Clocks in the Wild: Entrainment to Natural Light

Abstract

Entrainment denotes a process of coordinating the internal circadian clock to external rhythmic time-cues (Zeitgeber), mainly light. It is facilitated by stronger Zeitgeber signals and smaller period differences between the internal clock and the external Zeitgeber. The phase of entrainment ψ is a result of this process on the side of the circadian clock. On Earth, the period of the day-night cycle is fixed to 24 h, while the periods of circadian clocks distribute widely due to natural variation within and between species. The strength and duration of light depend locally on season and geographic latitude. Therefore, entrainment characteristics of a circadian clock vary under a local light environment and distribute along geoecological settings. Using conceptual models of circadian clocks, we investigate how local conditions of natural light shape global patterning of entrainment through seasons. This clock-side entrainment paradigm enables us to predict systematic changes in the global distribution of chronotypes.

Keywords: Arnold onion; chronotype; circadian clock; entrainment; photoperiodism; seasonality.

Figure 1

Entrainment ranges vary with photoperiod. (A) A certain circadian phenotype with free running period τ is entrained to Zeitgeber cycles of varying T. (B) Entrainment region (green) of a self-sustained oscillator under entrainment with varying photoperiods ϰ and Zeitgeber periods T. The circadian free-running period under constant darkness (ϰ = 0) has been set to τ = 24 h. (C) Different circadian phenotypes with varying free-running period τ, due to variation in a population of organisms or genetic background (e.g., clock mutants), entrained by a Zeitgeber signal with a fixed period T. (D) For a given Zeitgeber period T = 24 h, the entrainment region (green) can be determined for varying free-running periods τ and different photoperiods ϰ. Simulations underlying panels (B) and (D) have been done by simulating Equations (4) and (5), applying a square-wave Zeitgeber signal of peak-trough amplitude 0.1 to the x-variable and by using oscillator parameters A = 1 and λ = 0.5 h⁻¹ as described in Schmal et al. (2015).

Figure 2

Schematics of entrainment range prediction through geophysical considerations. (A) Phase plane representation of the conceptual clock model, the Poincaré oscillator. While the black circle denotes the limit cycle of steady state amplitude A, dashed lines show transient dynamics for different values of the radial relaxation rate λ, starting from an initial condition r₀ and φ₀ or x₀ = (r₀cos(φ₀), r₀sin(φ₀)). Larger values of λ lead to shorter transient dynamics after perturbations. (B) Illustration of the celestial constellation between the Earth and Sun during summer in the northern hemisphere. Here, ε denotes the obliquity of the ecliptic while δ is the declination of Sun. (C) Global distribution of daylengths on a summer day in the northern hemisphere, with geographical locations of representative cities used in this study: Kampala (equator), Taipei (sub-tropics), Berlin (temperate), and Ny-Alesund (above arctic circle). Note that the irradiance is lower in the northern extreme despite the longer daylength. LD, long daylength; 12:12, equinox; SD, short daylength. (D) Combining a conceptual oscillator model with a celestial mechanics based approximation of the light intensity, we compute entrainment ranges and phases of entrainment ψ under different seasons and latitudes. By this we can study impact of season and latitude on the distribution of chronotypes.

Figure 3

Impact of season and latitude on daylength and light intensity. (A) Solar irradiance on the Earth's surface for three different latitudes [ϕ = 10° (blue), ϕ = 60° (orange), ϕ = 80° (green)] on day N = 172 which corresponds to the 21st of June, i.e., summer solstice in the northern hemisphere. Here, time point zero corresponds to solar noon, i.e., when the Sun reaches the highest point at sky. Dashed lines correspond to sunrise and sunset as given by condition (8). (B) Daylength for different latitudes and times of year are depicted as given by Equation (9) in section 4. The daylength corresponds to the distance between sunrise and sunset as exemplified in (A) for three different latitudes. (C) Daily insolation for different latitudes and times of year. The daily insolation is defined as the daily sum over the solar irradiance as depicted in (A) for three exemplary latitudes. Please note that the solar irradiance in (A) and solar insolation in (C) have been calculated for sun rays reaching a horizontal plane. The direction of the surface (e.g., of plant leaves, the retina, etc.) has an impact on the solar irradiance perceived. (D) Standard deviation of the solar irradiance, calculated over a complete day (24 h), for different latitudes and times of year.

Figure 4

Entrainment ranges narrow down at high latitudes. Depicted are the entrainment regions and color-coded phases of entrainment ψ for different times around the year and varying intrinsic free-running periods τ at four exemplary latitudes ϕ, namely, ϕ = 0.3° (A), ϕ = 25.03° (B), ϕ = 52.51° (C), ϕ = 78.92° (D), corresponding to the latitudes of Kampala, Taipei, Berlin, and Ny-Alesund, respectively. Regions without color-coded phases correspond to parameter combinations of N and τ where the circadian clock is unable to entrain to the external Zeitgeber cycle. Other intrinsic oscillator properties have been set to A = 6 and λ = 0.1 h⁻¹.

Figure 5

Entrainment depends on latitude and time of year. Entrainment regions and color-coded phases of entrainment ψ for different latitudes and internal free-running periods τ at three different times of the year, namely March equinox (A), summer solstice (B) and winter solstice (C). Other internal clock parameters have been set to A = 6 and λ = 0.1 h⁻¹. Color coding of ψ is the same as in Figure 4.

Figure 6

High sensitivity of phases during winter and at high latitudes. (A) Phase of entrainment ψ at the latitude of Taipei (ϕ = 25.03°) vs. time of the year (N) for three different internal free-running periods τ = 23 h (dashed line), τ = 24 h (bold line), and τ = 25 h (dotted line), respectively. The gray shaded area depicts the duration of darkness. (B) Entrainment ranges with respect to variations in τ as determined by the difference between the upper and lower limits of entrainment in Figures 5A–C for different latitudes, respectively. (C) Dependency of the phase of entrainment ψ on internal free running period τ on March equinox for the latitudes of Kampala (blue), Taipei (orange), Berlin (green), and Ny-Alesund (red). (D) Dependency of the phase of entrainment ψ on internal free running period τ at winter solstice (dashed line), March equinox (bold line), and summer solstice (dotted line) at the latitude of Berlin (ϕ = 52.51°). All simulations have been performed for light perceived by a horizontal plane. Phases of ψ = ±12 h and ψ = 0 h correspond to midnight and solar noon, respectively.

Figure 7

The distribution of chronotypes broadens in winter and at higher latitudes. (A) Distribution of free-running periods τ sampled from a normal distribution of mean μ_τ = 24.09 h and standard deviation σ_τ = 0.2 h, following what has been described for female human using a forced desynchronization protocol (Duffy et al., 2011). (B) Distribution of entrainment phases ψ (i.e., of chronotypes), that follows from a period distribution as depicted in (A) under natural light entrainment in winter (Jan 1st) and summer (Jun 20th) at the latitude of Berlin (ψ = 52.51°). Approximately 12% of the (most extreme) sampled periods τ do not properly entrain under the conditions on Jan 1st. (C) Distribution of entrainment phases ψ during winter (Jan 1st) in Berlin (blue) and Taipei (orange). (D) Systematic analysis of the standard deviation in the distribution of ψ throughout the year at the latitude in Berlin (blue) and Taipei (orange). In the computations underlying panels (B–D), oscillator parameters others than τ are held constant to A = 6 and λ = 0.1 h⁻¹.

Figure 8

Global entrainment patterns across seasons. Entrainment ranges at different latitudes ϕ are depicted on a globe map for three different times of the year, namely winter solstice (A), March equinox (B), and summer solstice (C). This representation follows the scheme outlined in Figure 2, where the entrainment ranges correspond to those depicted in Figure 6B.