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* Instituto de Medicina Molecular, Faculdade de Medicina, Universidade de Lisboa, Lisbon, Portugal; and Laboratory of Receptor Biology and Gene Expression, National Cancer Institute, National Institutes of Health, Bethesda, Maryland
Correspondence: Address reprint requests to José Braga, Institute of Molecular Medicine, Faculty of Medicine, Av. Prof. Egas Moniz, 1649-028 Lisbon, Portugal. Tel.: 351-21-7999411; Fax: 351-21-7999412; E-mail: josebraga{at}fm.ul.pt.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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)). Inside the living cell, RNA molecules move between subcellular compartments and assemble into distinct macromolecular complexes that are highly dynamic over time and space. A variety of time-lapse microscopy techniques are currently available to track these movements using fluorescent tags that bind to the RNA (2).
In particular, fluorescence recovery after photobleaching (FRAP) is widely used as a tool to study molecular dynamics in vivo (3–6). FRAP is based on the local perturbation of the fluorescence steady state by inducing irreversible photobleaching with an intense light source, usually a laser. Then, due to the motion of unbleached molecules from regions not affected by bleaching, fluorescence relaxes to a new steady state. The rate by which this relaxation occurs is related to the overall mobility of the molecule: a higher mobility implies a faster recovery of fluorescence inside the bleached region. By bleaching a specific cellular region, FRAP experiments can be used to assess whether a fluorescently tagged molecule is either in constant exchange between two different pools or stably immobilized in a compartment (7–11). For mobile species, a simple step in extracting quantitative information from FRAP experiments is to calculate the half-time of recovery (12). Another approach is to fit recoveries to one exponential (13,14) or a sum of exponentials (15). However, care must be taken, as diffusion-like recoveries are apparently properly fitted with two exponentials, but this type of fitting gives incorrect information about the underlying process. Recent theoretical work (16) shows that recovery curves that seem to contain two recovery phases cannot necessarily be separated into two distinct processes occurring at different timescales (17). Estimating quantitative parameters, such as diffusion coefficient, immobile fractions, or binding rates from FRAP experiments, is a complex task for which several methods have been proposed (16,18–22). Of these, the most widely used in cell biology was the one developed by Axelrod for measuring diffusion rates (18,23–25).
Currently, a large body of evidence indicates that eukaryotic mRNAs form large ribonucleoprotein particles (RNPs) that are transported from the sites of transcription to the nuclear pores by random Brownian motion (18,26–31). However, estimates of the corresponding diffusion rate yielded values ranging from 0.03–0.04 µm2 s–1 (29–31) to 0.6 µm2 s–1 (18,27,32). In most of these experiments, the mRNAs were tagged with a small fluorescent probe, either an oligonucleotide or an RNA-binding protein. Since these probes are themselves mobile in the cell and they fluoresce regardless of whether or not they bind to the RNA, it was proposed that nonbound probe molecules may contribute to the fluorescence recovery after photobleaching and consequently lead to an overestimation of the mRNA diffusion rate (29). Here, we tested this possibility, and we show numerically how the binding affinity of a fluorescent probe to its substrate affects the measurement of the effective diffusion coefficient of the resulting complex. We also show that binding information can be obtained provided that the diffusion coefficients of the two species are known.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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Confocal microscopy and FRAP image analysis
Live-cell microscopy was performed on a confocal microscope (Axiovert 100 M with LSM 510 scanning module, Zeiss, Jena, Germany) using the PlanApochromat 63x/1.4 objective. EGFP fluorescence was detected using the 488-nm line of an Ar laser (25 mW nominal output) and a LP 505 filter. Cells were maintained at 37°C on a heating frame (LaCon GbR, Staig, Germany), in conjunction with an objective heater (PeCon GmbH, Zurich, Switzerland).
FRAP experiments were performed as previously described (18). Bleaching beam parameters were obtained from immobilized molecules, as described in Braga et al. (22). During scanning, the transmission of the acoustic optical tunable filter was set to 1% of laser power. Bleaching was performed at maximum transmission of the laser. The bleaching time was 110 ms for a circular bleach region of interest (ROI) of 0.71-µm radius. The bleached ROI was scanned, on average, 39 ms after the end of bleaching, and images were acquired with intervals of 78 ms.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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 | (1) |
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8 s) is much shorter than the time required to synthesize new fluorescent proteins (it takes 30–60 s to synthesize an average-sized eukaryotic protein). Consequently, the total amount of each species (including visible plus bleached molecules) is constant during the experiment. We assumed that bleached and unbleached molecules have exactly the same kinetic behavior and that bleaching does not affect chemical equilibrium, as it only disturbs the spatial distribution of the visible part of the system. Finally, we assumed that bleaching is a first-order linear process taking a finite amount of time. As previously shown, the axial extension of the bleached volume, even for a high NA objective, is larger than the cell thickness (22). This is consistent with a recent study showing that, with high illumination intensities, the dimensions of the bleaching beam can be larger than the theoretical expectations (33,34). Thus, as in other studies (16,35), we will consider that recovery of fluorescence is essentially two-dimensional. The limitations of this approximation are explored elsewhere (19)
The resulting reaction-diffusion system is mathematically translated as
 | (2) |
)
 | (3) |
The compartment under study, the nucleus, is considered to have a finite size with circular geometry (with radius Rnucleus). Bleaching is performed at the center of the circle. The boundary and initial conditions are then
 | (4) |
The dissociation constant of a chemical reaction (KD) is defined as
 | (5) |
 | (6) |
For some of our analysis, it is useful to eliminate 
and Seq from Eq. 2. For this, we denote the percentage of fluorescent molecules bound to the substrate before bleaching by the letter p. The equilibrium concentration of nonbound substrate species is thus given by
 | (7) |
) by
 | (8) |
In previous works, researchers quantitatively analyzed FRAP experiments by investigating the contribution of binding interactions to immobile substrates (DC = 0 µm2 s–1) (16,19,36,37). In this work, we aim to extend analysis of previous studies by considering the case of diffusing substrates (DC > 0 µm2 s–1). In this case, the expected FRAP behaviors over a broad range of reaction parameters will be analyzed in detail (see below). Equation 2 becomes
 | (9) |
To the best of our knowledge, no general analytical solutions are available to solve Eq. 9 with the boundary conditions given by Eq. 4. Thus, simulated FRAP curves are generated from the numerical solution of Eq. 9. The solutions were computed using the function NDSolve of Mathematica 4.0 (Wolfram Research, Champaign, IL). We compared simulated FRAP curves obtained for different radii of the nucleus (last line of Eq. 4.) and we found that if the size of the cellular compartment being analyzed is at least fourfold larger than the bleach spot, the differences are not significant. These computer-generated FRAP recovery curves were obtained using the actual experimental values for fluorophore and bleaching beam parameters. Assuming a Gaussian bleach, we determined the beam width in the radial direction, the size of the bleach ROI, and the bleach efficiency (K) according to Braga et al. (22). We also took into account the duration of bleach phase and acquisition parameters (such as the time between images, the starting time of the imaging phase, and the total duration of the FRAP experiment).
Considering that GFP (27 kDa) diffuses in cells at 33 µm2 s–1 (22), unbound GFP-PABPN1 molecules (60 kDa) are expected to diffuse at 25.3 µm2 s–1 (16).
Parameters that were specifically optimized for the simulations were the off-rate constant and the fraction of the tag bound to the substrate molecules (p). The optimization space was comprehensively explored by computing FRAP curves for many points in this space. The numerical solutions obtained were compared with the experimental data, and we selected the solution that minimized the sum of the squares of the residuals (
).
Fitting of experimental curves with a numerical simple diffusion model was performed by setting the fraction of bound molecules to zero. The parameter to be optimized was the diffusion coefficient of the free species.
Effective diffusion regimes for the reaction-diffusion model
For the special case when DC = 0, several studies (16,38) have demonstrated that the FRAP equations can be simplified under certain conditions. One simplification arises when the binding reaction is fast compared to the diffusion time. In this case, the FRAP equations can be reduced to a simple diffusion equation. The effective diffusion constant (Deff), however, is smaller than the free diffusion constant (DF) and given by Deff = DF(1 – p). This effective diffusion behavior has been observed for several different nuclear proteins, and so we investigated what the analog of this behavior would be for the case where DC is nonzero.
After bleaching, Eq. 2 becomes
 | (10) |
 | (11) |
is very fast relative to the time to diffuse at rate 
across the bleach spot, there is a local chemical equilibrium that arises throughout the FRAP recovery. In a local instantaneous chemical equilibrium, we have 
As a result,
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 | (13) |
 | (14) |
 | (15) |
Thus, in the case of fast reactions, the effective diffusion coefficient depends on the ratio 
Using the fact that 
Eq. 15 can be written as
 | (16) |
and the bound fraction diffusing at 
Note that this is a generalized effective diffusion coefficient and reduces to the "classical" one when 
Equation 16 shows that when effective diffusion is occurring, 
can be directly measured only if the vast majority of molecules are bound. In such a case, p = 1, and so 
Fitting FRAP experiments with simple diffusion and reaction-diffusion models
We previously reported the use of GFP-tagged PABPN1 to estimate the diffusion coefficient of poly(A)-RNA in the nucleus (18). Fitting the experimental FRAP recovery curves with a simple diffusion model according to Axelrod's method (18,25,39) resulted in an estimated diffusion coefficient of 0.6 µm2 s–1 and no immobile fraction (18). Similar values were obtained when the fitting was performed according to the method described by Braga et al. (22), which is also based on a simple diffusion model but takes into account diffusion of fast-moving molecules during the bleach period (Fig. 1 C). The sum of the square of the residuals () was found to be 0.0123, indicating that the fitting curve follows accurately the average values of the FRAP curve. Residual differences between the model and experimental data were at most 8%. Application of a numerical method considering simple diffusion yielded similar results (D = 0.56 µm2 s–1 (Fig. 1 D)), with a slightly higher value of 
and also low values of residuals (7%). Note that the only parameter being optimized in all of these cases is the diffusion coefficient.
Consistent with published fits (18,32), the data in Fig. 1, C and D, could mean that polyA RNA diffuses at a rate given by D 
0.6 µm2 s–1. However, in vitro assays reveal that binding of PABPN1 to poly(A)-RNA is a reversible process (40). Such binding interactions are also likely to occur in vivo, and therefore, rather than a pure diffusion model, a reaction-diffusion model is necessary to accurately describe the FRAP recovery of GFP-PABPN1.
To address this, we developed the reaction-diffusion model described above. In the model, GFP-PABPN1 is presumed to bind reversibly to poly(A)-RNA, and the two species, GFP-PABPN1 (the fluorescent, nonbound molecular form, F) and poly(A)-RNA (the substrate, S), are assumed to diffuse, though at different rates. The system is considered to be at a steady state by the time the FRAP experiment is performed, and all molecular species are homogenously distributed throughout most of the nuclear volume.
As a first test, we asked whether this reaction-diffusion model could explain the FRAP data if the diffusion constant for poly(A)-RNA was much smaller (DC = 0.04 µm2 s–1), namely, equal to the estimates obtained by other (non-FRAP) techniques (29,31). We also fixed the diffusion constant of the unbound GFP-PABPN1 at DF = 25.3 µm2 s–1, based on the measured diffusion constant of free GFP and the increased size of PABPN1. Then, the FRAP data were fit with two free parameters, the bound fraction (p) and the off rate (koff). Fitting was performed by choosing the numerically simulated recovery curve that minimized the sum of squares of the residuals () to the experimental curve (see Fig. 1 E). The minimum for 
occurred for koff = 22.2 s–1 and p = 97.9% (or 
), with residuals similar to the previous fit (Fig. 1 B). Thus, a reaction-diffusion model with a much slower diffusion constant for poly(A)-RNA can also account for the FRAP recovery.
To determine the sensitivity of the preceding fit to koff and p, we tested the behavior of in the neighborhood of the minimum (Fig. 2 A). We found that this function varies rapidly in the direction of p, but varies very smoothly in the direction of koff. For example, values of 
(a value only slightly higher than the absolute minimum) are tightly concentrated around p = 97.9% but the corresponding koff values spread widely from 14 s–1 to 30 s–1. In fact, Fig. 2 suggests that any koff value >10 s–1 will yield a good fit. Thus when the diffusion constants DF and DC are fixed, the FRAP data will yield a good estimate for the bound fraction p, but not for the off rate koff.
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is 0.0137 in all cases.
The results in Fig. 2 indicate that the reaction-diffusion model provides a good estimate of the bound fraction p, but not of koff, DF, or DC. The results in Fig. 1 show that good fits to the FRAP data can also be obtained with a simple diffusion model that does not incorporate a binding interaction at all, even though such interactions are likely to be occurring. This combination of circumstances has been observed before in the case of immobile binding sites (16,19), and is characteristic of a simplified form of the reaction-diffusion model known as effective diffusion. Thus, we wondered whether a comparable scenario could explain our observations with a model incorporating mobile binding sites.
To test this, we derived the effective diffusion simplification for the case of mobile binding sites (see previous section). We found that FRAP recoveries could also mimic pure diffusion under certain conditions, but with a generalized effective diffusion constant of Deff = (1 – p)DF + pDC. This result implies that the estimates for D in Fig. 1 could in fact reflect an effective diffusion constant, not pure diffusion of the poly(A)-RNA. If so, then the estimates obtained for D in Fig. 1 should also be consistent with the estimates obtained for p in Fig. 2 and the preceding equation for Deff. Using 
(as estimated by the numerical diffusion fit (Fig. 1 D)) and the corresponding DF and DC for each graph in Fig. 2, we obtain, using the effective diffusion equation above, p = 97.9% (Fig. 2 A), p = 98.2% (Fig. 2 B), p = 97.8% (Fig. 2 C), and p = 94.8% (Fig. 2 D). Thus, the predicted values for p correspond exactly to the estimates for p obtained from the full reaction-diffusion model, and the effective diffusion equation above also explains why the only parameter well determined by the fit is p. Together, these results strongly argue that effective diffusion is a reasonable explanation for the FRAP recovery of GFP-PABPN1.
Naturally, for the fit to be possible, DC should be lower than the effective diffusion coefficient measured with a simple diffusion model. However, 2.5-fold changes in DC or DF do not result in large changes in the estimates of p (Fig. 2). The values found by the optimization procedure in these different situations are in agreement and demonstrate that a large percentage of PABPN1 is bound to the mRNA (>94%) and that koff is >10 s–1. The reaction is too transient for koff to be accurately measured by photobleaching techniques. This situation was also found in previous studies (19). In fact, Eq. 16 shows that in an effective diffusion regime, only p significantly contributes to the effective mobility measurements.
In conclusion, our results show that FRAP recovery curves obtained with GFP-PABPN1 can be equally fitted by two distinct models. Assuming that the recovery after photobleaching reflects exclusively the dynamics of GFP-PABPN1 molecules bound to poly(A)-RNA, the estimated diffusion coefficient of the complex is 0.6 µm2 s–1, as previously reported (18). However, introducing the new model that takes into account that binding of GFP-PABPN1 to poly(A)-RNA is reversible in the cell, the experimental data becomes compatible with a diffusion coefficient of the complex an order of magnitude slower (0.04 µm2 s–1).
Influence of binding affinity on estimates of diffusion coefficient
The preceding analysis suggests that for the case of PABPN1 the full reaction diffusion equations for FRAP can simplify to effective diffusion behavior. To investigate more generally when this reduction to the simpler effective diffusion model may occur, we varied and koff over a large range while holding DF and DC constant, and compared a diffusion model solution to the full model solution by computing the between the two curves.
We found reasonably good agreement for most values of and koff (Fig. 3 A), with only a small subset of these values giving rise to clear differences between the FRAP curves (Fig. 3 A, red-outlined area). To determine whether these regions of good agreement could be explained by the effective diffusion theory, we plotted the difference between the predicted Deff (Eq. 16) and the Dest obtained from the diffusion model fit. This difference was negligible over a large region of the space (Fig. 3 B, area outside the blue curve), indicating that Eq. 16 for effective diffusion could account for all of the FRAP curves in this large region.
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1 (and 
is large) and, by Eq. 16, Dest = Deff DC. This situation arises in the area above the blue line in Fig. 3 C. Second, when virtually no molecules are bound to the mobile substrate, then p0 ( is small) and, by Eq. 16, Dest = Deff DF. This situation arises in the area beneath the green line in Fig. 3 C. The latter region is the analog of the pure diffusion domain in which binding interactions are negligible, as previously identified for the case of an immobile substrate (16). Note that although the region between the blue and red contour lines in Fig. 3 C was well fit by a diffusion model, Dest did not agree with Deff, as predicted by Eq. 16. We call this region the pseudoeffective diffusion domain (Fig. 3 D). Good fits of the diffusion model in this domain may be fortuitous, or they may reflect another, not yet identified, simplification of the full model equations.
In sum, several regions have been empirically identified by the preceding analysis (Fig. 3 D):
), occurs when binding is weak. In this case, a diffusion model can be used to fit FRAP data, and diffusion estimates are compatible with Eq. 16 and a negligible bound fraction (p 0). ), occurs when is larger than the characteristic diffusion time (
). In this case, a diffusion model can also be used to fit FRAP data and diffusion estimates are compatible with Eq. 16 and a measurable bound fraction (p > 0).
The preceding analysis demonstrates that a simple diffusion model will often yield a good fit to the full reaction diffusion model. However, Dest, as obtained from this simple diffusion fit, will in general not be the same as DC, the diffusion constant of the mobile substrate, which is actually the quantity of interest. As an example, if the fluorescent probe binds to its substrate with koff = 22.2 s–1 (a residence time of 50 ms), the effective diffusion coefficient measured will be at least an order of magnitude higher than DC, even if the bound fraction is as high as 98.5% (Fig. 4 A).
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and koff (Fig. 4 B) for points where the simple diffusion model yields a good fit (i.e., outside the red contour line). This plot shows that for very high binding affinities (large ), the value of Dest obtained from the simple diffusion fit is a very good estimate of DC. This occurs only when virtually all of the fluorescent tag is bound to the complex. As the binding affinity of the fluorescent molecule for the substrate decreases, the difference between Dest and DC increases. This is because an increasing fraction of fluorescent molecules become unbound, and instead freely diffuse, confounding the direct estimate of DC. In the extreme, virtually no fluorescent molecules are bound to the substrate, leading to Dest = DF, and thus Dest/DC = DF/DC = 25.3/0.04 625, which is the limiting case for the overestimate of DC with the values of DF and DC used here (Fig. 4 B, green contour line for pure diffusion regime).
As seen in Fig. 4 B, the amount that DC is overestimated changes as a function of the ratio When 
a 15-fold overestimate of DC occurs. This corresponds to the case for PABPN1 binding to RNA, in which the simple diffusion fit yielded Dest = 0.6 µm2 s–1, 15-fold larger than DC = 0.04 µm2 s–1 obtained by single-molecule tracking (29). In that study, mRNA diffusion rates were also estimated by FRAP using a different GFP-tagged mRNA binding molecule, namely MS2. The resultant FRAP curves for GFP-MS2 were then fitted with a simple diffusion model, yielding a value of Dest = 0.09 µm2 s–1. Viewed in light of our current analysis, this value may also be an overestimate due to the fact that MS2 may not be permanently bound to RNA. For MS2, Dest/DC = 2.25, corresponding to a predicted in vivo 
Thus, our reaction-diffusion model predicts that the in vivo affinity of MS2 for RNA (
) is significantly higher than that of PABPN1 for RNA (
), a result consistent with measurements showing that the in vitro affinity for RNA of MS2 (41) is much higher than the in vitro affinity for RNA of PABPN1 (40).
In sum, our analysis shows that the discrepant values for the mRNA diffusion rate previously reported based on PABPN1 and MS2 can be reconciled if these FRAP results are interpreted using a reaction-diffusion model.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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By exploring extensively the reaction parameter space, we studied the possible FRAP behaviors in the case of two interacting mobile species. We found that only in a small subset of points are the full reaction-diffusion equations required to fit FRAP data, and that for the majority of the cases, simple diffusion models yield good fits. However, extraction of binding information from fits should be done cautiously, as experiments performed in a pseudoeffective diffusion regime could be confused with proper effective diffusion behavior and lead to erroneous conclusions. In the effective diffusion regime, we show that, provided the diffusion coefficients of the free molecules and the complex are known, it is possible to accurately determine the percentage of fluorescent proteins bound to the substrate.
We have previously reasoned that if practically all nuclear mRNAs contain a poly(A) tail that is specifically covered by the PABPN1 protein, a fusion of PABPN1 to GFP could be used to analyze the dynamics of mRNPs in the nucleus (18). Because biochemical data and fluorescence loss in photobleaching experiments indicated that most (>95%) of the GFP-PABPN1 molecules expressed in cells were actually bound to poly(A)RNA (32), we considered that the complexes formed by GFP-PABPN1 and poly(A)RNA represented a single population. Accordingly, we used a simple single-component effective diffusion model to fit the experimental FRAP data, and we calculated a diffusion coefficient of 0.6 µm2 s–1. In this study, we show that the same experimental FRAP recovery curves can be equally fitted with an alternative model that considers the binding of GFP-PABPN1 to poly(A)-RNA as a reversible and not very tight process. The model assumes that, although at any given moment most GFP-PABPN1 molecules are bound to RNA, there is a constant exchange between bound and nonbound GFP pools. Free molecules were considered to diffuse at 25 µm2 s–1, according to their molecular weight, whereas the complexes were assumed to move at 0.04 µm2 s–1 (29). The results show that the effective diffusion coefficient measured by FRAP is much higher than the mobility of the mRNP complex, because fluorescent molecules not bound to RNA diffuse at rates 500-fold faster than bound molecules.
A recent study has elegantly avoided the problem of measuring the mobility of nonbound fluorescent probes through the use of molecular beacons that only fluoresce when hybridized to the specific target RNA substrate (31). Using this approach, mRNPs were found to diffuse at an average rate of 0.033 µm2/s (31). A very similar value (0.04 µm2 s–1) was reported for the diffusion of an mRNA tagged with multiple GFP-MS2 molecules, and measured by single-particle tracking (29,30). In this case, where the movement of a cluster of GFP-MS2 molecules is directly tracked, the reversible binding of the fluorescent tag to the mRNP is not a problem, because only fluorescent mRNPs are detected as single particles, whereas single, unbound GFP-MS2 molecules are not.
Theoretical approaches and computational simulations have been used to predict the time that an mRNA takes to reach a pore (32,42). Assuming that mRNPs move according to a three-dimensional Pearson-type random walk inside a spherical nucleus of 8-µm radius; that the mRNP particles do not interact with each other and cannot enter inside the volume occupied by nucleoli; and that the nucleus contains 2000 randomly scattered pores at the surface, each pore with a functional diameter of 40 nm (43), then the time an average mRNP particle takes to move from a random position in the nucleus to a nuclear pore by Brownian motion is 6.1 min for D = 0.033 µm2 s–1 and 20 s for D = 0.6 µm2 s–1 (32). Taking into account the results of classical pulse-chase experiments indicating that radioactively labeled mRNAs were detected in the cytoplasm 5–10 min after synthesis (44), we consider that a value within the range 0.02–0.04 µm2 s–1 most likely reflects an accurate estimate of the diffusion rate of an average mRNP in the nucleus.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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This work was supported by Fundação para a Ciência e a Tecnologia (Portugal) and the European Commission (LSHG-CT-2003-503259).
Submitted on September 11, 2006; accepted for publication January 5, 2007.
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REFERENCES |
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). In D, we used a numerical approach (see Results), which optimizes only for a single parameter, Deff. According to the reaction-diffusion model (E), the bound and nonbound pools of GFP-PABPN1 molecules are in constant exchange. The diffusion coefficients of the free and bound species (DF and DC, respectively) are fixed parameters (see Results for justification of values) and the fitting procedure yields the off rate of the reaction (koff) and the fraction of molecules bound to the mRNA (p). All fits (red lines) follow closely the experimental data and give similar minima for 
res2. Below each fit are plots of the residuals showing that experimental values differ from the estimates by <10% in all cases.
and the blue line the boundary of the region where 
(within a 5% tolerance). (D) Based on the findings from A–C, several regions can be identified: on the bottom right corner, the pure diffusion region; the full model region, where fits with a simpler diffusion model fail; the effective diffusion region,, in the upper part of the graph, where the diffusion model yields good fits, and simultaneously 
is constant, located in the positions predicted by