Photosynthetic Entrainment of the Circadian Clock Facilitates Plant Growth under Environmental Fluctuations: Perspectives from an Integrated Model of Phase Oscillator and Phloem Transportation

Plants need to avoid carbon starvation and resultant growth inhibition under fluctuating light environments to ensure optimal growth and reproduction. As diel patterns of carbon metabolism are influenced by the circadian clock, appropriate regulation of the clock is essential for plants to properly manage their carbon resources. For proper adjustment of the circadian phase, higher plants utilize environmental signals such as light or temperature and metabolic signals such as photosynthetic products; the importance of the latter as phase regulators has been recently elucidated. A mutant of Arabidopsis thaliana that is deficient in phase response to sugar has been shown, under fluctuating light conditions, to be unable to adjust starch turnover and to realize carbon homeostasis. Whereas, the effects of light entrainment on growth and survival of higher plants are well studied, the impact of phase regulation by sugar remains unknown. Here we show that endogenous sugar entrainment facilitates plant growth. We integrated two mathematical models, one describing the dynamics of carbon metabolism in A. thaliana source leaves and the other growth of sink tissues dependent on sucrose translocation from the source. The integrated model predicted that sugar-sensitive plants grow faster than sugar-insensitive plants under constant as well as changing photoperiod conditions. We found that sugar entrainment enables efficient carbon investment for growth by stabilizing sucrose supply to sink tissues. Our results highlight the importance of clock entrainment by both exogenous and endogenous signals for optimizing growth and increasing fitness.

w i (t) depends on the difference between the turgor pressure and the osmotic pressure: where m is per-area permeability, and A i is the surface area of component i. Under the assumption of dilute solution, the osmotic pressure is calculated as g i (t)RT by the van't Hoff equation, where R and T are the gas constant and absolute temperature, respectively.
Because we assume that every cylinder is rigid, water efflux should be equal to water influx at each component (Fig. 1D). The conservation law of water volume is therefore held: Turgor pressure p i (t) is algebraically represented by solving a set of simultaneous linear equations (Eqs. (S1)-(S6)). p i (t) depends solely on the sucrose concentration in the phloem, the dynamics of which is formalized below. Sucrose flux J i (t) is then calculated by substituting p i (t) into Eq. (S1).

Sucrose dynamics in the phloem tube
To describe the sucrose dynamics in the phloem, we introduce the symbol [x] + meaning max{0, x}. Sucrose in component k is transported to component i when J i (t) is positive. The amount of sucrose transported from k to i per unit time is given by g k (t)J i (t). In contrast, when J i (t) is negative, sucrose is transported from component i to k at an amount given by g i (t)( J i (t)). These two cases are integrated into the term , which denotes the rate of the sucrose change due to the flux J i (t) at component i (respectively, component k). By applying this consideration to each component, sucrose dynamics in the phloem are described by the following equations: where V i represents the volume of the source (i ∈ G), the tubes (i ∈ T), and the sinks (i ∈ Y). The dynamics of g i (t) (i = 0, 4, and 5) are described by Eqs.

Parameter estimation of the sugar input function
Similar to a previous study (Seki et al., 2017), we define the sugar input function f S by the Hill equation: where sgn is a sign function, and n and K are constants. The parameter values of K and n are determined for various values of the subjective dusk φ * using the procedure described in a previous study (Seki et al., 2017).

Alternative formalization of growth rate
In the main text, we define growth rate λα(S Y ) of the sink tissues as a nonlinear function of sucrose concentration (Eqs. (7) and (8)). Here we test the alternative formalization of growth rate as a linear function of sucrose. We assume that sucrose is consumed for growth at a rate α 1 and that growth is repressed when sucrose falls below a threshold level. Based on these assumptions, the dynamics of sucrose S Y and plant fresh weight W Y are described by where α 2 is a parameter, S Y * is a threshold constant, and the symbol [x] + is defined as in Section 1.3 (S11) is considered as the growth rate ( Fig. S4A) and is referred to as "linear growth rate". The growth rate in the main text is correspondingly referred to as "nonlinear growth rate". When sucrose concentration is relatively low, the linear growth rate becomes negative (Fig. S4A), which corresponds to death of living tissues due to insufficient maintenance respiration. We estimated the values of α 1 and α 2 using procedures similar to those explained in Section 2.2.2 of the main text. We set S Y * = 0.9 .
Similar to the results in the main text assuming the nonlinear growth with sucrose ( Fig. 3), 10-day growth of the mutant is lower than the wild type and homeostatic plant under constant photoperiod conditions (Fig. S4B). However, the growth of all three plants in long photoperiods are closer using the linear growth rate than when computed using the nonlinear growth rate. Growth patterns are significantly altered by the change in formalization of the growth rate (Figs. S4C and D; Figs. 4C and 5C), while the sugar dynamics are not (data not shown). Under a long day (16 L/ 8 D) with the linear growth rate, the very high sucrose level around dawn (Fig. 4B) strongly contributes to growth of the mutant, in contrast to the case using the nonlinear growth rate ( Fig. S4C; Fig. 4C), because growth rate keeps increasing with sucrose concentration. This reduces the growth difference between the mutant and the others. Under a short day (8 L/ 16 D) with the linear growth rate, the overall growth trend is unchanged, but its baseline is higher compared to that with the nonlinear growth rate ( Fig.  S4D; Fig.5C) simply because of the higher growth rate at a sucrose level of about 0.8 ( Fig. S4A and Fig. 5B).
When the plants are transferred from a short to a long photoperiod, the growth of the wild type and homeostatic plant are still greater than the mutant (Fig. S5A) as in the case for the nonlinear growth model. However, the mutant grows faster than the others when the plants are transferred from a long to a short photoperiod (Fig. S5B). Transient dynamics of the growth rates immediately after the photoperiod change from short to long (Fig. S5C) are similar to those assuming the nonlinear growth rate (Fig. 8B). On the other hand, when photoperiod changes from long to short (Fig. S5D) the growth rate is transiently higher in the mutant (around t = 0), lower in the wild type (around t = 24), and almost the same in the homeostatic plant compared to the dynamics assuming the nonlinear growth rate (Fig. 8B). These changes in transient growth dynamics of the plants exposed to the longto-short day transition, in addition to the small growth difference among the plants in constant conditions, distinguish the overall growth from those described in the main text ( Fig. S5B and Fig.  8A).     Table S1. Summary of parameters, variables, and functions.