Interactive Steady-State
Exploring the closed-form steady-state behavior
In the main text of the paper, we discuss that inclusion of the dilution in our expression for the precursor concentration allows us to calculate analytical expressions for a variety of steady-state properties of the ribosomal allocation model. Their complete derivations are provided in the Supplementary Information but we state their forms here. The steady-state precursor concentration is defined as \(c_{pc}^* = \frac{\mathsf{N}}{\lambda} - 1, \tag{1}\) where $\mathsf{N}$ is the maxium metabolic output, $\mathsf{N} = \nu_{max}(1 - \phi_{Rb} - \phi_O)$. With knowledge of the precursor concentration, we can then easily calculate the steady-state translation rate \(\gamma(c_{pc}^*) = \gamma_{max}\frac{c_{pc}^*}{c_{pc}^* + K_M^{c_{pc}}} \tag{2}.\)
With these in hand, we can then calculate an expression for the steady-state growth rate, which is a quadratic equation with only one physically meaningful root of \(\lambda = \frac{\mathsf{N} + \Gamma - \sqrt{(\mathsf{N} + \Gamma)^2 - 4\mathsf{N}\Gamma(1 - K_M^{c_{pc}})}}{2(1 - K_M^{c_{pc}})}, \tag{3}\) were we have introduce the notation of $\Gamma = \gamma_{max}\phi_{Rb}$, which is the maximum translational output.
A property evident in these expressions is that they all depend on the ribosomal allocation parameter, $\phi_{Rb}$. In the figure below, we plot the behavior of these three expressions as a function of the ribosomal allocation $\phi_{Rb}$, and allow the user to tune the remaining model parameters to gauge the importance of each parameter. The parameter $\nu_{max}$ can be thought of as a proxy for the quality of the nutrients in the environment with low values (light green in the figure) corresponding to poor environments and large values (dark blue in the figure) corresponding to rich conditions.