Mathematical modeling of preadipocyte fate determination

ARTICLE IN PRESS Journal of Theoretical Biology 265 (2010) 87–94

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Mathematical modeling of preadipocyte fate determination Huseyin Coskun a,, Taryn L.S. Summerfield b, Douglas A. Kniss b,c, Avner Friedman a,d a

Mathematical Bioscience Institute, Ohio State University, Columbus, OH, USA Obstetrics and Gynecology, (Laboratory of Perinatal Research), Ohio State University, Columbus, OH, USA Departments of Biomedical Engineering, Ohio State University, Columbus, OH, USA d Department of Mathematics, Ohio State University, Columbus, OH, USA b c

a r t i c l e in fo

abstract

Article history: Received 23 November 2009 Received in revised form 17 March 2010 Accepted 30 March 2010 Available online 10 April 2010

White adipose tissue is the major energy storage depot for neutral lipids and is also a key endocrine regulator of a host of homeostatic activities, including metabolism, feeding behaviors, cardiovascular functions and reproduction. Abnormal fat accretion in the setting of obesity can lead to insulin resistance and type 2 diabetes, and has been linked to some cancers and arteriosclerosis. Thus, a thorough appreciation of the intricate signaling events that must take place as quiescent adipocyte precursors are recruited into the proliferating cell population that then must ‘decide’ to differentiate into fully functional fat cells is critical to our understanding of diseases related to excess adipogenesis. We are beginning to gain insights into the molecular regulators of adipocyte differentiation. A significant advance would be to construct mathematical modeling tools that would assist cell biologists in predicting how environmental and/or intrinsic inputs could influence preadipocyte fate decision making. We have developed a model of how preadipocytes may use the dynamic interplay of two transcription factors, nuclear factor-k B ðNF-kBÞ and peroxisome proliferator-activated receptor-g (PPAR-g) in early proliferation and differentiation events in vitro. Critical to the model is the feedback signaling between NF-kB and its inhibitor, IkB. The model is based on differential equations that describe the interactions of stimuli for NF-kB activation and mitogen concentrations and allows one to make predictions about how mouse 3T3-L1 preadipocytes choose between proliferation, differentiation or quiescence. Those predictions are supported by experiments on mouse 3T3-L1 cells. Published by Elsevier Ltd.

Keywords: Proliferation Differentiation PPARg NF-kB Cyclin D1

0. Introduction Historically, it was thought that adipose tissue served merely as a energy storage depot by sequestering esterified fatty acids in large lipid droplets (Ailhaud et al., 1992; Bernlohr et al., 2004). More recently, many studies have demonstrated that white fat is a major source of adipocyte-specific adipokines (leptin, adiponectin, resistin) that orchestrate a complex communication system leading to energy homeostasis during health and disease states such as type 2 diabetes mellitus and metabolic syndrome (Miner, 2004; Scherer, 2006). In addition, pro-inflammatory cytokines (interleukin-6, IL-6; tumor necrosis factor-a, TNF-a) secreted by adipocytes and macrophages that invade adipose tissues are profoundly up-regulated in disease states such as arteriosclerosis, type 2 diabetes, obesity and the metabolic syndrome (Shoelson et al., 2006). Thus, it is now recognized that white fat is a highly

 Corresponding author. Tel.: + 1 614 688 3493.

E-mail addresses: [email protected] (H. Coskun), summerfi[email protected] (T.L. Summerfield), [email protected] (D.A. Kniss), [email protected] (A. Friedman). 0022-5193/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.jtbi.2010.03.047

complex tissue comprised of many cell types and pleiotropic metabolic functions. Moreover, the recruitment and differentiation of preadipocytes into fully functional adipocytes is a remarkably coordinated process involving several transcription factors and their target genes (Farmer, 2006). Specifically, in vitro studies have shown that proliferating preadipocytes undergo a one or two final rounds of clonal expansion prior to their initiation of the differentiation program led by a complex network of C/EBP-a and-b and the master adipogenic regulator peroxisome proliferator-activated receptorg (PPAR-g) (Tang et al., 2003a, b; Rangwala and Lazar, 2000). In recent years, there has been a recognition that local inflammatory events within adipose tissue may have profound consequences in obesity, insulin resistance and type 2 diabetes (Wellen and Hotamisligil, 2003). In addition, the central transcription factor in inflammatory cells, nuclear factor-kB (NFkB) is up-regulated in metabolic dysfunction such as type 2 diabetes and insulin resistance, and, in fact, may be a valuable target of therapeutic intervention (Yuan et al., 2001). In many cell types NF-kB is an important regulator of cell proliferation and survival (Karin and Lin, 2002). As such we postulated that during the final stages of preadipocyte cell growth preceding early differentiation events,

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NF-kB and PPAR-g interact to govern the state of cell quiescence, proliferation and initiation of differentiation. Importantly, we have approached this problem from a theoretical perspective and have constructed a simple mathematical model to predict how environmental and/or intrinsic molecular mechanisms might regulate these processes. We also conducted experiments with results that indirectly support our model. The differentiation of preadipocyte is regulated by a complex network of transcription factors and other proteins. The master transcription factors in adipogenesis are PPARg and C/EBPa (Farmer, 2006; Rosen et al., 2002; Rosen and Spiegelman, 2000). Agonistic ligands for PPARg are also shown to inhibit proliferation of T helper cell clones and B cells (Clark et al., 2000; Yang et al., 2000; Setoguchi et al., 2001). It is known that there is a mutual inhibition between NF-kB and PPARg (Beg, 2004; Chung et al., 2000; Ricote et al., 1998). PPARg, for example, inhibits cytokineinduced cytotoxicity via NF-kB (Kim et al., 2007). NF-kB in turn, through induction of TNF-a, is able to antagonize the synthesis of PPARg, block adipocyte differentiation, and contribute to insulin resistance (Jiang et al., 1998). The role of NF-kB in these processes is actually quite complex. It controls indirect auto-inhibition and auto-activation pathways, and thus has a dual role. Through induction of COX2, NF-kB positively effects production of the prostaglandins (Xie et al., 2006; Kniss et al., 2001; Kniss, 1999). The prostaglandin 15d-PGJ2 is considered to be the endogenous ligand and activator of PPARg (Forman et al., 1995). As a result, NF-kB plays a role, although indirectly, in the activation of PPARg. 15d-PGJ2 and IkB, a protein which is induced by NF-kB, are both known inhibitors of NF-kB (Ackerman et al., 2005; Ting and Endy, 2002; Hoffmann et al., 2002). Since, as mentioned above, NF-kB induces the potent activator, TNF-a, this process can be considered as auto activation (Beinke and Ley, 2004). NF-kB has positive influence on the induction of Cyclin D1 through TNF-a (Cawthorn et al., 2007; Kikuchi, 2000). Similarly, Cyclin D1 has a positive influence on the induction of PPARg through activation of E2F1, but it also acts as a transrepressor by recruiting HDACs to the PPARresponse element (Farmer, 2006; Fu et al., 2005). The model is formulated as a system of nonlinear ordinary differential equations, where most of the parameters are experimentally unknown at this time. A critical assumption of our model is that each state of the cell (quiescence, proliferation, early differentiation events) is represented by the steady-state solution of the equations contained within the model. Because of the complexity of the molecular phenomena, our initial mathematical model considers only three principal components: NF-kB, PPAR-g and cyclin D (Xie et al., 2006; Abella et al., 2005). Our equations predict that the most important parameter in the fate decision making by the preadipocyte is the rate of autoinhibition of NF-kB through the induction of IkB (NF-kB autoinhibitor). The parameters of secondary importance are the induction or degradation of PPAR-g and cyclin D1. In the interest of simplicity, the induction rate of cyclin D1, k8, was selected as the second parameter for the analysis. Bifurcation diagrams based on the model equations demonstrate how the relative strengths of these

Fig. 1. Biological model of indirect molecular interactions.

Table 1 References for Fig. 1. # Relations

References

1 Induction 2 Autoactivation 3 Autoinhibition 4 Activation

Cawthorn et al. (2007), Kikuchi (2000) Beinke and Ley (2004)

5 Induction 6 Transrepression 7 Mutualinhibition

Ackerman et al. (2005), Ting and Endy (2002), Hoffmann et al. (2002) Forman et al. (1995), Kniss et al. (2001), Xie et al. (2006), Kniss (1999) Farmer (2006) Fu et al. (2005) Kim et al. (2007), Beg (2004), Ricote et al. (1998), Chung et al. (2000)

two parameters determine the fate of the precursor cell. This simplified, first principles model may be useful in predicting how cells orchestrate the complex decision making of remaining quiescent, undergoing proliferation or differentiate into functional tissue cells.

1. The model We denote by x1, x2, and x3, the concentrations of NF-kB, Cyclin D1, and PPARg, respectively. The diagram in Fig. 1 illustrates the relations described in the introduction and summarized in Table 1. A single line in the figure represents activation or inhibition; a double line, induction or production; an arrow head stands for positive interaction, and a blunt head stands for negative interaction. Since the model does not distinguish between active and inactive forms of the proteins, single and double lines are considered to be the same in the mathematical formulation of the model. The relations indicated in Fig. 1 are mainly indirect mechanisms of molecular interactions. The diagram in Fig. 1 is expressed by the following system of differential equations x2 dx1 k2 k3 ¼ k1 2 1 2 k4 x1 þ k5 dt j1 þ x1 k2 þ x1 k3 þ x3

ð1Þ

dx2 x1 ¼ k6 k7 x2 þ k8 dt j2 þ x1

ð2Þ

dx3 x1 k10 x2 k12 ¼ k9 þk11 k13 x3 þ k14 dt j3 þ x1 k10 þ x1 j4 þ x2 k12 þ x2

ð3Þ

For NF-kB dynamics in Eq. (1), two step Hill kinetics is used for auto-activation in the first factor of the first term; the second and third factors represent auto-inhibition and inhibition by PPARg, respectively. Eq. (1) also includes a degradation term,  k4x1, and an induction term, k5. For Cyclin D1 dynamics in Eq. (2), the first term represents induction by NF-kB, the second term,  k7x2, accounts for degradation, and the last term, k8, is induction by factors other than NF-kB. For PPARg dynamics in Eq. (3), the first term accounts for activation (through COX2 and 15d-PGJ2) and inactivation by NF-kB. Similarly the second term represents activation and inactivation of PPARg by Cyclin D1. As before, a degradation,  k13x3, and an induction term, k14, are assumed. It is known that the inactivation of NF-kB is mainly due to the IkB concentration (Hoffmann et al., 2002; Ting and Endy, 2002), and induction of Cyclin D1 is due to mitogenic stimulation (Farmer, 2006). Based on these facts, it is reasonable to assume

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that the auto-inactivation parameter k2 for NF-kB is a decreasing function of IkB concentration, and the parameter of Cyclin D1 induction, k8, is an increasing function of mitogen concentration in the cell.

2. Results It is assumed that a decision about the state of the cell, i.e., whether to differentiate along the adipocyte lineage, proliferate, or remain in a quiescent state, is expressed by the equilibrium state of the system of Eqs. (1)–(3), that is, by dx1 dx2 dx3 ¼ ¼ ¼ 0: dt dt dt

ð4Þ

Based on this assumption, a jump from one branch to another at a bifurcation point along the curve of equilibrium points is interpreted as a change in the state of the cell. Eq. (4) are numerically solved for the equilibrium points (x1, x2, x3) for varying values of the parameters k2 and k8, using the software XPPAUT. The choice of k2 as a bifurcation parameter is due to an important model prediction that, bifurcation phenomena (the change in the state of the cell) occurs for only for a certain range of k2, as explained below. The other bifurcation parameter k8 is arbitrarily chosen from the set of degradation and induction parameters, namely, {k4, k5, k7, k8, k13, k14}. For the purpose of illustration, all parameters are fixed as in Table 2, except k2 and k8. It should be noted however that although the parameter set was arbitrarily chosen, the subsequent analysis does not appear to depend on the choice of the particular parameters. The dependence of the state of the cell on the parameters k2 and k8 is analyzed in what follows. In the figures below, the states of quiescence, differentiation, and proliferation are represented by Q, D, and P in the given order. The stable and unstable states are indicated by superscripts s and u, respectively. The continuation of stable equilibrium points are plotted with solid lines and that of the unstable equilibrium points with dashed lines. Fig. 2(a) profiles the x1, x2, and x3 components of the equilibrium points as functions of the parameter k2 when k8 ¼1, and Fig. 2(b) profiles these components as functions of the parameter k8 for k2 ¼6. In Fig. 2(a), the stable upper and lower branches are connected by the unstable middle branches at the bifurcation points V1 and V2, represented by red, solid circles. The fact that PPARg is the master regulator of adipocyte differentiation is used for the determination of the state of the cells: on the (x3, k2) curve, (and consequently on the corresponding branches of the other curves of Fig. 2(a)) the stable upper branch is identified by the differentiation state, Ds, because of the high equilibrium concentration level of PPARg, and the stable lower branch is identified by either proliferation, P, or quiescence, Q, states, (Q,P). Apparently k2 does not distinguish between P and Q states. Additional information through analysis of the system with respect to the parameter k8 will be employed on, in Fig. 4, in order to distinguish between the states P and Q. As k2 increases from 0 to k2,2 ¼6.947, where the bifurcation point V2 resides, the cell assumes Ds state. Upon surpassing k2,2, the cell’s state changes to (Q,P)s. The cell remains in this state as long as k2 remains larger than k2,1 ¼5.981 where the bifurcation Table 2 An arbitrary set of parameter values. k1 ¼1.6 k11 ¼ 5

k3 ¼ 5 k12 ¼ 5

k4 ¼0.1 k13 ¼0.5

k5 ¼0.05 k14 ¼ 0.2

k6 ¼ 1 j1 ¼ 4

k7 ¼ 1 j2 ¼ 0.5

k9 ¼1 j3 ¼0.5

k10 ¼ 0.5 j4 ¼ 4

89

point V1 resides. If k2 is further decreased, the cell goes back to the Ds state. This phenomena of coexistence of the states Ds and (Q,P)s in an overlapping interval, in this case k2,1 ok2 o k2,2 , is called hysteresis. In Fig. 2(b), where the profiles of the xi are shown as functions of k8 when k2 ¼ 6, there are four bifurcation points, denoted by Y1, Y2, Y3, and Y4 and represented by blue, solid diamonds on the graphs. In order to use these profiles to determine the cell state, the following facts were used: (i) NF-kB concentration decreases (most likely its active form, see Discussion) during differentiation, (ii) increased Cyclin D1 level is associated with proliferation, and (iii) as mentioned previously, PPARg is the master regulator of adipogenesis. The low equilibrium concentration of PPARg and Cyclin D1 for small values of k8 suggest the identification of the first, stable branches ending in Y2 by quiescence, Qs. The high concentration level of PPARg on (x3, k8), and low concentration of NF-kB on (x1, k8) graph suggest identification of middle, stable branches (Y1Y4) by differentiation, Ds. High equilibrium concentration of Cyclin D1 and low concentration of PPARg imply the identification of the third, stable branches starting at the bifurcation point Y4 by proliferation, Ps. Fig. 3 depicts (x1, k8) curves for three different values of k2. The behavior of the graphs indicates that the bifurcation points Y1 and Y3 are approaching each other as k2 increases, and eventually coalesce at the point Z1 for k2 ¼7.6017 and k8 ¼ 3.6435 on the parameter space (k8, k2) (see Fig. 3(a)–(c)). This point Z1 will be called as point of uncertainty. For increasing values of k2, points of uncertainty, Z, define the curve of uncertainty, which will be denoted by f on Figs. 4 and 3(d). The graphs of Figs. 2 and 3 are prototypical examples for specific values of k2 and k8. Fig 4, however, summarizes the correspondence between the curves of bifurcation points and different states of the cell together with their stability on the (k8, k2) parameter space. There are several observations to be made on Fig. 4. First, the model predicts that, bifurcation phenomena occur only when k2 is within a certain range, more specifically, k2,min ¼ 3:7787o k2 o 12:5837 ¼ k2,max : The parameter k2 also determines the number of bifurcation points. There are four bifurcation points along the parameter k8 when 4:3768 ok2 o 7:6017. For 7:6017o k2 o 12:5837 two of these bifurcation points, Y2 and Y4, coalesce into one equilibrium point, Z (see Fig. 3). So, in this interval, and also when 3:783 o k2 o 4:3768, there are only two bifurcation points along k8. The number of bifurcation points along the parameter k2 is always two within the range k8,min ¼ 0:1614o k8 o28:48 ¼ k8,max . There are no bifurcation points outside of the ranges specified above. The region enclosed by the dashed, closed bifurcation curves c in Fig. 4 is the region of three state coexistence. This region will be referred to as region of uncertainty, and will be denoted by U. The curve of uncertainty, f, separates this region into two parts; in the left half subregion Qs, Ds, Qu or Du coexist, while in the right subregion the existing states are Ds, Ps, Du or Pu. The importance of the curve of uncertainty is in that for 7:6017 o k2 o12:5837, and ðk2 ,k8 Þ A U, a cell may immediately transit from Qs to Ps state by crossing f with increasing values of k8, without a differentiation stage. The cell may assume Ds state in the region of uncertainity. The region left and right to the uncertainty region are predominantly single state regimes of stable quiescence, Qs, and proliferation states, Ps, respectively. These regions may be called regions of stable quiescence and stable proliferation, respectively. In a similar manner, the single state region of stable differentiation lies below the uncertainty region, and is characterized by relatively small values of both k2 and k8 (see Fig. 4).

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Fig. 2. The components of the equilibrium points along the parameters (a) xi’s vs k2 for k8=1 and (b) xi’s vs k8 for k2=6.

For the specific parameters chosen in Table 2, the quantitative characterization of U (enclosed by c), f, and of Z1 are given as follows Let n1 and d1 are given by pffiffiffi pffiffiffi pffiffiffi n1 ¼ ð10k28 þ ð1040 5Þk8 þ 980 52077Þð65k28 þ ð128260 5Þk8 pffiffiffi pffiffiffi pffiffiffi 256 5 þ1364Þðk8 2 5Þð2k8 þ 14 5Þ, pffiffiffi pffiffiffi d1 ¼ 2ð12 5 þk8 Þð1300k58 þð451013000 5Þk48 pffiffiffi 3 pffiffiffi þ ð31040 þ93920 5Þk8 þð4087317þ 1547240 5Þk28 pffiffiffi pffiffiffi þ ð21864672þ 10248468 5Þk8 41472068 þ18250144 5Þ: Then, the curve of uncertainty, f, as a function of k8, is

fðk8 Þ ¼

n1 , d1

3:6469ok8 o 3:7779:

ð5Þ

The points, Z, on f are not bifurcation points. The end points of f are Z1 ¼(7.6016,3.6469), and Z2 ¼(12.5836, 3.7779). It should be noted that on f, a cell may assume one of the three stable states while the number of possible states along c, including Z1 and Z2, is only two. Let K2 be a function of k8, and depend on x1 as a parameter, defined by K2 ðk8 ,x1 Þ ¼

n2 , d2

where n2 ¼ x1 ð2x1 1Þðx21 þ 16Þðð432x41 þ 944x31 þ 728x21 þ236x1 þ 27Þk28 þ ð8752x41 þ 17952x31 þ 13280x21 þ 4232x1 þ493Þk8 þ 16960x41 þ 31944x31 þ 20944x21 þ 5706x1 þ 540Þ,

d2 ¼ ð11264x51 þ 3840x41 þ1610x31 11344x61 2912x1 4923x21 432 þ864x71 Þk28 þ ð19872x51 þ 237616x41 788851936x1 84749x21 þ73682x31 þ 17504x71 113648x61 Þk8 þ12078x31 þ 267716x41 144936x51 337072x61 þ 33920x71 74016x1 169052x21 8640: For Lx,y being the region enclosed by two curves K2(k8,x) and K2(k8,y) when 1:1350 ox,y o 2:3599,x ay, the region of uncertainty U can then be described as [ ð6Þ U ¼ Lx,y , 1:1350 ox,y o 2:3599: x,y xay

In other words, there are two intersection points of the graphs of K2(k8,x) and K2(k8,y) with the conditions on x and y as given in Eq. (6), and these intersection points determine the curve c. Although the above formulas for U and f were obtained for the specific parameters given in Table 2, similar formulas can be determined for any other set of parameters.

3. Discussion A comprehensive understanding of adipose tissue development and its role in energy metabolism and endocrine signaling is of paramount importance given that molecular derangements during adipogenesis can have profound pathophysiological consequences as seen in obesity, insulin resistance and type 2 diabetes (Waki and Tontonoz, 2007; Lefterova and Lazar, 2009). The present work addresses in a predictive fashion the question of preadipocyte cell fate determination, and how fat cell precursors

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Fig. 3. Formation of the point (Z1 ) and the curve (f) of uncertainty. (a) x1 vs k8 for k2 ¼6, (b) x1 vs k8 for k2 ¼7.60, (c) x1 vs k8 for k2 ¼ 8 and (d) curve of uncertainty.

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utilize intrinsic and extrinsic information to ‘‘decide’’ whether to proliferate, differentiate into mature adipocytes, or remain in a quiescent state. For simplicity, in this initial modeling endeavor,

we did not consider the fourth decision, that is, apoptosis. Further refinements of the model will have to take this latter cellular state into consideration. The paper develops, for the first time, a

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mathematical model which predicts how a preadipocyte determines whether to proliferate, differentiate, or remain in quiescent state. Only three main components were considered in this initial modeling effort: NF-kB, cyclin D1, and PPAR-g. To our knowledge, this is the first paper to directly address the potential role of NF-kB in the proliferative and differentiation phases of preadipocytes. However, the role of NF-kB in cell proliferation and differentiation was considered in several studies in other cell systems, including vascular smooth muscle cells, breast cancer cells, and B lymphocytes (Hsieh et al., 2005; Hoshi et al., 2000; Setoguchi et al., 2001). As summarized in Fig. 4, the model provides a quantitative measure in predicting the adipocyte precursor fate and a tool in predicting the behavior of preadipocytes during the early stages of adipogenesis. The region of uncertainty, U, enclosed by c, is the position at which the preadipocyte can assume two different stable states: to the left of the curve of uncertainty, f, stable differentiation and quiescence states, and to the right of f stable differentiation and proliferation states. The region left of, right of, and below the region of uncertainty, U, are single stable quiescence, proliferation, and differentiation states, respectively. Although the parameters are currently largely unknown experimentally, the results obtained in Fig. 4 suggest novel approaches for new experiments to investigate cell fate determination. For example, based on the qualitative nature of the region of uncertainty U shown in Fig. 4, one could modulate the parameters k2 and k8 in order to arrest the cell in a quiescent state as a potential means of controlling obesity and its consequences (type 2 diabetes, insulin resistance, metabolic syndrome). The parameter, k2, which represents IkB concentration, or NF-kB inactivation (autoinhibition), and k8, which represents the strength of mitogenic stimulus are important elements of the model. Future experiments will test whether this prediction is empirically the case. That is, to validate the importance of k2 (i.e., the extent of NF-kB activity), it will be critical to know if the inhibition rate of NF-kB is vital for cell proliferation, differentiation or quiescence during the early stages of adipogenesis. In fact, there is some evidence in the literature that supports a role for NF-kB in cell proliferation. Vascular smooth muscle cell proliferation, for example, a crucial event in the formation of atherosclerotic plaques, is regulated by NF-kB and its inhibitor IkB-a (Hoshi et al., 2000). Hsieh et al. (2005) demonstrated that NF-kB was critically involved in controlling the growth of several breast cancer cell lines in response to Chinese herbal formulations. = mice, Ren et al. (2002) reported that Finally, using IkB kinase b B cell proliferation and subsequent antibody production following exposure to lipopolysaccharide was severely diminished, implicating NF-kB activation in lymphocyte cell cycle control. Our experimental data (see the Experimental Data section) demonstrated that, in preadipocytes stimulated with TNF-a, there was a time-dependent diminution in IkB-a and a concomitant increase in p65 activation, which is presumptive evidence in support of our simplified model. Interestingly, we detected no substantial difference in undifferentiated 3T3-L1 cells compared to cells undergoing early stages of differentiation (i.e., within 2 days following addition of the IDX cocktail), that is, following the upregulation of PPAR-g. It was recently reported that adipogenesis is associated with changes in amount and subunit composition of the NF-kB complexes. NF-kB subunits p65/RelA, RelB, and IkBa are up-regulated during fat cell differentiation (Berg et al., 2004). We did not detect such a dramatic increase in the absolute amount of p65, RelB or IkBa (data not shown). Rather, cells retained their ability to respond to TNF-a by enhanced IkBa phosphorylation and degradation without a major impact on the steady-state level of IkBa. Since localized inflammation within adipose tissue can be elicited by infiltrating macrophages in response to cytokines (TNF-a

and IL-6), it is likely that the case that the activation status of NF-kB is crucial the recruitment and differentiation of preadipocytes within white fat in pathological settings (e.g., obesity-related insulin resistance) (Rosen and MacDonald, 2006). Since our initial model does not specifically delineate between active and inactive forms of NF-kB, this result does not contradict the prediction of low levels of NF-kB concentration in (x1, k8) graph of Fig. 2. There is some experimental support for the possibility that NF-kB function decreases as differentiation proceeds (Chung et al., 2006). Thus, the steady-state concentration of NF-kB is consistent with the model predictions. Fig. 4 displays the steady state of the model system Eq. (3) for preadipocyte fate determination in terms of the parameters k2 and k8. A similar figure can be obtained in terms of k2 and k14. Here k14 is a parameter representing induction of PPARg or, equivalently, the effect of insulin on PPARg induction (Klemm et al., 2001). A recent paper by Guo et al. (2009) provides evidence that the early events in preadipocyte commitment toward the adipogenesis pathway can be segregated into two distinct stages, i.e., licensing of adipocyte formation and execution of the terminal differentiation program. The licensing stage can be identified with the quiescence stage in our paper. Their results suggest that differentiation licensing and differentiation execution can be uncoupled and separately linked to cell proliferation. As a matter of fact, Fig. 4 with (k2, k14) instead of (k2, k8), provides a quantitative measure for mutual relations among Qs, Ps and Ds based on the NF k B inhibition (k2) and PPARg induction or insulin administration (k14) coefficients. Their result that, when the licensed 3T3-L1 cells were treated only with insulin, the adipogenesis commitment could be maintained from one cell generation to the next, whereby the licensed program could be activated in a cell-cycle-independent manner once these cells were subjected to adipogenesis-inducing conditions, can be translated into the terminology of our results by stating that cells can switch from Qs to a stable differentiation stage Ds across the curve of uncertainty, f, based on the strength of insulin treatment k14 (when k14 is used instead of k8). Moreover, cell contact inhibition and subsequent reentry into mitotic clonal expansion are pivotal for this biphasic process. Now, our model that includes a description of the potential role of NFkB in these events can be tested further. As mentioned earlier, we selected only three principal molecular factors for the sake of simplicity, but we are fully cognizant that as the model evolves an increasing degree of complexity will be introduced. For example, we have only considered the canonical NF-kB pathway (Ghosh and Karin, 2002) in this model. A greater appreciation of the subunit contribution of NF-kB (i.e., p65, RelB, c-Rel, and p50) (Hayes and Ghosh, 2008; Lipniacki, 2004; Cheong and Levchenko, 2008), both isoforms of PPAR-g (PPAR-g1 and -g2) (Tontonoz and Spiegelman, 2008), and eventually different external agents (cytokines (NF-kB activators) and mitogens (insulin, insulin-like growth factor-I) (Gregoire, 2001) need to be considered separately for their importance and function in adipocyte biology. Finally, extensive experimental validation of the simplified and future generations of the model must be undertaken to assess the utility of this mathematical approach in predicting preadipocyte fate decision making. It will also be interesting to employ the model in other cell types, such as various stem cell populations.

4. Materials and methods 4.1. Cell culture Mouse 3T3-L1 fibroblasts were obtained from the American Type Culture Collection and cultured in Dulbecco’s minimal essential medium (DMEM) supplemented with 2 mM L-gluta-

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mine, 10% bovine calf serum (FBS) and antibiotics as previously described (Green and Kehinde, 1974; Xie et al., 2006). After 2 days in vitro, cells were rinsed with serum-free DMEM and then transferred to differentiation medium (insulin, dexamethasone and isobutylmethylxanthine, IDX) as previously described (Xie et al., 2006). 4.2. Immunoblotting Cellular proteins were extracted using RIPA lysis buffer supplemented with a protease and phosphatase inhibitor cocktail (Sigma) and quantified by the Biorad assay using bovine serum albumin (BSA, Sigma) as standard. Approximately 30 m g/lane were loaded onto 10% sodium dodecyl sulfate-polyacrylamide gels (SDS-PAGE) followed by transfer to nitrocellulose membranes (Hybond, Amersham). After thorough blocking of non-specific binding with Tris-buffered saline/Tween-20 (TBST) containing 5% nonfat dry milk for 1 h, membranes were incubated overnight at 4C, followed by washing with TBST/5% nonfat dry milk and incubation with secondary antibodies diluted 1:2000 with TBST/ 5% nonfat dry milk for 1 h, and visualized using a VersaDoc Imaging System (Biorad). 4.3. Experimental data To investigate the elements of the mathematical model presented here, we conducted immunoblotting studies to determine the relative time-course and signal strength of IkBa phosphorylation and degradation as a surrogate for NF-kB activation. Mouse 3T3-L1 cells were incubated in the presence or absence of IDX differentiation cocktail 2 days after plating, and after an additional 2 days they were exposed to 5 or 30 min with TNF-a (50 ng/ml) or lipopolysaccharide (LPS, 10 mg=ml). Immunoblots prepared from total cell extracts were fractionated by SDS-PAGE, transferred to nitrocellulose membranes and probed with antibodies directed against IkBa, p65, IkB kinase (IKK), or PPAR-g. As shown in Fig. 5, treatment with TNF-a led to a rapid (within 5 min) increase in IkBa phosphorylation and a concomitant diminution in total IkBa protein. In addition, TNF-a stimulation prompted the appearance of phospho-p65 subunit, a marker of NF-kB activation. When cells were treated with TNF-a there was also a rapid increase in phosphorylation of the a and b subunits of IKK, the heterodimeric kinase responsible for IkBa phosphorylation and subsequent degradation leading

Fig. 5. Mouse 3T3-L1 cells were incubated for 2 days in the presence or absence of IDX cocktail, and then challenged with TNF-a (50 ng/ml) or serum-free DMEM alone for 30 min. Cell extracts were fractionated by SDS-PAGE and immunoblots were probed with antibodies directed against phospho- or dephospho- IkBa. The blot is representative of two similar experiments.

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