Plants are inevitably exposed to daily and seasonal variations in light environments. To continuously grow in fluctuating environments, it is crucial for plants to stably supply carbon resources for respiration and growth. Plant growth in the day is supported by the supply of photosynthates, particularly soluble sugars such as sucrose that are transported from photosynthetic leaves (source tissues) to sink tissues (e.g., roots). Plants grow even in nighttime using carbon resources accumulated during the preceding daytime. Arabidopsis thaliana, a model plant, partitions a large fraction of assimilated carbon into insoluble starch, which is degraded at night to produce sucrose (Smith and Stitt, 2007; Stitt and Zeeman, 2012). Because early exhaustion of starch results in carbon starvation and ensuing growth inhibition (Graf et al., 2010; Yazdanbakhsh et al., 2011), careful management of starch metabolism is essential to cope with daily and seasonal fluctuations of light conditions.

In A. thaliana, starch amount increases during the day at an almost constant rate and decreases almost linearly at night (Caspar et al., 1985; Gibon et al., 2004; Smith et al., 2004; Lu et al., 2005). Plants in shorter photoperiods accumulate starch more rapidly during the day and degrade it more slowly at night than in longer photoperiods (Lu et al., 2005). Plants also adjust the rate of starch degradation immediately in response to an unexpectedly early or late onset of night (Lu et al., 2005; Graf et al., 2010; Scialdone et al., 2013). The circadian clock underlying the approximately 24-h cycle of biological processes is implicated in the control of starch metabolism (Graf and Smith, 2011). Wild type A. thaliana (Ws) exhausts starch reserves about 24 h after the last dawn even under non-24 h light/dark cycles (T-cycles) (Graf et al., 2010), indicating that the timing of starch exhaustion is programmed by the circadian clock. In the circadian clock mutant cca1/lhy, in which the functional clock has a period of about 17 h (Locke et al., 2005), the depletion of starch occurs prematurely under a 24-h T-cycle but coincides with dawn under a 17-h T-cycle (Graf et al., 2010). These studies suggest that coordination of the internal timing of starch turnover with environmental cycles is necessary to avoid carbon starvation.

Phase adjustment of the circadian clock to external stimuli such as light or temperature is fundamental for synchronizing biological processes with environments (Johnson et al., 2003). In addition to signals from the external environment, endogenous signals such as photosynthates are also important regulators of the circadian phase in A. thaliana. We previously reported that phase adjustment by sugar is necessary for plants to flexibly regulate carbon metabolism in fluctuating light environments (Seki et al., 2017). We developed a phase oscillator model describing phase regulation of the circadian clock by sucrose. This model predicted that phase adjustment of the circadian clock by sucrose is crucial for homeostatic regulation of carbon resources. These theoretical predictions were confirmed by physiological experiments using the mutant pseudoresponse regulator 7–11 (prr7–11), the circadian clock of which does not show clear phase response to sucrose pulse (Haydon et al., 2013).

Whereas, clock entrainment by exogenous signals such as light has been shown to be advantageous for competition and survival in several organisms (Woelfle et al., 2004; Dodd et al., 2005), the advantages of clock entrainment by photosynthetic products remain elusive. Here we theoretically evaluate the effects of clock entrainment by sugar on plant growth, a good proxy for plant fitness (Younginger et al., 2017). We extended the phase oscillator model for the circadian clock (Seki et al., 2017) by incorporating growth dynamics of sink tissues, including shoot apical meristems and roots. Growth dynamics are described by modeling phloem transportation of sucrose from source to sink tissues (Seki et al., 2015; Satake et al., 2016). We demonstrate that plant growth is facilitated by endogenous sugar entrainment under long photoperiods because the entrainment enables the stable supply of sucrose from source to sink tissues irrespective of light fluctuation. In short photoperiods, however, the effect of sugar entrainment on growth is negligibly small. Our results provide important theoretical evidence that circadian-phase adjustment by endogenous signals is advantageous for plant growth.

To investigate the effect of clock regulation by sugar on plant growth, we integrated two previously developed models, one describing the dynamics of starch and sucrose metabolism in source leaves (Seki et al., 2017) and the other growth of sink tissues dependent on phloem transportation of sucrose from source tissues (Seki et al., 2015; Satake et al., 2016). Photosynthetic products in the source leaf are partitioned into sucrose and starch (Seki et al., 2017; Figure 1A). Sucrose in the source leaf is loaded into the phloem, moves through the phloem tube, and is unloaded at sink tissues where it is used for respiration and growth (Seki et al., 2015; Satake et al., 2016; Figures 1B–D). These two models were coupled by incorporating a term that represents sucrose translocation from source to sink. We explain the detailed structures of each component of our new integrated model in the following sections. We consider two sink tissues, the shoot and root apical meristems (SAM and RAM, respectively), because plant growth and development mainly occur in these organs (Figure 1C). Our model can be extended to the structure including multiple sinks in the complex phloem network as studied previously (Seki et al., 2015; Satake et al., 2016).

When the plants are grown in a constant photoperiod, the growth of the sink (SAM or RAM) of the all three types increases as the photoperiod is lengthened (Figure 3A). The sugar-sensitive wild type and homeostatic plant grow significantly faster than the sugar-insensitive mutant under long days (Figure 3B). When photoperiod is shorter than 12 h, the growth difference between the mutant and the others almost disappears. Growth of the wild type is slower than that of the homeostatic plant in long photoperiods (Figure 3B), suggesting that the minimization of sucrose fluctuation is the most effective strategy for efficient growth under these conditions. We confirmed the robustness of our results under different parameter values for carbon metabolism (Figure S2) as well as for the phloem tube network (Figure S3).

Under a long day (16 L/ 8 D), sucrose levels in the source and sink (SAM or RAM) are highly variable in the sugar-insensitive mutant, moderately variable in the wild type, and almost constant in the homeostatic plant (Figures 4A,B). Diel patterns of growth reflect sucrose profiles in the sink, revealing a substantially lower growth rate during the evening in the mutant and a moderately lower growth rate during the evening in the wild type compared to the homeostatic plant (Figure 4C). The difference in evening growth rate is the major reason for differential growth among the three plant types in a long photoperiod. The mutant accumulates the largest amount of starch, with some amount unused even at the end of night (Figure 4D), indicating inefficient translocation of carbon. In contrast, the wild type accumulates less starch than the mutant and invests a larger amount of photoassimilate for growth (Figure 4D).

The different sucrose, starch, and growth profiles between the wild type and sugar-insensitive mutant are caused by the clock plasticity in response to sugar. The decrease in sucrose levels at dawn (Figure 4A) is sensed by the wild type as a negative sugar signal, driving a phase delay of the circadian oscillator (Figure 4E). This phase delay in the morning increases the starch degradation rate during the light period (Figure 4F), resulting in elevation of the sucrose level (Figure 4A) and decrease of the starch level (Figure 4D). Such a phase shift in the circadian oscillator does not occur in the mutant (Figure 4E). Excessively high sucrose in the mutant around dawn (Figure 4B) does not significantly contribute to growth (Figure 4C) due to the saturating property of the growth-related sucrose consumption rate α (Figure 2A), while the sucrose decrease around dusk substantially reduces growth rate since at this low level α is almost linearly dependent on sucrose.

Under a short day (8 L/16 D; Figure 5), the wild type and sugar-insensitive mutant show the similar sugar and growth dynamics as reported previously (Seki et al., 2017). The elevation of sucrose at dawn (Figure 5A) gives rise to a positive sugar signal and resultant phase advance in the wild type (Figure 5E). However, this phase shift does not substantially change the starch degradation rate (Figure 5F), so the sugar and growth dynamics are almost the same for the wild type and mutant (Figures 5A–D). The homeostatic plant displays distinct growth dynamics (Figure 5C). Nevertheless, the difference in 10-day growth among plants is very small in the short photoperiod (Figure 3B). Since all three plants fix less carbon per day compared to under long day conditions, sucrose concentration is low both in the source and sink (Figures 5A,B). This leads to a slow growth rate due to low supply of sucrose. Because all three plants are limited by sucrose under short day conditions, the growth difference among them is indistinguishably small.

We examined the effect of the timing of the subjective dusk ϕ^* on growth dynamics of the wild type (Figures 6, 7). Changes in ϕ^* do not markedly alter the values of 10-day growth of the sink and its difference among plant types (Figure 6). On the other hand, growth rate is strongly dependent on the timing of both the internal and external dusk. When the external dusk occurs later than the internal dusk (i.e., the length of the light period τ_L is larger than ϕ^*), the growth rate peaks around dawn and decreases around dusk (Figures 4C, 7). As the photoperiod is shortened, amplitude of growth rate decreases and eventually becomes almost zero. When the value of τ_L is smaller than ϕ^*, the opposite growth pattern is observed, with a peak around dusk and a trough around dawn (Figures 5C, 7).

When the plants are transferred from long to short or short to long days, the growth of the wild type and homeostatic plant are consistently higher than that of the sugar-insensitive mutant (Figure 8A). All three plants are able to restore normal sugar and growth dynamics in about 2 days after the transfer, although the dynamics are complex immediately after the photoperiod change (Figure 8B). Therefore, the growth differences under these conditions may reflect the results under the constant photoperiod conditions that growth of the mutant is inferior to the others especially under a long day (Figure 3). Note that the dynamics of other variables such as sucrose in source were reported in a previous study (Seki et al., 2017).

Our results provide the first theoretical evidence that clock entrainment by photosynthetic products improves the fitness of higher plants. The sugar-sensitive wild type is predicted to grow faster than the sugar-insensitive mutant under long day conditions (Figure 3) because the phase shift of the circadian oscillator by sugar signals enables efficient sugar allocation for growth, while the mutant accumulates carbon as insoluble starch unusable for growth. The growth of the wild type is also higher than the mutant under changing photoperiod conditions (Figure 8A). Since the wild type and mutant are assumed to possess the same entrainment property to light, the lower growth in the mutant stems solely from the lack of sugar-induced phase adjustment. These results suggest that clock entrainment by endogenous sugar signals, in addition to entrainment by exogenous light signals, optimizes plant growth in nature, where the day length gradually changes.

Our results also provide important information about the internal timing of plants. Our model predicts that the growth pattern of the wild type displays a maximum around dawn and a minimum around dusk in a 16-h photoperiod (Figure 4C) or more generally under conditions where the value of τ_L (length of the light period) is larger than ϕ^* (timing of the subjective dusk) (Figure 7). When the value of τ_L is smaller than ϕ^*, the pattern is reversed (Figure 5C where τ_L = 8 h and ϕ^* = 10 h; Figure 7). Yazdanbakhsh et al. (2011) reported, in both 16-h and 8-h photoperiods, a similar diel pattern in root elongation growth of A. thaliana (Col-0), which possibly correlates with fresh biomass change since both growth measures consider dry matter production and water content. These findings imply that ϕ^* of this accession is about 8 h. This idea is consistent with the PRC of the same accession to a sucrose pulse showing a transition from phase advance to delay, which corresponds to the phase ϕ^*, at 6–10 h zeitgeber time (Haydon et al., 2013; Seki et al., 2017). These data suggest that a relatively short light period is subjectively anticipated by this accession. Other ecotypes of A. thaliana that anticipate later subjective dusk (i.e., large ϕ^*) will display the inversed growth pattern in a short photoperiod and later phase advance-to-delay transition of the sucrose-pulsed PRC.

Numerous mathematical models on growth of various plants have been developed (e.g., Thornley and Johnson, 1990; Chew et al., 2014, 2017; Feller et al., 2015; Barillot et al., 2016a,b). Among them, two multiscale models of A. thaliana include the circadian clock sub-model and are able to quantitatively predict growth (Chew et al., 2014, 2017). Because the phase of the clock is dynamically regulated by sugar signals in our model and such phase adjustment is not considered in the previous models (Chew et al., 2014, 2017), only our model enables us to evaluate the impact of phase regulation by sugar on plant growth. Our model correctly predicts the empirical finding that wild type A. thaliana (Col-0) and the mutant prr7–11 in which phase response to sugar is abolished grow similarly in a 12-h photoperiod (Chew et al., 2017; Figure 3). We suggest that this is due to the relatively weak impact of sugar entrainment on growth, as our modeling results show that the sugar-induced phase shift has less effect on growth dynamics in shorter photoperiods (Figure 5). These considerations are reminiscent of the empirical finding that response of A. thaliana to induced increases in trehalose-6-phosphate, a potential signal metabolite of sucrose status, was more intense in a 16-h photoperiod than in a 12-h photoperiod (Figueroa et al., 2016). Our model also predicts significantly faster growth of the wild type than the sugar-insensitive mutant under long day conditions (Figure 3), which will be experimentally tested using wild type A. thaliana and prr7–11. Wild type is likely to grow faster than prr7–11 under long day conditions because the latter accumulates more starch than the former in a 16-h photoperiod (Seki et al., 2017) which implies that prr7–11 can utilize less sucrose for growth than wild type. However, it should be noted that mutation of PRR7 affects not only sugar signaling but also light signaling in the plant circadian system (Kaczorowski and Quail, 2003; Farré et al., 2005), and thus the growth characteristics of prr7–11 can be influenced by variations in light as well as sugar entrainment. Nevertheless, sugar entrainment is likely to be a dominant factor because our previous study has theoretically shown that the deficient in sugar signaling only is sufficient to correctly predict patterns of starch turnover of prr7–11 (Seki et al., 2017).

There are several possibilities to expand our model. We used a constant rate a for carbon capture, which was determined in a previous study based on experiments where nutrient levels were controlled (Gibon et al., 2004; Feugier and Satake, 2013). In higher plants, however, it has been demonstrated that sugar accumulation downregulates photosynthesis, possibly through decreased activity of Rubisco (Araya et al., 2006; Ribeiro et al., 2012; Quentin et al., 2013; Lobo et al., 2015). Formalizing the carbon capture rate as a decreasing function of sucrose concentration could reduce the growth of the sugar-insensitive mutant under long day conditions due to the very high sucrose level around dawn (Figures 4A,B). Regarding circadian clock properties, we considered a circadian oscillator with a period of 24 h (i.e., ω = 1). A recent study has reported a positive correlation between the free-running period of natural populations of Mimulus guttatus and the latitude of their geographic origin (Greenham et al., 2017). In A. thaliana, a similar but weak correlation has been found (Michael et al., 2003), although advantages of such variation in the period remain elusive. Growth simulations of plants with fast- and slow-running clocks could potentially reveal the adaptive value of natural variation in the circadian period in terms of growth optimization. Moreover, we can consider the possibility that two or more clock genes are involved in the phase response to sugar, where each gene can have differential responsiveness. Although the crucial role of PRR7 in the response has been established (Haydon et al., 2013), other clock genes could participate in it; for instance, CCA1 will be a good candidate, mutation of which disrupts the dependency of the circadian period on sucrose concentration in growth media (Haydon et al., 2013). To examine this possibility, the single phase oscillator will be expanded to coupled two or multiple oscillators model, in which each phase oscillator is defined with different PRC to sucrose and sugar input function (Z_S and f_S in Equation (4), respectively). Alternative way is to use the clock gene-regulatory network model (Fogelmark and Troein, 2014; De Caluwé et al., 2016) with the explicit formalization of the interaction between clock genes and sugar signals.

We conclude that plants optimize growth by monitoring nutrient status and utilizing endogenous sugar signals as circadian-phase regulators. Photosynthetic products act not only as direct growth substances but also as feedback signals to the clock to realize the efficient carbon-usage for growth. It is plausible that plants as sessile organisms utilize signals from metabolism because they seem more controllable than environmental signals. In this sense, the phase regulation by endogenous signals is potentially even more important for plants than animals that also utilize metabolic signals as zeitgebers (Woller et al., 2016). Our data strongly support the concept that the circadian clock improves the fitness of organisms by forming complex feedback loops with the signaling pathways it controls (Sanchez and Kay, 2016).

TO and AS developed the model and wrote the manuscript. TO analyzed the model.