Reentrainment of the circadian pacemaker during jet lag: East-west asymmetry and the effects of north-south travel

Highlights

• We use a mathematical model of the human circadian clock to study jet lag.

• Jet lag severity depends on the traveler’s internal clock and the season of the year.

• Our analysis explains when eastward travel is worse than westward and vice versa.

• We predict north-south travel can cause jet lag even when no time zones are crossed.

• Methods can be used by travelers visiting multiple destinations to minimize jet lag.

Abstract

The normal alignment of circadian rhythms with the 24-h light-dark cycle is disrupted after rapid travel between home and destination time zones, leading to sleep problems, indigestion, and other symptoms collectively known as jet lag. Using mathematical and computational analysis, we study the process of reentrainment to the light-dark cycle of the destination time zone in a model of the human circadian pacemaker. We calculate the reentrainment time for travel between any two points on the globe at any time of the day and year. We construct one-dimensional entrainment maps to explain several properties of jet lag, such as why most people experience worse jet lag after traveling east than west. We show that this east-west asymmetry depends on the endogenous period of the traveler’s circadian clock as well as daylength. Thus the critical factor is not simply whether the endogenous period is greater than or less than 24 h as is commonly assumed. We show that the unstable fixed point of an entrainment map determines whether a traveler reentrains through phase advances or phase delays, providing an understanding of the threshold that separates orthodromic and antidromic modes of reentrainment. Contrary to the conventional wisdom that jet lag only occurs after east-west travel across multiple time zones, we predict that the change in daylength encountered during north-south travel can cause jet lag even when no time zones are crossed. Our techniques could be used to provide advice to travelers on how to minimize jet lag on trips involving multiple destinations and a combination of transmeridian and translatitudinal travel.

Introduction

Circadian clocks have evolved to align biological functions with the 24-h environmental cycles conferred by the rotation of the earth (Johnson et al., 2003). In humans, a central circadian pacemaker coordinates various physiological rhythms so that they peak at the appropriate time of the day, such as the release of the sleep-promoting hormone melatonin in the evening and the wake-promoting hormone cortisol in the morning (James et al., 2007). The endogenous period of the human circadian oscillator in the absence of external time cues is not exactly 24 h (Czeisler et al., 1999). The period of the oscillator becomes 24 h under normal circumstances when exposed to natural environmental cycles, and a stable phase relationship between the oscillator and its environment is established: the oscillator is phase-locked or entrained to the external cycles (Wright et al., 2013). For circadian oscillators, the strongest entraining signal is the daily light-dark (LD) cycle (Duffy and Wright, 2005). If entrainment is disrupted by a sudden shift in the phase of the LD cycle, for example due to rapid travel across time zones, then the phase of the circadian oscillator undergoes adjustments until phase-locking is reestablished and the oscillator is reentrained (Aschoff et al., 1975).

Jet lag is a collection of symptoms experienced after rapid transmeridian travel. These symptoms—such as insomnia, excessive daytime sleepiness, gastrointestinal disturbances, and general malaise—are not simply due to travel fatigue following a long flight, but rather are caused by misalignment of the traveler’s internal circadian clock with the environmental cycles in the new time zone (Sack, 2009). Each year about 30 million US residents fly to overseas destinations (U.S. Citizen Travel to International Regions, 2017). For international business travelers, athletes, or government and military personnel, jet lag can impair judgment, hinder performance, or threaten public safety (Eastman and Burgess, 2009). Most travelers experience more severe jet lag after flying east than after flying west (Waterhouse et al., 2007), and a recent analysis of over 20 years of data from Major League Baseball games found that jet lag impairs performance moreso after eastward than westward travel (Song et al., 2017). The conventional explanation for this directional asymmetry in jet lag severity is that since the human circadian clock typically has an endogenous period of greater than 24 h, it is easier to phase delay the clock in response to the phase delay of the LD cycle caused by westward travel than it is to phase advance the clock in response to the phase advance of the LD cycle caused by eastward (Eastman and Burgess, 2009). Reentraiment though phase adjustment in the same direction as the shift of the LD cycle is referred to as orthodromic. After long trips, some travelers reentrain antidromically or through phase adjustments in the opposite direction of the phase shift of the LD cycle, i.e. phase delays after traveling east and phase advances after traveling west (Arendt, Aldous, English, Marks, Arendt, Folkard, 1987, Klein, Wegmann, 1977, Takahashi, Sasaki, Itoh, Yamadera, Ozone, Obuchi, et al., 2001).

In this paper we use a mathematical model of the human circadian pacemaker, the Forger–Jewett–Kronauer (FJK) model (Forger et al., 1999), to explain the existence of east-west asymmetry in jet lag severity and the antidromic mode of reentrainment. The FJK model is a widely accepted model in the circadian literature that captures both phase and amplitude dynamics of daily core body temperature oscillations. It has been fit to experimental data on how light affects human circadian rhythms and has been used in several studies to design schedules that minimize jet lag (Dean, Forger, Klerman, 2009, Serkh, Forger, 2014, Zhang, Qiao, Wen, Julius, 2016). Consistent with a recent study employing a phase-only model (Lu et al., 2016), we find that the endogenous period of the circadian oscillator does influence east-west asymmetry. Differently than (Lu et al., 2016), however, we find that the period being greater than or less than 24 h is not the critical factor. Furthermore we show that daylength, and therefore the season of the year, affects whether eastward or westward travel is worse.

The medical definition of jet lag requires travel across time zones, implying that strictly north-south or translatitudinal travel within the same time zone cannot cause jet lag. We take a broader view of jet lag as symptoms resulting from any travel-induced misalignment of the circadian clock and the external LD cycle, and argue that the change in daylength experienced when traveling across latitudes (for example between the northern and southern hemispheres) in the summer or winter may disrupt entrainment. The question of whether purely north-south travel can result in significant misalignment has received very little attention in the literature. We find that in the FJK model, a difference in the daylength between departure and destination cities is enough to cause jet lag on the order of several days (depending on parameters) even with no change in time zone. Combining our findings on east-west travel with those on north-south travel, we also investigate travel that incorporates both of these directions. By considering a hypothetical case study involving travel between four cities located in North America, South America, Asia, and Australia, we show that the north-south component of travel can significantly add to or reduce reentrainment times even in cases where strict north-south travel itself incurs no jet lag.

The main tool we use to gain insights into the properties of jet lag is the entrainment map, a technique we recently introduced for calculating the LD-entrained solution of an oscillator subjected to external periodic forcing consisting of N hours of light and $24-N$ h of darkness (Diekman and Bose, 2016). The method involved deriving a one-dimensional map, Π(x), whose fixed points corresponded to stable or unstable entrained periodic solutions. We showed that the entrainment map yields more accurate predictions about the phase of the stable entrained solution than methods based on phase response curves. In Diekman and Bose (2016), we showed how the entrainment map for the two-dimensional Novak-Tyson model of the Drosophila molecular clock (Tyson et al., 1999) depends on parameters of the model and how it can be used to determine regimes over which solutions entrain through phase advance or phase delay. The entrainment map was then applied to higher dimensional systems such as the three-dimensional Gonze et al. (2005) and the 180-dimensional Kim and Forger (2012) models of the mammalian molecular clock.

Here we build entrainment maps for the FJK model to explore various facets of reentrainment after travel. Travel can involve a change of time zone, such as eastward or westward travel, a change in photoperiod, such as northward or southward travel, or a combination of both, such as travel from North America to Australia. We show that reentrainment properties depend both quantitatively and qualitatively on key parameters including the endogenous period of the oscillator, the daylength, and the intensity of light. Using our methods, we can calculate reentrainment times for travel between any two locations on the globe, at any time of the year, and for any departure or arrival time. In doing so, we are able to explain that the east-west asymmetry of jet lag is a generic feature of the FJK model that is highly dependent on both the endogenous period of the traveler as well as the daylength. Using a generalization of the concept of neutral period introduced by Aschoff et al. (1975), we show that for different combinations of these two parameters, travel to the east can incur more jet lag than travel to the west or vice versa. In fact, because of seasonal changes in the daylength, for the same traveler a journey in one direction may be harder in the winter, while a journey in the opposite direction may be harder in the summer. Our findings are related, in part, to those of Herzel and collaborators (Bordyugov, Abraham, Granada, Rose, Imkeller, Kramer, et al., 2015, Granada, Herzel, 2009, Schmal, Myung, Herzel H. Bordyugov, 2015) who have characterized the phase of entrainment as a function of endogenous period, zeitgeber (external stimulus) strength, and photoperiod for several different circadian models using Arnold tongues and Arnold onions. The analysis of the entrainment map also provides insight into the different modes of reentrainment. Prior work using a model of the mammalian molecular clock identified a threshold separating orthodromic and antidromic modes of reentrainment, but did not explain what mathematical object might act as the threshold (Leloup and Goldbeter, 2013). Here we show that the unstable fixed point of the entrainment map can be used to predict the threshold that separates the two modes of reentrainment.

Contrary to what one might naively expect, we find that reentrainment time is relatively independent of departure or arrival time, and that the longest trips do not necessarily give rise to the longest reentrainment times. Instead, the worst-case trip is determined by the ordering and magnitude of the distance between the stable and unstable fixed points of the entrainment map, which themselves are dependent on the internal body clock and daylength. We find that for low light intensities, trips that place the traveler in a neighborhood of the unstable fixed point of the map will give rise to the longest reentrainment times. For higher light intensities, the longest reentrainment times still occur in a neighborhood of the unstable fixed point, but there is also the potential for dramatically short reentrainment times for certain trips within this neighborhood. These dramatically short reentrainment times are associated with amplitude suppression and a phase singularity, and have been observed previously in the FJK model at high light intensity (Serkh and Forger, 2014).

In this study, we consider the light level to be fixed at either low or high intensity (lux) across the entire photoperiod. Admittedly, this is not a light protocol that a traveler is likely to experience. However, the main purpose of our study is to provide a mathematical explanation for why certain features of jet lag arise, such as east-west asymmetry and different modes of reentrainment. This is most easily explained using single lux levels. As further discussed throughout the paper, we expect the mechanisms that underlie our findings to continue to exist under more realistic light schedules.