We vary Dicer cleavage rates as representative for silencing strength to study the effect of RNA silencing on the spread of virus particles over the tissue.
Within the cell three different behaviors can be observed [14]. High Dicer cleavage rate results in fast clearance of the virus. Low Dicer cleavage rate delays viral growth but hardly decreases the virus levels in equilibrium. Intermediate cleavage rate results in oscillating virus levels.
Infection patterns without RNA silencing
We first study virus spread without silencing present. We initialize the tissue with healthy cells and infect one cell in the center with 10 viral plus strands. After initiation the virus starts to produce virions that spread from cell-to-cell. We fix the fraction of virions leaving a cell to 1% per hour.
The virus spreads rapidly over the entire area and 75 hours post infection (hpi) a circular area of the grid has become infected with the virus. In Figure 2A we show the number of virions, the number of siRNAs and the number of plus strands in separate screen-shots. When virions, virus plus-stranded RNA or siRNAs are absent from the cell, it is shown in green. Cells that have virions, plus-stranded RNA or siRNAs are shown in a color ranging from black to yellow (via red): the color ramps are shown in Figure 2. Also shown is a space-time plot, that is, horizontal cross-sections of the grid every hour post infection (1 hpi top row, 75 hpi bottom row). In Figure 2B we show time-series of plus-strand accumulation in adjacent cells. The curve starting at 0 hpi is an initially infected cell. The other cells in turn each become infected by the neighboring cell(s) and in each cell the virus expands to the equilibrium.
Infection patterns with silencing, without siRNA movement
When RNA silencing is active, we initialize the cells with RISC and Dicer. We first assume that Dicer is only capable of cleaving siRNA from viral dsRNA, and that siRNAs cannot move from cell to cell. Even without siRNA movement RNA silencing slows down virus spread over the area (Figure 2C). In the space-timeplot can be seen that the infection advances much slower than without the presence of the silencing response. A stronger silencing response slows down the spread of the virus more than a weaker response. We here show results for an intermediate Dicer cleavage rate, that results in oscillatory behavior within the cells (Figure 2D). For high Dicer cleavage rate the virus is cleared immediately after introduction, and is not able to spread from cell to cell, because it is eliminated before virions could be produced.
Infection patterns with siRNA movement
As shown above, without siRNA movement the infection spreads homogeneously over the area. When siRNAs do move from cell to cell, they can limit virus growth in neighboring cells, resulting in viral growth in some cells and suppression in others. This results in patterns that can spread over the entire area or stay localized to the area around the inoculated site. We observe somewhat different patterns for low and intermediate Dicer cleavage rates and we will discuss results from both possibilities.
Low dsRNA cleavage rate by Dicer
We first study the effects of a low Dicer cleavage rate for all possible siRNA movement rates. Varying the rate of siRNA movement we observe that the virus does not spread uniformly over the area and that patterns are formed.
Low siRNA movement results in a circular pattern that is shown in Figure 3A. The virus reaches a high equilibrium in the cells close to the initiation site. The siRNAs produced by these cells inhibit viral growth in a ring of cells around the center. Virions are unpacked in these cells, but the viral RNA is silenced with the siRNAs from the neighbors. At sufficient distance from the siRNA producing cells, the virus will be able to grow, and these cells will in their turn inhibit viral growth in the next ring. Once a ring is formed it remains stable.
In the time-series (Figure 3A) it can be seen that the virus can expand to an equilibrium with high RNA levels in some cells, while in other cells the virus decreases to low RNA levels. We will refer to these cells as "silenced cells". The silenced equilibrium is maintained by siRNA movement: the virus grows to the high equilibrium when siRNA movement is stopped. In silenced cells the siRNA influx suppresses virus replication completely: no dsRNA is formed, and the observed plus-strand RNA levels result from the virion influx from neighboring cells.
For a higher siRNA movement rate a different pattern occurs (Figure 3B). The circular pattern breaks and protrusive waves occur that resemble mosaic-like symptoms. We can distinguish three cell types in this pattern: cells with high virus levels, silenced cells and healthy cells. The healthy spots between the infected patches are surrounded by the silenced cells. Apart from low virus levels in the silenced cells at the edge of the spots there are no virions or other virus particles present in the healthy spots. The siRNAs present in the edge cells seem to "guard" the edges of the healthy spots as described by Hirai et al. [13]. These siRNAs, however, are not produced by the silenced cells in the edges, but move there from the infected neighboring cells.
For the maximum siRNA movement rate (10% of the available siRNAs in a cell) the infection is confined to the area around the inoculated site (Figure 3C).
Concluding, siRNA movement creates silenced cells in which the virus is suppressed, and cells in which the virus grows to high values. Increasing siRNA movement increases the number of silenced cells, rather than decreasing virus load in all cells.
Intermediate dsRNA cleavage rate by Dicer
When Dicer cleavage rate is intermediate, similar patterns can be observed (Figure 3D–G). However, a lower siRNA movement rate is sufficient to generate them. For low siRNA movement rate a speckle pattern occurs. Initially the virus is able to expand considerably in all cells (Figure 3D). However, some time after infection virus levels drop in some cells and reach the silenced equilibrium. After 300 hours the pattern is almost completely stable, either cells are at the high or at the low equilibrium. This speckle pattern distinguishes itself from the others by the high initial growth of the virus in all cells: in other patterns virus levels never reach this high levels before declining to the silenced state.
When siRNA movement increases concentric circles are formed as is the case for the lower silencing efficiency (Figure 3E). The number of silenced cells increases compared to the speckle pattern. A relatively low siRNA movement rate results in broad bands of virus infection, a higher rate results in very thin bands.
With still higher siRNA movement the thin bands in which the virus reaches high levels break, and a growing ice crystal-like pattern is observed (Figure 3F). The pattern consists of protrusive waves, and in only a small number of cells the virus reaches the high equilibrium. Because the infected spots are much smaller compared to the similar mosaic pattern for low silencing strength there are not enough siRNAs to create truly healthy spots.
With even higher siRNA movement rate the protrusive waves are reduced to a local spot, with the possibility of one or two small outbreaks (Figure 3G). For low silencing efficiency we also observed a spot pattern (Figure 3C). Low silencing efficiency results in a completely infected spot, while higher silencing efficiency results in a small ring-like pattern.
Alternative equilibria depend on influx of siRNAs
As shown in the previous section, virus levels in the spatial model can reach two different equilibria, a high and a low one, while in the intracellular ODE model only a single equilibrium exists. The patterns disappear when siRNA movement is stopped, therefore siRNA movement maintains the silenced equilibrium. To analyze the influence of the tissue dynamics on the cellular dynamics mathematically we add in- and efflux of siRNAs and virions to the intracellular model. Efflux is fixed to the movement parameters used in the tissue model. We measure average virion and siRNA influx in the equilibrium for 4 cells from the center of the tissue and use these as influx values in the intracellular model.
As an example case we take the parameters and measurements for the mosaic-like pattern for low and intermediate silencing strength shown in Figure 3B and 3F. Without in- and efflux virus levels reach a high equilibrium for low silencing strength, and for intermediate silencing strength intracellular oscillations occur (black lines Figure 4A and 4D). Efflux of siRNAs and virions results in an increase in virus levels and the oscillations disappear (red lines in Figure 4A and 4D). When virion influx is fixed to the average virion influx measured and siRNA influx is added we observe a bifurcation: Depending on siRNA influx two equilibria occur, a high and a low one, corresponding to the equilibria in the spatial model (Figure 4B and 4E). A low siRNA influx results in virus growth to the high equilibrium, high siRNA influx results in growth to the silenced equilibrium.
We calculate bifurcation diagrams as a function of the siRNA influx. In Figure 4C we show the bifurcation diagram for low silencing strength. Each line in the bifurcation diagram is calculated with measurements from different patterns. For all measurements only the high equilibrium exists for low siRNA influx, while for higher siRNA influx there are three equilibria: two stable states separated by an unstable state (Figure 4C). Depending on the initial conditions and the siRNA influx the virus can either grow to the high or the low equilibrium. The number of virus RNAs in the high and low equilibrium are similar for the bifurcation lines calculated for different siRNA movement rates. However, the siRNA influx needed to reach the low equilibrium (that is, the bifurcation point) shifts to lower siRNA influx values for lower siRNA movement rate. Low siRNA movement rate increases the effect of silencing within the cell, because less siRNAs leave it and a lower siRNA influx is sufficient to reach the silenced state.
In the spatial model both virion and siRNA influx change over time. Therefore not the final siRNA influx but the combined virion and siRNA influx during the growth phase of the virus determine which equilibrium is reached. Once a stable state is reached a change in siRNA influx has no effect on equilibrium values. Only when siRNA influx is dramatically lowered or when a very large amount of virions would be introduced it would be possible to pass the bifurcation point and move from the silenced to the high state.
For intermediate silencing strength the bifurcation diagram is similar, however due to the lower siRNA efflux and the stronger silencing response within the cell, it is shifted toward lower siRNA influx. The high and the low equilibrium are connected, meaning that an increase in siRNA influx results in a shift from the high equilibrium to the low one. This is the case in the speckle pattern: all cells initially expand to the high equilibrium, but some cells decline to the silenced state later in infection.
To indicate the measured siRNA influx from the patterns shown in Figure 3, we placed circles in the bifurcation diagrams for two high and two low cells (Figure 4C and 4F). For most patterns the equilibrium values are far from the bifurcation points, resulting in a stable pattern unaffected by noise. For the speckle pattern the values are close to the bifurcation point. A slight increase in siRNA influx can push a cell with high virus levels to the silenced state. On the other hand a slight decrease in siRNA influx can result in growth from the silenced state to the high equilibrium. The resulting pattern is at a delicate balance: cells that reach the high equilibrium suppress virus growth in neighboring cells, and the silenced cells cause a low enough siRNA influx into the cell to stay at the high state.
Concluding, the effect of siRNA influx from neighboring cells is two-fold. Firstly, siRNA influx creates two coexisting equilibria; secondly, siRNA influx during the initial growth phase of the virus largely determines which equilibrium is reached.
Combined single- and double-stranded RNA cleavage by Dicer
It has been proposed that Dicer can cleave ssRNA in addition to dsRNA [2]. On the cellular level combined single- and double-stranded cleavage by Dicer results in the diverse skewed siRNA ratios that have been observed [2, 14, 18, 19]. Although combined Dicer cleavage of single- and double-stranded RNA can clear the virus for lower Dicer cleavage rates, it can also be less efficient. For the same total Dicer cleavage rate dsRNA cleavage results in somewhat lower plus-strand RNA levels then combined single- and double-strand cleavage [14]. This is due to the skewed siRNA ratio: many siRNAs are produced that target the minus-strand, but few siRNAs are produced that target the plus-strand. This bias results in slightly less efficient silencing, however when the minus strand is completely degraded, the virus is cleared despite the less efficient response.
The complex feedbacks between the intracellular and the spatial model result in an unexpected effect of ssRNA cleavage in the spatial model. While keeping total Dicer cleavage rate constant, increasing ssRNA cleavage by Dicer shifts the silenced equilibrium to higher RNA levels. An increase of virus levels in the silenced equilibrium results in less pronounced patterns and it can even result in disappearance of the silenced state.
In Figure 5D–F we show patterns for increasing rates of ssRNA cleavage by Dicer 300 hpi. The patterns change from circles to mosaic-like, and the silenced cells change from a dark to a red color, indicating higher virus levels in silenced cells. This effect becomes more clear later in infection (Figure 5H). Moreover, the infection pattern shown in Figure 5F fades out (at 450 hpi, shown in Figure 5I), and eventually the entire tissue becomes fully infected.
In Figure 5A–C we show how the equilibria change when changing ratio's of double-to single-stranded RNA cleavage by Dicer. We keep total Dicer cleavage rate constant at 10 cleavages per Dicer per hour. To explain the observed siRNA ratios approximately 0–20% of Dicer cleavages should take place on ssRNA [14]. In Figure 5A 5% of Dicer cleavages is on ssRNA and in Figure 5B 15%. The silenced state clearly shifts to higher virus levels when ssRNA is cleaved. When more ssRNA is cleaved the equilibrium shifts further upwards until it disappears. In Figure 5C we show the behavior for 19.5% ssRNA cleavage by Dicer, at 20% ssRNA cleavage the silenced state has completely disappeared. The upwards shift of viral RNA levels implies that virus replication is not completely silenced. Indeed dsRNA is produced in the silenced cells.
These results are explained further in the bifurcation diagram in Figure 5G. Shown are the equilibrium values of plus-strand RNA as a function of siRNA influx. The different lines represent a varying fraction of ssRNA cleavage by Dicer. The virion influx increases for increasing ssRNA cleavage by Dicer and the silenced state shifts upwards. Additionally, an increasingly higher siRNA influx is needed to reach this equilibrium. The dots indicate the location of cells from the spatial model that reach the high and the low equilibrium. Due to increased siRNA levels the siRNA influx is increased when ssRNA is cleaved by Dicer. However, due to the less efficient silencing within cells, an increasingly higher siRNA influx is needed to reach the silenced state.
This means that, although combined single- and double-stranded RNA cleavage by Dicer can clear the virus at lower Dicer cleavage rate [14], the slight increase in virus equilibrium is disadvantageous at the tissue level as RNA silencing can no longer fully suppress virus replication in silenced cells. However, an advantage of single strand cleavage by Dicer is the slower rate of spread over the tissue.
Effect of amplification
Amplification of silencing through RDR can decrease viral levels in plants [20–22]. It also affects symptoms observed: plants without a functional RDR become fully infected or die, while plants with RDR show mild chlorosis or mosaic [23, 24].
To study the effect of the secondary response on pattern formation and virus levels we add amplification to the model. At the cellular level RNA silencing with unprimed amplification can clear the virus for much lower silencing strength. It has to be noted however, that unprimed amplification can be triggered by any RNA, and will lead to responses against host RNA. Therefore, a mechanism has to be included to protect the host against auto-immunity [25, 26]. Primed and guided amplification can increase oscillations and create a new region of behavior, in which the virus is degraded after an initial growth peak. Primed amplification can only be beneficial when there is a net siRNA gain [14]. At low Dicer cleavage rate the virus equilibrium is slightly decreased on the cellular level. In the spatial model the addition of amplification can result in a change of patterns (Figure 6A). Without amplification almost the complete area becomes infected with the virus, with amplification circles or mosaic-like patterns can occur.
Concluding, for low silencing strength amplification has a similar effect to increasing silencing strength or siRNA movement.
New patterns can be observed when the virus is cleared after an initial growth peak at the cellular level. Because the virus can expand and produce virions initially, it is able to spread over the tissue. However, in each cell the virus will be completely degraded by the RNA silencing response. We therefore observe transient patterns that leave healthy tissue behind. These patterns are shown in Figure 6B and 6C. Without siRNA movement a single growing ring occurs, low siRNA movement can lead to a disappearing spot near the infected site. These results indicate that a strong feedback can result in transient infection patterns. When siRNA movement is higher, the feedback becomes smaller due to the efflux of siRNAs, and infection patterns as shown in Figure 3 and 6A occur.