(573b) Circadian Phase Entrainment Via Nonlinear Model Predictive Control

Oscillatory processes are omnipresent in nature and govern many organisms' behavior. Recent biological studies suggest that phase-related phenomena such as entrainment may be of greater importance than a system's endogenous period. In this study, a real-time method to optimize phase-resetting properties of robust nonlinear biological oscillators is developed. This nonlinear model predictive control algorithm is applied to a circadian system, resetting phase in under 3-days while minimizing transient effects (Fig. 1).

Figure 1: Recovering a 12 hour phase difference. Through application of a sequence of various intensity light pulses, we are able to recover a 12 hour initial phase difference in 63 hours. The nonlinear model predictive control algorithm is tested under four different conditions, using a single move, 2 moves, 4 moves, and 6 moves. Results show that increasing the number of control moves does not necessarily decrease the recovery time, though it does minimize transient effects.

A well-studied example of a biological oscillator is the circadian clock. The term circa- (about) diem (a day) describes a biological event that repeats every 24-hours. Such rhythms are possessed by most organisms, acting as a biological clock. They are observed at all cellular levels since oscillations in enzymes and hormones affect cell function, cell division, and cell growth [1]. Circadian rhythms serve to impose internal alignments between different biochemical and physiological oscillations. Their ability to anticipate environmental changes enables organisms to organize their physiology and behavior such that they occur at biologically advantageous times during the day [1]: visual and mental acuity fluctuate, for instance, affecting complex behaviors. An inability to entrain circadian phase to the environment or anticipate change causes many functional disorders.

Circadian disorders include non-24-hour sleep-wake syndrome (often due to blindness), rapid time-zone change syndrome (jet lag), work-shift syndrome (impaired sleep and alertness due to unusual work times), advanced or delayed phase sleep syndrome, and irregular sleep-wake pattern syndrome [2]. Such disorders are often caused by circadian oscillators that are out of phase with the environment, and thereby hinder one's performance. Many researchers have studied the clock in an attempt to both understand and resolve existing phase discrepancies. Daan and Pittendrigh, for instance, discussed light-induced phase shifts as a function of circadian time and the role phase response curves play in achieving entrainment [3]. Watanabe et al. confirmed Daan and Pittendrigh's work through experimental procedures proving that the basis for phase adjustment involves rapid resetting of both advance and delay components of the phase response curve [4]. Boulos et al. performed similar experiments establishing bright light treatment as a means to accelerate circadian re-entrainment following transmeridian travel [5]. Despite the decades of work put forth in understanding circadian phase and entrainment properties, the idea of optimally controlling phase via a closed-loop control algorithm is a recent area of interest.

In this study, the Drosophila melanogaster (fruit fly) 10-state mathematical model [6] demonstrates the utility of nonlinear model predictive control specific to nonlinear oscillators as it must track an oscillatory reference. This moderately complex system consists of two coupled negative feedback loops that model the transcription, translation, phosphorylation, and effective delays associated with period and timeless genes, and their protein counterparts (Fig. 2). Experimental data proves that a change in light pattern controls phase-resetting properties of the Drosophila melanogaster circadian clock [7]. Admitting light pulses under free-running (dark:dark) conditions resets the oscillator by inducing a phase delay or advance [8]. This light input to phase shift relationship illustrates time-dependence and asymmetry as the maximum phase advance of 3.2-hours occurs when light is flashed in the early subjective morning, while the maximum phase delay of 4.6-hours is due to a light pulse in the late subjective evening.

Figure 2: The 10-state circadian model (adapted from [6]). Negative regulation of per and tim gene expression occurs via the nuclear PER/TIM complex. per and tim genes are transcribed in the nucleus, after which their mRNAs are transported into the cytosol where they undergo protein synthesis. The newly formed PER and TIM proteins are phosphorylated, yielding P₁/T₁ and P₂/T₂ protein elements. The doubly phosphorylated proteins (P₂/T₂) form a PER/TIM complex, C, that enters the nucleus and closes the feedback loop by suppressing gene expression. Bright light doubles the TIM-PP degradation rate and serves as a control input for phase resetting.

Mott et al. uses model-based predictive control to find the optimal set of light pulses necessary to maintain and shift the biological clock within a constrained environment (i.e. maintaining an astronaut's rhythm in space) [9]. Their methods are applied to a modified Van der Pol oscillator with a natural period just over 24 hours. The Van der Pol system is transformed into a linear model through use of both a nonlinear state feedback compensation block and a nominal linear approximation. Both the Mott et al. work and our previous studies show that optimal control inputs for a system in a continuously dark (free-running) environment may be determined a number of ways. In one case, we calculated model-specific data (a priori) and used it in combination with a cost function to determine the next control move, simulating an iterative closed-loop look-up table problem [10]. Unfortunately, the algorithm introduces error as it relies solely on the predictive model and may not be applicable to systems in natural light:dark environments. In this study, there are no such approximations. We use a genetic evolutionary algorithm that serves as our global optimizer, choosing a set of control moves within a specified move horizon (Fig. 3). Although Mott et al. equate the move horizon with the prediction horizon, we require that the prediction horizon be longer than the move horizon to minimize transient-effects in the cost function. Each move is implemented in the real-time model for a 3-hour time interval, simulating a 15-minute light pulse [6].

Figure 3: Nonlinear model predictive control of oscillatory systems. This Model predictive control algorithm makes use of move horizons that range between a single control move and 6 sequential control moves. The optimizer determines the optimal control inputs by minimizing the difference between the system and reference output trajectories over a prediction horizon of 2 days.

While our control algorithm is designed to be general, its application to the nonlinear circadian network is key in resetting the clock's phase, optimizing performance and alertness, and minimizing the effects of circadian disorders. Preliminary data suggests that nonlinear model predictive control is effective at providing optimal control inputs (light pulses) that reset phase differences with minimal a priori information (Fig. 1). Furthermore, the algorithm minimizes transient effects and may be applied to systems in both constant darkness and natural light:dark environments, providing greater efficacy in its experimental use. Our control algorithm provides certain degrees of freedom - move horizon, prediction horizon, control boundaries, time steps, optimizer, cost function - that may be customized per application. Although we use the Drosophila melanogaster model to prove optimal phase resetting, our algorithm allows researchers to use any oscillatory model without introducing additional errors due to linearizing/approximating, providing a more accurate and robust application of nonlinear model predictive control.

References

[1]

Edery, I. 2000. Circadian rhythms in a nutshell. Physiol. Genomics 3:59-74.

[2]

Moore, R. Y. 1997. Circadian rhythms: Basic neurobiology and clinical applications. Annu. Rev. Med. 48:253-266.

[3]

Pittendrigh, C. S. and S. Daan. 1976. A functional analysis of circadian pacemakers in nocturnal rodents. J. comp. Physiol. 106:223-252.

[4]

Watanabe, K., T. Deboer, and J. H. Meijer. 2001. Light-induced resetting of the circadian pacemaker: Quantitative analysis of transient versus steady-state phase shifts. J Biol Rhythms 16:564-573.

[5]

Boulos, Z., M. Macchi, M. Stürchler, K. Stewart, G. Brainard, A. Suhner, G. Wallace, and R. Steffen. 2002. Light visor treatment for jet lag after westward travel across six time zones. Aviat. Space Environ. Med. 73:953-963.

[6]

Leloup, J.-C. and A. Goldbeter. 1998. A model for circadian rhythms in Drosophila incorporating the formation of a complex between the PER and TIM proteins. J. Biol. Rhythms 13:70-87.

[7]

Myers, M. P., K. Wager-Smith, A. Rothenfluh-Hilfiker, and M. W. Young. 1996. Light-induced degradation of TIMELESS and entrainment of the Drosophila circadian clock. Science 271:1736-1740.

[8]

Winfree, A. 2001. The Geometry of Biological Time. Springer, New York, NY, USA.

[9]

Mott, C., D. Mollicone, M. van Wollen, and M. Huzmezan. 2003. Modifying the human circadian pacemaker using model based predictive control. In Proc. Amer. Control Conf. 453-458.

[10]

Bagheri, N., J. Stelling, and F. J. Doyle III. 2005. Optimal phase-tracking of the nonlinear circadian oscillator. In American Control Conference. Portland, OR.