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

Model Validation

Results

Headers