The dynamic behavior of methionine metabolism cycle in human liver cell.
Metabolism of methionine is essential to the optimal functioning of the cardiovascular, skeletal and nervous system in mammal and constitutes a methionine cycle with major intermediates of methionine (Met), S-adenosylmethionine (AdoMet), S-adenosylhomocysteine (AdoHcy) and Homocysteine (Hcy). The metabolic pathway map of the methionine cycle is shown in Fig.1.(a). The extracellular Met is set to come into the Met pool at a constant flux, [v.sub.Metin] in order to treat the Met concentration as a variable. Met is converted into AdoMet by catalytic actions of two isoenzymes, MATI and MATIII, of methionine adenosyl transferase [2, 3]. In these actions, AdoMet inhibits MATI while it activates MATIII. AdoMet thus formed becomes AdoHcy in two different enzymatic reactions. One of the reactions for the AdoHcy formation proceeds parallel to methylation reaction with other compounds by catalytic action of several S-adenosylmethionine-dependent methyltransferases (METH). In this case, AdoMet plays a role of methyl donor . The other reaction of the AdoHcy formation is catalyzed by glycine N-methyltransferase (GNMT) . These two enzymatic reactions are inhibited by the product of the reactions, AdoHcy and the AdoHcy formed is decomposed to Hcy and adenosine by S-adenosylhomocystein hydrolase (AH). Subsequently, Hcy is converted into two metabolites in different pathways. Firstly, Hcy react with serine to form Ctn by catalytic action of cystathionine [beta]-synthase (CBS) through the transsulfuration pathway. Hcy is converted into Met in the remethylation reaction. Two enzymes are responsible for this reaction. One is methionine synthase (MS), by which Hcy react with 5-methyl tetrahydrofolate (5mTHF) to become Met. The other is the betaine homocystein methyltransferase (BHMT), by which Hcy reacts with betaine to become Met  and the reaction is inhibited by AdoMet and AdoHcy. Thus, the dynamics of the methionine cycle is rather complicated. Addressing the correct functioning of the methionine-homocysteine pathway is essential for the normal growth and development and on the other hand, excess Hcy accumulation implicates several toxic conditions in mammal. Investigation of the full cycle through a mathematical model therefore could be a useful approach. Reed et al. (2004) has presented a differential equation model for the methionine cycle which can be used for the steady state and dynamic analysis so as to understand the regulation mechanism of the cycle more explicitly. In the first paper  we applied Biochemical Systems Theory (BST) [6-8] to the model of Reed et al.  and characterized the methinine cycle model by steady state sensitivity analysis The model on the other hand, can efficiently be characterized by dynamic sensitivities and in this second paper, therefore, we transform the mathematical model into the power-law equations and compute dynamic logarithmic gains (time courses of the normalized local sensitivities) to elucidate the dynamic properties of the system disturbed infinitesimally for disease and mutation.
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.  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 .
2.2 Dynamic analysis in BST
BST approximates a given process by products of power-law functions. In addition to the S-system form , 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)
[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.
3.1 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.  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.
Figure 3 shows the time courses of the metabolite concentrations after the methionine cycle with [v.sub.Metin] or [X.sub.5] = 50 - 200 [micro]M [h.sup.-1] at the nominal steady state loses the function of the CBS activation by AdoMet at t=0. In these simulations, the steady state values when the CBS activation by AdoMet is present at [X.sub.5]=200 [micro]M [h.sup.-1] were employed as initial values. When [X.sub.5] is less than and equal to 138, the metabolite concentrations exhibit damped oscillation, whose numbers of oscillation increase with increasing [v.sub.Metin]. However, the steady state values are less influenced by [v.sub.Metin] (Table 1). These findings are entirely contrasted with the case for AdoMet to have the CBS activation function, where no [X.sub.1], [X.sub.3], and [X.sub.4] are influenced but [X.sub.2] changes remarkably. When [v.sub.Metin] is greater than 138, the methionine system behaves in a limit cycle, in which the metabolite concentrations finally take the same trajectories even when starting from different initial values and continue to exhibit oscillation behaviors, whose numbers of oscillation and amplitudes increase with increasing [v.sub.Metin]. Therefore, there is no steady state under this condition. A loss of allosteric regulation on the CBS enzyme activity by AdoMet was found to increase the homocysteine level . This experimental fact can be confirmed in Table 1, where the Hcy concentration is remarkably increased as a result of the loss of the CBS activation function. Also, a complete loss of the allosteric regulation was found to cause a little change in the Met concentration but a marked decrease in the AdoMet, AdoHcy, and Hcy concentrations. Moreover, it was found that the influence of extracellular Met concentration on the metabolite concentrations in the cycle is increased under the condition of a loss of allosteric regulation of CBS by AdoMet.
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.
 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.
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 Reed, M. C., Nijhout, H.F., Sparks, R., Ulrich, C.M., 2004. A mathematical model of methionine cycle. J.Theor. Biol., 226, 33-43.
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* (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: firstname.lastname@example.org
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 [v.sub.Metin] Variable Steady-state value 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 [v.sub.Metin] Steady-state value 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
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|Author:||Akhter, Jarin; Shiraishi, Fumihide|
|Publication:||International Journal of Computational and Applied Mathematics|
|Date:||Jan 1, 2013|
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