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Molecular and Cellular Endocrinology 303 (2009) 13–24

Contents lists available at ScienceDirect

Molecular and Cellular Endocrinology

journa l homepage: www.e lsev ier .com/ locate /mce

odeling of glucose regulation and insulin-signaling pathways

in Hoon Chewa,c, Yoke Lin Shiaa, Chew Tin Leea,∗, Fadzilah Adibah Abdul Majida,ee Suan Chuab, Mohamad Roji Sarmidib, Ramlan Abdul Azizb

Department of Bioprocess Engineering, Faculty of Chemical and Natural Resources Engineering, Universiti Teknologi Malaysia, 81310 Skudai, Johor, MalaysiaChemical Engineering Pilot Plant (CEPP), Universiti Teknologi Malaysia, 81310 Skudai, Johor, MalaysiaFaculty of Science, Engineering and Technology, Perak Campus, Universiti Tunku Abdul Rahman, Jalan Universiti, Bandar Baru Barat, 31900 Kampar, Perak, Malaysia

r t i c l e i n f o

rticle history:eceived 9 May 2008eceived in revised form2 December 2008ccepted 26 January 2009

a b s t r a c t

A model of glucose regulation system was combined with a model of insulin-signaling pathways in thisstudy. A feedback loop was added to link the transportation of glucose into cells (by GLUT4 in the insulin-signaling pathways) and the insulin-dependent glucose uptake in the glucose regulation model using theMichaelis–Menten kinetic model. A value of Km for GLUT4 was estimated using Genetic Algorithm. Theestimated value was found to be 25.3 mM, which was in the range of Km values found experimentally from

eywords:lucose regulation

nsulin signalinglucose uptakeLUT4 translocationetabolic modeling

in vivo and in vitro human studies. Based on the results of this study, the combined model enables us tounderstand the overall dynamics of glucose at the systemic level, monitor the time profile of componentsin the insulin-signaling pathways at the cellular level and gives a good estimate of the Km value of glucosetransportation by GLUT4. In conclusion, metabolic modeling such as displayed in this study provides agood predictive method to study the step-by-step reactions in an organism at different levels and shouldbe used in combination with experimental approach to increase our understanding of metabolic disorders

such as type 2 diabetes.

. Introduction

In the human body, blood glucose level is regulated by hor-onal control to bring it back to the normal range whenever

here is a perturbation. This system is known as glucose home-stasis, where cellular processes of glucose-related pathways in thehole body are linked together by hormones such as insulin and

lucagon.However, in a diabetic individual, the hormonal control system

ails to bring the blood glucose level back to the normal range.he cause of type 2 diabetes is still not clearly known because its a complex disease. However, many reports linked this disease toenetic (O’Rahilly et al., 1988; Barnett et al., 1981) and environmen-al factors (Bennett et al., 1992; Helmrich et al., 1991). These twoactors also interact with each other to cause both the initiation androgression of the disease (Jun et al., 1999).

According to DeFronzo (1988) and Yki-Järvinen (1994), type 2iabetes is a result of a variable combination of insulin resistance

nd insulin deficiency. In the case of insulin deficiency, the amountf insulin produced by the beta-cells in the pancreatic islets is notnough to control the blood glucose level. On the other hand, insulinesistance indicates that problems occur during the insulin sig-

∗ Corresponding author. Tel.: +60 7 5535538; fax: +60 7 5581463.E-mail address: ctlee@fkkksa.utm.my (C.T. Lee).

303-7207/$ – see front matter © 2009 Elsevier Ireland Ltd. All rights reserved.oi:10.1016/j.mce.2009.01.018

© 2009 Elsevier Ireland Ltd. All rights reserved.

nal transduction pathways starting from the binding of insulin toinsulin receptor to the translocation of GLUT4.

Many models have been developed throughout the years todescribe quantitatively the whole-organism glucose metabolismand its hormonal regulation (Cobelli and Bergman, 1981). Thesemodels can be categorized into two distinct classes, which are thesimple models and the comprehensive models. Simple models werepioneered by Bolie (1961). These models have been used for thequantification of specific metabolic functions through time-courseanalysis of glucose-insulin kinetics (Ackerman et al., 1965; Segreet al., 1973). Simple structures are usually used in these modelsto ensure that they can be solved easily. However, these simpli-fications sometimes fail to represent metabolic network of highcomplexity. In order to solve this problem, Bergman et al. (1981)introduced the “minimal model” approach, which combined a fewinsulin-dependent/independent tissue compartments. The “mini-mal model” is very simple but it manages to capture the plasmaglucose and insulin dynamics during an intravenous glucose toler-ance test (IVGTT). The Bergman minimal model has been used bymany as a starting point for modifications and extension to includeplasma free fatty acid dynamics (Roy and Parker, 2006), separate

the effects of glucose production from utilization (Cobelli et al.,1986, 1999), and capture the observed absorption, distribution, anddisposal dynamics (Hovorka et al., 2002).

While minimal models are mainly used to computate glucoseand insulin indexes in glucose tolerance test, comprehensive mod-

14 Y.H. Chew et al. / Molecular and Cellular Endocrinology 303 (2009) 13–24

F m; (ii)a antity( rstitiaa n gluc

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ig. 1. Block diagram of the Cobelli model (Cobelli et al., 1982): (i) glucose subsystend extracellular fluids (mg); u1p = quantity of pancreatic stored insulin (�U); u2p = qu�U); u12 = quantity of insulin in the liver (�U); u13 = quantity of insulin in the intend F1–F7 represent non-linear functions while Ix and Iu represent the test inputs o

ls have been mainly developed for predicting the overall systemehaviour under a variety of perturbations and for the interpreta-ion and design of experiments (Cobelli et al., 1982). One such models the whole-body glucose regulation model developed by Cobelli etl. (1982). This model was later validated and modified to considerral glucose tolerance test input (Cobelli and Mari, 1983). A moreecent comprehensive model of whole-body glucose homeostasisas been developed by Kim et al. (2007), which claimed to be therst mechanistic model of glucose homeostasis that links cellularetabolism to whole-body responses and incorporates effects of

ormonal control on fuel metabolism of various tissues/organs.Even though many models are available to describe the inter-

ction between blood glucose and hormonal signals, most of theseodels only allow us to monitor the dynamics of glucose, insulin,

lucagon and other main metabolites at systemic level. The detailednzymatic reactions that take place at the cellular level such as thensulin-signaling pathways are not displayed in the models. Thesetep-by-step reactions are crucial as they could be the ‘bottleneck’hich brings upon type 2 diabetes.

In this study, two independent mathematical models of differ-nt organizational levels were combined. A mathematical model ofhole-body blood glucose regulation system (systemic level) was

ombined with a mathematical model of insulin-signaling path-ays (cellular level). The insulin level simulated in the glucose

egulation system was used as an input for the insulin-signalingathways. Besides that, a feedback loop was added to send the out-ut from the signaling pathways – in this case, GLUT4 translocationback to the glucose regulation system.

. Theory

.1. Mathematical model of glucose homeostasis

The mathematical model used in the present study was devel-ped and validated by Cobelli and Mari (1983). This model was aodification from an earlier compartmental model developed by

insulin subsystem; (iii) glucagon subsystem. u1 = quantity of glucose in the plasmaof immediate releasable pancreatic insulin (�U); u11 = quantity of insulin in plasma

l fluids (�U); u2 = quantity of glucagon in the plasma and interstitial fluids (pg). Wose and insulin, respectively.

Cobelli et al. (1982) to consider oral glucose tolerance test input.Even though this model has been developed more than 20 yearsago, it is still referred to in current studies like Ward and Martin(2006). This model describes the interactions between glucose,insulin, and glucagon. In reality, glucose homeostasis is more com-plex and involves other components such as cytokines and free fattyacids (Kitano et al., 2004). However, the main focus of this study isthe insulin-signaling pathways and its interaction with the overallinsulin-dependent glucose uptake. Therefore, the Cobelli model issufficient for this study.

Compared to other models of similar complexity, the Cobellimodel was developed to deal with different kinds of insulin andglucose inputs under normal and pathological conditions. Othermodels such as the one describing the whole-body metabolism andhormonal control by Kim et al. (2007) concentrated on hormonalregulation during exercise while Dalla Man et al. (2007) focused onthe oral route of a mixed meal.

The Cobelli model used in the present study is also capable ofdescribing human experimental data accurately. Besides, distinctinsulin granule pools were included in the model and thereforethe model is capable of simulating the biphasic patterns of insulinsecretion in humans and animals (DeFronzo et al., 1979; Curry etal., 1968). The biphasic kinetics is essential as the first phase insulinsecretion in response to glucose has been found to be lost in diabeticpatients (Efendic et al., 1988; Temple et al., 1992).

The Cobelli model is shown in Fig. 1. It is divided into threesubsystems. The glucose subsystem consists of one compartment,which represents the extracellular fluids. This subsystem describesthe distribution and metabolism of glucose that cause changesto the glucose concentration in the compartment. The glucose-controlled unit processes being considered in this subsystem are

net hepatic glucose balance (NHGB) (the difference between glu-cose production and glucose uptake in the liver), renal excretion ofglucose, insulin-dependent glucose utilization (mainly by musclesand adipose tissues), and insulin-independent glucose utilization(mainly by the central nervous system and the red blood cells).

Y.H. Chew et al. / Molecular and Cellular Endocrinology 303 (2009) 13–24 15

F e symf te reaA ). The

tcapoTseA

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ig. 2. Complete model of metabolic insulin signaling pathways with feedback. Thor the reaction as denoted by the full-line arrows. Arrows with dashed lines indicarrows with dash-and-dot lines mark the feedback pathways (Sedaghat et al., 2002

The dynamics of glucose is regulated by two hormonal con-rollers, insulin and glucagon. The insulin subsystem has fiveompartments, which are the stored pancreatic insulin, immedi-te releasable pancreatic insulin, liver and portal plasma insulin,lasma insulin, and insulin in the interstitial fluids (ISFs). On thether hand, the glucagon subsystem only has one compartment.he three subsystems interact with one another through controlignals, as shown by the dashed lines in Fig. 1. The mathematicalquations and parameter values for the Cobelli model are shown inppendix A.

.2. Mathematical model of insulin-signaling pathways

The signal transduction pathways are a combination of a fewomplex subsystems. Each subsystem is made up of many biochem-cal syntheses and reactions involving many molecules. Therefore,

athematical modeling could provide a theoretical frameworkhich enables us to understand cell behaviour based on experi-ental data (Sedaghat et al., 2002). A comprehensive model should

e able to explicitly represent many known signaling componentsnd their behaviours towards stimulants.

Many models that describe the pathways related to signal trans-uction are available in the literature. Most of these models focusn specific subsystems such as insulin receptor binding kineticsWanant and Quon, 2000; Schäffer, 1994), protein kinase cascadeHeinrich et al., 2002) and GLUT4 exocytosis (Holman et al., 1994;uon, 1994). There are also spatio-temporal models which study

he diffusion of protein kinases and intracellular enzymes duringignal transduction (Haugh, 2002; Kholodenko et al., 2000).

Mathematical models that describe the insulin-signaling path-ays comprehensively, beginning from the binding of insulin to

ell surface receptor until the translocation of GLUT4 are still

bols k1–k14 and k−1–k−14 represent the association and dissociation rate constantsctions where the value of a state variable influences the value of the rate constant.symbol for each component is shown in Appendix B.

scarce. One such model is presented in Sedaghat et al. (2002). Itis a combination of previous models of insulin receptor bindingkinetics (Wanant and Quon, 2000), receptor recycling (Quon andCampfield, 1991a) and GLUT4 translocation (Quon, 1994; Quon andCampfield, 1991b), with an addition of postreceptor signaling sub-system (Sedaghat et al., 2002). The model has been validated withexperimental data and is therefore selected for this study. It is illus-trated in Fig. 2.

All the equations, initial conditions and parameter values arelisted in Appendix B. The initial conditions were determined by run-ning a steady-state analysis on the insulin-signaling model using acontinuous insulin input at the basal value in the interstitial fluidof the Cobelli model. This is because the original initial values usedby Sedaghat et al. (2002) were for an initial insulin concentrationof zero. For further details on the derivations and parameters used,please refer to Sedaghat et al. (2002).

2.3. Combined model and feedback loop

The glucose regulation model as presented in Section 2.1 wascombined with the model of insulin-signaling pathways describedin Section 2.2. The insulin level generated from the glucose regu-lation model was used as input to the insulin-signaling pathways.According to Bergman et al. (1979), Sherwin et al. (1974) and Insel etal. (1975), the insulin which controls the peripheral glucose utiliza-tion is that in the interstitial fluid of muscle and adipose tissue. Astudy was conducted by Poulin et al. (1994) to investigate the cor-

relation between glucose uptake and the insulin level in plasma,hindlimb lymph (interstitial fluid) and thoracic lymph. Their datashow that glucose uptake rate is strongly correlated with interstitialinsulin and that the rate of increase of glucose uptake is determinedby transendothelial insulin transport into insulin-sensitive tissue.

1 ellular Endocrinology 303 (2009) 13–24

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(Grodsky, 1972; Cerasi et al., 1974; O’Connor et al., 1980; Nesher andCerasi, 2002), such as the existence of different insulin granule poolsin the pancreatic beta-cells. Even though the mechanisms respon-sible for the biphasicity are only partially understood, the biphasic

6 Y.H. Chew et al. / Molecular and C

herefore, in the present study, the insulin input in the insulin-ignaling model, x1, was set to be equal to the concentration ofnsulin in the interstitial fluid, y4 in the glucose regulation models follows:

1 = (6.622 × 10−12)y4 (1)

Eq. (1) shows the unit conversion of y4 from �U/ml to mol/l for1.

A feedback loop was also added to send the output from thensulin signaling back to the glucose regulation system. The rela-ionship identified for the feedback loop was the dependence ofnsulin-dependent glucose uptake, F4 (glucose regulation model),n the concentration of plasma glucose, y1 (glucose regulationodel), and the amount of GLUT4 translocated to the cell sur-

ace, x21 (insulin-signaling pathways). A Michaelis–Menten kineticodel was used to describe the rate of glucose uptake into cells byLUT4 transporter. The Michaelis–Menten model is usually used touantify glucose transport because its molecular mechanism is stillot sufficiently understood to support a more sophisticated modelRoussel et al., 1998). The kinetic model for the feedback loop ishown as follows:

4 = Vmaxy1

Km + y1(2)

here y1 is the concentration of plasma glucose in mM (Eq. (A.9) inppendix A after unit conversion) and Km is the Michaelis–Mentenonstant for the transportation of glucose by GLUT4. Vmax is theaximal uptake rate given by:

max = TN[GT] (3)

TN is the turnover number and [GT] is the number of cell surfaceransporters (Nishimura et al., 1993). The number of cell surfaceransporter is always changing, depending on the stimulation levely insulin. In the insulin-signaling model (Section 2.2) by Sedaghatt al. (2002), GLUT4 translocation is expressed in terms of per-entage distribution in between cell surface and intracellular poolhere x21 is the percentage of cell surface GLUT4. Therefore

max = TN[TCT][TNC]x21

100(4)

here [TCT] is the total number of GLUT4 per cell while [TNC] is theotal number of cells in the human body containing GLUT4, assum-ng that the expression level of each cell is the same. Combiningqs. (2) and (4) yield

4 = Cx21y1

Km + y1(5)

here

= TN[TCT][TNC]100

(6)

The values of Km and C were determined by Genetic AlgorithmGA) using Eq. (5) and the F4 values generated in the glucose reg-lation model (Eq. (A.24) in Appendix A). The computed values ofm and C were later used to calculate a new glucose uptake rate,4MM

, which served as a feedback replacement of the original F4 inhe glucose regulation model.

. Methodology

The ordinary differential equations were solved using theon-linear curve-fitting function in the Matlab 7.0 programming

anguage. The ordinary differential equation (ODE) solver usedas Runge–Kutta 4,5. The glucose regulation model and model of

nsulin-signaling pathways were first solved separately and com-ared to the literature to ensure reproducibility of the simulationesults reported in the literature. Both models were later combined

Fig. 3. Biphasic insulin response of the Cobelli model at a glucose input of 7000 mgfor 3 min.

by setting the insulin input in the insulin-signaling model, x1, to beequal to the concentration of insulin in the interstitial fluid, y4 inthe glucose regulation model (Eq. (1)). The values of Km and C werethen determined using GA as described in the previous section.

4. Results and discussions

The combined model was simulated by inducing a glucose inputof 7000 mg for 3 min. This input is the same as used in Ward andMartin (2006) and the simulation results for the glucose, insulin andglucagon dynamics are also the same, as discussed in the following.

Fig. 3 shows the plasma concentration–time profile of insulin,which is as produced in Ward and Martin (2006). According to Fig. 3,the insulin was secreted in two phases. This biphasic response hasbeen reported in vivo in humans since 1963 (Cerasi and Luft, 1963).Various models have been proposed to explain the biphasic patterns

Fig. 4. The concentration of insulin in the interstitial fluids as simulated using theCobelli model.

Y.H. Chew et al. / Molecular and Cellular Endocrinology 303 (2009) 13–24 17

Fig. 5. Simulation results of the insulin signaling model in the present work with a continuous insulin input.

18 Y.H. Chew et al. / Molecular and Cellular Endocrinology 303 (2009) 13–24

Fig. 6. Simulation results of the insulin signaling model after a step input of 1 × 10−7 M insulin for 15 min (Sedaghat et al., 2002).

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Y.H. Chew et al. / Molecular and C

inetics is very important. This is because a reduction in the firsthase is seen early in the development of type 2 diabetes (Pigon etl., 1996; Dornhorst, 2001). Therefore, it has been suggested thathe biphasic pattern be used to predict the disease (Knowler et al.,995).

Fig. 4 is the concentration–time profile of the insulin in thenterstitial fluid. Under the stimulation of glucose for 3 min, the con-entration of insulin in the interstitial fluid increased rapidly fromts basal value, reaching its peak after 50 min, and then decreasedlowly. This profile was used as the input to the insulin-signalingodel. In most studies, a transient step input of insulin is used.owever, by combining the insulin signaling with the whole-bodylucose regulation model as shown in the present work, the effectf a continuously changing insulin input could be studied.

The simulation results of the insulin-signaling model are illus-rated in Fig. 5 (results of the present work with a continuous inputf Fig. 4) and Fig. 6 (results produced in Sedaghat et al. (2002) withtransient step input of 1 × 10−7 M for 15 min).

Comparing Figs. 5A and 6A, a step input caused a steep con-entration decrease in the unbound receptor and a rapid increasefter insulin was removed. On the other hand, a continuous insulinnput induced a smooth-curve response. Fig. 5B and C also showmooth curves as compared to Fig. 6B and C. The sudden short risen the concentration of once-bound phosphorylated receptors afternsulin removal in Fig. 6B (observed because the twice-bound phos-horylated receptors had to pass through the once-bound state toeturn to the unbound state) is not observed in Fig. 5B. Instead thence-bound receptors continued to increase in a slow manner evenfter the amount of insulin started declining at minute 50 beforeescending. This is due to a continuous and slow decline in thewice-bound receptors.

In Fig. 5D, unphosphorylated IRS-1 decreased when intersti-ial insulin increased, and slowly climbed up when the amount ofnsulin was reduced. At first, the reduction in unphosphorylatedRS-1 gave rise to an increase in the tyrosine-phosphorylated IRS-. However, after some time, most of the unphosphorylated IRS-1as converted to serine-phosphorylated IRS-1 due to the negative

eedback by the downstream PKC-� (Sedaghat et al., 2002). The tran-ient oscillatory behaviour as observed in Fig. 6D is not significantn Fig. 5D as the insulin input was continuous and varied slowly.

The concentration–time profiles in Fig. 5E, 5F and 5G forctivated PI 3-kinase, PI(3,4,5)P3, PI(3,4)P2, and activated PKC-� dis-layed a qualitatively similar dynamics. This trend has also beeneported in Sedaghat et al. (2002). According to the experimentalata in Standaert et al. (1997, 1999), PKC-� displays biphasic activa-ion in rat adipocytes in response to insulin. This biphasic responseould also be seen in this study, with the first peak at minute 25 andhe second peak in minute 50 in Fig. 5G. However, the biphasicitys not obvious, which is again due to the effect of a continuous andlow-changing insulin input.

Referring to Fig. 5H, the percentage of GLUT4 translocation atn insulin input of about 0.20 nM (maximum value of Fig. 4) was1.5%. Sedaghat et al. (2002) reported that the GLUT4 transloca-ion for maximal glucose uptake is at 38% (as shown in Fig. 6H).herefore, 21.5% out of the maximum 38% is about 56.6%. Thisesult matches well with the experimentally generated insulinose–response curve of Stagsted et al. (1993), which has an EC50alue for glucose uptake at 0.17 nM insulin.

.1. Kinetic parameters of GLUT4

The kinetics of GLUT4-facilitated glucose transport has receivedlot of attention due to the importance of GLUT4 in controlling glu-ose uptake in skeletal muscles and adipose tissue. Two importantinetic parameters for estimating the ‘intrinsic activity’ of GLUT4re the Km constant and the turnover number. The Km constant

Endocrinology 303 (2009) 13–24 19

indicates the affinity of the transporter for the molecule to be trans-ported. The lower the constant, the higher is the affinity. Turnovernumber, on the other hand, indicates the maximal rate of trans-portation for each cell surface transporter.

Nishimura et al. (1993) reviewed that some of the difficulties inmeasuring the kinetic parameters of GLUT4 are the lack of mam-malian cells that express exclusively one single isoform and theproblem in determining the number of cell surface transporters.Nevertheless, many groups have attempted to determine the val-ues of Km for glucose transport by GLUT4. There is a vast differencein the values reported, ranging from 2 to 10 mM, which is due todifferent cell source (e.g. 3T3-L1 cells, Xenopus oocytes) and glu-cose analog (e.g. 2-deoxy-d-glucose, 3-O-methyl-d-glucose) used(Bell et al., 1990, 1993; Thorens, 1993; Shepherd and Kahn, 1999).Roussel et al. (1998) attempted to measure the Km constant in thehuman body. They conducted an NMR study of glucose transportin human skeletal muscles of normal volunteers under hyperin-sulinemic clamp conditions. Both their in vivo and in vitro studiesreported Km values in the range of 15–25 mM for all three conditionsof euglycemia, mild hyperglycemia and high hyperglycemia.

In the present study, Genetic Algorithm was used to determinethe value of Km. The value was found to be 25.3 mM. Even thoughthis value is not in the range reported for transporters expressedin mammalian cells as mentioned earlier, it matches well with thehuman experimental results of Roussel et al. (1998). The fact thatthe glucose regulation model and insulin-signaling model are twoindependent models and yet produces results close to experimentsafter combination is a positive sign of the accuracy of the combinedmodel.

The value of C was found to be 4.56 × 10−6. Referring to Eq.(6), the turnover number can be determined if the total numberof GLUT4 per cell and the total number of human cells containingGLUT4 are known. The total amount of GLUT4 expressed per cell isusually estimated in vitro using immunoblotting for the determina-tion of kinetic parameters such as turnover number (Nishimura etal., 1993) or transport capacity (Palfreyman et al., 1992). However,the expression level of GLUT4 varies depending on the types of tis-sue and their physiological states (Richardson et al., 1991; Cusin etal., 1990), as well as exercise training (Ezaki et al., 1992). Besidesthat, the number of cells that make up a human body is also notconstant, even within the same individual. Therefore, the turnovernumber of GLUT4 would not be estimated in this study as it wouldnot have any significant meaning.

4.2. Glucose uptake

The values of Km and C as found in this study were used to gener-ate a new profile of glucose uptake, F4MM

, using Eq. (5). The profilesof the original glucose uptake, F4, and F4MM

are shown in Fig. 7A.The horizontal axis is displayed as data number. A total of 4200data were built and used in the computation. The sum square oferror was 2.12 × 10−9.

Referring to Fig. 7A, it can be seen that the overall profiles ofboth F4 and F4MM

are similar. One significant difference is the sud-den small decrease in the beginning of the F4MM

profile beforeit continued to increase, which can be explained by consideringthe plot as a time profile. Based on the profiles of blood glucoselevel (Fig. 7B) and GLUT4 translocation (Fig. 7C), at the beginningafter glucose input, blood glucose level shot up rapidly. Thus glu-cose uptake increased. However, there was a short lag phase in thetranslocation of GLUT4. Therefore, all the cell surface transporters

were occupied immediately in the beginning. Blood glucose con-tinued to increase but no free surface transporter was availableand this became the rate-limiting step, causing a small decreasein glucose uptake rate. When the translocation of GLUT4 startedto catch up, glucose uptake began to rise again, reached a peak,

20 Y.H. Chew et al. / Molecular and Cellular

F(

aldCFkigcatmrtiici

k02, which is a function of u1. As for the non-linear functions, F3

ig. 7. Profiles of glucose uptake (A), blood glucose level (B) and GLUT4 translocationC) using the Km and C values determined from Genetic Algorithm.

nd then decreased slowly when translocation and blood glucoseevel dropped. There are many possible explanations why the sud-en drop was not displayed in the original F4 generated by theobelli model. F4 considered the human body as a whole while4MM

was generated on a single-cell basis using Michaelis–Menteninetic analysis. Due to spatio-temporal differences in the level ofnsulin and glucose, there would be small time difference in thelucose uptake profile of each single cell unless the cells are syn-hronized. Intercellular synchronization have been reported suchs the synchronization of pancreatic islets during insulin secre-ion (Pedersen et al., 2005), synchronized contraction of cardiacyocytes (De Melo, 1982) and circadian synchronization of cellular

hythm (To et al., 2007). These synchronies are important for sys-emic regulation in response to environmental changes. However,

t does not seem to serve any purpose to synchronize the first peakn the F4MM

profile in Fig. 7A and to the best of our knowledge, noonfirmative report has been made on it so far. Thus, the first peaks attenuated when glucose uptake is measured as a whole. Another

Endocrinology 303 (2009) 13–24

possible explanation is that the sampling frequency of in vivo and invitro studies for the measurement of whole-body glucose uptake isnot high enough that the little peak is not observed. Nevertheless,until further studies are conducted, the first explanation seems tobe more valid.

5. Conclusion

Type 2 diabetes is a fast growing health problem attacking thewhole world. It is a disorder in the blood glucose regulation systemand is caused by many factors. In this study, the insulin-signalingmodel was combined with the whole-body model of blood glu-cose regulation system. A feedback loop was also added to linkthe output from the insulin-signaling pathways back to the glucoseregulation system. The simulated results of the combined modelmatched well with various experimental data, which is a positivesign on the accuracy of the model. In conclusion, modeling couldprovide an alternative method for metabolic analysis and should beused in tandem with experimental work for better understandingon the causes and potential treatment of type 2 diabetes.

Acknowledgements

The authors wish to thank the Ministry of Science, Technologyand Innovation of Malaysia (MOSTI) for providing financial supportthrough the eScienceFund Grant No. 79045, Research ManagementCentre, Universiti Teknologi Malaysia (UTM) and also ChemicalEngineering Pilot Plant (CEPP), UTM for their technical support.

Appendix A

The ordinary differential equations and their initial values forthe compartments in the modified model are given as follows:

du1

dt= NHGB(u1, u12, u2) − F3(u1) − F4(u1, u13) − F5 + Ix(t),

u1(0) = u10 (A.1)

du1p

dt= −k21u1p + k12u2p + W(u1), u1p(0) = u1p0 (A.2)

du2p

dt= k21u1p − (k12 + k02(u1))u2p, u2p(0) = u2p0 (A.3)

du11

dt= −(m01 + m21 + m31)u11 + m12u12 + m13u13 + Iu(t),

u11(0) = u110 (A.4)

du12

dt= −(m02 + m12)u12 + m21u11 + k02(u1)u2p,

u12(0) = u120 (A.5)

du13

dt= −m13u13 + m31u11, u13(0) = u130 (A.6)

du2

dt= −h02u2 + F7(u1, u13), u2(0) = u20 (A.7)

where mij, hij, and kij are constant rate parameters (min−1), except

represents renal excretion of glucose, F4 is the peripheral insulin-dependent glucose utilization, while F5 is the peripheral insulin-independent glucose utilization. Ix(t) is the rate of glucose input.W(x1) is a function representing insulin synthesis while Iu(t) is the

ellular

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Y.H. Chew et al. / Molecular and C

nsulin test input. F7 is the endogenous release of glucagon, whichs dependent on blood glucose and interstitial-fluid insulin.

The net hepatic glucose balance, NHGB, is given by the followingquation:

HGB = F1(u1, u12, u2) − F2(u1, u12) (A.8)

here F1 is the glucose production in the liver while F2 is the glucoseptake in the liver.

As experimentally measured variables are expressed in terms ofoncentration, the concentrations of glucose, insulin and glucagonre denoted as follows:

1 = u1

V1(A.9)

2 = u11

V11(A.10)

3 = u12

V12(A.11)

4 = u13

V13(A.12)

5 = u2

V2(A.13)

here y1 is the concentration of plasma glucose (mg/100 ml), while2, y3 and y4 are the concentrations in �U/ml of insulin in plasma,iver and interstitial fluids respectively. y5 is the concentration oflood glucagon (pg/ml). V1, V11, V12, V13 and V2 are the distributionolumes of various compartments, with their assumed values takenrom the literature (Norwich, 1977; Sherwin et al., 1974; Insel et al.,975) as 20, 4.5, 3.1, 10.6 and 20% of body weight, respectively.

The metabolic unit processes in the model consist of manynzyme-catalyzed biochemical reactions. Cobelli et al. (1982)dopted an aggregated approach to provide an overall mathe-atical description for their input–output behaviour. They used

yperbolic-tangent function to represent threshold and saturationhenomena, which are typically observed in the processes of pro-uction of material from one or more substrates (Charette, 1969).his form of function allows the effect of the controlling variablesuch as hormones to be easily incorporated into these descriptionss a change in the saturation value and in the slope of the character-stic. The functions for each subsystem in the model are illustrateds follows:

Liver glucose production:

1(u1, u12, u2) = a11G1(u2)H1(u12)M1(u1) (g/ min) (A.14)

here

1(u2) = 0.5{1 + tanh[b11(e21 + c11)]} (A.15)

1(u12) = 0.5{1 − tanh[b12(e12 + c12)] + [1 − tanh(b13e12 + c13)]}(A.16)

1(u1) = 0.5{1 − tanh[b14(ex + c14)]} (A.17)

e21, e12 and ex are the differences between the actual plasmaoncentrations of glucagon, insulin, and glucose and their basaloncentration values, respectively.

Liver glucose uptake:

2(u1, u12) = H2(u12) + M2(u1) (g/min) (A.18)

here

2(u12) = 0.5a21{1 + tanh[b21(e12 + c21)]} (A.19)

2(u1) = 0.5a22{1 + tanh[b22(ex + c22)]} (A.20)

Endocrinology 303 (2009) 13–24 21

Renal glucose excretion:

F3(u1) = M31(u1)M32(u1) if u1 > 2.52 × 104 (g/min)= 0 if u1 ≤ 2.52 × 104 (g/min)

(A.21)

where

M31(u1) = 0.5{1 + tanh[b31(u1 + c31)]} (A.22)

M32(u1) = a31u1 + a32 (A.23)

Insulin-dependent peripheral glucose utilization:

F4(u1, u13) = a41H4(u13)M4(u1) (g/min) (A.24)

where

H4(u13) = 0.5{1 + tanh[b41(e13 + c41)]} (A.25)

M4(u1) = 0.5{1 + tanh[b42(ex + c42)]} (A.26)

In this case, e13 is the difference between the actual and basalvalues of interstitial-fluid insulin concentration.

Insulin-independent glucose uptake:

F5(u1) = M51(u1) + M52(u1) (g/min) (A.27)

where

M51(u1) = a51tanh[b51(ex + c51)] (A.28)

M52(u1) = a52ex + a53 (A.29)

Insulin synthesis:

W(u1) = 0.5aw{1 + tanh[bw(ex + cw)]} (U/min) (A.30)

Insulin secretion:

F6(u2p, u1) = 0.5a6{1 + tanh[b6(ex + c6)]}u2p (�U/min) (A.31)

Glucagon secretion:

F7(u1, u13) = a71H7(u13)M7(u1) (�g/min) (A.32)

where

H7(u13) = 0.5{1 − tanh[b71(e13 + c71)]} (A.33)

M7(u1) = 0.5{1 − tanh[b72(ex + c72)]} (A.34)

Initial conditions (Ward and Martin, 2006):

u1(0) = ubasal (�g)

u11(0) = u11(basal) (�U)

u13(0) = m31 × u11(0)m13

(�U)

u12(0) = (m01 + m21 + m31) × u11(0) − m13 × u13(0)m12

(�U)

F6(0) = (m02 + m12) × u12(0) − m21 × u11(0) (�U/min)

u2p(0) = F6(0)0.5(1 + tanh(b6 × c6))

(�U)

u (0) = F6(0) + k12 × u2p(0)(�U)

1p k21

u2(0) = u2(basal) (pg)

All the parameter values are listed in Table A1.

22 Y.H. Chew et al. / Molecular and Cellular Endocrinology 303 (2009) 13–24

Table A1Baseline parameter values for a normal state.

Glucose subsystem

Liver production Liver uptake Renal excretion

a11 = 6.727 a21 = 0.56 a31 = 7.14 × 10−6

b11 = 2.1451 a22 = 0.105 a32 = −9.15 × 10−2

b12 = 1.18 × 10−2 b21 = 1.54 × 10−2 b31 = 0.143b13 = 0.15 b22 = 0.1 c31 = −2.52 × 104

b14 = 5.9 × 10−2 c21 = −172c11 = −0.854 c22 = −25c12 = −100c13 = 9.8c14 = −20

Glucose subsystem

Insulin-dependent peripheral utilization Insulin-independent peripheral utilization

a41 = 3.45 a51 = 8.4 × 10−2

b41 = 1.4 × 10−2 a52 = 3.22 × 10−4

b42 = 1.4 × 10−2 a53 = 3.29 × 10−2

c41 = −146 b51 = 2.78 × 10−2

c42 = 0 c51 = 91.5

Insulin subsystem

Beta-cell secretion Distribution and metabolism

k12 = 3.128 × 10−2 m01 = 0.125k21 = 4.34 × 10−3 m02 = 0.185aw = 0.74676 m12 = 0.209a6 = 1 m13 = 0.02bw = 1.09 × 10−2 m21 = 0.268b6 = 5 × 10−2 m31 = 0.042cw = −175c6 = −44

Glucagon subsystem

Alpha-cell secretion Distribution and metabolism

a71 = 2.3479 h02 = 0.086643b71 = 6.86 × 10−4

b72 = 3 × 10−2

c71 = 99.246c

A ncentrg Mari

A

•••

•

•

•

•

•

•••••

•

72 = 40

normal individual of 70 kg body weight is considered, with plasma glucose colucagon concentration y5 = 75 pg/ml. For further details, please refer to Cobelli and

ppendix B

The meaning of each symbol is shown below:

x1 = free insulin concentrationx2 = concentration of unbound surface insulin receptorsx3 = concentration of unphosphorylated once-bound surfacereceptorsx4 = concentration of phosphorylated twice-bound surface recep-torsx5 = concentration of phosphorylated once-bound surface recep-torsx6 = concentration of unbound unphosphorylated intracellularreceptorsx7 = concentration of phosphorylated twice-bound intracellularreceptorsx8 = concentration of phosphorylated once-bound intracellularreceptorsx9 = concentration of unphosphorylated IRS-1x10 = concentration of tyrosine-phosphorylated IRS-1

x10a = concentration of serine-phosphorylated IRS-1x11 = concentration of unactivated PI 3-kinasex12 = concentration of tyrosine-phosphorylated IRS-1/activated PI3-kinase complexx13 = percentage of PI(3,4,5)P3 out of the total lipid population

ation y1 = 91.5 mg/100 ml, plasma insulin concentration y2 = 11 �U/ml, and blood(1983).

• x14 = percentage of PI(4,5)P2 out of the total lipid population• x15 = percentage of PI(3,4)P2 out of the total lipid population• x16 = percentage of unactivated Akt• x17 = percentage of activated Akt• x18 = percentage of unactivated PKC-�• x19 = percentage of activated PKC-�• x20 = percentage of intracellular GLUT4• x21 = percentage of cell surface GLUT4

The differential equations describing the mass balance of eachcomponent in Fig. 2 are as follows:

x1 = insulin input (B.1)

dx2

dt= k−1x3 + k−3 [PTP] x5 − k1x1x2 + k−4x6 − k4x2 (B.2)

dx3

dt= k1x1x2 − k−1x3 − k3x3 (B.3)

dx4

dt= k2x1x5 − k−2x4 + k−4′ x7 − k4′ x4 (B.4)

dx5

dt= k3x3 + k−2x4 − k2x1x5 − k−3[PTP]x5 + k−4′ x8 − k4′ x5 (B.5)

dx6

dt= k5 − k−5x6 + k6[PTP](x7 + x8) + k4x2 − k−4x6 (B.6)

ellular

••••••••••••••••••••••

Y.H. Chew et al. / Molecular and C

dx7

dt= k4′ x4 − k−4′ x7 − k6[PTP]x7 (B.7)

dx8

dt= k4′ x5 − k−4′ x8 − k6[PTP]x8 (B.8)

dx9

dt= k−7[PTP]x10 − k7x9(x4 + x5)/(IRp)

+k−7′ x10a − k7′ [PKC]x9 (B.9)

dx10

dt=k7x9(x4 + x5)/(IRp) + k−8x12 − (k−7[PTP] + k8x11)x10 (B.10)

dx10a

dt= k7′ [PKC]x9 − k−7′ x10a (B.11)

dx11

dt= k−8x12 − k8x10x11 (B.12)

dx12

dt= k8x10x11 − k−8x12 (B.13)

dx13

dt= k9x14 + k10x15 − (k−9[PTEN] + k−10[SHIP])x13 (B.14)

dx14

dt= k−9[PTEN]x13 − k9x14 (B.15)

dx15

dt= k−10[SHIP]x13 − k10x15 (B.16)

dx16

dt= k−11x17 − k11x16 (B.17)

dx17

dt= k11x16 − k−11x17 (B.18)

dx18

dt= k−12x19 − k12x18 (B.19)

dx19

dt= k12x18 − k−12x19 (B.20)

dx20

dt= k−13x21 − (k13 + k13′ )x20 + k14 − k−14x20 (B.21)

dx21

dt= (k13 + k13′ )x20 − k−13x21 (B.22)

Initial conditions:

x1(0) = 6.49 × 10−11 Mx2(0) = 8.79 × 10−13 Mx3(0) = 1.37 × 10−18 Mx4(0) = 3.45 × 10−18 Mx5(0) = 1.77 × 10−14 Mx6(0) = 1.06 × 10−13 Mx7(0) = 1.64 × 10−20 Mx8(0) = 8.44 × 10−17 Mx9(0) = 9.19 × 10−13 Mx10(0) = 5.66 × 10−14 Mx10a(0) = 2.43 × 10−14 Mx11(0) = 9.96 × 10−14 Mx12(0) = 3.98 × 10−16 Mx13(0) = 0.767%x14(0) = 98.5%x15(0) = 0.718%x16(0) = 98.4%x17(0) = 1.61%x18(0) = 98.4%

x19(0) = 1.61%x20(0) = 88.2%x21(0) = 13.4%

Model parameters:

Endocrinology 303 (2009) 13–24 23

• k1 = 6 × 107 M−1 min−1

• k−1 = 0.20 min−1

• k2 = k1• k−2 = 100k−1• k3 = 2500 min−1

• k−3 = k−1• k4 = k−4/9• k−4 = 0.003 min−1

• k4′ = 2.1 × 10−3 min−1

• k−4′ = 2.1 × 10−4 min−1

• k5 = 10k−5 M min−1 if (x6 + x7 + x8) > 1 × 10−13

• =60k−5 M min−1 if (x6 + x7 + x8) ≤ 1 × 10−13

• k−5 = 1.67 × 10−18 min−1

• k6 = 0.461 min−1

• k7 = 4.16 min−1

• k−7 = (2.5/7.45)k7• k7′ = ln (2)/2 min−1

• k−7′ = k7′ [(2.5/7.45)(3.70 × 10−13)]/[(6.27 × 10−13) − (2.5/7.45)(3.70 × 10−13)]

• k8 = k−8(5/70.775) × 1012

• k−8 = 10 min−1

• k9 = (k9(stimulated) − k9(basal))(x12/PI3K) + k9(basal)• k9(stimulated) = 1.39 min−1

• k−9 = (94/3.1)k9(stimulated)• k9(basal) = (0.31/99.4)k−9• k10 = (3.1/2.9)k−10• k−10 = 2.77 min−1

• k11 = (0.1k−11)(x13 − 0.31)/(3.10 − 0.31)• k−11 = 10 ln(2)min−1

• k12 = (0.1k−12)(x13 − 0.31)(3.10 − 0.31)• k−12 = 10 ln(2)min−1

• k−13 = 0.167 min−1

• k13 = (4/96)k−13• k13′ = [(40/60) − (4/96)]k−13 (Effect)• k14 = 96k−14• k−14 = 0.001155 min−1

• Effect = (0.2x17 + 0.8x19)/(APequil)• IRp = 8.97 × 10−13 M• [SHIP] = 1.00• [PTEN] = 1.00• APequil = 100/11• [PTP] = 1.00[1 − 0.25(x17/(100/11)] for x17 ≤ (400/11), else [PTP] = 0• PI3K = k8(3.70 × 10−13)(1 × 10−13)/[k8(3.70 × 10−13) + k−8]• [PKC] = Vmax[PKC]x19(t−�)nPKC

[KnPKCd

+x19(t−�)nPKC ]• Vmax = 20• Kd = 12• n = 4• � = 1.5

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