Team:Rice/Model/Expression

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Host Expression

Overview

Expression begins when mRNAs are transcribed at an energy-dependent rate and then form complexes with ribosomes. Since most of the energy in a bacterial cell is used in translation [4], our model assumes that transcriptional energy use is negligible. Ribosome complexes translate mRNAs into protein molecules, using up cellular energy in the process. After translation, the complexes unbind and can be reused. rRNAs are also transcribed, though at a higher rate, and they bind to ribosomal proteins to form functional ribosomes.

Explanation

Note: if the equations below seem messed up, please follow this link to a simplified version of this page.

Expression starts with transcription; in our model, transcription is a spontaneous, energy-dependent process. Each mRNA is produced at a rate of $\frac{dm_i}{dt} = w_i \frac{E_c}{E_c+o_i}$, where $E_c$ is the cellular energy, $w_i$ is the maximum transcription rate for mRNA (with $i = E,T,R,H$ for the metabolic enzymes, transport enzymes, ribosomal proteins, or other host proteins respectively), and $o_i$ is the threshold for transcription. In addition, for $m_H$, the rate is multiplied by a regulatory constant $ \frac{1}{(1+p_H/k_H)^{h_H}}$, where $p_H$ is the number of other host proteins in the cell and the remaining values are constants. This factor approximates the normal host regulation of its own proteins. All RNAs slowly degrade in the cell, with a degradation constant of $d_r$. For translation to occur, functional host ribosomes need to be present. These are composed of ribosomal protein ($p_R$) and rRNA ($r_R$). Binding is a reversible process with binding constant $b_r$ and unbinding constant $u_r$, but once bound, the functional ribosomes ($R_h$) can bind with an mRNA ($m_i$) to form a complex $c_i$. This second binding step is also reversible. Finally, translation can occur and the corresponding protein ($p_i$) is produced. Translation occurs at a rate of $\gamma \frac{c_i}{n_i}$, where $\gamma$ is the global translation rate and equals $\gamma_m \Big( \frac{E_c}{k_\gamma + E_c} \Big)$. Here $\gamma_m$ is the maximum translation rate, $k_\gamma$ is the energy threshold for translation, and $n_i$ is the length of the protein to translation in amino acids.

Equations

\begin{equation*} \begin{aligned} \frac{dm_T}{dt} &= w_T \frac{E_c}{E_c+o_T} + \gamma \frac{c_T}{n_T} - b_r R_h m_T + u_r c_T - (d_r+\lambda) m_T \\ \frac{dm_E}{dt} &= w_E \frac{E_c}{E_c+o_E} + \gamma \frac{c_E}{n_E} - b_r R_h m_E + u_r c_E - (d_r+\lambda) m_E \\ \frac{dm_R}{dt} &= w_R \frac{E_c}{E_c+o_R} + \gamma \frac{c_R}{n_R} - b_r R_h m_R + u_r c_R - (d_r+\lambda) m_R \\ \frac{dm_H}{dt} &= w_H \frac{1}{(1+p_H/k_H)^{h_H}}\frac{E_c}{E_c+o_H} + \gamma \frac{c_H}{n_H} - b_r R_h m_H + u_r c_H - (d_r+\lambda) m_H \\ \frac{dc_T}{dt} &= b_r R_h m_T - u_r c_T - \gamma \frac{c_T}{n_T} - \lambda c_T \\ \frac{dc_E}{dt} &= b_r R_h m_E - u_r c_E - \gamma \frac{c_E}{n_E} - \lambda c_E \\ \frac{dc_R}{dt} &= b_r R_h m_R - u_r c_R - \gamma \frac{c_R}{n_R} - \lambda c_R \\ \frac{dc_H}{dt} &= b_r R_h m_H - u_r c_H - \gamma \frac{c_H}{n_H} - \lambda c_H \\ \frac{dp_T}{dt} &= \gamma \frac{c_T}{n_T} - \lambda p_T \\ \frac{dp_E}{dt} &= \gamma \frac{c_E}{n_E} - \lambda p_E \\ \frac{dp_R}{dt} &= \gamma \frac{c_R}{n_R} - b_r p_R r_R + u_r R_h - \lambda p_R \\ \frac{dp_H}{dt} &= \gamma \frac{c_H}{n_H} - \lambda p_H \\ \frac{dr_R}{dt} &= w_r \frac{E_c}{E_c+o_r} - b_r p_R r_R + u_r R_h - (d_r + \lambda) r_R \\ \frac{dR_h}{dt} &= b_r p_R r_R - u_r R_h +\gamma \sum \frac{c_i}{n_i} + u_r \sum c_i - b_r R_h \sum m_i - \lambda R_h \\ \end{aligned} \end{equation*}