Model equations

The models are built up by standard form ordinary differential equations (ODEs). All of equations are given in the Additional file material, both as equations and as simulation files. Below we only describe the equations that were added to the multi-level model in this article, specifically those of the insulin resistance model, the weight-meal response interconnection, the phenomenological energy intake, and the drug response model for topiramate.

Insulin resistance on organ/tissue leve

The insulin resistance part of the model is inspired by the similar insulin resistance equations implemented for mice in [12]. The equations used herein are:

$$__ }= 1 + _ \cdot \mathrm(xF) \cdot scale$$

(3)

$$f_ = 1 + b_ \cdot \log (xF) \cdot scale$$

(4)

$$__}= 1 + _ \cdot \mathrm(xF) \cdot scale$$

(5)

where \(xF\) is the relative change in fat mass from initial fat mass \(_\), \(F\) is the current fat mass, \(xL\) is the relative change in lean tissue mass from initial lean tissue mass \(_\), \(L\) is the current lean tissue mass, \(__}\) is the insulin resistance effect on hepatic and muscle glucose uptake, \(__}\) is the insulin resistance effect on endogenous glucose production, \(__}\) is the insulin resistance effect on insulin secretion, \(scale\) scales all the insulin resistance effects from mice to humans, and \(_\), \(_\), \(_\) are parameters.

The effect of the insulin resistance on insulin secretion is described by

$$\frac\left(_\right)= \left(\left(-\gamma \cdot _\right)+_\right)\cdot __ }\cdot _$$

(6)

where \(_\) is the amount of insulin in the portal vein, \(\gamma\) is the transfer rate constant between portal vein and liver, \(_\) is the insulin secretion into the portal vein, \(__}\) is the insulin resistance effect on insulin secretion, and \(_\) is a parameter for time conversion between the body composition model, defined in the time units days, and the other models, defined in minutes. Note that this is different from the previously reported mouse model, where the insulin resistance effect is directly on \(_\). One of the reasons for this difference is that insulin in the portal vein, \(_\), is not explicitly modelled in the mouse model.

The effect of insulin resistance on endogenous glucose production, \(EGP\), is described by

$$EGP = _ -(_\cdot _+_\cdot _+_\cdot _)) \cdot __}$$

(7)

where \(_\) is the extrapolated \(EGP\) at zero glucose and insulin, \(_\) is liver glucose effectiveness, \(_\) governs the amplitude of insulin action on the liver, \(_\) is a delayed insulin signal, \(_\) governs the amplitude of portal insulin action on the liver.

The effect of insulin resistance on glucose utilization in the liver, \(_\), and muscle tissue, \(_\), is affected by insulin resistance as follows:

$$_= \frac_\cdot \frac_}_+_\right)}}__ }}$$

(8)

$$_= xL\cdot \frac_\cdot \frac_}_+_\right)}}__ }}$$

(9)

where \(_\) is the maximum rate of glucose utilization in the liver, \(_\) is the glucose in tissue, \(_\) is a Michaelis–Menten parameter, \(_\) is the maximum rate of glucose utilization in muscle, and \(_\) is a Michaelis–Menten parameter. In our model, insulin resistance does not directly influence the glucose utilization in fat tissue, \(_\), since it has been observed in diabetics that glucose uptake is significantly changed in muscle and liver but not in fat tissue [13, 14].

Insulin resistance on cell level

The insulin resistance on the cell level is implemented as a gradual transition between the different parameter sets for non-diabetics and diabetics from the previous version of the model. The effect of diabetes was, as in the previous model, implemented on three different places in the model: \(IR\), \(GLUT4\), and \(diabetes\). The diabetes effect on \(IR\) decreases the total amount of \(IR\), and with less insulin receptors, less insulin can bind to the cell, i.e. the cell is less sensitive to insulin. The diabetes effect on \(GLUT4\) decreases the amount of \(GLUT4\), which means that less \(GLUT4\), can be taken up by the cell. The parameter named \(diabetes\) reduces the positive feedback from \(mTORC1\) to \(IRS1\) (Fig. 3E). All these diabetes effects result in an increase in insulin sensitivity and a decrease in glucose uptake in the model. The gradual transition of these diabetes effects was, as with previous insulin resistance equations, dependent on the change in fat mass as follows:

$$__ } = 1- _\cdot \mathrm\left(xF\right)$$

(10)

$$__}= 1- _\cdot \mathrm(xF)$$

(11)

$$__}= 1- _\cdot \mathrm(xF)$$

(12)

where \(_\), \(_\), and \(_\) are parameters.

As mentioned, the diabetes effect was static in the previous model – the model could either be diabetic, non-diabetic, but could not transition from one to the other. A transition between non-diabetic and diabetic version of the model was not possible since the total amount of \(IR\) and \(GLUT4\) could not change. To make the gradual transition to diabetes possible, equations that could change the total amount of \(IR\) and \(GLUT4\) was therefore added. Specifically, degradation and protein expression of \(IR\) and \(GLUT4\) was added (Fig. 4C). The protein expressions of \(IR\) and \(GLUT4\) are then influenced by the insulin resistance functions \(__}\) and \(__}\) to achieve the gradual decrease of \(IR\) and \(GLUT4\) that is part of the gradual transition to diabetes (Eqs. 15 and 19). For \(IR\), the following equations were changed:

$$\frac\left(IRm\right)= -v1a-v1basal+v1g+v1r+vIR$$

(13)

$$\frac\left(IRi\right)= v1e-v1r-vIRdeg$$

(14)

where \(IRm\) is the insulin receptors (\(IR\)) found in the cell membrane, \(v1a\), \(v1basal\), \(v1g\), and \(v1r\) are the unchanged reaction rates describing the transition of \(IRm\) to and from other \(IR\)-forms (see the Additional file 1 materials and [7]), and \(vIR\) is the new reaction rate describing the protein expression of \(IRm\), \(IRi\) is the internalized form of \(IR\), \(v1e\) and \(v1r\) are the unchanged reaction rates describing the transition of \(IRi\) to and from other \(IR\)-forms (see the Additional file material and [7]), \(vIR\) is the new reaction rate describing the protein expression of \(IRm\), and \(vIRdeg\) is the new reaction rate describing the degradation of \(IRi\). The reaction rates \(vIRdeg\) and \(vIR\) are defined as:

$$vIR=kIR \cdot __}$$

(15)

where \(kIR\) is a parameter. For \(GLUT4\), the following equations where changed:

$$\frac\left(GLUT4m\right)= v7f-v7b-vGLUTdeg$$

(17)

$$\frac\left(GLUT4\right)= -v7f+v7b+vGLUT$$

(18)

where \(GLUT4m\) and \(GLUT4\) are the two forms of \(GLUT4\), the first associated with the cellular membrane and the other inside the cell cytosol, \(v7f\) and \(v7b\) are the unchanged reaction rates describing the transition between \(GLUT4m\) and \(GLUT4\), \(vGLUTdeg\) is the new reaction describing the degradation of \(GLUT4m\), and \(vGLUT\) is the new reaction rate describing the protein expression of \(GLUT4\).

$$vGLUT=kGLUT \cdot __}$$

(19)

where \(kGLUT\) is a parameter. The membrane form, \(GLUT4m\), then effects the inflow of glucose to the cell, which is upscaled to \(_\) as described in [15].

The now gradual adiposity driven effect of insulin resistance on the positive feedback from \(mTORC1\) to \(IRS1\), \(v2c\), was applied in the same way as the parameter \(diabetes\) was in the previous model:

$$v2c = IRS1p \cdot k2c \cdot mTORC1a \cdot __}$$

(21)

where \(IRS1p\) is the amount of phosphorylated form of \(IRS1\), \(k2c\) is a parameter, \(mTORC1a\) is the amount of \(mTORC1a\).

Weight-meal response interconnection

As shown in Eq. 9, the change in lean tissue mass, \(xL\), has a direct effect on the glucose utilization in muscle tissue. This effect is a part of the connection between the whole-body weight model and the meal response model. The glucose utilization in fat tissue, \(_\), is also affected by the weight model, specifically by the change in fat mass:

$$U_ = xF \cdot V_ \cdot \left( }} + G_ }}} \right)$$

(22)

where, similarly to the utilization in the other tissues, \(_\) is the maximum rate of glucose utilization in muscle, and \(_\) is a Michaelis–Menten parameter.

Equations 9, 22 also show the connection between the whole-body and the organ/tissue level: the glucose uptake in muscle and fat tissue changes with the change in lean and fat mass respectively. Furthermore, the glucose rate of appearance, \(Ra\), changes with the total body weight (\(BW\)):

$$Ra = f\cdot _\cdot \frac_}$$

(23)

where \(f\) is the fraction of intestinal glucose absorption which appears in plasma, \(_\) is the absorption rate, \(_\) is the glucose content in the gut, and \(BW\) is the body weight. In the earlier model, \(BW\) was a constant, while here it is a variable in the whole-body level as described in [3].

To merge the different models, a parameter \(_=24\cdot 60\) was introduced to the models corresponding to time expressed in minutes, i.e., the organ/tissue level model and the cell model, to change the unit for time into days.

Phenomenological energy intake

We added an equation for accounting for differences in energy intake throughout the study period:

$$EI_ = EI_ - \Delta EI_ + (\Delta EI_ - \Delta EI_ )\,\,\frac }}^ + t^ }}$$

(24)

where \(E_\left(t\right)\) is the energy intake over time, \(_\) is the energy intake at baseline, \(\Delta _\) is the maximum change in energy intake, here fixed at the change in energy intake that the participants were asked to follow, \(\Delta _\) is the change in energy intake at steady state, \(t\) is the time, \(h1\) is the hill coefficient, and \(_\) is the timepoint where half of \(EI_ \,\,(t)\) has been reached.

Drug response model for topiramate

The energy intake was also altered with respect to the drug topiramate according to

$$EI\left(t\right)=_\cdot \left(1-_\cdot \frac^}^+_^}\right)$$

(25)

where \(EI\left(t\right)\) is the energy intake that influenced by topiramate, \(h2\), \(_\), and \(_^\) are parameters, and \(C\) is the concentration of topiramate in plasma. To get \(C\), we adopted the standard two-compartment pharmacokinetic model with first-order absorption from [15]

$$\frac(A) = -Ka\cdot A$$

(26)

$$\frac\left(C\right)= Ka \cdot \frac-K23 \cdot C+K23 \cdot C2-K10 \cdot C$$

(27)

$$\frac\left(C2\right)= K32 \cdot C-K32 \cdot C2$$

(28)

Here \(A\) is the absorption compartment, into which the daily dosages of topiramate are administered, \(Ka\), \(V\), \(K23\), \(K10\), and \(K32\) are parameters, and \(C2\) is the topiramate concentration in tissue.

Parameter estimation

Almost all of the 146 parameters in this multi-scale model were fixed at their values obtained from previous studies. These fixed parameters on the whole-body model were determined from literature values that have been validated on weight-loss data for both obese and nonobese women and men [16], and some parameters are determined from demographics (e.g. height, age, etc.). The organ/tissue level has been trained and validated on both healthy and type 2 diabetics with normal weight, and we used the parameters for the healthy group herein [4]. The cell-level model was also trained on both type 2 diabetic and healthy subjects, more specifically on data obtained from experiments on their subcutaneous fat cells, and the parameters for the healthy subjects were used herein [6]. The parameters estimated in this article are one scaling parameter of the insulin resistance model, the scaling parameter of the diabetes effects in the cell-level model, the new parameters in the cell-level model, those parameters corresponding to the phenomenological energy intake equation, and finally the parameters of the meal response model. The different parameters are estimated using different data and in different ways.

Most parameters were optimized using an optimization algorithm. Specifically, the parameters were estimated by minimizing the difference between model simulations, denoted, and experimental data, denoted \( \). The cost function used is the conventional weight least square, i.e.,

$$\mathrm(\theta )=\sum_^(_-}}}_}}}_(t)})}^$$

(29)

where the subscript \(i\) denotes the data point, where \(N\) denotes the number of data points, and where \(SEM\) denotes the standard error of the mean for the data uncertainty [17]. In practice, this parameter estimation was accomplished using the enhanced scatter search (eSS) algorithm from the MEIGO toolbox [18]. The optimization was restarted multiple times, run in parallel at the local node of the Swedish national supercomputing center (NSC). The parameter estimation was allowed to freely find the best possible combinations of parameter values within boundaries.

We use a \(^\)-test to evaluate the agreement between model simulations and data. To be more specific, we use the inverse of the cumulative \(^\)-distribution function for setting a threshold, \(_^}^\), and then compare the cost function \(V\left(\theta \right)\) with this threshold:

$$T_ }}^ = F_ }}^ \left( \right)$$

(30)

where \(F_ }}^\) us the inverse density function, \(\alpha\) is the significance level, and \(v\) is the degrees of freedom, which was the same as the number of data points in the training data sets. The model is then rejected if the model cost is larger than \(_^}^\).

The parameters that were not estimated using an algorithm were estimated manually due to simplicity, but the fit to data was assessed in the same way as for the optimization algorithm, i.e. with a \(^\)-test (Eq. 2). Note that apart from these explicitly mentioned parameters, all other parameters were optimized using an optimization algorithm (Eq. 29).

The scaling parameter of the insulin resistance model (scale, Table 1), which accounts for the scale difference in fat tissue between mice and humans, was estimated by hand. The data used for this manual fitting was the fasting insulin data from the Fast-food study [17, 18] (Fig. 3D).

Table 1 Parameters on the whole-body level that were estimated to both the Fast-food study and Topiramate data

The scaling of the three diabetes effects—IR, GLUT4, and diabetes—(bIR, bGLUT, and bdiabetes, Table 2) were adjusted by hand to fit to the level of diabetes seen in the cellular data from the Fast-food study (Fig. 3G). The three diabetes effects have their own range of diabetic to non-diabetic values (Fig. 3E, F) – 55–100 for IR, 50–100 for GLUT4, and 15.5–100 for diabetes. These ranges of diabetes effects were then scaled using one scaling parameter, scaling them towards a percentage of diabetes that corresponded to an acceptable fit to the cellular data after the fast-food diet.

Table 2 Parameters on the cell level that were estimated to the cell-level Fast-food study-data

The parameters added to the cell model to enable a gradual change due to insulin resistance, kGLUT4 and kIR (Eqs. 13, 14, 15, 16, 17, 18, 19, 20) (Table 2), were also adjusted manually. These parameters were adjusted so that the initial values of total IR and total GLUT4 had a steady state at 100%.

The last parameters to be adjusted manually were the parameters of the insulin resistance equations (Eqs. 10, 11, 12), bIR, bGLUT4, and bdiabetes (Table 2). These parameters were adjusted so that the initial values of total IR and total GLUT4 reached the scaled values from the estimation to cellular data within the time span of the Fast-food study (Fig. 4D).

Two sets of parameters were adjusted using an optimization algorithm: the energy intake parameters (Table 1) and the meal response parameters (Table 3). The parameters relating to the energy-intake equation were estimated using data from the Topiramate study. This estimation data consists of body-weight time-course data, which is denoted BW. The meal-response parameters were estimated using the baseline values of fasting plasma insulin and glucose from the Fast-food study, and were only changed when used in the training and predictions relating to the Fast-food study (i.e., the training and prediction related to the Topiramate study used the parameters from the original article [4]). These parameters were kept within tight bounds (a factor of 1) of the parameter values from the original model [4].

Table 3 Parameters on the organ/tissue level that were estimated to the Fast-food study-data (specifically the initial values of glucose and insulin). For the Topiramate study, the values from [4] where used

A further set of parameters were determined by the population demographics, and those as such function as a possible personalization. These parameters include the initial values for weight, fat mass, fat free mass or lean mass, age, height, change in energy intake, and topiramate dosage. Equations giving an estimate for the fat free mass and fat mass are included in the model, that can be used if these measurements are not readily available. Some further initial values could also be used for personalization but was instead estimated by the model in this work since values for them were not available in data, and these include the initial values of resting metabolism, extracellular fluid, glycogen mass, rate of glucose appearance, endogenous glucose production, glucose utilization and insulin secretion.

For detailed description of all parameters, see the Additional file material. All parameters not changed were fixed and set to values used in Nyman et al. (2011), and these values are also listed in the Additional file 1. We exploited the modular structure of the model by fitting the weight model on its own. In the final simulation with the multi-level model, all aspects of the model are simulated at the same time.

Model simulation

We exploited the unidirectional structure of the multi-level model to only simulate those parts of the model that are needed. In other words, the whole-body part of the model is not impacted by other parts of the model and could therefore be simulated on its own, for instance when estimating the parameters in that part of the model to only the weight data. In contrast, the entire multi-level model was simulated for the tissue- and cell-levels.

The initial values used in the simulations can be found in the Additional file material.

We used MATLAB R2020b (MathWorks, Natick, MA) and the IQM toolbox (IntiQuan GmbH, Basel, Switzerland) for the entire modelling work performed [19].

Uncertainty estimation

The uncertainty of both the parameters and the model simulations for estimation, validation, and predictions were gathered as proposed in [20] and as implemented in [21]. In short, the desired property (i.e., the fasting plasma glucose and insulin levels in the Fast-food study (Fig. 3) and the weight data in the Topiramate study (Fig. 4)) were either maximized or minimized, while requiring the cost to be below the \(^\)-threshold. See [21] for more details on how the uncertainty estimation was done.

Data

No new experimental data was collected in this study. We therefore refer to the methods sections in the original articles [22,23,24,25,26] for the corresponding details experimental methods. Information on the population demographics and what data that was used for estimation and validation respectively are given in the results section. All data besides that cell data from the Fast-food study was digitized from figures or read out from tables in their respective articles. The digitization was done using the WebPlotDigitizer [27]. All data are shown as means with SEM shown as error bars. For the Topiramate study, no SEM values were given. Therefore, the SEM values used in the statistical analysis (Eq. 1) were estimated to be 30% of the corresponding mean value.