# Analysis of Hepatic Lipid Metabolism Model: Simulation and Non-Stationary Global Sensitivity Analysis

^{*}

## Abstract

**:**

_{B}) and plasma triacylglyceride (TAG) concentrations according to proposed menus for different meals (breakfast, lunch, snack and dinner). A non-stationary Fourier amplitude sensitivity test (FAST) was applied to analyze the effect of 78 kinetic parameters on 24 metabolite concentrations and 45 reaction rates of the biological part of the hepatic lipid metabolism model at five time points (t

_{f}= 10, 50, 100, 250 and 500 min). This study examined the total influence of input parameter uncertainty on the variance of metabolic model predictions. The majority of the propagated uncertainty is due to the interactions of numerous factors rather than being linear from one parameter to one result. Obtained results showed differences in the model control regarding the different initial concentrations and also the changes in the model control over time. The aforementioned knowledge enables dietitians and physicians, working with patients who need to regulate fat metabolism due to illness and/or excessive body mass, to better understand the problem.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.1.1. Recommendations for Menu Planning

#### 2.1.2. Mathematical Model of Hepatic Lipid Metabolism

#### 2.2. Methods

#### 2.2.1. Hepatic Lipid Metabolism Model Simulation

_{B}) and plasma endogenous lipoprotein triglyceride concentrations (T

_{LB}) based on proposed menus (Table 4). Initial values of other variables were as given by Pratt et al. [35] (Table 2).

_{B}), plasma insulin concentration (I), plasma non-esterified fatty acid concentration (A

_{NB}) and plasma endogenous lipoprotein triglycerides (T

_{LB}) were observed over a time period of 500 min.

#### 2.2.2. Non-Stationary Global Sensitivity Analysis

_{i}are randomly chosen frequencies and φ

_{i}are randomly chosen phase angles. The phase angles are generated at random from the range φ

_{i}[−π, π], but the frequencies are integers that constitute an incommensurate set. The frequencies and phase angels are selected to provide uncorrelated parameter variations when the s variable is scanned throughout the limits from −1 to 1. The input variables (transformed model parameters) were sampled at frequencies ranging from 5 to 159 Hz, with the maximum frequency in the Fourier analysis set at 500 Hz. The selected range of frequencies was tested by a covariance matrix of the parameters, which showed that all parameter correlations were of order 10

^{−3}or smaller, which is an indication of an almost independent parameter.

_{ω}and B

_{ω}(Equations (3) and (4)):

_{i}are calculated using these Fourier coefficients. The partial sum D

_{i}of the harmonics is given by:

## 3. Results and Discussion

#### 3.1. Meal Plans

#### 3.2. Hepatic Lipid Metabolism Construction and Simulation in CellDesigner

_{NB}). Initially, in response to food intake and insulin action, there is a decrease in A

_{NB}concentration. Insulin inhibits the action of the enzyme lipase, which is responsible for the formation of A

_{NB}[49]. Figure 4(c1–c4) clearly shows that after food intake, there is a decrease in A

_{NB}concentration. By comparing subfigures in Figure 4(b1–b4), which present changes in insulin concentration and changes in A

_{NB}concentration, the mentioned dependence of A

_{NB}concentration on insulin concentration can be observed. When the insulin concentration decreases, the A

_{NB}concentration increases and vice versa. This is clearly visible in the examples of Breakfast 1, Lunch 1 and Snack 1.

#### 3.3. Non-Stationary Global Sensitivity Analysis

_{f}= 10, 50, 100, 200 and 500 min) to define the control change over time. The global sensitivity coefficients are marked with colors: green indicating low values of sensitivity coefficients, and red indicating high values of global sensitivity coefficients based on the data distribution at the percentile curve.

#### 3.3.1. Global Sensitivities of the Model Metabolites

^{−5}to 0.895 for Menu 1 and in the range from 6.000 × 10

^{−5}to 0.986 for Menu 2. Furthermore, changes in the model control over time can be observed. Throughout time, model parameters’ sensitivity coefficients change with different dynamics.

_{L}(liver glycogenolysis rate) and k

_{11}(basal insulin secretion rate) for both combinations of initial conditions. β

_{L}is a kinetic constant included in the description of liver glycogen balance. It describes the rate of liver glycogen release and conversion to glucose-6-phosphate inhibited by insulin. k

_{11}is included in insulin balance and it describes the glucose-stimulated insulin production. Using initial concentrations based on Menu 1 and Menu 2 for model simulation, high values of sensitivity coefficients were calculated for free fatty acid concentration in the liver (A

_{L}), free fatty acid concentration in muscle (A

_{M}), free fatty acid concentration in adipose tissue (A

_{A}), glucose concentration in adipose (G

_{A}), triglycerides in liver secretion pool (T

_{L}), triglyceride concentration in blood plasma (T

_{CB}), free fatty acid concentration in plasma (A

_{NB}) and endogenous triglyceride concentration in plasma (T

_{LB}) at t

_{f}= 10, 50, 100 and 250 min. At t

_{f}= 500 min, changes in β

_{L}and k

_{11}values mostly affected T

_{CB}and T

_{LB}concentrations for initial conditions based on the keto diet menu (Figure 5e) and additional T

_{L}for initial conditions based on the general population menu (Figure 6e). It is also important to mention that results show global sensitivity coefficients for Y

_{M}and T

_{M}over the 50th percentile of calculated coefficients (marked yellow) for all parameters at t

_{f}= 10, 50, 100 and 250 min for Menu 1 (Figure 5a–d). For Menu 2, global sensitivity coefficients for Y

_{M}, T

_{M}and T

_{A}were over the 50th percentile of calculated coefficients for all analyzed model parameters at all five time points, indicating that those variables are the most sensitive part of the model (Figure 6a–e).

_{NB}and T

_{LB}were mostly sensitive to variations in β

_{L}and k

_{11}values for both menus at all five analyzed time points. Furthermore, changes in control were noticed for G

_{B}and I. For Menu 1 (keto diet), G

_{B}concentration is mostly sensitive to β

_{L}, k

_{11}and k

_{aa}(rate of adipose free fatty acid esterification to triglycerides) values at t

_{f}= 10 min (Figure 5a). Prolonging the simulation time to t

_{f}= 50, 100, 250 and 500 min, results showed that G

_{B}concentration is mostly sensitive to β

_{6}(rate of liver de novo lipogenesis form pyruvate), β

_{M}(muscle glycogenolysis rate), k

_{10}(affinity for hydrolysis of triglycerides to secretory pool) and k

_{14}(basal very low-density lipoprotein 1 secretion fraction) (Figure 5b,e). A similar observation was noted for I concentration: at t

_{f}= 10 min (Figure 5a), the I concentration is mostly sensitive to β

_{6}(rate of liver de novo lipogenesis from pyruvate), β

_{L}, β

_{M}(muscle glycogenolysis rate), k

_{10}(affinity for hydrolysis of triglycerides to secretory pool), k

_{11}and k

_{14}(basal very low-density lipoprotein 1 secretion fraction), while at t

_{f}= 50, 100, 250 and 500 min, results showed that the I concentration was most sensitive to β

_{6}, β

_{M}, k

_{10}and k

_{14}(Figure 5b,e). Moreover, for Menu 2 (general population diet) at t

_{f}= 10 min (Figure 6a), G

_{B}concentration is mostly sensitive to β

_{L}, and k

_{11}values, but prolonging the simulation time to t

_{f}= 50, 100, 250 and 500 min, results showed that G

_{B}concentration becomes mostly sensitive to β

_{6}, β

_{M}, k

_{10}and k

_{14}(Figure 6b,e). At t

_{f}= 10 min, the I concentration is mostly sensitive to β

_{6}, µ

_{3}(rate of muscle glucose-6-phospahe usage) and k

_{cm}(muscle free fatty acid uptake of chylomicron triglycerides) (Figure 6a), while results showed that the I concentration was the most sensitive to β

_{6}, β

_{M}, k

_{10}and k

_{14}at t

_{f}= 50, 100, 250 and 500 min (Figure 5b,e).

#### 3.3.2. Global Sensitivities of the Model Reactions

^{−7}to 0.986 for Menu 1 (Figure 7) and in the range from 6.000 × 10

^{−7}to 0.986 for Menu 2 (Figure 8). As for the model metabolites, model reaction changes in the model control were observed over time. Similar to the model metabolites, numerically, the largest values of sensitivity coefficients (values over 0.75) were obtained for β

_{L}and k

_{11}and, additionally, for β

_{6}for both initial conditions (Figure 7 and Figure 8). Results showed that the largest global sensitivities were obtained for variations in β

_{L}, k

_{11}and β

_{6}values for both menus for the following reactions: (i) v

_{16}(uptake of exogenous plasma triglycerides into liver free fatty acids), (ii) v

_{17}(free fatty acid transport from plasma to liver), (iii) v

_{18}(free fatty acid synthesis form plasma endogenous triglycerides), (iv) v

_{24}(rate of triglyceride release into plasma), (v) v

_{32}(uptake of exogenous plasma triglycerides into muscle free fatty acids), (vi) v

_{34}(rate of endogenous triglyceride release into plasma), (vii) v

_{38}(uptake of plasma exogenous triglycerides by adipose tissue), (viii) v

_{39}(uptake of plasma endogenous triglycerides into adipose free fatty acids), (ix) v

_{44}(rate of complete uptake of plasma exogenous triglycerides by adipose tissue) and (x) v

_{45}(adipose tissue uptake of exogenous triglycerides from plasma). Additionally, a sensitivity coefficient over 0.75 was obtained for the influence of β

_{6}on v

_{43}(fat input form the diet). The obtained result confirms the statement by Pratt et al. [35] where they describe that under conditions of heightened insulin, the liver accumulates glucose in form of glycogen (glycogenesis) for later use, and during times of low insulin, the liver decomposes glycogen to glucose (glycogenolysis).

_{19}were over the 50th percentile of calculated coefficients (marked yellow) for all parameters at t

_{f}= 100 min (Figure 7c). With described initial conditions at t

_{f}= 100 min, the highest sensitivities for reaction v

_{19}were estimated for β

_{L}(0.617), k

_{a}(0.603) and β

_{M}(0.157), indicating that triglyceride storage conversion to free fatty acids in the liver mostly depends on the release of liver glycogen, uptake of plasma triglycerides by adipose tissue and conversion of glycogen into glucose-6-poshate in muscles. Obtained results present complex inter-relations that cannot be noticed by analyzing model balances alone, since global sensitivity analysis revealed important influences of specific parameters that are directly included into selected balances. In case of simulations with initial conditions according to Menu 2, global sensitivity coefficients for v

_{1}and v

_{2}were over the 50th percentile of calculated coefficients at t

_{f}= 10 min (Figure 8a) and for v

_{19}at t

_{f}= 100 min (Figure 8c). Results showed that v

_{1}and v

_{2}, reactions describing glucose-stimulated insulin production and insulin degradation, were mostly sensitive to β

_{6}, β

_{M}, k

_{10}and k

_{14}, while v

_{19}was mostly sensitive to changes in β

_{M}and k

_{a}, which is similar for the simulations with initial conditions according to Menu 1 (0.157).

_{4}), glucose flux from plasma to the liver (v

_{5}), insulin-stimulated glucose transport between plasma and muscle (v

_{25}) and glucose transport between plasma and adipose tissue (v

_{42}). For all four listed reactions, with both initial conditions, changes in control were noticed over the time frame. For example, v

_{4}for both menus was mostly sensitive to k

_{22}, β

_{L}and k

_{11}at t

_{f}= 10 min (Figure 7a and Figure 8a), while at t

_{f}= 50, 100, 250 and 500 min, β

_{6}, β

_{M}, k

_{10}and k

_{14}(Figure 7b,e and Figure 8b,e) became the most important parameters. Results also showed that the highest global sensitivities for v

_{5}, v

_{25}and v

_{42}were obtained for β

_{L}and k

_{11}. Insulin concentration change was described by glucose-stimulated insulin production (v

_{1}) and insulin degradation (v

_{2}). Gradually, both reactions with both indicial concentrations were mostly sensitive to β

_{6}, β

_{M}, k

_{10}and k

_{14}. Plasma free fatty acid concentration was specified by free fatty acid uptake from plasma into muscle (v

_{33}), free fatty acid transport from plasma to the liver (v

_{17}), uptake of plasma free fatty acids into adipose free fatty acids (v

_{40}), release of adipose triglycerides to plasma free fatty acids (insulin-inhibited) (v

_{37}) and adipose tissue uptake of exogenous triglycerides from plasma (v

_{45}). Global sensitivity analysis showed that for both menus, the highest sensitivities for v

_{33}, v

_{17}, v

_{40}and v

_{45}were obtained for β

_{L}and k

_{11}at all times point (Figure 7 and Figure 8). However, it is also important to mention that values of sensitivities for v

_{33}, v

_{17}and v

_{40}decrease over time, while those for v

_{45}were constant. Furthermore, endogenous plasma triglyceride concentration was modeled, taking into account the secretion of triglycerides from the liver (v

_{24}), export of triglycerides from the secretory pool to plasma (v

_{23}), liver uptake of triglycerides as free fatty acids (v

_{18}), uptake of plasma endogenous triglycerides into muscle free fatty acids (v

_{34}) and uptake of plasma endogenous triglycerides into adipose free fatty acids (v

_{39}). Obtained results showed that all reactions except v

_{23}were mostly sensitive to changes in values of β

_{L}and k

_{11}for both menus and at all time intervals.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Heer, M.; Egert, S. Nutrients other than carbohydrates: Their effects on glucose homeostasis in humans. Diabetes. Metab. Res. Rev.
**2015**, 31, 14–35. [Google Scholar] [CrossRef] - van Ommen, B.; van den Broek, T.; de Hoogh, I.; van Erk, M.; van Someren, E.; Rouhani-Rankouhi, T.; Anthony, J.C.; Hogenelst, K.; Pasman, W.; Boorsma, A.; et al. Systems biology of personalized nutrition. Nutr. Rev.
**2017**, 75, 579–599. [Google Scholar] [CrossRef] - Panagiotou, G.; Nielsen, J. Nutritional systems biology: Definitions and approaches. Annu. Rev. Nutr.
**2009**, 29, 329–339. [Google Scholar] [CrossRef] - Badimon, L.; Vilahur, G.; Padro, T. Systems biology approaches to understand the effects of nutrition and promote health. Br. J. Clin. Pharmacol.
**2017**, 83, 38. [Google Scholar] [CrossRef][Green Version] - Hood, L.; Heath, J.R.; Phelps, M.E.; Lin, B. Systems biology and new technologies enable predictive and preventative medicine. Science
**2004**, 306, 640–643. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mc Auley, M.T.; Proctor, C.J.; Corfe, B.M.; Cuskelly, G.C.; Mooney, K.M. Nutrition Research and the Impact of Computational Systems Biology. J. Comput. Sci. Syst. Biol.
**2013**, 6, 271–285. [Google Scholar] [CrossRef] - De Graaf, A.A.; Freidig, A.P.; De Roos, B.; Jamshidi, N.; Heinemann, M.; Rullmann, J.A.C.; Hall, K.D.; Adiels, M.; Van Ommen, B. Nutritional systems biology modeling: From molecular mechanisms to physiology. PLoS Comput. Biol.
**2009**, 5, e1000554. [Google Scholar] [CrossRef][Green Version] - Jolicoeur, M.; Jolicoeur, M. Modeling cell behavior: Moving beyond intuition. AIMS Bioeng.
**2014**, 1, 1–12. [Google Scholar] [CrossRef] - Yasemi, M.; Jolicoeur, M. Modelling Cell Metabolism: A Review on Constraint-Based Steady-State and Kinetic Approaches. Processes
**2021**, 9, 322. [Google Scholar] [CrossRef] - Strutz, J.; Martin, J.; Greene, J.; Broadbelt, L.; Tyo, K. Metabolic kinetic modeling provides insight into complex biological questions, but hurdles remain. Curr. Opin. Biotechnol.
**2019**, 59, 24–30. [Google Scholar] [CrossRef] [PubMed] - Nijhout, H.F.; Best, J.A.; Reed, M.C. Using mathematical models to understand metabolism, genes, and disease. BMC Biol.
**2015**, 13, 79. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ederer, M.; Gilles, E.D. Thermodynamically Feasible Kinetic Models of Reaction Networks. Biophys. J.
**2007**, 92, 1846–1857. [Google Scholar] [CrossRef] [PubMed][Green Version] - Félix, G.; Ancheyta, J.; Trejo, F. Sensitivity analysis of kinetic parameters for heavy oil hydrocracking. Fuel
**2019**, 241, 836–844. [Google Scholar] [CrossRef] - Wang, J.; Ye, J.; Yin, H.; Feng, E.; Wang, L. Sensitivity analysis and identification of kinetic parameters in batch fermentation of glycerol. J. Comput. Appl. Math.
**2012**, 236, 2268–2276. [Google Scholar] [CrossRef][Green Version] - Link, K.G.; Stobb, M.T.; Di Paola, J.; Neeves, K.B.; Fogelson, A.L.; Sindi, S.S.; Leiderman, K. A local and global sensitivity analysis of a mathematical model of coagulation and platelet deposition under flow. PLoS ONE
**2018**, 13, e0200917. [Google Scholar] [CrossRef][Green Version] - López-Cruz, I.L.; Rojano-Aguilar, A.; Salazar-Moreno, R.; Ruiz-García, A.; Goddard, J. A comparison of local and global sensitivity analyses for greenhouse crop models. Acta Hortic.
**2012**, 957, 267–273. [Google Scholar] [CrossRef] - Lin, S.; Xie, M.; Wu, M.; Zhou, W. Global Sensitivity Analysis of Large Reaction Mechanisms Using Fourier Amplitude Sensitivity Test. J. Chem.
**2018**, 2018, 1–8. [Google Scholar] [CrossRef] - Feng, K.; Lu, Z.; Yang, C. Enhanced Morris method for global sensitivity analysis: Good proxy of Sobol’ index. Struct. Multidiscip. Optim.
**2019**, 59, 373–387. [Google Scholar] [CrossRef] - Ge, Q.; Menendez, M. Extending Morris method for qualitative global sensitivity analysis of models with dependent inputs. Reliab. Eng. Syst. Saf.
**2017**, 162, 28–39. [Google Scholar] [CrossRef] - Silva, A.S.; Ghisi, E. Evaluation of capabilities of different global sensitivity analysis techniques for building energy simulation: Experiment on design variables. Ambient. Construído
**2021**, 21, 89–111. [Google Scholar] [CrossRef] - Zhang, X.Y.; Trame, M.N.; Lesko, L.J.; Schmidt, S. Sobol sensitivity analysis: A tool to guide the development and evaluation of systems pharmacology models. CPT Pharmacomet. Syst. Pharmacol.
**2015**, 4, 69–79. [Google Scholar] [CrossRef] [PubMed] - Carta, J.A.; Díaz, S.; Castañeda, A. A global sensitivity analysis method applied to wind farm power output estimation models. Appl. Energy
**2020**, 280, 115968. [Google Scholar] [CrossRef] - Xu, C.; Gertner, G. Understanding and comparisons of different sampling approaches for the Fourier Amplitudes Sensitivity Test (FAST). Comput. Stat. Data Anal.
**2011**, 55, 184. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ryan, E.; Wild, O.; Voulgarakis, A.; Lee, L. Fast sensitivity analysis methods for computationally expensive models with multi-dimensional output. Geosci. Model Dev.
**2018**, 11, 3131–3146. [Google Scholar] [CrossRef][Green Version] - Sobol’, I.M.; Kucherenko, S. Derivative based global sensitivity measures. Procedia Soc. Behav. Sci.
**2010**, 2, 7745–7746. [Google Scholar] [CrossRef][Green Version] - Nandi, S.; Singh, T.; Singla, P. Derivative based global sensitivity analysis using conjugate unscented transforms. Proc. Am. Control Conf.
**2019**, 2019, 2458–2463. [Google Scholar] - Fang, S.; Gertner, G.Z.; Shinkareva, S.; Wang, G.; Anderson, A. Improved generalized Fourier amplitude sensitivity test (FAST) for model assessment. Stat. Comput.
**2003**, 13, 221–226. [Google Scholar] [CrossRef] - Uys, L.; Hofmeyr, J.H.S.; Rohwer, J.M. Coupling kinetic models and advection–diffusion equations. 2. Sensitivity analysis of an advection–diffusion–reaction model. Silico Plants
**2021**, 3, 14. [Google Scholar] [CrossRef] - Saltelli, A.; Bolado, R. An alternative way to compute Fourier amplitude sensitivity test (FAST). Comput. Stat. Data Anal.
**1998**, 26, 445–460. [Google Scholar] [CrossRef] - Saltelli, A.; Tarantola, S.; Chan, K.P.S. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics
**1999**, 41, 39–56. [Google Scholar] [CrossRef] - Tušek, A.J.; Jurina, T.; Čurlin, M. Global Sensitivity Analysis of the Biological Part of the Integrated BTEX Bioremediation Model. Environ. Eng. Sci.
**2016**, 33, 404–422. Available online: https://home.liebertpub.com/ees (accessed on 20 June 2022). [CrossRef] - Tušek, A.; Kurtanjek, Ž. MATHMOD Vienna 09 Proceedings; ARGESIM: Vienna, Austria, 2009. [Google Scholar]
- Jurinjak Tušek, A.; Čurlin, M.; Jurina, T.; Landeka Dragičević, T.; Kurtanjek, Ž. Parameter Sensitivity Analysis of Activated Sludge Models for Wastewater Treatment. Available online: https://www.researchgate.net/publication/268502503_Parameter_sensitivity_analysis_of_activated_sludge_models_for_wastewater_treatment (accessed on 13 October 2022).
- Phinney, S.D. Ketogenic diets and physical performance. Nutr. Metab.
**2004**, 1, 2. [Google Scholar] [CrossRef][Green Version] - Pratt, A.C.; Wattis, J.A.D.; Salter, A.M. Mathematical modelling of hepatic lipid metabolism. Math. Biosci.
**2015**, 262, 167–181. [Google Scholar] [CrossRef] [PubMed][Green Version] - Godinez, H.C.; Rougier, E.; Osthus, D.; Lei, Z.; Knight, E.; Srinivasan, G. Fourier amplitude sensitivity test applied to dynamic combined finite-discrete element methods–based simulations. Int. J. Numer. Anal. Methods Geomech.
**2019**, 43, 30–44. [Google Scholar] [CrossRef][Green Version] - Bender, D.A. Introduction to Nutrition and Metabolism; CRC Press: Boca Raton, FL, USA, 2014. [Google Scholar]
- Scientific Opinion on Dietary Reference Values for carbohydrates and dietary fibre. EFSA J.
**2016**, 8, 1462. - EFSA. Trusted Science for Safe Food. Available online: https://www.efsa.europa.eu/en (accessed on 12 October 2022).
- Funahashi, A.; Morohashi, M.; Kitano, H.; Tanimura, N. CellDesigner: A process diagram editor for gene-regulatory and biochemical networks. BIOSILICO
**2003**, 1, 159–162. [Google Scholar] [CrossRef] - Funahashi, A.; Matsuoka, Y.; Jouraku, A.; Morohashi, M.; Kikuchi, N.; Kitano, H. CellDesigner 3.5: A versatile modeling tool for biochemical networks. Proc. IEEE
**2008**, 96, 1254–1265. [Google Scholar] [CrossRef] - Berg, J.M.; Tymoczko, J.L.; Stryer, L. Good Hardcover 5th or Later Edition. In Biochemistry, 6th ed.; Palgrave MacMillan: London, UK, 2007; Available online: https://www.abebooks.com/Biochemistry-6th-ed-Berg-Jeremy-Tymoczko/14974066950/bd#&gid=1&pid=1 (accessed on 12 October 2022).
- Cruz, M.L.; Evans, K.; Frayn, K.N. Postprandial lipid metabolism and insulin sensitivity in young Northern Europeans, South Asians and Latin Americans in the UK. Atherosclerosis
**2001**, 159, 441–449. [Google Scholar] [CrossRef] - Yoshizane, C.; Mizote, A.; Yamada, M.; Arai, N.; Arai, S.; Maruta, K.; Mitsuzumi, H.; Ariyasu, T.; Ushio, S.; Fukuda, S. Glycemic, insulinemic and incretin responses after oral trehalose ingestion in healthy subjects. Nutr. J.
**2017**, 16, 9. [Google Scholar] [CrossRef][Green Version] - Sarabhai, T.; Koliaki, C.; Mastrototaro, L.; Kahl, S.; Pesta, D.; Apostolopoulou, M.; Wolkersdorfer, M.; Bönner, A.C.; Bobrov, P.; Markgraf, D.F.; et al. Dietary palmitate and oleate differently modulate insulin sensitivity in human skeletal muscle. Diabetologia
**2022**, 65, 301–314. [Google Scholar] [CrossRef] - Campbell, J.E.; Newgard, C.B. Mechanisms controlling pancreatic islet cell function in insulin secretion. Nat. Rev. Mol. Cell Biol.
**2021**, 22, 142–158. [Google Scholar] [CrossRef] [PubMed] - Eliasson, L.; Abdulkader, F.; Braun, M.; Galvanovskis, J.; Hoppa, M.B.; Rorsman, P. Novel aspects of the molecular mechanisms controlling insulin secretion. J. Physiol.
**2008**, 586, 3313–3324. [Google Scholar] [CrossRef] [PubMed] - Bermudez, B.; Ortega-Gomez, A.; Varela, L.M.; Villar, J.; Abia, R.; Muriana, F.J.G.; Lopez, S. Clustering effects on postprandial insulin secretion and sensitivity in response to meals with different fatty acid compositions. Food. Funct.
**2014**, 5, 1374–1380. [Google Scholar] [CrossRef] [PubMed][Green Version] - Iannello, S.; Milazzo, P.; Belfiore, F. Animal and human tissue Na,K-ATPase in normal and insulin-resistant states: Regulation, behaviour and interpretative hypothesis on NEFA effects. Obes. Rev.
**2007**, 8, 231–251. [Google Scholar] [CrossRef] [PubMed] - Sun, L.; Tan, K.W.J.; Lim, J.Z.; Magkos, F.; Henry, C.J. Dietary fat and carbohydrate quality have independent effects on postprandial glucose and lipid responses. Eur. J. Nutr.
**2018**, 57, 243–250. [Google Scholar] [CrossRef] - Nguyen Quang, M.; Rogers, T.; Hofman, J.; Lanham, A.B. Global Sensitivity Analysis of Metabolic Models for Phosphorus Accumulating Organisms in Enhanced Biological Phosphorus Removal. Front. Bioeng. Biotechnol.
**2019**, 7, 234. [Google Scholar] [CrossRef] - Kent, E.; Neumann, S.; Kummer, U.; Mendes, P. What Can We Learn from Global Sensitivity Analysis of Biochemical Systems? PLoS ONE
**2013**, 8, e79244. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jurina, T.; Tušek, A.J.; Čurlin, M. Local sensitivity analysis and metabolic control analysis of the biological part of the BTEX bioremediation model. Biotechnol. Bioprocess Eng.
**2015**, 20, 1071–1087. [Google Scholar] [CrossRef] - Zi, Z. Sensitivity analysis approaches applied to systems biology models. IET Syst. Biol.
**2011**, 5, 336–346. [Google Scholar] [CrossRef]

**Figure 1.**Research flowchart [35].

**Figure 2.**Percentage of macronutrients in daily energy intake (%). (CHO—carbohydrates, F—fats, P—proteins).

**Figure 4.**Results of hepatic lipid metabolism model simulations in CellDesigner software. (

**a1**–

**a4**) plasma glucose concentration, (

**b1**–

**b4**) plasma insulin concentration, (

**c1**–

**c4**) plasma non-esterified fatty acid concentration, (

**d1**–

**d4**) plasma endogenous lipoprotein triglycerides. Initial concentration for different meals (1) breakfast, (2) lunch, (3) snack and (4) dinner according to different menus (

**⎯**) Menu 1, (

**⎯**) Menu 2, (

**⎯**) Menu 3, (

**⎯**) Menu 4. (♦) experimental data for plasma glucose concentration from Yoshizame et al. [44], (▪) experimental data for plasma insulin concentration from Yoshizame et al. [44] (•) experimental data for plasma concentration of triacylglycerol from Sarabhai et al. [45].

**Figure 5.**Global sensitivities of hepatic lipid model metabolites (initial conditions for simulations according to Menu 1): (

**a**) t

_{f}= 10 min, (

**b**) t

_{f}= 50 min, (

**c**) t

_{f}= 100 min, (

**d**) t

_{f}= 250 min, (

**e**) t

_{f}= 500 min. Rows: β

_{G}, β

_{6}, β

_{f}, β

_{l}, β

_{m}, µ

_{AMP}, µ

_{e}, µ

_{s}, µ

_{1}, µ

_{2}, µ

_{3}, µ

_{4},c

_{0}, c

_{c}, d

_{BA}, k

_{10}, k

_{11}, k

_{12}, k

_{13}, k

_{14}, k

_{22}, k

_{5}, k

_{6}, k

_{6l}, k

_{6p}, k

_{7}, k

_{8}, k

_{9}, k

_{9a}, k

_{a}, k

_{aa}, k

_{ai}, k

_{al}, k

_{ba}, k

_{bl}, k

_{bm}, k

_{cl}, k

_{cm}, k

_{d}, k

_{dl}, k

_{dy}, k

_{ft}, k

_{ga}, k

_{gi}, k

_{gl}, k

_{gl2}, k

_{gm}, k

_{gm2}, k

_{gp}, k

_{lp}, k

_{LG}, k

_{LH}, k

_{MH}, k

_{na}, k

_{p}, k

_{p6}, k

_{pp}, k

_{r}, k

_{re}, k

_{t}, k

_{yl}, k

_{ym}, l

_{max}, m

_{max}, v

_{min}, v

_{10,}v

_{12}, v

_{6}, v

_{8}, v

_{9}, v

_{LG}, v

_{LH}, v

_{MH}, y

_{0}, α

_{G}, α

_{F}, m

_{s}, m

_{e}. Columns: I, G

_{L}, Y

_{L}, P

_{L}, R

_{L}, A

_{L}, S

_{L}, T

_{L}, G

_{M}, Y

_{M}, P

_{M}, R

_{M}, A

_{M}, T

_{M}, P, T

_{A}, A

_{A}, L

_{A}, G

_{A}, T

_{CB}, A

_{NB}, T

_{LB}, G

_{B}. The global sensitivity coefficients are marked with colors: green indicating low values of sensitivity coefficients, and red indicating high values of global sensitivity coefficients based on the data distribution at the percentile curve.

**Figure 6.**Global sensitivities of hepatic lipid model metabolites (initial conditions for simulations according to Menu 2): (

**a**) t

_{f}= 10 min, (

**b**) t

_{f}= 50 min, (

**c**) t

_{f}= 100 min, (

**d**) t

_{f}= 250 min, (

**e**) t

_{f}= 500 min. Rows: β

_{G}, β

_{6}, β

_{f}, β

_{l}, β

_{m}, µ

_{AMP}, µ

_{e}, µ

_{s}, µ

_{1}, µ

_{2}, µ

_{3}, µ

_{4},c

_{0}, c

_{c}, d

_{BA}, k

_{10}, k

_{11}, k

_{12}, k

_{13}, k

_{14}, k

_{22}, k

_{5}, k

_{6}, k

_{6l}, k

_{6p}, k

_{7}, k

_{8}, k

_{9}, k

_{9a}, k

_{a}, k

_{aa}, k

_{ai}, k

_{al}, k

_{ba}, k

_{bl}, k

_{bm}, k

_{cl}, k

_{cm}, k

_{d}, k

_{dl}, k

_{dy}, k

_{ft}, k

_{ga}, k

_{gi}, k

_{gl}, k

_{gl2}, k

_{gm}, k

_{gm2}, k

_{gp}, k

_{lp}, k

_{LG}, k

_{LH}, k

_{MH}, k

_{na}, k

_{p}, k

_{p6}, k

_{pp}, k

_{r}, k

_{re}, k

_{t}, k

_{yl}, k

_{ym}, l

_{max}, m

_{max}, v

_{min}, v

_{10,}v

_{12}, v

_{6}, v

_{8}, v

_{9}, v

_{LG}, v

_{LH}, v

_{MH}, y

_{0}, α

_{G}, α

_{F}, m

_{s}, m

_{e}. Columns: I, G

_{L}, Y

_{L}, P

_{L}, R

_{L}, A

_{L}, S

_{L}, T

_{L}, G

_{M}, Y

_{M}, P

_{M}, R

_{M}, A

_{M}, T

_{M}, P, T

_{A}, A

_{A}, L

_{A}, G

_{A}, T

_{CB}, A

_{NB}, T

_{LB}, G

_{B}. The global sensitivity coefficients are marked with colors: green indicating low values of sensitivity coefficients, and red indicating high values of global sensitivity coefficients based on the data distribution at the percentile curve.

**Figure 7.**Global sensitivities of hepatic lipid model reactions (initial conditions for simulations according to Menu 1): (

**a**) t

_{f}= 10 min, (

**b**) t

_{f}= 50 min, (

**c**) t

_{f}= 100 min, (

**d**) t

_{f}= 250 min, (

**e**) t

_{f}= 500 min. Rows: β

_{G}, β

_{6}, β

_{f}, β

_{l}, β

_{m}, µ

_{AMP}, µ

_{e}, µ

_{s}, µ

_{1}, µ

_{2}, µ

_{3}, µ

_{4},c

_{0}, c

_{c}, d

_{BA}, k

_{10}, k

_{11}, k

_{12}, k

_{13}, k

_{14}, k

_{22}, k

_{5}, k

_{6}, k

_{6l}, k

_{6p}, k

_{7}, k

_{8}, k

_{9}, k

_{9a}, k

_{a}, k

_{aa}, k

_{ai}, k

_{al}, k

_{ba}, k

_{bl}, k

_{bm}, k

_{cl}, k

_{cm}, k

_{d}, k

_{dl}, k

_{dy}, k

_{ft}, k

_{ga}, k

_{gi}, k

_{gl}, k

_{gl2}, k

_{gm}, k

_{gm2}, k

_{gp}, k

_{lp}, k

_{LG}, k

_{LH}, k

_{MH}, k

_{na}, k

_{p}, k

_{p6}, k

_{pp}, k

_{r}, k

_{re}, k

_{t}, k

_{yl}, k

_{ym}, l

_{max}, m

_{max}, v

_{min}, v

_{10,}v

_{12}, v

_{6}, v

_{8}, v

_{9}, v

_{LG}, v

_{LH}, v

_{MH}, y

_{0}, α

_{G}, α

_{F}, m

_{s}, m

_{e}. Columns: I, G

_{L}, Y

_{L}, P

_{L}, R

_{L}, A

_{L}, S

_{L}, T

_{L}, G

_{M}, Y

_{M}, P

_{M}, R

_{M}, A

_{M}, T

_{M}, P, T

_{A}, A

_{A}, L

_{A}, G

_{A}, T

_{CB}, A

_{NB}, T

_{LB}, G

_{B}. The global sensitivity coefficients are marked with colors: green indicating low values of sensitivity coefficients, and red indicating high values of global sensitivity coefficients based on the data distribution at the percentile curve.

**Figure 8.**Global sensitivities of hepatic lipid model reactions (initial conditions for simulations according to Menu 2): (

**a**) t

_{f}= 10 min, (

**b**) t

_{f}= 50 min, (

**c**) t

_{f}= 100 min, (

**d**) t

_{f}= 250 min, (

**e**) t

_{f}= 500 min. Rows: β

_{G}, β

_{6}, β

_{f}, β

_{l}, β

_{m}, µ

_{AMP}, µ

_{e}, µ

_{s}, µ

_{1}, µ

_{2}, µ

_{3}, µ

_{4},c

_{0}, c

_{c}, d

_{BA}, k

_{10}, k

_{11}, k

_{12}, k

_{13}, k

_{14}, k

_{22}, k

_{5}, k

_{6}, k

_{6l}, k

_{6p}, k

_{7}, k

_{8}, k

_{9}, k

_{9a}, k

_{a}, k

_{aa}, k

_{ai}, k

_{al}, k

_{ba}, k

_{bl}, k

_{bm}, k

_{cl}, k

_{cm}, k

_{d}, k

_{dl}, k

_{dy}, k

_{ft}, k

_{ga}, k

_{gi}, k

_{gl}, k

_{gl2}, k

_{gm}, k

_{gm2}, k

_{gp}, k

_{lp}, k

_{LG}, k

_{LH}, k

_{MH}, k

_{na}, k

_{p}, k

_{p6}, k

_{pp}, k

_{r}, k

_{re}, k

_{t}, k

_{yl}, k

_{ym}, l

_{max}, m

_{max}, v

_{min}, v

_{10,}v

_{12}, v

_{6}, v

_{8}, v

_{9}, v

_{LG}, v

_{LH}, v

_{MH}, y

_{0}, α

_{G}, α

_{F}, m

_{s}, m

_{e}. Columns: I, G

_{L}, Y

_{L}, P

_{L}, R

_{L}, A

_{L}, S

_{L}, T

_{L}, G

_{M}, Y

_{M}, P

_{M}, R

_{M}, A

_{M}, T

_{M}, P, T

_{A}, A

_{A}, L

_{A}, G

_{A}, T

_{CB}, A

_{NB}, T

_{LB}, G

_{B}. The global sensitivity coefficients are marked with colors: green indicating low values of sensitivity coefficients, and red indicating high values of global sensitivity coefficients based on the data distribution at the percentile curve.

**Table 1.**Designed meal plans with different proportions of macronutrients with an emphasis on the carbohydrates and fats.

Meal | Components with Corresponding Masses | |
---|---|---|

Menu 1 | Breakfast | 2 egg yolks (30 g), bacon (50 g) and gouda cheese (80 g) prepared in olive oil (5 g) |

Lunch | Salmon (100 g) prepared in olive oil (10 g) and green salad (100 g) with flax seeds (10 g) and olive oil (5 g) | |

Snack | Handful of walnuts (30 g) | |

Dinner | Chicken (120 g) prepared with cooking cream (150 mL) and broccoli (100 g) with olive oil (5 g) | |

Menu 2 | Breakfast | Cornflakes (80 g) with 2.8% m.f. yogurt (200 mL) and a cup of chamomile tea with sugar (5 g) |

Lunch | Plate of vegetable soup (200 mL), chicken breast (200 g) prepared in olive oil (10 g) with couscous (50 g), tomato salad (100 g) with flax seeds (5 g), 2 slices of graham bread and a glass of orange juice (200 mL) | |

Snack | Banana (100 g) and a handful of almonds (30 g) | |

Dinner | Tuna steak (100 g) with potatoes (100 g) prepared with olive oil (10 g) and a cup of apple compote (200 mL) | |

Menu 3 | Breakfast | 2 egg yolks (30 g) and bacon (50 g) prepared in olive oil (5 g) and a slice of bread (25 g) |

Lunch | Plate of vegetable soup (200 mL), tuna steak (150 g) with potatoes (100 g) and Swiss chard prepared (100 g) in olive oil (10 g) | |

Snack | 3.2% m.f. yogurt (150 mL) and mixed nuts (40 g) | |

Dinner | Chicken (100 g), rice (60 g) with vegetable salad (150 g) and olive oil (15 g) | |

Menu 4 | Breakfast | 2 slices of graham bread (50 g) with butter (15 g) and honey (20 g) and chamomile tea (200 mL) with sugar (5 g) |

Lunch | Plate of vegetable soup (200 mL), beef stew (150 g) prepared with olive oil (10 g) with pasta (80 g) and vegetable salad (100 g) with olive oil (5 g) | |

Snack | Sliced apple (100 g) with peanut butter (10 g) | |

Dinner | Salmon (100 g) prepared in olive oil (5 g) with bulgur (50 g), vegetable salad (100 g) with olive oil (5 g) and a glass of orange juice (200 mL) |

**Table 2.**List of hepatic lipid metabolism model balances proposed by Pratt et al. [35].

Variable | Balance | Initial Conditions |
---|---|---|

plasma insulin | $\frac{dI}{dt}={k}_{11}+{k}_{22}\xb7\mathrm{erf}\left(\frac{{G}_{B}-v}{cc}\right)-{k}_{d}\xb7I$ | 60 pmol/L |

liver glucose | ${\alpha}_{L}\xb7\frac{d{G}_{L}}{dt}={S}_{G}\left(t\right)-{k}_{gl}\xb7{G}_{L}+{k}_{gl2}\xb7{G}_{B}-\frac{{v}_{LG}\xb7{G}_{L}}{{k}_{LG}+{G}_{L}}-\frac{{v}_{LH}\xb7{G}_{L}}{{k}_{LH}+{G}_{L}}\xb7\left(\frac{1}{1+{k}_{rep}\xb7{P}_{L}}\right)\xb7\phantom{\rule{0ex}{0ex}}{k}_{61}\xb7{P}_{L}$ | 8 mmol/L |

liver glucose-6-phospahte | ${\alpha}_{L}\xb7\frac{d{P}_{L}}{dt}=-\frac{1}{2}\xb7{k}_{yl}\xb7I\xb7{P}_{L}\xb7\left(1+tanh\xb7\left(\frac{{l}_{max}-{Y}_{L}}{{c}_{0}}\right)\right)+\frac{{\beta}_{L}}{1+{k}_{p6}\xb7I}\xb7\left(\frac{{Y}_{L}}{{Y}_{L}+{y}_{0}}\right)-\phantom{\rule{0ex}{0ex}}{k}_{p}\xb7I\xb7{P}_{L}+{k}_{gp}\xb7{L}_{A}+\frac{{\beta}_{6}\xb7{R}_{L}}{1+{k}_{p6}\xb7I}+\frac{{v}_{LG}\xb7{G}_{L}}{{k}_{LG}+{G}_{L}}+\frac{{v}_{LH}\xb7{G}_{L}}{{k}_{LH}+{G}_{L}}\xb7\left(\frac{1}{1+{k}_{rep}\xb7{P}_{L}}\right)-{k}_{61}\xb7{P}_{L}$ | 2.06 mmol/L |

liver glycogen | ${\alpha}_{L}\xb7\frac{d{Y}_{L}}{dt}=\frac{1}{2}\xb7{k}_{yl}\xb7I\xb7{P}_{L}\xb7\left(1+tanh\xb7\left(\frac{{l}_{max}-{Y}_{L}}{{c}_{0}}\right)\right)-\frac{{\beta}_{L}}{1+{k}_{dl}\xb7I}\xb7\left(\frac{{Y}_{L}}{{Y}_{L}+{y}_{0}}\right)$ | 50 mmol/L |

liver pyruvate | ${\alpha}_{L}\xb7\frac{{R}_{L}}{dt}={k}_{pp}\xb7{R}_{M}+{\mu}_{B}+{k}_{p}\xb7I\xb7{P}_{L}-\frac{{\beta}_{6}}{1+{k}_{p6}\xb7I}\xb7{R}_{L}-{k}_{al}\xb7I\xb7{R}_{L}+{\mu}_{B}$ | 0.37 mmol/L |

free fatty acids in liver | ${\alpha}_{L}\xb7\frac{d{A}_{L}}{dt}=3\xb7{k}_{cl}\xb7{T}_{CB}+{k}_{bl}\xb7{A}_{B}+3\xb7{k}_{r}\xb7{T}_{LB}+{k}_{al}\xb7I\xb7{R}_{L}-\frac{3\xb7{v}_{6}\xb7{A}_{L}}{{k}_{6}+{A}_{L}}+\phantom{\rule{0ex}{0ex}}\frac{3\xb7{v}_{10}\xb7{T}_{L}}{{k}_{10}+{T}_{L}}-\frac{3\xb7{v}_{8}\xb7{A}_{L}}{1+{k}_{5}\xb7I}-\frac{{k}_{7}\xb7{A}_{L}}{1+{k}_{5}\xb7I}$ | 0.57 mmol/L |

triacylglycerides secretory pool in liver | ${\alpha}_{L}\xb7\frac{d{S}_{L}}{dt}=\frac{{v}_{6}\xb7{A}_{L}}{{k}_{6}+{A}_{L}}-{k}_{9a}\xb7{S}_{L}$ | 0.0149 mmol/L |

triacylglycerol storage pool in liver | ${\alpha}_{L}\xb7\frac{d{T}_{L}}{dt}=\frac{{v}_{8}\xb7{A}_{L}}{{k}_{8}+{A}_{L}}-\left({k}_{12}\xb7tanh\left(\frac{{v}_{12}-I}{{k}_{13}}\right)+{k}_{14}\right)\xb7\frac{{v}_{9}\xb7{T}_{L}}{{k}_{9}+{T}_{L}}-\frac{{v}_{10}\xb7{T}_{L}}{{k}_{10}+{T}_{L}}$ | 40 mmol/L |

glucose in muscles | ${\alpha}_{L}\xb7\frac{d{G}_{M}}{dt}=\left(1+{k}_{gi}\xb7I\right)\xb7\left({k}_{gm}\xb7{G}_{B}-{k}_{gm2}\xb7{G}_{M}\right)-\frac{{v}_{MH}\xb7{G}_{M}}{{k}_{MH}+{G}_{M}}\xb7\left(\frac{1}{1+{k}_{rep}\xb7{P}_{M}}\right)$ | 0.5 mmol/L |

glycogen in muscle | ${\alpha}_{L}\xb7\frac{d{Y}_{M}}{dt}=\frac{1}{2}\xb7{k}_{ym}\xb7I\xb7{P}_{M}\xb7\left(1+tanh\xb7\left(\frac{{m}_{max}-{Y}_{M}}{{c}_{0}}\right)\right)-\frac{{\beta}_{M}}{1+{k}_{dy}\xb7I}\xb7\left(\frac{{Y}_{M}}{{Y}_{M}+{y}_{0}}\right)$ | 20 mmol/L |

glucose-6-posphate in muscle | ${\alpha}_{L}\xb7\frac{d{P}_{M}}{dt}=\frac{{v}_{MH}\xb7{G}_{M}}{{k}_{MH}+{G}_{M}}\xb7\left(\frac{1}{1+{k}_{rep}\xb7{P}_{M}}\right)-\frac{1}{2}\xb7{k}_{ym}\xb7I\xb7{P}_{M}\xb7\phantom{\rule{0ex}{0ex}}\left(1+tanh\left(\frac{{m}_{max}-{Y}_{M}}{{c}_{0}}\right)\right)+\frac{{\beta}_{M}}{1+{k}_{dy}\xb7I}\xb7\left(\frac{{Y}_{M}}{{Y}_{M}+{y}_{0}}\right)-{k}_{6p}\xb7I\xb7{P}_{M}$ | 0.133 mmol/L |

pyruvate in muscle | ${\alpha}_{L}\xb7\frac{{R}_{M}}{dt}={k}_{6p}\xb7I\xb7{P}_{M}-{k}_{pp}\xb7{R}_{M}-{\mu}_{3}\xb7{R}_{M}\xb7I\xb7P$ | 0.009 mmol/L |

free fatty acids in muscle | ${\alpha}_{L}\xb7\frac{{A}_{M}}{dt}=-3{m}_{s}\xb7I\xb7{A}_{M}+3{m}_{e}+3{k}_{cm}\xb7{T}_{CB}+{k}_{bm}\xb7{A}_{B}+3{k}_{t}\xb7{T}_{LB}-{\mu}_{4}\xb7{A}_{M}\xb7P$ | 0.53 mmol/L |

triacylglycerides in muscle | ${\alpha}_{L}\xb7\frac{d{T}_{M}}{dt}={m}_{s}\xb7I\xb7{A}_{m}-{m}_{e}$ | 14.8 mmol/L |

AMP in muscles | $\frac{dP}{dt}={\mu}_{amp}-{\mu}_{4}\xb7{A}_{M}\xb7P-{\mu}_{3}\xb7{R}_{M}\xb7I\xb7P$ | 0 mmol/L |

adipose triacylglycerides | ${\alpha}_{L}\xb7\frac{d{T}_{A}}{dt}={k}_{aa}\xb7I\xb7{A}_{A}\xb7{G}_{A}-\frac{{\beta}_{f}}{1+{k}_{ft}\xb7{I}^{2}}$ | 500 mmol/L |

adipose free fatty acids | ${\alpha}_{L}\xb7\frac{d{A}_{A}}{dt}=-3\xb7{k}_{aa}\xb7I\xb7{A}_{A}\xb7{G}_{A}+3\xb7{k}_{a}\xb7\left(1+{k}_{ai}\xb7I\right)\xb7{T}_{CB}+3\xb7{k}_{ba}\xb7{T}_{LB}+{k}_{na}\xb7{A}_{B}$ | 0.57 mmol/L |

adipose glycerol | ${\alpha}_{L}\xb7\frac{d{L}_{A}}{dt}=\frac{{\beta}_{f}}{1+{k}_{ft}\xb7{I}^{2}}-{k}_{gp}\xb7{L}_{A}$ | 0.17 mmol/L |

adipose glucose | ${\alpha}_{L}\xb7\frac{d{G}_{A}}{dt}={d}_{ba}\xb7\left(1+{k}_{ga}\xb7I\right)\xb7\left({G}_{B}\xb7{G}_{A}\right)-{k}_{aa}\xb7I\xb7{A}_{A}\xb7{G}_{A}$ | 2.53 mmol/L |

exogenous plasma triacylglycerides | $\frac{d{T}_{CB}}{dt}={S}_{F}\left(t\right)-{k}_{cm}\xb7{T}_{CB}-{k}_{cl}\xb7{T}_{CB}-{k}_{a}\xb7\left(1+{k}_{ai}\xb7I\right)\xb7{T}_{CB}$ | 0 mmol/L |

plasma free fatty acids | $\frac{{A}_{B}}{dt}=-{k}_{bm}\xb7{A}_{B}-{k}_{bl}\xb7{A}_{B}-{k}_{na}\xb7{A}_{B}+\frac{3\xb7{\beta}_{f}}{1+{k}_{ft}\xb7I}+3\xb7{k}_{a}$ | 0.5 mmol/L |

endogenous plasma triacylglycerides | $\frac{d{T}_{LB}}{dt}=F\left(I\right)\xb7\frac{{v}_{9}\xb7{T}_{L}}{{k}_{9}+{T}_{L}}+{k}_{9a}\xb7{S}_{L}-{k}_{r}\xb7{T}_{LB}-{k}_{t}\xb7{T}_{LB}-{k}_{ba}\xb7{T}_{LB}$ | Table 4 |

plasma glucose | $\frac{d{G}_{B}}{dt}={k}_{gl}\xb7{G}_{L}-{k}_{gl2}\xb7{G}_{B}-\left(1+{k}_{gi}\xb7I\right)\xb7\left({k}_{gm}\xb7{G}_{B}-{k}_{gm2}\xb7{G}_{M}\right)-{d}_{ba}\xb7\left(1+{k}_{ga}\xb7I\right)\xb7\left({G}_{B}-{G}_{A}\right)-{\mu}_{1}$ | Table 4 |

**Table 3.**List of hepatic lipid metabolism model parameters proposed by Pratt et al. [35].

Parameter | Value | |
---|---|---|

1. | α_{A} (adipose tissue volume) | 15.60 L |

2. | α_{L} (liver tissue volume) | 1.60 L |

3. | α_{M} (skeletal muscle volume) | 26.4 L |

4. | β_{6} (rate of liver de novo lipogenesis from pyruvate) | 31.6 L/min |

6. | β_{f} (adipose release of triacylglycerides to non-esterified fatty acids) | 0.117 mmol/min |

7. | β_{L} (liver glycogenolysis) | 12 L/min |

8. | β_{m} (muscle glycogenolysis) | 82.5 L/min |

9. | µ_{amp} (national adenosine monophosphate/adenosine diphosphate creation rate) | 1.8 |

10. | µ_{b} (lactate production by red blood cells) | 0.133 mmol/min |

11. | µ_{e} (muscle triglyceride breakdown to free fatty acids) | 0.420 mmol/min |

12. | µ_{s} (muscle free fatty acid esterification to triglycerides) | 7.19 × 10^{6} L mmol/min |

13. | µ_{1} (plasma glucose usage) | 0.588 mmol/min |

14. | µ_{3} (muscle glucose-6-phospahte usage) | 7.839 × 10^{7} L mmol/min |

15. | µ_{4} (muscle free fatty acid usage) | 100 L/min |

16. | c_{0} (small parameters) | 0.1 mmol/L |

17. | cc (range of glucose concentrations over which excess insulin secretion occurs) | 2.5 mmol/L |

18. | d_{ba} (adipose uptake of glucose) | 0.3 mmol/min |

19. | k_{10} (affinity for hydrolysis of triglycerides to secretory pool) | 0.625 mmol/L |

20. | k_{11} (basal insulin secretion rate) | 48 mmol/min |

21. | k_{12} (increased fraction of very low-density lipoprotein 1 secretion by insulin) | 0.2 |

22. | k_{13} (rate at which insulin modifies the fraction of very low-density lipoprotein 1 to very low-density lipoprotein 2 secretion) | 15 mmol/L |

23. | k_{14} (basal very low-density lipoprotein 1 secretion fraction) | 0.6 |

24. | k_{22} (excess insulin secretion rate due to glucose stimulation) | 48 mmol/min |

25. | k_{5} (flux control coefficient for insulin inhibition of free fatty acid oxidation) | 8.23 × 10^{7}/mmol |

26. | k_{6} (affinity for very low-density lipoprotein 2 triglyceride secretion through secretory pathway) | 0.3 mmol/L |

27. | k_{61} (liver glucose dephosphorylation rate) | 4 L/min |

28. | k_{6p} (muscle glucose-6-phospahte to pyruvate conversion rate) | 6.56 × 10^{8} L^{2}/mmol min |

29. | k_{7} (maximum rate of free fatty acid oxidation) | 0.759 L/min |

30. | k_{8} (affinity for esterification of free fatty acids to triglycerides) | 0.625 mmol/L |

31. | k_{9} (affinity of additional bulk lipidation) | 43.583 mmol/L |

32. | k_{9a} (release of very low-density lipoproteins from secretory pathway) | 1 L/min |

33. | k_{a} (adipose free fatty acid uptake of chylomicron triglycerides, insulin independent) | 0.1497 L/min |

34. | k_{aa} (adipose free fatty acid esterification to triglycerides) | 3.11 × 10^{5} L^{2}/mmol min |

35. | k_{ai} (adipose free fatty acid uptake of chylomicron triglycerides, insulin dependent) | 2.08 × 10^{6} 1/mmol |

36. | k_{al} (pyruvate to acetyl coenzyme A conversion rate) | 0.00002 L^{2}/mmol min |

37. | k_{ba} (adipose uptake of endogenous lipoprotein triglycerides) | 0.0104 L/min |

38. | k_{bl} (liver uptake of plasma non-esterified fatty acids) | 0.156 L/min |

39. | k_{bm} (muscle uptake of plasma non-esterified fatty acids) | 0.226 L/min |

40. | k_{cl} (liver free fatty acid uptake of chylomicron triglycerides) | 0.0075 L/min |

41. | k_{cm} (muscle free fatty acid uptake of chylomicron triglycerides) | 0.0449 L/min |

42. | k_{d} (insulin degradation rate) | 1.733 × 10^{14} L/mmol |

43. | k_{dl} (liver glycogenolysis; insulin-inhibited rate) | 3.5 × 18^{8} mmol/L |

44. | k_{dy} (muscle glycogenolysis; insulin-inhibited rate) | 4 × 10^{8} L/mmol |

45. | k_{ft} (adipose release of triglyceride to non-esterified fatty acids; insulin-inhibited rate) | 1.67 × 10^{14} L/mmol |

46. | k_{ga} (rate of glucose diffusion between plasma and adipose mediated by glucose 4 transporters) | 1.67 × 10^{6} |

47. | k_{gi} (glucose diffusion between plasma and muscles, insulin-mediated) | 2.632 × 10^{8} mmol/L |

48. | k_{gl} (plasma glucose diffusion rate to liver) | 0.9277 mmol/L |

49. | k_{gl2} (live glucose diffusion rate to blood) | 0.396 mmol/L |

50. | k_{gm} (plasma glucose diffusion rate to muscle) | 0.0380 mmol/L |

51. | k_{gm2} (muscle glucose diffusion rate to plasma) | 0.0380 mmol/L |

52. | k_{gp} (glucose-6-phospahte uptake from adipose glycerol) | 0.311 L/min |

53. | k_{lp} (rate of plasma triglyceride uptake by adipose tissue) | 0.25 |

54. | k_{LG} (Michaelis–Menten constant of glucokinase in liver) | 8.95 mmol/L |

55. | k_{LH} (Michaelis–Menten constant of hexokinase in liver) | 0.0115 mmol/L |

56. | k_{MH} (Michaelis–Menten constant of hexokinase in muscle) | 8.98 mmol/L |

57. | k_{na} (rate of plasma free fatty acid uptake into adipose free fatty acids) | 0.0697 L/min |

58. | k_{p} (rate of insulin-mediated glucose-6-phoshate to pyruvate) | 1.41 × 10^{7} mmol/L |

59. | k_{p6} (constant of pyruvate conversion to glucose-6-phospate) | 6.56 × 10^{8} L^{2}/mmol min |

60. | k_{pp} (rate of muscle pyruvate transport to liver) | 0.5 |

61. | k_{r} (rate of endogenously derived lipoprotein triglycerides by liver as free fatty acids) | 0.00058 mmol |

62. | k_{rep} (glucose-6-phospahte inhibition constant of hexokinase in muscle) | 2.98 mmol/L |

63. | k_{t} (uptake rate of plasma endogenous triglycerides into muscle free fatty acids) | 0.00348 mmol/L |

64. | k_{yl} (rate of the glycogen synthesis stimulated by insulin) | 1.28 × 10^{6} |

65. | k_{ym} (glycogen synthesis rate) | 21.3641 mmol/L |

66. | l_{max} (maximum glycogen store of liver) | 400 mmol |

67. | m_{max} (maximum glycogen concentration) | 100 mmol |

68. | v (rate of glycogen transport) | 7 mmol |

69. | v_{10} (rate of triglyceride storage conversion to free fatty acids) | 0.1 mmol/min |

70. | v_{12} (constant in triglyceride release into plasma) | 40 mmol L^{−1} |

71. | v_{6} (rate of liver free fatty acid input to secretory pool) | 0.0158 mmol/L |

72. | v_{8} (rate of free fatty acid input to storage pool) | 0.333 mmol/min |

73. | v_{9} (rate of triglyceride release into plasma)) | 0.0159 mmol/L |

74. | v_{LG} (maximum rate of glucokinase in liver) | 14.3 mmol/min |

76. | v_{LH} (maximum rate of hexokinase in liver) | 5.57 mmol/min |

76. | v_{MH} (muscle hexokinase maximum rate) | 54.288 mmol/min |

77. | y_{0} (range of liver glycogen concentration over which the release drops to zero) | 0.1 |

78. | α_{G} (rate of glucose change in diet) | 2 mmol |

79. | α_{F} (rate of fructose change in diet) | 2 mmol/L |

80. | m_{s} (insulin-dependent rate of skeletal muscle storage of free fatty acids in the form of triglycerides) | 0.8 |

81. | m_{e} (rate of skeletal muscle triglyceride breakdown to free fatty acids) | 0.9 |

**Table 4.**Initial concentrations for the hepatic lipid metabolism model simulation based on developed menus.

Menu 1 | Menu 2 | Menu 3 | Menu 4 | |
---|---|---|---|---|

breakfast | G_{B0} = 1.5 mmol/LT _{LB0} = 23 mmol/L | G_{B0} = 44.7 mmol/LT _{LB0} = 1.6 mmol/L | G_{B0} = 7.7 mmol/LT _{LB0} = 16.1 mmol/L | G_{B0} = 23.6 mmol/LT _{LB0} = 4.7 mmol/L |

lunch | G_{B0} = 3.2 mmol/LT _{LB0} = 9.1 mmol/L | G_{B0} = 49.8 mmol/LT _{LB0} = 9.5 mmol/L | G_{B0} = 14.0 mmol/LT _{LB0} = 7.7 mmol/L | G_{B0} = 43.2 mmol/LT _{LB0} = 14.5 mmol/L |

snack | G_{B0} = 2.3 mmol/LT _{LB0} = 5.4 mmol/L | G_{B0} = 16.3 mmol/LT _{LB0} = 4.3 mmol/L | G_{B0} = 8.3 mmol/LT _{LB0} = 9.0 mmol/L | G_{B0} = 13.5 mmol/LT _{LB0} = 1.6 mmol/L |

dinner | G_{B0} = 7.0 mmol/LT _{LB0} = 12.4 mmol/L | G_{B0} = 21.7 mmol/LT _{LB0} = 4.2 mmol/L | G_{B0} = 26.6 mmol/LT _{LB0} = 5.1 mmol/L | G_{B0} = 30.8 mmol/LT _{LB0} = 5.8 mmol/L |

**Table 5.**Energy and macronutrient content (carbohydrates and fats) per meals according to meal plans with daily proportion of saturated fatty acids (SFA).

Meal | Carbohydrates/g | Fats/g | SFA/% | Energy/kcal | |
---|---|---|---|---|---|

Menu 1 | Breakfast | 2.7 | 82.4 | 859.2 | |

Lunch | 5.7 | 32.6 | 408.6 | ||

Snack | 4.1 | 19.5 | 209.9 | ||

Dinner | 12.6 | 44.4 | 584.0 | ||

Per day | 25.1 | 178.9 | 37.0 | 2061.7 | |

Menu 2 | Breakfast | 80.4 | 5.9 | 421.9 | |

Lunch | 89.7 | 33.9 | 904.7 | ||

Snack | 29.3 | 15.3 | 284.1 | ||

Dinner | 39.1 | 15.1 | 395.1 | ||

Per day | 238.5 | 70.2 | 2.7 | 2005.8 | |

Menu 3 | Breakfast | 13.9 | 57.7 | 612.9 | |

Lunch | 25.2 | 27.7 | 517.3 | ||

Snack | 14.9 | 32.4 | 403.6 | ||

Dinner | 47.9 | 18.2 | 461.8 | ||

Per day | 101.9 | 136.0 | 27.1 | 1995.6 | |

Menu 4 | Breakfast | 42.5 | 16.7 | 343.9 | |

Lunch | 77.8 | 52.0 | 971.2 | ||

Snack | 24.3 | 5.6 | 164.4 | ||

Dinner | 55.4 | 20.8 | 533.6 | ||

Per day | 200 | 95.1 | 9.8 | 2013.1 |

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**MDPI and ACS Style**

Kosić, M.; Benković, M.; Jurina, T.; Valinger, D.; Gajdoš Kljusurić, J.; Tušek, A.J.
Analysis of Hepatic Lipid Metabolism Model: Simulation and Non-Stationary Global Sensitivity Analysis. *Nutrients* **2022**, *14*, 4992.
https://doi.org/10.3390/nu14234992

**AMA Style**

Kosić M, Benković M, Jurina T, Valinger D, Gajdoš Kljusurić J, Tušek AJ.
Analysis of Hepatic Lipid Metabolism Model: Simulation and Non-Stationary Global Sensitivity Analysis. *Nutrients*. 2022; 14(23):4992.
https://doi.org/10.3390/nu14234992

**Chicago/Turabian Style**

Kosić, Martina, Maja Benković, Tamara Jurina, Davor Valinger, Jasenka Gajdoš Kljusurić, and Ana Jurinjak Tušek.
2022. "Analysis of Hepatic Lipid Metabolism Model: Simulation and Non-Stationary Global Sensitivity Analysis" *Nutrients* 14, no. 23: 4992.
https://doi.org/10.3390/nu14234992