Why meals during resting time cause fat accumulation in mammals? Mathematical modeling of circadian regulation on glucose metabolism

Abstract

Synchronization of metabolic rhythms regulated by circadian clock and meal timing is essential for maintaining nutrient homeostasis in response to fluctuating food intake in animals. Despite numerous experimental findings on the involvement of circadian regulation of glucose and lipid metabolism, the optimal regulatory strategy for the maintenance of energy homeostasis remains poorly defined. A mathematical framework is useful to assess the circadian regulation of glycogen production/breakdown and de novo lipogenesis/lipolysis by evaluating the contribution of time of the day-dependent activation or the repression of each metabolic process in the maintenance of energy homeostasis. Here, we present a mathematical model that describes the dynamics of glycogen and triglyceride contents, two major forms of energy storage in the body that provide the fuel needed during different phases of food deprivation. By changing peak phases of glycogenesis and fat synthesis, we searched for the optimal phase set that minimizes the risks of two types of possible metabolic dysfunctions: (1) high blood glucose and (2) energy exhaustion. Based on the optimal phase set, we compared the level of fat accumulation between meal timing in the active and resting periods. Our results showed that an increased fat accumulation by food intake in the resting period can be the byproduct of minimizing energy homeostasis risks in the synchronized feeding schedule that animals adopt in nature. Our finding will be useful to schedule an optimal meal timing to prevent metabolic diseases caused by misalignment of biological and social time in modern society.

Appendices

Appendix A: Dependence of costs on the threshold of hyperglycemia and energy depletion

In this Appendix, we discuss the dependency of energy depletion costs on the values of thresholds $x_{G}^{U}$ and E_L. In Sects. 3.2 and 3.3, we calculated the risks of energy exhaustion (C_E) and hyperglycemia (C_G) setting $x_{G}^{U} = 2.0$ and E_L = 4.0 in Eq. (9a) and (9b), based on the two regulations. First, we set thresholds that did not result in C_E = 0 or C_G = 0 in multiple conditions of (φ_F, φ_L) so as to choose one condition that achieves the lowest risk. Second, we did not consider the cases where the energy levels were below the threshold throughout the period. We considered that if a chronic lack of energy occurs, the animal cannot survive and the amount of food intake L(t) will increase. Varying $x_{G}^{U}$ and E_L, these regulations are violated under certain conditions as follows.

Threshold of energy exhaustion

We calculated the risks of energy exhaustion by varying the threshold to E_L = 2.0, 4.0, 8.0 (Fig. 5a). The result of E_L = 4.0 was the same as the result shown in Sect. 3.1.

Fig. 5

Threshold dependences of risks of energy exhaustion and excess glucose levels. We calculated two possible risks, denoted by Eq. (9b) and (9c) for 24 × 24 = 576 combinations of (φ_F, φ_L). The light-colored cells represent a higher level of risk, as shown in the scales. The condition achieved the lowest risk in the white frames. The white and black bars along the axis represent the resting and active times, respectively. (a) Risk of energy exhaustion. (i) E_L = 1.0. The risks become zero in the multiple conditions represented by the black area. (ii) E_L = 2.0. The lowest risk is achieved at (φ_F, φ_L) = (9, 23). (iii) E_L = 4.0. The lowest risk is achieved at (φ_F, φ_L) = (1, 0). (b) The risk of hyperglycemia. (i) $x_{G}^{U} = 1.0$. The lowest risk is achieved at (φ_F, φ_L) = (19, 3). (ii) $x_{G}^{U} = 2.0$. The lowest risk is achieved at (φ_F, φ_L) = (19, 0). (iii) $x_{G}^{U} = 4.0$. The risks become zero in the black area

When we set E_L = 2.0, the threshold was too low to evaluate the best conditions. This is because some conditions of (φ_F, φ_L) resulted in C_E = 0 (black cells in Fig. 5b) and we could not find the optimum (φ_F, φ_L).

In contrast, when we set E_L = 8.0, the threshold was too high because the peaks of energy levels were below the threshold E_L = 8.0 in some (φ_F, φ_L). Furthermore, energy exhaustion occurred throughout the day, even in the feeding phases. Under such conditions, the lowest risk was achieved (φ_F, φ_L) = (0, 21) in the condition where φ_F was during the active period, which is differs from E_L = 4.0. Therefore, the threshold of energy exhaustion has an effect on the result of searching for the condition of lowest risks.

Threshold of hyperglycemia

We calculated the risks of hyperglycemia by varying the threshold to $x_{G}^{U} = 1.0, 2.0, 4.0$ (Fig. 5b). The result of $x_{G}^{U} = 2.0$ is shown in Sect. 3.2 [the lowest risk was achieved in (φ_F, φ_L) = (18, 1)].

When we set $x_{G}^{U} = 1.0$, the condition of lowest risk was achieved at (φ_F, φ_L) = (19, 4). When we set $x_{G}^{U} = 4.0$, many conditions of (φ_F, φ_L) achieved C_G = 0 (Fig. 5b) and the threshold was too low to find the best condition for the lowest C_G. The conclusion did not depend on $x_{G}^{U}$ much compared to E_L. This is because the glucose levels declined to near zero during fasting phases, and the lack of glucose occurred in all of the conditions during the resting periods. Then, chronic hyperglycemia did not occur where $x_{G}^{U} = 1.0, 2.0, 4.0$, unlike in the case of varying E_L.

Appendix B: Dependence of costs on relative energy production efficiency

In this Appendix, we discuss the dependency of energy depletion costs on the relative energy production efficiency, denoted by f in Eq. (8b). A larger f represents a higher fat efficiency as an energy resource compared to the efficiency of glucose. From one molecule of glucose, about 30 ATPs were obtained as a result of aerobic respiration. On the other hand, from one molecule of palmitic acid, 106 ATPs were produced through β-oxidation. If we consider that one molecule of triglyceride, consists of three palmitic acids, 318 ATPs can be obtained from the triglyceride. Thus, the standard relative ATP production efficiency is set $f = \frac{318}{{30}} = 10.6$. We calculated the costs for energy depletion to vary f = 5.0, 10.6, and 15.0 (Fig. 6). In Sect. 3.1, we set f = 10.6 as a standard value (Fig. 6b)). We examined the smaller value f = 5.0 (Fig. 6a) and the larger value f = 15.0 (Fig. 6c).

Fig. 6

Relative energy production efficiency depends on the risk of energy depletion. We calculated the risks of energy depletion, denoted by Eq. (8a) for 24 × 24 = 576 combinations of (φ_F, φ_L) changing the relative energy production efficiency (f). The light-colored cells represent a higher level of risk, as shown in the scales. The condition achieved the lowest risk in the white frames. The white and black bars along the axis represent the resting and active times, respectively. (a) f = 5.0. The lowest risk is achieved at (φ_F, φ_L) = (23, 23). (b) f = 10.6. The lowest risk is achieved at (φ_F, φ_L) = (11, 22). (c) f = 15.0. The lowest risk is achieved at (φ_F, φ_L) = (9, 23)

The optimal timing of glycogenesis (φ_L) did not depend on f and the risks of energy depletion were minimum around φ_L = 23 (Fig. 6a–c). To maintain the glycogen level during the resting periods, the peaks at the end of the resting periods were optimal, as explained in Sect. 3.2.

The optimal conditions for fat production (φ_F) were varied f. If f = 5.0, the value of risk did not depend on φ_F so much (Fig. 6a), and increasing f, the optimal conditions became dependent on φ_F (Fig. 6b, c), since fat had a greater impact on the energy level, and the production phase became more important with increasing f. The φ_F minimum risk was achieved at the end of the active periods when f = 5 (Fig. 6a), but the optimal φ_F was shifted to the end of the resting period when f = 10.6 and f = 15.0 (Fig. 6a, b).

As discussed in Sect. 3.1, fat can compensate for the lack of glucose at the end of the resting periods when the efficiency of fat as an energy resource is large enough (e.g. f = 10.6 and f = 15.0). Thus, producing fat at the end of the resting period can be an optimal strategy. However, when f is small (f = 5.0), the fat is unable to overcome the lack of glucose and energy depletion tends to occur when either fat or glucose is at a low level. Therefore, synchronized production of both fat and glycogen was a better strategy than shifted production timing.

Appendix C: Circadian oscillation is necessary to maintain robust peak phases of substances against timings of food intakes

The equilibria of the model (Eqs. 1, 5, and 6) can be derived by calculating $x_{G}^{*} ,x_{F}^{*}$, and $x_{L}^{*}$ that satisfy $\dot{x}_{G} = \dot{x}_{F} = \dot{x}_{L} = 0$, as follows:

$$ \begin{array}{*{20}c} {x_{G}^{*} = \frac{aL\left( t \right)}{{\overline{{\gamma_{F} }} + d_{G} }},\quad x_{F}^{*} = \frac{{\overline{{\gamma_{F} }} }}{{d_{F} }}\frac{aL\left( t \right)}{{\overline{{\gamma_{F} }} + d_{G} }},\quad x_{L}^{*} = \frac{{\overline{{\gamma_{L} }} }}{{\overline{{\gamma_{\beta } }} }}\frac{aL\left( t \right)}{{\overline{{\gamma_{F} }} + d_{G} }},} \\ \end{array} $$

(C1)

where circadian oscillation is not considered, $\overline{{\gamma_{F} }} ,\overline{{\gamma_{L} }}$, and $\overline{{\gamma_{\beta } }}$ can be assumed as constant values. The equilibria change in response to L(t), represented in Eq. (2a) and (2b), d_G and d_F are represented in Eq. (3a) and (3b), respectively. There are two equilibria corresponding to the feeding (L(t) = 1) and fasting (L(t) = 0) periods as follows:

$$ \begin{array}{*{20}c} {x_{G}^{*} = x_{F}^{*} = x_{L}^{*} = 0,\quad where\,L\left( t \right) = 0\quad \left( {i.e.\, fasting\,period} \right),} \\ \end{array} $$

(C2)

and

$$ \begin{array}{*{20}c} {x_{G}^{*} = \frac{a}{{\overline{{\gamma_{F} }} + d_{G} }}, \quad x_{F}^{*} = \frac{{\overline{{\gamma_{F} }} }}{{d_{F} }}\frac{a}{{\overline{{\gamma_{F} }} + d_{G} }}, \quad x_{L}^{*} = \frac{{\overline{{\gamma_{L} }} }}{{\overline{{\gamma_{\beta } }} }}\frac{a}{{\overline{{\gamma_{F} }} + d_{G} }},\quad where\,L\left( t \right) = 1 \,\left( {i.e. \,feeding\,period} \right). } \\ \end{array} $$

(C3)

Without the circadian oscillation of γ_F, γ_L, and γ_β, the peak time of x_G, x_F, and x_L in the two schedules were shifted by 12 h (lower row in Fig. 7), and the timings of the food intakes were directly reflected in the peak times of glucose, fat, and glycogen. In contrast, the peak phases did not dramatically shift depending on the two feeding schedules when the circadian oscillation was considered, where φ_F = 6, φ_L = 18 (upper row in Fig. 7). Considering the circadian regulation of the production rates, the peak times of glucose, fat, and glycogen were robust to changes of the timing of food intake.

Fig. 7

Daily patterns of variables with and without circadian oscillation of γ_L, γ_F, and γ_β. Upper row: dynamics of glucose, fat, and glycogen with circadian oscillation, where φ_F = 6 and φ_L = 18. Lower row: dynamics without circadian oscillation, where the production rate of fat, glycogen, and glycogenolysis are fixed at the mean values of the maximum and minimum levels of the production rates (γ_F = 0.11, γ_L = 1.025, and γ_β = 0.25; the maximum and minimum levels are shown in Table 1). The white and black bars along the axis represent the resting and active periods, respectively. The bold and dotted lines show the dynamics of food intake in the active period and in the resting period, respectively

Circadian regulation on the production rates also contributed to reduction of the risks of energy depletion and hyperglycemia. With the production rates fixed at the mean values of the maximum and minimum levels (γ_F = 0.11, γ_L = 1.025, and γ_β = 0.25), the risks of energy depletion and high glucose level were 289.90 and 369.60, respectively, which were larger than the minimum risks under the optimal phase conditions: 35.33 at (φ_F, φ_L) = (11, 22) and 255.29 at (φ_F, φ_L) = (18, 1).