A Computational Model of the Hepatic Lobule

Naito Y.

While many inter-organ and intra-organ gene regulations have been found recently, raison d'être of such regulations are hardly explicated. We aimed liver ammonia detoxification as a prospective target because of its simple histological structure and adopted systems biology approach to elucidate the question. In the mammalian liver, many metabolic systems including ammonia metabolism are heterogeneously processed among hepatocyte position in the lobule.1-5 Three enzymes that are incorporated in ammonia metabolism are expressed gradually between the periportal zone (influx side) and the pericentral zone (efflux side) in the lobule.6,7 To investigate the cause of the heterogeneous gene expression, a simple eight-compartments model, in which each compartment represented hepatocellular ammonia metabolism by largely enzyme kinetics equations, was developed as a lobule model.8 In silico simulation indicated that regulated enzyme gradient reduced ATP requirement for ammonia detoxification, suggesting that these enzyme gradients by gene regulations improve the fitness of organism by saving energy (ATP consumption).


Gene regulation seems to be an important device for functional specialization among cells, tissues and organs. Detection of numerous inter/intra-organ differential gene regulations (e.g., the expression rate and the alternative splicing, etc.) support this argument. Many technological innovations (e.g., genome sequencing, microarray, full-length cDNA library, histochemistry, etc.) accelerate the accumulation of these discoveries, but the raison dêtre of such regulations remains almost unclear. Regulation of gene expression is a highly energy consuming process because many macromolecules are involved. Therefore, it is thought that the gene regulation improves the fitness of the host individual as a payment of the energy cost. Various explanations for each regulation are proposed, but most of them are no better than the thought experiments despite the fact that many data that back up the explanations are quite solid and accurately quantitative.

At present, the greater part of the biological data and information is limited to the molecular and cellular (microscopic) level. While the explosive development of molecular and cellular biology has yielded both copious and precise information at the subcellular level, biology for higher-level (mesoscopic or macroscopic) structures has lagged far behind. Anatomy and histology illustrate a multicellular individual in a hierarchical classification scheme, namely of, tissues, organs and individual, going from the microscopic to macroscopic. The store of knowledge built up at each level of the hierarchy is at present excessively disproportionate. The knowledge accumulated in the last decade at higher-levels than the cell is undoubtedly less than that at the cellular and subcellular levels. A major constraint is the currently limited technology, which for the tissue or organ level presents greater difficulties in all aspects of sample preparation, cultivation and measurement than required for the single-cell level.

To overcome this situation, we adopted systems biology approach and focused on the ammonia detoxification in the liver as a competent model. In the mammalian hepatic lobule, many metabolic systems including ammonia metabolism are processed heterogeneously. This zonation of function seems to depend on the gradual existences of metabolic substrates, oxygen and hormones and the structural factors such as nerves, biomatrix and receptors. Some enzymes are not distributed uniformly in the lobule. Among ammonia metabolism relating enzymes, carbamoylphosphate synthetase (CPS), glutamine synthetase (GS) and ornithine aminotransferase (OAT) have steep gradual expression in the lobule. Its significance is unclear now.

It is assumed that gradual gene expressions of the three enzymes in the lobule affect the efficiency of ammonia metabolism in the liver. To elucidate the significance of gradual gene expressions in the lobule, it should be evaluated that how strong the differential expressions of enzymes affect the metabolic dynamics in each hepatocyte and what the total effects of heterogeneous hepatocytes on the liver is. Both of the amount of ATP required and the velocity for ammonia detoxification are useful as indices to evaluate the metabolic efficiency.

The liver has relatively simple histological structure among mammalian organs. However, it is still a complex system, since hepatocytes queue along the sinusoidal capillary in the lobule then metabolic change in sinusoidal upstream perturbs downstream hepatocytes. In vivo and in vitro investigations have hardly techniques to measure the fraction of ATP consumed for ammonia detoxification exactly among whole ATP consumption. Moreover, it is almost impossible to grasp the metabolic states of both of the whole liver and each hepatocyte in it. Meanwhile, approximately 15% of ATP is consumed by urea cycle in rat hepatocyte,9 indicating that a stir in urea cycle can enormously affect the energetics not merely of hepatocyte but also of the liver and the individual. In silico study using mathematical models suggested that gradual gene regulations of ammonia metabolism relating enzymes save the energy required for the metabolism.

The Single Hepatocyte Model

We started by construction of a model for ammonia detoxification in single hepatocyte and asking how well the mathematical model simulates the actual hepatocyte. We then assemble multiple cell models to a sinusoid model, which is used to address the advantage of the gene regulations in hepatic ammonia metabolism.

The single hepatocyte model included 67 substances and 29 reactions (Fig. 1A, see Appendix for details), most reactions are related with urea cycle and largely reproduced the metabolic states described in previous reports for mammalian hepatocyte. The liver is a giant bunch of the lobules and the lobule is a cluster of the sinusoids, which connect in parallel each other. Thus, we assumed that an appropriate model for sinusoidal metabolism linearly approximates the liver metabolism.

Figure 1. Schematic representation of the model.

Figure 1

Schematic representation of the model. A) The model describing ammonia metabolism in single zone model. Filled circles, open round rectangles and open circles represent substances, enzymes and transporter, respectively. Solid line arrows represent reactions (more...)

The Sinusoid Model

To construct the sinusoidal metabolism model, no difference among hepatocytes along sinusoid without the gradual gene expressions of CPSI, GS and OAT was assumed and single hepatocyte models were joined up in series along a model of sinusoidal material flow (Fig. 1B). A simple model that consists of eight hepatocellular compartments roughly reproduced the metabolic zonation of ammonia detoxification in the lobule.8

The active pathways were quite different between the periportal and the pericentral zones in the model (Fig. 2). Urea production, urea exportation and creatine generation were pronouncedly predominant in the periportal zone (light gray/red arrows in Fig. 2), while glutamine formation and exportation were predominantly seen in the pericentral zone (dark gray/blue arrows in Fig. 2). Because mitochondrial ornithine aminotransferase is mainly expressed in the pericentral zone, the concentration of glutamate, which is a reaction product of ornithine aminotransferase, was higher in the pericentral than the periportal zone. The glutamate concentration in mitochondria was increased from 6.97E-3 M to 8.70E-2 M along the porto-central axis. Consequently, the velocity of glutamate dehydrogenase, which catalyzes glutamate, was larger in the pericentral than the periportal zone. Glutamate-aspartate translocase and mitochondrial GOT also exhibited higher activities in the pericentral zone, while cytoplasmic GOT showed an opposite trend of flux between the periportal and the pericentral zone (Fig. 2).

Figure 2. Flux disparities between both ends of the porto-central axis.

Figure 2

Flux disparities between both ends of the porto-central axis. The width of each arrow proportionally reflects the flux ratio at both ends (the first and the eighth compartment). The thickest line indicates the flux disparity to be ten-fold or more. Light (more...)

To address the functional meaning of the gradual gene expressions, we developed a model without the enzyme gradients. In the no-gradient model, the lobule consumed more ATP to detoxify ammonia than the original model with gradual expressions. Most of the chemical reactions and transportation exhibited larger fluxes in the periportal zone than the pericentral zone. Due to the high affinity for ammonia of glutamine synthetase, i.e., 1/10 of Km of carbamoylphosphate synthetase, ammonia predominantly converted to glutamine in the periportal zone. The fluxes gently changed from the periportal to the pericentral zone while dramatic alterations were seen in the sixth or seventh compartment in the model with the gradual gene expressions, revealing that the pericentral hepatocytes played a lesser role in metabolism in the no-gradient model.

GS and OAT are co-expressed in hepatocyte and no co-expression of these genes is seen in other tissues. This raises a hypothesis that the co-expression of GS and OAT benefits the liver metabolism. Activities of both enzymes, which are parallel to gene expressions, were variously perturbed in the mathematical model to enquire whether the co-expression advantages ammonia detoxification in the lobule. Consequently, cooperative gradient of GS and OAT was suggested to improve the energy efficiency of ammonia detoxification.

Four intermediate models were constructed. The first intermediate model included only GS gradient, the second one included GS and CPS gradients, the third one included only OAT gradient and the last one included GS and OAT gradients. Our study using mathematical models suggested that the gene regulation in hepatic ammonia metabolism contributed for energy conservation. Although regulation of gene expression itself should spend much energy, the returning energy save is conceivable to overcome the break-even point. Co-expression of GS and OAT was also suggested to improve the energy efficiency of ammonia metabolism. These support the aspect that intra-tissue gradual gene expression has evolved in a direction to upgrade metabolic efficiency. To compare the metabolic aspects in the periportal zone and that in the pericentral zone, the flux distributions were examined. To evaluate the effects of the enzyme slope along the lobule metabolic state, the following rates were calculated and used as the indexes.

Image ch4097eqn01.jpg

RNH+4,detox and νNH44-tp are nearly equal under the assumption of steady-state. The model with all of gene expression gradients ranks higher among six models in the rate and energy efficiency of ammonia detoxification. The mean rate of elimination of ammonia from the sinusoid was 11.8% faster in the model with full-gradients than the no-gradient model. Although the rate of degradation of ammonia in all eight compartments was 20.0% slower in the with full-gradients model than the no-gradient model, the rate of ammonia generation was also 53.8% slower than the control model, showing that the full-gradients model was able to remove ammonia more efficiently than the control. The mean rate of ATP consumption in the full-gradients model was 9.5% less than the no-gradient model. Energy efficiency η, which means the number of consumed ATP molecules required for the elimination of one ammonia molecule, was smaller in the full-gradients model than the no-gradient model (3.59 ± 0.22 vs 4.47 ± 0.49). Two intermediate models also demonstrated smaller η while the other intermediate models demonstrated greater η than the control.

On the other hand, the urea cycle in the liver also strongly contribute to maintain acid-base balance, but our model did not considered it this time. Since available quantitative data on acid-base balance was fewer and less precise than that on ammonia detoxification, we judged that a model of just ammonia metabolism could be more effective than that including both of ammonia metabolism and acid-base balance. For further discussion about the raison dêtre of gene regulations, the mechanism of acid-base balance should be implemented to the model.


We evaluated that how strong the totality of heterogeneous metabolic changes in each cell influence the fitness of tissue and organ through systems biology techniques. Such approach can link molecular processes at subcellular level and macroscopic functions at tissue, organ, or individual level together, not by statistical correlation but by concrete causal relationship. Coming biosciences should piece together the huge data (e.g., genome sequence, expression profile, protein structure, etc.,) to gain integrated comprehension of life. Generally, in vivo and in vitro approaches do not have integrability but precision and in silico approach has vice versa, thus they surely function complementally and accelerate the progress of biological sciences.


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Appendix: Details of the Mathematical Model

1. Carbamoylphosphate Synthetase (EC.

The enzyme catalyzes 2ATP + NH+4 + HCO3-3 → 2AMP + 2Pi + CP in mitochondria. The kinetic model was obtained from previous literature.10,11

Image ch4097eqn02.jpg
Image ch4097eqn03.jpg

N-Acetylglutamate Synthetase (EC.

The enzymes catalyze AcCoA + Glu → CoA + NAG.

The reaction mechanism is a nonreversible rapid equilibrium random bi-bi mechanism.12

Image ch4097eqn04.jpg
Image ch4097eqn05.jpg

3. Glutamine Synthetase (EC.

The enzyme catalyzes ATP + Glu + NH+4 → AMP + Pi + Gln.

Image ch4097eqn06.jpg

4. Phosphate-Dependent Glutaminase (EC.

The enzyme catalyzes Gln + Pi → Glu + NH4+. It is activated by the product: ammonia.13 Cooperativity of glutamine and Pi, which is an essential activator for phosphate-dependent glutaminase, were modeled by the Hill equation.14

Image ch4097eqn07.jpg

5. Ornithine Carbamoyltransferase (EC.

The enzyme catalyzes CP + Orn ↔ Pi + Cit. The reaction mechanism is an ordered bi-bi sequential mechanism.15

Image ch4097eqn08.jpg
Image ch4097eqn09.jpg

6. Argininosuccinate Synthetase (EC.

The enzyme catalyzes ATP + Cit + Asp↔ AMP + Pi + ASA. The reaction mechanism is an ordered ter-ter mechanism.15

Image ch4097eqn10.jpg


Image ch4097eqn11.jpg

7. Argininosuccinate Lyase (EC.

The enzyme catalyzes ASA↔ Fum + Arg. The reaction mechanism is an ordered uni-bi mechanism.

Image ch4097eqn12.jpg

8. Arginase (EC.

The enzyme catalyzes Arg → urea + Orn. The reaction is an irreversible process and inhibited by ornithine.

Image ch4097eqn13.jpg

9. MetaNet Model

OTL, GTL, GATL, OAT, GOTm, GOTc, GDH, GAT and GAMT were modeled using MetaNet. 16 Reaction stoichiometries were defined as follow:

Image ch4097eqn14.jpg

Although MetaNet is not guaranteed to accurately reproduce enzyme kinetics, it was used in our model with the expectation it would roughly estimated the rates of reactions. Velocities of reactions were calculated as follows:

Image ch4097eqn15.jpg
Image ch4097eqn16.jpg

Ks,x and ns,x is the binding constant and the cooperativity index (essentially a Hill exponent) of substance (or effecter) s of enzyme x (equilibrium constant for dissociation of the enzyme-ligand complex), respectively and Ks,x is the former's effective binding constant, which reflects the activities of the competitive activators kx and the competitive inhibitors hx. cs is the concentration of substance s. k ′s and h′x are noncompetitive activators and noncompetitive inhibitors of the reaction catalyzed by enzyme x, respectively.13

10. System N

Glutamine is transported into the cytoplasm by a sodium-dependent transport mechanism. This process is inhibited by histidine.17

Image ch4097eqn17.jpg

11. System L

Glutamine is transported into the cytoplasm by a sodium-independent transport mechanism. This process is inhibited by tryptophan.17

Image ch4097eqn18.jpg

12. Ammonia Transport between Sinusoid and Cytoplasm

Ammonia transport between the sinusoid and cytoplasm was modeled based on the general mass action law.

Image ch4097eqn19.jpg

13. Transportation of Glutamine, Arginine and Ammonia between Cytoplasm and Mitochondria

Transports of glutamine, arginine and ammonia across the mitochondrial membrane were presumed to rapidly attain equilibrium.

Image ch4097eqn20.jpg

14. Urea Transport to Sinusoid

Excretion of urea in the sinusoidal space was modeled based on the general mass action law.

Image ch4097eqn21.jpg

15. Glutamate Transport between Sinusoid and Cytoplasm

Glutamate transport between the sinusoid and cytoplasm was modeled as Michaelis-Menten reversible kinetics.

Image ch4097eqn22.jpg

16. Glutamate Flux from the Outside Pathways

Glutamate flux from the outside pathways of the model was represented by the difference between zero-order influx and efflux based on the general mass action law.


17. Degradation of Metabolites

Degradation of N-acetylglutamate, Pi and CoA were modeled based on the general mass action law under the assumption of steady-state.

vdeg,s = kdeg,s[S] where s is a substance.

18. Ornithine Inflow from Other Reactions

To hold the steady-state, ornithine inflow from other reactions was presumed to be equal to the flux of ornithine aminotransferase, vOAT.

19. Metabolites Flows in Sinusoid

Flows of ammonia, glutamine, glutamate and urea from nth sinusoidal compartment to n + 1th compartment, ve,sn were modeled based on the general mass action law.

Je,sa = ke[Sn]e

where sn represents a substance in the nth compartment of the sinusoid.

20. Heterogeneous Gene Expression in Hepatic Lobule

To describe the regulated gene expression of three enzymes, carbamoylphosphate synthetase, glutamine synthetase and ornithine aminotransferase along the porto-central axis, we adopted the mechanistic model proposed by Christoffel et al.18 The model is based on simple receptor-ligand kinetics and the parameters are fitted by experimental values. [F*x] is the concentration of the active transcription factor F of enzyme x and assumed as follows18:

Carbamoylphosphate synthetase: [F*CPS] 0.2 - 0.01X

Glutamine synthetase and ornithine aminotransferase: [F*GS] = [F*OAT] 0.1X.

X is the radius of the hepatic lobule: X = 0 corresponds to the portal tracts and X = 10 corresponds to the central vein. Thus, X was defined as follows in our model:

Image ch4097eqn23.jpg
n is the number of compartment out of eight compartments: n = 1 corresponds to the compartment adjacent to the portal tracts and n = 8 corresponds to the compartment adjacent the central vein. The total of sinusoidal compartments is eight in our model. RGX,x is the relative rate of transcription, assumed to correspond to the transcription rate in our model. RGX,x is calculated using the fractional saturation YGX,x, the discussion constant KGX,x and the Hill coefficient nGX,x as follows:18
Image ch4097eqn24.jpg

Carbamoylphosphate synthetase was fitted with high-affinity (YGX,CPS,h) and low-affinity (YGX,CPS,l) units as follow:18.