Team:Waterloo/Modeling/CaMV Replication
CaMV Replication
This page contains all our information about the CaMV replication model.
Model Formation
Viral Life Cycle
Patrick and Mark please review
There are a few stages, we are focused on the replication one.
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
Cell concentrations are continuous Molecules No outside infectionThis 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 mass-actionThis assumption is made in the paper by Nakabayashi
Rate of P6 knockdown follows mass-actionE
DNA, RNA, proteins, and complete virions degrade, incomplete virions do notE
We are only targeting 19S RNAE
RNA production follows mass-actionE
Concentration of spliced/unspliced RNA is at rapid equilibriumE
Spliced and unspliced 35S RNA degrade at the same rateE
Packaging occurs in one stepE
Only unspliced RNA is packagedE
P1 and P2 do not affect replication dynamicsE
P6 is only translated from 19S RNA, translation rate is constantE
P3, P4, and P5 are translated from 35S RNA, translation is activated by P6E
Ignore Inclusion Body FormationE
Ignore P6 in nucleusE
Mass-action anchoring of P3 to virionsE
All P4 is instantaneously splicedE
Virions may reinfect nucleusE
Virions leave the cell at a constant rateE
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. |
Results
Model Validation
Include notes on how the model matches reality/our expectations of reality in this section.