Rotavirus VPs are complex signaling hubs composed of viral and cellular proteins, packed together with viral RNAs. By TEM, they roughly resemble circular electrodense structures whose internal components lack an obvious degree of spatial organization (Altenburg et al., 1980; Eichwald et al., 2012). In this work, we determined the relative spatial distribution of VPs components by immunofluorescence and SRM in MA104 cells infected with the rhesus rotavirus strain RRV at 6 hr post-infection (hpi), using protein-specific antibodies. Due to their important role as nucleating factors during VP biogenesis, we selected either NSP2 or NSP5 as spatial relative reference for the distribution of the VP1, VP2 and VP6 proteins. VPs were optically sectioned through total internal reflection fluorescence microscopy (TIRF), with an excitation depth of field restricted to 200 $nm$ from the coverslip. This approach avoids excitation of fluorophores marking structural components located away from this plane, that is towards the inner cellular milieu. Additionally, NSP2 was also co-immunostained with the viral outer layer protein VP4 as well as with the ER resident proteins NSP4 and VP7, all of which have been reported to form separate ring-like structures that closely associate with VPs (González et al., 2000). In order to gain more insight into the morphogenesis of rotavirus, we analyzed the distribution of both VP7 monomers (VP7-Mon) and trimers (VP7-Tri) since this protein is assembled into virus particles in the latter form (Kabcenell et al., 1988). The nanoscale distribution of VPs was then analyzed through 3B-SRM, with improvements in the technique, developed in the present work, to solve nanoscopic structures (‘Stochastic model fitted for 3B super resolution microscopy’Appendix 1). By different methods of analysis VPs exhibit roughly a circular shape (Figure 1A–E). However, unlike the diffraction-limited image (Figure 1B), in super-resolution microscopy structural details of VP are appreciated, like the different layer distributions of viral components with respect to NSP2 (Figure 1C–E). In addition to VPs, by diffraction-limited TIRF microscopy we detected in the cytoplasm several small and dispersed puncta of fluorescence (Figure 1B), and in these images it is also sometimes possible to differentiate the distribution of NSP2 from that of VP4, a closely viroplasm-associated viral protein (see also González et al., 2000); in this case, VP4 is detected as a ring-like structure that surrounds the VP. Nevertheless, the small size of the VPs effectively precludes measurement of component distribution for the majority of its structural elements, as their separation is below the spatial resolution of typical optical microscopes. In contrast, images obtained by 3B-SRM do allow the study of the relative distribution of the VP components (Figure 1C–E). In the case of SRM images of VP4 (Figure 1C), we observed that this protein forms a ring-like structure that does not colocalize with NSP2, and also ribbon-like projections that extend towards the cytoplasm, details that were not apparent in images captured with conventional fluorescence microscopy (Figure 1B). Additionally, we observed that the small puncta of proteins detected in the cytoplasm were in fact ribbon-like structures composed of various viral proteins that may represent different organization forms of the viroplasmic proteins (Figure 1C). In this regard, it is interesting to note that both NSP2 and VP4 have been reported to have at least two different intracellular distributions (González et al., 2000; Nejmeddine et al., 2000; Criglar et al., 2014). An examination of 3B-SRM images of VPs (Figure 1C–E) revealed that the viral components form ring like structures within the VPs and are arrayed as rather discrete concentric layers. As seen in Figure 1C–E, we find that although the structural proteins VP1, VP2 and VP6 partially overlap in position with NSP2, the bulk of the proteins form separate and distinct layers. Also, the monomeric as well as the trimeric forms of VP7 are clearly distinguished from NSP2, forming an outer ring. Of interest, the spatial distribution of NSP4 colocalized with that of NSP2, an unexpected result since, as mentioned, NSP4 is an ER integral membrane protein (see the Discussion section), and as such it was expected to colocalize with VP7 rather than with an internal viroplasmic protein (Petrie et al., 1984). With regard to NSP5, it was observed distributed inside the ring formed by NSP2 (Figure 1E).

Figure 1

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A qualitative analysis of the distribution of the VP components through 3B-SRM suggested that these are arranged as concentric spherical shells; thus, we set out to quantitatively validate the circularity of the VP shape. For this, we developed a segmentation algorithm based on a least squares approach, which we called ‘Viroplasm Direct Least Squares Fitting Circumference’ (VP-DLSFC) (see ‘Segmentation Algorithm’ in Appendix 1), to measure the spatial distribution of the components within individual VPs by adjusting concentric circumferences. This method is automatic, deterministic, easy to implement, and has a linear computational complexity. The performance of VP-DLSFC was tested on approximately 40,000 ‘ground truth’ (GT) synthetic images, showing a high robustness to noise and partial occlusion scenarios. Additionally, we compared our method with two other alternative methods (Gander et al., 1994), and our approach displayed an improved performance (see ‘Algorithm Validation’ in Appendix 1). Based on this new algorithm, we find that the mean radius of the NSP5 distribution was smaller than that of NSP2, suggesting that NSP5 is located in the innermost section, as a component of the core of VPs (Figure 2A). On the other hand, the distribution of the structural proteins VP1, VP2 and VP6 exhibit slightly larger mean radii than that of NSP2, and are thus primarily localized in a zone surrounding NSP2. Continuing further towards the outer regions of the VP, we observed a region occupied by the spike protein VP4. Finally, the two different forms of VP7 (VP7-Mon and VP7-Tri) were located together, close to the most external region of the VPs (Figure 2A). The distribution of the glycoprotein NSP4 showed a similar mean radius to that of NSP2 (around $0.4\mu m$) suggesting, as described above, that these two proteins are located in the same structural layer of the VP (Figure 2A).

Figure 2

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In order to confirm our preliminary observations and clarify the nanoscopic organization of the VPs, we evaluated the relative separation between NSP2 and each accompanying protein. Again, the results show a remarkable degree of organization in the structure of the VP (Figure 2B). As predicted from Figure 2A, we found that NSP5 is located in the internal part of the VP, in close proximity ($\approx 0.05\mu m$) to the area occupied by proteins NSP2 and NSP4, which themselves show the closest association. After the NSP2-NSP4 region, VP6 occupies a middle region at $\approx 0.05\mu m$ from NSP2, followed by the VP4 protein, which were located at a distance of $\approx 0.18\mu m$. Finally, the VP7-Mon and VP7-Tri were situated at $\approx 0.38\mu m$ from NSP2 (Figure 2B). A Mann-Whitney test showed that the distances of the various viral components in relation to NSP2 were significantly different (Figure 2B), suggesting that they are situated in specific areas of the VPs. The two forms of VP7 were located at the same distance to NSP2, suggesting that the formation of trimers of VP7 takes place at the ER membrane, where the VP7 monomers should also be located. Note that in Figure 2B the relative distance of VP1 and VP2 to NSP2 was not included, since the radii obtained for NSP2 in these two combinations were significantly smaller than those found when it was determined in combination with the other VP components (see ‘Supplementary Exploratory Analysis’). In addition to this, we found no significant differences between the distance of both VP1 and VP2 to NSP2 (Figure 2C). Nonetheless, based on the inferential analysis, we could place these two proteins in the same layer as VP6 (see below).

Next, through a hierarchical cluster analysis, we studied the relationship between the components of the VP, taking into account multiple variables at the same time, like the mean distance to NSP2 [‘Mean(Dist)”], the standard deviation of the distance to NSP2 [‘Std(Dist)”], the mean radius of NSP2 [‘Mean(NSP2)”], and the radii of the other proteins [‘Mean(Other)”] (Figure 2D). Note that the proteins within a cluster should be as similar as possible and proteins in one cluster should be as dissimilar as possible from proteins in another. Because our variables are related with the distance to NSP2 and the radii of the proteins, this is a no-parametric analysis that should provide evidence about the spatial distribution/order of the viral proteins into the VP. As we are considering the distance to NSP2, VP1 and VP2 were not included in this analysis. The first level of the hierarchical agglomerative cluster (Figure 2D, left) partitioned the VPs and the surrounding proteins in five clusters, composed by NSP4, NSP5, VP6, VP4 and {VP7-Mon, VP7-Tri}, which suggest that these five proteins compose different layers of the VP. The second agglomerative level merged into the same group the proteins NSP4 and NSP5, meanwhile VP6 and VP4 continue as independent clusters, which indicate that NSP5 and NSP4 are closer to each other than to VP6 and VP4 in the VP. In the third level, VP6 and VP4 are clustered in the same group, and as consequence are more related between them than with the others viral proteins. The subsequent groups in the clustering analysis indicate that VP7 remains as an independent layer with respect to the other proteins. Based on this analysis, the viral proteins seem to be highly organized, with VP7 conforming the most external layer, while NSP5, NSP4, VP6 and VP4 are distributed very close but as independent layers. The clusters between NSP5-NSP4 and VP6-VP4 suggest that these two pairs of proteins (in each cluster) conform continuous layers in the VP.

The scatterplot between the radius of the spatial distribution of NSP2 (independent variable, x-axis) and the radius of the distribution of other viral components (response variable, y-axis) showed a strong linear relationship (Figure 3A). The distribution of NSP5 grows $0.87\mu m$ for each $1\mu m$ increase in the radius of NSP2 (slope interpretation), whereas the radius of the distribution of NSP4 increases $0.99\mu m$ (Figure 3B). These findings indicate that NSP5 is distributed in a proportionally smaller region than NSP2 regardless of the absolute size of the VP, supporting our observation that NSP5 is a constituent of the core of the VP. Moreover, the fact that the increase in the radius of the fitted distribution of NSP4 is directly proportional to the same parameter measured for NSP2 supports the idea that these proteins are both constituents of a putative second layer. VP1, VP2 and VP6 exhibit similar slopes which diverge between 0.03 and 0.05 μm (Figure 3B and Appendix 1—table 6); thus, these results confirm that VP1, VP2 and VP6 are components of the same layer in the VPs which, from the data in Figure 3, is located just after the layer of NSP2 and NSP4. Finally, as noted in our quantitative analysis, VP4 and VP7 form consecutive external layers with a slope of 1.39 and 1.94 μm, respectively (Figure 3B and Appendix 1—table 6). These findings indicate that the spatial distribution of the viral components in the VPs and in the surrounding areas is conserved regardless of their absolute size, and also form the basis of a predictive model, where, for a given radius of distribution of NSP2, it is possible to predict the radii of the remaining VP components (NSP5, NSP4, VP1, VP2, VP6) and of VP4 and VP7 proteins. This predictive model is available as a web app at https://yasel.shinyapps.io/Nanoscale_organization_of_rotavirus_replication_machineries/. The mathematical details and the residual analysis that validate these linear models are available in the Appendix 1, section ‘Linear dependency between the viral components’, Appendix 1—table 6 and Appendix 1—figure 9.

Figure 3

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In order to confirm the observed structural organization of VPs, we analyzed two more experimental conditions in which we chose a different reference protein for pairwise comparisons. The first was based on the distribution of NSP5 and its comparison with the relative localizations of VP6 and VP4, and the second considered NSP4 as the reference protein to compare with the distribution of VP6. We found that both analyses produced an identical structural organization for the VPs, with a comparative localization error of approximately $0.05\mu m$ between models (close to the effective resolution limit of the 3B algorithm; see ‘NSP5 and NSP4 as reference proteins’ in Appendix 1). An extensive quantitative validation regarding the congruence between the NSP2, NSP5 and NSP4 models is available in the Appendix 1.

Based on our extensive quantitative, descriptive and inferential statistical analyses, we propose that the VP and the surrounding viral proteins form an ordered biological structure composed of at least five concentric layers organized as depicted in Figure 4. In this structure, NSP5 constitutes the innermost layer, followed by a {NSP2-NSP4} layer. Then, there is a layer composed by {VP1-VP2-VP6} and two consecutive external layers formed by VP4 and VP7. The different layers of proteins are most likely highly porous to allow the entry of positive-sense single-stranded viral RNA (+RNA) during genome replication and also of the antibodies used for VP staining.

Figure 4

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The rhesus monkey kidney epithelial cell line MA104 (ATCC) was grown in Dulbecco’s Modified Eagle Medium-Reduced Serum (DMEM-RS) (Thermo-Scientific HyClone, Logan, UT) supplemented with 5% heat-inactivated fetal bovine serum (FBS) (Biowest, Kansas City, MO) at 37°C in a 5% CO₂ atmosphere. The cells were confirmed to be free of mycoplasm by testing with the INTRON Mycoplasma PCR Detection Kit (#25234). Rhesus rotavirus (RRV) was obtained from H. B. Greenberg (Stanford University, Stanford, Calif.) and propagated in MA104 cells as described previously (Pando et al., 2002). Prior to infection, RRV was activated with trypsin (10 μg/ml; Gibco, Life Technologies, Carlsbad, CA) for 30 min at 37°C.

Monoclonal antibodies (MAbs) to VP2(3A8), VP4 (2G4), VP6 (255/60), VP7 (60) and VP7 (159) were kindly provided by H. B. Greenberg (Stanford University, Stanford, CA) (Shaw et al., 1986; Greenberg et al., 1983). The rabbit polyclonal sera to NSP2, NSP4 and NSP5, and the mouse polyclonal serum to NSP2 were produced in our laboratory (González et al., 1998). The hyperimmune serum to NSP4 (C-239) was generated in our laboratory by immunizing New Zealand white rabbits with a recombinant protein expressed in E. coli with a histidine-tail, representing the carboxy-terminal end (amino acids 120 to 175) of the rhesus rotavirus RRV NSP4 protein; see also Maruri-Avidal et al. (2008), in which this serum was used. The hyperimmune serum to VP1 was also generated in our laboratory by immunizing BALB/c mice with a recombinant protein expressed in E. coli with a histidine-tail, representing amino acids 227 to 539 of the rhesus rotavirus RRV VP1 protein. Goat anti-mouse Alexa-488- and Goat anti-rabbit Alexa-568-conjugated secondary antibodies were purchased from Molecular Probes (Eugene, Oreg.).

MA104 cells grown on glass coverslips were infected with rotavirus RRV at a multiplicity of infection (MOI) of 1. Six hours post infection, the cells were fixed with and processed for immunofluorescence as described (Silva-Ayala et al., 2013). Finally, the coverslips were mounted onto the center of glass slides with storm solution (1.5% glucose oxidase $+100$ mM $\beta$-mercaptoethanol) to induce the blinking of the fluorophores (Dempsey et al., 2011; Heilemann et al., 2009).

Cells grown in 75-$c{m}^2$ flasks were infected with rotavirus RRV at an MOI of 3 as described above. Six hours postinfection the cells were fixed in 2.5% glutaraldehyde-0.1 M cacodylate (pH 7.2), postfixed with 1% osmium tetroxide, and embedded in Epon 812 resin. The ultrathin sections obtained were stained with 2% uranyl acetate-1% lead citrate (Reynolds mix). The grids were examined with a Zeiss EM-900 electron microscope at 80 kV.

All super-resolution imaging measurements were performed on an Olympus IX-81 inverted microscope configured for total internal reflection fluorescence (TIRF) excitation (Olympus, cellTIRFTM Illuminator). The critical angle was set up such that the evanescence field had a penetration depth of ~200 nm (Xcellence software v1.2, Olympus soft imaging solution GMBH). The samples were continuously illuminated using excitation sources depending on the fluorophore in use. Alexa Fluor 488 and Alexa Fluor 568 dyes were excited with a 488 nm or 568 nm diode-pumped solid-state laser, respectively. Beam selection and modulation of laser intensities were controlled via Xcellence software v.1.2. A full multiband laser cube set was used to discriminate the selected light sources (LF 405/488/561/635 A-OMF, Bright Line; Semrock). Fluorescence was collected using an Olympus UApo N $100x/1.49$ numerical aperture, oil-immersion objective lens, with an extra 1.6x intermediate magnification lens. All movies were recorded onto a 128 × 128-pixel region of an electron-multiplying charge coupled device (EMCCD) camera (iXon 897, Model No: DU-897E-CS0-#BV; Andor) at 100 nm per pixel, and within a 50 ms interval (300 images per fluorescent excitation).

Sub-diffraction images were derived from the Bayesian analysis of the stochastic Blinking and Bleaching of Alexa Fluor 488 dye (Cox et al., 2011). For each super-resolution reconstruction, 300 images were acquired at 20 frames per second with an exposure time of 50 ms at full laser power, spreading the bleaching of the sample over the length of the entire acquisition time. The maximum laser power coming out of the optical fiber measured at the back focal plane of the objective lens, for the 488 nm laser line, was 23.1 mW. The image sequences were analyzed with the 3B algorithm considering a pixel size of 100 nm and a full width half maximum of the point spread function of 270 nm (for Alexa Fluor 488), measured experimentally with 0.17 μm fluorescent beads (PS-SpeckTM Microscope Point Source Kit, Molecular Probes, Inc). All other parameters were set up using default values. The 3B analysis was run over 200 iterations, as recommended by the authors in Cox et al. (2011), and the final super-resolution reconstruction was created at a pixel size of 10 nm with the ImageJ plugin for 3B analysis (Rosten et al., 2013), using parallel computing as described in Hernández et al. (2016). The resolution increase observed in our imaging set up by 3B analysis was up to five times below the Abbe’s limit (~50 nm). The resolution provided by 3B was improved by computing the photo-physical properties of Alexa Fluor 488, and Alexa Fluor 568 dyes, which were provided to 3B algorithm, as an input parameter which encompass the probability transition matrix between fluorophore’s states. The method was validated with 40 nm gattapaint nanorules (PAINT 40RG, gattaquant, Inc) labeled with ATTO 655/ATTO 542 dyes (see ‘3B Algorithm’ in Appendix 1).

The segmentation algorithm (VPs-DLSFC) was developed in Matlab R2018a (9.4.0.813654) software. A detailed explanation of each the developed methods is available in Appendix 1. Statistical analysis were performed using R version 3.4.4 (2018-03-15) software. All the codes are available at https://github.com/Yasel88/Nanoscale_organization_of_rotavirus_replication_machineries (Garcés Suárez, 2019; copy archived at https://github.com/elifesciences-publications/Nanoscale_organization_of_rotavirus_replication_machineries).

The use of primitive models for the segmentation of the SRM images has many benefits, like computational efficiency, simple programmation, and understandable information about the objects that were segmented. In this regard, we developed a simple method for fitting a circumference to scattered data, which we called ‘Direct Least Squares Fitting Circumference’ (DLSFC). This approach considers basic mathematical analysis tools for the computation of the extreme value of a continuous function with N variables.

The spatial distribution of a viral proteins into the VP can be seen as a set of points ${\displaystyle P=\left\{(x_i,y_i)|i=\overline{1:N}\right\}\in {\mathbb{R}}^2}$ (scattered data in the plane). Taking into account the implicit equation of a circumference ${(x-x_c)}^2+{(y-y_c)}^2=r^2$, where $(x_c,y_c)$ is the center, and $r$ is the radius. The problem of adjust a circumference to $P$, is a optimization problem given by:

The center of mass of $P$ is given by:

Then, considerering Equation (2), the problem (Equation (1)) can be rewritten as:

where $u_i=x_i-\overline{x}$, $v_i=y_i-\overline{y}(\text{for all}i=1,2,\mathrm{\dots },N)$, $\alpha =r^2$, and $(u_c,v_c)$ is the center of the circumference after the change of variable.

For simplicity, we will define as $f_{u_c,v_c,\alpha }(u_i,v_i):={(u_i-u_c)}^2+{(v_i-v_c)}^2-\alpha$ the distance function, and by $F(u_c,v_c,\alpha ):=\sum _{i=1}^N{f}_{u_c,v_c,\alpha }{(u_i,v_i)}^2$ the objective function of our minimization problem.

Note 1 Note that, with the new convention of notation, the problem (Equation (3)) is equivalent to:

A simple alternative to resolve Equation (4) is to consider the extreme values of $F(u_c,v_c,\alpha )$, this is:

The partial derivates of the function $F(u_c,v_c,\alpha )$ are:

1. For $\alpha$:

Taking into account the condition given in Equation (5) for ${\displaystyle \frac{\mathrm{\partial }F}{\mathrm{\partial }\alpha }}$, we obtain that:

2. For $u_c$:

3. For $v_c$: It is the same process than $u_c$, the final result is,

Expanding the Equation (7) we obtain:

and taking into account that ${\displaystyle u_i=x_i-\overline{x}\Rightarrow \sum _{i=1}^N{u}_i=\sum _{i=1}^N{x}_i-N\overline{x}=\sum _{i=1}^N{x}_i-N{\displaystyle \frac{\sum _{i=1}^N{x}_i}{N}}=0}$, we obtain that:

Let $C_{(u^k,v^t)}:=\sum _{i=1}^N{u}_i^k{v}_i^t$, then the equation Equation (9) can be written as:

and considering the extreme condition $\frac{\partial F}{\partial u_c}=0$ (see Equation (5)) we obtain:

Following the same process, but now considering the Equation (8), we obtain:

The extreme values $\widehat{u_c}$ and $\widehat{v_c}$ of the function $F(u_c,v_c,\alpha )$ are obtained as solution of the linear system formed by Equation (10) and Equation (11).

Note 2 The coordinate system translation to the mass center of the original data, allow the simplification of the equation (Equation (9)).

The radius of the circumference is obtained developing the Equation (6):

and writing the equation in function of $\alpha$, we obtain that:

The point $(\widehat{u_c},\widehat{v_c},\widehat{\alpha })$ is a critical point of the function $F(u_c,v_c,\alpha )$. Finally, we need to compute the Hessian matrix of $F(u_c,v_c,\alpha )$ to test if $(\widehat{u_c},\widehat{v_c},\widehat{\alpha })$ is a maximum, a minimum, or an inflection point:

The principal minors of ${\displaystyle H}$ are:

Note 3 The Cauchy-Schwarz inequality is equal to zero if and only if exist a real value $r$ such that $u_i\mathit{}r\mathrm{+}{v}_i\mathrm{=}\mathrm{0}$, for all $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots }\mathrm{,}N$, that is, all the points are in a straight line. In our problem, the proteins do not follow a straight line distribution in the space, then, we can guarantee the strict inequality. Note that $H_{\mathrm{2}}\mathrm{=}\mathrm{0}$ if and only if $u_i\mathrm{=}\mathrm{0}\mathrm{,}\mathit{\text{for all}}\mathit{}i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots }\mathrm{,}N$, which is a particular case of a straight line.

The principal minors of the matrix $H$ are greater than zero, then, the point $(\widehat{u_c},\widehat{v_c},\widehat{\alpha })$ is a local minimum, but this point is the only critical point of the function $F(u_c,v_c,\alpha )$, and as consequence it is a global minimum.

The center of the fitted circumference in the original coordinate system is $(x_c,y_c)=(\widehat{u_c},\widehat{v_c})+(\overline{x},\overline{y})$, and the radius is $R=\sqrt{\widehat{\alpha }}$.

Note 4 This method is based on a simple and very popular mathematical techniques (specifically Differential Calculus). Even when we do not found a published paper with this approach, we want to take cares and not adjudicate this technique to our work.

Note 5 Note that, by definition, the distance function $f_{u_c\mathrm{,}{v}_c\mathrm{,}\alpha }\mathit{}\mathrm{(}{u}_i\mathrm{,}{v}_i\mathrm{)}$ is not a geometric distance to the circle, which is probably more intuitive to use in this kind of problems. Let consider for a second the geometric distance of a set of points $P$ to a circle, which is given by:

where $\mathrm{(}{x}_c\mathrm{,}{y}_c\mathrm{)}$ is the center of the circumference and $r$ is the radius. Considering the extreme condition for the function $G\mathit{}\mathrm{(}{x}_c\mathrm{,}{y}_c\mathrm{,}r\mathrm{)}$, it is obtained that:

where $\overline{x}$ and $\overline{y}$ are the mean values of the variables $x$ and $y$ respectively. Simultaneously equating these partials to zero does not produce closed form solutions for $x_c\mathrm{,}{y}_c\mathrm{,}$ and $r$. With numerical methods it is possible to carry out the optimization of these parameters, but these alternatives are iterative, no-deterministic process, which also are highly sensitives to the initial values, while our proposal not have these kind of unwished characteristics. In any case, in Appendix 1 Section –Algorithm validation– we compared our approach with an algorithm based on geometric distance and explained in details the benefits of each one.

The final segmentation algorithm is composed by three steps (Appendix 1—figure 1). The first is a manual pre-segmentation that guarantees the existence of one and only one viroplasm per image (Appendix 1—figure 1A). In the second step, we adjust a circumference to the reference protein (remember that these are paired experiments, where NSP2 is present in all the combinations as reference protein) through the algorithm DLSFC (Appendix 1—figure 1B), and finally the radius of the accompanying protein is adjusted using the Equation (12). Note that, the center of the accompanying protein is the same that the center of the reference protein (concentric model). Out of the manual pre-segmentation step, this algorithm is automatic, deterministic, not iterative and with a linear computational complexity.

Appendix 1—figure 1

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Note 6 The hypothesis that the viral proteins in the viroplasm are distributed like as concentric circumferences was proved in Appendix 1 Section – Model Considerations– (‘Model Considerations’).

The algorithm DLSFC was compared with other two approaches proposed by Gander et al. (1994). For a more comprehensive and clear comparison we named these others two methods as ‘Algebraic Least Square Fitting Circle’ (ALSFC) and ‘Geometric Least Square Fitting Circle’ (GLSFC).

The algorithm ALSFC considers the algebraic representation of a circle in the plane:

where $p=(x,y)$ is a point in ${ℝ}^2$, $B=(B_1,B_2)\in {ℝ}^2$, $A\ne 0$, and $C$ is the independent term. Then, let $p_i=(x_i,y_i)\in {ℝ}^2,i=1,2,\mathrm{\dots },N$ the set of points that we want to adjust. The objective function for this minimization problem is $\sum _{i=1}^{n}F(p_i)$, which can be represented as matrix form:

The final optimization problem is given by:

where $\parallel u\parallel =1$ is a contrain to avoid the trivial solution $u=(0,0,0,0)$. The notation ${\displaystyle \Vert v\Vert =\sqrt{v_1^2+v_2^2+\dots +v_m^2}}$ represent the Euclidean norm. Finally, the problem (Equation (16)) is solved considering the right singular eigenvector associated with the smallest singular eigenvalue of $\mathrm{\Psi }$ (Gander et al., 1994).

On the other hand, the algorithm GLSFC deals with the minimization of the geometric distance:

Note that ${\left(\parallel (x_c,y_c)-p_i\parallel -r\right)}^2$ is the geometric distance of the point $p_i$ to the circle ${(x-x_c)}^2+{(y-y_c)}^2=r^2$. This problem is nonlinear, and in Gander et al. (1994), the authors consider to resolve it trought a Gauss-Newton algorithm. The Gauss-Newton method is an iterative algorithm that depends of an initial approximation, and in this case, Gander and collaborators consider the solution of the problem (Equation (16)) as the initial value for the iterative process. More details about the algorithms ALSFC and GLSFC can be consulted in Gander et al. (1994). In summary, the method ALSFC is very simple and computational efficient, but the algebraic distance might not reflect with accuracy the real distance between the points and the circumference and, as consequence, the parameters of the model might be biased. On the contrary, the algorithm GLSFC is highly precise, but it is an iterative process (demand more computation time), and depends of good initial values to achieve the convergence. The Matlab codes for the algorithms GLSFC and ALSFC were obtained from Brown (2007).

For the validation, we simulated the spatial distribution of viral elements as circumferences, for which we know their parametric form (‘ground truth’ dataset) (Appendix 1—figure 2A). The use of ‘ground truth’ allows us to quantify the error in the adjustment of the algorithms DLSFC, ALSFC and GLSFC at different noise levels (Appendix 1—figure 2B) and partial occlusion conditions (Appendix 1—figure 2C). We generated over 40 000 images (size $512\times 512$ pixels) taking into account different levels of additive white Gaussian noise (AWGN) in order to consider auto-fluorescence and the error in the localization of the fluorophores by the algorithm 3B-ODE (see Appendix 1 Section –3B Algorithm–), and partial occlusion angles. Several radii and position of the synthetic viroplasms were generated randomly through a uniform distribution function. The codes for this simulation are availables at https://github.com/Yasel88/Nanoscale_organization_of_rotavirus_replication_machineries (Garcés Suárez, 2019; copy archived at https://github.com/elifesciences-publications/Nanoscale_organization_of_rotavirus_replication_machineries).

Appendix 1—figure 2

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Let $ℐ$ a synthetic image and $p_i\in {ℝ}^2,i=1,2,\mathrm{\dots },N$ points of the ‘ground truth’ circunference $𝒞\in ℐ$, the AWGN over ${\displaystyle p_i,\mathrm{\forall }i=\overline{1,N}}$ was generated as:

where $p_i^c$ is the corrupted point after the contamination with AWGN, $\sigma$ is a scalar that represents the level of noise, and $X_i$ is a trial of a random variable $X\sim N(0,1)$ (standard gaussian distribution). In our experimental design we included 20 different levels of noise $\sigma =0:0.5:10$ (increase by 0.5 from 0 to 10). Note that the noise was added to the points of the circumferences and not to the images. This is because we are working with SRM images, and as consequence does not exist background noise.

The partial occlusion can be seen as the existence of incomplete information of the objects of interest, for example, an image where we have only information about the half of the VPs. In our perspective, it is important that a segmentation algorithm have a good performance even in these situations.

Definition 1 We called an angle $\theta$ as a partial oclussion angle for a set of points $p_i\mathrm{=}\mathrm{(}{x}_i\mathrm{,}{y}_i\mathrm{)}\mathrm{\in }{\mathrm{R}}^{\mathrm{2}}\mathrm{,}i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots }\mathrm{,}N$, if not exist $j\mathrm{\in }\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots }\mathrm{,}N\mathrm{\}}\mathrm{,}r\mathrm{\in }\mathrm{R}$ and $\beta \mathrm{\in }\mathrm{[}\alpha \mathrm{,}\alpha \mathrm{+}\theta \mathrm{]}$ such as:

where $\alpha$ is an angle in $\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{2}\mathit{}\pi \mathrm{]}$, and $\mathrm{(}\overline{x}\mathrm{,}\overline{y}\mathrm{)}\mathrm{=}\mathrm{(}\frac{\mathrm{1}}{N}\mathit{}\mathrm{\sum }_{i\mathrm{=}\mathrm{1}}^N{x}_i\mathrm{,}\frac{\mathrm{1}}{N}\mathit{}\mathrm{\sum }_{i\mathrm{=}\mathrm{1}}^N{y}_i\mathrm{)}$ is the center of mass of the points $p_i$.

Note 7 Note that, in the Definition 1 we considered the interval $\mathrm{[}\alpha \mathrm{,}\alpha \mathrm{+}\theta \mathrm{]}$ as a partial oclusion angle $\theta$, but it is possible to obtain the same oclussion angle with $\mathrm{[}\alpha \mathrm{,}\alpha \mathrm{-}\theta \mathrm{]}$. For our validation we took $\alpha \mathrm{=}\mathrm{0}$ for simplicity, but this has not consequences in our experiments, because we generate the points as a circumference.

The relationship between cartesian and polar coordinates is given by:

where, (a, b) is the circumference center, $r$ is the radius and $\beta$ is the angular coordinate. Then, for example, to provoke a partial occlusion of $\theta$, it is just needed to avoid the generation of points in the angle $[0,\theta ]$ or $[2\pi ,2\pi -\theta ]$ through the Equation (19). Appendix 1—figure 3 shows this strategy, in this example, a half of the information is removed from the ‘ground truth’ circumference.

Appendix 1—figure 3

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In general, the average of the points that will be conserved after the generation of partial occlusion is given by:

In the validation of the algorithm DLSFC, we considered four different partial occlusion conditions: 1- Not partial occlusion: Conserves all the points of the ‘ground truth’ circumference; 2– 25% of partial occlusion: Partial occlusion angle $\theta =\pi /2$; 3– 50% of partial occlusion: Partial occlusion angle $\theta =\pi$; 4– 75% of partial occlusion: Partial occlusion angle $\theta =3\pi /2$.

In the experiments, we represent the noise through the mean and the standard deviation of the distance of the points $p_i^c,i=1,2\mathrm{\dots },n$ to the ‘ground truth’ circumference $𝒞$, this is:

where $d_{i𝒞}=\left|\parallel \mu -p_i^c\parallel -r\right|$ is the geometric distance between the point $p_i^c$ and the circumference $𝒞$, $|\cdot |$ represent the absolute value, $\mu =(x_c,y_c)$ is the center of the circumference and $r$ is the radius. The mean and standard deviation are an useful alternative to understand the amount of dispersion generated by a noise $\sigma$ in a circumference $𝒞$.

The Appendix 1—figure 4 shows the adjustment through the algorithms DLSFC, GLSFC and ALSFC of one ‘ground truth’ circumference (solid black line) that was corrupted by different levels of AWGN and partial occlusion angles. As was expected, the performance of the algorithms seems to get worse when increase the level of partial occlusion, but the three algorithms shows good results for high AWGN levels. This example suggests that the algorithm GLSFC (solid red line) has a better performance than DLSFC (solid blue line) and ALSFC (solid green line) when the partial occlusion is less than $\pi$ (columns A-C), but when $\theta =3\pi /2$, this method breaks down, and this observation is more relevant when increase the level of noise. The algorithm ALSFC shows relative good results for all the experimental conditions, but it is interesting to note that, in most cases, the adjustment is worse than in the DLSFC and GLSFC alternatives, probably as consequence of the algebraic distance (see Appendix 1 Section –Algorithm validation–). This example is just for a first visual analysis, but a complete study of the results obtained by the three algorithms over the 40 000 synthetic images will be presented below.

Appendix 1—figure 4

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A circumference can be represented as a vector $𝒞=(r,x_c,y_c)\in {ℝ}^3$, where $r$ is the radius, and $(x_c,y_c)$ the center. Let $\widehat{𝒞}=(\widehat{r},\widehat{x_c},\widehat{y_c})$ and $𝒞=(r,x_c,y_c)$ the ‘ground truth’ and the adjusted circumference, respectively. The fit error committed by the algorithms can be quantified as:

Now, thanks to the use of ‘ground truth’ circumferences, it is possible to compare the obtained outcome with the ideal result. We consider that an adjustment is good enough if the estimation error of each circumference’s component $𝒞$ is less than 0.1 μm. This value is approximately two-fold smaller than the theoretical Abbe’s limit ($\approx 250nm$), and taking into account that the synthetic images resolution vary between 4.6 and 25.8 μm in mean (see Appendix 1—figure 5), we are demanding that the algorithm has a nanoscopic precision (less than 100 nm) even when the images have a very bad resolution. Combining the threshold explained above with Equation (21), we obtain the final condition based on the euclidean norm between these two vectors:

Appendix 1—figure 5

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The Appendix 1—figure 5(a) shows the descriptive analysis (through boxplots) of the errors generated by the algorithms DLSFC, GLSFC and ALSFC over the 40000 synthetic images. In some experiments we obtained extremely good adjusts ($\epsilon \approx {10}^{-15}\mu m$), but in general the 75% of the errors are in the interval $[{10}^{-3},1]\mu m$. For all occlusion angles, the algorithm GLSFC had a bad behaviour in a wide range of experiments with an error of $\approx {10}^{10}\mu m$. This kind of situation was expected for big partial occlusion angles (for example $\theta =3\pi /2$), but in conditions where does not exist partial occlusion ($\theta =0$), the bad adjustment of GSFC is evidence that this algorithm is not a good alternative in this kind of problems. Note that a similar situation happens with the algorithm ALSFC, but in this case the maximum error is $\approx {10}^3\mu m$, which is high too, but significantly less than that obtained by GLSFC. The algorithm DLSFC does not show the previous problems; observe that the maximun error is under $1\mu m$ for all experimental conditions, even when $\theta =3\pi /2$. Now, if we forget the outliers and take into account only the 75% of the experiments (boxes), a direct relationship between the error of each algorithm and the partial occlusion angles (increase occlusion angle imply a major error) is observed. However, the raise of the mean error is small in relationship with the occlusion angle, which suggests that these three methods are robust to partial occlusion. The blue shadow band highlights the occlusion angle $\left(\theta =3\pi /2\right)$ for which the algorithms have a median error greater than $0.1732\mu m$. Therefore, it is necessary at least the 50% of the data to obtain a good adjustment (with an error under $0.1732\mu m$) in the 75% of the experiments.

The Appendix 1—figure 5(b) displays the mean error obtained by each algorithm in relationship with the resolution of the images (mean and the standard deviation of the points distance to the ‘ground truth’ circumference, Equation (20)). This graph shows again that the method GLSFC did not reach the convergence in some experiments (remember that GLSFC is based on an iterative optimization algorithm), even for small partial occlusion angles, but it is interesting to point out that in all these cases (at least for the occlusion angles $\theta =\{0,\frac{\pi }{2},\pi \}$), these high errors ($\approx {10}^{10}\mu m$) seem to be related with bad initial values. Note that in these experiments the algorithm ALSFC do not has a good performance (remember that the result of ALSFC is the initial value for GLSFC), which is an evidence of the sensibility of the GLSFC to the initial values. For the occlusion angle $\theta =\frac{3\pi }{2}$, a lot of bad fits were obtained through the algorithm GLSFC. In our opinion, even when the algorithm GLSFC can reach an extremely good fit, its sensitivity to initial values, the computational cost (iterative process), and the possibility of not convergence, are strongs reasons to consider that this alternative is not viable to use in this kind of problems.

In the Appendix 1—figure 5(c), the mean error of the algorithms DLSFC and ALSFC was splitted to see more clearly the performance of each one. For small partial occlusion angles $\left(\theta =\{0,\frac{\pi }{2}\}\right)$ the algorithm DLSFC has errors ${\displaystyle \le 0.1732\mu m}$, while ALSFC, in high noise scenes, get a poor adjustment (black arrows). When $\theta =\pi$, the method ALSFC has errors $\ge 0.1732\mu m$ in many cases, that increase when the level of noise grows. Both algorithms generate an error $\ge 0.1732\mu m$ when $\theta =\frac{3\pi }{2}$, but in the case of DLSFC, in some experiments had errors $<0.1732\mu m$, even for high noise levels. The Appendix 1—figure 5(d) show only the performance of the algorithm DLSFC. As noted, the method DLSFC shows a behaviour extremely robust and stable to noise and partial acclusion; note that for $\theta =\{0,\frac{\pi }{2},\pi \}$ does not exist an increment in the mean error when the noise level is incremented. In extreme experimental conditions, when only the 25% of the data remain in the image $\left(\theta =\frac{3\pi }{2}\right)$, DLSFC has a mean error between ${\displaystyle [0.3,0.5]\mu m}$ with a 95% confidence interval equal to ${\displaystyle [0.1,0.9]\mu m}$.

So far we have shown the behaviour of the three algorithms considering several noise levels and partial occlusion angles conditions. However, it is interesting to know through a hypothesis test (inferential statistics) the performance of DLSFC and ALSFC taking into account the experiments we developed (samples). Note that we are not going to study the algorithm GLSFC, because, as we mentioned above, its sensitivity to initial values and its computational cost (iterative process) are strong reasons to consider that this algorithm is not viable to use in this kind of problems. Taking into account the threshold in Equation (22), for each experiment we have two possible outcomes:

From this point of view, each experiment follows a Bernoulli’s distribution, and then, all the experiments for the same experimental condition (partial occlusion angle) follows a Binomial distribution. The generated samples are independent (random generation), the decision rule (Equation (23)) is dichotomous, and our experimental design is a fair representation of the population (we considered many experimental conditions in more than 40000 images); then, it is possible to develop a ‘Binomial Test’ (Prybutok, 1989; Howell, 1982) to evaluate the population proportion of success, which we will denote by $p$. The null (H0) and alternative (H1) hypothesis for this test are:

The null hypothesis is that the population proportion with an error less that 0.1732 is 0.7, and the alternative hypothesis is that the proportion is more that 0.7. The significance level for these tests is fixed at $\alpha =0.05$. For more information about this test and its implementation review (R Development Core Team, 2017; Prybutok, 1989; Howell, 1982) Online documentation is available in url https://stat.ethz.ch/R-manual/R-devel/library/stats/html/00Index.html.

The Appendix 1—table 1 shows the results of the ‘Binomial Test’ for the algorithms DLSFC and ALSFC. For small occlusion angles $\theta =\{0,\frac{\pi }{2}\}$, the results of both methods are very similar and indicate a high performance. Note that in these cases the null hypothesis was rejected with a probability error of type I of $\approx {10}^{-16}\le \alpha$, and the 95% confidence intervals are equal to $[0.99,1]$, which means that, at a population level, we hope that at least the 99% of the adjustments had an error $\le 0.1732\mu m$. For $\theta =\pi$, the algorithm DLSFC has a probability of success greater than 0.7 with a p-value of $\approx {10}^{-16}\le \alpha$. On the contrary, for the same oclussion angle, the test revealed that the null hypothesis can’t be rejected in the case of the algorithm ALSFC. Finally, as was analyzed above, the performance of the algorithms decrease significally for $\theta =\frac{3\pi }{2}$, and we can’t reject the null hypothesis, in others words, we don’t have enough evidence to assume that in this case the probability of success is greater than 0.7.

The study carried up in this section shows that the algorithm GLSFC has a good performance and is capable to obtain a low error in the adjustment, but this alternative is sensitive to the initial values (Appendix 1—figure 5 (a,b)), and as any iterative process, consumes more computational resources. The method ALSFC proved to have a good behavior in many experimental conditions, although in diverse occasions a bad fit was obtained. Also, the behaviour of this algorithm was always inferior than that of DLSFC (Appendix 1—figure 5(C) and Appendix 1—table 1). Finally, the algorithm DLSFC had a great performance and stability in all the experimental conditions, even in extreme partial occlusion angles (Appendix 1—figure 5 and Appendix 1—table 1). This method proved to be robust in high noise conditions and it is a deterministic and not an iterative algorithm. For all these reasons we consider that the algorithm DLSFC is the best choice in comparison with GLSFC and ALSFC, to fitting the spatial distribution of the viral elements.

To proof the hypothesis that the viral elements of the VPs can be approximated through circumferences, we carried up a series of experiments based on the comparison between the circumference obtained by the algorithm DLSFC, and a least squared ellipse resulting of the ‘Direct Least Square Fitting Ellipse’ (DLSFE) algorithm (Fitzgibbon et al., 1999). The algorithm DLSFE has been used previously to adjust the viral replication centers of adenoviruses in fluorescence images (limited by diffraction) (Garcés et al., 2016), and has also been shown to be robust to noise, computationally efficient, and easy to implement (Fitzgibbon et al., 1999).

Let us denote by $(x_i,y_i),i=1,2,\mathrm{\dots },N$ the set of N points relative to the protein $𝒫$, and by $C_{𝒫}$ and $E_{𝒫}$ the adjusted implicit form of the conics obtained by the algorithms DLSFC and DLSFE respectively. The variable $C_{𝒫}=(r,x_c,y_c)$ is a vector where $r$ is the radius, and $(x_c,y_c)$ is the center of the circumference. On the other hand, the implicit form of the ellipse can be represented as $E_{𝒫}=(a,b,x_c^E,y_c^E,\theta )$, where $\{a,b\}$ are the semi-major and semi-minor axis respectively, $(x_c^E,y_c^E)$ is the center of the ellipse, and $\theta$ is the rotation angle. Under the assumption that the protein $𝒫$ can be approximate by a circumference, we expect that does not exist significative statistical differences between the radius of the circumference $(r)$ and each one of the ellipse semi-axis $(a,b)$.

Note 8 Note that this approach is independent of which protein we choose to test the circularity hypothesis.

For each protein combination, we accomplish a Shapiro-Wilk hypothesis test to study if any of the three distributions functions (circumference radius ($r$), semi-major axis ($a$) and semi-minor axis ($b$)) came from a normally distributed population (Shapiro and Wilk, 1965). The results revealed that, our data does not seem to follow a Gaussian distribution, and as consequence, it is impossible to use a parametric test (for example t-student). Based on our data conditions, we considered the Mann-Whitney test (Mann and Whitney, 1947; Hollander et al., 2013) to evaluate the differences between the radii of the circumferences and each of the ellipses semi-axis. This is a nonparametric test that can be applied when the observations are independent (our variables are independent because the radius and the semi-axes were obtained by different algorithms), the variables are ordinal (also our variables meet this requirement because they are numeric), and finally, we want to test if the radius of the circumferences and each one of the ellipses semi-axis, are equal (null hypothesis) or are not (alternative hypothesis).

The statistical analysis reveal that does not exist a significative statistical differences between $R_C$ and $a$ (see Appendix 1—table 2) with a significance level of $\alpha =0.05$. The same results were obtained for $R_C$ and $b$ (see Appendix 1—table 3), and as consequence not matter if we use circumferences or ellipses to fitt the spatial distribution of the viral proteins.

On the other hand, we carried up a study to validate the viral elements concentricity hypothesis based on the distance between the centers of the adjusted circumferences of both proteins. For all the proteins, the median value is very close or smaller than $0.05\mu m$ (Appendix 1—figure 6 and Appendix 1—table 4 (Column 2)), which is in accordance with the resolution limit of the algorithm 3B-ODE ($0.04-0.05\mu m$) (Cox et al., 2011). Also, the 95% confidence interval (Appendix 1—table 4, Column 3) shows that the distance between the centers in each case is, 95% of the times in the order of ${10}^{-2}\mu m$, which we consider a small error taking into account the limitations of the algorithm 3B-ODE.

Appendix 1—figure 6

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Note 9 The Mann-Whitney test was used to study the differences in population medians (difference in location) (Mann and Whitney, 1947).

The algorithm VPs-DLSFC allows the quantification of the radius of two proteins in the same VP, and also the relative distance between them. At a first stage, we analyzed approximately 590 images (see histogram in Appendix 1—figure 7A) that were obtained by the pre-segmentation step of the algorithm VPs-DLSFC (Appendix 1—figure 1). In agreement with previous reports (Eichwald et al., 2012; Eichwald et al., 2004; Carreño-Torres et al., 2010), we found a wide heterogeneity in the size of VPs measured by NSP2, which exhibited radii ranging from 0.2 to 0.75 μm. In order to compare the distribution of each protein taking NSP2 as reference, we selected a subset of the individual VPs, in a way that there were no significant statistical difference between the radii of NSP2 from each condition.

Appendix 1—figure 7

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In accordance with this criterion, we were able to adjust the size of NSP2, such that the mean in the distribution of the radius was similar when this protein was paired with NSP5, NSP4, VP6, VP4 and VP7 (Appendix 1—figure 7B). Unfortunately, in the case of NSP2~VP1 and NSP2~VP2, even though we could selected a group of VPs in which the mean radius of NSP2 (0.35 μm) was closer to the size of the NSP2 from the other conditions ($0.4\mu m$), the analysis of the distributions in the radius of NSP2 by the Mann-Whitney hypothesis test (Appendix 1—figure 7B), revealed that the VPs of VP1 and VP2 were significative smaller than the mean size of viroplasms of the others proteins. As consequence, was not possible to compare directly the position of VP1 and VP2 with the others viral elements through an exploratory analysis. However, this problem did not limit the inferential study that we developed.

The use of super-resolution microscopy and the algorithm VPs-DLSFC allows the detection of small differences (resolution of nanometers) in the relative position of the viral elements. To validate that the different viral elements were in fact located in a different region from NSP2, we developed a Mann-Whitney test to study the existence of significative differences between the radii of NSP2 and the radii of each of the other viral elements. According with our previous results (see main text), this test revealed that only NSP4 has a similar spatial distribution that of NSP2 (Appendix 1—figure 7C and Appendix 1—table 5), while the others viral elements have different radii. Moreover, the small negative differences in location of NSP5 turned out to be significative different with respect to NSP2 (the p-values of the test are available in the (Appendix 1—figure 7C and Appendix 1—table 5). These results support that NSP5 is located in the innermost part of the viroplasm and that NSP4 shares the same layer with NSP2. The structural proteins VP1, VP2 and VP6 seem to be in the same layer of the viroplasm, with similar values, but with a significative statistical difference in location with NSP2 at a level $\alpha =0.01$. Also, this test suggests that VP4 and VP7 form independent layers in the most external zone of the VPs (Appendix 1—figure 7C and Appendix 1—table 5), which is in accordance with our proposed model.

Appendix 1—figure 8

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The spatial distribution of the viral elements in the VPs is conserved regardless of their size; that is, the radius of each protein shows a linear dependency with the radius of NSP2.

Let $P_i=(x_i,y_i)\in {ℝ}^2,i=1,2,\mathrm{\dots },N$ a set of points, where $x_i$ and $y_i$ are the radii of NSP2 and the acompayning viral element Y, respectively. The adjustment of the linear model can be solved through the following optimization problem:

The model (Equation (25)) does not include an offset term because we don’t have any quantification about the radius of the accompanying viral element when NSP2 have been silenced.

The solution of this problem is given by Kiefer (1987):

where $cor(x,y)$ is the linear correlation between x and y, $cov(x,y)$ is the covariance between x and y, Var(x) is the variance of x, and $\overline{x}$ is the mean value of the variable $X$.

The ‘Residual Standard Error’ (RSE or $\left({\sigma }^2\right)$) is a measure of fit of the linear regression model.

where ${ϵ}_i=y_i-\widehat{\beta }{x}_i$. The ‘Standard Error’ $\left({\sigma }_{\widehat{\beta }}^2\right)$ in the estimation of the slope $\widehat{\beta }$ is:

Consider the statistic $t_{\beta }=\frac{\widehat{\beta }-\beta }{{\sigma }_{\widehat{\beta }}}$ and the null (H0) and alternative hypothesis (H1):

Under the null hypothesis, $t_0=\frac{\widehat{\beta }}{{\sigma }_{\widehat{\beta }}}$ follow a t-Student distribution with N-1 degrees of freedom. The p-value of this tests is computed as:

where $P(T\le t_0)$ is the probability that a t-Student variable with N-1 degrees of freedom be less than $t_0$. Note that through this hypothesis test it is possible to evaluate if the slope $\beta$ is significative to explain the linear relationship between the variables.

The variance of the dependent variable $y$ is:

where, ${\widehat{y}}_i=\widehat{\beta }{x}_i$. In accordance with the Equation (31), the variance rate that could be explained by the regression model is the regression variance divided by the full variance, this is:

Let $\mathrm{\Psi }$ be the lineal regression model that adjusts the data $(x_i,y_i)$, and be $R^2$ the associated r-square value, then, $100\times R^2$ is the percent of the data variance that it was explained by the model. For strong linear dependencies, the expected $R^2$ value should be close to 1. For more information about linear regression models consult (Kiefer, 1987).

The results of the regression models are available in Appendix 1—table 6. In all regression models, the standard errors are in the order of ${10}^{-2}\mu m$, with a $p-value<{10}^{-33}$ which is a strong evidence of the linear relationship that exist between NSP2 with the other seven viral proteins. Also, the RSE have values in the order of ${10}^{-2}\mu m$, except for VP7 which have a RSE of $0.15\mu m$, which could suggest that exist a greater dispersion of VP7 in comparison with the others viral proteins, something expected since, according to our model, VP7 is the most external protein.

The average errors are under 8% in all models (Appendix 1—table 6), which confirm the capacity of our linear models to predict with high accuracy the radius of the described viral elements. On the other hand, the percentage variance rate that could be explained by the regression models ($100\times R^2$, see Equation (32)), was greater than the 96% in all cases (Appendix 1—table 6), demonstrating again the high linear dependency between NSP2 and the others viral elements into the viroplasm.

Appendix 1—figure 9

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As was mentioned in the main text, we carried out a similar study using NSP5 and NSP4 as reference proteins. Since these experiments were developed to validate our previous results with NSP2, they were not as deep and extensive as in the case of the NSP2 model.

The number of VPs studied was 38 and 23 for NSP5~VP6 and NSP5~VP4 respectively (Appendix 1—figure 10A.1). The distribution of NSP5 relative to VP6 and VP4 did not show significative statistical differences according to the Mann-Whitney test, and the radius of VP6 was smaller than VP4, which is in accordance with our model of the viroplasm (Appendix 1—figure 10A). Furthermore, the distance of VP6 and VP4 towards NSP5, reflects that VP6 is closer to NSP5 than VP4, with a significance level $\le 0.001$ (Appendix 1—figure 10B). On the other hand, we found that exists a linear relationship between the radii of NSP5 and {VP6, VP4} (Appendix 1—figure 10 C-E). The linear models of both, VP6 and VP4, explain approximately 97% of the data variability and exhibit a RSE in the order of ${10}^{-2}\mu m$ (see Appendix 1—figure 10 (D,E) and Appendix 1—table 7). Additionally, we performed a Mann-Whitney test to study the differences of the radii of NSP5 and each of the accompanying proteins (VP6 and VP4). This test demonstrated that the differences in location between NSP5 and VP6 is ${\displaystyle \approx 0.1\mu m}$, while in the case of VP4 is ${\displaystyle \approx 0.23\mu m}$, with a p-value in the order of ${10}^{-7}$ and ${10}^{-8}$ respectively (see Appendix 1—table 8), indicating that although these protein are near to each other, they occupy different layers on the VPs.

Appendix 1—figure 10

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In the case of the model based on NSP4, we studied 60 viroplasms through the algorithm VPs-DLSFC. The results show that the radius of VP6 were bigger than the radius of NSP4 in approximately 0.04 μm (Appendix 1—figure 11 A-B). This difference is statistically significative ($p-value\le 0.05$) according with the Mann-Whitney test (Appendix 1—figure 11A). Like the models based on NSP2 and NSP5, we found a linear relationship between NSP4 and VP6 (Appendix 1—figure 11C). This linear model was validated through a residual analysis (Appendix 1—figure 11D and Appendix 1—table 9), obtaining a $RSE=0.079\mu m$, and a $R^2=0.97$.

Appendix 1—figure 11

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As was described, our study involves three differents reference proteins (NSP2, NSP5 and NSP4). Since the three models should respond to an unique spatial organization of the viral elements into the VP, it is expected to obtain similar results independenly of the selected model.

The equivalence of the differents models (models based on NSP2 and NSP5) can be proven through a comparison of the distances between NSP5 and VP6. If the two models describe the same distribution of the VPs, we expect that the distance of NSP5 to VP6 computed by the NSP5 model be similar to the the distance between NSP5 and NSP2, plus the distance of NSP2 to VP6 that were computed by the model based on NSP2. Hence:

The subindex (‘name’) in ${\displaystyle d_{name}(\cdot )}$ specifies the model employed to compute the distance. Therefore, replacing the corresponding values (available in Appendix 1—table 5 and Appendix 1—table 8) in the above equation, we obtain that: