Autophagy dynamics at the level of a single cell This research is a collaboration between Los Alamos Laboratory (Bill Hlavacek, Yen-Ting Song, Song Chen) and UNM (Francesco Sorrentino, Afroza Shirin, Isaac Klickstein). Autophagy is a cellular housekeeping and digestive process where misfolded or aggregated proteins, damaged organelles, and intracellular pathogens are identified by the cell, trapped in double-membrane vesicles, called autophagosome vesicles (AVs), and then delivered to the lysosome in the cell for degradation. Autophagy is also induced in conditions of nutrient starvation, during which it provides cells with additional nutrient supplies. The motivating consideration for mechanistically investigating how to optimize the level of autophagy is to better understand cell survival and death. Autophagy, as a mechanism for cell survival, can promote the growth of established tumors and increase their resistance to treatments [1, 2, 3, 4]. Therefore, efforts to inhibit autophagy to improve cancer therapy are a critical next step. We attempt to control the number of AVs in a cell by supplying different types of pharmacological reagents (PRs). With the help of our collaborators at Los Alamos, we have developed a simplified 5-dimensional dynamical model, which is schematically represented in the figure and is described by the following set of ODEs: \dot{x}_1(t) = (1-x_1)g_1 (1 - u_1) (1 - u_2)- x_1 h(x_2) h(x_3) x_1: $[mTORC]$ \dot{x}_2(t) = (1-x_2) h(x_3) (1 - u_3) -x_2 h(x_1) x_2: $[ULK1]$ \dot{x}_3(t) = (1-x_3)k_1 (1 - u_4) - g_2 x_3 x_2 (1 - u_5) x_3: $[AMPK]$      (1) \dot{x}_4(t) = (1-x_4)k_2 h(x_2) - k_3 x_4 x_4: $[Vps34]$ \dot{x}_5(t) = k_4 x_4 - k_5 x_5 (1- u_6) x_5: $\%23 Autophagic Vesicle/10$ subject to the following (possible) control inputs/PRs: u_1: Rapamycin u_2: Taselisib; u_3: Sbi0206965; u_4: Dorsomorphin; u_5: A769662; u_6: Bafa1. In Eq. (1) the h(x_i) = \frac{6 x_i^{n_i}}{x_i^{n_i}+\vartheta_i^{n_i}} are scaled Hill functions, where n_i is an even integer and \vartheta_i is a constant for each x_i, k_i, i = 1,\cdots,5 are the rate constants; g_1 and g_2 are the amount of nutrients and energy available in a cell. Nutrients are building blocks of proteins, amino acids, and nucleotides. Energy is provided by Adenosine triphosphate (ATP) molecules. When these two constants are changed we see qualitative changes of the dynamics [5]. This model is derived from a higher-dimensional model for autophagy regulation [5, 6] that will be used to validate the control solution derived for the low dimensional model. The figure is a network representation for the system of equations (1), where each node corresponds to one of the states of the system and a directed edge connects node a to node b if a appears in the equation for the time evolution of b. The goal is to affect the number of AVs (target) by injecting selected control inputs/PRs into certain network nodes. Recent studies (see e.g. [7, 8, 9, 10]) have identified several proposed strategies for autophagy inhibition. These involve different PRs. Among these: (1) {Rapamycin} (an mTORC kinase inhibitor) prevents cell growth; (2) Bafa1 (a VPS34-complex inhibitor) impairs phagophore (the isolation membrane that delimits the part of cytoplasm that the cell eats) nucleation and (3) {Dorsomorphin} (an AMPK inhibitor) blocks cellular energy homeostasis. These three PRs are shown in blue in the figure, affecting three different nodes, i.e., `a cocktail of drugs'. Note that Eq. 1 is more general and considers a total of 5 possible PRs, which provides the possibility for many more PR combinations. This research incorporates a novel approach as it includes a comparison of several strategies in terms of controllability and control energy (amounts of drug needed and its dosage over time). The timescale for autophagy dynamics is on the order of hours [6]. Here we assume to be able to affect the system with control inputs (PRs) that evolve on a timescale comparable with that of the system's dynamics. The control strategy is open loop, as real-time information on the time-evolution of the activity of a cell is typically unavailable. We have computed optimal control trajectories that provide optimal dosage functions for several PRs. The results of some of our computations are shown below for Rapamycin, ... In our simulations, we set the initial condition for each state to be equal to the steady state solution for the system of Eqs. (1) in free evolution (i.e., when u_i(t)=0, i=1,...,6) and set the amount of nutrition and energy g_1 = g_2 = 0.5. The goal of the control action is to upregulate/downregulate the final number of autophagy vesicles at the end of a fixed time period x_5(t_f), while minimizing the control effort, i.e., the overall quantity of each drug that is provided during the time period. The motivation for this choice of the objective function is to use the least quantity of a drug and to reduce unwanted off-target effects. The chosen control timescale $t_f$ is consistent with experimental observations for the time that it takes for autophagosomes to form in a cell [6]. In each of the figures below, panel (c) shows a curve of the minimum control energy versus the target quantity of AVs. These curves show how changing the final requirements in terms of target AVs affects the minimum required expenditure of each individual drug. Figure: Rapamycin Figure: Taselisib Figure: BafA1 References [1]  Zhineng J Yang, Cheng E Chee, Shengbing Huang, and Frank A Sinicrope. 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