CaMV DNA

CaMV DNA is present in both gapped and ungapped forms in the nucleus and may be converted to "modified" form by CRISPR/Cas9, which removes its P6 activity. There is assumed to be a carrying capacity for viruses within the nucleus because of limited cellular resources and the nucleus may be reinfected by virions packaged in the cytoplasm.

$$\frac{d d_g}{dt} = k_v V (d_{max} - d_{total}) - \alpha_c d_g - k_g d_g - \gamma_d d_g$$ $$\frac{d d_{cc}}{dt} = \alpha_c d_g - k_c d_{cc} - \gamma_d d_{cc}$$ $$\frac{d d_{m.g}}{dt} = k_v V_m (d_{max} - d_{total}) - \alpha_c d_{m.g} + k_g d_g - \gamma_d d_{m.g}$$ $$\frac{d d_{m.cc}}{dt} = \alpha_c d_{m.g} + k_c d_{cc} - \gamma_d d_{m.cc}$$

RNA

We consider three transcripts, all assumed to be of importance only in the cytoplasm. The 19S mRNA transcript, the 35S transcript and the modified 35S transcript. These transcripts are degraded both passively, according to the $\gamma_{19S}$ and $\gamma_{35S}$ parameters, and actively by the plant RNAi defenses, captured in the $\gamma_{r}$ term. The 35S transcripts are also packaged into virions by a P4/P5 complex present on the inclusion bodies.

$$\frac{d r_{19S}}{dt} = \alpha_{19S} d_{cc} - (\gamma_{19S}+\gamma_{r}) r_{19S}$$ $$\frac{d r_{35S}}{dt} = \alpha_{35S} d_{cc} - k_p p_4 p_5 f_u r_{35S} - (\gamma_{35S}+\gamma_{r}) r_{35S}$$ $$\frac{d r_{35Sm}}{dt} = \alpha_{35S} d_{m.cc} - k_p p_4 p_5 f_u r_{35Sm} - (\gamma_{35S}+\gamma_{r}) r_{35Sm}$$

Protein

The concentrations of four proteins are modelled. Translation of P3, P4 and P5 from the either of the 35S transcripts occurs because of activation by P6, while P6 is translated from the 19S transcript. All transcripts proteins are degraded and P3, P4 and P5 participate in viral packaging.

$$\frac{d p_3}{dt} = \beta_3 \left( \frac{p_6}{p_6+K_6} \right) (r_{35S} + r_{35Sm}) - k_a p_3 (V_i+V_{im}) - \delta_3 p_3$$ $$\frac{d p_4}{dt} = \beta_4 \left( \frac{p_6}{p_6+K_6} \right) (r_{35S} + r_{35Sm}) - k_p p_4 p_5 f_u (r_{35S} + r_{35Sm}) - \delta_4 p_4$$ $$\frac{d p_5}{dt} = \beta_5 \left( \frac{p_6}{p_6+K_6} \right) (r_{35S} + r_{35Sm}) - k_p p_4 p_5 f_u (r_{35S} + r_{35Sm}) - \delta_5 p_5$$ $$\frac{d p_6}{dt} = \beta_6 r_{19S} - \delta_6 p_6$$

Virions

The concentrations of four proteins are modelled. Translation of P3, P4 and P5 from the either of the 35S transcripts occurs because of activation by P6, while P6 is translated from the 19S transcript. All transcripts proteins are degraded and P3, P4 and P5 participate in viral packaging. Due to the complicated nature and relative lack of characterization of viral packaging, we decided in our model to combine the packaging reaction into one term, dependent on P4, P5, and unspliced 35S RNA concentration.

$$\frac{d V_i}{dt} = k_p p_4 p_5 f_u r_{35S} - k_a p_3 V_i$$ $$\frac{d V}{dt} = k_a p_3 V_i - k_v V (d_{max} - d_{total}) - v_e V - \delta_v V$$ $$\frac{d V_{im}}{dt} = k_p p_4 p_5 f_u r_{35Sm} - k_a p_3 V_{im}$$ $$\frac{d V_m}{dt} = k_a p_3 V_{im} - k_v V_m (d_{max} - d_{total}) - v_e V_m - \delta_v V_m$$

Algebraic Equations

Not all equations pertinent to the dynamics are differential: in particular, the carrying capacity of genomes in the nucleus and effect of RNAi are modelled using algebraic equations.

$$d_{total} = d_g + d_{cc} + d_{m.g} + d_{m.cc}$$ $$\gamma_r = \frac{L}{1+e^{k (p_6-x_0)}}$$ $$x_0 = \frac{1}{2} p_6 ^{ss}$$ $$p_6 ^{ss} = \frac{\beta_6}{\delta_6} \frac{\alpha_{19}}{\gamma_{19}} d_{max}$$

• Cell concentrations are continuous
• Molecules in the cytosplasm are well-mixed
• No outside infection
• This ODE model only tracks replication within one cell, it cannot track multiple cells. This is handled by the viral spread model instead
• Limited number of genomes in nucleus
• "Another pool of viral genomes, in the order of 10-100 copies of minichromosomes comprising supercoiled circular viral DNA and host histones, accumulates in the nucleus."
• Rate of repair of gapped DNA follows law of mass-action
• Rate of P6 gene mutation is proportional to number of wild-type genomes
• DNA, RNA, proteins, and complete virions degrade, incomplete virions do not
• We assume P3 binds to virions rapidly enough for incomplete virion degradation to be negligible
• We are only targeting 19S RNA
• RNA production follows mass-action
• Concentration of spliced/unspliced RNA is at rapid equilibrium
• Spliced and unspliced 35S RNA degrade at the same rate
• Encapsidation and reverse transcription occur simultaneously
• Information for CaMV is lacking, but for HBV (another pararetrovirus) reverse transcription is initiated during encapsidation and encapsidation is initiated by RT . The complexity of this packaging process is not fully captured in this model
• Only unspliced RNA is packaged
• Spliced RNA lacks P1 and P2 and so, although able to replicate within the cell, virions which package spliced RNA will be unable to propagate
• P1 and P2 do not affect replication dynamics
• P1 is primarily involved in cell-to-cell movement and P2 with host-to-host movement. Neither have a significant impact on the replication process
• P6 is only translated from 19S RNA, translation rate is proportional to cccDNA copy number
• P3, P4, and P5 are translated from 35S RNA, translation is activated by P6
• This is very well established
• All P6 is incorporated in inclusion bodies
• Mass-action anchoring of P3 to virions
• All P4 is instantaneously spliced
• As discussed above the subtleties of encapsidation are not captured by this model
• Virions may reinfect nucleus
• This is very well established
• Virions leave the cell at a constant rate