To assess the probability of cryptic pocket opening in the three filoviruses, we first used molecular dynamics simulations guided by adaptive sampling. We first ran 8 μs of simulation of each variant using a goal-oriented adaptive sampling algorithm called FAST (Zimmerman and Bowman, 2015) that balances between broad exploration of conformational space and focusing data acquisition on conformations with more open pockets. We then built a Markov State Model (MSM) of each homolog from these simulations and used the centers as seeds for long MD simulations on the Folding@home distributed computing platform (Voelz et al., 2023). We collected a total of 126 and 124 μs of data for Reston and Marburg IIDs. Then we built a new MSM for each variant that incorporates the data from Folding@home. The cryptic pocket observed in Zaire IID occurs as the helix spanning residues 305–309 moves away from the alpha helical domain. Therefore, we characterize pocket opening based on the distance between residues 236 and 306 in Zaire, which corresponds to residues 225 and 295 in Reston and Marburg. For the sake of brevity, unless mentioned otherwise, we will use Reston and Marburg numbering going forward. We thus obtain the probability distribution of the distance between residues 225 and 295 based on the equilibrium probability of each structure in the MSM (Figure 2).

We find that Marburg has a higher probability of opening than Zaire, but that Reston has a significantly lower probability (Figure 2). Furthermore, Marburg opens more widely than the other variants, as judged by reaching larger distances between residues 225 and 295 (Figure 2 insets). We also performed CARDS analysis (Singh and Bowman, 2017) on the Marburg and Reston IID simulations to compare to the allosteric network obtained for Zaire IID using this analysis (Cruz et al., 2022; Figure 2—figure supplement 2). Similar to Zaire IID, we observe a community of residues around the pocket. We also observe that this community is strongly connected to the green and orange communities that contain residues that interact with dsRNA, indicating that the pocket opening could be affecting dsRNA binding in these homologs as well. However, we also note that the strength of communication both within and between these communities differs between the homologs suggesting that pocket opening has varying influence on the dihedrals of the RNA binding residues in the homologs. Furthermore, the differences in pocket opening have an interesting correspondence to the fact that Marburg prefers to bind the dsRNA backbone, while Reston and Zaire bind both blunt ends and the backbone. We, therefore, hypothesized that open pocket conformations preferentially bind the backbone while closed conformations preferentially bind blunt ends.

As a first test of our computational predictions, we used thiol labeling to experimentally measure the probability that the cryptic pocket is open in each IID homolog. In these experiments, we measure the rate of covalent modification of a cysteine in the cryptic pocket, as we have done previously with β-lactamases (Knoverek et al., 2021; Porter et al., 2019a) and Zaire IID (Cruz et al., 2022). In these experiments, 5,5’-dithiobis-(2-nitrobenzoic acid), or DTNB, is added to the protein sample. In the presence of an oxidized (solvent-exposed) thiol group of a cysteine residue, the disulfide bond between the two TNB molecules breaks, and one TNB molecule attaches to the cysteine via a disulfide bond. The free TNB molecule left from this reaction absorbs light at 412 nm. Therefore, when a cysteine buried inside a pocket is exposed to solvent as the pocket opens, we observe an exponential increase in absorbance at a rate that depends on the opening and closing rates of the pocket. We quantify this using the Linderstrøm-Lang model (see Materials and methods). We focus on thiol labeling of cysteine 296 as its solvent exposure is correlated with the extent of pocket opening (as measured by the distance between residue 225 and 295) in our simulations (Figure 3A, B and C), whereas the solvent exposures of other cysteines are not correlated with pocket opening (Figure 3, Figure 3—figure supplement 1).

As expected from the MSMs, we observe the highest probability of pocket opening in Marburg, followed by Zaire and Reston in that order. Thiol labeling of Reston IID and Marburg IID fit to one exponential per cysteine, each with the same amplitudes, indicating that all the cysteines in each variant get labeled (Figure 3—figure supplement 2). We performed point mutations of individual cysteines to serines to assign observed labeling rates to the cysteines as we have previously done for Zaire IID (Figure 3—figure supplement 3). We compare these results to the labeling rates we previously observed for Zaire (Cruz et al., 2022). The cysteines in Reston’s cryptic pocket label far more slowly than those in Zaire, while the labeling rates of the cysteines in Marburg are intermediate between Zaire and Reston (Figure 3D and H). These data were fit to the Linderstrøm-Lang model (see Materials and methods) to quantify the probability and kinetics of pocket opening (Supplementary file 1). In all three homologs, C296 labels faster than expected from labeling of the unfolded fraction (Figure 3—figure supplement 4), suggesting that the pocket does exist in all three homologs. The equilibrium constants for C296 exposure (Equation 3 in Materials and Methods) for Reston, Zaire, and Marburg IIDs are 0.078±0.001, 0.413±0.009, 5.2±0.5. These correspond to probabilities of pocket opening of 0.072, 0.292, and 0.839 for Reston, Zaire, and Marburg, respectively. Therefore, in agreement with our computational prediction, we observe the highest probability of pocket opening in Marburg, followed by Zaire and Reston in that order.

To test our prediction that more open homologs prefer the backbone while more closed ones preferentially bind the blunt ends, we used a fluorescence polarization assay to quantify the affinity of the homologs to different length dsRNA substrates, thereby varying the number of backbone binding sites available. Briefly, we use dsRNA labeled with fluorescein on one 5’ end and titrate in varying concentrations of the proteins into a fixed concentration of the RNA (100 nM). Free RNA emits depolarized light upon excitation with polarized light due to its fast rotation, whereas bound RNA emits polarized light. Recording the fluorescence polarization throughout the titration gives us the fraction of RNA bound to the protein. Particularly, we used a short 8 bp RNA to get an accurate measure of binding to the blunt ends and the same 8 bp RNA with a 2-nucleotide overhang on the 3’ ends to get an accurate measure of the binding to the backbone alone. Structural studies on these homologs suggest that the backbone binding mode has a footprint of three nucleotides (Leung et al., 2010a; Leung et al., 2010b; Ramanan et al., 2012). Previous RNA binding studies also suggest multiple IID molecules bind to each RNA molecule in solution (Leung et al., 2010a; Edwards et al., 2016). Therefore, we also performed measurements with a longer 25 bp RNA with and without a 2 nucleotide overhang on the 3’ ends to get an accurate measure of any cooperativity in the backbone binding. To accurately account for multiple protein molecules binding to a single molecule of dsRNA, we used a one-dimensional lattice binding model to fit the experimental data (see Materials and methods). This model allows nearest-neighbor cooperativity between all binding modes. Globally fitting these four binding curves for each homolog to this model (Figure 4A, B and C) generates estimates for the equilibrium constant for opening the pocket (Figure 4—figure supplement 1A), the dissociation constants of the closed and open conformations to each site on the backbone (KD,backboneclosed\begin{document}$K_{D,backbone}^{closed}$\end{document} and KD,backboneopen\begin{document}$K_{D,backbone}^{open}$\end{document}, respectively), the dissociation constant of closed conformations to the blunt end of the RNA (KD,endclosed\begin{document}$K_{D,end}^{closed}$\end{document}), and the cooperativity between these interactions (Figure 4D, Figure 4—figure supplement 1B).

In support of our proposed model, we find that the equilibrium constant for pocket opening (Koc\begin{document}$K_{oc}$\end{document}, where oc stands for open vs closed) from our fits to dsRNA-binding data are in good agreement with those from our thiol labeling experiments (KeqC296)\begin{document}$\left (K_{eq}^{C296}\right)$\end{document}. If the pocket opening observed in the thiol-labeling assays is the same conformational change affecting the RNA binding, we would expect the Koc\begin{document}$K_{oc}$\end{document} obtained from the fits to be comparable to KeqC296\begin{document}$K_{eq}^{C296}$\end{document}. Indeed, Koc\begin{document}$K_{oc}$\end{document} for all three homologs obtained from the fits to the binding model agree very well with the equilibrium constant for the exposure of C296 (C307 in Zaire IID) obtained from the thiol labeling assay (Figure 4—figure supplement 1A).

In support of our hypothesis, our global fits show that the probability of pocket opening is the main difference between the variants. As described above, the equilibrium constants for pocket opening vary over a range of about 80-fold between the three homologs. In contrast, each of the dissociation constants between either open or closed protein and the blunt ends or backbone of dsRNA varies by no more than a factor of about sevenfold between the homologs. We also do not observe large differences in the magnitude of the cooperativity parameters between the homologs (Figure 4—figure supplement 1B). The interaction between the closed state and the blunt ends of dsRNA is far stronger than the interaction of either the closed state or the open state for the backbone (Figure 4D). There is also negative cooperativity between two closed structures binding the backbone at adjacent positions (Figure 4—figure supplement 1B). As a result, homologs with low probabilities of pocket opening predominantly bind the blunt ends and their binding curves are left-shifted compared to more open homologs. Homologs with a higher probability of pocket opening are less likely to bind the blunt ends, despite the high affinity of the closed state for blunt ends, since the open state is incompatible with blunt end binding. The interaction between the open state and the backbone (KD,backboneopen\begin{document}$K_{D,\,backbone}^{open}$\end{document}) is about two orders of magnitude stronger than that of the closed state (KD,backboneclosed\begin{document}$K_{D,backbone}^{closed}$\end{document}) for all three homologs (Figure 4D). There is also strong positive cooperativity between a protein in the open state binding the backbone alongside either a closed or open state (Figure 4—figure supplement 1B). As a result, homologs with high probabilities of pocket opening, like Marburg, predominantly bind the backbone and have right-shifted binding curves compared to more closed variants. Interestingly, most of the proteins that do bind the backbone are in the open state even for homologs where pocket opening is rare (Figure 4E and F). For example, about 80% of the backbone is covered by Reston proteins in the open state even though the equilibrium constant for pocket opening (Koc\begin{document}$K_{oc}$\end{document}) is 0.061±0.008, which corresponds to a probability of cryptic pocket opening of 0.057, the lowest of the three homologs. It is also noteworthy that crystal structures capture the closed state bound to the backbone but not the open state, suggesting that the crystallization conditions may shift the equilibrium in favor of the closed state.

As a further test of our model, we next sought to identify point mutations that modulate the probability of pocket opening and assess if they have the expected impact on backbone versus blunt end binding. While the sequence of Marburg IID differs significantly from Reston and Zaire IIDs with a sequence identity of 42% and 45%, respectively (Figure 1—figure supplement 1), the sequences of Reston and Zaire IID are 88% identical and 94% similar. Particularly, substitutions between these homologs are all distal to the RNA-binding interfaces and all the residues known to make contacts with dsRNA from structural studies are identical. Therefore, we reasoned that comparing these two homologs would help us identify minimal substitutions that control pocket opening probability and allow us to study its effect on dsRNA binding with minimal perturbation of other factors. Of these substitutions, one causes an interesting structural difference between Zaire and Reston IIDs (Leung et al., 2010b). Residue P280 in Reston IID results in a formation of an alpha helix, where an alanine in the same structural position (A291) in Zaire IID results in a disordered loop. This disordered loop forms a hinge that rotates as the pocket opens, and we reasoned this motion may be inhibited by the structure the proline induces in Reston. Therefore, we hypothesized that mutating the alanine in Zaire IID to a proline would result in a reduced probability of opening the pocket and a stronger preference for binding dsRNA blunt ends. This hypothesis is supported by our past work showing that the substitution A291P in Zaire IID reduces the probability of pocket opening (Cruz et al., 2022). Furthermore, we propose that mutating the proline in Reston to an alanine should increase the probability of pocket opening and enhance the binding to the dsRNA backbone.

As expected, introducing A291P into the Zaire IID leads to a reduced probability of pocket opening in both simulations and thiol labeling experiments, while introducing P280A into the Reston IID leads to an increased probability of pocket opening. FAST simulations performed on these variants indeed showed an increased probability of opening the pocket in Reston IID P280A and a decreased probability in Zaire IID A291P (Figure 5A). The pocket also opens more widely in Reston IID P280A than Wild Type (WT) Reston IID. As before, examining the solvent exposure of the cysteines observed in simulation shows that while exposure of C296 (C307 in Zaire) is correlated with pocket opening, that of other cysteines is not (Figure 5—figure supplement 1). We followed this up with the thiol labeling assay and assigned labeling rates to cysteines by labeling point mutations of individual cysteines to serines (Figure 5—figure supplement 2). We observe that both cysteines in Reston IID P280A labeled faster than WT Reston IID (Figure 5B, Figure 5—figure supplement 3A). In comparison, both cysteines in Zaire IID A291P labeled slower compared to WT Zaire IID (Figure 5B, Figure 5—figure supplement 3B). Fits to the Linderstrøm-Lang model show a 26-fold increase in the equilibrium constant for C296 exposure in Reston IID P280A compared to Reston IID WT and a 36-fold decrease in Zaire IID A291P compared to Zaire IID WT (Figure 5B). We observed only moderate differences in the stability of Reston IID WT and Reston IID P280A under urea denaturation (Figure 3—figure supplement 4). Taken together, this indicates that the mutation modulates pocket opening without significantly impacting the stability of the protein.

In further support of our model, the point mutation that increases Reston’s probability of pocket opening also shifts the protein towards backbone binding while the mutation that reduces the probability of pocket opening in Zaire shifts the balance towards blunt end binding. We performed fluorescence polarization binding experiments to the four RNA substrates we used for the WT homologs (Figure 6A–D). We analyzed these experiments using global fits of the binding of each mutant to all four RNAs to the one-dimensional lattice model described for the WT homologs (Figure 6A-D, Figure 6—figure supplement 1). Importantly, the equilibrium constants for pocket opening (Koc)\begin{document}$(K_{oc})$\end{document} obtained from the fits to the binding model continue to agree very well with the equilibrium constants for the exposure of C296 obtained from the thiol labeling assay (Figure 6E). While there are large differences between the equilibrium constant for pocket opening amongst these variants, the rest of the fit parameters are very similar to those from the WT proteins. We observe that the dissociation constant of the closed conformations to each site on the backbone (KD,backboneclosed)\begin{document}$(K_{D,backbone}^{closed})$\end{document} is two orders of magnitude weaker than that of the open conformations (KD,backboneopen)\begin{document}$(K_{D,backbone}^{open})$\end{document} for the mutants as well (Figure 6F), further increasing our confidence in the model. We do not observe large differences in the magnitude of the binding and cooperativity parameters between the WTs and the mutants, except for the cooperativity between two neighboring molecules in the open state binding to the backbone σoo\begin{document}$\sigma _{oo}$\end{document} Figure (6 F, G, and H). This parameter seems to buffer the effect of the increased affinity of the open state to the backbone. This inherent trade-off also suggests that shifting the equilibrium too drastically in favor of the open conformation would not be beneficial for binding to the backbone. Particularly, Reston P280A is more open than WT Reston, but its binding curves for the overhang substrates are right-shifted compared to WT. This suggests that though the open pocket conformations bind better to the backbone, increasing the fraction of open conformation in solution has a negative effect on the overall backbone binding. To understand this, we calculated the fraction of the RNA backbone covered by the open and closed states of the various homologs and mutants for different lengths of RNA based on the binding parameters obtained from the fit (Figure 6—figure supplement 2). We see that in all cases, a much larger proportion of the backbone is covered by the open states than by the closed states. This suggests that while it is necessary to have some ability to open the pocket, increasing the probability of opening has no added benefit for backbone binding. Meanwhile, Zaire A291P is more closed and prefers to bind the blunt ends more than the WT protein. This is reflected by its binding curve for the blunt ended substrates being left-shifted compared to WT. Overall, these results show a strong correspondence between the probability of pocket opening and the relative affinities of the VP35 IID for different binding sites on dsRNA.

Simulations for Reston IID and Marburg IID were initiated from the apoprotein models of PDB 3L2A (Reston IID Leung et al., 2010b) and 4GHL (Marburg IID Ramanan et al., 2012) and run with Gromacs (Van Der Spoel et al., 2005) using the amber03 force field (Duan et al., 2003) and TIP3P explicit solvent (Jorgensen et al., 1983) at a temperature of 300  K and 1  bar pressure, as described previously (Hart et al., 2016). We first applied our FAST-pockets algorithm (Zimmerman and Bowman, 2015) to replicate the simulation protocol of Zaire IID in Cruz et al., 2022. As done previously, we performed ten rounds of FAST simulations with 10 simulations/round and 80 ns/simulation. We then performed an RMSD-based clustering using a hybrid k-centers/k-medoids algorithm (Beauchamp et al., 2011) implemented in Enspara (Porter et al., 2019b) to divide the data into 1000 clusters. Then we ran three simulations initiated from each cluster center on the Folding@home distributed computing environment, resulting in an aggregate simulation time of 126 μs for Reston IID and 124 μs for Marburg IID. We calculated the Solvent Accessible Surface Area (SASA) of all atoms in each frame using a 2.8 Å probe. We summed up the SASA of all the atoms of the side chain of each residue and clustered this sidechain SASA using k-centers upto a cluster radius of 3.5 nm² for Reston IID and 2.7 nm² for Marburg IID followed by five rounds of k-medoids. We then built Markov State Models (MSMs) at multiple lagtimes and used the implied timescales test (Figure 2—figure supplement 2) to decide on a 6 ns lag time for the final MSMs of all homologs.

All variants of VP35’s IID were purified from the cytoplasm of E. coli BL21(DE3) Gold cells (Agilent Technologies) following the protocols detailed in Leung et al., 2010b; Ramanan et al., 2012; Leung et al., 2009. Variants were generated using the site-directed mutagenesis and confirmed by DNA sequencing. Cells were transformed using heat shock at 42 °C. Transformed cells were grown in LB media (Fisher Scientific or Lambda Biotech, Ballwin, MO) at 37 °C until OD 0.3 then grown at 18 °C until induction at OD 0.6 with 1  mM IPTG (Gold Biotechnology, Olivette, MO). Cells were grown for 15  hr then centrifuged. The pellet was resuspended in 20  mM sodium phosphate pH 8, 1  M sodium chloride, 5 mM Imidazole with 5.1  mM β-mercaptoethanol. Resuspended cells were subjected to sonication at 4 °C followed by centrifugation. The supernatant was then subjected to Ni-NTA affinity (BioRad Bio-Scale Mini Nuvia IMAC column) and eluted with 20  mM sodium phosphate pH 8, 1  M sodium chloride, 250 mM Imidazole with 5.1  mM β-mercaptoethanol. This was dialyzed to a final buffer of 20  mM sodium phosphate pH 8, 50 mM NaCl with 5.1  mM β-mercaptoethanol. The dialyzed sample was subjected to TEV digestion (proTEV plus, Promega) at room temperature for 24–48 hr. We followed this by cation exchange (BioRad UNOsphere Rapid S column) and eluted using a slow gradient of 20  mM sodium phosphate pH 8, 1 M NaCl with 5.1  mM β-mercaptoethanol. Cleaved VP35 with the sequence shown in Figure 1—figure supplement 1 elutes at around 180 mM NaCl. We follow this by size-exclusion chromatography (BioRad Enrich SEC 70 column or Cytiva HiLoad 16/600 Superdex 75) into 10  mM HEPES pH 7, 150  mM NaCl, 1  mM MgCl₂, 2  mM TCEP.

We monitored the change in absorbance over time of 5,5’-dithiobis-(2-nitrobenzoic acid) (DTNB, Ellman’s reagent, Thermo Fisher Scientific). Various concentrations (100–1000 μM) of DTNB were added to the 5 μM protein and change in absorbance was measured in an SX-20 Stopped Flow instrument (Applied Photophysics, Leatherhead, UK) at 412  nm until the reaction reached a steady state (300–1200  s). Each time course was fit with as many cysteines being labeled for that protein to obtain an observed labeling rate (k_obs) for each cysteine at each DTNB concentration. The k_obs as a function of DTNB concentration for each cysteine were fit with a Linderstrøm–Lang model, shown below, to extract the thermodynamics and/or kinetics of pocket opening, as described in detail previously (Porter et al., 2019a).(1)Closed⇌kclosekopenOpen→kint[DTNB]Labeled\begin{document}$$\displaystyle \text{Closed} \xrightleftharpoons[k_{close}]{k_{open}} \text{Open} \xrightarrow{k_{int}[DTNB]} \text{Labeled} $$\end{document}

Which leads to,(2)kobs=kopen∗kint[DTNB]kclose+kopen+kint[DTNB]\begin{document}$$\displaystyle k_{obs}=\frac{k_{open}\ast k_{int}[DTNB]}{k_{close}+k_{open}+k_{int}[DTNB]} $$\end{document}

From this fit, we report the equilibrium constant for cysteine exposure calculated as:(3)Keq= kopenkclosed\begin{document}$$\displaystyle K_{eq}=\ \frac{k_{open}}{k_{closed}}$$\end{document}

As a control, the equilibrium constant for folding and the unfolding rate were measured (Figure 3—figure supplement 4) and used to predict the expected labeling rate from the unfolded state. The equilibrium constant was inferred from a two-state fit to urea melts monitored by fluorescence and unfolding rates were inferred from exponential fits to unfolding curves monitored by fluorescence after the addition of urea, as described previously (Bowman et al., 2015; Porter et al., 2019a; Zimmerman et al., 2017). Fluorescence data were collected using a Jasco FP-8300 Spectrofluorometer with Jasco ETC-815 Peltier and Koolance Exos2 Liquid Coolant-controlled cuvette holder.

Binding affinities between homologs and variants of VP35’s IID and dsRNA were measured using fluorescence polarization in 10  mM HEPES pH 7, 150  mM NaCl, 1  mM MgCl₂. 8 bp, and 25 bp FITC-dsRNA (Integrated DNA Technologies) substrates with or without a 2-nucleotide 3’ overhang of the following sequences were included at 100  nM.

The sample was equilibrated for 1 hr before data collection. Data were collected on a BioTek Synergy2 Multi-Mode Reader as polarization and were converted to anisotropy (r) as in Keck, 2012:r=2p3−p\begin{document}$$\displaystyle r= \frac{2p}{3-p} $$\end{document}

Where p\begin{document}$p$\end{document} in the polarization calculated from the observed parallel and perpendicular intensities, I∥\begin{document}$I_{\parallel }$\end{document} and I⊥\begin{document}$I_{\bot }$\end{document} as:p= I∥−GI⊥I∥+GI⊥\begin{document}$$\displaystyle p=\ \frac{I_{\parallel }-\mathrm{G}I_{\bot }}{I_{\parallel }+\mathrm{G}I_{\bot }}$$\end{document}

The total fluorescence intensities were checked at each well for fluorescence anomalies such as quenching.

To obtain the contributions from the end-binding and backbone-binding to the increase in anisotropy observed in the fluorescence polarization experiments, we used a one-dimensional lattice model for proteins competing to bind to nucleic acids. Here, we consider our labeled dsRNA to be the macromolecule and the IIDs to be the ligands. Specifically, our model has the open and the closed conformations of the IIDs competing for binding to the dsRNA. The total concentration of the ligand is known for every point in the titration and relates to the relative concentration of the open and the closed states as:(4)[IID]total=(1+Koc)[IID]closed\begin{document}$$\displaystyle [IID]_{total}=\left (1+K_{oc}\right )\left [IID\right ]_{closed}$$\end{document}(5)[IID]open = Koc[IID]closed\begin{document}$$\displaystyle [IID]_{open\ }=\ K_{oc}\left [IID\right ]_{closed}$$\end{document}

To treat the simplest case first, we assume that only the closed state binds to the ends of the dsRNA. We refer to the intrinsic association constant for this interaction as KAend\begin{document}$K_{A}^{end}$\end{document}. We assume that the open and closed states bind to the dsRNA backbone with intrinsic association constants KAopen\begin{document}$K_{A}^{open}$\end{document} and KAclosed\begin{document}$K_{A}^{closed}$\end{document}.

We performed the fits at varying binding site sizes for the three interactions described above. Specifically, we varied the end binding site size between 1–8 nucleotides and the backbone binding site sizes between 3–8 nucleotides, choosing the lower limits in both cases to be the binding site size observed in the available crystal structures. We analyzed the sum of the residuals squared obtained from these fits across all five variants used in this study to obtain the binding site sizes that produce the best fit (Figure 6—figure supplement 3). This analysis showed that the best fit was obtained for the blunt end and both backbone binding site sizes all being three nucleotides each. Using a threshold of five times the minimum sum of the residuals squared, we note that binding site sizes ranging from one to four nucleotides for the end binding, 3–5 nucleotides for the closed state binding to the backbone, and three to four nucleotides for the open state binding to the end produce comparable fits.

Furthermore, we found that the major conclusions of the study hold true for varying binding site sizes in this range (Figure 6—figure supplement 4; Figure 6—figure supplement 5 and Figure 6—figure supplement 7). Future RNA binding studies with a technique with a higher sensitivity for binding site sizes will help in confidently estimating these parameters.

The dissociation constants referred to in the text are inverses of the association constants described above. Therefore,(6)KD, backboneclosed = 1KAclosed\begin{document}$$\displaystyle K_{D,\ backbone}^{closed}\ =\ \frac{1}{K_{A}^{closed}}$$\end{document}(7)KD, backboneopen = 1KAopen\begin{document}$$\displaystyle K_{D,\ backbone}^{open}\ =\ \frac{1}{K_{A}^{open}}$$\end{document}(8)KD, endclosed = 1KAend\begin{document}$$\displaystyle K_{D,\ end}^{closed}\ =\ \frac{1}{K_{A}^{end}}$$\end{document}

We treat the backbone binding of the closed state, the backbone binding of the open state, and the end-binding of the closed state as three ligand species competing for binding the dsRNA. We denote the ligand type by s=1,2 or 3\begin{document}$s=1,2\ or\ 3$\end{document} for each of those cases, respectively.

We use the transfer matrix method detailed in Nilsson et al., 2014 to calculate the probability, ps(i)\begin{document}$p_{s}\left (i\right)$\end{document}, that a base pair i\begin{document}$i$\end{document} is occupied by a ligand of type s\begin{document}$s$\end{document} given by:(9)ps(i)= Zs(i)Z\begin{document}$$\displaystyle p_{s}\left (i\right )=\ \frac{Z_{s}\left (i\right )}{Z}$$\end{document}

Where Z\begin{document}$Z$\end{document} is the partition function and Zs(i)\begin{document}$Z_{s}(i)$\end{document} is a sum over all allowed Boltzmann-weighted states consistent with base pair i\begin{document}$i$\end{document}, for i ∈(1,N)\begin{document}$i\ \in (1,N)$\end{document} where N is the length of the dsRNA, being covered by a type s\begin{document}$s$\end{document} ligand. Zs(i)\begin{document}$Z_{s}(i)$\end{document} and Z\begin{document}$Z$\end{document} are calculated using transfer matrices.

To construct the transfer matrix for each nucleotide i\begin{document}$i$\end{document}, we enumerate all the possible states a given nucleotide i\begin{document}$i$\end{document} can be in and assign each state a statistical weight according to the method in Nilsson et al., 2014. We need to make two major changes to this model to describe our system.

The first is to account for the fact that the bulk concentrations of the ligand species in our system are not independent. In the model, the statistical weight for the binding of ligand of type s\begin{document}$s$\end{document} is given by csKs\begin{document}$c_{s}K_{s}$\end{document} where cs\begin{document}$c_{s}$\end{document} is the bulk concentration of the ligand and Ks\begin{document}$K_{s}$\end{document} is its intrinsic association constant. In our system, the bulk concentrations of the ligands are the concentration of the open and closed conformations of the protein. These concentrations are related as shown in Equations 4; 5. Therefore, the statistical weights for each of the interactions in our system are given by Keffs\begin{document}$K_{eff}^{s}$\end{document} where,(10)Keff1= KAclosed[IID]total(1+Koc)\begin{document}$$\displaystyle K_{eff}^{1}=\ \frac{K_{A}^{closed}[IID]_{total}}{(1+Koc)}$$\end{document}(11)Keff2= Koc KAopen[IID]total(1+Koc)\begin{document}$$\displaystyle K_{eff}^{2}=\ \frac{Koc\ K_{A}^{open}\left [IID\right ]_{total}}{\left (1+Koc\right )}$$\end{document}(12)Keff3= (KAend−KAclosed)[IID]total (1+Koc)\begin{document}$$\displaystyle K_{eff}^{3}=\ \frac{\left (K_{A}^{end}-K_{A}^{closed}\right )\left [IID\right ]_{total}\ }{\left (1+Koc\right )}$$\end{document}

The second change we need to make is to account for the fact that end-binding can only occur at nucleotides i=1 or N−Nend+1\begin{document}$i=1\ or\ N-N_{end}+1$\end{document}, where Nend\begin{document}$N_{end}$\end{document} is the end binding site size. Therefore, the value of Keff3\begin{document}$K_{eff}^{3}$\end{document} given by Equation 12 is only included in transfer matrices for nucleotides 1orN−Nend+1\begin{document}$1\,or\,N-N_{end}+1$\end{document} and we set Keff3=0\begin{document}$K_{eff}^{3}=0$\end{document} for all other nucleotides.

Furthermore, we allow cooperativity between all binding modes denoted by the constants σss`\begin{document}$\sigma _{ss'}$\end{document} where s\begin{document}$s$\end{document} and s`\begin{document}$s'$\end{document} can be 1,2or3\begin{document}$1,2\,or\,3$\end{document} denoting the open state binding to the backbone, closed state binding to the backbone and the closed state binding to the ends, respectively. For example, the cooperativity between one closed conformation binding to the backbone and an open conformation binding to the backbone to its right is given by σ12\begin{document}$\sigma _{12}$\end{document}. For ease of understanding, we refer to ligand types s=1,2and3\begin{document}$\rm s=1,2\,and\,3$\end{document} as o, c, and e in the text. Therefore, the cooperativity between one closed conformation binding to the backbone and an open conformation binding to the backbone to its right is given by σco\begin{document}$\sigma _{co}$\end{document}. Examples of these statistical weights are shown in Figure 6—figure supplement 8.

With these statistical weights, we construct the transfer matrices for each nucleotide i\begin{document}$i$\end{document} as below, with Keff3=0\begin{document}$K_{eff}^{3}=0$\end{document} for all nucleotides other than 1 and N−Nend+1\begin{document}$N-N_{end}+1$\end{document}.(1000001001100σ1100σ1200σ13010000000000Keff10000000100σ2100σ2200σ23000010000000000Keff20000100σ3100σ32000000000010000000000Keff30)\begin{document}$$\displaystyle \left (\begin{array}{cccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1\\1 & 0 & 0 & \sigma _{11} & 0 & 0 & \sigma _{12} & 0 & 0 & \sigma _{13}\\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & K_{eff}^{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\1 & 0 & 0 & \sigma _{21} & 0 & 0 & \sigma _{22} & 0 & 0 & \sigma _{23}\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & K_{eff}^{2} & 0 & 0 & 0 & 0\\1 & 0 & 0 & \sigma _{31} & 0 & 0 & \sigma _{32} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & K_{eff}^{3} & 0\end{array}\right )$$\end{document}

We use these to calculate the probability, ps(i)\begin{document}$p_{s}\left (i\right )$\end{document}, that a base-pair i\begin{document}$i$\end{document} is occupied by a ligand of type s\begin{document}$s$\end{document} given by Equation 9.

From this, we calculate the average probability of the backbone being bound as(13)pb= ∑i=1N(p1(i)+ p2(i))N\begin{document}$$\displaystyle p_{b}=\ \frac{\sum \limits_{i=1}^{N}\left (p_{1}\left (i\right )+\ p_{2}\left (i\right )\right )}{N}$$\end{document}

and the average probability of the ends being bound as(14)pend=p3(1)+ p3(N)2\begin{document}$$\displaystyle p_{end}=\frac{p_{3}\left (1\right )+\ p_{3}\left (N\right )}{2}$$\end{document}

We then convert this average probability to observed anisotropy (robs\begin{document}$r_{obs}$\end{document}) as follows:(15)robs=(r0−rmaxend)pend+ (r0−rmaxb)pb\begin{document}$$\displaystyle r_{obs}=\left (r_{0}-r_{max}^{end}\right )p_{end}+\ \left (r_{0}-r_{max}^{b}\right )p_{b}$$\end{document}

Where, ro\begin{document}$r_{o}$\end{document} is the anisotropy of the free RNA.

rmaxend\begin{document}$r_{max}^{end}$\end{document} is the maximum anisotropy for blunt end binding,

  rmaxb\begin{document}$r_{max}^{b}$\end{document} is the maximum anisotropy for backbone binding,

Here, rmaxend\begin{document}$r_{max}^{end}$\end{document} and rmaxb\begin{document}$r_{max}^{b}$\end{document} are parameters obtained from the fit and are shown in Figure 6—figure supplement 9.

Fits were performed using SciPy 1.8.0, NumPy 1.22.2, and lmfit 1.2.2.