized by a χ2/ν close to 1.

the tissue balanced by a localized sink at the opposite end [36]. He showed that the solution of the corresponding diffusion-reaction equation was a stable linear concentration gradient [36]. Such a purely diffusive transport model was criticized at the time, in view of the (perceived) “extreme rarity with which sheer diffusion processes occur in living systems” [80]. The idea of a diffusive morphogen, however, has endured, although Crick's model evolved into the synthesis-diffusion-degradation (SDD) model where, instead of a localized sink, diffusion is balanced by a firstorder kinetics degradation process occurring uniformly throughout the organism [81]. More precisely, in its current form (as formulated for example in [82–85]) the SDD model is based on three assumptions: (i) a localized source producing the morphogen with rate K, (ii) morphogen diffusion with a diffusion coefficient D, and (iii) uniform morphogen degradation at a rate 1/τ (τ is the half-life of the morphogen). The concentration of the morphogen c in space (x) and time (t) away from the source is therefore regulated by the following reaction-diffusion equation (assuming a one-dimensional geometry):

of the embryo during early development, which allows quantitative fluorescence experiments. Yet, in some ways, it is also an unusual system, because gradient formation and early interpretation take place in a syncytium, that is an embryo in which, until nuclear cycle 14 (2 h after egg laying), nuclear divisions occur without being accompanied by cellular division. Cellularization only takes place at cycle 14, with plasma membrane furrows closing in around each nucleus. 3. Bicoid gradient formation 3.1. The synthesis-diffusion-degradation model Evidence of a “dispersion” process associated with morphogens has been accumulating since the 60 s, when transplantation experiments showed that pieces of tissue grafted at a different position in the embryo caused alterations of the developmental pattern in the area surrounding the grafted tissue. It was immediately proposed that morphogen transport might in fact be diffusion [34,35,79]. It was also recognized at that time that, in order for a morphogen gradient to reach a steadystate, another process needed to balance the effect of diffusion, and it was initially proposed that this process could be active transport of the morphogens against the gradient [34]. A related observation was made by Wolpert, namely that the scale of morphogen gradients is in general 1 mm or less, and that they usually require a time on the order of hours to be established [28]. This led Francis Crick to propose a model for morphogen gradient formation based on diffusion, with a localized source of morphogen at one end of

∂c (x , t ) ∂2c (x , t ) =D − τ −1c (x , t ), ∂t ∂x 2

(1)

Two boundary conditions must be added to Eq. (1), D∂c/∂x|x=0 = −K to account for the influx of morphogen at one end of the organism (x = 0), and ∂c/∂x|x =L = 0 to account for the fact that morphogens cannot exit at the other end (x = L). In steady-state (that is for t ≫ τ), and for τD ≪ L (which corresponds to a gradient with characteristic length significantly shorter than the length of the organism), the 1680

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Bicoid, effectively resulting in a capture of the proteins in nuclei during interphases, could potentially alter the dynamics of Bicoid spreading at the large scales relevant to gradient formation [94–96]. Neither nuclei nor membrane furrows, however, would have any significant influence before cycle 10, that is during the first hour of embryo development while the gradient is forming. Indeed nuclei have been shown to play little or no role in Bcd gradient formation [44].

solution of Eq. (1) is [83]:

c (x ) =

Kλ −x / λ . e D

(2)

It describes a stationary exponential gradient with characteristic length λ = τD . When Driever and Nüsslein-Volhard first observed the exponential shape of the Bicoid gradient, they immediately thought it might be formed by diffusion [52]. However, until recently, lack of quantitative data on the diffusive properties and the half-life of Bicoid made it difficult to test this assumption.

3.3. Bicoid degradation As the process counterbalancing the effect of Bicoid spreading, Bicoid degradation plays an important role in influencing the size, shape, establishment time and robustness of the Bicoid gradient. Bicoid contains a PEST sequence, thought to be a signal for degradation [52]. However, the exact mechanism for Bicoid degradation remains unknown. A few models of uneven degradation have been considered for Bicoid, for example degradation occurring inside nuclei as a way to achieve scaling of the gradient with embryo length [55,57]. Uneven degradation along the AP axis has also been proposed as a general way to modulate the shape of morphogen gradients [97], and could potentially apply to the Bicoid gradient. However, there is no experimental evidence so far that the degradation process differs from a spatially uniform first-order kinetic process fully characterized by the average half-life of the protein. A first estimate of Bicoid's half-life in D. melanogaster embryos was made based on the lag time between the disappearance of the bcd mRNA and that of the Bicoid protein, and evaluated to be less than 30 min [52]. Another approximate estimate for Bicoid's half-life is given by the time needed for the gradient to stabilize, which is about 90 min, and is approximately equal to τ [82,84]. More precise measurements, based on the repeated photoactivation of a fusion of Bicoid to the photoconvertible fluorescent protein Dronpa, showed that τ = 50 min in regular embryos before nuclear cycle 14 (and decreasing after cellularization) [89] and τ > 70 min in unfertilized embryos where the gradient length is also increased, lending further credit to the suitability of the SDD model to explain Bicoid gradient formation [98].

3.2. Bicoid cytoplasmic diffusion Precise measurements of Bicoid's diffusion coefficient became possible with the preparation of D. melanogaster lines expressing a fluorescent Bcd-EGFP N-terminal fusion protein replacing the endogenous Bcd [57]. A first measurement was performed by Gregor et al. during the mitosis following nuclear cycle 13, using FRAP. The analysis of the fluorescence recovery curves obtained yielded a surprisingly low value for Bidoid's diffusion coefficient, D = 0.3 μm2/s, seemingly invalidating the SDD model [57]. This led to several alternative propositions for Bicoid gradient formation, including active transport mechanisms [57], Bicoid mRNA transport [86] and advective transport [87]. However, due to a long photobleaching step, the FRAP measurements from [57] suffered from the “halo effect” discussed in Section 1, with the consequence that the presence of fast diffusive species was missed. For this reason, we used the same fly line as in [57] to measure the diffusion coefficient of Bicoid in the cortical cytoplasm during nuclear cycles 12 to 14, using FCS instead of FRAP (Fig. 4) [58]. We analyzed our data using several different transport models, and we found that, no matter what the model was, the extracted average diffusion coefficient for Bicoid was between 3 and 10 μm2/s. Higher values for D than that initially reported in Ref. [57], but still lower than those we had measured, were subsequently also reported by other groups. A re-analysis of the FRAP data from Ref. [57], correcting for the halo effect, led to the estimate D ≃ 1 μm2/s [88], while indirect measurements by the Wieschaus group, relying on the analysis of the long-range shape of photoactivated Dronpa-Bcd gradients, led to D = 1 to 4 μm2/s [89]. What can explain the discrepancy between these different results? Interestingly, the diffusion of Bicoid clearly showed subtle signs of complexity (see the shape of the FCS autocorrelation function shown in Fig. 4), in contrast with the free diffusion observed for the control NLSEGFP protein (for which we measured D = 25 μm2/s, corresponding to diffusion in a medium with viscosity 4 times that of water) [58]. The large discrepancy in diffusion coefficient between Bcd-EGFP and the control protein, whereas they have similar sizes, points towards a slowing down of the overall mobility of the Bcd due to specific interactions with different binding partners. It has been in fact proposed that interactions with binding sites could explain the different estimates for D obtained using different methods, since some techniques are more sensitive to mobile particles (FCS) while others are more sensitive to immobile interacting particles (FRAP) [90,91]. The existence of two (or more) Bicoid populations with different mobilities and different capacities to act as transcription factors would have an effect not only on gradient formation (both its stationary profile and formation dynamics) but also on the transcriptional response [92]. Taken together, these results suggest that Bicoid molecules diffuse relatively rapidly over short distances (such as the ones probed by FCS, that is just under a μm), although they may be slowed down by molecular interactions. In addition, their motion might be impeded at larger scales, for example due to the presence of membrane furrows in the syncytial embryo [93]. This could explain why “large-scale” measurements (such as FRAP or photoactivation measurements) return smaller diffusion coefficients than “small-scale” ones (FCS). Finally, it is also worth noting that the active nuclear import of

3.4. Morphogen gradients formation After correction for experimental misinterpretation, we see that the values of Bicoid's diffusion coefficient (D ≃ 7 μm2/s in our best estimate at the small 0.5 μm scale of FCS experiments [58], and between 1 and 4 μm2/s at larger scales [88,89]) are perfectly consistent with the SDD model, which for a Bcd half-life τ = 50 min [89] predicts an exponential gradient with characteristic length λ = 100 μm, as observed, for D = 5 μm2/s. Thus quantitative measurements of Bcd mobility provide in this particular case a very strong support for the simplest diffusive model of gradient formation. But what of other gradients? Whereas the Bicoid gradient is formed in a syncytium, other morphogen gradients form in tissues where cells are densely packed, i.e. under a very different set of constraints. So it is fair to question whether the diffusion-driven Bcd gradient formation is in fact a universal mechanism for gradient establishment, as envisioned by Francis Crick, or just an exception. A number of other morphogen gradients have been studied in details, and the mobility of the corresponding morphogen quantified, as reviewed for example in [99]. In all cases, a close and sometime critical examination of the experimental results available show that passive diffusion is the most likely mechanism for gradient formation [100,101]. Most informative of all is maybe the case of Dpp, since it tells a comparable cautionary tale as Bicoid's about the use of FRAP. Decapentaplegic (Dpp) is a morphogen that drives pattern formation in the developing fly wing. Large-scale raster-scan FRAP measurements seemed to show that Dpp had an effective mobility corresponding to D = 0.1 μm2/s, and detected the presence of a large immobile fraction [102]. It was therefore concluded that an active process for 1681

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provide error correction mechanisms, thus improving robustness as embryogenesis progresses [61,113]. In the case of the Bicoid/hunchback system, one can see both the possibility of feedback loops, since Hb can help activate its own transcription (something that can explain why the sharpness of the Hb response increases with time during nuclear cycle 14 [76]), as well as the possibility of interactions between two gradients (the Bcd gradient and a second, opposite gradient), something which has been shown to provide scaling [114]. In conclusion, the picture that emerges is that of morphogen gradients formed by diffusion, and whose early readout is progressively refined through a number of feedback processes. Importantly, it seems that most, if not all, studied morphogen gradients are in fact formed by mechanisms based on passive diffusion [100,101]. If additional mechanisms (e.g. spread of the morphogen source, active diffusion or transport or mechanical contractions, as reviewed for example in [115]) contribute to the spreading of morphogens, they are likely to be only secondary fine-tuning mechanisms.

spreading was required, which in this case was proposed to be endocytosis. However, the question of Dpp mobility was later revisited using several different techniques, namely spatial FRAP (a variant of FRAP where the full spatial profile of the fluorescence is considered, allowing a more stringent characterization of the diffusion process than with regular FRAP), photoconversion and two variants of FCS [103]. It was found in this later study that Dpp was split into two different pools, one large pool trapped within cells (and therefore appearing as immobile or very slow in FRAP experiments), and a very small extracellular fraction diffusing with D = 20 μm2/s and proposed to drive gradient formation. Other notable examples of morphogens whose gradients have been convincingly shown to form by diffusion are that of Fgf8, Lefty and Nodal. Fibroblast growth factor 8 (Fgf8) is a morphogen involved in the development of the zebrafish embryo's neural tube. FCS measurements performed in blastula-stage embryos have shown that the Fgf8 gradient is formed via extracellular diffusion [104]. As for Dpp, the opposing force for gradient stabilization in this case is cellular endocytosis [104,105], although compared to Dpp, the balance between the intra-cellular and extracellular pools of Fgf8 is inverted, with only small amounts of Fgf8 present within cells. The diffusion of Nodal and Lefty, which are respectively an activator and a repressor working in tandem during zebrafish embryogenesis, was measured by FRAP. It was found that there was a correlation between the effective diffusion coefficient of these proteins (D ≃ 15 μm2/s for Lefty, D = 3 μm2/s for Nodal) and the range of their respective gradients (λ = 100 μm for Lefty, λ = 40 μm for Nodal), lending support to a diffusion-driven mechanism [106]. Notably, although the FRAP experiments in that case were spatial FRAP experiments performed over very large photobleaching areas (resulting in very long recovery time, even for fast diffusing proteins), the data was analyzed taking into account the halo effect (as done for Bicoid in [88]). This allowed measuring diffusion coefficients as high as 20 μm2/s. One argument often raised against the SDD model, and diffusiondriven gradient establishment models in general, is that it neither provides a precise mechanism, nor does it allow for adjustment in response to perturbations [32,107]. However, the formation of morphogen gradients consistently lead to responses that are both precise (reproducible from embryo to embryo) and robust (stable against all kinds of perturbations, e.g. variation in embryo length), leading to wellproportioned organisms in a variety of conditions [108]. A number of points can be made in answer to this criticism of diffusion-driven gradients. First, it is possible that gradients are being read before they stabilize, and it has been argued that such a pre-steady state decoding might lead to an increase in robustness [82], including the possibility of gradient scaling with embryo length [109]. Second, several theoretical studies have shown that if a gradient is driven by anomalous diffusion, it is likely to be more robust in response to perturbations than a gradient driven by normal diffusion [97,110]. Given the ubiquity of anomalous diffusion in cells (and the evidence for non-simple diffusion we obtained for Bicoid, Fig. 4), this could provide a general mean to increase gradient robustness. Third, since diffusion-driven gradient formation is always balanced by a degradation or trapping process, it is possible that the robustness comes from this second process. It has been proposed, for example, in the case of Bicoid, that if degradation occurs only in nuclei, scaling with embryo length for different species would automatically follow [55,57]. However, nuclei have since then been shown to play no role in gradient formation [44]. Along the same lines, it has been proposed that the size of the Dpp gradient may scale with the increasing size of the tissue during development through a gradual decrease of the degradation rate [111], and that non-linear degradation process (that is, enhanced by the morphogen itself) can increase the robustness of gradients [112]. Fourth, and maybe most importantly, it has become increasingly clear that secondary mechanisms act to finetune the interpretation of primary morphogen gradients, and provide ways to increase the precision of the gradient readout, as well as

4. Bicoid gradient interpretation 4.1. Diffusion limits the precision of morphogen gradient interpretation For all morphogen gradients, the readout process involves cells or nuclei “sensing” the concentration of the morphogen. This implies that morphogen molecules are being “counted”, via their interaction with a number of binding sites (for example the different binding sites on the enhancer region of a target gene, or receptors on the cell surface). Berg & Purcell recognized that in the case of the random arrival of N signalling molecules at their binding site, the precision of the counting process was limited, like all Poisson processes, by statistical fluctuations, such that for an optimal counting process the error would be δN = N1/2 [116]. Thus the limit on the precision of the concentration measurement, known as the Berg-Purcell limit, is δc/c = δN/N = N −1/2. At the same time, the rate at which freely diffusing molecules reach their target site cannot exceed the diffusion-limited rate calculated by Smoluchowski, kDL = 4πDac, which predictably depends on the concentration (c) and diffusion coefficient (D) of the morphogen, as well as on the size of the binding site (assimilated in Smoluchowski's calculation to a perfect sphere of radius a) [117]. In a more general case, one has kDL = ϵDa*c, where ϵ is a pre-factor accounting for geometrical effects (the actual solid angle available to the ligand to approach the target, and the probability that the ligand has the correct orientation to bind its target) and where a* is the effective size of the target (accounting for the presence of long-range interactions) [118]. The average number of molecules reaching the target site over a period of time T can then be estimated as N = ϵDa*cT. Combining the idea of number fluctuations (i.e. the Berg-Purcell limit) with the diffusionlimited rate (to calculate the maximum average value of N), one sees that any concentration measurements relying on the diffusion of the detected molecules has a relative error:

δc / c ≥ (ϵDa*cT )−1/2 .

(3)

In the context of the Bicoid gradient readout by the hb enhancer region, Eq. (3) means that the decision to express the hb gene can be “taken” according to the local concentration of Bicoid (c) only with a certain statistical error (δc). The minimum value of δc is limited by the number of Bicoid molecules that reach the hb promoter region. This in turn depends on the concentration and diffusion coefficient of Bicoid, as well as on the time available to “count” Bcd molecules. If a nucleus is exposed to a certain Bicoid concentration for only a limited time T, as happens during early fly development when mitoses follow in close succession, the precision that this nucleus is able to achieve during a single interphase in terms of producing the correct amount of hb expression is limited according to Eq. (3). Taking into account the details of the binding process (i.e. the facts that binding is affected by the affinity between the receptor and the ligand [116], and that the binding 1682

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Fig. 5. (a) Average autocorrelation functions obtained from multiple FCS measurements performed in anterior nuclei of D. melanogaster embryos (nuclear cycles 12 and 13) expressing either Bcd-EGFP (black line) or NLS-EGFP (grey line) [77]. Error bars represent the standard error of the mean. The data is shown twice to present two different fits, one with a two-component model (purple line) and the other with a stick-and-diffuse model (blue line, curve shifted upwards). (b) Residuals (normalized by the standard error) of three different fits performed on the BcdEGFP data shown in (a). The anomalous diffusion and twospecies diffusion models are as explained in the caption of Fig. 4. The stick-and-diffuse model refers to a diffusion process that is interrupted by intermittent binding to immobile structures, for example chromatin. Note that the residuals have been normalized by the standard error of the mean of the data, in order to better show the regions of the autocorrelation function which contributes to an elevated normalized chi-squared.

diffuse quite efficiently through the nuclear space, a non-negligible fraction of Bicoid is constantly interacting with nuclear structures [120]. Given the constraint imposed by Eq. (3), what then is the limit on gradient readout precision that can be expected? In early fly development, only a short time is available during each interphase for nuclei to “read” Bicoid concentration, where T increases from 3 to 8 min between cycle 11 and 13. The concentration of Bicoid in anterior nuclei, cM, has been evaluated by different methods that gave comparable estimates: cM = 55 nM according to quantification of the fluorescence of Bcd-EGFP using for reference a solution of known EGFP concentration [38], and cM = 50 to 100 nM based on immunofluorescence measurements calibrated by single particle detection during nuclear cycles 10 − 14 [66]. In agreement with these results, we observed using FCS measurements that cM increased from 50 nM at nuclear cycle 10 to 100 nM at nuclear cycle 13 [58]. Considering a gradient with characteristic length λ = 100 μm, Bicoid's concentration at the hb expression border is therefore cT = 4 − 8 nM. Systematically using the highest reasonable estimate for these different variables (D = 5 μm2/s, T = 8 min, cT = 8 nM) and choosing ϵ = 2π and a* = 0.34 nm (the size of a single base pair), we thus estimate using Eq. (3) that the precision on gradient readout should not be better than δc/c = 20%. Although different choices can be made for ϵ and a*, no reasonable choice can explain the actual experimentally measured precision of hb transcriptional response to Bicoid, which is δc/c = 10% (Fig. 3c) [77].

process is itself a source of statistical noise [37]), leads to slight corrections to the Berg-Purcell formula (Eq. (3)), and a slightly higher limit on the error of the concentration measurement [119]. However, given a certain Bicoid concentration and readout time, the diffusion of the morphogen remains the principal limiting factor controlling the precision of the gradient readout. 4.2. Bicoid nuclear diffusion In light of the Berg-Purcell limit to gradient readout precision, it was important to measure the actual diffusion coefficient of Bicoid inside nuclei, where the protein performs its search for target sites such as, for example, the hb gene. We therefore used a similar strategy to the one we had used for measuring cytoplasmic diffusion, and performed singlepoint FCS experiments within anterior nuclei of D. melanogaster embryos expressing Bcd-EGFP (nuclear cycle 12 and 13) [77,120]. These measurements showed two things, as highlighted in Fig. 5. First, the mobility of Bicoid inside nuclei was comparable to its mobility in the cortical cytoplasm, with an effective diffusion coefficient (estimated from the characteristic time of the average correlation function) D ≃ 5 μm2/s. Second, the behaviour of Bicoid in nuclei seemed much more complex than in the cytoplasm, as evidenced by the shape of the autocorrelation function (Fig. 5). Of all the models that we fitted to the data, the most satisfactory one was the “stick-and-diffuse” model, which assumes that the particles binds to immobile obstacles between diffusion periods [13]. Our data thus suggests that although it is able to 1683

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even say the symbiosis -, between noisy small-scale processes such as diffusion, and exquisite large-scale structural order such as that observed for example in developing embryos. The fact that morphogen behaviour is time, position and scale-dependant, combined with their high activity and the inherent complexity of the biological medium, make measurements of Bicoid mobility very challenging. Combination of several techniques with different dynamic range, or use of new techniques such as for example recently proposed FCS variants (variable-lengthscale FCS [129], STED-FCS [130], inverted FCS [20,131], SPIM-FCS [132], spatio-temporal FCS [133]) allowing to characterize diffusion processes as a function of scale, could help resolve some issues and would permit a more complete characterization of morphogen diffusion.

4.3. Resolving the precision paradox: memory and facilitated diffusion There are at several possible resolutions of the paradox highlighted in the previous section. Gregor et al. proposed that a high precision might be recovered through a mechanism allowing for a spatial averaging of the readout effectuated by different nuclei [38]. Another possible explanation would be that nuclei are able to keep a memory of their hb transcriptional status during the previous nuclear cycle, either by retention of proteins of the transcription machinery or epigenetic modification [77]. Evidence for such a process for a different gene, snail, expressed at a slightly later stage than hb in the Drosophila embryo, has very recently been found [68]. Yet another possible explanation for the very high precision achieved by the hb border is that Bicoid molecules may use facilitated diffusion to search for their target. The idea that a reduction in the dimension of the explored space would result in a more efficient search can be traced back to Delbrück [121]. It was later developed in the context of nuclear proteins searching for a DNA target [122,123], after it was shown that the lac repressor could bind target DNA faster than allowed by the diffusion-limit [124]. Since then, there has been a growing body of evidence suggesting that many DNA-binding proteins are undergoing a combination of three-dimensional diffusion in the nucleoplasm and one-dimensional diffusion along the DNA (a process driven by non-specific electrostatic interactions between the protein and the charged phosphate backbone of the DNA, as the structure shown in Fig. 2b suggests could happen for Bicoid). This process allows, within some parameter range, to speed up the rate at which proteins can find their targets. It can be understood as an “antenna effect”, where the size of the target, instead of being the size of a single base pair, effectively becomes the size of the region explored during a 1D diffusion event, that is na if n base pairs are being explored on average [125,126]. The “facilitated diffusion”-limited rate at which proteins find their targets then become: kFDL = 4πf3DD(na)c, where f3D represents the fraction of the time spent diffusing in 3D [125]. In turn, the limit on precision must then become δc/c ≥ (ϵf3DD(na)cT)− 1/2. Based on our FCS measurements, and the fit to the data done with the stick-and-diffuse model, which assumes that molecules bind and unbind from immobile structures with Poisson statistic, we estimate that in anterior nuclei Bicoid molecules spend on average τu = 240 ms in an unbound diffusive state with D = 7.7 μm2/s, before spending on average τb = 120 ms in a bound state (presumably non-specifically bound to DNA). The (non-specific) binding rate to DNA extracted from our FCS experiments is thus kon = 1/τu = 4.1 s− 1 (note that this effective rate will depend on available DNA concentration), while the unbinding rate is koff = 1/τb = 8.2 s− 1. Very recent single particle tracking experiments in live embryos have demonstrated the existence of a population of Bicoid molecules consistent with proteins non-specifically bound to DNA (with τb on the order of a few 100 ms), confirming the pertinence of using the stick-and-diffuse model to interpret our FCS data [127]. The parameters extracted from our FCS measurements allow to calculate an estimate for f3D = τu/(τu + τb) = 0.67 and n = 2D1D τu / a = 325, if we assume D1D = 0.05 μm2/s (as measured for LacZ [128]). In this case, the limit on precision becomes δc/c = 11%, very close to the measured experimental value. Although further studies will be necessary to strengthen this hypothesis, this first estimate shows that facilitated diffusion provides, in the case of the Bcd/hb system, a possible resolution to the border precision paradox.

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