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

This ODE model only tracks replication within one cell, it cannot track multiple cells. This is handled by the viral spread model instead.

"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."

This assumption is made in the paper by Nakabayashi

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Parameters

Here are the parameters for the model

Results

Model Validation

Include notes on how the model matches reality/our expectations of reality in this section.

Discussion

References