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Carbohydrate‐active enzymes exemplify entropic principles in metabolism

2011; Springer Nature; Volume: 7; Issue: 1 Linguagem: Inglês

10.1038/msb.2011.76

1744-4292

Önder Kartal, Sebastian Mahlow, Alexander Skupin, Oliver Ebenhöh,

Microbial Metabolic Engineering and Bioproduction

Article25 October 2011Open Access Carbohydrate-active enzymes exemplify entropic principles in metabolism Önder Kartal Önder Kartal Max Planck Institute of Molecular Plant Physiology, Potsdam, Germany Department of Plant Physiology, Institute of Biochemistry and Biology, University of Potsdam, Potsdam-Golm, Germany Search for more papers by this author Sebastian Mahlow Sebastian Mahlow Department of Plant Physiology, Institute of Biochemistry and Biology, University of Potsdam, Potsdam-Golm, Germany Search for more papers by this author Alexander Skupin Alexander Skupin Max Planck Institute of Molecular Plant Physiology, Potsdam, Germany Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Luxembourg Search for more papers by this author Oliver Ebenhöh Corresponding Author Oliver Ebenhöh Institute for Complex Systems and Mathematical Biology, Department of Physics, SUPA, University of Aberdeen, Aberdeen, UK Institute of Medical Sciences, University of Aberdeen, Aberdeen, UK Search for more papers by this author Önder Kartal Önder Kartal Max Planck Institute of Molecular Plant Physiology, Potsdam, Germany Department of Plant Physiology, Institute of Biochemistry and Biology, University of Potsdam, Potsdam-Golm, Germany Search for more papers by this author Sebastian Mahlow Sebastian Mahlow Department of Plant Physiology, Institute of Biochemistry and Biology, University of Potsdam, Potsdam-Golm, Germany Search for more papers by this author Alexander Skupin Alexander Skupin Max Planck Institute of Molecular Plant Physiology, Potsdam, Germany Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Luxembourg Search for more papers by this author Oliver Ebenhöh Corresponding Author Oliver Ebenhöh Institute for Complex Systems and Mathematical Biology, Department of Physics, SUPA, University of Aberdeen, Aberdeen, UK Institute of Medical Sciences, University of Aberdeen, Aberdeen, UK Search for more papers by this author Author Information Önder Kartal1,2,‡, Sebastian Mahlow2,‡, Alexander Skupin1,3,‡ and Oliver Ebenhöh 4,5,‡ 1Max Planck Institute of Molecular Plant Physiology, Potsdam, Germany 2Department of Plant Physiology, Institute of Biochemistry and Biology, University of Potsdam, Potsdam-Golm, Germany 3Luxembourg Centre for Systems Biomedicine, University of Luxembourg, Luxembourg 4Institute for Complex Systems and Mathematical Biology, Department of Physics, SUPA, University of Aberdeen, Aberdeen, UK 5Institute of Medical Sciences, University of Aberdeen, Aberdeen, UK ‡These authors contributed equally to this work *Corresponding author. Institute for Complex Systems and Mathematical Biology, Department of Physics, SUPA, University of Aberdeen, Meston Building, Meston Walk, Aberdeen AB24 3UE, UK. Tel.: +44 (0)1224 272520; Fax: +44 (0)1224 273105; E-mail: [email protected] Molecular Systems Biology (2011)7:542https://doi.org/10.1038/msb.2011.76 PDFDownload PDF of article text and main figures. Peer ReviewDownload a summary of the editorial decision process including editorial decision letters, reviewer comments and author responses to feedback. ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InMendeleyWechatReddit Figures & Info Glycans comprise ubiquitous and essential biopolymers, which usually occur as highly diverse mixtures. The myriad different structures are generated by a limited number of carbohydrate-active enzymes (CAZymes), which are unusual in that they catalyze multiple reactions by being relatively unspecific with respect to substrate size. Existing experimental and theoretical descriptions of CAZyme-mediated reaction systems neither comprehensively explain observed action patterns nor suggest biological functions of polydisperse pools in metabolism. Here, we overcome these limitations with a novel theoretical description of this important class of biological systems in which the mixing entropy of polydisperse pools emerges as an important system variable. In vitro assays of three CAZymes essential for central carbon metabolism confirm the power of our approach to predict equilibrium distributions and non-equilibrium dynamics. A computational study of the turnover of the soluble heteroglycan pool exemplifies how entropy-driven reactions establish a metabolic buffer in vivo that attenuates fluctuations in carbohydrate availability. We argue that this interplay between energy- and entropy-driven processes represents an important regulatory design principle of metabolic systems. Synopsis Statistical thermodynamics and in vitro experimentation demonstrate that metabolic enzymes can be driven by an increase in the entropy of a reaction system, and point to a role for entropy gradients in the emergence of robust metabolic functions in vivo. By analyzing the equilibrium distributions of glycans in vitro, we demonstrate that several carbohydrate-active enzymes are driven by an increase in mixing entropy of the reaction system. We present a novel formalism for these 'entropic enzymes' that allows biochemical processes in glycobiology to be described by concepts from statistical thermodynamics, thereby establishing a sound theoretical framework for polymer-active enzymes in general. Our interdisciplinary study reveals a new role of entropy in metabolism and promotes a novel view of metabolism as an intricate interplay between energy- and entropy-driven processes. We demonstrate by stochastic modeling how the concerted action of entropic enzymes in vivo results in a robust and adaptive buffering function needed to ensure a constant provision of carbohydrates for downstream processes. Introduction Glycans, comprising polysaccharides and oligosaccharides, constitute the most abundant polymers found in nature but are far less investigated than proteins and nucleic acids (Seeberger, 2005; BeMiller, 2008). They govern a remarkably wide range of biological functions, including carbon and energy storage (Ball and Morell, 2003; Zeeman et al, 2010), mechanical stabilization of cells or tissues (Cosgrove, 2005), cell–cell or cell–protein interactions (Seeberger, 2005; Finkelstein, 2007) and organelle division (Yoshida et al, 2010). Moreover, they have attracted considerable interest as renewable energy source (Himmel et al, 2007; Zeeman et al, 2010) and starting materials or additives for many technological applications (Takaha and Smith, 1999). Glycans can possess complex chemical structures and often occur as a polydisperse mixture of compounds with different molecular weights (BeMiller, 2008). Their biosynthesis and degradation involves the concerted action of numerous carbohydrate-active enzymes (CAZymes) (Davies and Henrissat, 2002; Coutinho et al, 2003; Kobayashi and Ohmae, 2006; Cantarel et al, 2009), which can repeatedly act on sugar donors and acceptors to generate polydispersity. Hence, two aspects complicate the description and characterization of CAZyme-mediated systems. First, polymer-active CAZymes typically do not catalyze only a single reaction and consequently rate laws with the usual kinetic (Km and Vmax) and thermodynamic (Keq) parameters are insufficient to appropriately characterize their dynamics. Second, polydispersity implies that a huge number of variables are required to precisely describe the system. A model based on differential equations describing the temporal change of each individual species would be impractical due to potentially infinite numbers of reactants and conversions. Numerical tractability may be increased by novel rule-based approaches (Feret et al, 2009) or by replacing individual chemical species by a continuous mixture, leading to integro-differential equations (Aris, 1989). However, despite many experimental studies (Jones and Whelan, 1969; Lin and Preiss, 1988; Kakefuda and Duke, 1989; Colleoni et al, 1999; Steichen et al, 2008) and some attempts to model (Thoma, 1976; Nakatani, 1999) the kinetics of CAZymes, a generally applicable theoretical description is still lacking. Our aim is to provide a general understanding of enzymes acting on polydisperse substrates. For our approach, we employ statistical thermodynamics and represent polydisperse mixtures of substrates as statistical ensembles. The thermodynamic theory allows characterizing systems with a huge number of particles by a small number of state variables, such as temperature, pressure, internal energy or entropy. We develop an analogous description to show that the same principle holds for polydisperse reactant mixtures. In these systems, the state variable entropy has a particularly important role. If energy is neither added nor removed from the system, the equilibrium state is characterized by maximal entropy (Alberty, 2003). Conceptually, we thus follow early approaches describing chemical systems of polymers (Flory, 1944; Tobolsky, 1944). Whereas in these early studies the entropy was introduced for specific idealized conditions, we provide here a rigorous deduction from fundamental principles (Landau and Lifschitz, 1979) to arrive at a generally applicable expression of the mixing entropy (Box1; Supplementary information). A further critical advancement of our theory is the inclusion of enzymatic reactions. Enzymes catalyze changes in the polydisperse mixture. However, these changes are not completely arbitrary, but are limited to those that are in accordance with the underlying enzymatic mechanisms. Consequently, enzymatic systems acting on polydisperse mixtures are described as constrained statistical ensembles (Box 1). Box 1 Enzymatic reactions on polydisperse substrates Generation of polydisperse mixtures by CAZymes: α-1,4-glucans, linear polysaccharides consisting of glucose residues that are linked by α-1,4-glucosidic bonds, are important intermediates in carbohydrate metabolism. Any such glucan can be characterized by its number of residues or degree of polymerization (DP). Glucanotransferases, such as DPE1, transfer glucosyl residues between α-1,4-glucans of any DP. Panel A illustrates the action of DPE1 for the pure initial substrate maltotetraose (DPini=4). All possible products of the first reaction step and a representative second step with a single pair of substrates are shown, indicating the strong diversification of the glucan pool generated by the huge number of possible reactions. Every transfer reaction conserves the number of molecules present in the reaction mixture as well as the total number of glucose residues distributed in the polydisperse pool. As a consequence, the average DP maintains the constant value DPini, which is in general determined by the average DP of the initially applied mixture of glucans and can assume also non-integer values. Therefore, at any time the relationships hold, where xDP describes the molar fraction of the glucan with the respective DP. This illustrates how polydisperse pools of glucans are generated by enzymatic action. Thermodynamic description of polydisperse reactant mixtures: As any reaction, the DPE1-mediated disproportionation must proceed in the direction in which the Gibbs free energy, G=H−TS, decreases, where H is the enthalpy, T the temperature and S the entropy. The enthalpy H measures the energy contained in the reactants. Since the energy contained in the bonds of a glucan increases with increasing DP, the different DPs can be considered as different energy levels. The molar fractions xDP can be interpreted as the occupation of the corresponding energy states. Thus, the distribution {xDP} represents a statistical ensemble which fully characterizes a polydisperse reactant mixture (Flory, 1944; Landau and Lifschitz, 1979; Alberty, 2003). The entropy S measures the dispersal of energy within the ensemble. It is defined as where R is the universal gas constant. The DPE1-mediated transfer reactions occur without net enthalpy change, ΔH=0 (Goldberg et al, 1991; Tewari et al, 1997). Consequently, decrease in Gibbs free energy is equivalent to entropy increase, which results exclusively from changes in the composition of the glucan pool. This is illustrated in panel B. At t=0, the reactant mixture is monodisperse and contains only maltotetraose molecules. In this case x4=1 and xDP=0 for DP≠4, resulting in S=0. For t → ∞, the equilibrium distribution in which the energy (or interglucose bonds) is maximally dispersed is reached. Determination of the equilibrium distribution: The equilibrium distribution can be computed by identifying those values of xDP, which maximize entropy (2) under the constraints imposed by the enzymatic mechanisms. For DPE1, the constraints are given by the relationship (1) and the resulting equilibrium distribution reads with the characteristic exponent β. For DPE1, the exponent assumes the particularly simple form demonstrating how β fully characterizes the equilibrium in dependence on the initial substrates (see the derivation of Supplementary Equation S45 in Supplementary information). While exponential distributions as in Equation (1) are also found for the other examples considered in the text, the specific expressions for β differ due to additional constraints on the enzymatic transitions. Inserting the distribution (3) and the expression (4) for β in the expression for the entropy yields the corresponding equilibrium entropy for DPE1 The characteristic exponent is a generalization of the equilibrium constant: The equilibrium constant Keq for the single reactions can be calculated from the equilibrium concentrations (3), resulting in Keq=(xn−qxm+q)/(xnxm)=1 for every individual reaction. The functional form of β, given by Equation (4), provides additional information by revealing the dependence on the initial conditions. The exponent β is predicted to decrease when the average DPini increases. Apparently, β serves as an appropriate descriptor of equilibria of polydisperse mixtures and, since it entails the equilibrium constants of the individual reactions, it can be considered as a generalization of the mass action ratio in equilibrium. To develop and experimentally validate our concept, we focus here on CAZymes catalyzing interconversions of α-1,4-glucans, intermediates in the metabolism of the most common storage polysaccharides, starch and glycogen (Ball and Morell, 2003). This class of polysaccharides consists exclusively of glucose residues linked by α-1,4 glucosidic bonds. Each distinct substrate can thus be characterized by the number of glucose residues, denoted as degree of polymerization (DP). Two CAZymes, disproportionating enzyme 2 (DPE2) and the cytosolic phosphorylase (Pho) mediate the turnover of the soluble heteroglycan (SHG) pool (Fettke et al, 2006, 2009b) in the cytosol of plant cells. We use this system to exemplify the utilization of polydisperse systems for metabolic regulation, thus presenting a novel interpretation for the SHG pool. This system shows how a metabolic function (carbohydrate provision and allocation) can be achieved robustly by CAZymes without requiring any additional control mechanisms such as allosteric regulations. Results Disproportionating enzymes increase the entropy of reaction systems Disproportionating enzyme 1 (DPE1; Jones and Whelan, 1969; Lin and Preiss, 1988; Kakefuda and Duke, 1989; Colleoni et al, 1999; Critchley et al, 2001) is a plastidial 4-α-glucanotransferase (EC 2.4.1.25; GH77) (Takaha and Smith, 1999) catalyzing readily reversible reactions according to the equation Gn+Gm↔Gn−q+Gm+q, where Gx denotes an α-1,4-glucan with DP x, and q=1,2,3 is the number of transferred glucosyl residues (Supplementary Figure S1). All reactions occur without noticeable net enthalpy change since for every intersugar linkage cleaved another one is formed and every linkage contains approximately the same enthalpy (Goldberg et al, 1991), raising the question of the reaction's driving force. To the best of our knowledge, Nakatani (1999) was the first to propose, based on stochastic simulations, that in equilibrium the DP distribution has maximal entropy. Within the framework of statistical thermodynamics, it can be proven rigorously that this must indeed be the case (see Supplementary information). Moreover, our theoretical approach allows predicting the exact form of the equilibrium distribution. In order to understand the action of enzymes like DPE1, it is helpful to interpret the distinct chemical species as different energy levels. This allows the reactant mixture to be described as a statistical ensemble (Flory, 1944; Landau and Lifschitz, 1979; Alberty, 2003). Enzymes catalyze transitions between the energy levels and the enzymatic mechanisms define which transitions are possible (Box 1; Supplementary Figure S1). Thus, enzymatic action results in a dispersal of energy. If, as is the case for DPE1, there is no net change in enthalpy, then this dispersal of energy is the only driving force of the reactions. Thus, according to the second law of thermodynamics (Landau and Lifschitz, 1979), the equilibrium distribution of the glucan mixture can be calculated by determining the maximum entropy under the constraints defined by the enzymatic mechanisms (Box 1). At equilibrium, the DPs are exponentially distributed (see Equation (3) and Box 1 Figure). The distribution is fully characterized by the exponent β (Equation 4), which depends on the average DP of the initially supplied glucans, DPini. The characteristic exponent β can be interpreted as a generalization of the equilibrium constant Keq for polydisperse mixtures (Box 1). We have experimentally tested our predictions by incubating DPE1 with defined maltodextrins. The reactions were followed until no change in the glucan patterns was detectable and the reaction system apparently reached equilibrium. The glucan patterns confirm the prediction that an exponential distribution is approximated and that the characterizing factor β depends only on the average initial DPs, DPini (Figure 1A–C; Supplementary Figure S2). Furthermore, the observed distributions quantitatively confirm the predicted decrease of β with increasing DPini (Figure 1D and E). From the observed glucan patterns, we determined the experimental entropy, which also is in accordance with the predicted entropy (Equation 5), in equilibrium (Figure 1F). Figure 1.DPE1 maximizes entropy in vitro. (A–C) Experimentally determined equilibrium distributions depend only on the average degree of polymerization of the initial substrates, DPini. All DP patterns obey the theoretically expected exponential distribution where the exponential factor β depends on the initial substrates, as demonstrated by maltotriose G3 (black in A), maltotetraose G4 (red in B) and maltopentaose G5 (blue in C). The distributions are independent of how DPini is realized. In each panel, the distributions obtained from two different initial conditions with identical DPini are indistinguishable (G3 and a 1:1 mixture of G2 and G4 (gray) in (A), G4 and a 1:1 mixture of G3 and G5 (orange) in (B), G5 and a 1:1 mixture of G3 and G7 (cyan) in (C)). (D) Comparison between the experimental results (dots) and the theoretical predictions (solid lines) in a semi-log plot demonstrates the differences of the coefficients β (corresponding to the slopes) for different initial substrates. (E) Agreement of observed and predicted β demonstrates the entropic mechanism of glucanotransferases. (F) The experimentally determined equilibrium entropies Seq (dots) in dependence on the average initial degree of polymerization (DPini) match with the values predicted by Equation (5), indicated by the solid line. Distributions for DPini=2 and DPini=7 are shown in Supplementary Figures S2B and S5B, respectively. (All error bars denote standard deviation of three independent experiments.) Source data is available for this figure in the Supplementary information. Source data for Figure 1A [msb201176-sup-0001-SourceData-S1.xls] Source data for Figure 1B [msb201176-sup-0002-SourceData-S2.xls] Source data for Figure 1C [msb201176-sup-0003-SourceData-S3.xls] Download figure Download PowerPoint Similar to the plastidial DPE1, the cytosolic 4-α-glucanotransferase DPE2 (Chia et al, 2004; Fettke et al, 2006) mediates a randomization of α-1,4-glucans. In contrast to DPE1, DPE2 transfers single glucosyl residues only and neither utilizes maltotriose as a donor nor maltose as an acceptor (Steichen et al, 2008). This means that maltose molecules cannot be elongated and maltotriose molecules cannot be further shortened. Thus, whenever maltose donates a glucose residue a glucose molecule is released, and whenever glucose acts as acceptor a maltose molecule is formed. As a result, DPE2 effectively obeys an additional constraint: the conservation of the sum of glucose and maltose molecules (x1+x2=const., see Supplementary Figure S3). Again, an exponential equilibrium distribution is predicted but the additional constraint leads to a different functional dependence of β on DPini, which is experimentally confirmed (see Supplementary Equation S57 and Supplementary Figure S4). The example of DPE2 shows the importance of recognizing constraints resulting from enzymatic mechanisms and illustrates how polydisperse mixtures relax to different equilibrium distributions when subjected to enzymes with different action patterns. Stochastic simulations and time-resolved experiments reveal different time scales of DPE1 According to previous reports on DPE1 from white potato (Jones and Whelan, 1969), maltose is neither formed nor utilized as a glucosyl donor. These findings established the idea, that there are 'forbidden linkages' which cannot be cleaved, a rule that was later applied to DPEs from other species, such as Arabidopsis thaliana (Lin and Preiss, 1988) and Chlamydomonas reinhardtii (Colleoni et al, 1999). However, our measurements (Figure 2) clearly demonstrate that this rule is not valid at least for recombinant DPE1 from A. thaliana. Presumably, this discrepancy is due to differences in the length of the incubation period. As revealed by our measurements, ∼10 min after incubation a quasi-stationary equilibrium is reached in which maltose is undetectable. Subsequently, maltose levels rise and approach the theoretically predicted equilibrium with a much lower rate after several days (Figure 2A). These data are consistent with the assumption that 4-α-glucanotransferases prefer distinct glucan binding modes (Suganuma et al, 1991; Nakatani, 1999, 2002; Takaha and Smith, 1999). Based on this view, we developed a stochastic model which reproduces the observed time-resolved glucan patterns under the sole assumption that glucosyl transfers occur with an 800-fold smaller probability than transfers of maltosyl or maltotriosyl residues (Figure 2A). The two time scales can be identified by following the change in mixing entropy (Figure 2B). The quasi equilibrium entropy (dotted line in Figure 2B) is theoretically calculated by excluding maltose from the ensemble representing the polydisperse mixture (see Supplementary Equation S48 in Supplementary information). In the vicinity of the quasi equilibrium, the mixing entropy increases more slowly, while steadily evolving toward the predicted maximum entropy state. Our simulations demonstrate that three kinetic constants are sufficient to characterize the DPE1-mediated system: one rate constant reflecting maximal turnover and two constants describing different transfer probabilities reflecting different subsite affinities at the substrate binding domain (Thoma et al, 1971; Suganuma et al, 1991; Nakatani, 1999). Experimentally, these values are not accessible through simple incubation experiments in analogy to the classical treatment of enzymes catalyzing single reactions, but rather require monitoring of the entire reactant mixture. An alternative description based on two maximal turnover constants can reproduce the same kinetics but is biochemically less plausible. Glycoside hydrolase domains usually possess several binding subsites which allow for different alignments of the substrate formed with different probabilities (Thoma et al, 1971). In contrast, the transfer step always acts between well-defined subsites irrespective of the actual alignment of the substrate (Barends et al, 2007). Figure 2.Low binding affinity for maltose induces a quasi equilibrium distribution. (A) The experimental time course (dots) shows the generation of the different glucans for DPE1 incubated with maltotriose, demonstrating that maltose is produced on a slower time scale compared with the other glucans. Stochastic simulations (solid lines) assuming an 800-fold reduced probability for the transfer of single glucosyl residues compared with maltosyl and maltotriosyl residues accurately reproduce the data. (B) The increase in entropy exhibits two time scales. In the first phase, the entropy rapidly increases toward a quasi equilibrium state without detectable maltose. The dotted line at Sqeq indicates the predicted equilibrium entropy for a constrained system not capable of producing maltose (see Supplementary information). The second phase is characterized by a much slower relaxation towards the real equilibrium Seq (dashed line). The corresponding temporal DP distributions are shown in Supplementary Movie. (All error bars describe standard deviation of three independent experiments.). Source data is available for this figure in the Supplementary information. Source data for Figure 2A [msb201176-sup-0004-SourceData-S4.xls] Download figure Download PowerPoint Generalization for energetically open systems The interpretation of the distinct reactants as different energy states offers a straightforward generalization to reaction systems in which bond enthalpy is not conserved. Taking into account the sum of energies of formation gf, the equilibrium is determined by a minimum in Gibbs free energy (Alberty, 2003) given by g=gf−TS, where T is the temperature and S the entropy (cf. Supplementary Equation S31 in Supplementary information). First, we consider the reaction system catalyzed by α-glucan phosphorylase (Pho; EC 2.4.1.1; GT35). Reversibly transferring terminal glucosyl residues from the non-reducing ends of soluble glucans to orthophosphate, this CAZyme does not conserve the bond enthalpy since it replaces a glucosidic by an ester bond. The resulting equilibrium distribution for different initial conditions, predicted by minimizing the Gibbs free energy, is described by an implicit equation, , which additionally depends on the difference in the enthalpies of bond formation, Δg (see Supplementary Equation S68 in Supplementary information). The predictions are experimentally confirmed by in vitro experiments (Figure 3). Figure 3.Equilibrium distributions of the degree of polymerization for phosphorylase experiments. (A) Schematic representation of the mechanism of phosphorylase. From an α-1,4-linked glucan chain, one glucose residue is reversibly transferred to orthophosphate (red), producing glucose-1-phosphate (G1P). The gray box represents any primer molecule, which can also be a glucan. (B) The experimental distribution (red bars) for a 1:50 mixture of DPini=7 and orthophosphate exhibits a steep decrease. From the theoretical prediction (See Supplementary Equation S68) shown in blue, we can fit the unknown parameter as 0.19. The inlet shows the logarithmic data (red) and further predictions for k0=0.1 (dashed) and k0=0.4 (solid). (C) Comparison between G4 (black) and G7 (red) incubated with G1P in a 1:4 ratio demonstrates the dependence on the initial substrate. (D) Both distributions obey an exponential distribution as shown by the logarithmic plot. The agreement of experiments (dots) and theoretical predictions (lines) validates the theoretical approach. (Error bars indicate standard deviation of three independent experiments.). Source data is available for this figure in the Supplementary information. Source data for Figure 3B [msb201176-sup-0005-SourceData-S5.xls] Source data for Figure 3C [msb201176-sup-0006-SourceData-S6.xls] Download figure Download PowerPoint Exothermic reactions shift equilibrium distributions As a prototype of a multi-enzyme system, we consider the action of DPE1 in the presence of hexokinase (HK, EC 2.7.1.1), which phosphorylates glucose at the expense of ATP to produce glucose-6-phosphate and ADP. In this direction, the reaction is exothermic and its equilibrium is experimentally controlled by the ATP level. Since the HK reaction diminishes the glucose pool accessible to DPE1 but keeps the number of interglucose bonds constant, the equilibrium pattern of the glucans is shifted toward larger DPs (Figure 4; Supplementary Figure S5). This result concurs with earlier findings (Walker and Whelan, 1959; Kakefuda and Duke, 1989) showing the capability of DPE1 to synthesize amylose, which now experience a quantitative theoretical explanation. The predicted exponential distribution of the equilibrium pattern is again accurately described by the parameter β. Here, Figure 4.Energetically unbalanced reactions affect the equilibrium parameter β. In the presence of hexokinase (HK), the DPE1-mediated equilibrium distribution depends on the applied amount of ATP, the Gibbs energy of the HK reaction and the average initial degree of polymerization, DPini. (A) The equilibrium distribution of DPE1 incubated with 500 nmol maltotriose (G3) and 125 nmol ATP (red) is shifted toward longer DPs compared wi