A multi-scale in silico mouse model for diet-induced insulin resistance

insulin production, which


Introduction
The link between obesity, insulin resistance and T2D is commonly studied in murine animal models, utilizing both genetic and dietinduced perturbations, such as high-fat diet (HFD)-feeding [1].Importantly, murine models allow for investigations of multi-scale mechanisms, i.e., mechanisms at several different timescales and biological organization levels (Fig. 1A.i).The top biological level is the whole-body level, with for example bodyweight and fat tissue mass.The next level, the tissue and organ level, covers the interplay between different tissues such as muscle and fat tissue, which primarily interact via the circulatory system, i.e., blood.One such interplay is the relationship between glucose uptake and insulin secretion, commonly measured by glucose tolerance tests (GTTs).Finally, the cellular level includes cellular aspects, such as intracellular signaling, gene expression, and metabolism, which are usually measured in vitro, using tissue biopsies or isolated cells.The dynamics of biochemical processes at these three levels occur on timescales which differ by orders of magnitude.The slower dynamics often takes place at the whole-body and tissue levels and can be monitored in vivo.More rapid dynamic events primarily occur at the cellular level and can often only be studied in isolated tissue in vitro.To interpret these complex multi-level and multi-timescale processes, all data should be integrated into a cohesive framework.The arguably most powerful and rigorous way to do that analysis is to use systems biology and mechanistic mathematical modelling.
This type of mechanistic mathematical modeling has been used extensively to study different aspects of insulin resistance and T2D, but not all aspects in a combined multi-level and multi-timescale model.For instance, obesity-and weight-related mathematical models for rodents describe weight changes but do not propagate these effects to other metabolic parameters, such as insulin resistance [2][3][4].Instead, insulin resistance is commonly described in mathematical models at an exclusively organ and tissue level, dissecting interactions during a meal-response (measuring e.g.glucose and insulin levels) [5].Further, for the cellular level, there are in silico murine models describing insulin signaling in adipocytes [6], and regulation of insulin secretion in β-cells [7].However, these models are also single-level models that have not been connected to corresponding mathematical models of other levels.
Despite important advancements in understanding the progression of insulin resistance, there thus exists no multi-level and multi-timescale in silico model that integrates all three biological organization levels of insulin resistance etiology (Fig. 1 Ai).In this work, we therefore aim to create a model integrating multi-level and multi-timescale murine data and modelling in a new cohesive framework: a multi-scale in silico mouse model (Fig. 1 Aii).The in silico mouse model connects three different levels using three different models (Fig. 1B): bodyweight and bodycomposition, and their changes in response to energy intake and expenditure (Guo et al. model [2]), meal-response glucose and insulin dynamics (Alskär et al. model [5]), and adipocyte intracellular insulin signaling (Bergqvist et al. model [6]).The three levels are connected by a new model for long-and short-term changes in insulin resistance (Fig. 2).The in silico mouse model was trained on our own previously published multi-level data from C57BL/6J mice (Fig. 3A, D, Fig. 5, training data) [8,9].Validation of the model predictions was done by collecting new experimental data generated for a longer feeding protocol (Fig. 3B, E, validation data).The validated model was used to predict organ-specific changes in insulin-dependent glucose uptake during insulin resistance progression (Figs.4,6), as well as other organ-specific changes, regarding e.g.insulin production and endogenous glucose production (EGP) (Fig. 6).Finally, we exemplify the usability of our validated in silico mouse model by including additional β-cell dynamics to simulate a translation from mouse data to human T2D progression (Fig. 7).

Assembling of a mechanistic, multi-level, and multi-timescale model
Our multi-level model is comprised of three interconnected, previously published, dynamic models, which here are linked together using a new sub-model for insulin resistance (Fig. 2A).The whole-body model (Fig. 2A) describes body-composition changes [2], the tissue-level model (Fig. 2B) describes organ-specific glucose uptake and insulin regulation [5], and the cellular level (Fig. 2C) describes intracellular insulin signaling [6].The body and organ levels are trained and validated on data from mice undergoing various HFD schemes, and the intracellular model is based on data from primary rodent adipocytes, isolated from rodents undergoing HFD-feeding.The new insulin resistance model is assumed to be driven by two known risk factors for insulin resistance: adiposity and diet [10].These two factors are quantified by the time spent on HFD (t HFD ), and by the relative increase in fat mass (FM) above baseline (FM 0 ), i.e. ( FM FM0 ).The variable t HFD counts the number of days spent on an HFD, and FM comes from the whole-body level.All model equations and parameter values, with their biological explanations, and a description of all changes made to the constituent models are reported in Appendix A (Section 2).All simulation files can be found in our GitHub repository DOI: 10.5281/zenodo.6619762.

Multi-level model simultaneously fitted and tested to data from all three levels
We have merged new and previously generated data to a combined dataset obtained from our laboratory, which measures the processes occurring on all three levels (Fig. 3A, B).More specifically, we have induced different degrees of bodyweight and fat mass changes, and insulin resistance in mice, using various combinations of control chow diet and HFD.During these experiments, we measured bodyweight, and performed an intraperitoneal glucose tolerance test (IPGTT) to measure insulin secretion and glucose clearance.In parallel experiments, primary adipocytes were isolated to study insulin signaling.
While the model was iteratively developed via numerous tests and modifications in its different parts (Fig. 3C), the final model could describe all the data with one and the same simulation which provides an acceptable agreement with all data.The parameter estimation process is detailed in Appendix A section 1.2.This is confirmed by a χ 2 test (0.95, 35), Materials and methods), which statistically confirms that the model should not be rejected.The parameter set, θ best , producing the simulation with the best agreement to data is presented (Fig. 3D-E, Fig. 5B, grey line).In practice, the model was estimated to some of the data.The remaining data were used in a subsequent validation step and these validation data come from new experiments performed by us (Fig. 3E).Let us now look at this estimation and validation data and simulations more in detail.

Model agreement with data for bodyweight and IPGTT data for mice on various HFD schedules
We have previously performed dietary alteration experiments according to the setup shown in Fig. 3A-B, where male C57BL/6J mice were switched from a chow diet to an HFD at different times during a 2week period [9], resulting in a control group (control chow diet) and groups of mice fed HFD for 2, 4, 6 days and 14 days (Fig. 3A).
Almost all model parameters for the bodyweight model have previously been experimentally validated [2] for male C57BL/6J mice.Similar to Gennemark et al. [4], we let only the physical activity parameter (referred to as λ) be re-estimated.The model agrees well with training data (Fig. 3D-E, grey line).We also show the associated model uncertainty, which has been obtained by maximizing the model uncertainty at each data-point with the constraint that we must have agreement with all training data (Fig. 3D-E, grey area) (Materials and methods).
In the same dataset [9], IPGTTs were performed at day 14 of each diet: control with chow diet, and initial chow diet followed by 2,4, 6 days, and 2 weeks of HFD (Fig. 3A).The data indicates that exposure to  HFD is correlated to higher fasting glucose concentrations (first time-point in Fig. 3Dii), higher fasting insulin concentrations (first time-point in Fig. 3Diii), and lower glucose clearance (lower peak and faster decline in Fig. 3Dii), which all unanimously indicates that a longer time spent on HFD gives increasingly higher degrees of insulin resistance.As can be seen in Fig. 3D, the model simulations (grey lines and areas) agree with all these experimental observations (black error bars).In other words, our tissue-level model provides an acceptable explanation to the IPGTT data, collected at different stages of insulin resistance.

Prediction of new validation data not used during estimation
We further validated the trained model by predicting new previously unpublished data.Here, C57BL/6J mice were fed either control (n = 10) or HFD (n = 10) for 8 weeks with IPGTT carried out at 6 weeks (Fig. 3B).

Fig. 4.
Model predictions for HFD-induced changes of total and organ-specific glucose uptake (GU).A) Model-predicted total insulin-glucose uptake (orange area) during an IPGTT performed after different durations of HFD or control, i.e., after 2 weeks (first five responses) or at 6 weeks (last two responses) B) The model predicted fat mass increase for the 2-week intervention (control and different duration of HFD) and after 6-week HFD with corresponding control.C) Ratio in insulindependent glucose uptake between the different tissues (adipose, muscle, and hepatic tissue).The total insulin-dependent uptake is the height of the bar (sum of all tissues).The ratio and total uptake are plotted for different durations on HFD during 2 or 6 weeks, compared to corresponding controls.D) Simulation of IPGTTs performed every day during an 8-week period, to show the effect of fat mass gain on the adipose tissue glucose uptake.E) Each dot represents the AUC for adipose tissue glucose uptake during an IPGTT, after a certain time on HFD.The individual adipocyte glucose uptake is shown with blue dots, and the total adipose tissue glucose uptake is shown with white dots.E) The AUC for the adipose glucose uptake during each IPGTT is plotted as individual dots against the time-axis.Here, the individual adipocyte glucose uptake is shown with blue dots, and the total adipose tissue glucose uptake is shown with white dots.GU=glucose uptake.
The model was trained on the bodyweight data for the first two-week duration (Fig. 3Ei, black error bars) to account for the varying experimental conditions from experiment to experiment.The model can predict the new data for the remaining duration of the 8 weeks' period (Fig. 3Ei, blue error bars).As can be seen, the experimental data lie within the model prediction uncertainty (green area).Also, the trained model could predict the new plasma glucose IPGTT data for HFD (blue error bars, Fig. 3Eii).The data for the corresponding control was not used for validation (Fig. 3Eii).The blue line shows the simulation which generated the lowest total cost when compared with these validation data (cost = 20.51< 27.58 = F cdf− inv χ2 (0.95, 17)).The model can thus produce predictions that explain validation data at a satisfactory level.

New predictions of missing experiments
To illustrate the usability of the model, we predicted experiments not available in our dataset.First, we predicted the bodyweight curve after 2 days of HFD (Fig. 3Di) and the plasma glucose during the IPGTT after 4 days of HFD (Fig. 3Dii).The latter prediction seems to follow the insulin resistance progression trend seen in data.These two predictions are in the same timeframe as the estimation dataset, and thus have narrow uncertainty.The model was also fitted to fasting insulin concentrations (from our own dataset) after the 2-week control, 2, 6 days and 2-weeks of HFD (black error bars, Fig. 3Diii).Second, we used the model to predict the plasma insulin responses during all IPGTTs (Fig. 3Diii, 3Eiii), and qualitatively compared them to data if available (green error bars, Fig. 3Diii,).As can be seen, the model predicts increased insulin response with prolonged HFD, and predictions seem to agree with the majority of experimental data points.However, due to the model's simple description of the changes in insulin secretion, and not being trained to any dynamic insulin data, the uncertainty becomes larger, compared to, e.g., dynamic plasma glucose.For example, the model fails to predict the last data-point of the insulin response during the IPGTT performed after 2 weeks of HFD, likely due to a slight overestimation of the insulin clearance (Fig. 3Diii).

Prediction of changes in total and organ-specific glucose uptake during the development of insulin resistance
To look further into the process of how the model explains the data, we have for the same set of simulations (Fig. 3D-E), looked at model predictions of glucose uptake.Fig. 4A shows the model prediction (line, simulation with best parameter set θ best ) and its uncertainty (area) for the total insulin-dependent glucose uptake, during the same IPGTTs shown in Fig. 3D-E (IPGTTs after 2 weeks, and after 6 weeks).Here we can see that the insulin-dependent uptake decreases with increasing time on HFD (2 days -2 weeks of HFD).Apart from predicting this overall insulin-dependent glucose uptake, we can also predict the timevarying contributions of the different organs.
The total insulin-dependent uptake is the sum of the uptake from adipose, hepatic and muscle tissue, and we can simulate each of these uptake-rates separately (Fig. B2, in Appendix B).Fig. 4B shows modelpredicted changes in the glucose uptake ratio between the different tissues.The glucose uptake was assessed by calculating the area under curve (AUC), during each IPGTT (corresponding bar plot in Fig. 4B).As can be seen, the uptake from adipose tissue becomes increasingly larger (Fig. 4B, blue), whereas the uptake from liver and muscle become increasingly smaller (Fig. 4B, red, green).In the model, the decrease in liver and muscle uptake is due to the increasing increased insulin resistance (Fig. 2).The increase in adipose tissue glucose uptake is due to the increase in adipose tissue volume, whichjust as insulin resistanceincreases with increasing time on HFD (Fig. 4C).
To further investigate the mechanisms underlying these long-term changes in adipose tissue glucose uptake, we performed an in silico experiment, using the best parameter set θ best .Here, we simulated IPGTTs each day for an 8-week HFD intervention (Fig. 4D).The model predicts that the ability of individual adipocytes to take up glucose will slowly be reduced by ~40 % after 8 weeks of HFD (Fig. 4E, blue dots).This decrease is due to the increasing insulin resistance.However, due to the large increase in fat-mass (Fig. 4C), the adipose tissue glucose uptake will increase (Fig. 4E, white dots).The model also predicts that the glucose uptake from adipose tissue will not increase forever, but will saturate after ~6 weeks.This prediction is an example of a prediction that you only can do by combining the previously existing weight and glucose uptake models with each other, using our new model for insulin resistance.

Agreement with phosphorylation data for key adipose insulin signaling intermediates
Let us also go through the model agreement to training data for the intracellular insulin signaling model for primary adipocytes.Almost all parameters were held constant at the previously published values [6].Note that these values were validated using independent validation data, not used for estimating the parameters, but that both that estimation and validation were done in rats.To fit the model to new data from mice, some parameters were re-estimated: the parameter scaling the insulin input, the parameter scaling the glucose output, the parameter governing insulin receptor substrate 1 IRS production and degradation, and the newly added insulin resistance parameters (Table A6, in Appendix A).The model simulations, shown in Fig. 5, were produced with the same set of parameters used for simulations in Figs.3-4.The cell-level model was fitted to our published data, Hansson et al. [8], which have not previously been used for modelling.More specifically, we use data obtained from primary adipocytes isolated from diet-challenged C57BL/6 mice which were fed either control diet or HFD for 4 weeks (Fig. 5A) [8].The two experiments were compared, and the protein expression levels were normalized to the expressions in the control data (Materials and methods).The model was trained to phosphorylation fold data of PKB pS473, AS160 pT642 as well as to glucose uptake data from adipocytes incubated with 0.1 nM insulin for 30 min (Fig. 5B).The model uncertainty is shown as the grey box, with the best model fit shown as the dark-grey line.The large model uncertainties are both due to limited data in relation to model complexity, and that data on all levels are fitted at the same time thus allowing for better agreement with some of the data and worse agreement with some of the data.This is also the reason the best model fit for PKB pS473 is underestimated.As can be seen, the uncertainty of the model is overlapping with the uncertainty of the data for all three experiments (Fig. 5B).

Unravelling the differentiated actions of insulin resistance, in the different organs and biological levels
We have now shown that the model can describe and predict data from all three biological organization levels during the progression of insulin resistance.Next, we illustrate how the newly developed insulin resistance sub-model works.For this, we simulated a 4-week HFD intervention with corresponding control, with IPGTTs performed every week (Fig. 6A).Such an experiment would be costly, and require many animals, if done in vivo, but is fast in a computer.
As before, the model predicts a larger glucose peak and a slower glucose clearance for prolonged HFD (Fig. 6B, orange areas), compared to corresponding control (Fig. 6B, grey).This increase is largely due to the increasing insulin resistance.
The components of this insulin resistance model are shown in Fig. 6C.The model consists of five functions: F IR− GU which impacts the glucose uptake in muscle and liver, F IR− PMA which impacts the plasma membrane associated (PMA) state in the insulin signaling in the adipose tissue, F IR− IRS which impacts the ability of the insulin receptor (IR) to activate the IRS in the adipose tissue; F IR− EGP which impacts the endogenous glucose production (EGP) from the liver; and F IR− IS , which impacts the insulin secretion.All these different functions are governed by the same two inputs: increase in fat-mass and the duration spent on HFD.These expressions are either linear or logarithmic regressions functions which scale a corresponding model rate expression (Fig. 2B, and Appendix A for more details), and either increase or decrease during the disease progression.More specifically, for all acceptable parameters, glucose uptake decreases in muscle and liver, and insulin secretion increases from the pancreas (Fig. 6D).In contrast, glucose uptake in the adipose tissue typically increases because of the growing adipose tissue (Fig. 4C), and EGP typically increases by 2-20 fold (Fig. 6D, blue box).These changes together lead to the observed dynamics in glucose and insulin (Fig. 6D, green box).

Illustration of potential model usability: adding a β-cell model with optional glucotoxicity
To further exemplify the potential usability of our new model we added a previously published β-cell model created by Topp et al. [11], and re-scaled it to fit our in silico mouse model (Fig. 7A) The addition lets us tweak both β-cell mass and function simultaneously.The β-cell function is determined by our insulin resistance model (F IR− IS ), and the β-cell mass is governed by fasting plasma glucose as described in the Topp et al. model.The β-cell mass can be affected by glucotoxicity, and the impact is determined by the parameter r2 in the model.This makes it possible to re-scale the model from having zero glucotoxicity like in C57BL/6J mice used in this study, to higher degrees of glucotoxicity which can induce loss of β-cell mass.To illustrate this, we simulated a 10-week HFD intervention with corresponding control, using the lowest cost parameter-set (θ best ).Fig. 7B illustrates four scenarios: control (black line), 'normal' HFD with low glucotoxicity and increasing insulin resistance (yellow line), HFD with higher glucotoxicity and increasing insulin resistance (red line) and HFD with 'humanized glucotoxicity' (green line).For the scenario of humanized glucotoxicity, the β-cell parameters were tweaked so that the β-cell mass would initially increase and then slowly decrease.This behavior is more in accordance with the human pre-diabetes etiology [12], at least compared to the other presented scenarios.Simulations are shown for β-cell mass, fasting plasma glucose and insulin (Fig. 7B).The control remains unchanged, and the normal HFD (yellow) has a small increase with subsequent decrease in β-cell mass.Fasting plasma glucose and insulin are consistent with our previous simulations.The HFD with higher glucotoxicity (red) has a fast reduction in β-cell mass due to the increase in fasting glucose level.The fasting plasma insulin level is initially increased due to increase in β-cell function (week 0-2) but then plateaus (week 2-4) and subsequently decreases due to the loss of β-cell mass.For the HFD with humanized glucotoxicity, an initial increase in β-cell mass is seen (week 0-4) in response to increased fasting glucose level.As the glucose concentration continues to increase, the β-cell mass will slowly begin to decrease (week 4-10).The fasting plasma insulin concentration increases due to the increase in both β-cell function and β-cell mass.When the β-cell mass begins to decrease the insulin concentration first plateaus (week 4-6) and then slowly starts to decrease.

Sensitivity and uncertainty analysis to identify key targets for improved glucose control
Another possible usage of our model is to examine which parts of the model that are most well-determined, which are most influential to key outcomes, and to use these results to identify possible targets for future drugs and other interventions.We thus did a Markov chain Monte Carlo (MCMC) analysis (Materials and methods, Fig. B5, in Appendix B) to identify the uncertainty of each fitted model parameter (Fig. B4, in Appendix B).This shows that e.g., the parameter CL GI is one of the more well-determined parameters (value range from 4.94e-5-9.28e-5(L • h − 1 ), Fig. B4, Table A4, in Appendix A and B).This parameter is a lead parameter in the insulin-stimulated glucose uptake in the liver and muscle.Furthermore, a sensitivity analysis based on simulations of the sensitivity equations show that this parameter is associated with the

6.
Simulations illustrating the functionality of the new insulin model.A) Schematic for the in-silico experiment which simulates a 4-week HFD intervention with corresponding control, with IPGTTs performed at day 1 and at end of week 1, 2, 3, and 4. B) The plasma glucose during each performed IPGTT demonstrates the progression of insulin resistance during the prolonged HFD.C) Schematic over the new insulin resistance model.i) The input to the new insulin resistance model is the duration spent on HFD and the relative increase in fat mass ii) The input is used in five different scaling functions.These functions either increase or decrease model fluxes that are relevant to the insulin resistance disease progression.The fluxes being scaled are the following: iii) hepatic and muscle insulin dependent glucose uptake (F IR− GU ) and adipose insulin dependent glucose uptake (, F IR− PMA , F IR− IRS ), iv) endogenous glucose uptake (F IR− EGP ), v) and insulin secretion (F IR− IS ).The changes induced in these fluxes will in turn affect vi) fasting plasma glucose and vii) fasting plasma insulin levels.D) Model simulations for the 4-week HFD intervention with corresponding control.Shown is the predicted fat mass gain which is used as model input.The values for each of the five-scaling function are as also shown as well as their propagated affect to corresponding model variable (hepatic, muscle and adipose GU, EGP and insulin secretion).These variable in turn affect and the fasting plasma glucose and insulin values, these are shown in the green box.

C. Simonsson et al.
high sensitivity for several key states (e.g., glucose, during parts of the time, Fig. B7, in Appendix B).We therefore also did a perturbation of this parameter, ranging from 0 % to 60 %, to see the impact on e.g., glucose control during the 14 days on HFD.As can be seen, these perturbations almost completely restore the glucose control back to normal (orange) during the first days, but the improvements are smaller and smaller over time (Fig. B8.A, in Appendix B).The opposite can be seen when changing only adipose tissue glucose uptake (green) (Fig. B8.A), the effect on glucose control become larger at later IPGTTs.The adipose tissue glucose uptake was increased by increasing the parameter GluProp 0-60 %.The same pattern can be seen when looking at AUC for plasma glucose for IPGTTs at different durations (Fig. B8.B, in Appendix B).This is due to the fact that the muscle and liver contributions to glucose uptake decrease over time (Fig. 4C).This indicates that a drug that targets glucose control in highly obese subjects probably would do well to target the adipose tissue, even though the adipose tissue normally is considered as a small player in normal glucose control.

Discussion
Herein, we have presented the first multi-scale mechanistic model of insulin resistance progression: an in silico mouse model.The model can simultaneously find agreement to our multi-scale dataset from C57BL/ 6J mice, describing insulin-resistance development on short-, and longterm timescales, and on three different biological organization levels: body composition (Fig. 3Di), plasma glucose and insulin (Fig. 3Dii-iii), and intracellular adipocyte insulin signaling (Fig. 5B).The model can both describe estimation data and correctly predict independent validation data (Fig. 3E).The model can also predict and shed light on the change and behaviors in organ-specific insulin-dependent glucose uptake and insulin-secretion necessary to find agreement to data (Figs.4,6).This shows that the adipose tissue, which initially is a small player in glucose uptake, becomes the dominant organ at the end of the HFD, even though the glucose uptake in each individual adipocyte is decreased by ~40 % (Fig. 4E).Lastly, to exemplify the potential usability of our model, we implemented a previously published β-cell model and rescaled it to fit our model (Fig. 7).With the expansion, we could change, in silico, the level of β-cell sensitivity to glucotoxicity, and thus simulate more human-like scenarios of T2D progression.All in all, this means that our in-silico mouse model could be used to integrate a wide variety of data, and be used to design new experiments, understand nonmeasured processes, and help identify potential drug targets.

The in silico model highlights key changes necessary to describe insulin resistance data
The model needs to produce two effects of the insulin model, to be able to describe the multi-scale training data: 1) the observed changes in fasting plasma glucose and fasting insulin levels, and 2) the dynamic plasma glucose IPGTT responses.In the data, fasting plasma concentration rapidly increases and then plateaus following initiation of HFD.More specifically, mice fed HFD for two days showed a 2 mmol increase in fasting plasma glucose, as well as a 4-fold increase in fasting insulin (Fig. 3Dii), but no change between 2 and 6 weeks (9.64 ± 0.54 mmol and 10.15 ± 0.50 mmol; non-significant difference).In contrast, the short-term IPGTT glucose response changes continuously with prolonged HFD.In other words, there is a small increasing difference for 2, 4 and 6 days to 2 weeks on HFD, both in peak value and glucose clearance (same for plasma insulin response in Fig. 3Diii).There also seems to exist a difference in the IPGTT response for the chow control between 2 and 6 weeks (Fig. 3Dii and 3Eii).This could be due to variability between cohorts, stress-levels or ageing, or a combination of those.Also, there is a clear difference between IPGTT response for 2 and 6 weeks, with a greater and delayed peak, as well as a lower clearance for the 6-week response (Fig. 3Dii-Eii).This data indicates that there might be different mechanisms in play; a fast increase in fasting-levels with a small change in the IPGTT response (for 2 days HFD compared to control) and increasing differences in the dynamic IPGTT response (for 2 weeks compared to 6 weeks HFD).In the model, the variable that mainly dictates the dynamics of the IPGTT response is the insulindependent glucose uptake (Fig B2).Thus, because there is only a small change in the dynamic IPGTT response between the control and 2-days HFD, the insulin-dependent glucose uptake cannot be responsible for this fast change in fasting plasma concentrations in the model.Hence, instead the scaling of EGP and insulin secretion had to be initially faster, for the model to describe the fast initial increase in fasting concentrations (Fig. 4D, orange line).This is in coherence with studies showing that early onset of systemic insulin resistance could be linked to impaired hepatic insulin action [13].
The model is simultaneously fitted to corresponding bodyweight data.The body-composition model predicts the fat mass in the model, which is the main driving force for insulin resistance.The predicted fatmass after 6 weeks is almost 2-fold compared to the 6-week control.The fat-mass affects two major parts of the model; 1) it is the main input to our disease-progression model, and 2) it determines the size of the available pool of fat for adipose glucose uptake.Therefore, adipose tissue glucose uptake is predicted to become dominant during prolonged HFD (Fig. 4C, blue bars).However, we also show that manifestation of insulin resistance in the adipose intracellular insulin signaling model leads to a saturated uptake in adipocytes (Fig. 4E, white circles), even though the fat mass is increasing (Fig. 4B).The assumption to scale fatmass linearly might not be correct.Nevertheless, in human studies it has been shown that obese individuals have a larger adipose tissue glucose uptake compared to corresponding lean controls [14].However, mice gain significantly more relative fat mass during a short period of time.
The model in its current state has a too simple description of insulin secretion to describe the behavior shown by insulin during (Fig. 3Diii).To show the possibility of expanding the insulin secretion part of the model, we exemplify an integration of a published β-cell model.

Why is it interesting to add a β-cell model? Our in silico model, unlike available in vivo mice, can display diet-induced β-cell failure
Herein, we have studied C57BL/6J mice.In this mouse model, HFDfeeding induces insulin resistance, which in turn triggers a compensatory increase in insulin production [15], but this is in normal conditions not followed by β-cell failure [16].In other words, these mice only become pre-diabetic, i.e. insulin resistant, but they do not develop full β-cell failure and T2D as seen in humans [17,18].
Mice in vivo models have been developed to study β-cell failure and the hypo-insulinemic phenotype, either by surgically removing the mice β-cells [19], chemically inducing β-cell failure [20], or by genetically perturbing the β-cell function [1,[21][22][23][24].Indeed, in a study by Rezania et al., streptozotocin-induced diabetic mice displayed normalized glucose levels after receiving human β-cell implants [19], pin-pointing the lack of insulin resistance.In all these models, the low insulin production is not a consequence of the high insulin demand but is low already before the development of insulin resistance.In summary, there is no murine model that describes the initial increase in insulin production, due to insulin resistance, leading to β-cell dysfunction, followed by loss in β-cell mass and failure of the pancreas to produce insulin.
By introducing the re-scaled Topp et al. β-cell model (Fig. 7) we have shown the possibility to tweak properties, such as β-cell glucotoxicity, in silico.This tweaking allows our in silico C57BL/6J mice to express behavior which otherwise would require another animal murine model.Also, we show the possibility of our in silico model to simulate disease progression not possible using currently available in vivo mice models; diet-induced insulin resistance leading to loss in β-cell mass.
For future model development it would be of interest to re-estimate the model using human data on insulin resistance.

The power of a mechanistic, multi-level, and multi-timescale model
The novel model presented herein opens the door to many new possibilities.One such possibility is that it can serve as a knowledge base.In other words, even if the model would not have any predictive capabilities, it is still useful as a compact way of presenting and visualizing a wide variety of data and knowledge.However, a model goes beyond mere visualization -it is an explanation.An explanation is a statistically acceptable hypothesis, which also is compatible with mechanistic knowledge about the system [25].Thus, our model presents a simplified view of the likely mechanisms behind the many different datasets presented herein, and shows a reasonable way in which these data could have been generated.Because of the complex nature of the data, spanning different levels and timescales, such a testing of different tentative mechanistic explanations is almost impossible to do by mere reasoning around the data.Nevertheless, the arguably most important role of a mechanistic models is their ability to do simulations, i.e., predictions of what would happen in different scenarios.Our model could, e.g., be used to predict what would happen to these mice, if they would have been fed with a different diet protocol, or if they would have been exposed to a drug that modifies some of the processes described.In principle, the model can then describe how such modifications are propagated between the different levels, e.g.how an intracellular modification is propagated to the whole-body level.Such predictions can initially be used to test the current explanation for the data, by doing the same type of validation experiments that we have done here, comparing such predictions with new data (Fig. 3E).Note that such predictions are useful also if the prediction fails, since that means that some critical mechanism is missing in the model (or that already included mechanisms needs to be modified).However, once such a multi-level multi-timescale model has been sufficiently validated, it will become increasingly more useful for e.g., inspection of simulations of non-measured variables, for in silico drug screening, or for identification of new drug targets.

Limitations and assumptions
Some limitations to the current results should be mentioned.To be able to find agreement to all data in our multi-level dataset, we had to reestimate several of the model parameters (Table A3-5, in Appendix A).We believe this is reasonable since we describe experiments with different conditions compared to the original publications.Furthermore, our glucose load was in the form of IPGTT, and not IVGTT and OGTT used in the studies to infer the original model [5,26].Another limitation is that we do not know the true uncertainty of the data points, since variations in repeats may underestimate the uncertainty and not represent systematic errors.Therefore, within a dataset, all SEM values that were smaller than the mean SEM for the dataset, were replaced with mean SEM value, to avoid overfitting.Another limitation is the relatively large model prediction uncertainty i.e., Fig. 3E-Diii.The large model prediction uncertainty can be explained by a variety of reasons, e. g., the choice of cut-off and the choice of model prediction.The method for the collection of model uncertainty is detailed in the method Section 4.3.There exist other choices in cut-off value which would have resulted in a narrower uncertainty, however we chose to include all possible parameters.The choice of prediction is also a possible reason for the high uncertainty.For example, some of the model predictions (such as Fig. 3E) are at much later time-points compared to estimation data.Also, if more data was introduced into the model training, we could lower the model uncertainty.However, this does not change the fact that the broad uncertainty may question the validity of some of the model predictions.To further study the model uncertainty we also did a parameter sensitivity analysis (Figs. B4-7, in Appendix B).Also, in Table A3-6 parameter confidence intervals are given for the model prediction uncertainty.Another limitation is the sparse data for intracellular responses in mice adipocytes, and that the original intracellular model was based on data obtained from rats.The result of the data sparsity is seen in the higher model uncertainty for the model fit to the PKB pS473 data (Fig. 5B).An important future task is thus to adopt and refine the intracellular model to mouse data.Another limitation is that we decided not to include insulin meal-response data in the model training.The model's description of the insulin meal response is very simplified, and substantial model development would have been necessary to find agreement with such data, especially dynamic insulin data for the meal-response during different stages of insulin resistance.Finally, there are a lot of missing process in the model.Some examples of such processes are intracellular insulin signaling in muscle and liver, ectopic fat storage, and hormones such as glucagon and stress hormones.The model also lacks important metabolic pathways such as intracellular glucose metabolism for adipose tissue, muscle, and liver.The addition of intracellular metabolic pathways would be important to better describe the manifestation of insulin resistance and might thus be the most important possible new addition to the model.This would however require more data on these pathways and processes.The processes can iteratively be added to future versions of the model.
In conclusion, while there are limitations to the current model, it is the first mechanistic, multi-scale model for insulin resistance based on multi-scale data.We believe this model will become an important basis for future interpretation and integration of experimental data, which ultimately will help with the development of new treatments for insulin resistance and T2D.

Animals and high fat diet intervention
Short-term overfeeding was carried out as previously described [8,9,27].Male C57BL/6J mice (Taconic, Ry, Denmark) were used at 8-9 weeks of age.Animals were on a 12 h light cycle with non-restricted food and water.Groups of animals (n = 10 animals/group) were fed either chow or HFD (D12492 60 E % fat content; Research Diets, New Brunswick, NJ, USA) for 6-8 weeks.Bodyweight (BW) of individual animals were measured weekly.Data are presented as the mean value with standard error of the mean (SEM).

Intraperitoneal glucose tolerance test
A subset of animals (n = 10/group) were subjected to intraperitoneal glucose tolerance test (IPGTT).In short, mice fasted overnight (12 h) were injected intraperitoneally with glucose (50 mg/mouse) and serum samples were collected at indicated times.Blood glucose levels were measured (Accu-Chek, Lund, Sweden), and insulin levels were assayed using ELISA (Mercodia, Uppsala, Sweden).Data is presented as the mean value with standard error of the mean (SEM).The supplier´s protocol was followed with an adjustment of the amount of sample volume (5 µL of each sample) and the incubation time (30 min at room temperature).
All animal procedures were approved by the Malmö/Lund Committee for Animal Experiment Ethics, Lund, Sweden, and were carried out in accordance with the relevant guidelines and regulations.

Mathematical modelling
The mathematical analysis, model simulation, numerical optimization of model parameters, was carried out using MATLAB 2018b and the SBtoolbox2 package [28].The model is based on ordinary differential equations (ODEs).The parameters were estimated by minimizing a normalized least-square function, using the scatter search algorithm from the MEIGO toolbox [29].The quality of the resulting optimal simulations was evaluated using a χ 2 test [25].Finally, the model uncertainty was calculated using the core prediction algorithm proposed in Cedersund et al. [30] and implemented in Lövfors et al. [31], to find the parameter sets that either maximized or minimized the model simulation at all given data points (36 in total), with the condition that the model should pass a χ 2 test.Parameter uncertainty analysis using Markov Chain Monte Carlo sampling was done via the PESTO toolbox [32].MATLAB scripts for plotting the figures and all data used for training and validation are provided at https://github.com/chrsi30/MLMMand copied to Zenodo (DOI: 10.5281/zenodo.6619762).Detailed equations for the both the multiscale model and for the used methods are provided in the Appendix A.

CRediT authorship contribution statement
Christian Simonsson (CS) has done the model analysis, with support from William Lövfors (WL), and with initial analysis by Nicklas Bergqvist (NB).Primary supervision and original conceptions have been done by Gunnar Cedersund (GC), and additional supervision and inputs on parts of the modelling have been done by Karin Stenkula (KS), Elin Nyman (EN), and Peter Gennemark (PG).Experiments have been supervised and carried out in the lab of KS.The paper has been written primarily by CS and GC, with input from all authors.All authors have seen and approved of the final manuscript.

Fig. 1 .
Fig. 1.Overview of the development of the in silico model and its multiscale properties.A). i) Data describing changes in insulin resistance brought on by HFD feeding has been gathered on different biological organization levels and timescales.Using this multi-level type of dataset, we have created a multi-level and multitimescale in silico mouse model of insulin resistance induced by HFD feeding.ii) The model was trained and then validated on different dataset all from our own laboratory.The validated model could then be used to iii) predict new experiments and iv) predict non-measured variables.B) Schematic showing the connections between the different models constituting the in silico mouse and what parts of the multi-level and multi-timescale axes they describe.

Fig. 2 .
Fig. 2. The multi-level and multi-timescale model structure featuring all model reactions and dependencies.A) The structure of the body level model and the insulin resistance sub-model connecting the three levels.The color coding of the arrows of the insulin resistance model is used in B and C. B) The structure of the organ-level glucose and insulin model, with two blue and two red arrows indicating where insulin resistance is affecting this model C) The structure of the cell-level adipocyte intracellular insulin signaling model, with two red arrows indicating the insulin resistance interaction.

Fig. 3 .
Fig. 3. Results of model training and validation on bodyweight data and data of plasma glucose and insulin following an intraperitoneal glucose tolerance test (IPGTT).A) Experimental setup used for training dataset.B) Experimental setup used for the validation dataset.C) Schematic showing our modelling approach.D) Model training on and prediction of Hansson et al. (2019) data.The black error bars are training data, the blue error bars are validation data, and the green error bars were used for qualitative comparison.Grey areas are the model training uncertainty.The green areas are the model prediction uncertainty.The model simulation corresponding to the best parameter set θ best is shown as the grey line in all graphs.i) Model training to body weight data, and model prediction of the bodyweight after 2 days on HFD.ii) Model training to data for plasma glucose from the different IPGTTs.Missing from our dataset was IPGTT data after 4 days on HFD, we here instead show model prediction for said experiment.iii) Model training for fasting insulin levels (black error bar).Also, the model predictions for the dynamic insulin response during each IPGTT are shown and qualitatively compared to data (green error bars) E) Model training on and validation of the new dataset for the 8-week HFD intervention with corresponding control.i) Model training uncertainty (grey area) for the first 2 weeks for both control and HFD feeding.Model predication uncertainty compared against the validation dataset (blue error bars) for bodyweight data.ii) For the new IPGTT data, we first show the model prediction uncertainty (green area) qualitatively compared to IPGTT data after 6 weeks of control (green error bars).The model prediction uncertainty (green area) compared to corresponding IPGTT validation data (blue error bars) after 6 weeks of HFD.iii) Model prediction of the plasma insulin during the 6-week IPGTT both for HFD and corresponding control.

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Fig. 5 .
Fig. 5. Comparison of data and simulations of the trained model for the intracellular insulin signaling model.A) Schematic of the experimental setup.B) Result of model training to intracellular data of PKB pS473, AS160 pT6420 phosphorylation, and glucose uptake data after and 4-week duration of HFD or control diet.The data (black error bar and white bar) describes the decrease in amount of phosphorylated protein after HFD compared to control.The data comes from mice primary adipocytes stimulated with 0.1 nM insulin for 30 min.The grey line represents the parameter-set that found the best agreement to the whole training dataset, with the corresponding model uncertainty shown as the grey box.

Fig. 7 .
Fig. 7. Addition of β-cell model by Topp et al. to the in-silico mouse model.A) Schematic of the addition of the β-cell model.The addition makes is possible to change both β-cell function (F IR− IS ) and mass simultaneously.B) Simulations of four different control (black line), 'normal' HFD with low glucotoxicity and increasing insulin resistance (yellow line), HFD with higher glucotoxicity and increasing insulin resistance (red line) and HFD with humanized glucotoxicity where the β-cell mass initially increases and then slowly decreases (green line).