Zonation of hepatic fat accumulation: insights from mathematical modelling of nutrient gradients and fatty acid uptake

Abstract

Intrinsic of non-alcoholic fatty liver diseases is an aberrant accumulation of triglycerides (steatosis), which occurs inhomogeneously within lobules. To improve our understanding of the mechanisms involved in this zonation patterning, we developed a mathematical multicompartment model of hepatic fatty acid metabolism accompanied by blood flow simulations. A model analysis determines the influence of the uptake process of fatty acids, the porto-central gradient of plasma fatty acid concentration, and the oxygen supply via blood on the zonation of triglyceride accumulation. From this theoretical perspective, the plasma oxygen gradient, but not the fatty acid gradient, leads the way to a zonated triglyceride accumulation by its decisive role in oxidative processes. In addition, the uptake mechanism of fatty acids seems to be fundamental for a pericentral dominance of steatosis. However, the mechanism of cellular fatty acid uptake from the blood is still under debate. Our theoretical approach supports the transporter-mediated uptake mechanism and reveals that the maximal velocity of fatty acid uptake affects the switching between a periportal and a pericentral triglyceride accumulation. Further research on hepatic fatty acid uptake is needed to push forward our understanding of aberrant triglyceride accumulation in diet-induced steatosis.

1. Introduction

Our liver performs a plethora of essential metabolic tasks to keep the body in homeostasis. This range of functions is facilitated by the unique anatomy of the liver lobules, in which the position of a hepatocyte along the blood vessels (called sinusoids) dictates its metabolic set-up. Jungermann and colleagues termed this phenomenon ‘metabolic zonation’ and described it in detail decades ago [1–6]. Generally, three metabolic zones can be discriminated along the sinusoids: hepatocytes near the portal triad belong to the periportal zone, whereas hepatocytes near the central vein belong to the pericentral zone. A middle or intermediate zone is confined between these two areas [7]. The hepatic architecture combined with the unidirectional blood flow leads to a microcirculation, in which perfusion proceeds in a sequential manner facing the hepatocytes in the periportal zone with the highest supply of metabolites and signalling molecules [8]. The concentrations of these substrates decrease progressively for subsequent hepatocytes depending on the uptake rate of periportal hepatocytes. Hence, decreasing concentration gradients of metabolites, hormones and other signal molecules are established from zone 1 to zone 3, which determine the specific metabolic activity of each hepatocyte.

The zonation of hepatic metabolism plays a prominent role in liver pathologies precipitated by metabolic changes, such as non-alcoholic fatty liver diseases (NAFLD). These diseases are in the first instance characterized by an aberrant accumulation of triglycerides (TGs) within the cytosol of hepatocytes, called steatosis. One major cause of steatosis is an excessively high uptake of carbohydrates or fat from the diet, such that caloric intake exceeds the burning of calories. The excess of nutrients is stored in the form of TGs in adipose tissue and liver to protect the cells against the lipotoxicity of free fatty acids (FAs) [9,10]. In adult livers, TGs accumulate inhomogeneously within the lobules. Unfortunately, the amount of studies considering specifically the zonation pattern of steatosis in humans is still sparse and the situation is aggravated by a low number of studies reporting the zonation pattern of steatosis in animal feeding experiments. For patients with NAFLD or steatohepatitis (NASH) clinical studies mentioned an aberrant TG accumulation predominantly in hepatocytes located in the pericentral zone [11–14] and steatosis accentuated in the pericentral zone was declared as one major histopathologic feature of NASH [15]. But also azonal and panacinar steatosis was observed in adult NAFLD and NASH patients [11,13], whereas periportal steatosis is most common in paediatric NAFLD patients [14]. In rodent studies, the observed zonation pattern of steatosis depends strongly on the composition of the diet and on the animal species and strain [16]. Commonly reported, a carbohydrate-rich diet (i.e. high in sucrose or fructose) clearly initiated a periportal TG accumulation, e.g. in Sprague–Dawley rats [17] and Fischer-344 rats [18]. In contrast, a high-fat diet leads to TG accumulation restricted to hepatocytes around the central vein in C57BL/6 J mice [19,20] and Swiss mice [21]. A periportal TG accumulation was observed when feeding a high-fat diet with concomitant high carbohydrate content (i.e. a Western diet) in C57BL/6 J mice [22]. In some rodent studies, overfeeding or long-term feeding on a high-intake diet was accompanied by an expansion of TG storage along the whole sinusoid, ultimately leading to an azonal or panacinar steatosis pattern in the long run (e.g. [22,23]). Obviously, the amount and kind of nutrients taken up in excessive quantity seems to determine the pattern of fat accumulation during steatosis establishment [24].

The general question for key metabolic processes involved in steatosis development is a core area in past and present liver research. Thus, the metabolic processes involved in steatosis establishment (such as impaired β-oxidation and TG secretion or increased de novo lipogenesis) have been investigated for decades, but the impact of the zonation patterns of these processes on inhomogeneous TG accumulation are rarely explored in its full range. The lack of knowledge on the intertwined action of zonated metabolic processes on the establishment of a zonated steatosis hampers our progress in understanding fatty liver diseases. However, understanding the causes of aberrant TG accumulation in patients with diet-induced fatty liver is essential to develop novel therapeutic strategies.

In general, each hepatocyte has essentially a similar metabolic set-up. This is supported by cell culture experiments revealing a similar response of isolated periportal and pericentral hepatocytes to short-term stimuli [25–27]. The emergence of metabolic zonation results from different stimulating microenvironments in the hepatic zones, i.e. different concentrations of metabolites and signalling molecules. In the focus of interest is the porto-central gradient of oxygen tension because oxygen operates as an important cellular signal and messenger molecule [28,29]. De facto, the uptake of oxygen from the blood by periportal and mid-zone hepatocytes lowers the supply of oxygen to pericentral hepatocytes [28,30], hampering mitochondrial oxidation processes in the cells of this zone. On the other side, the uptake of FAs from blood by periportal hepatocytes reduces the load for posterior hepatocytes [31]; therefore the blood gradient of free FAs may be a further relevant factor in zonation patterning. Both oxygen and FA levels are important in steatosis development due to their link to FA oxidation, a process counteracting fat accumulation. But how do the plasma oxygen gradient and the plasma FA gradient work together in influencing the rate of mitochondrial FA oxidation in the hepatocytes along the sinusoid?

Additionally, an increased uptake of FAs from blood into hepatocytes has been reported to be relevant in steatosis development [32,33], because the intracellular concentration of (activated) FAs is central in regulating the rates of mitochondrial FA oxidation and the synthesis of TGs. But the uptake of FAs determines not only the fat load within the hepatocytes, it also influences the amount of FAs in blood delivered to pericentral hepatocytes, therefore determining the gradient of plasma FA concentration. It is unclear how the rate of FA uptake may influence the zonation of hepatic TG accumulation and, moreover, the mechanism of FA uptake into hepatocytes is subject of much debates [32,34,35], arguing for a passive diffusion or a transporter-mediated view.

As an opportunity to improve our understanding of zonated steatosis establishment, the up-to-date knowledge of the involved processes can be implemented in a computational model. Such a model is helpful (i) to test our current view on regulators of zonation and (ii) to elucidate promising determinants from a theoretical perspective to incite further laboratory experiments. In the past, computational models were used to understand the zonation pattern of processes involved in the detoxification of xenobiotics and drugs, e.g. [36–39]. However, only a few mathematical models exist, which consider explicitly aspects of the zonation of hepatic core metabolism. Based on the need of a fast and efficient elimination of toxic ammonia, the process of hepatic ammonia detoxification was at the centre of zonation research. Here, mathematical modelling was used to understand how a heterogeneous enzyme distribution along the sinusoid contributes to the high efficiency of hepatic ammonia detoxification [40–43]. Using computational modelling, Ghallab et al. [44], for example, disentangled the complexity in hepatic ammonia detoxification caused by the zonation of the involved processes and illustrated how concrete therapies can be derived from mathematical modelling. In contrast to the almost static zonation pattern of enzymes involved in ammonia metabolism, the distribution of enzymes involved in carbohydrate and lipid metabolism is more flexible in regard to their location and activity patterns along the porto-central axis. This flexibility is a prerequisite to respond to a wide range of dietary supply. Some computational models considered the zonation of carbohydrate and energy metabolism [45,46] and a promising new multiscale approach by Ricken et al. [47] illustrated the establishment of zonation patterns of hepatic glucose metabolism in lobules by integrating perfusion and hepatic cell metabolism in a multicomponent model. Moreover, Ashworth et al. [48] recently developed an ODE system of hepatic lipid, carbohydrate and energy metabolism. They included the fine-tuned regulation of metabolic pathways by insulin on the scale of a sinusoid and integrated up-to-date knowledge of zonated enzyme expressions and regulatory dependencies. They focused on elucidating the effect of insulin resistance and, additionally, evaluated by sensitivity analysis, which metabolic pathways are the most influencing ones in the determination of a zonated TG accumulation. Obviously, the complex nature of hepatic metabolism and the intertwined processes in zonation establishment brings in mathematical modelling as a well-suited approach in this research area.

In this paper, we applied mathematical modelling to simulate hepatic lipid metabolism under a high-fat diet with particular emphasis on the relevance of FAs and oxygen as substrates in the oxidation process. Our aim is to test potential mechanisms that may shape the zonation pattern of TG accumulation. In doing so, two factors are in focus: (i) the gradients of oxygen and FA concentrations in the blood along the sinusoids, which influence the rate of mitochondrial FA oxidation; and (ii) the uptake mechanism of FAs from blood.

2. Methods

A mathematical model of hepatic FA metabolism was developed to simulate the zonated accumulation of TGs along a hepatic blood vessel under a high-fat diet. The modelling of FA metabolism follows Schleicher et al. [49] with some modifications to include metabolic zonation and simplify blood flow. Essentially, the following pathways are part of our model: FA and oxygen uptake from blood into the metabolic compartment (representing the hepatocytes of one zone), mitochondrial FA oxidation, oxygen consumption by other oxidation processes, TG synthesis and TG secretion. These metabolic pathways were implemented as a system of ODEs in the software R (R Foundation for Statistical Computing 2015) and the R-packages ‘deSolve’ [50] and ‘FME’ [51] were used for analyses.

Of course, lipid metabolism is closely intertwined with carbohydrate metabolism and the dietary content of carbohydrates is a key determinant for the amount of stored TGs and the zonation pattern of steatosis (e.g. [48]). However, with our special focus on investigating the impact of dietary FAs on the zonation of lipid metabolism, we assume in our model that the carbohydrate portion in the simulated diet regime is in a normal, healthy range and that only the lipid content increases. Thus, in our model, we insulated the influence of an increased FA supply via diet from the effects of carbohydrates. The carbohydrate part is only indirectly represented in our model by the additional degradation rate for oxygen (equation (5) in the electronic supplementary material, S1). This represents the usage of oxygen for other oxidative processes, such as glucose oxidation.

We adopted the approach of multiple working hypotheses [52,53] by implementing different versions of the metabolic model. Therefore, the impact of both metabolite gradients and uptake of FAs on the establishment of a pericentral TG accumulation can be discriminated. All versions comprise the same basic metabolic pathways of hepatic FA metabolism as listed above, but differ in their implementation of oxygen and/or FA concentration gradients in the blood and uptake kinetics of FAs from the blood into hepatocytes (linear versus nonlinear kinetics). An overview and summary of our modelling approach is provided in table 1 and figure 1. Details on the implementation process for the different model versions and the calibration process are provided in the electronic supplementary material, S1 and S4.

Table 1.

Overview of model versions implemented to study the zonation patterning of triglyceride (TG) accumulation under a high supply of fatty acids (FAs) via blood. Based on the idea of multiple working hypotheses [52,53], the different model versions are used to focus separately on mechanisms that may be involved in hepatic TG accumulation.

model version	brief description	aim
model 1	one-compartment model with basic FA metabolism (a) linear FA uptake kinetics (b) nonlinear FA uptake kinetics	a minimal model to implement metabolic pathways and calibrate parameter values. A sensitivity analysis reveals the most important parameters. The effect of linear, non-saturable and nonlinear, saturable FA uptake kinetics should be revealed
model 2	three-compartment model with an oxygen gradient (a) linear FA uptake kinetics (b) nonlinear FA uptake kinetics	a compartment model to elucidate how the oxygen gradient influences the zonation pattern of steatosis. The effect of linear, non-saturable and nonlinear, saturable FA uptake kinetics should be revealed
model 3	three-compartment models with a FA gradient (a) linear FA uptake kinetics (b) nonlinear FA uptake kinetics	a compartment model to elucidate how the FA gradient influences the zonation pattern of steatosis. The effect of linear, non-saturable and nonlinear, saturable FA uptake kinetics should be revealed
model 4	three-compartment models with both gradients (FA and oxygen) (a) linear FA uptake kinetics (b) nonlinear FA uptake kinetics	a combined investigation of both gradients with a sensitivity analysis determining the most important parameters. The effect of linear, non-saturable and nonlinear, saturable FA uptake kinetics should be revealed

Figure 1.

The schematic overview shows each model version with its respective gradient(s) (see also table 1). FA, fatty acid; O₂, oxygen; TG, triglyceride; n, metabolic compartment (n = 1, 2, 3) representing the periportal, middle and pericentral zone, respectively.

Reactions and parameter values are similar in each compartment; preventing an a priori establishment of a zonation pattern due to differences in kinetic values. This means that the zonation pattern can only be influenced by the concentrations of the metabolites (here, FAs and oxygen) in the blood or the FA uptake kinetics. Thus, this approach is aimed to understand how these factors influence the establishment of a zonation pattern in the short-term, i.e. without zone-specific alterations in the level of enzyme amounts or activities. Of course, in the long run, the enzyme expressions in each hepatocyte are modulated by alterations in protein synthesis according to the local conditions [8]. This means that the amount and/or activity of relevant enzymes can be adjusted and, therefore, varies between hepatocytes located in the periportal, middle or pericentral zone. In our model, we do not account for such zone-specific adjustments in enzyme expressions (see ‘Discussion’).

Model version 1 represents a minimal model of hepatic FA metabolism in one (metabolic) compartment representing one hepatic zone along the sinusoid. The blood supply of FAs and oxygen is fixed to physiological values (for details, see electronic supplementary material, S1) and no blood flow is simulated, thus no gradients of metabolites are established in this model version. This minimal model serves for parameter calibration and for a first general evaluation of parameters that are influential for hepatic TG accumulation.

Model version 1 was extended by adding two more metabolic compartments and connecting the compartments by blood flow. Now, each compartment represents one zone along a hepatic blood vessel (i.e. the periportal, middle and pericentral zone). The compartments are connected by a directed blood flow, transporting FAs and/or oxygen from periportal to pericentral. Each metabolic compartment takes up FAs and oxygen from blood, thus substrate concentration gradients along the blood compartment are established. We implemented three different versions of this three-compartment model: model version 2 has only an oxygen gradient (declining from periportal to pericentral), model version 3 has only a FA gradient (declining from periportal to pericentral), and model version 4 merges both types of gradients. Each model version is implemented with a linear, non-saturable (mass action; equation (2.1a)) and a nonlinear, saturable (Michaelis–Menten; equation (2.1b)) function, respectively, to simulate the uptake of FAs from the blood into the metabolic compartment.

2.1a

and

2.1b

All model versions were challenged with a range of blood FA supply ([FA]_blood) from relatively low to high concentrations. This is based on the finding by Donnelly et al. [33] that the plasma FA supply to the liver is crucial for hepatic TG accumulation and that the serum concentration of FAs bound to albumin is elevated in NAFLD patients. Sources of this elevated serum FA concentration are an increased lipolysis of TGs in adipocytes and the release into circulation as well as an excess dietary supply.

We performed local sensitivity analyses (using the ‘sensFun’ function in the R package FME) for model version 1 and 4 to reveal the decisive parameters with regard to model output (here, TG concentration within a compartment). These analyses were conducted by calculating dimensionless sensitivity functions [51], suitable to evaluate the effect of small changes in parameter values on model output. To calculate sensitivity functions, each parameter is separately altered by a very small value, and then the change in model output is quantified. The absolute sensitivity value calculated for each parameter provides an impression on the importance of the parameter for the selected model output. The higher the value, the more influential is the parameter for the stored TG amount.

3. Results

3.1. A constitutive metabolic model without nutrient gradients (model version 1)

Model version 1 was used to find suitable parameter values (table 2; see electronic supplementary material, S4, for details on model calibration). After model calibration, a sensitivity analysis was performed to find the parameters that mostly affect the storage of TGs within a hepatic zone. The analyses revealed that parameters involved in the uptake of FAs are the most sensitive ones (figure 2). Under a low FA supply and linear FA uptake kinetics, the essential factors are FA supply by blood and the first order constant of FA uptake, k_FAup (figure 2a). But the supply of oxygen is also a crucial determinant for steatosis due to its role in mitochondrial FA oxidation. Similarly, under a nonlinear FA uptake, the TG concentration was also very sensitive to the parameters of FA and oxygen uptake (figure 2b). Under a high FA supply, the export of TGs becomes decisive (k_{ex_TG}; figure 2c,d).

Table 2.

Parameter values of the model. For details on the model parameter calibration process see electronic supplementary material, S4. dl, dimensionless.

parameter	description	value (unit)
kflow	flow constant for transport of metabolites (FAs and O2) along the blood compartment	0.2 (dl)
kFAup	rate constant for uptake of FAs from blood into metabolic compartments (linear kinetics)	0.078 (dl)
[FA]blood	fatty acid concentration of blood entering the liver	150–2100 µmol min−1
vmax_FAup	maximal velocity of FA uptake (nonlinear kinetics)	35 µmol min−1
KM_FAup	Michaelis–Menten constant of FA uptake (i.e. affinity for FAs; nonlinear kinetics)	300 µmol min−1
[O2]blood	oxygen concentration in blood	136.5 µmol min−1
	maximal velocity of O2 uptake	15 µmol min−1
	Michaelis–Menten constant of O2 uptake (i.e. affinity for oxygen)	100 µmol min−1
koxid	rate constant for FA oxidation	10 min−1
Koxid	Michaelis–Menten constant of mitochondrial FA oxidation (i.e. affinity for oxygen)	0.1 µmol
kfor_TGsyn	forward rate constant for TG synthesis	0.6 min−1
kback_TGsyn	backward rate constant for TG synthesis	0.01 min−1
kex_TG	rate constant for TG secretion into blood	0.2 min−1
	rate constant for other oxidation processes using O2 (degradation)	0.1 min−1

Figure 2.

Results of the sensitivity analysis of model version 1 (one compartment model without metabolite gradients). The dimensionless sensitivity functions provide a measure of how small changes in parameter values influence model output (here: TG concentration in the compartment). For each model run, only the four most influential parameters are plotted due to facility of inspection. Moreover, parameters with lower sensitivity values are not important for hepatic TG storage. The model was implemented with a linear (mass action) kinetics (a,c) and with a nonlinear (Michaelis–Menten) kinetics of fatty acid (FA) uptake (b,d). In (a) and (c) the two most sensitive parameters overlap in their sensitivity ([FA]_blood and k_FAup), because of the linear kinetics used to simulate FA uptake.

3.2. Zonated metabolic model with an oxygen gradient (model 2)

We consider here how an oxygen gradient may affect the zonation pattern of TG accumulation. In doing this, the values of FA supply by blood for each of the three metabolic compartments were fixed, i.e. there was no transport of FAs with blood flow. Thus, the metabolic compartments are challenged by a similar amount of FAs taken up, but a different supply of oxygen. A declining oxygen gradient is established from periportal (pp) to pericentral (pc) due to the uptake of oxygen from the blood into the metabolic compartments. In the model with a nonlinear FA uptake kinetics, a pericentral dominated TG accumulation (pp/pc ratio = 0.89, table 3) occurred. The TG accumulation pattern is influenced by the high oxygen supply for the first compartment facilitating a periportal dominant oxidation of FAs (pp/pc ratio = 1.6, table 3), whereas the lower supply with oxygen of the pericentral zone bolsters up TG synthesis (pp/pc ratio = 0.89, table 3). The ratio of periportal to pericentral oxidation rate is in agreement with experimental data in fed rats (pp/pv ratio = 1.5 [27]).

Table 3.

Ratio of periportal (pp) to pericentral (pc) steady state concentrations of triglycerides (TG), fatty acids (FA) and oxygen (O₂), as well as the ratios of FA oxidation rate and TG synthesis rate. Ratios are calculated from the simulation run of model version 2 (only oxygen gradient) under a high FA supply by blood with a linear (mass action) uptake kinetics and a nonlinear (Michaelis–Menten) uptake kinetics of FAs, respectively.

	pp/pc ratio
	linear model	nonlinear model
TG concentration	0.98	0.89
FA concentration	0.98	0.89
oxygen concentration	1.63	1.81
rate of FA oxidation	1.60	1.60
rate of TG synthesis	0.98	0.89

The zonation pattern was more pronounced in the simulation with nonlinear FA uptake kinetics, whereas the linear FA uptake kinetics showed only a slight zonated TG accumulation under high FA supply (pp/pc ratio = 0.98, table 3). A scan over the whole [FA]_blood range for the model with linear FA uptake showed that the pericentral steatosis dominance disappears with increasing FA supply (figure 3). The rate of oxidation is periportally dominant over the whole FA range due to the higher oxygen supply to this zone. However, with increasing FA supply via blood the concentration of FAs increases rapidly within the periportal compartment (figure 3). Because of the limitation of mitochondrial oxidation by the supplied oxygen amount, the process is not able to handle this high amount of FAs. The concentration of oxygen in the periportal compartment also decreases rapidly (figure 3), attenuating the periportal dominant gradient observable in oxygen concentration. Thus, with increasing supply of FAs by blood the periportal and pericentral compartment assimilates in their metabolite concentrations and the prior pericentral dominated TG accumulation propagates into the periportal zone.

Figure 3.

Scan over the whole range of [FA]_blood supply for model version 2 (only oxygen gradient, linear FA uptake kinetics). The periportal (pp) to pericentral (pc) ratio of stored triglycerides (TG), fatty acids and oxygen is shown, as well as the ratio for the rate of TG synthesis.

3.3. Zonated metabolic model with fatty acid gradient (model 3)

Next, we consider the effect of a FA gradient declining from periportal to pericentral, holding the oxygen supply for each compartment constant (i.e. no oxygen gradient). We observed a periportal TG accumulation with a linear FA uptake kinetics (pp/pc ratio = 2.09; see also figure 4a). This is not astonishing for the reason that compartment 1 (periportal zone) receives the greatest FA load, but is limited by the same amount of oxygen for mitochondrial FA oxidation as the other two compartments. Thus, the rate of FA oxidation is similar between the compartments (pp/pc ratio = 1.0), but TG synthesis dominates periportally (pp/pc ratio = 2.09) due to the higher FA uptake (pp/pc ratio = 1.93). Restricted by oxygen supply, mitochondrial oxidation of FAs has reached its maximal rate, therefore remaining non-oxidized FAs are directed to TG synthesis.

Figure 4.

Parameter scan over the whole fatty acid (FA) range supplied by blood ([FA]_blood) for model version 3 (porto-central FA gradient, but fixed blood oxygen supply) with a (a) linear and (b) nonlinear FA uptake kinetics, respectively. Blood flow was simulated from the periportal compartment (C1, solid line) to the middle (C2, dashed line) to the pericentral compartment (C3, dotted line) transporting FAs.

The implementation of a FA gradient with nonlinear FA uptake kinetics revealed only a slight periportal TG accumulation under high FA supply (pp/pc ratio = 1.04; see also figure 4b). The nearly similar TG accumulation in all zones is caused by the approaching of FA uptake to saturation, thus all compartments are challenged with almost the same amount of FAs (figure 5) in case of a high FA supply via the bloodstream.

Figure 5.

Parameter scan over the whole [FA]_blood range of model version 3 (porto-central FA gradient, but fixed blood oxygen supply) with nonlinear FA uptake rate. Inset: the ratio of periportal (pp) to pericentral (pc) rate of FA uptake.

3.4. Zonated metabolic model with both gradients (model 4)

Finally, the implementation of both gradients in one model with linear FA uptake kinetics showed a periportal TG accumulation (figure 6a) over the whole range of FA blood values. The high supply of FAs to the periportal compartment cannot be handled by mitochondrial oxidation, which has reached its maximal rate under the specific conditions (similar to model version 3 above), therefore non-oxidized FAs were directed to TG synthesis, supporting a periportal dominance of this process (pp/pc ratio = 1.96). The results of the sensitivity analysis are provided in the electronic supplementary material, S2.

Figure 6.

Scan for TG accumulation in each of the three compartments (C1—periportal, C2—middle, C3—pericentral) in model version 4 (FA and oxygen gradients) over the whole range of [FA]_blood supply. TG accumulation was evaluated under (a) linear and (b) nonlinear FA uptake kinetics, respectively.

Replacing the linear FA uptake kinetics by a nonlinear, saturable Michaelis–Menten function, TG accumulation shows a switch in its zonation pattern (figure 6b). Under a low FA supply, the FA uptake rate showed periportal dominance due to the location of the compartment facing it with higher FA blood values then the other two compartments. The FAs can be handled by mitochondrial oxidation and only a small part of FAs is directed to TG synthesis, predominantly in the periportal compartment. But with increasing FA supply via blood the pattern changed to a pericentral dominance of TG synthesis (pp/pc ratio under a high FA supply = 0.92). The rate of FA uptake gets more similar in the three compartments, based on the saturation of the process (pp/pc ratio under a high FA supply = 1.02; similar behaviour as in figure 5). Finally, the FA uptake rate approaches to saturation in all compartments, whereas the periportal dominance of mitochondrial oxidation remains (due to oxygen limitations). Therefore, an increasing pericentral dominance of TG accumulation is observable (figure 6b).

Which parameters are the most influential ones in determining TG accumulation in this model version? The sensitivity analysis showed that the supply of FAs via blood is an essential determinant of TG accumulation under low FA supply (figure 7a). Under a high FA supply, the maximal velocity of FA uptake (v_{max_FAup}) and the export of TGs (k_{ex_TG}) are critical parameters (figure 7b). We investigated the effect of these two parameters on the zonation pattern of TG accumulation by performing parameter scans for these two parameters. Only the maximal velocity of FA uptake influences the zonation pattern of steatosis (figure 8), whereas k_{ex_TG} limits the amount of TG storage but has no influence on the zonation pattern (see electronic supplementary material, S3). The smaller the maximal velocity of FA uptake is, the more pronounced is the pattern of pericentral TG accumulation, because even with a low FA blood supply the saturation of the uptake process in all three compartments is reached.

Figure 7.

(a–b) Sensitivity analysis of model 4, hepatic fatty acid (FA) metabolism with declining FA and oxygen gradients from C1 (periportal compartment, solid line (i)) to C2 (middle compartment, dashed line (ii)) to C3 (pericentral compartment, dotted line (iii)), nonlinear FA uptake under a low FA supply and a high FA supply, respectively. The dimensionless sensitivity functions provide a measure of how small changes in parameter values influence model output (here: TG concentration in the compartments). Only the three to four most influential parameters are plotted due to facility of inspection. Moreover, parameters with lower sensitivity values are not important for hepatic TG storage.

Figure 8.

Parameter scan for a range of v_{max_FAup} values under a high fatty acid (FA) supply by blood for model version 4 (hepatic FA metabolism with declining FA and oxygen gradients, nonlinear FA uptake kinetics). Blood flows from the periportal compartment (C1, solid line) to the middle (C2, dashed line) to the pericentral compartment (C3, dotted line).

4. Discussion

Steatosis, the overload of lipids within hepatocytes, establishes heterogeneously along the liver blood vessels under a high-fat diet, starting in the pericentral zone [19,21]. With our computational modelling approach, it could be revealed that the oxygen gradient—but not the FA gradient—and a saturable FA uptake kinetics dictates this pericentral TG accumulation. In our model framework, we considered oxygen and FAs acting as substrates in oxidative processes and not as signalling molecules inducing zone-specific enzyme expressions.

In detail, the simulation showed that the higher oxygen availability in the periportal zone promotes a high rate of FA oxidation, thus preventing the accumulation of TGs in this zone. This result is in line with the prevailing view in the research community of oxygen being a major determinant for metabolic zonation. Additionally, it highlights the importance of oxygen for the establishment of a zonated steatosis pattern. The in vivo observable oxygen gradient along hepatic blood vessels is established due to diffusion of oxygen and its uptake from the blood into liver cells. This gradient was discovered decades ago and since then its influence on hepatic metabolism has been continuously investigated [28,30,54–59]. The importance of the oxygen gradient is also underscored by the great attention it received in cell culture experiments [60,61]. For example, Sato et al. [61] developed a new cell culture microdevice to mimic the hepatic microenvironment in vitro and to establish an oxygen gradient within a cell culture of hepatocytes. With such a device, the zonation pattern of metabolism in response to the applied oxygen concentration can be investigated in vitro.

In contrast to the oxygen gradient, the simulation of a porto-central FA gradient revealed a periportal zonation pattern in steatosis establishment. The periportal compartment is faced with a high uptake of FAs leading the way to a high synthesis rate of TGs. And, although the rate of mitochondrial oxidation of incoming FAs is high, the process is insufficient to handle this demanding supply due to its limitation by the available oxygen. This observation supports the hypothesis that oxygen leads the way in the zonation patterning of steatosis by its decisive role in oxidation processes.

The particular value of oxygen lies in its function as a terminal electron acceptor in the respiratory chain, thus its intracellular concentration influences the activity of that chain. Closely linked to the respiratory chain activity is the rate of mitochondrial FA oxidation [62] by the delivery of reducing equivalents NAD+ and FAD+. Accordingly, a low oxygen supply to the pericentral zone can cause a shortage of reducing equivalents that would impair the rate of mitochondrial FA oxidation [63] in the hepatocytes located in this zone. To prevent an accumulation of cytotoxic FAs within the cells, non-oxidized FAs become esterified to TGs. It is well known that the pericentral zone is prone to hypoxic conditions [64,65] and to an augmented production of reactive oxygen species [66–68]. This oxidative stress drives, as one major factor, the progression of simple steatosis to more severe forms of liver tissue damage (e.g. fibrosis, cirrhosis) [69].

Besides the relevance of the oxygen gradient, our simulations also revealed the vital role of FA uptake kinetics in a zonated steatosis development. Donnelly et al. [33] showed already that the plasma FA supply drives mainly the hepatic TG accumulation and that its concentration is increased in NAFLD patients. In support for this, Hijmans et al. [70] found by computational modelling that the hepatic influx of FAs is a major contributor to hepatic TG accumulation in early phases. But how the uptake process may shape the establishment of the zonated TG accumulation is unknown. Our model simulations support these previous studies and, in addition, point out that the mechanism of FA uptake may be important for the zonated accumulation of TGs during early stages of steatosis. As mentioned above, the blood gradient of FAs along the porto-central axis seems to be unlikely to determine the establishment of pericentral TG accumulation. Instead, the uptake mechanism of FAs from blood into the cells dictates the zonation pattern.

The mechanism how hepatocytes take up FAs from blood and whether FA transporters are involved has been debated in the last years [71]. Based on the fact that the cell membrane is a lipid bilayer, it was concluded that FAs may simply diffuse passively through the cell membrane without the need of protein-mediated transport [34,72]. However, this view is questioned due to the insufficient experimental evidence of passive FA diffusion. Kleinfeld [73] and Bradbury [74], among others, provide a detailed discussion. In fact, long-chain FAs are the essential type of dietary FAs for TG accumulation and experimental data advocate a more complex biological process than passive diffusion involved in their uptake [75]. Only the assumption of a transporter-mediated uptake of long-chain FAs can adequately explain the saturation in uptake under a high FA supply. Such a transporter-mediated uptake process can be described by a Michaelis–Menten function [76–79]. Bradbury [74] proposed a combination of both, linear non-saturable kinetics with saturable Michaelis–Menten kinetics, but with the notion of only a minor contribution of linear FA diffusion. A saturable uptake implies that the transport of FAs into hepatocytes relies on transport molecules. Several of such transporter proteins were identified (reviewed in [80]), particularly relevant are fatty acid translocase/cluster of differentiation 36 (FAT/CD36) [81] and fatty acid transport proteins (FATPs) [82,83]. Moreover, the observation of an association between an increased expression of FA membrane transporters and the grade of steatosis further supports the hypothesis of a transporter-mediated uptake process [84,85].

Our model results reinforce the perception of a transporter-mediated uptake mechanism of FAs in liver cells: the model with saturable FA uptake kinetics induced a pericentral steatosis pattern, whereas linear uptake of FAs caused a periportal dominated pattern of fat accumulation. The Michaelis–Menten function results in low to medium concentration of FAs in a nearly linear uptake. This means that the hepatic FA uptake, described by this function, can handle the physiological normal range of plasma FA concentrations. This is in accordance with the theory that the K_M of substrate transporters fits well to the normal concentration of the substrate in the circulating blood [71]. For example, the K_M of the glucose transporter GLUT4 (K_M = 4.3 mM) matches to the normal circulating plasma glucose concentration of 4–5 mM in humans [71]. Of course, the relationship is more complex for plasma FAs, because the blood contains various types of FA [86], which can be taken up by the liver. It is likely that each may have its specific K_M value for uptake. In our model, the FA uptake process is a simplification of the real uptake process, considering FAs in a whole and not differentiating among types. The total plasma FA concentration in a healthy human is in a range of 100–400 µM [71] and we chose a K_{M_FAup} of 300 µM for the nonlinear uptake kinetics. However, this is also a simplification in our model, neglecting the fact that FA uptake into hepatocytes seems to be determined by the unbound (i.e. non-protein bound) concentration of FAs in blood [87]. The concentration of unbound FAs was found to be 7.5 ± 2.5 nM in humans [88] and increases exponentially with a rise in the total plasma FA concentration [89]. Unfortunately, no studies report on kinetics of hepatic FA uptake in humans, thus a comparison of the plasma (unbound) FA concentration to the K_M value of hepatic FA uptake is not possible at present.

For NAFLD patients, an increased total plasma FA concentration was reported, with values above 1000 µM [90]. Theoretically, this concentration may be considerably higher than the K_M value of FA uptake, i.e. if the plasma FA concentration raises the uptake into hepatocytes becomes saturated. This saturation of the FA uptake process seems to be important for the zonated accumulation of TGs: as a consequence of the saturated FA uptake of periportal hepatocytes, the blood level of FAs rises and a greater amount of FAs reaches the pericentral hepatocytes. Yet the zone in which oxygen supply is reduced, allowing only a low rate of FA oxidation. Of note, our model results revealed that the maximal velocity of FA uptake (v_{max_FAup}) determines the zonation pattern. A low maximal velocity (in regard to the blood FA level) leads to a pericentral TG accumulation even under a low FA supply via blood. Of course, we admit the necessity of experimental support to further reinforce our theoretical results of the importance of FA uptake on steatosis zonation patterning. However, the small size of the sinusoid makes it challenging to obtain small-scale data on zonation patterns in a quantitative way. Immunohistochemical studies allow for exploration of the zonation pattern of enzymes in a qualitative manner, assigning a metabolic pathway to be periportal, pericentral or azonal. But this allows no quantitative measurements of the differences between zones. Furthermore, the complex interaction of pathways of glucose and lipid metabolism makes it difficult to separate effects. In future, the organ-on-a-chip technology [91] or the next generation human Liver Acinus MicroPhysiolgy System [92] may be promising techniques to further explore the zonation of metabolic pathways in a more quantitative way and in dependence of local oxygen supply. These techniques allow mimicking the multicellular architecture and, therefore, hepatic microcirculation [91], which would facilitate the investigation of the effect of the physical microenvironment on a zone-dependent hepatocyte function not possible with conventional hepatocyte cultures. Notwithstanding, investigating the zonation of FA uptake could be an essential core area of future research. We recommend, in the field of steatosis research, directing further effort towards improving the understanding of the hepatic FA uptake mechanism and its link to steatosis zonation.

The sensitivity analysis additionally points out that the export of TGs from hepatocytes is a crucial determinant of TG accumulation, although it does not influence the zonation pattern. For the secretion process, TGs undergo a lipolysis/re-esterification cycle [93] with subsequent assembly to very-low-density lipoproteins (VLDLs). Interestingly, VLDL-TG secretion seems to be more or less independent from intracellular FA concentration, but is influenced by stored TGs [94]. Our result that TG export influences steatosis establishment is consistent with recent literature data: for example, the rate of VLDL-TG export is reduced in NALFD and NASH patients [95], resulting in a greater amount of stored TGs. The secretion of VLDL-TG is not specified to a certain hepatic zone, rather a substrate-dependent view is favoured. This means a higher secretion rate can be attributed to a higher TG synthesis rate rather than to a higher capacity in one zone [26]. This concept is supported by the observation that periportal and pericentral hepatocytes have a similar capacity of VLDL formation and secretion in cell culture [96].

In our modelling approach, we only considered oxygen and FAs as substrates for hepatic metabolism, neglecting the role of both substrates in modulating signalling pathways and influencing enzyme expressions. But gradients of substrate concentrations are only one aspect in the zonation pattern of steatosis. In the long run, the activities and expressions of enzymes of each hepatocyte are adjusted to the local physiological conditions the cell is faced with. This resulting functional heterogeneity is induced by local signals such as metabolite and hormone concentrations [8,97]. Beyond that, morphogens are important zonal determinants of the metabolic set-up of hepatocytes along the porto-central axis. For example, the Wnt/β-catenin signalling pathway induces the expression of pericentral genes but decreases periportal gene expression [98,99]. Thus, the zonal distribution of enzyme expressions is influenced by morphogens such as β-catenin [100], which obviously can also affect the zonation pattern of steatosis [98]. In sum, metabolite, hormone and morphogen concentration gradients lead to the establishment of a zonated metabolic activity along the porto-central axis. For example, the fatty-acid-binding protein (FABP), involved in intracellular FA trafficking [101], shows a periportal dominated expression [31,102]. This periportal dominance is considered as being a response of the cells to the physiological condition in this zone [31], i.e. to the high concentration of plasma FAs entering the liver through the portal vein. The gradient of FABP expression may be induced by an increased FA uptake and, therefore, be an indicator for a periportal dominated FA uptake [16,31]. Unfortunately, the zonal patterns of membrane proteins directly involved in FA uptake are not known at present, but an influence on the zonation of steatosis is without doubt. An increased amount of membrane transporter proteins in periportal hepatocytes would increase the velocity of the uptake process thus more FAs can enter these hepatocytes and become available for oxidation and/or TG synthesis. The pattern of steatosis may shift thereupon from pericentral dominance in the early phase of steatosis establishment to an azonal pattern, i.e. an extension of TG storage to the middle and the periportal zones, the longer the high-fat supply takes.

In accordance to the periportal dominance of FABP expression, some of the enzymes involved in mitochondrial β-oxidation also show an increased expression or activity level in this zone [26,27]. However, in isolated cell culture experiments, the higher concentration of FABP in zone 1 hepatocytes does not lead to an increased rate of oxidation in these cells compared to cells isolated from the pericentral zone [31]. Moreover, under fed conditions, male rats exhibited a homogeneous distribution of the enzyme 3-hydroxyacyl-CoA dehydrogenase [103], which is involved in mitochondrial β-oxidation. The understanding of zone-specific expression patterns of enzymes of mitochondrial β-oxidation is not yet clear. In our modelling framework, we do not consider the zone-specific expression of enzymes, because we want to focus on the effects of substrate concentrations. In contrast, Ashworth et al. [48] developed a comprehensive kinetic model of hepatic metabolism, in which the kinetic parameters in each metabolic compartment were adjusted to mirror the variation of enzyme expression along the sinusoid. Similar to our model results, they found that the zonation pattern of steatosis was most sensitive to FA uptake and TG secretion. Beyond that, our model allows a closer look at the FA uptake mechanism and the influence of the blood gradients of oxygen and FAs on zonation patterning.

We are aware that other processes than those referred in our model may be involved in the establishment of the zonation pattern of steatosis. One process, which may influence the zonation pattern of TG accumulation, is hepatic de novo lipogenesis (DNL), i.e. the production of new FAs mainly from acetyl-CoA derived from the degradation of carbohydrates and amino acids. DNL can contribute greatly to TG accumulation under certain circumstances [104] and is considered to take place in the pericentral zone [26,103]. This process however is critical only under a carbohydrate-rich diet [105,106] and tightly regulated by insulin and glucose [107,108]. In our modelling framework, we placed emphasis on a high-fat diet and the early stages of a zonated TG accumulation associated with it. For our model application with focus on the FA content of a diet, we assumed that the carbohydrate content in the diet is in a normal, healthy range and, thus, does not contribute to an increased DNL. In support of this procedure is the fact that at the beginning of a high-fat dietary intervention the excess supply of fat does, in fact, inhibit the process of DNL [106]. In accordance, Hijmans et al. [70] predicted with a novel computational modelling approach that not increased DNL but the amount of FA influx determines hepatic fat load. Beyond our modelling approach, the comprehensive modelling study by Ashworth et al. [48] factors in both carbohydrate and energy metabolism and shed light on the impact of insulin resistance and DNL for a zonated TG accumulation under a high-intake diet.

In conclusion, from a theoretical perspective, the porto-central oxygen gradient and the saturable FA uptake process are involved in the establishment of a pericentral dominant steatosis. Our results highlight the outstanding role of oxygen as a major determinant of a zonated fat accumulation by its role in mitochondrial FA oxidation. In contrast, the porto-central gradient of FAs along the blood vessels does not contribute to pericentral TG accumulation. Instead, the FA uptake process seems to be relevant; more specifically, the maximal velocity of FA uptake determines the switching between periportal and pericentral fat accumulation. Understanding the causes of aberrant TG accumulation (steatosis) in patients with diet-induced fatty liver is essential to develop novel therapeutic strategies. Although our study is a theoretical one, we are confident that it will encourage wet-lab research to push forward our understanding of hepatic TG accumulation in patients with NAFLD.

Data accessibility

A detailed model description and further data material supporting this article are provided as electronic supplementary material.

Authors' contributions

J.S., U.D., S.S. and R.G. were involved in the planning of the basic concept. The development of the model structure, based on biological knowledge, was conducted by J.S. and S.S. J.S. implemented the mathematical model and carried out model analyses. All authors participated in manuscript writing and the editing process. All authors gave final approval for publication.

Competing interests

We declare we have no competing interests.

Funding

This research was supported by the German Research Foundation (grant nos: SCHL 2130/1-1 (J.S.), DA 251/10-1 (U.D.)). We further gratefully acknowledge the previous support from the ‘Virtual Liver Network’ of the German Ministry of Education and Research (BMBF; S.S., R.G., U.D.).