Mathematical modeling informs the impact of changes in circadian rhythms and meal patterns on insulin secretion

Abstract

Disruption of circadian rhythms has been associated with metabolic syndromes, including obesity and diabetes. A variety of metabolic activities are under circadian modulation, as local and global clock gene knockouts result in glucose imbalance and increased risk of metabolic diseases. Insulin release from the pancreatic β cells exhibits daily variation, and recent studies have found that insulin secretion, not production, is under circadian modulation. As consideration of daily variation in insulin secretion is necessary to accurately describe glucose-stimulated insulin secretion, we describe a mathematical model that incorporates the circadian modulation via insulin granule trafficking. We use this model to understand the effect of oscillatory characteristics on insulin secretion at different times of the day. Furthermore, we integrate the dynamics of clock genes under the influence of competing environmental signals (light/dark cycle and feeding/fasting cycle) and demonstrate how circadian disruption and meal size distribution change the insulin secretion pattern over a 24-h day.

INTRODUCTION

Circadian rhythms, the biological time-keeping mechanisms that maintain 24-h oscillations in physiology and behavior, are ubiquitous in plants and animals (30). In mammals, the master clock in the suprachiasmatic nucleus (SCN) receives photic input via the eyes and synchronizes the network of peripheral clocks to the light/dark cycle (7, 8). In addition to the light/dark cycle, food intake also acts as a strong zeitgeber, or an environmental cue that resets the internal body clock. Especially in organs governing energy homeostasis, feeding rhythms can metabolically synchronize the peripheral clocks independent of the SCN (11, 40). It is well established that disruption of circadian rhythms increases the risk of metabolic syndrome, obesity, and type 2 diabetes. Loss and/or misaligned sleep increases the risk of obesity (39). Disruption in daily rhythms of food intake results in diabetes and cardiovascular diseases (9, 14, 17). To elucidate the mechanism of interplay between circadian rhythms and metabolism, we have previously modeled the effect of misalignment in feeding and light on peripheral clock genes and hepatic gluconeogenesis (2, 3). We attempt to broaden our understanding of the relationship between circadian rhythms and metabolism by modeling the circadian dynamics of glucose-stimulated insulin release.

Although many stages of insulin action exhibit circadian profiles, we focus on circadian rhythmicity of insulin secretion in this work. Rhythmicity in insulin level is observed in both animals and humans. An experiment with perifused rat pancreatic islets shows evidence for circadian rhythmicity of insulin release from pancreas (27). In healthy human volunteers, glucose-stimulated insulin secretion increased from a nadir between midnight and 6:00 AM and reached a peak between noon and 6:00 PM (5). There truly exists a circadian rhythmicity of insulin secretion in humans, as it is not related to the duration of prior fast (41).

As insulin secretion is under circadian modulation, functional clock genes are required for proper islet function. One of the points of evidence is that clock-mutant mice (Clock and Bmal1) suffer from hypoinsulemia and diabetes (20). For human pancreatic islet cells (34), Clock knockout cells showed a significant decrease in both acute and chronic glucose-stimulated insulin secretion. The synchrony of insulin secretion rhythm was also perturbed upon clock disruption. RNA sequencing results further elucidated that clock genes are involved in insulin secretion rather than production. Both global and β-cell-specific Bmal1 deletions result in β-cell dysfunction and diabetes attributable to insufficient glucose-stimulated insulin secretion (18, 33, 36). The isolated pancreatic islets from the mutant mice showed normal insulin content, whereas glucose-stimulated insulin secretion was defective. Therefore, Bmal1 may control the release of insulin from the pancreatic islets as opposed to the production of insulin.

From the above observations, we hypothesize that oscillations in Bmal1 drive the daily rhythms of glucose-stimulated insulin secretion and propose a mathematical model to study the effect of circadian rhythms on insulin secretion. Previous work on modeling glucose control of insulin secretion found it necessary to embed circadian modulation to adequately predict C-peptide concentrations over a 24-h period (21). Although this was instructive, a semimechanistic mathematical modeling approach allows testing the effects of life style patterns, such as meal size distribution and meal intake time, on insulin secretion. A recent breakthrough in the research of circadian insulin secretion is that the oscillations of insulin granule trafficking largely drive the secretion pattern (12). Therefore, we propose a model that can simulate the effect of circadian rhythms on the dynamics of insulin granule formation, translocation to the cell membrane, and release to the extracellular space. Of the previously developed models studying insulin granule trafficking (4, 25, 26), we chose the model of Bertuzzi et al. (4) to be combined with our light and feeding entrained clock gene model (2, 3, 22, 23, 28) because it is relatively simple yet provides the structure for integrating the effect of clock genes and meal intake. Briefly, glucose-stimulated insulin secretion is tightly related to the cellular ATP/ADP ratio and Ca²⁺ concentration (19). Increase in glucose concentration causes the cellular ATP/ADP ratio to increase, causing closures of ATP-dependent K⁺ channels (19). When cell membrane is depolarized, Ca²⁺ channels open, allowing the influx of Ca²⁺ ions. Increased concentration of Ca²⁺ ions mediates the exocytosis of insulin granules.

In this new model, the parameter describing the production of granule membrane material oscillates in a circadian manner, resulting in graded insulin secretion response to glucose stimulation over a 24-h period. The model predicts that there exists an optimal light-feeding phase relation that maximizes glucose sensitivity and that meal size distribution also plays an important role in determining the level of insulin secretion.

METHODS

We adapted and modified the insulin granule trafficking model by Bertuzzi et al. (4) to study the effect of circadian characteristics on glucose-stimulated insulin secretion. The kinetics of four intracellular pools of insulin granules from the original model, the reserve pool (R), the pool of docked granules (D), the pool of immediately releasable granules (D_IR), and the pool of granules fused with cell membrane (F), were retained. The schematic of the model is shown in Fig. 1, and the list of variables and descriptions are available in Table 1. The dynamics of the above four pools and the pool of proinsulin (I) are described by the following equations:

Fig. 1.

Schematic of the insulin model, adapted from the granule-trafficking model (4). The circadian rhythmicity is introduced to the model via granule membrane synthesis rate (b_V), the rate of biosynthesis of granule membrane material (per min), circled in gray. As for the state variables, I is pool of proinsulin aggregates, V is the pool of granule membrane material, R is reserve pool, D is the pool of docked granules, D_IR is the pool of immediately releasable granules, C is the pool of unbound Ca²⁺ channels, F is the pool of granules fused with plasma membrane, γ is the rate coefficient of granule externalization and priming related to ATP/ADP (per min), and ρ is rate coefficient of granule fusion with cell membrane related to [Ca²⁺] (per min). As for other parameters, b_I is the biosynthesis rate of proinsulin aggregates at a given glucose concentration (per min), k is the rate constant of formation of proinsulin-containing granules (per min), k₁⁺ is the rate constant of association for the binding between granule and Ca²⁺ channel (per min), k₁^- is the rate constant of dissociation for the binding between granule and Ca²⁺ channel (per min), σ is the rate constant of insulin release from granules fused with cell membrane (per min), and τ_V is the time delay related to recycling of granule membrane material (min).

Table 1.

List of variables

Variable	Description
C	Pool of unbound Ca2+ channels
D	Pool of docked granules
DIR	Pool of immediately releasable granules
F	Pool of granules fused with plasma membrane
I	Pool of proinsulin aggregates
R	Reserve pool
V	Pool of granule membrane material
γ	Rate coefficient of the granule externalization, related to ATP/ADP, per min
ρ	Rate coefficient of granule fusion to the cell membrane, related to [Ca2+], per min
ISR	Insulin secretion rate, pmol/min

$$\frac{\text{d}I}{\text{d}t}=-k\cdot I\left(t\right)\cdot V\left(t\right)-{\text{α}}_I\cdot I\left(t\right)+b_I$$ (1)

$$\frac{\text{d}V}{\text{d}t}=-k\cdot I\left(t\right)\cdot V\left(t\right)-{\text{α}}_{\text{V}}\cdot V\left(t\right)+b_{\text{V}}+\text{σ}\cdot F_2$$ (2)

$$\frac{\text{d}R}{\text{d}t}=k\cdot I\left(t\right)\cdot V\left(t\right)-\text{γ}\cdot R\left(t\right)$$ (3)

$$\frac{\text{d}D}{\text{d}t}=\text{γ}\cdot R\left(t\right)-k_1^{+}\cdot \left[C_T-D_{\text{IR}}\left(t\right)\right]\cdot D\left(t\right)+k_1^{-}\cdot D_{\text{IR}}\left(t\right)$$ (4)

$$\frac{\text{d}{D}_{\text{IR}}}{\text{d}t}=k_1^{+}\cdot \left[C_T-D_{\text{IR}}\left(t\right)\right]\cdot D\left(t\right)-k_1^{-}\cdot D_{\text{IR}}\left(t\right)-\text{ρ}\cdot D_{\text{IR}}\left(t\right)$$ (5)

$$\frac{\text{d}F}{\text{d}t}=\text{ρ}\cdot D_{\text{IR}}\left(t\right)-\text{σ}\cdot F\left(t\right)$$ (6)

$$\frac{\text{d}{F}_2}{\text{d}t}=\frac{1}{{\text{τ}}_{\text{V}}}\cdot \left(F-F_2\right)$$ (7)

In Eq. 1, the dynamics of proinsulin I is described. Here, k is the segregation rate constant for granule membrane material, α_I is the degradation rate of proinsulin, and b_I is a baseline production rate of proinsulin, described in Eq. 8. Equation 2 describes the granule membrane material V. In this equation, α_V is a degradation rate of granule membranes, b_V is the rate of production of the granule membranes, σ is the rate constant for the fusion of granule to cellular membrane. In the same equation, F₂ is the transit compartment added to F, the granules undergoing fusion with cellular membrane. The transit compartment F₂ was added to account for the membrane material recycling time τ_V in the original model and is presented in Eq. 7. In Eq. 3 that describes the insulin reserve pool R, γ is a rate constant that is dependent on the glucose concentration G and represents a main rate-limiting step in the response to glucose (described in Eq. 11 later). In Eqs. 4 and 5, the docked granules D and immediately releasable granules D_IR are represented. In these equations, C_T is the constant pool of total Ca²⁺ channels, and C_T-D_IR represents the pool of unbound Ca²⁺ channels represented by C in Fig. 1. Parameters $k_1^{+}$ and $k_1^{-}$ are the association and dissociation rate constants, and ρ is the rate coefficient that accounts for the factors that promote the fusion of granules with cell membrane (described in Eq. 14 later). Eq. 6 describes the pool of granule being fused with cellular membrane. The values of the rate constants in the above equations were retained from the original model (4) and are listed in Table 2.

Table 2.

Parameter values and descriptions

No.	Parameter	Value	Units	Description/Reference
1	k	0.01	per min	Rate constant of formation of proinsulin-containing granules (4)
2	αI	0.3	per min	Rate constant of degradation of proinsulin aggregates (4)
3	αV	0.6	per min	Rate constant of degradation of granule membrane material (4)
4	Σ	30	per min	Rate constant of insulin release from granules fused with cell membrane (4)
5	kIb	0.262		Constant describing the period of bV (4)
6	kIc	10		Constant describing the period and horizontal shift of bV (4)
7	kI1	10	nM−1·min−1	Activation rate for bV
8	kI2	9		Deactivation rate for bV
9	kI3	2.3	(l/mmol)3	Parameter for bI
10	kI4	4	(l/mmol)3	Parameter for bI
11	kI5	60	per min	Parameter for bI
12	kIn	3		Hill coefficient for bI
13	γ	1e-4	per min	Rate constant of granule externalization and priming related to ATP/ADP (4)
14	k1+	5.788e-5	per min	Rate constant of association for the binding between granule and Ca2+ channel (4)
15	k1–	0.255	per min	Rate constant of dissociation for the binding between granule and Ca2+ channel (4)
16	CT	300		Constant pool of total Ca2+ channels (4)
17	τV	5	min	Time delay of recycling of granule membrane material (4)
18	η	4	per min	Rate constant for γ (4)
19	γb	1e-4	per min	Basal value of γ (4)
20	τG	1	per h	Time delay for increase in glucose (4)
21	G*	4.58	mmol/l	Glucose concentration threshold for the activation of γ (4)
22	Ĝ	10	mmol/l	Glucose concentration over which hγ remains constant and equal to ĥ (4)
23	ĥ	3.93e-3	per min	Maximal value of hγ (4)
24	ζ	4	per min	Rate constant for ρ (4)
25	ρb	0.02	per min	Basal value of ρ (4)
26	kρ	350		Sensitivity of ρ on the activation of γ (4)
27	I0	1.6	amol	Insulin amount contained in a granule (4)
28	N	2.76e6		Total number of β cells in the pancreas (4)
29	fb	0.05		Basal value of the fraction f (4)
30	Kf	3.43	mmol/l	Parameter for f(G) (4)

$$b_I=k_{\text{I5}}\frac{k_{\text{I3}}\cdot G_2^{k_{\text{In}}}}{k_{\text{I4}}^{{\text{k}}_{\text{In}}}+G_2^{k_{\text{In}}}}$$ (8)

$$b_{\text{V}}=k_{\text{Ia}}\cdot \sin \left(k_{\text{Ib}}\cdot t-k_{\text{Ic}}\right)+k_{\text{Id}}$$ (9)

$$\frac{\text{d}{b}_{\text{V}}}{\text{d}t}=k_{\text{I1}}\cdot Bmal-k_{\text{I2}}\cdot b_{\text{V}}$$ (10)

In addition to the transit compartment for F, two important modifications were introduced to the model. Equation 8 was developed to make the model more realistic by modifying the proinsulin (I) synthesis rate. The synthesis rate b_I is modeled as a Hill function and is dependent on the glucose concentration around the β cells. G₂ is the transit compartment for smoothing G, the glucose concentration, later defined in Eq. 12. The parameters k_I3, k_I4, k_I5, and the Hill coefficient k_In were selected to fit the glucose-dependent proinsulin biosynthesis data from male Wistar rats (35). In Eqs. 9 and 10, the circadian oscillation of membrane material production is incorporated into the model.

The synthesis rate of V, or b_V, was simulated in two different ways. First, it was assumed to be a sinusoid that has a nadir at 4 AM and peak at 4 PM, with the parameters k_Ia, k_Ib, k_Ic, and k_Id describing the characteristics of the oscillation in Eq. 9. The nadir and peak of the oscillation were selected to follow the daily rhythm of Bmal1 mRNA (15, 42). Alternatively, it was modeled as an indirect response of Bmal1 mRNA in Eq. 10. The equation was constructed based on the observation that dysfunction of Bmal1 results in reduced secretion of insulin, as well as loss in rhythmicity of insulin secretion (33). The dynamics of Bmal1 mRNA is modeled by a previously built circadian model that takes in the light/dark cycle and feeding/fasting cycle as inputs and describes their intertwined effects on clock genes, cortisol, and gluconeogenesis (3). Although the model was developed to describe hepatic gluconeogenesis, the feeding entrainment of clock genes occurs ubiquitously in tissues with difference in its speed (11). Because we are interested in equilibrium behavior, we can use this model to describe the circadian oscillations in the pancreas. Briefly, the environmental signals are modeled as a step function that is either at on or off state, repeating every 24 h. The light/dark cycle is integrated through the hypothalamic-pituitary-adrenal (HPA) axis, which has a self-sustained oscillation entrained to the light signal. From the HPA axis, cortisol is secreted to the peripheral compartment and modulates the transcription of Per and Cry genes. The PER/CRY protein complex indirectly activates the transcription of Bmal1 while also inhibiting the CLOCK/BMAL1 heterocomplex from activating the transcription of Per and Cry genes. As described here, Bmal1 takes a core part in complex positive and negative feedback loops together with other clock genes. The feeding/fasting cycle is incorporated into the model via NAD⁺/NADH ratio, which influences the activation of SIRT1, a histone deacetylase that closely interacts with the clock genes. For a more complete description of the clock model, we guide the readers to Ref. 3. Relevant equations are available in the online repository.

$$\frac{\text{dγ}}{\text{d}t}=\text{η}\cdot \left\{-\text{γ}\left(t\right)+{\text{γ}}_b+h_{\text{γ}}\left[G_2\right]\right\}$$ (11)

$$\frac{\text{d}{G}_2}{\text{d}t}=\frac{1}{{\text{τ}}_G}\cdot \left(G-G_2\right)$$ (12)

$$h_{\text{γ}}\left(G\right)=\{\begin{array}{cc}0,& G\le G^{*}\\ \frac{\widehat{h}\left(G-G^{*}\right)}{\widehat{G}-G^{*}},& G^{*}<G\le \widehat{G}\\ \widehat{h},& G>\widehat{G}\end{array}$$ (13)

$$\frac{\text{ρ}}{\text{d}t}=\text{ζ}\cdot \left\{\text{ρ}\left(t\right)+{\text{ρ}}_b+h_{\rho }\left[\text{γ}\left(t\right)\right]\right\}$$ (14)

$$h_{\text{ρ}}\left(\text{γ}\right)=\hspace{0.17em}\{\begin{array}{cc}0,& {\text{γ\hspace{0.17em}<\hspace{0.17em}γ}}_b\\ k_{\text{ρ}}\cdot \left(\text{γ\hspace{0.17em}}-{\text{\hspace{0.17em}γ}}_b\right),& \text{γ}\ge {\text{γ}}_b\end{array}$$ (15)

Equations 11–15 describe the triggering of insulin secretion by glucose stimulation, adapted from Bertuzzi’s model (4). In Eq. 11, γ represents the rate coefficient of granule externalization, related to the ratio ATP/ADP, previously shown in Eq. 4 to describe the docked pool. In the same equation, η is a rate constant, and γ_b is the basal value of γ at low glucose. Equation 12 is a transit compartment for smoothing the glucose concentration. Here, τ_G, the time required for glucose metabolism, was selected to replicate the daily glucose concentration profile in healthy subjects (13). The glucose activation model h_γ is described by Eq. 13, where ĥ is the maximal value of h_γ and is reached at G = Ĝ. Below the threshold of G*, h_γ is 0, and between G* and Ĝ, the function increases linearly. Equation 14 describes the behavior of ρ, the rate coefficient of granule fusion with cell membrane, determined by the calcium concentration. In Eq. 14, ζ is a rate constant, and ρ_b is the basal value of ρ. The function h_ρ, which embodies the action of ATP on Ca²⁺, or the dependence of ρ on γ, is described in Eq. 15. If γ is less than the basal value γ_b, h_ρ is 0. Otherwise, h_ρ increases linearly with γ.

$$\text{ISR}\left(t\right)=I_0\cdot \text{σ}\cdot F\left(t\right)\cdot f\left(t\right)\cdot N$$ (16)

$$f\left(G\right)=\{\begin{array}{cc}{f}_b,& G<G^{*}\\ f_b+\left(1-f_b\right)\frac{G-G^{*}}{K_f+G-G^{*}},& G\ge G^{*}\end{array}$$ (17)

The insulin secretion rate (ISR) is defined in Eq. 16, where I₀ is the insulin amount contained in granule, and N is the total number of β cells in the pancreas. These two factors together with the granules being fused (F) and the rate of fusion (σ) to the cellular membrane, determine the ISR. The fraction of the total cell population, which responds to glucose, f, is defined in Eq. 17. Here, f_b is a basal value for glucose concentration less than G*, and K_f is a constant related to effectiveness of recruitment.

All simulations were performed with Matlab R2017a. Source codes and supplemental information are available in an online repository (https://github.com/AndroulakisGrp/SB_Insulin).

Sensitivity analysis.

We try to understand the effect of changes in oscillatory behavior of the system to the insulin secretion in multiple ways. First, we impose a sinusoid (Eq. 9) on b_V and examine the effect of its amplitude and vertical shift on ISR peak heights and area under the curve (AUC) of ISR over a 24-h period, when multiple glucose pulse stimulations occur in a day simulating three meals.

As for the case where light/feeding model is incorporated into the insulin granule model, much larger parameters from the light/feeding model affect the oscillation of b_V, through a complex network of transcription and translation of clock genes. To understand the effect of light/feeding model parameters on ISR, we employed the Morris method. The method computes and uses elementary effects or the changes occurring in an output attributable to changes in a particular input (24). The elementary effects (EE) were calculated with the following Eq. 6 for k = 20 selected peripheral parameters of the light/feeding model.

$${\text{EE}}_i=\frac{Y\left(x_1,\dots \hspace{0.17em},\hspace{0.17em}{x}_{i-1},x_i+{\text{Δ}}_i,x_{i+1},x_k\right)-Y\left(x_1,\dots \hspace{0.17em},x_k\right)}{{\text{Δ}}_i}$$ (18)

The 20 parameters of interest were selected because they appeared to have the highest effect on hepatic gluconeogenesis in our previous work (3). The list of these parameters, their descriptions, and nominal values are available in Supplemental Table 1 in the online repository. In the above equation, Y is a deterministic function of k parameters, which was the 24-h AUC of ISR in this study. Each input parameter x_i is scaled from 0 to 1, and the right quantity is divided by Δ_i, which is a value in {1/(p − 1),…,1 − 1/(p − 1)}, p representing the number of levels. The mean (Eq. 19) and variance (Eq. 20) of the elementary effects are considered together in ranking the parameters in the order of importance.

$${\text{μ}}_{\text{i}}=\frac{1}{r}{\displaystyle \sum }_{j=1}^r{\text{EE}}_i^j$$ (19)

$${\text{σ}}_{\text{i}}^2=\frac{1}{r-1}{\displaystyle \sum }_{j=1}^r{\left({\text{EE}}_i^j-\text{μ}\right)}^2$$ (20)

Sobol’s quasirandom numbers (37, 38) were used to generate radial sampling points between ± 20% of the nominal values, adopted from the methods presented previously (6). The repetition r = 60 was used for analyzing our model as suggested previously (31). With the use of this method, the mean and variance of the elementary effects for the peripheral parameters were considered together to select the important parameters.

RESULTS

Effect of circadian rhythms on ISR.

Our insulin secretion model aims to depict the dependence of glucose-stimulated insulin response on circadian time. To achieve this goal, we first tried to investigate the effect of imposing an oscillating production of granule membrane material V on the insulin granule dynamics under glucose stimulation. We first observed the system response for pulse glucose stimulations at different times of the day and reported the results in Fig. 2, A and B. The amplitude of oscillation of b_V (k_Ia) in Eq. 9 was set to 3, and the average level of the vertical shift (k_Id) was set to 6. The values of the rest of the parameters are shown in Table 2. The resulting V profile has a nadir at 4 AM and a peak at 4 PM, 2 h before the starting and ending time of the light (active) period, to simulate the effect of Bmal1 mRNA, which peaks in the afternoon. The system was first stabilized at a glucose concentration of 1 mmol/l for 24 h, below the glucose activation threshold of G*. Then a 1-h pulse at 16.7 mmol/l was applied at different times of the day in 2-h increments, returning to 1 mmol/l immediately after the pulse. This is intended to simulate sudden availability of nutrients (such as taking in a meal). The concentration of 16.7 mmol/l is the higher limit for glucose concentration in the original insulin granule trafficking model (4), selected because it corresponds to the higher glucose activation threshold for insulin packet model in Grodsky’s data (16). Example profiles of ISR, R, and D are shown in Fig. 2A. In Fig. 2B, the AUCs of ISR resulting from every glucose pulse were calculated and compared among different pulse start times. The baseline for percentage of change is the AUC of ISR with glucose pulse starting at 6:00 AM. From this figure, it is evident that identical pulse glucose stimulations at different times of the day can cause differences in ISR. ISR is the greatest if the pulse is applied at 6:00 PM, near the peak of b_V. Once glucose concentration is returned to 1 mmol/l, ISR also becomes 0. However, lasting effects of glucose stimulation are observed after pulse time in intermediates such as R and D (Supplemental Fig. 1 in the online repository).

Fig. 2.

A: sample simulation results for pulse glucose stimulation. The time profiles for insulin secretion rate (ISR) (μg/h), reserve pool (R), and pool of docked granules (D) are shown for the case of 1-h glucose pulse at 6:00 AM. R and D are in arbitrary units on the right-side axis. B: % difference in the area under the curve (AUC) of ISR resulting from each glucose pulse is plotted against the pulse start time in circadian hours. The baseline for % difference is the AUC of ISR when pulse start time is at 6:00 AM. C: sample simulation results for glucose infusion. The time profiles for ISR (μg/h), R, and D are shown for the case of glucose infusion starting at 6:00 AM. R and D are in arbitrary units on the right-side axis. D: % difference AUC of ISR (μg/h) over the firstst 24 h after each glucose infusion is plotted against the infusion start time in circadian hours. The basis for comparison in % change is the AUC of ISR when infusion starts at 6:00 AM.

We next wanted to determine whether the circadian dependence on ISR is observed for continuous glucose stimulation rather than pulse stimulations. Although it does not have a physiological meaning, continuous infusion simulation is still useful in determining model behavior at the extreme condition. Glucose concentration was initially set to 1 mmol/l for the first 24 h, then was raised to 16.7 mmol/l for the next 24 h. The simulation was repeated 12 times with infusion starting at different times of the day at 2-h increments. In Fig. 2C, a sample model simulation using continuous glucose infusion started at 6:00 AM is shown. The intermediates R and D oscillate with very low amplitude before glucose stimulation (Supplemental Fig. 2). Once glucose infusion is started, the amplitude of R and D increase, and insulin secretion begins. After the initial 24 h into glucose infusion (by day 3), the system reaches an equilibrium and all profiles oscillate with an identical amplitude and phase regardless of when glucose infusion started. However, the transient behavior for the initial 24 h into glucose infusion depends on stimulation start time. For example, the AUC of ISR over the first 24 h depends on the circadian time of infusion start (Fig. 2D). The percentage of change was based on the 24-h AUC of ISR when the glucose infusion is started at 6:00 AM.

Simulating multiple glucose stimulations in a day.

To test the system under a more realistic lifestyle pattern, we simulated three glucose pulses intending to replicate three meals. For every 24-h day, three 1-h pulses at 16.7 mmol/l are given to the system. The start time of the first pulse and the end time of the third pulse are separated by 12 h; therefore, each 1-h pulse is separated by 4.5 h of rest, in which glucose concentration drops to 1 mmol/l. The remaining 12 h in the 24-h day experience the glucose concentration of 1 mmol/l. The three pulses/day glucose stimulation pattern results in three distinct ISR peaks for each 24-h period. A sample ISR profile is shown in Supplemental Fig. 2. The three pulses/day pattern was simulated for 10 days to allow the system to reach an equilibrium, and only the data from the tenth day were further analyzed. To assess the effect on circadian rhythms on this glucose pattern, the system was simulated at multiple first pulse start times. Furthermore, we performed a parametric analysis on b_V to understand how the oscillatory characteristics influence glucose-stimulated insulin response. The three pulses/day pattern was simulated for b_V oscillation with various amplitudes and vertical shifts (average levels).

In Fig. 3, the percentage differences in ISR peak heights for individual pulses from the three pulses/day glucose stimulation pattern are plotted. The calculation was based on the first ISR peak when first glucose pulse is given at 6:00 AM, when b_V oscillation amplitude (k_1a) is equal to 3. We observe that the ISR peak level is higher for later pulses in most cases. If the glucose stimulation is started during the active period (between 6:00 AM and 6:00 PM), the first peak is the lowest and last peak is the highest. Interestingly, the difference between the first and second peaks is larger than the difference between second and third peaks, a phenomenon observed in human subjects (32). The increased insulin secretion for later peak under identical caloric intake was also confirmed by a study that provided identical meals in 6- or 12-h increments (41). The differences among ISR peaks within a day is greater if the first pulse is started during the light phase. If the glucose stimulation is started during the dark phase, the peak differences are reduced, especially between second and third peaks. Additionally, peak differences are greater if b_V rhythm has a higher amplitude (Fig. 3, B and C). However, for low-amplitude rhythms (Fig. 3A), the peak differences are consistent regardless of start time.

Fig. 3.

Relative insulin secretion rate (ISR) peak values for the 3 pulses. First peaks are depicted as squares, second peaks as circles, and third peaks as triangles. Data for low-amplitude rhythm (A), medium-amplitude rhythm (B), and high-amplitude rhythm (C) are shown separately. Within each figure, data for rhythms with low (open green), medium (hatched yellow), and high (filled red) vertical shifts are shown next to one another.

We performed a similar analysis on the effect of oscillatory characteristics of b_V on total insulin secretion over a 24-h period under three pulses/day glucose stimulation pattern and showed the results in Supplemental Fig. 3 in the online repository. In this figure, the AUC of ISR on the tenth day was calculated for each simulation, and the results were plotted against the first pulse start time. For each vertical shift, three different amplitudes of b_V were tested. In examination of the data together, a higher level of b_V leads to higher AUC of ISR, whereas greater amplitude of b_V leads to greater differences in AUC of ISR depending on the glucose stimulation start time. For all cases, the most amount of insulin is secreted if the first pulse is started at 10 AM, when the slope of b_V is positive.

We next tested whether having pulses of different height throughout a three-pulse day can have an effect on the overall insulin secretion and showed the results in Figs. 4 and 5. For the “small dinner” pattern, three glucose pulses occur during the 12-h active phase, starting with a 1-h pulse at 16.7 mmol/l. The second pulse is a 1-h pulse glucose stimulation at 75% of the first pulse concentration, and the third pulse is 1-h pulse at 50% of the first pulse concentration. For the “small breakfast” pattern, the pulses appear in the reverse order. For the “three equal meals” pattern, all pulses are at 75% of the maximum concentration. Therefore, for all three glucose stimulation patterns, the AUC of glucose concentration over a 24-h day remained identical. The system was allowed to reach an equilibrium, and the AUC of ISR on the tenth day was calculated and plotted in Fig. 6. To show the relative difference attributable to pulse start time, the data were normalized to the AUC of ISR for three equal meals when the first pulse is started at 6:00 AM. Our model predicts that, compared with three equal meals, small dinner meal pattern will produce less insulin if meals are started during late night morning time. In contrast, big dinner meal pattern will produce less insulin than the three equal meals in almost all cases although the difference between the two during the late night phase is small.

Fig. 4.

% Change in area under the curve (AUC) of insulin secretion rate (ISR) over a 24-h period for different glucose pulse size distributions. % Changes were calculated based on the AUC of ISR for 3 equal meals when the first pulse is started at 6:00 AM.

Fig. 5.

Relative insulin secretion rate (ISR) peak values for different pulse size distributions. % Changes were calculated in reference to the value of the first peak for 3 equal meals, when first meal is started at 6:00 AM.

Fig. 6.

The light/dark cycle and feeding/fasting cycle drive the oscillation of granule membrane synthesis rate (b_V). The area under the curve (AUC) of insulin secretion rate (ISR) (μg) is shown with amplitude of Bmal1 mRNA (A). The individual peak heights are shown in B, and the effect of constant light is shown in filled gray, denoted as LL.

The relative ISR peak values throughout the day depending on the pulse size distribution are presented in Fig. 5. For comparison, the peak values are normalized to the value of first ISR peak resulting from three equal meals starting at 6:00 AM. As described previously, providing three equal glucose pulses causes ISR peak level to increase toward the end of the day. For the big dinner pattern, the peaks still increase toward the later meals, with smaller first peaks and higher third peaks. For the small dinner pattern, peak values decrease toward the end of the day. Additionally, the difference between the second and third peaks is smaller than the difference between the first and second peaks for the three equal meals and small dinner pattern; however, this observation is not true for the big dinner meal pattern.

Effect of phase relationship between light and feeding periods on ISR.

To evaluate the role of light/feeding phase relations on insulin secretion, the circadian clock model from our previous work (3) was incorporated into the insulin model. The production rate of V was expressed as an indirect response from Bmal1 mRNA, as shown in Eq. 10. We first tested the model with three pulses/day glucose stimulation pattern, which consists of three 1-h pulses at 16.7 mmol/l spread out over 12 h, followed by 12 h of rest (1 mmol/l). When glucose stimulation was on, feeding signal was set to 4, so that the AUC of feeding matches the AUC of light, which was on at 1 from 6:00 AM to 6:00 PM. This set up was tested with the first meal starting at different times of the day in 2-h increments. The resulting change in the AUC of ISR on the tenth day is shown in Fig. 6A, comparing the reference AUC of ISR where the first pulse starts at 6:00 AM. On the same plot, the amplitude of Bmal1 mRNA is shown on the right-side axis. The increase in Bmal1 oscillation amplitude results in higher ISR, resulting in highest insulin secretion if the glucose stimulation starts at 8:00 AM, 2 h after the light phase starts. In Fig. 6B, the ISR peak heights are plotted, normalized to the first peak value where the first meal starts at 6:00 AM. We applied a constant light condition (LL, light is set to 1 constitutively) to test the effect of circadian disruption on ISR. The results show that insulin secretion is reduced if the system experiences constant light, consistent with data from rodents exposed to bright light (10, 29).

Figure 7 shows the effects of glucose pulse size distribution in the circadian insulin model. For the small dinner pattern, the first 1-h pulse was set at feeding signal of 4 and glucose concentration of 16.7 mmol/l. The second 1-h pulse was set to 75% of the first signals (glucose concentration and feeding signal), and the third 1-h pulse was set to 50% of the first signals. For the big dinner meal pattern, the pulses appeared in the reverse order. For three equal meals, all pulses were set at 75% of the maximum level to achieve identical AUC of feeding and glucose stimulation over a 24-h period. Light was on at 1 from 6:00 AM to 6:00 PM. For all glucose stimulation patterns, the system was allowed to reach an equilibrium, and the AUC over ISR on the tenth day was calculated and plotted on Fig. 7, along with the AUC of granule membrane V. For all patterns, AUC of ISR is the greatest when AUC of V is the greatest. For three equal meals and small dinner, insulin secretion is greatest if pulses start at 8:00 AM. However, for big dinner pattern, insulin secretion peaks if meals start at 10:00 AM. Additionally, two equal pulses result in the highest insulin secretion regardless of the start time. Big dinner pattern results in the lowest insulin secretion.

Fig. 7.

The effect of pulse size distribution and time on area under the curve (AUC) of insulin secretion rate (ISR) is shown for the insulin model in which granule membrane synthesis rate (b_V) is driven by light/dark cycle and feeding/fasting cycle. The % changes are calculated in reference to the AUC of ISR for 3 equal meals, in which the 1st pulse starts at 6 AM. AUCs of V (pool of granule membrane material) are plotted on the right-side axis in arbitrary units.

The results from sensitivity analysis of light/feeding parameters using the Morris method are presented in Fig. 8, plotting variance against μ*. The most sensitive parameters in terms of both primary and interaction effects are k_1i (inhibition coefficient for Per/Cry gene transcription) and k_4b (Michaelis constant for Bmal1 transcription).

Fig. 8.

Results from sensitivity analysis using Morris method performed on the 24-h area under the curve (AUC) of insulin secretion rate (ISR). 60 repetitions of sampling were used to generate this figure. The variance (σ²) is plotted against μ*.

DISCUSSION

Evidence for circadian secretion of insulin exists at multiple levels, from tissue cultures of rat pancreatic islets (27) to healthy human volunteers (5). RNA sequencing of human pancreatic islet cells with knockout of clock genes suggests that insulin secretion, rather than production, is under circadian control (34). As such, we have developed a model to study the circadian rhythmicity in glucose-stimulated insulin secretion, incorporating the dynamics of granule maturation and release. Previous models of insulin release successfully explain the insulin granule dynamics but do not address the effect of clock function (4, 25, 26). A more recent work utilized a Petri net model to explain the effect of sleep deprivation on glucose-stimulated insulin secretion (1). In this work, the degree of sleep deprivation was represented by altering the parameter set that governs the transcription and translation of core clock genes. Our contribution is to test specific oscillatory characteristics of the clock genes on glucose-stimulated insulin secretion and using the results to explain alterations to insulin secretion for various glucose stimulation patterns relative to the light/dark cycle.

Although our model is not optimized for quantitatively replicating experimentally or clinically observed data for glucose-stimulated insulin secretion, it aims to capture some key qualitative features in ISR during a 24-h day, giving confidence in the strategy for linking the clock gene oscillation to granule membrane material. Although the absolute effect size (i.e., ISR) for individual simulation is not physiologically meaningful, particular attention was paid to the relative differences among ISRs responding to glucose stimulations at different times of the day. Using this model, we investigated how the oscillatory characteristics of circadian rhythms can influence the time of the day dependence on insulin secretion rate. Furthermore, we studied how meal size distribution can change the insulin secretion rate throughout a 24-h day when the granule dynamics are under circadian control. Finally, we use these results to interpret the ISR resulting from the clock-integrated model to study the role of the light/dark cycle, feeding/fasting cycle, and their phase relations in glucose-stimulated insulin secretion.

A simple glucose pulse or a continuous infusion proved useful in demonstrating the circadian dependence of ISR, as shown in Fig. 2. In the case of pulse stimulation, higher ISR was achieved when the oscillatory component of the model (b_V) is near the peak. For continuous infusion of glucose, the ISR profiles show different transient behaviors depending on the infusion start time; however, all ISR profiles reach identical equilibrium within 24 h of infusion start. Although these two simple tests successfully demonstrated that glucose-stimulated insulin secretion can change at different times of the day, they were not successful in capturing the lasting effects of glucose stimulation on ISR. The levels of intermediates, such as the reserve pool (R) and docked pool (D), did not reach an equilibrium past the pulse duration or past 24 h of infusion. Repeated three pulses/day pattern can capture the lasting effects of glucose stimulation. However, it should be noted that the three pulses/day pattern used in this work is three 1-h pulses of glucose stimulation, which is not a physiologically accurate representation of actual meals. In reality, the varying level of ISR depending on the time of the day will have an impact on the glucose level, which is not presently accounted for. This is a limitation of our model that should be addressed in future work.

As shown in Figs. 4 and 6B, the model predicts that the first ISR peak of the day is the lowest peak of the day. The second and third peaks are higher than the first peak in most cases, consistent with experimental data (32), except for the case in which the granule membrane synthesis rate (b_V) oscillates with a high amplitude and low level and meals are started during the middle of the dark phase. Because we observe that this peak pattern appears regardless of when the first glucose pulse of the day is started, we can deduce that the increase in peak level for later meals is not due to the oscillations in b_V. Instead, the increase is due to the lasting effects of the meal on the granule dynamics. Supplemental Fig. 1 in the online repository shows that, although ISR returns to 0 immediately after the glucose pulse is over, the increased levels of the reserved pool (R) and docked granules (D) remain at higher levels, confirming our hypothesis. Additionally, the increase in levels of these intermediates is higher when b_V level is increased. Highest ISR peaks and AUC of ISR are achieved when b_V has a steep positive slope rather than at the peak of b_V. Another noteworthy observation is that, for the clock-integrated model (Fig. 6B), the ISR peak orders within a 24-h day for all meal start times are identical. Although this pattern looks similar to Fig. 3A in which b_V oscillation amplitude is low, it is unlikely that amplitude is the reason for consistent peak orders. Instead, the combined effect of light and feeding signals in the clock model causes phase shift of clock genes, eventually entraining the clock rhythms to feeding periods. Therefore, the oscillation phase of b_V, which follows the Bmal1 rhythm, is changed to follow the meal start time, allowing the first meal to occur near the trough regardless of the circadian start time. As observed in Fig. 3, the peak orders are conserved if the first meal starts during the active phase at every amplitude and vertical shift of b_V. This shows that overall insulin secretion over a 24-h day is dependent on the oscillatory characteristics of b_V. We observe a vertical shift in the positive direction for the AUC of ISR as the average level of b_V is increased. If the average level of b_V is identical but the amplitude of oscillation is increased, AUC of ISR also varies in greater magnitude depending on when the meals are started.

Figures 4, 5, and 7 show the effects of unequal meal sizes on ISR. Figure 7 shows that, when the clock model is integrated, maximum ISR is achieved if three equal glucose pulses are presented to the system. Having a big dinner and small breakfast will result in the lowest ISR, and the ISR from combination of small dinner and big breakfast will be in the middle. Additionally, the glucose pulse start time that results in the greatest ISR for small dinner is the earliest, followed by three equal meals, and then big dinner. The same glucose pulse start times also result in the greatest AUC of V for each meal pattern, as represented in the dashed lines in the same figure. This pattern is consistent in Fig. 4, when b_V is modeled as a sinusoid rather than an indirect response from the clock, suggesting that shift in peak time for ISR is a function of meal size distribution rather than changes in b_V oscillations. A simple explanation is that, if the third glucose pulse is higher than the first, an earlier glucose stimulation start time would allow the glucose pulses to appear when the slope of b_V is positive.

Figure 5 can provide more insight into the differences in ISR resulting from pulse size distributions. For three equal meals, ISR peaks increase toward the later pulses, but the three peaks are close together compared with the other two glucose stimulation patterns. In the case of big dinner, the peaks still increase toward the end of the day, but the differences between the peaks are greater compared with the equal meals. In the case of small dinner, the peaks decrease toward the end of the day. Examining Fig. 7 more closely helps elucidate the reason for observing lower AUC of ISR for big dinner meal pattern in Fig. 5. In Fig. 7, the second peaks for small dinner are always higher than the second peaks for big dinner although the meal sizes are identical. The above relationship occurs because a larger breakfast will cause the reserve pool (R) to be at a higher level when the system receives the lunch pulse. Because granule levels are already more elevated at the time of lunch, identical stimulation results in greater ISR for the small dinner case.

Perspectives and Significance

Our model shows that glucose-stimulated insulin release from the pancreatic β cells can change because of the oscillatory characteristics of circadian rhythms in insulin granule formation and fusion. Although ISR is elevated for later meals of the day, this pattern can be altered depending on the oscillation amplitude and level of granule membrane material. Furthermore, our model suggests that distributing the glucose stimulation differently throughout the day can alter the level of ISR, which can be helpful in developing a meal plan for patients suffering from metabolic diseases. Of course such a prediction would require a disease model to be first established. As an interesting future work, one approach is to induce a low-amplitude, low-level circadian oscillation by knockdown of a key clock gene or shining a constant light, then trying to rescue the ISR with a behavioral change involving meal time and size. Because previous works suggest that this will result in low-amplitude, low-level oscillation for the clock components, we can expect subdued rhythms for b_V and V as well. Results in Figs. 3 and 7 suggest that disrupted clock result in lower level of insulin secretion. The model can be further improved by incorporating the influence of insulin on clock genes and glucose level for a more accurate description of the interaction between insulin and circadian clocks.

GRANTS

This work was supported by the National Science Foundation Graduate Research Fellowship under grant number DGE-1433187 and the National Institute of General Medical Sciences of the National Institutes of Health under award numbers T32 GM008339 and GM024211.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

S.-A.B. and I.P.A. conceived and designed research; S.-A.B. and I.P.A. analyzed data; S.-A.B. and I.P.A. interpreted results of experiments; S.-A.B. prepared figures; S.-A.B. drafted manuscript; S.-A.B. and I.P.A. edited and revised manuscript; S.-A.B. and I.P.A. approved final version of manuscript.