Introduction

Diffusion is a key factor underlying many physiological processes among them, the formation of morphogen gradients. Having reliable estimates of diffusion rates in cells is thus of great relevance. Optical techniques provide a means to obtain such estimates. A difficulty with their direct application in cells and embryos is that free diffusion, as first considered by Einstein[1], [2], rarely occurs in living organisms [3], [4]. In particular, in many occasions binding/unbinding processes hinder transport. When the resulting net transport is observed over a long enough time it usually recovers the properties of (normal) diffusion but with a diffusion coefficient that depends on concentrations and on the rates of binding/unbinding as well. A single species, , that reacts with slowly diffusing or immobile binding sites, , to form a complex , (1)has two distinct diffusion coefficients: a "collective'' one, , that governs the rate at which concentrations spread, and a "single molecule'' one, , that governs the rate at which the mean squared displacement of the individual particles increase with time [5]. Both types of coefficients are weighted averages of the free diffusion coefficients, , of the molecules, , and that of the binding sites, , that depend on the concentrations of the species involved. In the case of the scheme given by Eqs. (1) or (5) they read:(2)where is the dissociation constant of the binding/unbinding process and and are the unbound and total binding sites concentrations, respectively. If the molecules do not react but only diffuse freely within a simple solvent, it is . If they also bind/unbind to/from slowly moving sites, the ratio can be arbitrarily small [5]. We illustrate the difference between a situation with freely diffusing particles and with particles that diffuse and react by means of Videos S1 and S2, respectively. In both these simulations a bolus of fluorescent particles is initially added to an equilibrium situation. The subsequent spread of the deviation of the concentration of all the particles with respect to equilibrium (left most panels), of the concentration of added particles (center panels) and of the added particles (right most panels) are shown in these Videos. The rate at which these three quantities spread out with time are characterized, respectively, by the second moment of the distribution of all particles, the second moment of the distribution of added particles and by the mean square displacements of the added particles (see Materials and Methods). These are shown in Figs. 1 and 2. In both simulations, the quantities shown in the figures eventually scale linearly with time and diffusion coefficients can be estimated from the slopes (see Materials and Methods and supplementary text S3). We observe that for the freely diffusing particles all three slopes yield the same diffusion coefficient to within a few percent which coincides with the free coefficient of the particles ( in the simulation). In the case in which the particles interact with the binding sites the coefficient derived from the slope of the second moment of the distribution of all particles (Fig. 2 A) is an order of magnitude larger than the other two which coincide between themselves. The former corresponds to (Eq. (2) gives for the simulation parameters) and the other two to (Eq. (2) gives for the simulation parameters). Video S2 and Fig. 2 show that the spreading of the individual particles and that of the deviation with respect to equilibrium of the total particle concentration are eventually diffusive but with two different (effective) diffusion coefficients in the presence of binding/unbinding (for more details see supplementary text S1). The existence of one coefficient ruling the diffusion of individual particles and another one ruling the decay of concentration gradients also occurs in the context of non-ideal solutions [6] particularly those involving polymers [3], [7]. The combination of free diffusion and binding/unbinding processes can also result in what is called anomalous diffusion [3], [4], [8], [9]. The defining property of this type of transport is that, differently from normal diffusion, the mean square displacement of a molecule is not proportional to the time elapsed. In this Introduction we will limit the description to situations in which the observed transport has the properties of normal diffusion. We discuss the properties of free, anomalous and effective diffusion in more detail in supplementary text S1.

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Figure 1. Second moments of particle distributions and mean square displacements obtained from a simulation of freely diffusing particles.

The data for this figure (shown in green and red) comes from the simulation of Video 0.1 which corresponds to a system of particles that diffuse with in the absence of binding sites. The simulation starts with an equilibrium situation that is perturbed by adding fluorescent particles to the the central cube of the simulation volume. In this figure we characterize the rate at which the deviations with respect to equilibrium of the concentrations of all particles and of the added ones spread out with time by means of second moments. We compare these second moments with the MSD of the added particles. Please notice that we are not plotting the mean square displacements in A) and B) but a quantity (the second moment) that depends linearly with the time lag with the same slope as the mean square displacement. For more details see Materials and Methods. A: (shown in green) computed using Eq. (12) with the number of all particles in the box. Linear fit (shown in black). B: (shown in red) computed using Eq. (12) with the number of fluorescent particles in the box. Linear fit (shown in black). C: The mean of the squared displacements of the added particles (shown in red) computed using Eq. (11). Linear fit (shown in black). As explained in supplementary text S3 the diffusion coefficient, , can be estimated by taking 1/6 of the slope of the fitting curves. In this case the three estimates yield (A), (B) and (C). The second moment shown in A) corresponds to the "collective diffusion coefficient'', the one in C) to the "single molecule diffusion coefficient'' and the one in B) could be called . According to the theory all three should coincide in the case of freely diffusing particles and this is reflected in this figure.

https://doi.org/10.1371/journal.pcbi.1003629.g001

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Figure 2. Second moments of particle distributions and mean square displacements obtained from a simulation of particles that diffuse and react with immobile binding sites.

Similar to Fig. 1 but for the simulation of Video S2 which corresponds to a system of particles that diffuse with and react with immobile binding sites according to Eq. (1). The simulation parameters are such that the effective diffusion coefficients defined in Eq. (2) are and . As in the simulation with the freely diffusing particles, the simulation starts when a bolus of fluorescent particles is added to a background of (non-fluorescent) particles that are initially spatially uniform and in chemical equilibrium with the binding sites. Also in this case we compute two second moments and the averaged mean square displacement of the added particles to quantify relevant properties of the simulation. For more details see Materials and Methods. A: (shown in gren) computed using Eq. (12) with the number of all particles in the box. Linear fit (shown in black). B: (shown in red) computed using Eq. (12) with the number of fluorescent particles in the box. Linear fit (shown in black). C: The mean of the squared displacements of the added particles (shown in red) computed using Eq. (11). Linear fit (shown in black). As explained in supplementary text 0.3, in this case, the diffusive behavior sets in after a transient. Once this behavior is reached, diffusion coefficients, , and , can also be estimated by taking 1/6 of the slope of the three fitting curves. Differently from the situation of freely diffusing particles, in this case, the estimates differ from one another. They yield A: , B: , C: . According to the theory (see supplementary text 0.3), and and should be given by Eqs. (2). In fact, the values derived from the simulation satisfy and are pretty close to the theoretical values, and .

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In the effective diffusion regime, the optical techniques, fluorescence recovery after photobleaching (FRAP), and fluorescence correlation spectroscopy (FCS), provide information on the effective diffusion coefficients. FRAP is an optical technique that is commonly used to estimate the diffusion rate of fluorescently labeled proteins in cells [10], [11]. In FRAP the fluorescence is photobleached inside a small region. By measuring the time it takes for the fluorescence to recover the transport rate of the fluorescent species can be estimated. When the fluorescent species diffuses and undergoes binding/unbinding reactions and the transport is effectively diffusive FRAP prescribes the single molecule coefficient [5], [6], [12]–[14]. We illustrate this in Video S3 where we show the simulation of a FRAP -like experiment. The interaction between the binding sites and the particles is the same in this simulation as in Video S2. The circle that is predominant at the beginning is the (projection into 2 space dimensions of) the FRAP volume. The particles in the FRAP volume that are bleached are shown as blue at all times. The unbleached particles inside the FRAP volume are shown as red. The unbleached particles outside the FRAP volume are not shown. Fig. 3 A shows the recovery curve and the time of recovery for this "experiment''. From the recovery curve we estimate . The MSD graph yields . We see that FRAP and particle tracking give diffusion coefficients in reasonable agreement with that of in Eq. (2) (). These estimated effective diffusion coefficients are in rough accord with the single molecule coefficients obtained in the particle bolus simulation (Figs. 2B, C) and are an order of magnitude smaller than the collective diffusion derived from the same simulation (Fig. 2A). FCS is also commonly used to estimate diffusion coefficients of fluorescently labeled proteins. When the fluorescent proteins diffuse and react with other species FCS can give information on both effective coefficients [12]. Both FRAP and FCS produce time-dependent data whose interpretation requires an underlying mechanistic model. By fitting the experimental data to functions derived from the model, one may obtain estimates of model parameters. The choice of the mechanistic model is especially important when diffusion and binding/unbinding processes are involved since the fitted parameters need not correspond to fixed model parameters (e.g., free diffusion coefficients), but instead may be functions of space or time dependent quantities (e.g. concentrations). FCS and FRAP have been used to estimate the diffusion coefficient of the morphogen Bicoid in Drosophila melanogaster embryos giving values such that the one obtained with FCS is an order of magnitude larger than the one obtained with FRAP [15], [16]. In this paper we analyze these results using an underlying mechanistic model that provides a clear distinction between fixed parameters and model variables (see Materials and Methods). In this way we determine a consistent set of model parameters that explains the difference in the Bicoid diffusion coefficients obtained with FRAP and FCS.

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Figure 3. Simulated FRAP-like experiment.

A: Recovery of relative fluorescence, , given by Eq. (14) (shown in red), obtained from the simulated FRAP experiment depicted in Video 0.3. For the initial conditions of the simulation (a totally bleached spherical volume of radius ) the half recovery time, , (i.e. the time at which ) is related to the diffusion coefficient by . From the simulation we obtain . Using we derive . B: Single molecule MSD, , computed using Eq. (11) for the bleached particles, as a function of time (shown in blue) and linear fit (shown in black). The slope of the fitting curve is which yields an estimated diffusion coefficient of .

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Bicoid (Bcd) is a key morphogen for the organization of the anterior-posterior axis in Drosophila embryos [17], [18]. The inhomogeneous distribution of its concentration induces the differential expression of certain genes determining the embryo body plan along the axis [18]. This patterning starts with the deposition of maternal cues, among them the transcription factor Bcd, into the developing egg. About 2 hours after egg deposition Bcd is unevenly distributed in the embryo with a gradient of concentration that decays exponentially from the anterior to the posterior end. This gradient becomes stable within 80 min (at ) after deposition and remains stable during the next several nuclei divisions [16]. The question naturally arises as to what are the mechanisms by which this gradient is established so rapidly and precisely. To answer this question it is necessary to determine how the Bcd distribution depends on the localization and dynamics of the underlying Bcd mRNA and how Bcd is transported and affected by other processes inside the cell. The exponential distribution of the Bcd concentration is consistent with the so-called SDD model in which the protein is synthesized at the anterior end and subsequently diffuses and is degraded throughout the embryo [17], [18]. Within this model the Bcd diffusion coefficient is key to set the timescale over which the Bcd gradient forms and becomes stable. Gregor et al. [16] estimated the Bcd diffusion coefficient using FRAP during mitosis in embryos that expressed Bcd-EGFP (Bicoid fused to eGFP). Surprisingly, their estimate ) was an order of magnitude lower than the value that is necessary in the SDD model to account for the formation of a stable gradient within the observed times. Consequently, Spirov et al. [19] suggested an alternative model for the Bcd gradient formation and stabilization. Namely, they argued that the Bcd gradient is the reflection of an underlying bcd mRNA gradient. Later on, the diffusion coefficient of Bcd was again estimated but using FCS in the cytoplasm [15] and inside nuclei [20] of Bcd-EGFP expressing embryos. These experiments yield a set of values one of which was as fast as needed by the SDD model to explain the establishment of a stable gradient within the experimentally observed time. In view of this new estimate, the SDD cannot be discarded without first reconciling the two contrasting measurements of Bcd diffusion. Further support for the SDD model came from the results of Little et al. [21] according to which 90% of the Bcd mRNA is located within the anterior 20% of the embryo at any given time. Furthermore, including the observed mRNA gradient in an extended version of the SDD model, these authors concluded that the mRNA gradient could not account by itself for the protein gradient dynamics so that Bcd movement was necessary for the formation of its gradient. In view of these results, having reliable estimates of the rate at which Bcd diffuses in embryos becomes again most relevant. Abu-Arish et al. [15] not only estimated this coefficient using FCS but also performed FRAP experiments which yield a value of the same order of magnitude as the one obtained by Gregor et al. [16]. The question then arises as to what is the rate at which Bcd diffuses, the one given by FRAP or the one given by FCS? In order to answer this question it is necessary to understand why these values are so different. Abu-Arish et al. argued that their FRAP estimate was only a lower bound of the actual Bcd diffusion coefficient since the FRAP recovery half-time, , they determined was of the order of the photobleaching time, . However, as discussed in supplementary text S2, we do not expect the estimate determined by FRAP to be so far off from the actual value only because . Our explanation of the discrepancy between the FCS and FRAP estimates is based on our demonstration that these two techniques report different effective coefficients (, or ) when probing the transport of a substance that does not diffuse freely but also undergoes binding and unbinding [12]. Since the collective, , and the single molecule, , coefficients can be very different for molecules that diffuse and interact with slowly moving partners [5], we explain the disparate Bcd diffusion estimates by hypothesizing the existence of a significant pool of Bcd interacting molecules at the cortex during embryo development. Given that Bcd has demonstrated physical interactions with several proteins [22]–[24], and that it is able to bind specific mRNA species in the cytoplasm [25] it is reasonable to assume that Bcd does undergo binding/unbinding processes as it diffuses within the embryo. One argument in favor of this assumption is that FCS experiments performed using NLS-EGFP (a construct with a short nuclear localization signal and a GFP tag identical to that in Bcd-EGFP but that should diffuse freely in the cytoplasm [15]) yielded a larger diffusion coefficient than the one obtained for Bcd-EGFP with a difference that cannot be accounted for by the smaller size of NLS-EGFP relative to Bcd-EGFP.

In the present work we combine various published experimental results and interpret them within a biophysical model in which Bcd molecules interact with a single type of binding sites and can be fluorescent or not depending on EGFP maturation. Building upon the results of Pando et al. [5] and Sigaut et al. [12] we obtain a consistent set of values for the free diffusion coefficients, concentrations and dissociation constant of the model species that explains the difference in the Bcd-EGFP (effective) diffusion coefficients determined with FRAP and FCS. In view of the different physical meanings of the coefficients reported by FRAP and FCS we also conclude that the experimentally observed time it takes for the Bcd gradient formation is compatible with the SDD model.

Results

We use the simple biophysical model described in Materials and Methods to interpret the results of the FCS experiments of Abu-Arish et al. [15] performed to probe the transport of Bcd-EGFP in the cortical cytoplasm of the anterior region of embryos during interphases of cycles 12–14 and those of Porcher et al. [20] performed in anterior nuclei during cycles 13 and 14. In our model, fluorescent and non-fluorescent Bcd-EGFP molecules coexist, diffuse with free coefficient, , and interact with dissociation constant, , with a single type of binding sites, , that diffuse with free coefficient (see Table 1 for a complete list of symbols of the model). For the analysis we map the correlation times derived from fits to the auto-correlation function (ACF) of the fluorescence fluctuations presented in Refs. [15], [20] to analytic expressions that we derive for our model in terms of model parameters. Abu-Arish et al. [15] tried fits with different numbers of correlation times (or components). Those that provided the best results had two or three. In Ref. [20] only the results of fits with two components were presented. Our analytic ACF also has two or three components depending on whether the traps are immobile () or not with times that correspond to the effective coefficients, and , defined in Eq. (2) and to the free coefficient of the traps, (in case it is not zero). The mapping between the parameters of our ACF and those of the fits of Refs. [15], [20] is done by associating the components depending on the relative ordering of the times which in our case is (see Materials and Methods). We also analyze the results of FCS experiments performed using NLS-EGFP [15], [20]. Assuming that this construct does not interact with binding sites, the free Bcd-EGFP diffusion coefficient, , can be derived from the fits to these experiments taking into account a conversion factor due to the different molecular weights of both molecules. Thus, from the analysis of the correlation times derived from FCS experiments we determine both free and effective diffusion coefficients and, using Eqs. (2) and other properties of the model, concentrations and the dissociation constant of the reaction between Bcd and its putative binding sites. Our approach allows us to separate fixed parameters and variables so that we can analyze experiments performed under other conditions for which the variables can take on other values. In particular, we analyze the fluorescence recovery time obtained in the FRAP experiments of Abu-Arish et al. [15] and of Gregor et al. [16] which were performed during the mitosis following nuclear cycles 12 or 13 and determine that they are consistent with the parameters derived from the FCS experiments. We show the results obtained and some consistency tests in what follows. For more details we refer the reader to supplementary text S2.

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Table 1. List of main symbols used in this paper.

https://doi.org/10.1371/journal.pcbi.1003629.t001

FCS and FRAP yield consistent estimates of Bcd effective diffusion

We first analyze the results derived from FCS experiments performed in the cytoplasm during interphase [15]. From the experiments performed using NLS-EGFP we estimate the free Bcd coefficient, . From the experiments performed using Bcd-EGFP we derive the estimates , , and if we use the results of the three component fit of Abu-Arish et al. [15] while we obtain , and if we use the two component fit instead. Thus, our interpretation of the FCS experiments performed in the cytoplasm during interphase is that Bcd-EGFP has a relatively large free diffusion coefficient, , but that inside the embryo it also binds to sites that diffuse very slowly (with or less). The net Bcd-EGFP transport that results from its free diffusion and binding and unbinding to is characterized by two effective diffusion coefficients that differ by an order of magnitude ( and or and according to the three or two component fit estimates). As we mentioned before, FRAP yields the value, . In fact, the value derived for this coefficient from the FCS experiments using the three component fit is of the same order of magnitude as the one derived using FRAP by Abu-Arish et al. [15] () and the one obtained using the two component fit is closer to the result obtained with FRAP by Gregor et al. [16] (). However, we must remember that the FRAP and FCS experiments that we analyze here were performed during mitosis and interphase, respectively. Thus, we can expect the relevant concentrations and, thus, the effective diffusion coefficient values to be different. Assuming that the free coefficients, and , and the dissociation constant, , do not change between mitosis and interphase, we conclude that a 20% change of the free binding site concentration can explain a variation of between during interphase and during mitosis (see supplementary text S2).

Our interpretation of the timescales derived from the FCS experiments enables us to determine the ratio of concentrations and of with respect to any concentration in the cytoplasm during interphase at the location where the FCS experiments are performed. Using the estimated values of , in the cytoplasm during interphase ( from FCS experiments) and during mitosis ( from FRAP [15]) and assuming that the ratio of total Bcd concentrations in both situations, , is the same as that of fluorescent Bcd (which we estimate from Ref. [16]), we can also derive the ratios of all the concentrations during mitosis with respect to the total cytoplasmic Bcd concentration during interphase at the location of the FCS experiments, . We show in Table 2 the ratios derived using the parameters of the two component fit of Abu-Arish et al.[15] (first value listed in each cell) and the three component fit (second value listed in each cell). In all cases, the values listed were derived from the mean values obtained with the fits.

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Table 2. Estimates of equilibrium concentrations and of model parameters derived from experiments performed in the cytoplasm.

https://doi.org/10.1371/journal.pcbi.1003629.t002

The concentration changes along the axis of the embryo. Thus, the concentration ratios, and, consequently, the effective diffusion coefficients, , and , given by Eq. (2) could vary along the axis as well. We do not know what the binding sites are. If we assume that they are uniformly distributed in the cortex along the axis of the embryo as nuclei are and that does not vary either then, , and , would only change along the axis due to changes in . Using Eq. (2) and the relations that the various concentrations satisfy at equilibrium we can rewrite the expressions for and in terms of the ratios , and (see supplementary text S2). In particular, setting equal to the value derived from the FCS or the FRAP experiments we can determine how and vary with cytoplasmic for the interphase () or the mitotic () conditions, respectively. We show in Figure 4 plots of (solid line) and of (dashed line) obtained in this way using the total concentration of binding sites derived for interphase () and a plot of (dashed-dotted line) using the total concentration of binding sites derived for mitosis (). Based on this Figure we conclude that the dissociation constant, concentrations and free diffusion coefficients of the species involved are such that for a wide range of values, which include both the ones probed with FRAP and FCS. Therefore, it is reasonable that the two techniques report widely different diffusion coefficient estimates.

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Figure 4. Dependence of the effective diffusion coefficients on the total cytoplasmic Bcd concentration,

, for fixed values of the total concentration of binding sites, , and of the dissociation constant, , as prescribed by our theory. The solid and dashed curves correspond, respectively, to and for the estimated value of at interphase ( inferred from FCS experiments). The dashed-dotted curve corresponds to for the estimated value of during mitosis ( inferred from FRAP experiments). We used in all cases. The symbols correspond to the situations probed with FCS (circles) and FRAP (triangle) experiments for which , is equal to 1 and 1.2, respectively.

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Consistency test I: The ratio of nuclear to cytoplasmic total Bcd concentration estimated from FCS fits is within the observed ratios of mature [Bcd-EGFP]

The ratio of the nuclear to cytoplasmic concentrations of mature Bcd-EGFP may be estimated from the data presented by Gregor et al. [16]. It can also be derived from the results of FCS experiments under some assumptions. In the absence of detailed experiments, we consider the simple assumption that the dissociation constant of the binding/unbinding processes that Bcd undergoes in the nuclei and in the cytoplasm is the same. In this way we can combine the analyses of the FCS experiments performed in the cytoplasm [15] and in the nuclei [20] to obtain the ratio of nuclear to cytoplasmic Bcd concentration. Here we analyze to what extent these two estimations match each other.

Porcher et al. [20] only report the parameters of a 2-component fit to the ACFs obtained using NLS-EGFP and Bcd-EGFP. Working as in the case of the experiments of Abu-Arish et al. [15], we obain the values listed in Table 3 where is the total Bcd concentration in nuclei at the location where FCS experiments were performed. Again, only the values derived using the mean fitting parameters are listed in the table. In order to check the compatibility of the results of Tables 2 and 3, we assume that the between Bcd and the putative binding sites is the same in the cytoplasm and in nuclei. We then combine the ratio of Table 3 with the value derived from the 2-component fit listed in Table 2 () to determine . We obtain , which is within the ratios of mature [Bcd-EGFP] reported by Gregor et al. [16]. This ratio is reduced by a half if we assume that the dissociation constant in the cytoplasm during interphase is twice as large as the one in nuclei.

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Table 3. Estimates of equilibrium concentrations and of model parameters derived from experiments performed in nuclei.

https://doi.org/10.1371/journal.pcbi.1003629.t003

From ratios to absolute concentration values

Using the relative weight of the various components of the ACF, the fraction, of fluorescent to total Bcd-EGFP molecules can be estimated. However, as described in supplementary text S2, there are some uncertainties regarding the correct expression for the weights. Using different expressions we estimate to be between 0.7 and 0.99. The total concentration of fluorescent Bcd, , is not very well known either and a wide range of values, , is given in Abu-Arish et al. [15]. The relationship between in nuclei and in the cytoplasm on the other hand varies along interphase which adds another uncertainty. We use , which is among the possible ones, and the rough estimate to convert ratios (Tables 2 and 3) to absolute values of concentrations and of the effective dissociation constant between Bcd and its binding sites. In particular, using the same value of when combined with the 2-component FCS fitting parameters obtained in nuclei and in the cytoplasm during interphase, , and , we obtain the values of Table 4. These values, however, should be considered with great care due to all the uncertainties involved in their derivation.

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Table 4. Absolute values of concentrations and of the binding/unbinding dissociation constant.

https://doi.org/10.1371/journal.pcbi.1003629.t004

Discussion

We have considered a simple biophysical model to analyze the different experiments that have been done to estimate the rate at which Bcd diffuses in Drosophila embryos. We have shown that the disparate estimates obtained using FRAP [16] and FCS [15] are perfectly consistent within this simple model. Namely, they can be explained in terms of two distinct effective diffusion coefficients, and [5]. In our simple biophysical model effective diffusion coefficients describe the net transport that results from the combination of free diffusion and binding/unbinding processes when this transport is observed over a long enough time. The collective diffusion coefficient, , describes the rate at which concentration inhomogeneities spread out with time while the single molecule one, , characterizes the distance that an individual molecule travels during a given time. As illustrated by Videos S1 and S2 and Figs. 1 and 2 both coefficients coincide in the absence of the inter-particle coupling that the binding/unbinding processes introduce but otherwise can be arbitrarily different between themselves, with always equal or smaller than . The existence of two different diffusion coefficients, one that describes the mean-square displacement of a molecule and another that gives the rate of decay of a concentration gradient also occurs in crowded, non-ideal solutions, particularly those involving polymers [3], [6], [7]. In our model the interaction between the molecules of Bcd that underlies the existence of the two disparate transport rates is provided by the presence of binding sites with which Bcd interacts. The time after which the net transport can be described by these effective coefficients depends on the relationship between the diffusive and reaction timescales [12]. Once this occurs, FRAP experiments give information on the single molecule coefficient [5], [6], [12]–[14] while those that use FCS can give information on both [6], [12]. Particle tracking experiments give the single molecule coefficient as well. In FCS the autocorrelation function of the observed fluorescence fluctuations is computed and subsequently fitted to determine correlation times, and, from them, diffusion coefficients. In this paper we have analyzed the experimental data of Abu-Arish et al. [15], Gregor et al. [16] and Porcher et al. [20] under the assumption that the timescales are such that the derived transport coefficients correspond to effective ones. One could wonder what conclusions would be drawn if this were not the case. It is under this assumption, however, that we can explain the disparity between the diffusion coefficients estimated using FCS and FRAP. It has been argued [15] that the disparity could be due to an experimental limitation of FRAP. Namely, the recovery time derived from FRAP by Abu-Arish et al. [15] is of the same order of magnitude as the time it takes to photobleach the observation volume. This means that once the photobleaching is over and the recovery is monitored there is a noticeable fraction of bleached molecules outside the observation volume. If the data is fitted as if this fraction were negligible the recovery time and, consequently, the diffusion coefficient, are understimated [26]. Numerical simulations of our simple model show that this effect cannot account for over an order of magnitude difference between the coefficients determined using FRAP and FCS (see supplementary text S2).

Our approach differs from fitting "blindly'' the experimental data since, by using explicit expressions for the correlation times (and the weights) in terms of the parameters of an underlying biophysical model, we can combine observations performed under different experimental conditions and, in this way, estimate free (instead of effective) diffusion coefficients, concentrations and the reaction dissociation constant. According to our analyses, the experiments of Abu-Arish et al. [15] and of Gregor et al. [16] are compatible with Bcd having a free diffusion coefficient and interacting with immobile or slowly moving sites (). The transport rate of Bcd is then limited by these two values and is larger the larger its concentration. We have also determined that in the region where FCS experiments are performed, the majority of Bcd (%) is bound to sites and that a similarly large fraction of sites is also bound. This implies that the affinity of Bcd for the binding sites is high. Although Bcd physically interacts with several proteins [22]–[24], it is probably its binding to mRNAs [25] that most significantly affects its diffusion. First, mRNAs are relatively large, and thus diffuse more slowly than proteins. Second, Bcd has already been shown to bind tightly to the homogeneously distributed caudal mRNA. In addition when in the nucleus, it might spend a significant amount of time bound to DNA. Actually, it has been determined that Bcd binds cooperatively to multiple sites of DNA and that this results in a higher affinity () [27]. Our model is very simplified regarding binding. It is implicit in the scheme (5) that the sites act independently of one another. If we replace this scheme by one with cooperative binding we expect the estimated values of and to be smaller than the ones derived using the simple model (5). This, in turn, would imply a larger value of (see supplementary text S2). Thus, the estimate of the relationship between the dissociation constant and the total concentration of Bcd listed in the Tables should be considered as some sort of effective value. A simple scheme like the one in (5) was used in the model introduced by Deng et al. [28] to study the dynamics of the Bcd gradient in Drosophila embryos. According to the analysis presented by these authors the stability of [Bcd] inside nuclei and other properties are dependent on the binding/unbinding equilibrium of Bcd molecules to DNA sites. Deng et al. [28] explore the parameter space of their model under the assumption that and determine that a guarantees the stability of the Bcd gradient along division cycles. Their estimate of , however, depends on the assumed value of . It would have been much smaller if they had assumed , the value that we deduce from our analysis of the experiments of Abu-Arish et al. [15].

Our interpretation of the experimental results of Abu-Arish et al. [15] and of Gregor et al. [16] can be probed with other experiments. In particular, the non-uniform distribution of Bcd along the embryo implies that the effective coefficients that may be estimated with FCS or FRAP could, in principle, vary with position too. We can compute by how much they should vary with the distance to the location where FCS and FRAP experiments are usually performed (the anterior pole) if we assume, as before, that the concentration of binding sites is spatially uniform. In particular, assuming that the total Bcd concentration decays exponentially with a characteristic lengthscale we can go from Figure 4, in which the coefficients are plotted as functions of to a figure in which they are plotted as a function of the distance to the typical FCS location, . We show the results obtained in Figure 5 where we have plotted and as functions of . There we observe that at , , is reduced to about 20% of its value at the anterior pole. Although this numerical estimate is rough, we expect that changes of along the embryo should be detectable using FCS.

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Figure 5. Effective diffusion coefficients and Bicoid concentration as functions of position along the embryo.

Top: Effective diffusion coefficients, with solid lines, with dashed lines, as functions of the distance to the location where FCS and FRAP experiments were performed, , for a uniform concentration of binding sites consistent with its estimated cytoplasmic value during interphase () and for a total concentration of Bcd that decreases exponentially with distance with a decaying length. Bottom: corresponding exponential profile of total Bcd.

https://doi.org/10.1371/journal.pcbi.1003629.g005

Albeit with its uncertainties, our approach provides a self-consistent picture of a variety of observations. The establishment of the Bcd gradient is a nonlinear process and an accurate estimate of the time it takes to develop should be obtained with a reaction-diffusion model in which diffusion, binding and unbinding are described separately. The actual transport of Bcd is not purely diffusive although it can be characterized by effective diffusion coefficients that are concentration-dependent and vary along a gradient. This means that the transport of Bcd involves a multiscale diffusion process, to some extent, similar to the process analyzed by Daniels et al. [29]. In any case, it is the collective, rather than the single molecule effective coefficient that gives a rough estimate of the (local) rate at which concentration inhomogeneities spread out with time. Within our interpretation of the results of Abu-Arish et al. [15] it is . Thus, a gradient over a lengthscale could be established within 48–70 min. Bcd, on the other hand, regulates the expression of various genes and its gradient plays a relevant role since certain proteins are synthesized at very specific locations along it. The very small number of Bcd molecules implies that fluctuations are important. The expression of these downstream genes, however, occurs with high precision. In particular, the analyses of Gregor et al. [30] have estimated this precision at 10%. As discussed by these authors, the physical limit to concentration measurements is determined by the dynamics of molecules arrivals at their targets. This, in turn, is determined by the diffusion coefficient of the molecules. Based on the estimate of this coefficient obtained using FRAP [16], the studies of Gregor et al. [30] concluded that the system would need a very long time ( hours) to average out the fluctuations to obtain the observed level of precision. The authors then invoked an average in space to reconcile the estimate of the diffusion coefficient of Gregor et al. [16] and the 10% precision of the read-out mechanism. We must recall that these computations used the estimate of the diffusion coefficient obtained with FRAP thus, the single molecule coefficient, . However, it is , not , that determines the mean time of separation between subsequent arrivals of the signaling molecules (Bcd) at their targets. According to our estimates, is at least an order of magnitude larger than (and it could be twice as large as the one used by Gregor et al. [30]) at the location where the FCS experiments were performed. In particular, computing the time, , it could take to achieve a 10% precision as done by Gregor et al. [30] before invoking the spatial averaging but with we obtain which fits within a nuclear cycle. We must recall, however, that both and would decrease with the distance to the anterior pole if the concentration of binding sites remains constant along the embryo (see Fig. 5). Thus, it is not certain that the value determined from the FCS experiments is the one that should be used. A more detailed study is necessary to address the problem of the read-out mechanism precision.

Our approach involves several simplifications, in particular, the assumption that Bcd interacts non-cooperatively with a single type of binding site. For the analysis of the FCS experiments of Abu-Arish et al. [15] we also assume that the system is in a regime such that the net transport of Bcd can be described in terms of effective diffusion coefficients. This is supported by the goodness of the fits presented by Abu-Arish et al. [15], although a picture including anomalous diffusion could also hold. In favor of our model we have shown that it is self-consistent, but because of all its simplifications the numbers we derive might not be completely accurate. More detailed studies are necessary to draw a more definitive picture. In any case we do think that it is the coupling between Bcd molecules that is introduced by the binding with almost immobile sites that can explain the disparate values of Bcd diffusion estimated with FRAP and FCS, solve the problems associated with the timescale of the Bcd gradient formation and help understand the precision of its read-out mechanisms.

Materials and Methods

Supporting Information

Acknowledgments

We acknowledge Cecile Fradin and Thomas Gregor for their careful reading of the manuscript and their useful comments. JEP would like to acknowledge useful conversations with Jeff Drocco, Nick Hengartner, Brian Todd, M. Malek Mansour and Ray Kapral. JEP also acknowledges the hospitality of the Kavli Institute for Theoretical Physics in Beijing, China. SPD would like to acknowledge the hospitality of Ian Parker at UC-Irvine where part of this work was written.