Single-lamella (SL) Chains

The SL CLD is governed by the actions of enzyme sets (i) and (ii). Two alternative models for this are presented here, with small but significant differences between them. One, termed the “substrate-competing model”, is consistent with the assumption that enzyme sets (i) and (ii) act on the same pool of substrate(s), perhaps as a result of enzyme complexes (see Model section for the evolution equation which describes the SL chains, Eqn 1). The alternative is termed the “independent substrates model”, when enzyme sets (i) and (ii) predominantly act on distinct populations of substrate(s), perhaps as a result of substrate specificities. The substrate-competing model is new; the independent substrate model is an extension of our earlier treatment [24]. Evidence will be provided to demonstrate that which of the two alternative models is applicable in a given case depends on the particular plant species; it is also possible that differing models may apply in different organs, under different environmental conditions, or during different stages of organ development.

The SL CLD can only reach a steady state with restricted values of certain SL kinetic parameters, as detailed in the Model section. These SL kinetic parameters for the SL enzyme sets (i) and (ii) in the substrate-competing model are denoted β_(i), X_0(i), X_min(i), β_(ii), X_0(ii) and X_min(ii). The quantities with the symbol β are ratios of enzyme activities, defined in Eqns 2–3. The actual values of the fitted parameters for the amylopectin CLD (Figure 2) of a rice variety (cv Nipponbare) are discussed in a later section. For the independent-substrate model, there is one additional parameter, which is the relative contributions of sets (i) and (ii), denoted h_(ii/i).

The substrate-competing model quantitatively reproduces Features A, B and C in the SL range of the rice amylopectin CLD (Figure 3A): (1) the shape and location of Features A and B. Feature A appears as part of the curve for the global maximum (Feature B). Feature B, the global maximum occurs between X_0(i) and X_0(i)+X_min(i) in the present case; (2) the shape and location of Feature C, which corresponds to X_0(ii) and X_min(ii); and (3) the steepness of the nearly linear slope of log N_de(X). Although enzymes sets (i) and (ii) both contribute significantly over the whole SL range, set (i) is largely responsible for the shape for the global maximum and the nearly linear slope, and enzyme set (ii) is largely responsible for the bump (Feature C) on the nearly linear slope.

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Figure 3. Components of the CLD, given in Figure 2, fitted with the model.

(A) Fitting (red filled circles) to single-lamella (SL) chains with the substrate-competing model. X = 32 marks the apparent length of chain needed for the chains to enter the contiguous amorphous lamella (Figure 5). Where the slope changes significantly in Feature F gives the range of presumed three-lamellae-spanning chains. (B) Fitting (blue filled circles) to the type-2 TL chains with the substarte-competing model, which is the difference between the experimental and the calculated SL CLD in (A) and displaced (details in the Model section). The actual X is shown on the upper abscissa of (B).

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As stated, the substrate-competing model provides an excellent fit to the rice SL CLD. The independent substrate model can also fit the SL range of the rice amylopectin CLD (Figure S1A), although the fit is not as good (see Discussion).

The substrate-competing model cannot quantitatively reproduce the prominent Feature C in the CLD of the Triticeae tribe, such as wheat (Figure S5A). A good fit is obtained by treating enzyme sets (i) and (ii) as not competing for the same substrate, which means obeying a separate evolution equation in the form of Eqn 1 for the CLD for the two additional enzyme sets. Having a given range (i.e. SL or TL ranges) of the CLD with contributions from enzymes sets which do not compete for the same substrates is the “independent substrate model”. This model fits the wheat data (Figure S2A). This difference in rice and wheat may lie within the mode of enzymatic actions in different species, as hypothesized earlier.

Chains Located in more than a Single Lamella

The calculated SL CLD for rice (Figure 3A) also indicates that there are long SL chains (X ≥32 for the present results). These long SL chains, which would protrude out of single lamellae, are denoted Type–1 trans-lamella (TL) chains. They contribute to a portion of the chains in the high X range (i.e. 32≤ X <60–70). While the fit for shorter SL chains is excellent, the calculated SL CLD at high X does not resemble the experimental one. The X at which the calculated CLD significantly deviates from the experimental CLD is taken to define the start of the trans-lamella (TL) range (Figure 3A; 32≤ X <60–70) where the chains are no longer quantitatively described by the SL enzyme sets alone. The CLD in the high X range must come from contributions from additional enzyme sets. It could be supposed that the additional enzyme sets also act in a substrate-competing manner with the SL enzyme sets. However, this is not the case: incorporating the contributions from additional enzyme sets in Eqn 1 is unable to reproduce the experimental CLD for X ≥32 (Text S2 and Figure S3).

The only way that the CLD can be quantitatively modeled for both rice and wheat for X ≥32 is if there are additional enzyme sets operating, not competing for the same substrate(s), but independently from the SL enzyme sets: the independent substrate model. The additional enzyme sets are termed the TL enzyme sets: these enzyme sets (iii) and (iv) for rice are again treated with the substrate-competing model as for the SL enzyme sets. The calculated CLD from the TL enzyme sets are denoted Type–2 TL chains.

The TL enzyme sets give the TL kinetic parameters denoted β_(iii), X_0(iii), X_min(iii), β_(iv), X_0(iv) and X_min(iv). The ratios of enzymatic activities for enzyme set (iii) and (iv) are defined analogously to Eqns 2–3. These parameters have the same steady-state restrictions to those for the SL kinetics parameters (see Model section). The calculated type-2 TL CLD is fitted to the corresponding experimental CLD (Figure 3B), obtained as the difference between the experimental CLD and the calculated SL CLD (detailed in the Model section). The fitted type-2 TL CLD quantitatively reproduces the observed features: Features D and E. The Feature A-equivalent feature, not apparent in the overall experimental CLD (Figure 2), is revealed in the type-2 TL CLD obtained by this subtraction. A significant change in the near-linear slope is seen at X = 60–70. Chains longer than this are presumed to span three lamellae and to be governed by additional independent enzyme sets. Accurate data on these chains cannot be obtained with current techniques, and thus are not treated here.

In the substrate-competing model, the SL and TL enzyme sets are independent, and thus a parameter h_(iii/i), defined in the Model section, is required to specify the relative abundance of the CLDs generated from both kinetics; this parameter is obtained by fitting.

In the independent substrate model, the kinetics of each enzyme set is independent of the others. Therefore, the relative abundance of the CLDs from enzyme sets (ii), (iii) and (iv) are ratios in relation to that of enzyme set (i) and are specified by the parameters h_(ii/i), h_(iii/i) and h_(iv/i), respectively. The quantity h_(ii/i) was denoted a₂/a₁ in our previous paper [24]; the current terminology is more appropriate.

The overall calculated CLD in the substrate-competing model is obtained by summing the calculated CLDs from both the SL and the TL enzyme sets. Note that it is the N_de(X) (not log N_de(X)) that are summed, although the data are best presented as log N_de(X). This overall calculated CLD accurately fits rice amylopectin CLDs (Figure 2).

Fitting Model Parameters

Non-linear least-squares fitting with the substrate-competing model was performed on the CLD of amylopectin from six replicates of a rice variety, Nipponbare [32]; results are in Figure 4. Despite the number of parameters involved in these fittings, the standard deviation in the fitted values of these parameters from the replicates is quite small, provided that the CLD data are sufficiently precise, as is the SL range in the present case. Replicate data beyond the SL range (X ≥32) show more scatter, as reflected in the range of fitted values (e.g. Figure 4, β_(iii) and β_(iv)). (It is noted that the fitted parameters from the substrate-competing model have no simple relationship to those from the independent-substrate model; Figure S4).

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Figure 4. Fitted parameters for six replicates of the CLD described in Figure 2.

(A) Averages and ± standard deviations (thin bars) of β (the relative branching activity from an enzyme set to that of the total propagation from set (i) and (ii) in the SL range, or (iii) and (iv) in the TL range) from different enzyme sets; h_(iii/i) is the ratio of the maximum of type-2 TL CLD to that of the SL CLD. (B) The modes (highest repeating values) of X₀ and X_min (which are discrete variables) from different enzyme sets. Where the thin bars are not seen means the standard deviation is small. The red and blue colors correspond to the calculated SL and type-2 TL CLD in Figure 2.

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Amylopectin CLD Synthesis is in a Steady State which Requires the Involvement of Debranching Enzymes

Amylopectin CLD is governed by at least four enzyme sets. In rice, the SL enzyme sets (i) and (ii), giving the SL CLD, appear to be substrate-competing; TL enzyme sets (iii) and (iv), giving the type-2 TL CLD, also appear to be substrate-competing. However, the SL enzyme sets are independent of the TL enzyme sets (see Results). In wheat, enzyme sets (i) and (ii) appear to be described by the independent substrate model while the rest are within the substrate-competing model, the same as in rice.

The modes of actions of the four enzyme sets quantitatively reproduce Features A–E in the rice amylopectin CLD (Figure 2). This methodology has also been applied with equal success to data for many other species and varieties (e.g. Figure S5B–D). This implies that there is no need to invoke further enzyme classes (i.e. other than SSs, SBEs and DBEs). Calculations show that Feature A is dependent on the values of X_0(i) and X_min(ii). It appears as a local maximum when the difference between X₀ and X_min is great (Figure S6). The local maximum merges with the global maximum (Feature B) when the values of X_0(i) and X_min(i) are close (Figure S6). Feature A cannot be distinguished in rice amylopectin CLD (Figure 2), which is reflected in the fitted values of X_0(i) and X_min(ii) being close (Figure 4). Potato amylopectin CLD (Figure S5B) shows Feature A as a shoulder, and the difference in the fitted vaues of X_0(i) and X_min(ii) is greater (Figure S5). These results are consistent with the in vitro characteristics of the SBEs from both rice and potato. Rice SBEs generate a single peak in the product CLDs after reaction with linear glucans, while multiple peaks are observed with potato SBEs [14], [33]. The difference in the X₀ and X_min is the cause of this behavior of peaks in the CLDs.

Even though either the substrate-competing or the independent substrate models yield good fits to rice amylopectin CLD (Figure 2 and S1), the substrate-competing model is favored because it is bound to produce a smooth Feature C, as seen in e.g. rice. The substrate-competing model also has one less parameter in the SL range (see Model section).

It is apparent that the type-2 TL chains come from independent TL enzyme sets. Within the TL range (X ≥32), however, a distinct Feature E from the TL enzyme sets (iii) and (iv) is not apparent (e.g. Figure 2, Figure S5 and Figure 3 in [24]). This is analogous to the smooth Feature C in rice. This favors the substrate-competing model for the type-2 TL CLD.

The substrate-competing and independent-substrate models are slightly different cases, and may be applicable to different ranges of CLDs and in different plant species. This suggests the following effects which have not been seen before. (1) Depending on the plant species, enzyme set (i) and (ii) may be described by either the substrate-competing or independent-substrate model. (2) A significant portion of the chains spanning beyond the SL range, the type-2 TL chains, arises from the TL kinetics, which are independent from the SL kinetics. The existence of these chains will be seen in a later section to suggest a new tool for biotechnology. (3) The enzyme sets (i.e (iii) and (iv)) governing the type-2 TL chains appears to be described by the substrate-competing model in all plant species studied.

It is also inferred from the good fit (Figure 2) that the contribution of primer glucans during the de novo synthesis of amylopectin molecules is insignificant for amylopectin CLD (i.e. r = 0 in Eqn 7), although the total mass of amylopectin changes in time, and this is governed by primer glucans.

The amylopectin CLD is effectively described by the enzymatic actions (i.e. N_de = N_de.NL in Eqn 7) in a steady state (dN_de/dt = 0 in Eqn 7), where crystallization does not influence dN_de/dt (F_cryst = 0 in Eqn 7). The mechanism for the steady-state amylopectin CLD formation is envisaged as follows. At the surface of a growing starch granule, the steady-state SL kinetics govern the amylopectin chains in the nascent SL space; these nascent chains are not yet arranged into a crystalline structure. Chains confined to the nascent SL space, amorphous in nature (Figure 5), are referred to as the non-lamellar chains. This is not to be confused with chains that reside in the amorphous lamellae in a grown starch granule. Crystallization of glucan chains is a series of complex physical processes. Deriving the exact expression for the rate of crystallization would require knowledge of both crystallization kinetics between the length and distance of the chains forming an α helix with another chain, and a knowledge of influences and interactions from other chains. Once crystallization occurs, the CLD is “frozen”: not susceptible to significant enzyme-induced changes. It is assumed that crystallization of the non-lamellar CLD proceeds in a non-selective manner with respect to the length of chains X. In other words, a chain of any length has the same chance of being frozen in the crystalline structure. It is argued that this freezing effect applies not only to the chains that are actually involved in α-helices, but also to the chains that reside in the amorphous lamellae. The latter are governed by the steady-state TL kinetics while a starch granule is growing, and become entrapped beneath layers of crystalline lamellae in a fully formed starch granule (Figure 5, where the amorphous lamella is sandwiched by two established crystalline lamella). Altogether, the crystallized chains and the frozen chains in the amorphous lamellae make up the steady-state amylopectin CLD in granular starch. This means the amylopectin CLD is simply that of the steady-state non-lamellar CLD and the steady-state TL CLD, which are governed by the enzymatic action, but then frozen irrespective of chain lengths in the lamella structure of starch granules. These conclusions are supported by the close fit of the model, which was developed for the CLD prior to crystallization, to amylopectin CLD in rice (Figure 2) and other species and varieties.

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Figure 5. A proposed mechanism for the formation of the arrays of semi-crystalline lamellae in amylopectin.

C and A indicate the crystalline and amorphous lamellae. The purple shaded regions indicate the nascent SL space where non-lamellar amylopectin chains, amorphous in nature, are being formed. Types of chains: (1) single-lamella (SL) chains that are (mostly) confined in crystalline lamellae (grey lines); (2) type-1 TL chains, crystalline-lamella-connecting chains, which span more than one crystalline lamellae (red lines); (3) type-2 TL chains, non-lamella-connecting chains, which protrude the SL space but remain in the amorphous lamellae (blue lines). The longest chain length confined to a crystalline lamella, predicted from Figure 2, is X ∼ 31 (i.e. chains ≥32 enters the contiguous amorphous lamella), while longer SL chains (red lines) with X ∼ 60–70 are sufficiently long to participate in crystalline formation in the subsequent crystalline lamella.

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Not all possible steady-state CLDs may be crystallization-competent. For example, Figure 5 of Ref. [24] showed a steady-state CLD can be generated with a large value of β >1, but this yields a drastically different CLD to that observed in nature. We have performed fitting of a large number and variety of CLDs from the literature and always found β ≪ 1. The steady-state amylopectin CLD is a well established feature of granular starches. However, this may or may not apply to all stages of granule development or the growth of individual starch granules. For example, the amylopectin CLD at an early developmental stage of maize endosperm shows a different CLD to that of the later stages [34]. The amylopectin CLDs in A- and B-granules of wheat, which are synthesized at different developmental stages of the grain, are also different, but only slightly so [35]; amylopectin CLDs in developing barley grains do not show significant differences [31]. Amylopectin in the core of maize [36] and potato [37] starch granules contain more long chains compared to the periphery. Nevertheless, these discrepancies are minor in terms of the CLD ensemble from all of the starch granules developed at different time in cereal endosperms. Systematic study using the model to fit these differences in CLDs may provide insight into the biosynthetic pathway at different developmental stages of grains and different size granules.

The amylopectin CLD also varies with the starch-accumulating species, implying that there is a range of CLDs (almost certainly associated with an appropriate distribution of spacing between branch points) which are crystallization-competent. This may be related to the forms of crystallization contribution, F_cryst, in a way which we do not yet understand.

Both SL and TL kinetic parameters have restrictions for a steady-state amylopectin CLD to form: the zero-eigenvalue requirement, as discussed in the Model section. The model implies that if either of SL or TL kinetic parameters do not obey these restrictions, the CLD will either proliferate indefinitely or disappear in time: a steady-state CLD will not be attained.

The steady-state condition provides a mathematical proof that precisely balanced ratios of the enzymatic activities are one of the requirements for the synthesis of an amylopectin CLD which is crystallization-competent. This is an independent confirmation of the conclusion, previously reached solely on the basis of genetic studies, of the absolute requirement for debranching enzyme in crystalline amylopectin synthesis (e.g. [2], [20], [38], [39], [40]). Loss of relative debranching activities will shift the parameters away from the steady state restrictions needed for crystallization-competent amylopectin CLD to form. We propose that this consequence of the model is entirely consistent with, and provides an explanation for, the experimental observation that in the complete absence of isoamylase-debranching enzyme activity, non-crystalline phytoglycogen is accumulated in the cereal endosperm [20], [38].

It is apparent from the steady-state surface showing relations between possible ratios of enzymatic activities (see Model section and Figure 6) that there is a less steep change of the DBE ratio γ_(i,ii) with respect to the SBE ratio β_(ii) than with β_(i): the steady state is less sensitive to β_(ii). This is consistent with the fact that enzyme set (i), which creates most of the short chains in amylopectin, is important for crystalline formation. The weaker β_(ii) dependency may explain why the appearance of Feature C (Figure 2) varies appreciably among different starch-accumulating plants (e.g. rice [41]) and wheat (e.g. [42]), noting that enzyme sets (i) and (ii) may be either substrate-competing or independent. The variation is also seen within species (e.g. rice [41]). It is speculated that enzyme set (ii) (i.e. SS(ii), SBE(ii), DBE(ii)), rather than being genetically defined by particular isoform of enzymes, might be a biochemically defined machinery through processes such as phosphorylation, enzyme complexation, or other regulatory mechanism [5]. These processes are likely to be less consistent across species and varieties, and also subject to the influence of environmental factors.

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Figure 6. The steady-state surface, which describes the restrictions on the parameters for amylopectin CLD to form.

Surface generated for X_0(i), X_min(i), X_0(ii) and X_min(ii) = 6, 7, 9 and 14 respectively. These values are the fitted parameters for Figure 2. The thick red line is an approximation to the steady-state line in our previous development (Figure 4 in [24]), where the exact steady-state line is when β_(ii) (as defined in Figure 4) = 0. γ_(i,ii) is the sum of relative debranching activities from set (i) and (ii) to that of propagation from set (i) and (ii).

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Inferences from the Fitted Parameters

One of the reductionist assumptions in the theoretical development is that enzymatic activities are represented by overall average rates (see the Model section). Now, chain-length-dependent activities certainly exist: Commuri and Keeling [43] have, for example, shown that SSI has different specific activity towards different chain lengths. However, there are insufficient data and as yet unresolved complexities in understanding the biosynthetic process. Examples of the concerns with respect to data availability are: (1) data are not available for all SS isoforms; (2) the relative activity of isoforms changes over time during organ development; (3) the available data are based on isolated enzymes in dilute solutions rather than at a soluble/insoluble interface zone; (4) the phosphorylation status of the enzymes is not yet known or described; (5) no account has yet been taken of possible conformation changes that may occur as part of enzyme complexation. Thus we consider that at this time it is more appropriate to use the reductionist assumption rather than incomplete data which may be subject to revision as the field matures. Once data to address the unknown parameters become available, extensions of the model can be made to include those. The fitted parameters, at present, may not reflect the true ratios of enzymatic activities; however, qualitative comparisons can be made between them.

The fitted parameters (Figure 4A) show the relative branching activities for the type-2 TL chains are lower than those of the SL. This is because long chains are relatively abundant in the type-2 TL CLD (Figure 3B). Lower relative branching activities means higher rate of chain elongation, and thus more long chains. The steady-state condition requires that lower β values in the TL kinetics must be accompanied by a lower relative debranching activity than in SL (Figure 6). It is reasonable to assume that this is because the branching points of the type-2 TL chains, embedded in the preceding crystalline lamella, are less accessible to DBEs.

The shortest X that SBEs can produce is X = 6, as determined by Arabidopsis leaves pulse-chased with ¹⁴CO₂ [16]. Rice (e.g. Figure 2) and many other plant species (e.g. Figure S5) also show the shortest possible amylopectin chain in significant abundance is X = 6. This is consistent with the fitted value X_0(i) for rice (Figure 4B). There is no definite way of distinguishing if X = 6 from the references above represents that of X₀ or X_min at this stage. Related experiments often examine the final CLD formed by subjecting glucan polymers to SBEs, and there is at present no unambiguous way to determine the transferred glucan minimum DP (X_min) from that for the remaining glucan (X₀).

The quantity giving the relative abundance of type-2 TL chains, h_(iii/i), is ∼10^–2 for rice (Figure 4A), with small values also obtained for many other species (Fig. S5). This implies that the contribution from enzyme sets (iii) and (iv), giving the type-2 TL chains that reside in the amorphous lamellae, is relatively small compared to the SL chains. This may be because the majority of SL chains, when aligned into crystalline lamella, are less likely to be elongated further. It is also possible that the physical environment of the amorphous lamellae may obstruct growth of the type-2 TL chains due to steric hindrance. Further values for h_(iii/i) were found by treating the CLDs obtained in [25] with the model (Figure S5 bottom panel). The resulting values h_(iii/i) are correlated with the botanical backgrounds: higher for storage starches in dicots (B-type) than monocots (A-type). High-amylose maize (B-type) shows elevated abundance of type-2 TL chains compared to normal maize (A-type). Together, these show the variability of h_(iii/i), which may be associated with the types of crystal forms. A higher h_(iii/i) is found for potato starch (B-type), which is consistent with its amorphous lamellae being denser (filled with more glucan materials) compared to the A-type starches, a conclusion based on the electron density profiles of stacked lamellae [44].

Formation of Successive Arrays of Crystalline Lamellae in the Semi-crystalline Layers of Starch Granules

Fitting various data sets with the substrate-competing model implies that long chains, 32≥ X 60–70 for the particular case fitted in detail here, comprise two types: types-1 and -2 TL chains. The former, being involved in crystalline lamellae, is governed by the SL enzyme sets (i.e. SS(i–ii), SBE(i–ii) and DBE(i–ii)). The only possible way for this to occur is that longer chains (X ≥32) originate from a crystalline lamella, protrude from that lamella, span across the contiguous amorphous lamella, and are then able to evolve further under SL kinetics and participate in crystalline formation in the subsequent crystalline lamella (Figure 5, red lines). The type-1 TL chains, connecting crystalline lamellae, correspond to the crystalline-lamellar connecting chains in the cluster model of amylopectin (e.g. [17]).

Type-2 TL chains are governed by the TL enzyme sets (i.e. SS(iii–iv), SBE(iii–iv) and DBE(iii–iv)) which are independent of the SL enzyme sets. They are likely to be produced when some of the long SL chains protrude from their parent crystalline lamella. It is envisaged that the crystalline lamella is established to a degree where the biosynthetic enzymes can no longer “sense” the glucans in the crystalline structure. It is then assumed that the protruding portion of these long SL chains behaves like short SL chains again. This assumption is taken into account in the model by displacing the starting X of the type-2 TL CLD to X = 1 when fitting (see Model section). The type-2 TL chains evolving under the TL enzyme sets are suggested to remain in the amorphous lamellae (Figure 5, blue lines), which can explain the different parameters in for the type-2 TL chains (Figure 4A). The existence of type-2 TL chains has not been distinguished previously in the cluster model of amylopectin. They are ∼ 2.2 times more numerous than the type-1 TL chains for the particular case fitted in detail here (the total number of chains of type-2 to type-1 TL chains for 32≥ X 60–70).

The formation of successive arrays of crystalline lamellae is envisaged as follows. Consider a randomly branched nascent SL space at the periphery of a growing starch granule. The CLD confined to this space is governed by the SL kinetics in a steady state. To facilitate crystallization, “improperly positioned” chains are removed by the action of DBEs. While chain removal is occurring, a steady-state CLD is always maintained because there are contributions from the SSs and SBEs with rates in a steady-state with that of the DBEs. For chains in the nascent SL space to crystallize, a steady-state CLD must be attained; other factors may include the branch spacing, which has been proposed to facilitate crystallization (e.g. [3], [19]). Crystallization freezes a distribution of chain lengths in the non-lamellar phase and in the amorphous lamellae beneath in a growing starch granule non-selectively (discussed earlier). This gives the steady-state amylopectin CLD in granular starch. Some longer SL chains (X ≥32) protrude their originating crystalline lamellae, are captured in the materializing subsequent crystalline lamella, and evolve to type-1 TL chains (Figure 5). The rest of the long SL chains remain in the contiguous amorphous lamella and evolve under TL kinetics to form the type-2 TL chains, which is a new type of branch both in the amylopectin cluster model [17] and in the “two-step branching and improper branch clearing model” proposed by Nakamura [3].

The semi-crystalline structure in starch granules is produced by a repetitive synthesis of the crystalline and amorphous lamellae as described above. This implies that amylopectin chains form semi-crystalline structures progressively in layers, through successive lamellar formation.

A New Parameterization Tool for Structure-property Relations

This CLD model provides a new parameterization, which reduces the amylopectin CLD to a small number of meaningful biosynthetic-based parameters (Figure 4) for comparisons between species and mutants. The actual parameters are β_(i), X_0(i), X_min(i), β_(ii), X_0(ii), X_min(ii), β_(iii), X_0(iii), X_min(iii), β_(iv), X_0(iv), X_min(iv) and h_(iii/i) (in practice, e.g. [45], not all of these are needed). This is an alternative to the widely used empirical approaches, such as difference plots, for comparisons between different CLDs (e.g. [3], [19], [20], [35], [42]).

It has been found (e.g. [45]) that this new parameterization provides an improved tool for a statistical identification of structurally important characteristics of a starch with regard to properties such as crystallinity and digestibility. This parameterization is physically based, and is able to represent the entire CLD accurately in terms of a few parameters; it thus encompasses all information in empirical representations such as difference plots.

Insight into Starch Biosynthesis for Plant Biotechnology

The significant implication for plant biotechnology from this work is that two types of TL chains are distinguished in the substrate-competing model as for rice and their relative abundance, h_(iii/i), may not be restricted by a steady-state condition as for the SL and TL kinetic parameters. It is not clear what controls h_(iii/i). It is possible that the variability of this quantity is associated with the nature of crystal forms as discussed earlier. This points to the need for understanding the factors influencing the synthesis of type-2 TL chains in the amorphous lamellae for developing starch with elevated abundance of long chains. There is evidence that slowly digestible starch is positively correlated with abundance in long and intermediate chains, which may have nutritional benefits [46], [47]. Of course, this may not be the only mechanism for quality foods: e.g., resistant starch, which has considerable health benefits, can be obtained from different starches and structures and processing treatments [48].

The modeling approach set out in this paper has the potential to aid in the understanding of starch structures and their manipulations.

Derivation of the Steady-state Solution

The concentration of the lamellar CLD, , is determined by the rate of crystallization of . Once the chains are arranged into lamellar structure in starch granules it is assumed that they are “frozen” and thus not susceptible to significant enzyme-induced changes [49]. Thus the time evolution of the absolute concentration of the lamellar CLD is given by:(6)

The lamellar CLD is approximated as independent of time as discussed in the Introduction. N_de(X) (the relative number of chains) is related to (the actual concentration of chains) by:where B(t) is related to the overall rate of synthesis/degradation of starch in time.

The rate of change of the SL lamellar CLD is then, in matrix notation:(7)

Eqn 7 is derived from Eqn 5, which describes the evolution of the SL CLD, by considering that the rate of change of the SL CLD in cereal endosperms, governed by the two SL enzyme sets (i.e. SS(i–ii), SBE(i–ii) and DBE(i–ii)), is at a steady state (i.e. the right hand side of Eqn 5 = 0); putting r = 0; F_cryst = 0; N_de = N_de.NL; and dividing Eqn 5 by â_SS(i) g(t)+â_SS(ii) g(t). These assumptions, necessary for solving for N_de, explained in the discussion, give a mechanism for the attainment of the steady-state lamellar CLD.

Eqn 7, a system of infinite linear equations, has the solution (see e.g. [24]) (Eqn 8):(8)(9)where λ_i is the i^th eigenvalue of Ω and u_i is the corresponding eigenvector; c_i is obtained by solving the appropriate relation (Eqn 9) as a system of simultaneous equations. N_de(t = 0) is a vector whose elements comprise all the elements of N_de(X, t = 0), which give the lamellar CLD at the beginning of the period in question. Practically, i is truncated at a sufficiently large value (∼110) so that N_de(t) converges to within a desirable tolerance.

The solution of Eqn 8 is dictated by the SL kinetic parameters: β_(i), X_0(i), X_min(i), β_(ii), X_0(ii), X_min(ii) and γ_(i,ii). The only way that a unique steady-state CLD is obtained at long times is if one eigenvalue is exactly zero and all others are negative. If one or more eigenvalues are positive, the CLD grows indefinitely, while the contributions from the negative eigenvalues disappear in a short induction time, leaving that for the zero eigenvalue. The steady-state CLD is then given by the eigenvector corresponding to the zero eigenvalue.

The steady-state SL CLD restricted to only a particular set of SL parameters which are conveniently presented by a steady-state surface for a given set of values of X_0(i), X_min(i), X_0(ii) and X_min(ii) (Figure 6). This surface gives γ_(i,ii) as a function of β_(i) and β_(ii), which then excludes the need to have γ_(i,ii) as an independent parameter: γ_(i,ii) is a function of β_(i), β_(ii), X_0(i), X_min(i), X_0(ii) and X_min(ii). The surface depends on the values of the various X₀ and X_min, but the basic shape of the surface is retained. The steady-state surface is analogous to the “steady-state line” depicted in Figure 4 of our previous paper [24], but with one extra dimension: β_(ii). The slice of the surface at β_(ii) = 0 gives the previous steady-state line. Any point on the steady-state surface gives a set of eigenvalues (Eqn 8) comprising exactly one zero eigenvalue and the rest negative.

Method of Fitting the Model to Experimental CLDs

A freeware Fortran program for least-squares fitting both the substrate-competing and independent substrate models to amylopectin CLD in cereal endosperms such as rice, implementing the mathematical treatment development given above, is available for download (Text S3, S4 and [50]). A detailed step-by-step guide and examples of fitting (Figure S7, S8, S9, S10, S11) are also provided.

Fitting is carried out by non-linear least squares fitting of Eqn 8 to the SL range of the experimental CLD with this Fortran program. As apparent from Figure 3A, SL chains alone do not reproduce the Features D and E, as discussed in the Results section. The calculated SL CLD is subtracted from the experimental CLD and the difference, referred to as the type-2 TL CLD. In order to treat the type–2 TL CLD the starting X of the type-2 TL CLD (X = 32 in rice amylopectin CLD (Figure 3A)) is displaced to X = 1 and then fitted as described for the SL CLD. The implication of the displacement treatment is given in the Discussion. It is assumed that the type-2 TL chains are governed by two TL enzyme sets (i.e. SS(iii–iv), SBE(iii–iv) and DBE(iii–iv)) acting in a substrate-competing manner, analogous to enzyme sets (i) and (ii), in a steady state.

The ratio of the maximum of the type-2 TL CLD to that of the SL CLD in the substrate-competing model is given by the quantity h_(iii/i). This is found by subtracting the calculated SL CLD, with a maximum of h_(i), from the experimental CLD and dividing the maximum of the calculated type-2 TL CLD, with a maximum of h_(iii), by h_(i). For convenience the SL CLD is always normalized to a maximum of 1 (i.e. h_(i) = 1). The ratio of h_(iii) to this h_(i) is referred to as h_(iii/i).

The overall fitting is obtained by adding the calculated SL CLD (normalized to a maximum of 1) and the calculated type-2 TL CLD (removing the displacement in X) normalized to a maximum of h_(iii/i). This is given by: