Mathematical model of fructan biosynthesis and polymer length distribution in plants

Background and Aims There are many unresolved issues concerning the biochemistry of fructan biosynthesis. The aim of this paper is to address some of these by means of modelling mathematically the biochemical processes. Methods A model has been constructed for the step-by-step synthesis of fructan polymers. This is run until a steady state is achieved for which a polymer distribution is predicted. It is shown how qualitatively different distributions can be obtained. Key Results It is demonstrated how a set of experimental results on polymer distribution can by simulated by a simple parameter adjustments. Conclusions Mathematical modelling of fructan biosynthesis can provide a useful tool for helping elucidate the details of the biosynthetic processes.


INTRODUCTION
Lolium perenne ( perennial ryegrass) plants containing unusually high concentrations of the fructose polymer fructan (high sugar grasses, HSGs) were obtained in the 1970s by Pollock and Jones (1979). In recent years there has been growing interest in the possible deployment of HSGs in grazed pastures (Turner et al., 2006;Edwards et al., 2007;Parsons et al., 2012) which may increase animal meat and milk production (Lee et al., 2001;Miller et al., 2001), decrease nitrogen (N) deposition to the soil and the generation by the soil of the greenhouse gas (GHG) nitrous oxide (Ellis et al., 2011), and have effects on methane (CH 4 ) emanations from grazing animals (Ellis et al., 2012). This could lead to substantial impacts on the environmental footprint of pastoral agricultural systems. For these and other reasons, it is important to understand fructan biosynthesis and degradation, including the dynamics and principal controls of the processes.
Recently, we have developed improved LC-MS (liquid chromatography-mass spectrometry) methods to analyse fructans in ryegrass (Harrison et al., 2009(Harrison et al., , 2011(Harrison et al., , 2012, and a series of observations prompted us to reinvestigate the 1-SST/ 1-FFT hypothesis [1-SST is the sucrose: sucrose 1fructosyltransferase enzyme, reaction (3) below; 1-FFT is the fructan: fructan 1-fructosyltransferase enzyme, reactions (4) and (5)]. Originally, the 1-SST/1-FFT model was proposed for the inulin-producing species Helianthus tuberosus (Edelman and Jefford, 1968), which has subsequently served as the conceptual basis for fructan biosynthesis in other higher plants. The 1-SST/1-FFT paradigm has been criticised, mainly because of discrepancies found between in vitro produced fructan oligomer profiles and those present in planta (Cairns, 1993;Cairns et al., 2008, and references therein). Fructans in grasses are much more complex than the linear inulin fructans found in, for example, H. tuberosus due to the activity of additional fructosyltransferases (FTs) such as 6G-FFT (Lasseur et al., 2006;Hisano et al., 2008) and 6-SFT (Lasseur et al., 2011). These enzymes produce the fructan trimers 6G-kestose and 6-kestose which, when polymerized by FTs, result in the synthesis of levans and the neoseries of inulins and levans. However, from a kinetic point of view all FTs catalyse the same reaction, i.e. the transfer of a fructose from a fructan oligomer to either sucrose or another fructan oligomer and can therefore be summarized into a single reaction equation.
We have constructed a mathematical model of possible pathways of fructan biosynthesis to test a range of assumptions and to compare predictions of the model with observations on fructan concentration and polymer length. Generally, the models are based on: (i) a constant supply of glucose (glc), (ii) the energetically neutral and reversible production of fructose (fru) from glucose [reaction (1)], (iii) the production of sucrose [reaction (2)], (iv) the production of 1-kestose (gf2) by the 1-SST reaction [reaction (3)], (v) the production of fructans of higher degrees of polymerization by successive transfer of fructose from kestose to the growing fructan polymer by FTs [reactions (4) and (5)], (vi) other possible transfers of fructose [reactions (6) and (7)] and (vii) the synthesis and transfer of small fructose polymers [reactions (8)].
Mathematical models of reaction kinetics are used to make predictions of fructan production, including polymer length distributions, which are compared with our own measurements. This may lead to a better understanding of this poorly understood but important area of research.

Plant materials and fructan extraction
Lolium perenne seeds from two lines ('Fennema', 'PG113') were germinated and seedlings transferred to pots containing nutrient-rich potting mix. Plants were grown in controlled environment chambers at two different temperature regimes with a 14-h light and a 10-h dark period. Temperatures were set to either 10 8C constant or to 20 8C during the light and 10 8C during the dark period. Plants were regularly cut back (every 3 weeks) and maintained as described previously (Rasmussen et al., 2009).
Blades (vegetative material above the ligule) were removed after a 3-week regrowth period, immediately frozen and ground in liquid nitrogen and freeze-dried. Ground plant powder (25 mg) was extracted twice in 1 mL 80 % ethanol and subsequently twice in 1 mL of water at 65 8C with constant shaking as described (Rasmussen et al., 2007). Extracts were centrifuged and supernatants combined for MS analysis of fructans. Combined supernatants were brought to dryness under vacuum, reconstituted into 1 mL of water, filtered through a 10-mm filter and transferred into high-performance liquid chromatography (HPLC) glass vials.

Fructan analysis
Filtrates (5 mL) were injected into the ultra-HPLC system and fructans separated on a Thermo Hypercarb column as described (Harrison et al., 2009). MS analysis was performed using an LTQ ion trap mass spectrometer (Thermo Fisher Scientific, Waltham, MA, USA) with electrospray ionization in negative mode, and data collection over the mass range of 300-4000 allowing the collection of MS data for fructans up to DP (degree of polymerization) 49 (Harrison et al., 2009). Here, we report data corresponding to fructan oligomers in the DP range of DP 3 -10 only. As reported previously (Harrison et al., 2011) ions of fructan oligomers with a DP of 3 -10 were predominantly singly charged, resulting in mass to charge ratios (m/z) for the deprotonated molecular ions [M-H] (Thornley and France, 2007, pp. 21, 24).
Throughout, it is assumed that glucose is provided at a constant concentration. Glucose (denoted by glc and also by g) is converted into fructose (denoted by fru and also by f ) reversibly according to eqns (A2) and (A14): (1) Next glucose and fructose are converted irreversibly into sucrose (glucose -fructose, also denoted by gf1) by means of eqn (A3): glucose + fructose sucrose gf1 (2) Energy (ATP) is generally required for this reaction (e.g. Thornley and Johnson, 2000, p. 301), although the possible modification of the kinetics by this requirement is not considered here. Any invertase or sucrase action, giving sucrose hydrolysis, is ignored.

Scheme (a): the basic scheme
Fructan biosynthesis in plants requires multiple, substratespecific FTs. The currently most widely accepted hypothesis for the biosynthesis of fructans in plants is the 1-SST/1-FFT model proposed by Edelman and Jefford (1968). This hypothesis was originally designed to represent inulin biosynthesis in Jerusalem artichoke (Asteraceae), but is now generally applied to other plant systems as well.
The first step in this reaction sequence is catalysed by the 1-SST enzyme, which facilitates the transfer of a fructose unit from a donor sucrose molecule to the 1 position of the fructose on a sucrose acceptor molecule, forming a b2-1 glycosidic bond and resulting in a 1-kestose molecule (gf2) and glucose (Koops and Jonker, 1996;Van den Ende et al., 1996;Lüscher et al., 2000a, b;Chalmers et al., 2003). This reaction is sucrose gf1 + sucrose gf1 kestose gf2 + glucose glc Here, sucrose is denoted by gf1 (glucose-fructose), kestose by gf2 (glucose-fructose-fructose) and the glucose monomer by glc. See eqn (A18) for the rate equation. The product of this reaction, kestose (gf2), can be used as donor and acceptor for a second fructosyl transfer reaction which is carried out by 1-FFT. This enzyme transfers a fructose unit from the donor 1-kestose molecule to the 1-position of the fructose on an acceptor molecule, forming a b1 -2 glycosidic bond, thus extending the fructose chain on the acceptor molecule by one fructose unit; glucose is always terminal. 1-FFT can only use fructans as donor molecules, but not sucrose, while sucrose, fructans and fructose itself [reactions (8)] can act as acceptor molecules (Jeong and Housley, 1992;Koops and Jonker, 1996;Lüscher et al., 1993a, b;Van den Ende et al., 1996). The reaction for kestose as the acceptor and donor of fructose is [eqn (A25)]: kestose gf2 + kestose gf2 sucrose gf1 + gf3 (4) where gf3 denotes the polymer glucose-fructose-fructosefructose. This reaction can be generalized [eqn(A26)] to kestose gf2 + gfn sucrose gf1 + gf n + 1 ( ).
Here n is an integer (n ¼ 2, 3, . . . , 9). gfn is a fructan with n fructose units attached linearly to glucose. Here we consider polymers up to gf10. Forage grasses such as Lolium perenne ( perennial ryegrass) accumulate a mixture of fructan types, namely the inulin and levan series, and the inulin and levan neoseries (e.g. Heldt, 1997, p. 241). Here we just deal with the inulin series, as in (5). These series differ in the glycosidic bonds employed (b2-1 in inulins; b2 -6 in levans), and also in the position of glucose in the fructan chain (terminal in the inulin and levan series; internal in the inulin and levan neoseries). The biosynthesis of these fructans requires additional enzyme activities such as 6G-FFT (synthesizes fructans with internal glucose), and 6-SFT (synthesizes levans). The general assumption is that these FTs, like 1-FFT, also transfer single fructose units and can only use fructans (gfn; n ≥ 2) as donors, but not sucrose.

Scheme (c): effect of a fast reaction
This scheme simulates the production of fructans as described under scheme (b) (Fig. 3), but where one of the reactions of reaction (6), namely that with n ¼ 4: has its maximum velocity (v 34_25 ) set to a high value, that is v 34_25 ¼ 10 [eqns (A35) with n ¼ 4]. The result of this tenfold increase is a strong depression of the concentration of acceptor gf4  (A26)]. In Scheme (a) transfer of a single fructose only occurs from kestose (gf2). glc denotes glucose; fru, fructose; gfn, a fructan with n fructose units attached to glucose; gf1 is sucrose; gf2 is kestose. Steady-state concentrations are given. glc denotes glucose; fru, fructose; gf1, sucrose; gf2, kestose; gfn (n ¼ 3, 4, . . . , ), a fructan with n fructose units attached to glucose. Default parameters (Table 1) ( Fig. 4). That is, a gap is generated in the distribution of fructans according to degree of polymerization. With constant glucose ([glc]) concentration of 0 . 1 mol L 21 , the steady-state concentrations (mol L 21 ) of the other variables are: fructose 9 (Fig. 4).

Scheme (d): fructose transfers to fructose polymers
The occurrence of oligomeric carbohydrates containing exclusively fructose units (without any glucose) has been described in early studies on the carbohydrate composition of monocotyledons (see Archbold, 1940, and references therein). We therefore added an additional scheme (d), with putative transferase enzymes catalysing the following reactions: In the first two reactions, kestose (gf2) donates a single fructose, which can be accepted by fructose (f ) or by f2 [eqns (A27)], yielding sucrose as a product. In the last reaction, an enzyme catalyses transfer of gf2 (kestose) to f3 [eqns (A28)], giving a gf5 molecule with no other product. The basic reaction set of scheme (a) [reactions (1)-(5), eqns (A2), (A14), (A3), (A18), (A25) and (A26)] is applied as usual but one of the reactions, namely reaction (4) catalysed by 1-FFT, is made zero by setting the velocity for this reaction to zero: There is no production of gf3 or therefore gf4 [reaction (5) with n ¼ 3].
At the steady state, concentrations of kestose (gf2) were very low and gf3 and gf4 were not produced at all, making a hole in the polymer distribution. Fructans with gfn (n ¼ 5 -9) and f2 and f3 reached high concentrations, comparable to those of sucrose (gf1). With constant glucose concentration, [glc] of 0 . 1 mol L 21 , steady-state concentrations of the other variables are (mol L 21 ): fructose Scheme (e): comparison with data As described above we grew ten genotypes from two L. perenne lines, the European cultivar 'Fennema' (F) and an experimental breeding line 'PG113' (P) in two temperature regimes (one at constant 10 8C -10/10, the other at 20 8C during the light and 10 8C during the dark period -20/10). In blades (tissue above the ligule) harvested after 21 d of re-growth after defoliation 'PG113' accumulated significantly higher concentrations of total water-soluble carbohydrates (mg g 21 d. wt) compared with 'Fennema' in both the 10/10 (means P ¼ 322 . 2, F ¼ 236 . 3; P , 0 . 0001) and the 20/10 (means P ¼ 341 . 1, F ¼ 258 . 6; P , 0 . 0001) treatments (Rasmussen et al., 2009). We chose 21 d post-defoliation blade material as it had been shown previously that at this time the expression and activity of exohydrolases is negligible and that fructan polymerization is prevalent in these tissues (Morvan et al., 1997;Lasseur et al., 2007;Lee et al., 2010;Tamura et al., 2011).
To determine the distribution of fructan oligomers with different DP, extracts of the above material were analysed by ion trap MS (Harrison et al., 2009). For most of the individual fructan oligomers, we detected several peaks corresponding to the m/z of the individual fructan oligomers. Specifically, we detected 5 peaks with the m/z of 503 . 3, two of which represent the nonfructan sucrosyl-galactosides raffinose and loliose (Amiard et al., 2003), the other three the fructan oligomers 1-kestose, 6-kestose and 6G-kestose (Harrison et al., 2012). The latter three are represented in Fig. 6 in reaction (6) (A35)]. However, now a high maximal velocity is assigned to reaction (7) (Fig. 6).
As can be seen in Fig. 6, the pattern of fructan oligomer distribution in blades followed a comparable trend for both L. perenne lines and at both temperature regimes (Fig. 6). The highest relative intensities of DP3 -10 fructans in blades were detected for tri-and tetrameric oligomers (DP3 and DP4), followed by approx. 4-fold lower intensities of m/z signals corresponding to DP5 -8 fructan oligomers. The lowest intensities were detected for DP9 oligomers with slightly higher concentrations of DP10 fructans.
Parameters of our model have been adjusted to see if it is possible to obtain agreement with our recent measurements. In fact, it is very easy to obtain such agreement, as can be seen when Fig. 7 (model predictions) is compared with observations in Fig. 6. In Fig. 7, minimal parameter changes have been made to the simplest model scheme with default parameters [( Table 1; scheme (a); Fig. 2; reactions (1) -(5), eqns (A2), (A14), (A3), (A18), (A25) and (A26)]. The velocity parameters alone were adjusted to give the results shown in Fig. 6, focusing on the relative amounts of fructan trimers (gf2, glucose-fructosefructose) through to gf9, a glucose with nine fructoses attached to glucose, and ignoring the concentrations of glucose, fructose and sucrose. We do not believe our particular parameterization (Fig. 7) is unique and there are likely to be other sets of parameter values that would do equally well.

DISCUSSION
This paper presents a 'proof-of-concept' position, demonstrating that constructing and simulating specific biosynthetic schemes for fructan biosynthesis may significantly assist in understanding these processes. The mathematical analysis and computations are straightforward and, arguably, such approaches could be part of the standard armoury of techniques which is brought to bear on these problems.
To date no such high DP FT has been isolated from L. perenne and our model and data might indicate that the identification of such an enzyme would help to explain the observed differences of in vitro synthesized fructan profiles obtained by assaying known native L. perenne FTs compared with plant fructan profiles (Cairns et al., 2008). In view of the possible importance of the high-sugar phenotype in grassland productivity, in mitigating GHG emissions and in possibly promoting carbon sequestration, we believe that a detailed mechanistic understanding given by analyses along the lines presented here may enable us to better understand the contributions of the high-sugar traits to grassland ecosystems. Aworthwhile aim is to represent such characteristics realistically in grassland ecosystem models (e.g. Thornley, 1998).
In conclusion, it has been demonstrated that mathematical analysis and simulation, based on well-established biochemical kinetics and calculus, is able to play a role in furthering our knowledge of fructan biochemistry, with its potential importance to the environmental and economic consequences of grassland agriculture.

APPENDIX. MATHEMATICAL DETAILS AND PARAMETERIZATION OF THE MODEL
The equations given below were programmed in ACSL (Advanced Continuous Simulation Language, Aegis Research, Huntsville, AL, USA; version 11 . 8 . 4), an ordinary differential equation solver, using fourth-order Runge -Kutta integration and a time step of 0 . 03125 (¼1/32) d. The time unit of the model is days (d). There were no problems in model implementation. We step through the biochemical state variables pool by pool, giving the outputs from, the inputs to, and the differential equation of each pool. A constant volume of reactant, 1 litre (L), is assumed. A molar concentration is 1 gram molecule (mol) per litre (mol L 21 ), and a flux of 1 mol glucose L 21 d 21 is 1 gram molecule of glucose per litre per day.
Fluxes are calculated as outputs O from a substrate pool, using a mechanistic equation, usually the Michaelis -Menten equation or something similar for bisubstrate reactions (Thornley and France, 2007, pp. 107 -109, 113-114). If there is no substrate in a pool, there can be no output. Apart from the system input [eqn (A8)], inputs I are the consequence of outputs O from other pools, sometimes applying a simple stoichiometry, e.g.
For a single substrate (x) single product ( y) reaction, fluxes are calculated and designated as outputs (O x_y ) from substrate pool x towards a product pool y with units of mol x per litre per day. Inputs to pool y from the x y reaction are designated I x_y (mol y L 21 d 21 ). Where two substrates (x and y) are required giving two products (z and u), the output flux is designated O xy_zu and is calculated as an output from one of the substrate pools (say x) which requires a stoichiometrically related flux from the second substrate pool ( y). The input flux I xy_zu is in units of mol u L 21 d 21 or mol z L 21 d 21 into the u and z pools. Parameter values are given where they are introduced and are listed in Table 1 for reference.

Synthesis of fructans is driven by maintaining a constant concentration of glucose [glc], namely
[glc] = 0·1 mol L −1 = 0·1g mol litre −1 . (A1) Outputs. The first output flux from the glucose pool is to fructose [the forward reaction (1) The second output flux is due to the two-substrate reaction (2) for sucrose synthesis (denoted by gf1, for glucose-fructose): Again, the parameters are: a maximum velocity, v . . . and a MM constant, K . . . , this time for a two-substrate reaction. This output flux is removed from the fructose pool [eqns (A15)] as well as equally from the glucose pool and it is put into the sucrose (gf1) pool [eqn (A19)].
Total output from the glucose pool is (mol glucose L 21 d 21 ) Inputs. There is an input from the fructose glucose reaction (1): This is calculated as an output from the fructose pool in eqn (A14). There is a second input to the glucose pool from the 1-SST reaction for kestose synthesis (3). This is calculated as (mol glucose L 21 d 21 ) O gf1gf1_gf2glc is an output from the sucrose pool (gf1) [eqn (A18)] below. Adding these two input fluxes gives a total input flux of The glucose concentration is (by assumption) held constant. This is achieved by adding an external ( In view of eqn (A9), d[glc]/dt is zero, and the glucose concentration remains at its initial value of 0 . 1 mol L 21 . Note that the total input to the system is (mol glucose L 21 d 21 ) There are no outputs from the system so the rate of change of the whole system variable (S ys mol glucose L 21 d 21 ) is dS ys dt = I ext glc (A12) A useful check on the formulation and programming of the problem is also to calculate the rate of change of the whole system by summing its parts: Equations (A12) and (A13) should be identically equal at all times.

Fructose (fru) pool
Outputs. The first output is to glucose via reaction (1) in the reverse direction: This is the mirror image of eqn (A2). The second output is to sucrose synthesis, O glcfru_gf1 , in eqn (A3) above. The third output, O gf2f1_gf1f2 , is for the synthesis of the fructose-fructose dimer (f 2 ) [first of reactions (8)], calculated as an output from the kestose (gf2) pool below [first of eqns (A27)]. Total output is Inputs. There is a single input to the pool from glucose, with where the output from the glucose pool is calculated in eqn (A2).
Differential equation. This is dt dt, and Outputs. It is assumed that there is a single output from the sucrose pool giving kestose (gf2) synthesis as in reaction (3) (invertase or sucrase activity, promoting sucrose hydrolysis, is ignored). This flux is Inputs. The principal input to the pool is due to sucrose synthesis from glucose and fructose, reaction (2), with [eqn (A3)] A second input is from the kestose pool, reaction (4), given by This output from the kestose pool is calculated in eqn (A25). There are also many similar inputs, arising from the outputs of reactions (5) with n ¼ 3, 4, . . . , 9: These outputs from the kestose pool are calculated in eqn (A26). There are two more inputs, arising from the synthesis of the fructose dimer f2, and the fructose trimer f3, shown in reactions (8): The outputs on the right of these equations are again calculated under the kestose pool, eqns (A27).
Total input to the sucrose pool is I gf1 = I glcfru gf1 + I gf2gf2 gf1gf3 + I gf2gf3 gf1gf4 + ... + I gf2gf8 gf1gf9 + I gf2gf9 gf1gf10 dt dt, and Outputs. There is an output [reaction (4)] giving gf3 synthesis: There are a series of similar outputs from reaction (5) (n ¼ 3, 4, . . . , 9): There can be outputs from kestose of a fructose monomer to a fructose monomer or to a fructose dimer, accomplishing the synthesis of fructose dimers and trimers [reactions (8)] : Note that these reactions are turned off in default giving just the basic reaction scheme [scheme (a), reactions (1)- (5)]. There is one further potential output to be considered, that of the third of reactions (8). This is Total outputs from the gf2 (kestose) pool are Inputs. The principal input to the kestose pool is via the 1-SST reaction from sucrose (3) with [see eqn (A18)] There is also a series of inputs from reaction (6) where gf3 can act as a fructose donor [this is outside the basic scheme of reactions (1) -(5)]: = O gf3gfn gf2gf (n+1) , n = 4, 5, 6, ..., 9 The right side outputs are calculated under the gf3 pool below [eqn (A35)]. Total input is I gf2 = I gf1gf1 glcgf2 + I gf3gf3 gf2gf4 + I gf3gf5 gf2gf6 + ...
dt dt, and Outputs. There is an output of reaction type (5) with n ¼ 3 giving gf4 synthesis: This is given by eqn (A26) with n ¼ 3. There is a series of outputs from reaction (6) (n ¼ 3, 4, . . . , 9): Total output from the gf3 pool is Inputs. The only input is via the 1-FFT reaction from kestose (4) with [see eqn (A25)] dt dt, and Outputs. There is an output of reaction type (5) and eqn (A26) with n ¼ 4 giving gf5 synthesis: There is a second output from reaction (6) and eqn (A35) with n ¼ 4: Total output from the gf4 pool is Inputs. There are two inputs [with eqns (A26) with n ¼ 3 and eqn (A35) with n ¼ 3]: Outputs. There is an output of reaction type (5) and eqn (A26) with n ¼ 5 to gf6 synthesis: There is a second output from reaction (6) and eqn (A35) with n ¼ 5: Total output from the gf5 pool is Inputs. The first two inputs are [reaction (5), eqns (A26) with n ¼ 4; and reaction (6), eqns (A35) with n ¼ 4]: There is a third input from the last of reactions (8)  Outputs. There is an output of reaction type (5) [eqn (A26) with n ¼ 6] to gf7 synthesis: There is a second output from reaction (6) and eqn (A35) with n ¼ 6: O gf3gf6 gf2gf7 (A53) Total output from the gf6 pool is Inputs. There are two inputs. The first is [reaction (5); eqns (A26) with n ¼ 5]: The second is [reaction (6) and eqn (A35) with n ¼ 5]: The equations are the similar to those for the gf6 pool [eqns (A52) to (A58) ]: for the gf7 pool, change 7 to 8, then 6 to 7 and 5 to 6; for the gf8 pool, change 7 to 9, 6 to 8 and 5 to 7.

gf9 pool
Outputs. There are two outputs: The first is an output of reaction type (5) with n ¼ 9 synthesizing gf10 [eqn (A26) ]; the second is from reaction (6) and eqn (A35) with n ¼ 9. Total output from the gf9 pool is Inputs. The two inputs are [reaction (5), eqns (A26) with n ¼ 8; and reaction (6), eqns (A35) with n ¼ 8]: The gf9 pool is (in the present formulation) the last pool to reach a steady state (Fig. 1). Therefore, the proportional or relative growth rate of gf9, R gf9 (d 21 ), is calculated to assess the accuracy of the final state. This is This is the last pool included in our analysis. It has inputs of [eqns (A26) and (A35) both with n ¼ 9] giving a total input of I gf10 = I gf2gf9 gf1gf10 + I gf3gf9 gf2gf10 (A66) There are no outputs so that gf10 increases linearly in the steady state, when all the pools except gf10 are constant [see eqn (A8) for the total system input, I ext_glc ]. Thus, in the steady state, the equation