Mathematical modeling of white adipocyte exocytosis predicts adiponectin secretion and quantifies the rates of vesicle exo- and endocytosis

Adiponectin is a hormone secreted from white adipocytes and takes part in the regulation of several metabolic processes. Although the pathophysiological importance of adiponectin has been thoroughly investigated, the mechanisms controlling its release are only partly understood. We have recently shown that adiponectin is secreted via regulated exocytosis of adiponectin-containing vesicles, that adiponectin exocytosis is stimulated by cAMP-dependent mechanisms, and that Ca2+ and ATP augment the cAMP-triggered secretion. However, much remains to be discovered regarding the molecular and cellular regulation of adiponectin release. Here, we have used mathematical modeling to extract detailed information contained within our previously obtained high-resolution patch-clamp time-resolved capacitance recordings to produce the first model of adiponectin exocytosis/secretion that combines all mechanistic knowledge deduced from electrophysiological experimental series. This model demonstrates that our previous understanding of the role of intracellular ATP in the control of adiponectin exocytosis needs to be revised to include an additional ATP-dependent step. Validation of the model by introduction of data of secreted adiponectin yielded a very close resemblance between the simulations and experimental results. Moreover, we could show that Ca2+-dependent adiponectin endocytosis contributes to the measured capacitance signal, and we were able to predict the contribution of endocytosis to the measured exocytotic rate under different experimental conditions. In conclusion, using mathematical modeling of published and newly generated data, we have obtained estimates of adiponectin exo- and endocytosis rates, and we have predicted adiponectin secretion. We believe that our model should have multiple applications in the study of metabolic processes and hormonal control thereof.

Adiponectin is a hormone secreted from white adipocytes and takes part in the regulation of several metabolic processes. Although the pathophysiological importance of adiponectin has been thoroughly investigated, the mechanisms controlling its release are only partly understood. We have recently shown that adiponectin is secreted via regulated exocytosis of adiponectincontaining vesicles, that adiponectin exocytosis is stimulated by cAMP-dependent mechanisms, and that Ca 2؉ and ATP augment the cAMP-triggered secretion. However, much remains to be discovered regarding the molecular and cellular regulation of adiponectin release. Here, we have used mathematical modeling to extract detailed information contained within our previously obtained high-resolution patch-clamp time-resolved capacitance recordings to produce the first model of adiponectin exocytosis/secretion that combines all mechanistic knowledge deduced from electrophysiological experimental series. This model demonstrates that our previous understanding of the role of intracellular ATP in the control of adiponectin exocytosis needs to be revised to include an additional ATP-dependent step. Validation of the model by introduction of data of secreted adiponectin yielded a very close resemblance between the simulations and experimental results. Moreover, we could show that Ca 2؉ -dependent adiponectin endocytosis contributes to the measured capacitance signal, and we were able to predict the contribution of endocytosis to the measured exocytotic rate under different experimental conditions. In conclusion, using mathematical modeling of published and newly generated data, we have obtained estimates of adiponectin exo-and endocytosis rates, and we have predicted adiponectin secretion. We believe that our model should have multiple applications in the study of metabolic processes and hormonal control thereof.
Adiponectin is a hormone exclusively secreted from white adipocytes with roles in regulation of glucose and lipid levels. Obesity and type 2 diabetes are associated with reduced circulating levels of adiponectin. Moreover, high adiponectin levels protect against development of type 2 diabetes (1). Although the pathophysiological relevance of adiponectin is clear, the molecular regulation of its release is inadequately studied. Similarly to other hormones (2), adiponectin has been proposed to be released via regulated exocytosis (3)(4)(5). However, the molecular mechanisms controlling adiponectin vesicle release are not fully resolved. Our previous work has demonstrated that adiponectin is released from an intracellular pool of adiponectincontaining vesicles (6,7). However, unlike most peptide/protein hormones (2), exocytosis of adiponectin is triggered by cAMP, and a combination of intracellular Ca 2ϩ and ATP potently augments the cAMP-stimulated secretion (6,7). Thus, the regulation of adiponectin secretion shows similarities with how hormone release is controlled in other endocrine cell types, although there are some fundamental differences (i.e. endocrine cell exocytosis is typically triggered by Ca 2ϩ , whereas cAMP has an amplifying effect on secretion) (2). Our results have clearly provided significant insight into the regulation of adiponectin exocytosis. However, the data contain information on mechanistic details that can only be fully retrieved using a mathematical modeling approach (8,9).
Several available mathematical models describe the regulation of Ca 2ϩ -triggered exocytosis in traditional endocrine cell types, such as insulin-secreting beta cells (10 -12) and catecholamine-releasing chromaffin cells (13)(14)(15). Additional models have defined disease mechanisms related to white adipocyte insulin resistance (16 -19). Clearly, a mathematical model specifically defining adiponectin exocytosis mechanisms is required to completely understand the mechanistic regulation of adiponectin release.  Here, we use modeling as a tool to qualitatively and quantitatively evaluate and revise our current cell physiological understanding of how adiponectin exocytosis is regulated. Our prevailing concept of regulatory mechanisms (Fig. 1A, adapted from Ref. 6) is updated in several steps resulting in a more detailed understanding of the adiponectin exocytosis process. Using the model to analyze data, we first show that ATP dependence of the functionally divided intracellular adiponectin vesicle pools needs to be formalized in a new way. We further show that endocytosis contributes to the measured capacitance signal, and we are able to quantify the endocytosis contribution. By refinement of the model in several steps, we finally provide a revised and validated model of the molecular and cellular regulation of white adipocyte adiponectin release.

Formal testing of hypotheses of adiponectin exocytosis regulatory mechanisms
To investigate the molecular and cellular mechanisms controlling adiponectin secretion, we have previously examined the role of the three intracellular mediators cAMP, Ca 2ϩ , and ATP in 3T3-L1 adipocyte adiponectin exocytosis (6). Those mediators are known to have important roles in regulation of (neuro)endocrine cell secretion (2). In Komai et al. (6), we performed whole-cell electrophysiological recordings of exocytosis by measuring the change in membrane capacitance, which corresponds to alterations of the cell-surface area (increases with exocytosis and decreases with endocytosis). Our key findings, confirmed in primary human subcutaneous adipocytes, were as follows: (i) adiponectin exocytosis is triggered by cAMP, and (ii) a combination of Ca 2ϩ and ATP potently augments the cAMP-stimulated adiponectin vesicle release (Fig.  1A). To formally test our previous interpretations of experimental data, we have here directly translated this hypothesis (Fig. 1A) into a system of ordinary differential equations (ODEs). 6 During the model development, all interactions were explicitly included, resulting in a model with three states corresponding to functionally diverse adiponectin vesicle pools (Fig.  1B): a reserve pool (Reserve), a releasable pool (Release), and a pool corresponding to vesicles undergoing exocytosis (Exocytosis). The original model contains 11 parameters that control the rate of the reactions between the states (see under "Experimental procedures"). These parameters are not possible to determine experimentally. We therefore used the time-series data in Ref. 6 with rates of exocytosis resulting from different inputs (i.e. perfusion of the interior of the patched cells with pipette solution containing or lacking cAMP, Ca 2ϩ , or ATP) to estimate parameter values. The pipette solutions contained the following: 1) a combination of cAMP, Ca 2ϩ , and ATP; 2) cAMP together with Ca 2ϩ ; 3) cAMP together with ATP; and 4) cAMP alone (Fig. 2, A-D). Despite extensive searches through the parameter space, we were unable to find agreement between the simulation of the original model (Fig. 2, A-D, solid gray lines) and experimental data (Fig. 2, A-D, black symbols Ϯ S.E.) that passed a 2 test. The model structure was therefore rejected (Exocytosis model -original, Table 1), and we concluded that the original model lacks important mechanisms and must be refined.

Refinement of the original hypothesis to include ATP-dependent replenishment of releasable vesicles
Analysis of the model simulations for the original hypothesis revealed that the dynamics of the cAMP, Ca 2ϩ , and ATP input data, with a high peak and slow decrease in exocytotic rate (⌬C/⌬t; Fig. 2A, gray line), were not possible to fit concurrently with the lower peak in the cAMP and Ca 2ϩ data set with ATP absent (Fig. 2B, gray line). This lack of agreement is due to that the refilling of the releasable vesicle pool in the original model

Modeling mechanisms of adiponectin exocytosis
depends on Ca 2ϩ alone, without a role for ATP (Fig. 1, A and B). As a consequence, releasable vesicles would be expected to be depleted at a faster rate after the high exocytosis peak attained in the presence of ATP. Fig. 2, E and F, shows model simulations of the remaining vesicles under the two conditions, as indicated. The observation that the releasable pool is rapidly depleted in the presence of ATP (Fig. 2E, gray lines) together with the lack of agreement in Fig. 2, A and B, indicates that ATP

Table 1 Summary of all model evaluations
The threshold for the 2 error for 95% significance is calculated using the number of data points (28 in the original data and 39 in the extended data) minus 1 normalization parameter minus the number of parameters in the model.

Modeling mechanisms of adiponectin exocytosis
is involved in vesicle replenishment. The original hypothesis was therefore modified to contain an ATP-dependent term affecting the transition of vesicles from the reserve pool to the releasable pool (Fig. 2G, indicated in blue). This adjustment of the model is not in conflict with our original observations in Ref. 6, but it merely represents another way to formalize the experimental findings. Inclusion of the ATP dependence resulted in improved agreement between model simulations and experimental data (Fig. 2, A-D, dashed blue lines). However, the model was still rejected by a 2 test (Exocytosis model -ATP, Table 1).
To quantify the improvement resulting from the addition of the ATP dependence, a likelihood ratio test was used to test the null-hypothesis that the models are equally good at describing the data. The extra parameter for the ATP dependence decreased the 2 error by 31 (from 62 to 31, Table 1). This should be compared with the threshold of four for 95% confidence and 1 degree of freedom (1 extra parameter added to the model), i.e. the agreement with data improved more than the threshold value, and the null-hypothesis was rejected. Thus, the added ATP dependence step significantly improved the fit between the model and recorded data.

Inclusion of the contribution of endocytosis to the measured signal significantly improves the model
A remaining qualitative shortcoming is the inability of the model to capture the experimentally attained 25% lower peak of exocytotic rate in response to a combination of cAMP and Ca 2ϩ (black symbols in Fig. 2B) compared with when exocytosis is triggered by cAMP alone (black symbols in Fig. 2D). This lower peak rate cannot be captured by a model structure, including exclusively positive effects of Ca 2ϩ on the measured signal. The membrane capacitance recordings denote the net result of cell exo-and endocytosis, and Ca 2ϩ has been shown to trigger both exo-and endocytosis in several peptide-secreting cell types (20 -24). A very high intracellular free concentration of Ca 2ϩ (Ն10 M) triggers endocytosis (i.e. it affects capacitance measurements negatively) in a variety of (neuro)endocrine cell types (23,24).
In Ref. 6, we showed that a non-stimulatory (cAMP-lacking) pipette solution containing 1.5 M free Ca 2ϩ and ATP did not induce significant endocytosis even in experiments lasting as long as 20 min. With a few example traces, we further demonstrated that infusion of adipocytes with a pipette solution containing Ն10 M free Ca 2ϩ (20 mM CaCl 2 in combination with 10 mM of the Ca 2ϩ chelator BAPTA) together with cAMP and ATP resulted in a lower exocytotic rate, interpreted as contribution of Ca 2ϩ -dependent endocytosis (6). Those findings suggest that, in white adipocytes, cAMP stimulation of adiponectin exocytosis is required in order for Ca 2ϩ -dependent endocytosis to occur.
Based on the above premises, we included a Ca 2ϩ -dependent negative effect of Ca 2ϩ on exocytosis in the form of endocytosis in the model (Fig. 3A). To experimentally quantify the effect of high Ca 2ϩ on white adipocyte exo-and endocytosis, we infused 3T3-L1 adipocytes with the cAMP-and ATP-containing pipette solution supplemented with Ն10 M free Ca 2ϩ . Exocytosis was stimulated at a peak rate of ⌬C/⌬t ϭ 14 Ϯ 1 fF/s (measured at 2 min; Fig. 3B), thus at a significantly lower rate than triggered by the same solution containing 1.5 M free Ca 2ϩ (⌬C/⌬t ϭ 21 Ϯ 7 fF/s; p Ͻ 0.01; Fig. 2A). Our results indicate that an unphysiologically high intracellular Ca 2ϩ concentration induces excessive endocytosis, manifested as a reduced measured exocytotic rate.
The positive effect of the increase from 1.5 to Ն10 M Ca 2ϩ does not display a linear relationship between Ca 2ϩ concentration and exocytotic response (Fig. 3B), and thus it needs to be saturated. Saturation was included in the model using a halfmaximal saturation constant (K m ) as shown.
Note that saturation was not required for comparison of the original data sets (Fig. 2, A-D) with a single free Ca 2ϩ concentration.
Testing of the endocytosis hypothesis confirmed agreement with data with high Ca 2ϩ (Fig. 3B, black symbols Ϯ S.E.) as well as data with 1.5 M Ca 2ϩ alone (Fig. 3C, black symbols Ϯ S.E.). The developed model was also conformable with previous data sets (Fig. 2, A-D), and we therefore gathered all parameters that passed a 2 test (Fig. 3, B-G, orange areas).

Validation of the Exocytosis model -ATP/endocytosis
To evaluate the model in Fig. 3A further, we used a new kind of experimental data consisting of accumulated adiponectin released into the media (measured by ELISA (6)). Intact 3T3-L1 adipocytes were incubated with a solution containing the cAMP-increasing agent forskolin in combination with the phosphodiesterase inhibitor IBMX (FSK/IBMX) under different intracellular Ca 2ϩ conditions (6). To ensure the agreement between the extracellular cAMP-elevating agents and infusion of cAMP via the patch pipette, we measured the effect of FSK/ IBMX on membrane capacitance. FSK/IBMX potently stimulated exocytosis, and the maximal ⌬C/⌬t averaged ϳ20 fF/s (Fig. 4). This rate is very similar to the exocytotic rate achieved upon infusion of cells with cAMP and 1.5 M free Ca 2ϩ (cf. Fig.  2A) thus justifying the usage of the secretion data for model validation.
The secretion data were attained using intact cells. Thus, cytoplasmic levels of intracellular mediators were, in contrast to the clamped conditions in the capacitance recordings, allowed to fluctuate. Consequently, the implementation of the secretion data in the adiponectin release model required an estimation of the intracellular levels of cAMP, ATP, and Ca 2ϩ under the new experimental conditions. By use of values in Refs. 6 (25)). In Komai et al. (6), we found Ca 2ϩ to increase 1.5-fold in the presence of FSK/IBMX. This combined information allowed us to limit the range of Ca 2ϩ concentrations in the model simulations.

Modeling mechanisms of adiponectin exocytosis
The basal cAMP levels in adipocytes are low due to spontaneous release of adenosine (26). Absolute concentrations of adipocyte intracellular cAMP have never been described, but Zu et al. (27) reports basal cAMP in primary rat adipocytes to be ϳ6 pmol/mg protein. However, it is difficult to estimate how this value translates into actual cytoplasmic concentration, as that depends on the volume of the fat-free mass and the protein content per cell. We therefore calculated the basal cAMP concentration in 3T3-L1 adipocytes from values reported in Ref. 6 and related those to the fat-free mass (measured by coherent anti-Stokes Raman scattering and multiphoton excitation fluorescence; Table 2). The average basal cAMP concentration was calculated to be 55 M. The lowest measured value was 2 M (Table 2), in the same range as concentrations reported in car-diomyocytes (1.2-1.3 M) (28,29). Our capacitance measurements show that 100 M cAMP stimulates adiponectin exocytosis (6). Thus, the basal intracellular cAMP concentration was allowed to range between 2 and 100 M when predicting the secretion experiments.
The intracellular ATP levels in fully differentiated 3T3-L1 adipocytes exposed to a physiological glucose concentration are unknown. Hresko et al. (30) have reported an ATP concentration of ϳ5 mM in 3T3-L1 adipocytes. However, this value was measured in cells exposed to high glucose (25 mM, likely

Modeling mechanisms of adiponectin exocytosis
resulting in enhanced ATP production) and is therefore difficult to compare with our experiments carried out at a basal (5 mM) glucose concentration. Due to lack of experimental constraints, and to avoid underestimation of the uncertainty of the prediction, we allowed for a wide variety of ATP levels in the model simulations (1 M to 1 mM) regardless of treatment conditions. The basal cytosolic ATP concentration has been reported to be 1 mM in clonal beta cells (31), thus in agreement with our range of cytosolic ATP.
We used the determined intervals of intracellular Ca 2ϩ , cAMP, and ATP, and an approximation of all found acceptable parameters to simulate the adiponectin accumulation after stimulation with FSK/IBMX alone, FSK/IBMX after pre-treatment with the Ca 2ϩ chelator BAPTA, and FSK/IBMX together with the Ca 2ϩ ionophore ionomycin (Fig. 5A, filled area). We compared the simulated 30-min time point (Fig. 5, A, black boxes, summarized in B) with real adiponectin secretion (Fig. 5C), measured as accumulated adiponectin in the media under the same experimental conditions (6). The simulations were found to closely resemble the data (cf. Fig. 5, B and C). To simulate the effect of ionomycin, we allowed the Ca 2ϩ concentration to increase at least 1.5 times with ionomycin, and maximally 10 times above basal levels with FSK/IBMX in combination with ionomycin.
To further validate the model, in a dynamic setting, we studied the time-resolved prediction of FSK/IBMX-stimulated adiponectin secretion at 15, 30, 45, and 60 min. The predictions with corresponding uncertainties show a clear no-linear behavior with saturation (Fig. 5D). We performed the corresponding experiments of FSK/IBMX-stimulated adiponectin secretion, measured at the simulated time points. As shown in Fig. 5E, there was a distinct resemblance between simulated and real data. It is noteworthy that although the model was developed using capacitance recordings restricted to a time period of 12 min, it is capable of simulating the dynamic adiponectin secretion at severalfold later time points.

Quantification of the contribution of Ca 2؉ -dependent endocytosis to the exocytosis measurements
Using the validated model, we predicted the contribution of endocytosis to the measured signal under different experimental conditions (Fig. 6A). To determine the interval of possible contribution using the uncertainty of data, we used two global optimization algorithms ("Experimental procedures") to collect different parameter sets (all yielding simulations in agreement with data within a 95% confidence interval) (8). The contribution of endocytosis to the measured signal was found to be as follows: (i) 21 Ϯ 12% with cAMP, 1.5 M Ca 2ϩ , and ATP; (ii) 29 Ϯ 13% with cAMP and 1.5 M Ca 2ϩ (ATP excluded); and (iii) 42 Ϯ 5% in experiments with Ն10 M Ca 2ϩ added together with cAMP and ATP (Fig. 6). As expected, the contribution of endocytosis to the capacitance measurements is most pronounced at exceedingly elevated Ca 2ϩ concentrations.

Discussion
Here, we use a combined experimental and mathematical modeling approach to investigate the mechanisms regulating white adipocyte adiponectin secretion. We present the first mathematical model of adiponectin exocytosis. By taking advantage of both previously published and newly generated data, we draw quantitative conclusions that would have been impossible to assess without modeling.

Ca 2؉ -dependent endocytosis ensues simultaneously with exocytosis
The final model equations can be translated back to a biological interaction graph that summarizes our findings (Fig. 7). We conclude that (i) the redistribution of vesicles from the Reserve pool to the Releasable pool includes an ATP-dependent component, and (ii) a negative effect of Ca 2ϩ , in the form of endocytosis, is needed to explain the capacitance measurements. The ability to quantify the contribution of endocytosis to recorded capacitance increases is essential for the interpretation of our adiponectin exocytosis studies. Importantly, the model presented here shows that we may underestimate the magnitude of adiponectin exocytosis in the presence of a physiologically relevant intracellular Ca 2ϩ concentration and that the potentiating effect of Ca 2ϩ on cAMP-stimulated adiponectin secretion is larger than displayed in our electrophysiological recordings (6). The finding that Ca 2ϩ -dependent endocytosis depends on the simultaneous presence of cAMP does not signify that endocytosis as such requires the presence of the cyclic nucleotide per se, but rather designates that endocytosis

Modeling mechanisms of adiponectin exocytosis
(plasma membrane retrieval) needs to be preceded by exocytosis (plasma membrane enlargement). The observation that the contribution of endocytosis is pronounced in the presence of Ն10 M free Ca 2ϩ (Fig. 6A) emphasizes the importance of performing the capacitance measurements under physiologically relevant conditions (thus in the intracellular presence of appropriate levels of mediators known to affect exocytosis). We found that the membrane retrieval was unaffected by intracellular ATP. This is in agreement with results in other peptidesecreting cell types such as rat corticotrophs (32) and mouse

Modeling mechanisms of adiponectin exocytosis
beta cells (33) where endocytosis has likewise been shown to proceed uniformly in the absence or presence of the nucleotide. Moreover, the lack of ATP dependence suggests that the endocytosis following cAMP-stimulated adiponectin secretion does not depend on protein phosphorylation. The magnitude of Ca 2ϩ -dependent endocytosis may appear rather large also under physiologically relevant conditions (Ն20% Fig. 6B). However, the percentage of the measured signal corresponding to endocytosis predicted using our model is low compared with the calculated ratio of endocytosis/exocytosis after stimulated exocytosis in rat corticotrophs (32).

The developed model can predict adiponectin secretion/exocytosis
In the validation of the final model structure, we used all acceptable parameters that were justified for the new experimental conditions (see under "Results") to include the uncertainty of the data in model predictions. This approach is important, because the optimal parameters and other acceptable parameters might give different but still plausible predictions for new data. We find that the developed model is able to predict adiponectin secretion under several different stimulatory conditions and at protracted time points in dynamic experiments (Fig. 5, D and E). This suggests that in the future the model can be used to predict whole-cell membrane capacitance responses at late time points that are methodologically challenging to measure experimentally, even though such analyses require further validation. Moreover, the finding that simulations of adiponectin secretion so accurately predict experimental results strengthens our proposal that adipocyte membrane capacitance increases in response to cAMP, with or without Ca 2ϩ , signify adiponectin vesicle fusion with the plasma membrane (6,7).
We want to emphasize that although the model is in agreement with both estimations and validation data, there could be other models able to explain the data equally well. The developed model is, however, useful because it is (i) a minimal model based on current biological knowledge, (ii) reached after several rejections of simpler models, and (iii) currently the only existing model of adi-ponectin secretion. We encourage that our model be further advanced, by us or by others, when more data become available.

Using models to understand the pathophysiology of adiponectin exocytosis in metabolic disease
Recent work from our group has enhanced our understanding of adiponectin exocytosis in health and in metabolic disease. Using subcutaneous white adipocytes isolated from lean or obese/diabetic mice, we have shown that adiponectin exocytosis is stimulated by catecholamines (elevates cytoplasmic cAMP) and that the stimulation mainly involves activation of ␤ 3 -adrenergic receptors. Moreover, adipocytes isolated from obese/diabetic mice display blunted catecholamine-stimulated adiponectin secretion in a state we define as "catecholamine resistance." The blunted release is due to a secretory defect and does not result from reduced intracellular adiponectin content (34). To improve our understanding of the mechanistic details involved in adrenergically stimulated adiponectin exocytosis under pathophysiological conditions, it is necessary to further analyze the experimental data by use of a modeling approach.
The adiponectin secretion model presented in Fig. 7 can in the future be combined with models describing other cellular processes and functions in white adipocytes, e.g. the detailed insulin signaling and glucose uptake model in Ref. 17. Moreover, our adiponectin secretion model can be extended and included in a state-of-the-art multilevel model where interactions are studied on cellular, organic, and systemic levels. Such a convoluted model, where metabolic signaling pathways are continuously added, has already been developed for the glucose and insulin system (16,35). Complex models of the regulation of whole body physiology, with ensuing additions of more regulators and sub-systems, are essential to enhance our understanding of pathophysiological metabolic conditions.

Mathematical modeling
All mathematical analyses, i.e. simulations and optimizations, were carried out in MATLAB R2016a using IQM tools (a  (Fig. 1A), adiponectin exocytosis is triggered by cAMP alone and augmented by the combination of ATP and Ca 2ϩ . As revealed by this study, ATP is also required in combination with Ca 2ϩ in order for adiponectin-containing vesicles to attain release competence (in our model, to be transferred from the reserve pool to the releasable pool). In addition to stimulating exocytosis, Ca 2ϩ also induces membrane retrieval (endocytosis). The Ca 2ϩ -triggered endocytosis needs to be preceded by cAMPtriggered exocytosis (cf. Fig. 3C). The contribution of Ca 2ϩ -dependent endocytosis to the measured signal is substantial at unphysiologically high intracellular Ca 2ϩ concentrations (see text for more details).

Modeling mechanisms of adiponectin exocytosis
continuation of the SBtoolbox2 package (36,37)) and the MATLAB Global Optimization toolbox. A combination of the Global Optimization toolbox algorithms, particleswarm, fmincon, and simulannealbnd, were used to estimate the parameters. Models were based on ODE, and reactions were assumed to mainly follow mass action kinetics (see exceptions below). Rate constants are referred to as parameters, i.e. values that change the model output. Evaluation of parameters was done by simulating the model using IQM tools and then calculating the 2 error according to Equation 1, Here, y denotes the measured data; ŷ denotes the model simulation to compare with data; S.E. denotes the standard error of the mean; t denotes the measured time points, and p denotes the model parameters. The sum is over all measured time points for all datasets. The calculated sum follows a 2 distribution. Therefore, 2 tests were used to determine model rejections (38). We used a significance level of 0.05, and the number of degrees of freedom equals the number of data points minus one parameter for normalization and minus the number of parameters in the model. For the time-series data (Fig. 2, A-D) with n ϭ 7⅐4 ϭ 28 data points, the border of rejections for the original model with 11 parameters is given by the inverse cumulative 2 (28 Ϫ 1 Ϫ 11, 0.05) ϭ 26.
All model files and code to simulate the models are available at github.com/elinnyman/adiponectin-secretion. 7

Detailed description of the developed model
We aimed to develop a mathematical model that describes all our data for adiponectin secretion in adipocytes in an acceptable manner. We limited the scope and the size of the model to the available data to keep the estimation of parameters as simple as possible. The developed ODE model consists of two parts: the input equations and the actual adiponectin secretion model. The input equations describe the intracellular dynamics of Ca 2ϩ , ATP, and cAMP after infusion via the pipette solution. These equations all have the same format.
The states of the model are marked in bold. All input equations have the same format. As an example, the details of the Ca equation are as follows. In the steady-state simulation, kattach is set to 0, and when the pipette is attached to the cell, kattach changes from 0 to 1. The state Ca corresponds to the intracellular concentration of Ca. basalCa is the desired concentration of Ca in steady state (before attachment of the pipette), and targetCa is the desired final concentration after attachment of the pipette.
The parameters for these equations were allowed to vary within certain limits during the parameter estimation, to give some flexibility in the input dynamics. kCa was limited to 0.0035-0.7, to give a half-maximal response time roughly between 1 and 200 s. kATP and kcAMP were allowed to vary 10% from kCa. basalCa was allowed to vary between 35  where Res is the Reserve Pool of vesicles, assumed to be in high abundance compared with the other pools of vesicles and thus assumed not to change over time. Rel is the Releasable Pool of vesicles, and Exo is the state of vesicles undergoing exocytosis. Res0 was chosen to be an arbitrary number, and the equations for Rel0 and Exo0 were derived by setting the ODEs to zero (i.e. force steady state) and then solving the system of equations. This gives the following.
The reaction rates for the original model are as shown. The reaction rate from the state Exo (vExo) corresponds to the measured rate of change compared with the capacitance measurements. A scaling parameter was used for transition to the unit-less model to the unit in the measurements (fF/s).
In the Exocytosis model-ATP, we modified the original model to include an extra ATP dependence with an ATP dependence of the reaction from Reserve pool to Releasable pool.
In the Exocytosis model-ATP/endocytosis model, the dynamics of the endocytotic vesicles are described: The corresponding reaction rates are as shown: The difference between the reaction rates (vExo-vEndo) was used to compare as the measured rate of change in the capacitance measurements in the endocytosis model. With a scaling parameter, the model output thus converts to To be able to fit the data with high concentration of Ca 2ϩ , a saturation effect was added to the positive Ca 2ϩ effects:

Model validation
In the model validation, we used the model to predict experimental data not used in the parameter estimation (measurements of accumulated released adiponectin). The model output that we compare with data is We simulated the final model using all acceptable parameters, i.e. all found parameters that gave model simulations that passed a 2 test. Instead of the previously used input equations for Ca 2ϩ , ATP, and cAMP, we used a wide range of values to simulate the predictions (see under "Results" for motivation of the intervals). We also added 25% uncertainty to the intervals.
For basal intervals For FSK/IBMX simulation For FSK/IBMX ϩ BAPTA simulation For FSK/IBMX ϩ ionomycin simulation To simulate the model predictions, we sampled from all acceptable parameters and used the maximum and the minimum values of the above intervals in all possible combinations to explore the entire interval of predictions.

Prediction of endocytosis contribution
In the evaluation of the contribution of endocytosis to the measured signal, the calculated ratio between the rate of endocytosis and the sum of endocytosis and exocytosis were multiplied times 100 to get the contribution in %.
To calculate the intervals of the contribution (Fig. 6, A and B), we performed parameter searches in the direction of this specific prediction (7). 2 error for each value of the prediction is shown in the figures (Fig. 6B).

Experimental methods
All experimental data, with the exception of that presented in Figs. 3, A and B, and 5E and Table 2, are adapted from Ref. 6. The data in Ref. 6 consist of whole-cell patch-clamp recordings
Electrophysiology-3T3-L1 adipocytes, grown in plastic (Nunc, Denmark) Petri culture dishes, were continuously superfused during the recordings with an extracellular solution (EC) containing (in mM): 140 NaCl, 3.6 KCl, 2 NaHCO 3 , 0.5 NaH 2 PO 4 , 0.5 MgSO 4 , 5 Hepes (pH 7.4 with NaOH), 2.6 CaCl 2 supplemented with 5 mM glucose. 3T3-L1 adipocyte exocytosis was measured as increase in membrane capacitance using an EPC-9 patch-clamp amplifier (HEKA Electronics, Lambrecht/Pfalz, Germany) and PatchMaster software. A fire-polished borosilicate glass pipette was attached to cells by gentle suction and electrical access attained by rupture of the plasma membrane within the patch. Consequently, the pipette filling solution (containing indicated concentrations of cAMP, ATP, and/or Ca 2ϩ ) equilibrates with the cytosol. Cells were voltage-clamped at Ϫ70 mV. The pipette-filling solutions contained (in mM): 125 potassium glutamate, 10 KCl, 10 NaCl, 1 MgCl 2 , 5 Hepes (pH 7.15 with KOH) and were supplemented with cAMP, ATP, and Ca 2ϩ as indicated. Exocytotic rates (⌬C/⌬t) were measured by applying a linear fit to the capacitance trace at different time points and calculating the slope of the fitted line. For more details, see Ref. 6.
Measurements of released adiponectin-Secreted total adiponectin was measured using 3T3-L1 adipocytes grown on 12-well plates (Sarstedt) stimulated with EC containing indicated secretagogues during 15-60-min incubations at 32°C, as described previously (6). EC aliquots and cell homogenates were stored at Ϫ80°C. Secreted adiponectin (measured with mouse ELISA DuoSets; R&D Systems) was analyzed in relation to total protein content (Bradford protein assay).
CARS and MPEF microscopy-Live differentiated cells were stained with 2 M CellTracker TM Red CMTPX Dye (Molecular Probes TM , Thermo Fisher Scientific) in serum-free medium for 30 min at 37°C. The lipid content of the living cells was determined ( Fig. 8) with combined CARS and MPEF microscopy on a custom-built setup (39). The 2845 cm Ϫ1 C-H stretching vibrations of triglycerides were driven with a spatially and temporal overlap of 2-ps pulsed laser beams (816.9 and 1064 nm, 7 ps, 76 MHz, Nd:vanadate, HighQ, pumped Levante Emerald OPO, APE Berlin, Germany) in the focal plane of an inverted microscope (Eclipse TE2000-E with a C2 confocal microscope scanning head, Nikon) using an oil immersion objective (Nikon Plan Fluor, ϫ40 NA 1.30). The cytosolic fluorescence probe was simultaneously excited with the 816.9 nm beam. Single photoncounting detectors (Becker & Hickl Gmbh) coupled with dichroic mirrors and bandpass filters (Semrock) were used to detect CARS signals in the forward direction (BP 661/20 nm) and MPEF signals in the epi-direction (609/57 nm). Five images with 5.04-s pixel dwell time were summed for each acquisition. Individual cells were identified from the combined MPEF and CARS images, and their size was measured. The lipid signal per cell was separated, thresholded (RenyEnthropie algorithm) using ImageJ (National Institutes of Health), and also measured in size. Assuming a spherical shape, the cell and lipid volume can be estimated from the area (20 cells ϫ3 technical repeats; Fig. 8).
Confocal microscopy-Fully differentiated 3T3-L1 cells were fixed in 4% paraformaldehyde, stained with Hoechst (34580, Molecular Probes), and imaged using a Nikon Ti-E microscope. The images were analyzed (nuclear count) using Fiji/ImageJ.
Author contributions-C. B. planned the study, performed the experiments, analyzed models and data, and wrote the paper. W. L. performed modeling and contributed to the writing of the manuscript. A. P. performed the experiments, analyzed the data, and contributed to the writing of the manuscript. A. M. K., M. F. E. H., and S. M. performed the experiments and analyzed the data. T. A. performed modeling. E. N. planned the study, performed modeling, analyzed the models, and wrote the manuscript. C. S. O. planned the experiments, analyzed the data, wrote the manuscript, and contributed reagents and material. All authors have read, edited, and approved the final version of the manuscript.