Team:Waterloo/Modeling/CaMV Replication
CaMV Replication
This page contains all our information about the CaMV replication model.
Model Formation
Background
Network
DNA
$$\frac{d d_g}{dt} = k_v V (d_{max} - d_{total}) - \alpha_c d_g - k_g d_g - \gamma_d d_g$$ $$\frac{d d_c}{dt} = \alpha_c d_g - k_c d_c - \gamma_d d_c$$ $$\frac{d d_{gm}}{dt} = k_v V_m (d_{max} - d_{total}) - \alpha_c d_{gm} + k_g d_g - \gamma_d d_{gm}$$ $$\frac{d d_{cm}}{dt} = \alpha_c d_{gm} + k_c d_c - \gamma_d d_{cm}$$
RNA
$$\frac{d r_{19S}}{dt} = \alpha_{19S} d_c - (\gamma_{19S}+\gamma_{r}) r_{19S}$$ $$\frac{d r_{35S}}{dt} = \alpha_{35S} d_c - k_p p_4 p_5 f_u r_{35S} - (\gamma_{35S}+\gamma_{r}) r_{35S}$$ $$\frac{d r_{35Sm}}{dt} = \alpha_{35S} d_{cm} - k_p p_4 p_5 f_u r_{35Sm} - (\gamma_{35S}+\gamma_{r}) r_{35Sm}$$
Protein
$$\frac{d p_3}{dt} = \beta_3 \left( \frac{p_6}{p_6+K_6} \right) (r_{35S} + r_{35Sm}) - k_a p_3 (V_i+V_{im}) - \delta_3 p_3$$ $$\frac{d p_4}{dt} = \beta_4 \left( \frac{p_6}{p_6+K_6} \right) (r_{35S} + r_{35Sm}) - k_p p_4 p_5 f_u (r_{35S} + r_{35Sm}) - \delta_4 p_4$$ $$\frac{d p_5}{dt} = \beta_5 \left( \frac{p_6}{p_6+K_6} \right) (r_{35S} + r_{35Sm}) - k_p p_4 p_5 f_u (r_{35S} + r_{35Sm}) - \delta_5 p_5$$ $$\frac{d p_6}{dt} = \beta_6 r_{19S} - \delta_6 p_6$$
Virions
$$\frac{d V_i}{dt} = k_p p_4 p_5 f_u r_{35S} - k_a p_3 V_i$$ $$\frac{d V}{dt} = k_a p_3 V_i - k_v V (d_{max} - d_{total}) - v_e V - \delta_v V$$ $$\frac{d V_{im}}{dt} = k_p p_4 p_5 f_u r_{35Sm} - k_a p_3 V_{im}$$ $$\frac{d V_m}{dt} = k_a p_3 V_{im} - k_v V_m (d_{max} - d_{total}) - v_e V_m - \delta_v V_m$$
Algebraic Equations
$$d_{total} = d_g + d_c + d_{gm} + d_{cm}$$ $$\gamma_r = \frac{L}{1+e^{k (p_6-x_0)}}$$ $$x_0 = \frac{1}{2} p_6 ^{ss}$$ $$p_6 ^{ss} = \frac{\beta_6}{\delta_6} \frac{\alpha_{19}}{\gamma_{19}} d_{max}$$
Assumptions
Parameters
Here are the parameters for the model
| Symbol | Value | Units | Description |
|---|---|---|---|
| $k_v$ | 0.1 | min$^{-1}$ | Rate at which virions produced by the cell reinfect the nucleus. |
| $d_{max}$ | 100/vol | molecules/volume | Maximum concentration of viral genomes in the nucleus. |
| $\alpha_c$ | 0.1 | min$^{-1}$ | Rate at which gaps are repaired in gapped DNA to form cccDNA. |
| $k_g$ | 0.01 | min$^{-1}$ | Rate at which the P6 gene on the gapped DNA is modified. |
| $k_c$ | 0.01 | min$^{-1}$ | Rate at which the P6 gene on cccDNA is modified. |
| $\gamma_d$ | 0.001 | min$^{-1}$ | DNA Degradation rate. |
| $\alpha_{19S}$ | 0.01 | min$^{-1}$ | Transcription rate of 19S RNA. |
| $\alpha_{35S}$ | 0.05 | min$^{-1}$ | Transcription rate of 35S RNA. |
| $\gamma_{19S}$ | 0.001 | min$^{-1}$ | Degradation rate of 19S RNA. |
| $\gamma_{35S}$ | 0.001 | min$^{-1}$ | Degradation rate of 35S RNA. |
| $f_u$ | 0.3 | unitless | Fraction of unspliced 35S RNA in the cell (assumed at equilibrium). |
| $\beta_3$ | 0.1 | min$^{-1}$ | Translation rate of P3. |
| $\beta_4$ | 0.1 | min$^{-1}$ | Translation rate of P4. |
| $\beta_5$ | 0.1 | min$^{-1}$ | Translation rate of P5. |
| $\beta_6$ | 0.1 | min$^{-1}$ | Translation rate of P6. |
| $K_6$ | 1000 | molecules/volume | Half-saturation constant for transactivation of P1-P5 production. |
| $\delta_3$ | 0.001 | min$^{-1}$ | Degradation rate of P3. |
| $\delta_4$ | 0.001 | min$^{-1}$ | Degradation rate of P4. |
| $\delta_5$ | 0.001 | min$^{-1}$ | Degradation rate of P5. |
| $\delta_6$ | 0.001 | min$^{-1}$ | Degradation rate of P6. |
| $k_p$ | 0.1 | ??? | Packaging rate. |
| $k_a$ | 0.1 | ??? | Rate of P3 anchoring to virions. |
| $v_e$ | 0.1 | min$^{-1}$ | Rate at which virions exit the cell. |
| $\delta_v$ | 0.001 | min$^{-1}$ | Rate of virion degradation. |