Mathematical analysis of the regulation of competing methyltransferases

Background Methyltransferase (MT) reactions, in which methyl groups are attached to substrates, are fundamental to many aspects of cell biology and human physiology. The universal methyl donor for these reactions is S-adenosylmethionine (SAM) and this presents the cell with an important regulatory problem. If the flux along one pathway is changed then the SAM concentration will change affecting all the other MT pathways, so it is difficult for the cell to regulate the pathways independently. Methods We created a mathematical model, based on the known biochemistry of the folate and methionine cycles, to study the regulatory mechanisms that enable the cell to overcome this difficulty. Some of the primary mechanisms are long-range allosteric interactions by which substrates in one part of the biochemical network affect the activity of enzymes at distant locations in the network (not distant in the cell). Because of these long-range allosteric interactions, the dynamic behavior of the network is very complicated, and so mathematical modeling is a useful tool for investigating the effects of the regulatory mechanisms and understanding the complicated underlying biochemistry and cell biology. Results We study the allosteric binding of 5-methyltetrahydrofolate (5mTHF) to glycine-N-methyltransferase (GNMT) and explain why data in the literature implies that when one molecule binds, GNMT retains half its activity. Using the model, we quantify the effects of different regulatory mechanisms and show how cell processes would be different if the regulatory mechanisms were eliminated. In addition, we use the model to interpret and understand data from studies in the literature. Finally, we explain why a full understanding of how competing MTs are regulated is important for designing intervention strategies to improve human health. Conclusions We give strong computational evidence that once bound GNMT retains half its activity. The long-range allosteric interactions enable the cell to regulate the MT reactions somewhat independently. The low Km values of many MTs also play a role because the reactions then run near saturation and changes in SAM have little effect. Finally, the inhibition of the MTs by the product S-adenosylhomocysteine also stabilizes reaction rates against changes in SAM. Electronic supplementary material The online version of this article (doi:10.1186/s12918-015-0215-6) contains supplementary material, which is available to authorized users.

In specifying the differential equations, we use lower case letters and simple abbreviations for the substrates; these abbreviations are indicated in Table 1, below. Velocities are always indicated by V X where the subscript X gives the name of the enzyme that catalyzes that particular velocity. Each velocity depends, of course, on the current values of various of the substrates. fgnmtf 5mTHF-GNMT-5mTHF The 13 differential equations are simply mass balance equations that say that the rate of change of the concentration of a substrate is the sum of the velocities of the reactions that make the substrate minus the sum of the reactions that use the substrate. The differential equations follow: Some of the reactions depend on the concentrations of other substrates that are not variable (in the model) and are assumed to be constant. These are give in Table S2. The details of the biochemistry and the biology are in the functional forms that show how each of the velocities depends on the current values of the variables that influence it. Many reactions have Michaelis-Menten kinetics in one of the following standard forms: for unidirectional, one substrate, unidirectional, two substrates, and bidirectional, two substrates, two products, respectively. For these reactions, Table S3 lists the K m and V max values. In general, we take K m values from the literature. V max values are extremely variable because they depend on enzyme expressions levels that vary in time and therefore experimental measurements in vivo are difficult and unreliable. We usually adjust the V max values so as to obtain the typical substrate concentration values that we find in the literature. Parameters have sometimes been chosen by comparing model outputs in various circumstances to qualitative and quantitative experimental data. Now we discuss in detail the methylation reactions, the more difficult modeling issues, and reactions with non-standard kinetics.
AS3MT. Inorganic arsenic is metabolized in two methylation steps catalyzed by AS3MT. The first step uses utilizes a methyl group from SAM and is followed by a reduction step to produce methylarsonic acid (MMA). The second step uses utilizes a methyl group from SAM and is followed by a reduction step to produce dimethyarsinic acid (DMA), which is readily exported from the liver and cleared in the urine. We have recently studied the biochemistry of these methylation steps that are quite complicated [22]. For, example the first step shows substrate inhibition by inorganic arsenic and product inhibition by MMA and glutathione (GSH) both sequesters the arsenic compounds and activates AS3MT. In our study here, we are mainly interested in studying the availability of methyl groups from SAM, so we take the arsenic concentrations and the GSH concentration to be constant, and model just the first methylation step. SAM shows substrate inhibition for AS3MT [23], but the effect is small and occurs only at very high SAM concentrations, so we ignore it. Thus, the velocity of methylation is taken to be: We take the K m of AS3MT for SAM to be 50µM as determined in [24] and the K m for iAs to be 4.6µM [25]. It is known that SAH inhibits AS3MT [26,27], but the nature of the inhibition and the K i are not known. We'll assume the inhibition is competitive and take K i = 10µM, which is typical of other methyltransferases. A high, but realistic arsenic load is 1 µM in liver [28] and a typical flux would be the order of magnitude of 1 µM/hr. So, we choose V max = 28µ/hr, which accomplishes this given that a typical SAM concentration is 24 µM.
BHMT. The kinetics of BHMT are Michaelis-Menten with the parameters K m,hcy = 12, K m,bet = 100, and V max = 502 [29] [30]. The form of the inhibition of BHMT by SAM and SAH was derived by non-linear regression on the data of [31] and scaled so that it equals 1 at the normal methionine input of 50µM/hr. V BHMT (hcy, bet, sam, sah) = V max (hcy)(bet) (K m,hcy + hcy)(K m,bet + bet) · e −.0021(sam+sah) e .0021 (28) Binding of 5mTHF to GNMT. In a series of papers, Wagner, Luka, and colleagues have studied the inhibitory effect of 5mTHF on the activity of GNMT [32,23,33,34,35]. GNMT has two binding sites for 5mTHF, so we assume the simple reversible reactions: with forward and backward rate constants, k 1 and k 2 , for the first reaction and k 3 and k 4 , for the second reaction. We choose the rate constants k 1 = 50, k 2 = 1, k 3 = 1, k 4 = 1.6 so that the K D values are those found in Table 2 of [33].
CBS The kinetics of CBS are standard Michaelis-Menten with K m,1 = 1000 for hcy taken from [36] and K m,2 = 2000 for ser taken from [37], with V max = 117,000. The form of the activation of CBS by sam and sah was derived by non-linear regression on the data in [38] and [39] and scaled so that it equals 1 when the external methionine concentration is 30 µM.
DNMT. The velocity of the DNMT reaction is given by .
The inhibition by SAH is competitive [40]. We choose K m = 1.4µM for SAM and K i = 1.4µM for SAH as indicated in [41]. The reaction depends on the cytosines available, but since we take their concentration to be constant we fold that dependence into the V max . The value V max = 12.5µM/hr was chosen so that the flux of the DNMT reaction is normally (when the cell is not dividing) much less than the fluxes of GNMT, PEMT, and GAMT.
GAMT. The velocity of the GAMT reaction is given by .
The inhibition by SAH is competitive [42,43]. We choose K m = 49µM for SAM and K i = 16µM for SAH as indicated in [44] and take K m = 13.3µM for gaa as in [45]. The value V max = 210µM/hr was chosen so that the flux of the GAMT reaction is comparable to the fluxes of the GNMT and PEMT reactions, the two other methyl transferases that carry much of the methylation flux.
GNMT. The kinetics of GNMT for SAM are cooperative and we take the Hill coefficient to be n = 2 as suggested in [23] and we use K m = 100µM as indicated in [44]. The inhibition by SAH is competitive [46] and has K i = 35µM [44]. The reaction has glycine as a substrate and we take K m = 12.2µM of GNMT for glycine as found in [?]. Thus, .
This formula for V max resulted from our in silico experiments described under Results 3.1.
The concentration of free GNMT, gnmt, is a variable in our model. GNMT can be bound by one or two molecules of 5mTHF. Our simulations and the data in [33] suggest strongly that once bound GNMT, namely gnmtf, retains 50% of it's activity. The factor 4026 is chosen so that GNMT has a normal reaction velocity comparable to the reaction velocities of PEMT and GAMT, the two other methyl transferases that carry much of the methylation flux.
MAT-I. The MAT-I kinetics are from [47], Table 1, and we take V max = 260 and K m = 41. The inhibition by SAM was derived by non-linear regression on the data from [47], Figure 5.
MAT-III. The methionine dependence of the MAT-III kinetics is from [48], Figure 5, fitted to a Hill equation with V max = 220, K m = 300. The activation by SAM is from [47], Figure  5, fitted to a Hill equation with K a = 360. We model the activation of MATIII by SAM by effectively changing the V max , but, in fact, SAM lowers the K m for methionine; for a detailed discussion, see [49].
MTHFR. The first factor in the formula for the MTHFR reaction velocity 3. * 10 10 + (sam − sah) is standard Michaelis-Menten with K m,1 = 50, K m,2 = 16, and V max = 5300 taken from [50][51] [52]. The inhibition of MTHFR by SAM, the second factor, was derived by nonlinear regression on the data of [53] [54] and has the form 10/(10+ sam). In addition, SAH competes with sam for binding to the regulatory domain of MTHFR. It neither activates nor inhibits the enzyme [54] but prevents inhibition by sam; thus, we take our inhibitory factor to be: 10 10 + (sam − sah) .
The factor 3 scales the inhibition so that it has value 1 when the external methionine input is 50 µM/hr.

NE.
The kinetics of the non-enzymatic reversible reaction between thf and ch2 are taken to be mass action, with rate constants are k 1 = 0.03, and k 2 = 22. hcho represents formaldehyde.
PEMT. The velocity of the PEMT reaction is given by The inhibition by SAH is non-competitive [55]. We choose K m = 18.2µM for SAM and K i = 3.8µM for SAH as indicated in [44]. The reaction depends on pe (phophatidylethanolamine) and we take K m = 5000µM of PEMT for pe as found in [55].The value V max = 49100µM/hr was chosen so that the flux of the PEMT reaction is comparable to the fluxes of the GNMT and GAMT reactions, the two other methyl transferases that carry much of the methylation flux.