# The dynamic behavior of methionine metabolism cycle in human liver cell.

1. Introduction

2. Theoretical

2.1 Mathematical model of methionine metabolism

The metabolic pathway map of the methionine cycle is shown in Fig.1a. To study the salient features of its regulation properties Reed et al. [5] formulated four differential equations (Eqs.(1)-(4)):

d[Met] / dt = [v.sub.MS] + [v.sub.BHMT] + [v.sub.Metin] - [v.sub.MATI] - [v.sub.MATIII] (1)

d[AdoMet] / dt = [v.sub.MATI] + [v.sub.MATIII] - [v.sub.METH] - [v.sub.GNMT] (2)

d[AdoHcy] / dt = [v.sub.METH] + [v.sub.GNMT] - [v.sub.AH] (3)

d[Hcy] / dt = [v.sub.AH] - [v.sub.CBS] - [v.sub.MS] - [v.sub.BHMT] (4)

All the rate equations on the right hand side and the values of all other parameters and constants included in the rate equations with their units are as in our first paper [1].

2.2 Dynamic analysis in BST

BST approximates a given process by products of power-law functions. In addition to the S-system form [1], another important variant within BST is the Generalized Mass Action (GMA)system form described as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [X.sub.i](i = 1,2,...,n) are the dependent variables, [X.sub.i](i = n+1,n+2,..., n+m) are the independent variables, n is the number of dependent variables, m is the number of independent variables, [v.sub.+ik] and [v.sub.-ik] (i = 1,2,...,n; k = 1, 2,..., p or q) are the local influx and efflux, respectively, [[gamma].sub.ik] is the rate constant, and [g.sub.iJk] is the kinetic order. Evidently, the rate constant for [v.sub.ik] is set as a positive value for the incoming flux and a negative value for the outgoing flux. Transforming Eq. (5) according to the definition of the logarithmic gain leads to the differential equations for the time derivatives of the logarithmic gains as:

[dL.sub.i,f] / dt = 1 / [X.sub.i] (N.summation over (l=1)] [[??].sub.il] [L.sub.l,f] + [J.sub.if] (i = 1,-,N;f - N +1,...,N + m) (6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The initial value of Eq. (6) is given as

[L.sub.i,f] (0) - 0 (i -1,-,N; f - N + 1,-,N + m) (8)

Therefore, Eq. (6) describes the functional relationship of the logarithmic gain with time.

3. Results

3.1 Behavior of Dynamic logarithmic gains

Figure 2 shows the dynamic logarithmic gains in responses of the metabolite concentrations to an infinitesimal change at t=0 in each independent variable in the methionine cycle at the nominal steady state under the regulation of CBS by AdoMet, i.e., when [X.sub.1]=53.5, [X.sub.2]=138, [X.sub.3]=12.8, and [X.sub.4]=0.884. It should be noticed here that there is no change in each metabolite concentrations.

After all the logarithmic gains start from a zero value and change in the positive and negative regions, they reach their respective steady state values. For example, the logarithmic gains, L([X.sub.1], [X.sub.9]), L([X.sub.3], [X.sub.9]), and L([X.sub.4], [X.sub.9]), expressing percentage responses of [X.sub.1], [X.sub.3], and [X.sub.4] to an infinitesimal increase in [X.sub.9], pass through their respective maximums and then approach their respective steady state values: 0.69, 0.68, and 0.56, respectively. On the other hand, L([X.sub.2], [X.sub.9]) decreases in the negative region and finally approaches its steady state value: 0.60. These behaviors imply that when [X.sub.9] is perturbed at t=0, [X.sub.1] , [X.sub.3], and [X.sub.4] are increased because the incoming fluxes at their pools are larger than the outgoing fluxes, whereas [X.sub.2] is decreased because there is a reverse relation between the fluxes at this pool. The logarithmic gains, L([X.sub.4], [X.sub.10]), L([X.sub.1], [X.sub.15]), and L([X.sub.1], [X.sub.16]), have distinct characteristics that they change in both the positive and negative regions. The logarithmic gain, L([X.sub.1], [X.sub.5]), takes a maximum of about 0.25 in the neighborhood of t=0.2 and then finally approaches a small value of 0.013. This would give a judgment that [X.sub.5] hardly influences [X.sub.1], if the sensitivity analysis is carried out only at the nominal steady state. However, this judgment is evidently not valid because L([X.sub.1], [X.sub.5]) takes a value of about 0.25 at maximum until the system is stabilized. This fact suggests that it is insufficient to carry out the sensitivity analysis of a system only at a steady state and the dynamic sensitivity analysis should be made aggressively.

The logarithmic gains in Fig.2 change smoothly, implying that the metabolite concentrations also change smoothly without oscillation and then reach their respective steady state values when a small perturbation suddenly occurs with an enzymatic activity.

3.2 Dynamic logarithmic gains on mutation or diseases

Several experimental studies on the patients of cardiovascular diseases have observed elevated levels of plasma Hcy concentration [9-11]. Prudova et al. [12] has analyzed the pathological defects in the methionine metabolism in liver cells and found that a loss in the activation of CBS by AdoMet increases the Hcy concentration. Since CBS is normally activated by AdoMet, it is natural that the steady-state of the system is entirely influenced by this allosteric regulation. In understanding of why this regulation is present, on the other hand, it is very useful to elucidate the system behavior when the CBS activation is completely lost.

4. Conclusions

The results of the dynamic sensitivity analysis revealed that the metabolites in the methionine cycle behave stably as a result of perturbation to the enzymatic activity but have a possibility to oscillate in the absence of the allosteric regulation of CBS by AdoMet. The dynamic logarithmic gain analysis also suggested the major independent variables that are responsible for the frequent variation in the metabolites. A further careful investigation based on the experimental observation is therefore necessary in this respect.

References

[1] Akhter, J. and Shiraishi, F., 2011. The methionine metabolism in human liver cell: Steady-state sensitivity analysis by Biochemical Systems Theory. Int. j. Compt and App. Mathem., 6 (3), 221-233.

[2] Finkelstein,J.D.,1990.Methionine metabolism in mammals. J. Nutr.Biochem,1,228- 237.

[3] Zubay, G. L., 1998. Biochemistry. McGraw-Hill, USA.

[4] Martinov, M.V., Vitvitsky, V.M., Mosharov, E.V., Banerjee R and Ataullakhanov F.I, 2000. A substrate switch: A new mode of regulation in the methionine metabolic pathway. J. Theor. Biol., 204,521-532.

[5] Reed, M. C., Nijhout, H.F., Sparks, R., Ulrich, C.M., 2004. A mathematical model of methionine cycle. J.Theor. Biol., 226, 33-43.

[6] Savageau, M.A., 1969a, "Biochemical systems analysis I. Some mathematical Properties of the rate law for the component enzymatic reactions," J. theor. Biol., 5, 365-369.

[7] Savageau, M.A., 1969b, "Biochemical systems analysis II. The steady state solutions for an n-pool system using a power-law approximation," J. theor. Biol., 25, pp.370-379.

[8] Savageau, M.A., 1970, "Biochemical systems analysis III. Dynamic solutions using a power law approximation," J. theor. Biol., 26, pp.215-226.

[9] Refsum, H., Ueland, P.M., Nygard, O. and Vollset S.E. 1998. Homocysteine and cardiovascular diseases. Annu.Rev.Med. 49, 31-62.

[10] Bots, M.L. Launer L.J, and Lindemans J., 1999. Homocysteine and short-term risk of myocardial infarction and stroke in the elderly. Arch. Intern. Med.159, 38-44.

[11] Schnyder, G., Roffi, M. and Pin, R. 2001. Decreased rate of coronary restenosis after lowering of plasma homocysteine levels. N. Engl. J. Med. 345, 1593-600.

[12] Prudova, A., Martinov, M.V., Vitvitsky, V.M. Ataullakhanov, F.I and Banerjee R., 2005. Analysis of pathological defects in methionine metabolism using a simple mathematical model. Biochim. et Biophys. Acta, 1741, 331-338.

* (1) Jarin Akhter and (2) Fumihide Shiraishi

(1) Institute of Forestry and Environmental Sciences, University of Chittagong, Chittagong 4331, Bangladesh.

(2) Dept. of Bioscience and Biotechnology, Graduate School of Bioresource and Bioenvironmental Sciences, Kyushu University, 6-10-1, Higashi-ku, Hakozaki, Fukuoka 812-8581, Japan.

* Corresponding author e-mail: jarinakhter@yahoo.com
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Table 1. New steady-state of the model at pathological consequences
(loss of [V.sub.CBS] sensitivity on AdoMet concentration) for three
different values of Metin concentration

at [V.sub.CBS] activation

50     [X.sub.1]           49.77732
[X.sub.2]            66.6479
[X.sub.3]            8.99838
[X.sub.4]            0.50749
100    [X.sub.1]           52.33299
[X.sub.2]           99.50118
[X.sub.3]           10.34429
[X.sub.4]            0.63663
120    [X.sub.1]           52.77676
[X.sub.2]           109.3545
[X.sub.3]           10.85539
[X.sub.4]           0.686708

at loss of [V.sub.CBS]
activation

50              51.9218
125.7268
29.23908
2.53724
100            51.51252
127.1569
35.95358
3.213257
120            51.30819
127.2470
38.18116
3.437574
```