Theoretical Biology and Medical Modelling Open Access in Silico Experimentation with a Model of Hepatic Mitochondrial Folate Metabolism

Background: In eukaryotes, folate metabolism is compartmentalized and occurs in both the cytosol and the mitochondria. The function of this compartmentalization and the great changes that occur in the mitochondrial compartment during embryonic development and in rapidly growing cancer cells are gradually becoming understood, though many aspects remain puzzling and controversial.

The model consists of 23 differential equations that express the rate of change of each of the substrates in the rectangular boxes in Figure 1. Each of the differential equations is a mass balance equation that says that the time rate of change of the particular metabolite is the sum of the rates at which it is being made minus the rates it is being consumed in biochemical reactions plus or minus net transport rates from other compartments. In order to display the differential equations in a coherent and understandable way, we have chosen notation for the variables and reaction rates that is both more uniform and more spare than some notation commonly in use, for example the concentration of 5-methyltetrahydrofolate is denoted 5mf instead of the usual [5mTHF]. All of this notation is described in Part A, below. In Part B we give the differential equations, which are written in terms of reaction and transport rates. In Part C the kinetic formulas and constants for these reaction and transport rates are given with some justifications. Part D describes how the model works, what is given and what is computed, in particular experiments. Part A: Notation.
The complete names of the enzymes indicated by acronyms in Figure 1 are as follows.  -I  methionine adenosyl transferase I  MAT-III  methionine adenosyl transferase III  GNMT  glycine N-methyltransferase  DNMT  DNA-methyltransferase  SAAH  S-adenosylhomocysteine hydrolase  MS  methionine synthase  BHMT  betaine-homocysteine methyltransferase  CBS cystathionine β-synthase We will use lower case three letter abbreviations for the concentrations of metabolites (µM). A prefix of m, c, or b, for mitochondria, cytosol, or blood, indicates the compartment. Metabolites occuring in only one compartment (like met), or metabolites whose concentrations are assumed equal in different compartments (like dmg and src) have no prefixes.  It is assumed that total cellular folate is equally divided between the mitochondria and the cytosol [1] and that the mitochondria occupy one quarter of the volume of the cell. Thus the total normal folate concentration in the mitochondria is 40 µM, and in the cytosol is 13.3 µM.
Part B: The Equations.
For each of the biochemical reactions indicated by a reaction arrow in Figure 1, we denote the velocity of the reaction (in µM/hr) by a capital V whose subscript is the acronym for the enzyme that catalyzes the reaction. Thus, for example, the velocity of the methionine synthase reaction is denoted by V M S . Each of these velocities depends on the current values one or more of the variables (metabolite concentrations) and possibly also on one or more of the constants. Velocities of reactions that occur in both the mitochondria and cytosol wiil be distinguished by c and m, for example, V cSHM T and V mSHM T .
The velocities of transport from blood to cytosol or mitochondria to cytosol are given by the transparent notation V bSERc and V mSERc , respectively. The units are in µM/hr increase in the cytosol. Since the cytosol is assumed to have three times the volume of the mitochondria, one µM increase in the cytosol due to transport from the mitochondria means a 3 µM decrease in the mitochondria. This is the reason for the 3's in the equations involving transport into and out of the mitochondria and in the transport kinetics below.
Part C: Kinetics.
For many of the velocities, we assume that their dependence on substrates has Michaelis-Menten form with one substrate or random order Michaelis-Menten form with two substrates: . Some reactions, for example V cSHMT , are assumed to have reversible random order Michaelis-Menten kinetics with two substrates in each direction. For all these velocities, the form of the kinetics is clear. The K m and V max values appear in Table S4 (modified from [2]), along with references. Enzymes that occur in both the mitochondria and the cytosol are assumed to have the same kinetics in both compartments. If the kinetic constants differ in the cytosol and the mitochondria that is indicated by the prefixes m and c, respectively. Reactions that do not have one of these simple forms and transport velocities are discussed individually after Table S4. We now discuss the reactions where the kinetics have a special form.
BHMT. The kinetics of BHMT are Michaelis-Menten with the parameters K m,1 = 12, K m,2 = 100, and V max = 2160 [31], [32]. The form of the inhibition of BHMT by SAM was derived by non-linear regression on the data of [33] and scaled so that it equals 1 when the external methionine concentration is 30 µM.
The inhibition of BHMT is controversial because Bose et al. [34] found that sam has no effect on recombinant BHMT's ability to remethylate homocysteine. It is possible that sam affects the expression of BHMT rather than BHMT itself in which case the results of [33] and [34] would not be contradictory. In any case, the inhibition that we use has significant effects on the BHMT reaction only for sam concentrations well above normal. Thus, only the methionine and protein loading experiments would be affected by removing the inhibition of BHMT by hcy, in which case hcy does not rise as much during loading (simulations not shown). Similarly, we found in [23] that the presence or absence of this inhibition had little effect on the stabilization of DNA methylation.
CBS. The kinetics of CBS are standard Michaelis-Menten with K m,1 = 1000 for hcy taken from [35] and K m,2 = 2000 for cser taken from [36], with V max = 402, 000. The form of the activation of CBS by sam and sah was derived by non-linear regression on the data in [37] and [38] and scaled so that it equals 1 when the external methionine concentration is 30 µM.
DNMT. The DNA methylation reaction is given as a uni-reactant scheme with sam as substrate. That is, the substrates for methylation are assumed constant. Their variation can be modeled by varying the V max . The kinetic constants, V max = 180, K m = 1.4, and K i = 1.4 are from [39].
GNMT. The first factor of the GNMT reaction is standard Michaelis-Menten with V max = 245, and K m,1 = 63 for sam and K m,2 = 130 for cgly estimated from [41]. The second term is product inhibition by sah from [40] with K i = 18. The third term, the long-range inhibition of GNMT by 5mf, was derived by non-linear regression on the data of [42], Figure 3, and scaled so that it equals 1 when the external methionine concentration is 30 µM.
MAT-I. The MAT-I kinetics are from [43], 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 [43], Figure 5.
MAT-III. The methionine dependence of the MAT-III kinetics is from [44], Figure 5, fitted to a Hill equation with V max = 220, K m = 300. The activation by sam is from [43], Figure  5, fitted to a Hill equation with K a = 360.
The inhibition of MTHFR by sam, the second factor, was derived by non-linear regression on the data of [47] [48] 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 [48] but prevents inhibition by sam; thus, we take our inhibitory factor to be: The factor 60 scales the inhibition so that it has value 1 when the external methionine concentration is 30 µM.
NE. The kinetics of the non-enzymatic reversible reaction between thf and 2cf are taken to be mass action, The rate constants are k 1 = 0.03, and k 2 = 22 in the cytosol and k 2 = 20 in the mitochondria.
We now discuss the kinetics of transport between the compartments. The general formula for the kinetics of transport between the blood and the cytosol is taken to be where AA stands for an amino acid and the prefixes b and c refer to the blood and cytosolic compartments, respectively. Thus the kinetics are Michaelis-Menten coming into the cell and linear going out. We take the kinetics of transport of serine and glycine between the cytosol and mitochondria to be Michaelis-Menten in both directions and the transport of formate between these two compartments to be linear in both directions. Transport of amino acids into cells is accomplished by a relatively small number of transport systems each of which handles several amino acids. Each transporter is specialized to handle amino acids with particular ionic characteristics [49] [50]. The transporters are saturable and the K m values have been measured in many systems [51][52] [53]. Relatively little is known about the kinetics by which amino acids leak out of cells so we take this process to be linear. This linear rate also includes the loss of cytosolic amino acids to other metabolic processes not in the model (see Figure 1), for example use in protein synthesis.
Part C: In silico experimentation.
If one starts with any initial values for the variables and solves the differential equations when the velocities have the formulas given in Part C, one finds that the variables all approach steady concentrations after a few hours. The concentrations of the variables and the velocities of reactions at this "normal" steady state are given in Table 1 in the main body of the paper. Most of the in silico experiments reported in the paper were done by starting the system at this steady state, changing one or more parameters, and letting the system relax to a new steady state. For example, in Section A of Results, the external glycine concentration, bgly, was varied systematically from 50 to 900 µM, and for each such concentration the resulting values of various velocities and metabolite concentrations were computed at steady state. In Section B of Results, the total cellular folate, FOL, was changed from 20 µM to 10 µM. By solving the differential equations we could report the effect of this change on velocities and metabolite concentrations. Similarly, the effect of changing the expression of cSHMT was computed by changing the V max of the SHMT reaction and computing the new steady state. In other experiments, reactions, or whole groups of reactions, were removed entirely. Finally, in the methionine loading and protein loading experiemnts, the blood concentrations of glycine, serine, and methionine were allowed to vary in time. By solving the differential equations we saw how the various velocities and metabolite concentrations varied in time.