A systems theoretic approach to analysis and control of mammalian circadian dynamics

Human Alertness Optimization with a Three-Process Dynamic Model

Circadian rhythm is an important biological process for humans as it modulates a wide range of physiological processes, including body temperature, sleep-wake cycle, and cognitive performance. As the most powerful external stimulus of circadian rhythm, light has been studied as a zeitgeber to regulate the circadian phase and sleep. This paper addresses the human alertness optimization problem, by optimizing light exposure and sleep schedules to relieve fatigue and cognitive impairment, in cases of night-shift workers and subjects with certain mission periods based on dynamics of the circadian rhythm system. A three-process hybrid dynamic model is used for simulating the circadian rhythm and predicting subjective alertness and sleepiness. Based on interindividual difference in sleep type and living habits, we propose a tunable sleep schedule in the alertness optimization problem, which allows the appropriate tuning of sleep and wake times based on sleep propensity. Variational calculus is applied to evaluate the impacts of light and sleep schedules on the alertness and a gradient descent algorithm is proposed to determine the optimal solutions to maximize the alertness level in various cases. Numerical simulation results demonstrate that the cognitive performance during certain periods can be significantly improved by optimizing the light input and tuning sleep/wake times compared to empirical data.

A dual-feedback loop model of the mammalian circadian clock for multi-input control of circadian phase

The molecular circadian clock is driven by interlocked transcriptional-translational feedback loops, producing oscillations in the expressions of genes and proteins to coordinate the timing of biological processes throughout the body. Modeling this system gives insight into the underlying processes driving oscillations in an activator-repressor architecture and allows us to make predictions about how to manipulate these oscillations. The knockdown or upregulation of different cellular components using small molecules can disrupt these rhythms, causing a phase shift, and we aim to determine the dosing of such molecules with a model-based control strategy. Mathematical models allow us to predict the phase response of the circadian clock to these interventions and time them appropriately but only if the model has enough physiological detail to describe these responses while maintaining enough simplicity for online optimization. We build a control-relevant, physiologically-based model of the two main feedback loops of the mammalian molecular clock, which provides sufficient detail to consider multi-input control. Our model captures experimentally observed peak to trough ratios, relative abundances, and phase differences in the model species, and we independently validate this model by showing that the in silico model reproduces much of the behavior that is observed in vitro under genetic knockout conditions. Because our model produces valid phase responses, it can be used in a model predictive control algorithm to determine inputs to shift phase. Our model allows us to consider multi-input control through small molecules that act on both feedback loops, and we find that changes to the parameters of the negative feedback loop are much stronger inputs for shifting phase. The strongest inputs predicted by this model provide targets for new experimental small molecules and suggest that the function of the positive feedback loop is to stabilize the oscillations while linking the circadian system to other clock-controlled processes.

Reconciling periodic rhythms of large-scale biological networks by optimal control

Periodic rhythms are ubiquitous phenomena that illuminate the underlying mechanism of cyclic activities in biological systems, which can be represented by cyclic attractors of the related biological network. Disorders of periodic rhythms are detrimental to the natural behaviours of living organisms. Previous studies have shown that the state transition from one to another attractor can be accomplished by regulating external signals. However, most of these studies until now have mainly focused on point attractors while ignoring cyclic ones. The aim of this study is to investigate an approach for reconciling abnormal periodic rhythms, such as diminished circadian amplitude and phase delay, to the regular rhythms of complex biological networks. For this purpose, we formulate and solve a mixed-integer nonlinear dynamic optimization problem simultaneously to identify regulation variables and to determine optimal control strategies for state transition and adjustment of periodic rhythms. Numerical experiments are implemented in three examples including a chaotic system, a mammalian circadian rhythm system and a gastric cancer gene regulatory network. The results show that regulating a small number of biochemical molecules in the network is sufficient to successfully drive the system to the target cyclic attractor by implementing an optimal control strategy.

Time optimal entrainment control for circadian rhythm

The circadian rhythm functions as a master clock that regulates many physiological processes in humans including sleep, metabolism, hormone secretion, and neurobehavioral processes. Disruption of the circadian rhythm is known to have negative impacts on health. Light is the strongest circadian stimulus that can be used to regulate the circadian phase. In this paper, we consider the mathematical problem of time-optimal circadian (re)entrainment, i.e., computing the lighting schedule to drive a misaligned circadian phase to a reference circadian phase as quickly as possible. We represent the dynamics of the circadian rhythm using the Jewett-Forger-Kronauer (JFK) model, which is a third-order nonlinear differential equation. The time-optimal circadian entrainment problem has been previously solved in settings that involve either a reduced-order JFK model or open-loop optimal solutions. In this paper, we present (1) a general solution for the time-optimal control problem of fastest entrainment that can be applied to the full order JFK model (2) an evaluation of the impacts of model reduction on the solutions of the time-optimal control problem, and (3) optimal feedback control laws for fastest entrainment for the full order Kronauer model and evaluate their robustness under some modeling errors.

Compensating for Sensor Error in the Model Predictive Control of Circadian Clock Phase

The circadian oscillator regulates many critical biological functions; misalignment between the phase of this oscillator and the environment has been linked to adverse health outcomes. Thus, shifting the circadian phase of the oscillator to align with the environment using either light or small molecule pharmaceuticals as control inputs is desired. One challenge to controlling circadian phase is that the magnitude and direction of the phase shift caused by these inputs is dependent on the phase at which the input is delivered. Simulations show that model predictive control (MPC) can successfully shift the phase of the circadian clock using perfect knowledge of the current phase of the system. However, methods to assess circadian phase continuously in real time, as would be needed to implement MPC in vivo, are limited in their accuracy. Here, we explore the impact of imperfect sensing on our ability to control circadian phase. While some pathological patterns of sensor error can make control impossible, we show that by assuming errors in the phase sensor are bounded to be sufficiently small, we can bound the error of our MPC algorithm. We propose using the expected phase response curve to improve control when sensor error is present.

Mathematical modeling of circadian rhythms

Circadian rhythms are endogenous ~24-hr oscillations usually entrained to daily environmental cycles of light/dark. Many biological processes and physiological functions including mammalian body temperature, the cell cycle, sleep/wake cycles, neurobehavioral performance, and a wide range of diseases including metabolic, cardiovascular, and psychiatric disorders are impacted by these rhythms. Circadian clocks are present within individual cells and at tissue and organismal levels as emergent properties from the interaction of cellular oscillators. Mathematical models of circadian rhythms have been proposed to provide a better understanding of and to predict aspects of this complex physiological system. These models can be used to: (a) manipulate the system in silico with specificity that cannot be easily achieved using in vivo and in vitro experimental methods and at lower cost, (b) resolve apparently contradictory empirical results, (c) generate hypotheses, (d) design new experiments, and (e) to design interventions for altering circadian rhythms. Mathematical models differ in structure, the underlying assumptions, the number of parameters and variables, and constraints on variables. Models representing circadian rhythms at different physiologic scales and in different species are reviewed to promote understanding of these models and facilitate their use. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Pharmaceutical-based entrainment of circadian phase via nonlinear model predictive control

The widespread adoption of closed-loop control in systems biology has resulted from improvements in sensors, computing, actuation, and the discovery of alternative sites of targeted drug delivery. Most control algorithms for circadian phase resetting exploit light inputs. However, recently identified small-molecule pharmaceuticals offer advantages in terms of invasiveness and potency of actuation. Herein, we develop a systematic method to control the phase of biological oscillations motivated by the recently identified small molecule circadian pharmaceutical KL001. The model-based control architecture exploits an infinitesimal parametric phase response curve (ipPRC) that is used to predict the effect of control inputs on future phase trajectories of the oscillator. The continuous time optimal control policy is first derived for phase resetting, based on the ipPRC and Pontryagin's maximum principle. Owing to practical challenges in implementing a continuous time optimal control policy, we investigate the effect of implementing the continuous time policy in a sampled time format. Specifically, we provide bounds on the errors incurred by the physiologically tractable sampled time control law. We use these results to select directions of resetting (i.e. phase advance or delay), sampling intervals, and prediction horizons for a nonlinear model predictive control (MPC) algorithm for phase resetting. The potential of this ipPRC-informed pharmaceutical nonlinear MPC is then demonstrated in silico using real-world scenarios of jet lag or rotating shift work.

Nonlinear Model Predictive Control For Circadian Entrainment Using Small-Molecule Pharmaceuticals

Recent in vitro studies have identified small-molecule pharmaceuticals effecting dose-dependent changes in the mammalian circadian clock, providing a novel avenue for control. Most studies employ light for clock control, however, pharmaceuticals are advantageous for clock manipulation through reduced invasiveness. In this paper, we employ a mechanistic model to predict the phase dynamics of the mammalian circadian oscillator under the effect of the pharmaceutical under investigation. These predictions are used to inform a constrained model predictive controller (MPC) to compute appropriate dosing for clock re-entrainment. Constraints in the formulation of the MPC problem arise from variation in the phase response curves (PRCs) describing drug effects, and are in many cases non-intuitive owing to the nonlinearity of oscillator phase response effects. We demonstrate through in-silico experiments that it is imperative to tune the MPC parameters based on the drug-specific PRC for optimal phase manipulation.

Computational modeling of the cell-autonomous mammalian circadian oscillator

This review summarizes various mathematical models of cell-autonomous mammalian circadian clock. We present the basics necessary for understanding of the cell-autonomous mammalian circadian oscillator, modern experimental data essential for its reconstruction and some special problems related to the validation of mathematical circadian oscillator models. This work compares existing mathematical models of circadian oscillator and the results of the computational studies of the oscillating systems. Finally, we discuss applications of the mathematical models of mammalian circadian oscillator for solving specific problems in circadian rhythm biology.