Generalized Inverse Optimal Control and its Application in Biology

TL;DR

This paper addresses the challenge of discerning the optimization principles that govern biological dynamics by proposing generalized inverse optimal control (gIOC), a bi-level framework that simultaneously infers objective functions, constraints, and dynamics from time-series data. By accommodating multi-criteria objectives, nested scales, active constraints, switches in optimality principles, and robustness considerations, gIOC extends traditional IOC/IRL to a data-driven regime suitable for molecular and cellular systems. The authors illustrate the approach with forward-control examples (e.g., brachistochrone and metabolic pathways) and discuss practical considerations, including model identification, interpretability, identifiability, persistent excitation, and algorithmic strategies for solving inner and outer problems. They argue that, despite current computational and data challenges, gIOC offers a principled path to uncovering the objectives naturally shaping biology and to enabling predictive forward-control applications in bioengineering and medicine, provided interdisciplinary collaboration drives theory and numerical tooling forward.

Abstract

Living organisms exhibit remarkable adaptations across all scales, from molecules to ecosystems. We believe that many of these adaptations correspond to optimal solutions driven by evolution, training, and underlying physical and chemical laws and constraints. While some argue against such optimality principles due to their potential ambiguity, we propose generalized inverse optimal control to infer them directly from data. This novel approach incorporates multi-criteria optimality, nestedness of objective functions on different scales, the presence of active constraints, the possibility of switches of optimality principles during the observed time horizon, maximization of robustness, and minimization of time as important special cases, as well as uncertainties involved with the mathematical modeling of biological systems. This data-driven approach ensures that optimality principles are not merely theoretical constructs but are firmly rooted in experimental observations. Furthermore, the inferred principles can be used in forward optimal control to predict and manipulate biological systems, with possible applications in bio-medicine, biotechnology, and agriculture. As discussed and illustrated, the well-posed problem formulation and the inference are challenging and require a substantial interdisciplinary effort in the development of theory and robust numerical methods.

Figures (9)

• Figure 1: Examples of optimality principles observable in real life. \ref{['fig:subfigB']}: Already the Roman scholar Varro hypothesized that bees use a regular hexagonal lattice because it is the most efficient way to store honey while using the least wax. In 1999, Thomas Hales gave a mathematical proof for this. \ref{['fig:subfigC']}: Optimality principles have also been investigated for the movement of animals and humans. Shown is the example of undulatory movements of snakes, which were conjectured to minimize drag forces Astley2015.
• Figure 2: Optimal control in a simple three-step linear metabolic pathway, where substrate $x_1$ has a buffered concentration and is converted into product $x_4$ in three steps catalyzed by three enzymes $u_i$. The upper plots show the optimal controls (concentrations $u_i(t)$) that minimize transition time (i.e., time to convert a certain amount of product). Due to a path constraint on the total amount of enzyme at any given time, the optimal enzyme profiles show a pulse-like behavior. This pattern agrees with experimental results of temporal gene expression Klipp2002-sgZaslaver2004-rd. Details of the formulation are given in Tsiantis2020-nf, based on previous works de2014globalBartl2010-vtKlipp2002-sg. Variants of this problem are discussed in Section \ref{['sec_metabolic']}.
• Figure 3: Optimality principles in biology and their embedding into environmental, genetic, and physical constraints. Constraints (physical, genetic and environmental) and nested multi-criteria trade-offs shape the outcomes of evolutionary processes. Organisms are limited by physical laws, their genetic makeup (inherited traits), and the environment they live in (resources, predators, etc.). These constraints influence the "optimal" solution that can evolve. But such optimality always involves trade-offs. Balancing trade-offs is essential for the overall fitness and functionality of organisms.
• Figure 4: Optimal control using a minimalistic car model in 1D. The car is at rest ($v=0$) at $t=0$ at position $x=A$. How would you accelerate/brake the car to get from $A$ to $B$ in minimum time and stop at $B$? The control $a(t)$ is bounded by $-1 \le a(t) \le 1$. The optimal control is of bang-bang type, i.e., the maximum acceleration until the midpoint is followed by maximal braking.
• Figure 5: Compared to Figure \ref{['fig_carSimple']}, a more detailed model of a racing car (Porsche CS) on the Hockenheim ring as studied in Kehrle2011 is considered. Shown are the topology of the race track and optimal control functions calculated in Kehrle2011 that involve \ref{['fig:carAdvancedA']} gear choices and \ref{['fig:carAdvancedB']} steering angle velocity, braking pedal position, and acceleration pedal position. One observes that the optimality principle (here: minimum time driving) encapsulates the regulation of the system (here: the control actions of the driver) and that path constraints have an impact on the optimal controls in comparison to the simplified setting in Figure \ref{['fig_carSimple']}.