Optimal control of ribosome population for gene expression under periodic nutrient intake

Abstract

Translation of proteins is a fundamental part of gene expression that is mediated by ribosomes. As ribosomes significantly contribute to both cellular mass and energy consumption, achieving efficient management of the ribosome population is also crucial to metabolism and growth. Inspired by biological evidence for nutrient-dependent mechanisms that control both ribosome-active degradation and genesis, we introduce a dynamical model of protein production, that includes the dynamics of resources and control over the ribosome population. Under the hypothesis that active degradation and biogenesis are optimal for maximizing and maintaining protein production, we aim to qualitatively reproduce empirical observations of the ribosome population dynamics. Upon formulating the associated optimization problem, we first analytically study the stability and global behaviour of solutions under constant resource input, and characterize the extent of oscillations and convergence rate to a global equilibrium. We further use these results to simplify and solve the problem under a quasi-static approximation. Using biophysical parameter values, we find that optimal control solutions lead to both control mechanisms and the ribosome population switching between periods of feeding and fasting, suggesting that the intense regulation of ribosome population observed in experiments allows to maximize and maintain protein production. Finally, we find some range for the control values over which such a regime can be observed, depending on the intensity of fasting.

Keywords: dynamical system; optimal control; protein translation; ribophagy; ribosome; systems biology.

Figure 1.

Mathematical model of protein production dynamics with resource and ribosomal population. A fraction V of ribosomes (R) is allocated to produce other ribosomes (ribosome biogenesis), while the remaining fraction 1 − V is responsible for the remaining protein synthesis (P). Biosynthesis rates depend on the amount of cellular resources (E) injected in the system (resource intake). Through the control U, the ribophagy process contributes to the pool E by recycling resources.

Figure 2.

Phase portrait, time trajectories and convergence speed of the system, with resources E, ribosome population R and protein production P. Nullclines are represented in dotted black and blue lines (see also appendix A), with some trajectories in black, green and blue in the non-oscillatory case (a), and in the oscillatory case (b). (c) The asymptotic convergence speed of the system $\Psi$ as a function of the ribophagy rate U and the ribogenesis rate V, for nutrient intake α = 2 × 10⁸ h⁻¹ μm⁻³. The dotted box represents an example of the optimization domain. (d) Vertical and horizontal slices from (c).

Figure 3.

Example of an optimal control solution obtained from solving equation (3.8) (proposition 3.2) with maximal constant protein production and maximal convergence rate (to enforce the quasi-static approximation, see §3.4). Plotted variables are the nutrient intake α, ribosome concentration R and a resource concentration E, with the system producing proteins at a rate ρ under the ribophagy U and ribogenesis V controls. Optimization was performed with the Newton method of the scipy.optimize.minimize python function. We used parameters from table 1, as described in appendix E. Simulations were performed using the solve_ivp function of the python scipy package [26].

Figure 4.

Representation of regions of admissible parameters in two different settings, as a function of the minimum resource intake ${\alpha }_{min}$. The grey, red and black lines represent different inequalities as a boundary of the parameter domain. The other coloured curves represent, for different values of ${\alpha }_{min}$ displayed on the colour bar, the value of controls such that the protein production stays constant during low intake period (starvation). In (a), the ribogenesis control value range $(V_{min},V_{max})$ is fixed and the value range of ribophagy $(U_{min},U_{max})$ is studied, whereas in (b), the protein production is fixed (ρ₀), and the values of the controls $(U_{max},V_{min})$ are studied at low intake period (starvation). The maximal resource intake ${\alpha }_{max}$ was set to 2.5 × 10⁻⁸ h⁻¹ μm⁻³.

Figure 5.

Vector field associated with the dynamical system. The R- and E-nullclines (equations (A 5) and (A 6)) are represented in blue and black dotted lines, respectively. They partition ${\mathbb{R}}_{>0}^2$ into four regions ${\mathcal{Q}}_i$, 1 ≤ i ≤ 4 (equation (A 8)), and we also represent the section S, introduced in equation (A 9), to define the Poincaré map and prove the asymptotic convergence of trajectories starting from S.