Single-particle studies of the effects of RNA–protein interactions on the self-assembly of RNA virus particles

Significance RNA viruses first inspired the term “self-assembly.” Yet much is still not understood about how even the simplest such viruses assemble or if different viruses assemble in similar ways. Theoretical models suggest many possible assembly pathways, with many different roles for RNA, but until recently measuring these pathways has not been possible. We use a sensitive microscopy technique to follow the assembly of individual particles of brome mosaic virus (BMV), a plant virus. We find evidence of an RNA-mediated nucleation-and-growth pathway that is strikingly similar to that of MS2, a bacterial virus. The last common ancestor of BMV and MS2 existed only in ancient times, suggesting that their assembly pathway might be evolutionarily conserved and that other viruses might follow a similar pathway.

For our experiments, we quantify the spread in start times by fitting each measured trace to a piecewise exponential and determining the MAD of the fitted start times. Figure 6A in the main text shows how the MAD of start times depends on protein concentration. We simulate the concentration dependence of the MAD of start times by fitting an ensemble of simulated traces to the same piecewise exponential function and calculate the MAD of fitted start times at different concentrations. We perform simulations with ! values of 0.7 and 1 because these are the best fit value from our experiments and the expected value from Langmuir kinetics, respectively. We choose !,# values of 12 s and 5.5 s for ! values of 0.7 and 1, such that the growth time for each concentration is similar to the experimentally measured median growth times. We choose a noise standard deviation of 0.05 relative to the maximum intensity to visually match variations in the measured traces. Figures S1A-S1C show all measured traces at 84 mmol/L NaCl along with an ensemble of five simulated traces. Figure S1D shows the MAD values extracted from fits to the simulated curves along with power-law fits for simulations with different protein concentrations. The fit power-law exponent from the simulations, -./ , is 0.46 and 0.68 for ! values of 0.7 and 1, respectively. We note that the -./ does not depend on the simulated noise magnitude (data not shown). Both values of -./ are comparable with the measured exponent of = 0.5 +-0.1 at 84 mM NaCl, so at this salt concentration, the concentration exponent of the MAD of start times (  10,000 traces were simulated at each protein concentration. The lines are best fits to a power law function and )12 is the negative of the exponent from the fit.

Appendix B: The effects of non-instantaneous protein introduction
Here we estimate how the non-instantaneous introduction of protein affects the measured starttime concentration-scaling exponent.
In our iSCAT experiments, we introduce protein solution into the sample chamber by pumping it in over the course of 20 seconds. The flow is at the low Reynold's number of 0.5, so we expect laminar flow within the chamber. Thus, with no slip boundary conditions, we expect a parabolic flow profile, as illustrated in Figure S2A, top. After pumping in a protein solution with concentration # , we expect the protein solution to be centered in the chamber such that the protein must diffuse a distance to reach the chamber surface where the RNA is tethered. In our experiments, the field of view is approximately 9.3 mm from the chamber entrance, and given the volume of protein solution we introduce, the tip of the parabola would be approximately 45 mm beyond the chamber entrance. Thus, above the small, 10 μm by 10 μm iSCAT field of view, we expect the protein solution forms a nearly flat planar layer once pumping stops.
After pumping stops, the protein then diffuses to fill the chamber. To model how the protein diffuses to the surface-bound RNA, we solve the 1D diffusion equation. We assume the initial condition illustrated in Figure S2A, bottom, where the protein and buffer solutions form infinite layers. The chamber has a height of 2 = 0.75 mm and the protein must diffuse a distance − to reach the chamber surface. We believe the assumption of planar layers is justified because the protein solution is pumped well beyond the iSCAT field of view, and the 0.75-mm channel height is small compared to the distance the protein solution is pumped. We impose no-flux boundary conditions at the chamber walls and solve the 1D diffusion equation to find the protein concentration at the surface as a function of time. The concentration of protein at the surface is where > = 5 ⁄ is a characteristic diffusion time and is the diffusion coefficient of a protein dimer. For BMV protein dimers, we assume = 91 μm 2 /s based on measurements (1) of the hydrodynamic radius of MS2 CP2 (2.5 nm).

Figure S2B
plots the protein concentration at the surface relative to # for different distances that the protein must diffuse, assuming that it starts diffusing at = 20 s. Assuming parabolic flow and given the volume of protein we flow into the chamber, we calculate that the distance the protein must diffuse to reach the surface is 40 μm ( / = 0.89). We experimentally verify this distance using bright-field microscopy to image a solution of 1 μm-diameter polystyrene particles that we flow into the imaging chamber. With bright-field microscopy, we find that the distance protein must diffuse ranges from 20 μm to 50 μm ( / = 0.95 to 0.87) which is consistent with that calculated based on parabolic flow. Based on the diffusion model, the time required for the protein concentration at the surface to reach half of the bulk protein concentration ranges from 6 s to 39 s.
To model how protein diffusion affects the measured nucleation kinetics, we couple the protein diffusion model to a classical nucleation model. In the classical model, is the faction of RNA strands on which a nucleus has formed, is a rate constant determining the nucleation rate, and the exponent A describes how the nucleation time scales with protein concentration. By the law of mass action, A is the number of proteins in the critical nucleus, so we expect A > 1. The solution to this differential equation is an exponential distribution of nucleation times. We extend this classical nucleation model by allowing the protein concentration to vary with time, as determined by our diffusion model. The differential equation becomes , and # is the time at which protein starts to diffuse. Figure S2C plots ( ) for varying values of / with A = 2, A,# = 50 s, # = 20 s, and # = 1 μmol/L . Note that if / = 1, that is, the protein solution instantaneously fills the entire chamber, * = − # , and classical exponential nucleation kinetics(2, 3) are recovered.
Our goal is to use the model of nucleation kinetics to determine how protein diffusion affects the experimentally measured nucleation concentration scaling exponent. To do this, we perform a least-squares fit on measured cumulative distributions of start times. We simultaneously fit the cumulative distributions from all experiments at a given salt concentration to determine the best fit to A at each salt concentration. At both 84 mmol/L and 167 mmol/L salt, we perform 6 experiments each (two experiments at three protein concentrations). At each salt concentration, there are 14 fitting parameters: A , A,# , six values of the ratio / (one for each experiment), and six values of # (one for each experiment). Figures S2D and S2E show the best fit to our measurements at 84 mmol/L NaCl and 167 mmol/L NaCl, respectively.
The best fit value of A depends strongly on bounds limiting the fit value of the ratio / . With lower bound on / , at 84 mmol/L NaCl, the fit converges to A = 3.3 ± 1.4, as indicated in Figures  Similarly, at 167 mmol/L NaCl, the fit converges to A = 2.8 ± 0.4, but again, two experiments have fit values of / below 0.87. If a lower bound is placed on / , the fit value of A decreases. Thus, we use our fits with no lower limit on / to set an upper bound on A , and we perform additional fits where we limit / to be no less than 0.87 to set a lower bound on A . The lower bound on A , as determined from these fits is A = 1.2 ± 0.4 at 84 mmol/L NaCl, and

Appendix C: Intensity fluctuations prior to the start of assembly
One can get a feel for the relevant intensity fluctuations prior to the start of assembly by taking a closer look at the trace shown in Figure 4 of the manuscript, which is recorded with 1-s frame time (reproduced here in Figure S3A). The portion of the trace between 60 and 120 s is taken after the coat proteins are injected but before the start time of assembly ( Figure S3B). To better visualize these fluctuations, we plot the frame-to-frame changes in intensity (Figure S3C), and the distribution of frame-to-frame changes as a kernel density plot ( Figure S3D). To get a feel for the magnitude of the fluctuations, we compare them to the measurement noise. We estimate the noise level by plotting the intensity traces from 22 regions of the field of view where no RNA molecules are tethered and where no particles assemble over the course of the experiment. These traces and their intensity distributions are shown in color in Figure S3E and S3F. We see that the black (assembly) trace and the colorful (background) traces have similar fluctuations, and we conclude that the fluctuations prior to assembly are roughly equal to the noise level of the measurement. If we assume that proteins bind only transiently to the RNA prior to nucleation, then the magnitude of the fluctuations gives an upper limit on the nucleus size. For example, if the critical nucleus size were 10 dimers, then we would expect to see fluctuations of up to 10 dimers prior to nucleation. That we see no evidence of fluctuations larger than 6 dimers suggests that the apparent nucleus size is equal to or smaller than that.

Appendix D: Plots of all assembly traces and fits
In the following pages, all of the assembly traces analyzed in the main text are plotted in various colors, which designate the experimental conditions used, and fitted with a piecewise function, which is shown in red. As described in the main text, the piecewise function has the form where I(t) is the intensity as a function of time, # is the initial intensity (which can be offset from zero because of microscope drift), @ is the final intensity, ) is the start time -the time at which the trace begins to rise from its initial value, and C is the growth time-the time for the trace to reach (1-1/e) = 0.63 of its final value once it has started increasing. The gray bar to the right of each trace shows the size of a full BMV capsid.
Some of the traces are shorter than the full measurement time of 600 s. The reason for these shortened traces is that these particles show an instantaneous drop in intensity during the measurement. Such intensity drops reflect a detachment event: either the RNA tether has detached from the surface or a cluster of assembling proteins have detached from the RNA. If the drop in intensity is smaller than the intensity associated with an RNA molecule, then we assume that only proteins have detached or that the dip in intensity was the result of microscope drift, and we continue recording the trace. Otherwise, we assume the RNA has detached, and we stop recording the trace at the time of the drop.
Some of the traces with detachment events are fit poorly by the exponential function. But because these traces are rare, they do not affect the median or MAD values that we report in the main text. They might impact the mean and standard deviation of the values we measure, but in the main text we report and discuss the median and MAD values because they are robust to these rare traces.
Dataset S1. The mass photometry data from iSCAT experiments measuring the sizes of coat-protein clusters at different concentrations of NaCl. The data is saved in an .xlsx file with each NaCl concentration in a separate sheet. This data is shown in Figure 2A.
Dataset S2. The time-series data from all iSCAT assembly kinetics experiments. The intensity as a function of time for each particle is recorded at 1,000 Hz and reported with a 1,000-frame average. The measurement duration of most of the particles extends to 600 s, but the duration of some particles is shorter because these particles detached from the surface. The data is saved in an .xlsx file with each experiment in a separate sheet. This data is shown in Figures 1, 3