Morrison and Allegra (1989), Folate Cycle
June 2009, model of the month by Lukas Endler
Original model: BIOMD0000000018
Folate, also known as Vitamin B7, and its derivatives are important coenzymes in the metabolism of all organisms. They function as donors or acceptors of one-carbon units and play essential roles in methionine, purine and pyrimidine synthesis. Its metabolism is of great interest in humans, as vitamin B7 deficiency can lead to anaemia and, in pregnant women, may result in birth defects. Furthermore it has been implicated in some kinds of cancer and, leading to elevated levels of homocysteine, in cardiovascular disease.
The different forms of folate are interconverted in a cyclic fashion with tetrahydrofolate (THF) as a the central carbon unit acceptor leading to the derivatives containing an additional carbon in various oxidation states (see Figure 1). Due to its vital role in metabolism and its implication in diseases the folate cycle has been intensely investigated and modelled . As it plays a key role in nucleotide synthesis, it is also targeted for cancer therapies. Methotrexate (MTX), a structural analog of folic acid, was one of the first drugs used in chemotherapy against cancer. Its mode of action mainly seems to lie in an inhibition of Dihydrofolate reductase (DHFR), the enzyme regenerating THF from DHF after thymidine synthesis. MTX has been indicated to be, like the folates, polyglutamated in the cell increasing its affinity to DHFR and cytotoxic efficiency .
Figure 1. The general polyglutamated structures of folates (I) and MTX (II). After .
Figure 2. A depiction of the folate cycle. Abbreviations: FH4: tetrahydrofolate (THF); CH2FH4: methylene THF; MeFH4: 5 methyl THF; FFH4: formyl THF; FH2: dihydrofolate (DHF); FFH2: formyl DHF; GAR: glycinamide rionucleotide; AICAR: aminoimidazol carboxamide ribonucleotide; FGAR: formylglycinamidine ribonucleotide; FAICAR: formamidoimidazole carboxamide ribonuclease. Enzymes: FA: FGAR amidotransferase; GT: GAR transformylase; FDS: FFH2 synthase; FTS: FFH4 synthase; AT: AICAR transformylase; SH: serine hydroxymethyl transferase; MS: Met synthase; MTD: CH2FH4 dehydrogenase; MTR: CH2FH4 reductase; DHFR: FH2 reductase; TS: thymidylate synthase. From .
As the folate cycle consists of a number of intertwined loops with many of the enzymes allosterically regulated (see Figure 2), it is not straightforward to predict its dynamical or steady state behaviour. The activity of MTX additionally depends strongly on its polyglutamation state, adding another layer of complexity to the system. Morrison and Allegra (BIOMD0000000018, ) set out to develop the first comprehensive mathematical model of both the folate cycle and the effects of MTX and its polyglutamated forms on it. One of the biggest problems for developing a complex model like that are the values for parameters and approximate metabolite concentrations. While some models had created before, which the authors could build on, most of the parameters used and estimated for those stemmed from different sources and none were specific for human cells. For parameterising their model for the human breast cancer cell line MCF-7 Morrison and Allegra could draw on a number of reaction rate and folate concentration measurements performed in their laboratory. Furthermore, the authors had studied the kinetics of polyglutamation in that cell line and developed a mathematical model thereof .
The model is similar to the comprehensive one by Jackson and Harrap (1973)  but, while omitting the complex pyrimidine and salvage pathways, includes some important enhancements. First the complex polyglutamation kinetics of MTX lead to more intricate regulations of inhibited enzymes and a different retention of MTX in the modelled system. It also includes the 10-formyl-dihydrofolate (FFH2) cycle as an alternative reaction for purine synthesis under MTX inhibition of DHFR. As MCF-7 cells growing in a full medium were modelled, serine and glycine could be assumed to be available in constant concentrations. For the different enzymes only few mechanisms were exactly determined, leading to the use of generic random order rapid equilibrium rate laws for most of them; more elaborate rate laws were only chosen for DHFR and thymidylate synthase (TS). For the folylpolyglutamate synthases and hydrolases simple pseudo first order rate laws could be assumed, as those reactions should be far from saturation with the concentrations of MTX used in the experiments. While the efflux of all modifications of MTX had to be modelled, due to the large culture volume only inflow of the monoglutamate form was taken into account. The efflux was assumed to be first order with different rate constants depending on the glutamation state, thereby coupling MTX retention in the cell to the polyglutamation state.
More complex and difficult to derive were the inhibitory influences of the different folates and forms of MTX. For example, for TS under high availability of dUMP, dihydrofolate (FH2) and formyl-dihydrofolate (FFH2) show competitive inhibition, whereas monoglutamated MTX and higher glutamated forms show uncompetitive and noncompetitive inhibition, respectively. This is assumed to be due to the different affinities of the poly and monoglutamated forms and was modelled by a reaction mechanism with 14 assumed intermediary enzyme complexes. DHFR also needed a special treatment as the binding strength of MTX has been shown to change with its glutamation state. To model this, the various MTX-DHFR complexes were explicitly modelled. As MTX is also known to induce DHFR levels in MCF-9 cells, DHFR expression and degradation were included in the model, with an expression rate depending on presence or absence of the drug.
Most of the parameters could be taken from measurements directly performed on MCF-9 cells. The maximal velocities of many reactions, though, had to be derived from measured steady state concentrations and balancing of reaction fluxes. Additionally, some estimates from other models and comparisons to in vitro measurements were needed to fully parametrise the system. In the end the model could reproduce the steady states with maximal velocities within a factor two of measured values.
With experimentally derived parameter values, the model could quite accurately simulate the dynamics of some of the folate concentrations after addition of 1 μM MTX over a period of 21 h (see Figure 3). Contrary to previous models in which the whole folate pool ended up as dihydrofolate, methylene- and formyl-THF concentrations declined rapidly but did not tend to zero, which better fits experimental results.
Figure 3. Time courses of various folate species after the addition of 1μM MTX. The bars denote experimental results, squares, circles and diamonds simulation results of models of cell cycle independent, the cell cycle dependent and the S phase cells respectively. The cell cycle dependent results are the averages over a population of cells. From .
Another layer of complexity comes from cell cycle dependent regulation and expression of some of the key enzymes. TS and DHFR activities especially have been found to increase 20-fold during S phase, which the authors included in their model by adapting the Vmax of TS and DHFR and the expression rates of DHFR depending on whether a cell was in G1 or G2 or S phase. Cell cycle progression was included using a slightly altered cell cycle model  and using a normally distributed starting population. The cell cycle dependent model, averaged over a population of cells, showed little differences to the cycle independent one, which corroborates the assumed alterations.
The model allows for the determination of potential key effectors of folate dynamics under MTX action and different assumptions. For instance, it corroborates inhibition of TS by the di- and triglutamated forms of MTX and the assumed mechanism and inhibition pattern of TS, including the non-competitive inhibition by FFH2.
While of course a useful tool to analyse effects of drugs on or find new potential targets in the folate cycle, the model also shows the power of combining measurements and experimentally derived mechanisms to recreate a complex reaction network. Apart from the estimated and fitted maximal velocities, the enzyme parameters were either identical to their measured values or, for only some of the inhibition constants, varied approximately 2-fold. The article describes the whole and very thorough modelling process employed, from the gathering of the known parameters, to deducing the rate laws and estimating the maximal velocities from the reaction network and steady state concentrations to validating and refining the model.
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