User:Johnsy/Lipoprotein Modelling/Bile Acid Biosynthesis

Background

We first start off with a diagram of the model that we wish to use for the bile acid biosyntheis pathway.

In hepatocytes, cholesterol is converted to bile acids to be excreted into the intestine via the action of the enzyme cholesterol 7α hydroxylase (C7H). The bile acid is excreted and stored in the gall bladder and is pooled until hormonal changes within the body signal to contract and excrete the bile acid into the gut during feeding. Bile acids are key to increasing the solubility of ingested lipids. Approximately 95% of the bile acids are returned to the circulation through the ileum (termed enterohepatic circulation). The returned bile acids then inhibit the gene expression of C7H through a complex unknown pathway, but modeled simply as a direct inhibitor as seen below. The returned bile acids are returned to the bile acid pool in the hepatocytes with a certain rate constant.

The Model

The equation governing the enzymatic degradation of cholesterol by the enzyme Cholesterol 7α Hydroxylase (C7H) is modeled using a simple Michaelis-Menten type equation as shown below. This equation will be combined with that from the de novo synthesis pathway to complete our hepatocyte model.

[math]\displaystyle{ \frac{d[IC]}{dt} = - \frac{k_3[C7H][IC]}{k_{m2} + [IC]} }[/math]

The equation for the synthesis of bile acids follows from the degradation of cholesterol assuming that in this model cholesterol is only degraded to bile acid.

[math]\displaystyle{ \frac{d[BA]}{dt} = \frac{k_3[C7H][IC]}{k_{m2} + [IC]} - d_3 \eta[BA] - r_1(1-\eta)[BA] + k_5[RBA] }[/math]

Individual terms are described below:

• [math]\displaystyle{ d_3 \eta[BA] }[/math] - This term describes the excretion of bile acids of which a fraction η is not returned to the liver (η is usually about 5% of the excreted bile acids).

• [math]\displaystyle{ r_1(1-\eta)[BA] }[/math] - This term describes the excretion of bile acids which are recycled to the blood stream and are converted to "Returned Bile Acids" (RBA). This was done for the convenience since the gene expression of C7H is dependent on the returned bile acid pool and not the bile acid pool already present in the hepatocyte.

• [math]\displaystyle{ k_5[RBA] }[/math] - This term describes the conversion of Returned Bile Acids to the Bile Acid pool.

The equation governing the production of the enzyme C7H is described below. This is a genetically controlled production and will be justified and described further below. The justification follows similar lines to the genetic justification for the production of HMGR in the de novo synthesis pathway.

[math]\displaystyle{ \frac{d[C7H]}{dt} = \frac{k_4}{b_2 + [RBA]} - d_4[C7H] }[/math]

And finally, the equation relating to the returned bile acids is shown below.

[math]\displaystyle{ \frac{d[RBA]}{dt} = r_1(1-\eta)[BA] - k_5[RBA] }[/math]

Parameters

• k₃ - rate constant for the conversion of cholesterol to bile acids via C7H (equivalent to k_cat value)
• k_m2 - Michaelis-Menten constant for C7H
• d₃ - degradation rate of bile acids which are not returned
• r₁ - "degradation" rate of bile acids to recycled bile acids
• k₅ - rate constant for the conversion of returned bile acid to bile acid
• η - percentage of bile acids not recycled back to the liver
• k₄ - transcription rate
• b₂ - attenuation factor for the regulation of C7H by RBA
• d₄ - degradation rate of C7H

Assumptions

Cholesterol Conversion to Bile Acid

In the conversion of cholesterol to bile acid, there are several enzymes involved, but the rate limiting step is the conversion of cholesterol to 7α-hydroxy cholesterol by the action of a P450 cytochrome reductase [1]. We assume that all the other steps in this conversion are fast and the rates are negligible compared to that of the rate limiting step. Hence, we can assume that the overall rate of production of bile acids from cholesterol is defined by the kinetics of this step.

Furthermore, we also assume that the concentration of C7H is not constant and will vary over time defined by the equation above governing production of C7H. Although the derivation of the Michaelis-Menten equation assumed that the concentration of enzyme was kept constant, we assume that the steady state value of the enzyme will be reached before any significant changes in the level of enzyme. This is supported by the fact that there is a delay of approximately 40 minutes between regulation of gene expression and the presence or reduction of protein in the cell.

We currently assume that the only degradation by cholesterol is through bile acids, but we will couple the cholesterol governing equation with the same from the de novo synthesis pathway and from the lipoprotein pathway. By separating each component, we can compartmentalize and have modularity of the model.

Unlike the conversion of HMG-CoA to cholesterol, we cannot assume that the enzyme will be rate limiting. Although studies that have probed the kinetic parameters of C7H have assumed cholesterol levels that are much higher than enzyme levels, in our model, there is a dynamical equilibrium set up between the amount of cholesterol in the cell and the amount converted to bile acids.

Enterohepatic Circulation

A complex pathway is required for the return of bile acids from the ileum to the liver through chylomicrons and portal vein circulation. Unfortunately, the exact mechanism for this return is unknown [2] and we just assume a linear relationship between the rate of return and the amount of bile acid present in the intercellular bile acid pool. Furthermore, we also assume a linear relationship in the rate of conversion from the recycled bile acid pool to the intercellular bile acid pool, since literature review has not given us any significant insight into this metabolic pathway in hepatocytes.

We have also assumed that only the returned bile acid pool regulates the gene expression of C7H. In fact, there is a complex negative feedback through multiple nuclear hormone receptors from the returned bile acid which regulates the production of transcription factors that regulate the transcription of the genes required for bile acid synthesis. [2] As usual for modeling, we are neglecting these complications and simplifying our model as much as possible yet attempting to maintain the key biochemical elements involved.

References

1. Russell DW and Setchell KD. Bile acid biosynthesis. Biochemistry. 1992 May 26;31(20):4737-49. DOI:10.1021/bi00135a001 |
2. Redinger RN. The coming of age of our understanding of the enterohepatic circulation of bile salts. Am J Surg. 2003 Feb;185(2):168-72. DOI:10.1016/s0002-9610(02)01212-6 |
3. Barrio M, Burrage K, Leier A, and Tian T. Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation. PLoS Comput Biol. 2006 Sep 8;2(9):e117. DOI:10.1371/journal.pcbi.0020117 |