Reduction of a biochemical model with preservation of its basic dynamic properties Sune Danø 1 , Mads F. Madsen 1 , Henning Schmidt 2 and Gunnar Cedersund 2 1 Department of Medical Biochemistry and Genetics, University of Copenhagen, Denmark 2 Fraunhofer Chalmers Research Centre for Industrial Mathematics, Gothenburg, Sweden Systems biology aims to understand the behaviour of biological systems, and in particular how the system’s behaviour emerges from the interactions among its components. Consequently, there is an increased focus on biochemical dynamics and its relation to the under- lying biochemical reaction network. This connection is most often described by means of mathematical models with varying degrees of detail. A primary advantage of a detailed, biochemically formulated model is that a one-to-one comparison can be made between model and biochemistry. A major disadvan- tage of such a full-scale model stems from its large number of parameters. A large number of parameters, compared with the information available from experi- ments, makes the model unidentifiable. This means that there are an infinite number of parameter combinations Keywords core model; glycolysis; Hopf bifurcation; model optimization; model reduction Correspondence H. Schmidt, Fraunhofer Chalmers Research Centre for Industrial Mathematics, Sven Hultins gata 9D, S-41288 Gothenburg, Sweden E-mail: henning@fcc.chalmers.se Note The mathematical models described here have been submitted to the Online Cellular Systems Modelling Database and can be accessed free of charge at http://jjj.biochem. sun.ac.za/database/hynne/index.html, http://jjj.biochem.sun.ac.za/database/dano1/ index.html, http://jjj.biochem.sun.ac.za/ database/dano2/index.html and http://jjj. biochem.sun.ac.za/database/dano3/index. html (Received 8 June 2006, revised 22 August 2006, accepted 31 August 2006) doi:10.1111/j.1742-4658.2006.05485.x The complexity of full-scale metabolic models is a major obstacle for their effective use in computational systems biology. The aim of model reduction is to circumvent this problem by eliminating parts of a model that are unimportant for the properties of interest. The choice of reduction method is influenced both by the type of model complexity and by the objective of the reduction; therefore, no single method is superior in all cases. In this study we present a comparative study of two different methods applied to a 20D model of yeast glycolytic oscillations. Our objective is to obtain bio- chemically meaningful reduced models, which reproduce the dynamic prop- erties of the 20D model. The first method uses lumping and subsequent constrained parameter optimization. The second method is a novel approach that eliminates variables not essential for the dynamics. The applications of the two methods result in models of eight (lumping), six (elimination) and three (lumping followed by elimination) dimensions. All models have similar dynamic properties and pin-point the same interactions as being crucial for generation of the oscillations. The advantage of the novel method is that it is algorithmic, and does not require input in the form of biochemical knowledge. The lumping approach, however, is better at preserving biochemical properties, as we show through extensive analy- ses of the models. Abbreviations ACA, acetaldehyde; ADH, alcohol dehydrogenase; BPG, 1,3-bisphosphoglycerate; DHAP, dihydroxyacetone phosphate; ENVA, elimination of nonessential variables; F6P, fructose 6-phosphate; FBP, fructose 1,6-bisphosphate; G6P, glucose 6-phosphate; GAP, glyceraldehyde-3- phosphate; GAPDH, glyceraldehyde-3-phosphate dehydrogenase; Glc, glucose; HK, hexokinase; LASCO, lumping and subsequent constrained optimization; ODE, ordinary differential equation; PFK, phosphofructokinase; PGK, phosphoglycerate kinase; PK, pyruvate kinase; Pyr, pyruvate; trioseP, triosephosphates. 4862 FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS which gives rise to virtually identical agreements with the data [1]. Therefore, it is impossible to decide which parts of the model’s predictions are well-supported, and which are more or less arbitrary. In this way, model complexity renders many of the advantages of the one-to-one correspondence useless [2]. Large num- bers of variables and reactions are also associated with problems regarding, for example, numerics and model analysis [3]. A number of model-reduction techniques for com- plex chemical kinetics have been developed in order to deal with such problems. As reviewed by Okino & Mavrovouniotis [3], most model-reduction techniques fall into three classes: lumping methods, techniques based on sensitivity analysis, and timescale-based tech- niques. Lumping is, probably, the most widely used technique. It returns reduced models with new varia- bles corresponding to pools of the original variables. The new model structure is usually formed by bio- chemical intuition of very fast or very slow reactions (e.g. [4]), and this is the main reason why it is so com- mon. However, pooling can also be based on some systematic analyses of, for example, the correlation between the variables [5]. Sensitivity-analysis-based methods use sensitivity analysis to identify those parts of a model that are (locally) unimportant for the prop- erty of interest, and these parts are then eliminated [3,6–8]. Timescale-based methods are applicable if there are processes in the model occurring at timescales widely different from the one of interest. If processes occur at considerably slower timescales they are neg- lected, and if they occur at considerably faster time- scales they are projected to low-dimensional manifolds [3,9–11]. An example of a model-reduction technique that does not fall into one of these three classes is bal- anced truncation. It is widely used within control engineering [12,13]. This method has the advantage that it is optimal for the preservation of a given input– output property. It is not used so much in biochemical modelling because the reduced models have state varia- bles, which lack a biochemical interpretation. One way around this problem is to apply the method to the per- ipheral parts of a model (‘the environment’), possibly using other methods to reduce the central part [14,15]. The existence of such widely different reduction methods is explained by the fact that no single method is superior in all cases. The applicability of a method depends on both the objective with the reduction, and on the type of complexity in the original model. There- fore, test case studies comparing the consequences of different model-reduction methods are of interest. We chose the cyanide-induced glycolytic oscillations observed in starved yeast cells, because this is a partic- ularly well-studied biochemical model system. The experimental system has been thoroughly characterized in terms of both biochemistry and dynamics [16–23], and this has led to a number of mathematical models of this system [4,24–30]. Our 20D model [30] is a full-scale model that des- cribes the system in detailed biochemical terms. It is in quantitative agreement with almost all experimental observations, but it suffers from the above-mentioned general problems of detailed models. In contrast, we have shown that the persistent oscillations can be des- cribed as a 2D phenomenon [20,31]. Even though the structure of the biochemical reaction network is not at all present in this 2D model (Eqn 1), it is possible to obtain biochemical interpretations of the two modes involved in the oscillatory dynamics [31]. This raises the general question to what extent can a full-scale model be reduced to a smaller biochemically meaning- ful model, with preserved basic dynamic properties? This study addresses this question for the specific test case of the 20D model developed in Hynne et al. [30]. The most basic dynamic property of the 20D model is oscillations. The dynamic structure underlying these oscillations has been characterized further both experi- mentally [20] and in the 20D model [30]. In both cases it has been found that the system is close to the onset of oscillations, and that this transition between station- ary and oscillatory behaviour is a supercritical Hopf bifurcation. The closeness to a Hopf bifurcation implies that the system’s persistent dynamics is gov- erned by the normal form of the Hopf bifurcation, also known as the Stuart–Landau equation [32,33]: dz dt ¼ðix 0 þ rlÞz þ gzjzj 2 : ð1Þ In Eqn (1), z is a complex variable that describes the state of the system in a local coordinate system of the oscillatory plane. The distance from the bifurcation point is given by the real parameter l; the bifurcation point is found at l ¼ 0 (When we use a nondimension- less parameter p as the bifurcation parameter, then the dimensionless bifurcation parameter l is calculated as l ¼ (p ) p 0 ) ⁄ p 0 with p 0 being the value of p at the bifurcation point.) The real parameter x 0 is the fre- quency of oscillations at the bifurcation point, and the imaginary part of the complex parameter r determines the l-dependency of the frequency at the stationary state. Re(r) determines the l-dependency of the linear stability, and hence the direction of the bifurcation. The complex nonlinearity parameter g determines the properties of the limit cycle, which is born in the Hopf bifurcation. In particular, the Hopf bifurcation is supercritical if Re(g) < 0 and subcritical if Re(g)>0. S. Danø et al. A case study in model reduction FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS 4863 The Stuart–Landau equation is the 2D description dis- cussed above, and the parameters x 0 , r and g can be calculated from a full-scale model at a Hopf bifurca- tion [34]. As such, it provides a firm connection between the full-scale model and its basic dynamic properties. In this study we evaluate two model-reduction meth- ods. The first is based on lumping and subsequent con- strained optimization (LASCO); it is the optimization step which involves new stages. The second is the elimination of nonessential varia- bles (ENVA). This is a new model-reduction method with a philosophy similar to that of sensitivity analy- sis-based methods: it eliminates the dynamics of varia- bles that are nonessential for the basic dynamic properties. Starting from the comprehensive and relatively com- plex 20D model, the two different methods result in two different reduced models. The lumped model is then further reduced, resulting in a total of three reduced models. These processes, and the resulting models, are described in the first part of the Results. Because we wish to investigate the consequences of model reduction, we compare the biochemical proper- ties of the models. This analysis constitutes the last part of the Results. Because the nature of the work, we have chosen to integrate the Experimental Proce- dures section with the description of the results. Addi- tional information is available in the Supplementary material. The mathematical models described here have been submitted to the Online Cellular Systems Modelling Database and can be accessed at http://jjj.biochem.sun. ac.za/database/hynne/index.html, http://jjj.biochem.sun.ac. za/database/dano1/index.html, http://jjj.biochem.sun.ac. za/database/dano2/index.html and http://jjj.biochem. sun.ac.za/database/dano3/index.html free of charge. Results Model reductions When performing the model reductions, we aimed to preserve the models’ dynamic properties. The main dynamic feature is the oscillations. Subsequently, we aimed to preserve the closeness to a supercritical Hopf bifurcation, when the mixed flow glucose concentration [Glc x ] 0 is used as bifurcation parameter [20] (Glc, glu- cose). If these two properties are preserved, the model is said to be in qualitative agreement with the 20D model (as well as the experimental observations). Good quantitative agreement is also said to be found when the Stuart–Landau parameters, i.e. parameters x 0 , r and g of Eqn (1), are in reasonable agreement with those of the 20D model. Figure 1 provides an overview of the models devel- oped here. Two model-reduction strategies are applied. The first, LASCO, is the elimination of nonessential reactions, commonly known as lumping. The other, ENVA, is the elimination of nonessential variables. Starting with the 20D model [30] (Fig. 2), we use LASCO to arrive at the 20L8D model, and ENVA to arrive at the 20E6D model. The 20L8D model was fur- ther reduced by ENVA, resulting in the 20LE3D model. We now describe the three model reductions in detail, and present the resulting models. Construction of the 20L8D model by LASCO A traditional approach to model reduction is lumping. In essence, a simpler model structure is obtained by lumping a number of reactions together and assuming some reasonable overall rate expression to describe the combined kinetics of the lumped reactions. We present the model-reduction method LASCO. It ensures that the dynamic properties of a lumped model are in good agreement with those of the parent model. The model structure is obtained from traditional lump- ing, and the parameters are subsequently optimized (‘fitted’) using a highly constrained optimization method [30,35,36]. Use of this powerful optimization strategy for model reduction is the novelty of our approach. In the context of glycolytic oscillations in yeast cells, Wolf & Heinrich proposed a biochemically formulated of variables elimination 20E6D model 20D model 20L8D model & fitting lumping of variables elimination 20LE3D model Fig. 1. Overview of the model reductions. The 20D model is the model described in Hynne et al. [30]. The 20E6Dmodel was con- structed from the 20D model using ENVA as described in the text. The 20L8D model was constructed using LASCO. For this purpose, we adopted a modified version (see text) of the 7D model by Wolf & Heinrich [4]. Subsequently, we adjusted the intrinsic parameters so that the dynamic properties of the 20L8D model are as similar as possible to those of the original 20D model. Details of this pro- cedure are given in the text. Application of ENVA reduced the 20L8D model to the 20LE3D model. A case study in model reduction S. Danø et al. 4864 FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS 7D model [4]. We have adopted this model structure here, as an example of a lumped model structure. (Brusch et al. [37] have previously developed a modi- fied version of the model by Wolf and Heinrich with the same purpose as we have here. We, however, find this model unsuitable due to problems concerning the formulation of the model.) In order to be able to map the 20D model onto the reduced model in a straight- forward manner, we made the following modifications to the model structure: extracellular glucose was intro- duced as an additional species, glucose transporter kin- etics (r ¼ GlcTrans) and glucose flows in and out of the reactor (r ¼ inGlc) were added, a glycogen-produ- cing side branch was added (r ¼ storage), and the removal of extracellular acetaldehyde (ACA) (r ¼ out- ACA) was changed so that it is now formally com- posed of the ACA leaving the reactor with the outflow, and the ACA being removed by reactions in the extracellular medium. This resulted in the 20L8D model structure shown in Table 1 and Fig. 3. The cor- responding ordinary differential equations (ODEs) are constructed from Table 1 according to y s dc s dt ¼ X r m sr v r ð2Þ where r and s denote reactions and species, respect- ively, and y s ¼ V extracellular ⁄ V cytosolic ¼ y vol (i.e. the ratio of the extracellular volume to the cytosolic) if s is an extracellular species and y s ¼ 1 for intracellular s. c s is the concentration of s, v r is the rate of reaction r and v sr is the stoichiometric coefficient of species s in reaction r. Neither the original 7D nor the new 8D model structure have dynamic properties similar to the 20D model (or the experiments) when the parameter values of Wolf and Heinrich are inserted. We therefore per- formed a parameter optimization in order to achieve this. The 8D model structure has a limited number of intrinsic parameters: K trans , q, K i , y vol and [Glc x ] 0 . (Intrinsic parameters are those that are not scalar mul- tipliers of the rate expressions [30,35,36]). This allows us to perform parameter optimization in an efficient and unique manner, which we now explain. We first parameterize the velocity parameters (i.e. the non- intrinsic parameters) k 0 , V 1 , V 2 , k 3 , , k 10 in terms of the stationary fluxes and concentrations of the desired Fig. 2. Reaction network of the 20D model described in Hynne et al. [30]. Extracellular species and reactions are shown in red. Table 1. Model structure of the 20L8D model. The two stoichio- metric constraints A tot ¼ [ATP] + [ADP] and N tot ¼ [NADH] + [NAD + ] reduce the dimension of the model to eight. The corres- ponding set of ODEs is constructed according to Eqn (2). Param- eter values are listed in Table S1. Reaction r Rate expression v r inGlc: Ð Glc x k 0 ([Glc x ] 0 ) [Glc x ]) GlcTrans: Glc x fi Glc V 1 ½Glc x  K trans þ½Glc x  HK–PFK: Glc + 2 ATP fi 2 trioseP + 2 ADP V 2 ½Glc½ATP 1þ ½ATP K i  q GAPDH: trioseP + NAD + fi BPG + NADH k 3 [trioseP] [NAD + ] lowpart: BPG + 2 ADP fi ACA + 2 ATP k 4 [BPG] [ADP] ADH: ACA + NADH fi NAD + k 5 [ACA] [NADH] ATPase: ATP fi ADP k 6 [ATP] storage: Glc + 2 ATP fi 2 ADP k 7 [Glc] [ATP] glycerol: trioseP + NADH fi NAD + k 8 [trioseP] [NADH] difACA: ACA Ð ACA x k 9 ([ACA] ) [ACA x ]) outACA: ACA x fi (k 0 + k 10 ) [ACA x ] S. Danø et al. A case study in model reduction FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS 4865 operating point and the intrinsic parameters [30,37]. For example, V 1 ¼ m GlcTrans ðK trans þ½Glc x ]Þ ½Glc x  . We then fix the concentrations, y vol , [Glc x ] 0 flux distribution, and flux magnitude at the corresponding values of the 20D model at the supercritical Hopf bifurcation point. This ensures that the 20L8D model has the same operating point and stationary state as the 20D model for any combination of the remaining free parameters in the optimization. These are K trans , q and K i . We then scan the Hopf bifurcation manifold in this 3D parameter space by means of the continuation software cont [38], and choose the parameter set which yields the best quantitative agreement between the dynamics of the 20D and the 8D models. Here we base this quanti- tative comparison of dynamic properties on the Stu- art–Landau parameters x 0 , r and g (Eqn 1). (For ease of comparison we choose [Glc x ] 0 as the bifurcation parameter as in Hynne et al. [30]). Because we are left with only three free parameters, and are constrained by a 2D Hopf manifold, it is possible to obtain a com- plete overview of the parameter space. This allows us to choose a unique parameter set which results in opti- mal agreement with the dynamic properties of the 20D model. In this sense, the resulting 20L8D model is unique. Details of the optimization are given in the supplementary material (Doc. S1). The final set of parameters (11 velocity parameters and seven intrinsic parameters) is given in Table S1 and Table 6. In the cause of the optimization we noticed a remarkable property of the reduced model. When con- strained by the operating point and the Hopf manifold, the frequency of oscillation and the right critical eigen- vector (i.e. the complex, right eigenvector associated with the complex conjugate eigenvalues which have zero real part) are both constant under the variation of the remaining free parameters (K trans , q and K i ). This implies that the constraints and the model structure in combination dictate the frequency of the oscillation as well as the relative amplitudes and phases of the species (properties of the right critical eigenvector). The frequency of oscillation does, however, change if the operating point is changed. The 20L8D model is constructed from the 20D model by lumping a number of reactions. Consequently, the variables of the 20L8D model refer to metabolite pools rather than to the act- ual metabolites. For the model developed here, a rea- sonable interpretation of this is [Glc] 20L8D ¼ [Glc] + [G6P] + [F6P], [trioseP] 20L8D ¼ [FBP] + [DHAP] + [GAP] and [ACA] 20L8D ¼ [Pyr] + [ACA] instead of the literal interpretation [Glc] 20L8D ¼ [Glc], [trioseP] 20L8D ¼ [DHAP] + [GAP] and [ACA] 20L8D ¼ [ACA]. The per- iod of the oscillations is 38 s in the 20D model, and the 20L8D model has a period of 7.2 s with the literal inter- pretation of concentrations and a period of 22 s with the concentrations pooled as indicated. Consequently, we performed the parameter optimization at the operating point with pooled concentrations (see Tables 6–8 of Hynne et al.) [30]. Construction of the 20E6D model by ENVA Model reduction by means of lumping often relies on ‘biochemical intuition’ to choose the relevant reduced- model structure, although this need not be the case [5,39,40]. We present ENVA as an alternative approach to model reduction, where the system’s basic dynamic properties are used as a guide for the elimin- ation of variables that are not essential for the dynam- ics. We eliminate a variable by fixing the metabolite concentration at its steady-state value at a particular operating point of the original model. A systematic search is performed among the possible models in order to identify the model of lowest dimension, which retains the basic dynamic properties of the full system. The basic dynamic properties of each of the reduced Fig. 3. Reaction network of the 20L8D model. Extracellular species and reactions are shown in red. A case study in model reduction S. Danø et al. 4866 FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS models are evaluated by calculating the eigenvalues of the Jacobian matrix at a particular stationary state, common to all models. In this case, the basic dynamic property is oscillation. Models with an oscillatory mode are identified as those with one or more sets of complex conjugate eigenvalues, and oscillatory models are identified as those with complex conjugate eigen- values with positive real parts. This is standard nonlin- ear dynamics theory [41]. Because we seek model(s) of the lowest dimension, which retains the basic dynamic properties of the full system, it is not necessary to search all 2 N possible models. Instead, we first search all 2D models, then all 3D models, etc. until one or more satisfactory n-dimensional models have been found. Hence P n k ¼2 ð N k Þ models are searched. In the construction of the 20E6D model N ¼ 20 and n ¼ 6, so the properties of 60 439 models were tested. This analysis was done at the stationary state defined by the mixed flow glucose concentration [Glc x ] 0 ¼ 24 mm and all other parameters as in Hynne et al. [30]. Calculations were performed using the program cont [38] and customized Perl scripts. Table 2 lists the 5D and 6D models with complex eigenvalues. None of the models with a lower dimension has complex eigen- values. The only model with complex eigenvalues with positive real parts, and hence the only one that shows oscillations at the chosen operating point, is the 6D model with [ATP], [ADP], [BPG], [FBP], [GAP] and [DHAP] as variables (BPG, 1,3-bisphosphoglycerate). We choose this model structure as our reduced model. Its structure consists of the reactions involving one or more of these species (Table 3, Fig. 4). The eliminated variables must be represented in the reduced model. This can be done in several ways. For example, the quasi steady-state approximation can be applied for each of the eliminated variables, or they can simply be fixed at their steady-state values. In this study we want the models to become as simple as poss- ible, and we therefore take the last approach, which results in simpler rate equations. When N ) n ¼ 14 species are fixed at their steady-state values, we intro- duce 14 conservation-of-mass relations. In order to ensure self-consistency, we must make sure that all these conservation-of-mass relations are fulfilled within the framework of the model. Some of these relations are external to the reduced model (Table 3) and do not call for any action. Others have both internal and external parts. For example, d½NADH dt ¼ 0 ¼ v glycerol þ v ADH À v GAPDH has the external part v ADH and the internal parts v glycerol and v GAPDH . We deal with these cases simply by assuming that the external parts balance the equa- tions. The remaining two relations Table 2. Results of our search for minimal, oscillatory models with nonessential variables eliminated. The model structure is described by a sequence of 1s and 0s. 1 indicates that the corresponding metabolite is a dynamic variable in the model, 0 that it is fixed at its steady-state concentration. The corresponding ordered sequence is {ADP, ATP, BPG, FBP, G6P, F6P, NADH, DHAP, GAP, PEP, ACA, Glc, ACA x , Pyr, EtOH, EtOH x , glycerol, glycerol x , Glc x ,CN À x }. All 1D to 6D models, that have oscillatory modes, are shown; the first two models are 5D, the remaining 30 are 6D. The model in bold is the only one with complex eigenvalues with positive real parts, and hence the only one which shows oscillations at this operating point. The operating point is defined by [Glc x ] 0 ¼ 24 mM and all other parameters as Hynne et al. [30]. Model structure Complex eigenvalues Model structure Complex eigenvalues 11110000100000000000 )6.28 ± 4.98 i 11110000100000000001 )6.28 ± 4.98 i 11101100000000000000 )3.25 ± 9.86 i 11101110000000000000 )0.42 ± 7.53 i 11111100000000000000 )3.25 ± 9.86 i 11101101000000000000 )3.25 ± 9.86 i 11111000100000000000 )6.08 ± 4.42 i 11101100100000000000 )2.73 ± 9.57 i 11110100100000000000 )7.12 ± 5.32 i 11101100010000000000 )5.08 ± 10.6 i 11110001100000000000 0.95 ± 7.91 i 11101100001000000000 )3.25 ± 9.86 i 11110000110000000000 )7.23 ± 12.9 i 11101100000100000000 )3.27 ± 9.84 i 11110000101000000000 )6.28 ± 4.98 i 11101100000010000000 )3.25 ± 9.86 i 11110000100100000000 )6.28 ± 4.97 i 11101100000001000000 )3.25 ± 9.86 i 11110000100010000000 )6.28 ± 4.98 i 11101100000000100000 )3.25 ± 9.86 i 11110000100001000000 )6.28 ± 4.98 i 11101100000000010000 )3.25 ± 9.86 i 11110000100000100000 )6.28 ± 4.98 i 11101100000000001000 )3.25 ± 9.86 i 11110000100000010000 )6.28 ± 4.98 i 11101100000000000100 )3.25 ± 9.86 i 11110000100000001000 )6.28 ± 4.98 i 11101100000000000010 )3.25 ± 9.86 i 11110000100000000100 )6.28 ± 4.98 i 11101100000000000001 )3.25 ± 9.86 i 11110000100000000010 )6.28 ± 4.98 i 01110011010000000000 )168 ± 0.197 i S. Danø et al. A case study in model reduction FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS 4867 d½PEP dt ¼ 0 ¼ v lpPEP À v PK ð3Þ d½G6P dt ¼ d½F6P dt ¼ 0 ¼ v HK À v storage þ v PFK ð4Þ need more careful attention. Equation (3) demands that v lpPEP is substituted by v PK or vice versa, and Eqn (4) demands substitution of v HK ,v storage or v PFK . We tested the six possible combinations of solutions to Eqns (3) and (4). The choice of solutions is unique, because only one of the combinations retains the ability to oscillate. This combination is substitution of v PK with v lpPEP and of v HK with v storage +v PFK . The resulting model is the 20E6D model defined by Table 3 and the rate expres- sions in Table 4. The corresponding ODEs are con- structed from the tables according to dc s dt ¼ P r m sr v r . Elimination of variables allowed us to lump a number of parameters as indicated in Table 4. All the underly- ing parameter values are the same as in the 20D model, i.e. no parameter optimization was done in the elimin- ation of variables approach. The model’s parameters are listed in Table S2. With a 20D model as the starting point it is feas- ible to do a complete scan of all the possible reduced models that can be constructed by elimination of vari- ables. This will generally not be the case, however, because the number of possible combinations grows exponentially with the number of variables. As des- cribed in Schmidt & Jacobsen [42], interaction analy- sis can be used to rank the interactions among the chemical species in a full-scale model according to their importance for the occurrence of oscillations. As such, the ranking identifies the oscillating core of the model [42]. This ranking can be used to restrict the combinatorial search to the most relevant species. This is done simply by sequentially fixing the least important species at their steady-state concentrations until the point at which the oscillations are lost upon further sequential elimination. The combinatorial search need now only be performed for this reduced model, where most of the nonessential species have already been eliminated. For the 20D model, we find oscillations when the nine most important species are retained. The ordered list of Table 2 corresponds to the ordering according to decreasing importance of the species (Fig. 7, upper left). It is seen from Table 2 that the unique oscillatory 6D model is indeed found within the subset of the nine most important species. (This would have reduced the number of model evaluations in the search from 60 439 to 445). Construction of the 20LE3D model by ENVA We use ENVA to construct the 20LE3D model from the 20L8D model at the operating point of the 20L8D model defined by [Glc x ] 0 ¼ 24 mm. Of the reduced models with complex eigenvalues, those of lowest dimensionality are 3D; we find one with positive real parts of the complex eigenvalues and one with Table 3. The model structure of the 20E6D model. The stoichio- metric constraint A tot ¼ [ATP] + [ADP] + [AMP] reduces the dimen- sion of the model to six. Reaction r HK a : ATP fi ADP PFK: ATP fi ADP + FBP ALD: FBP Ð GAP þ DHAP TIM: DHAP Ð GAP GAPDH: GAP Ð BPG lpPEP ADP þ BPG Ð ATP PK b : ADP Ð ATP glycerol: DHAP fi storage: ATP fi ADP ATPase: ATP fi ADP AK: ATP þ AMP Ð 2ADP a The rate expression of the hexokinase reaction has been substi- tuted according to vi HK ¼ v storage +v PFK . b The rate equation of the pyruvate kinase reaction has been substituted according to m PK ¼ m lpPEP . This makes the reaction reversible. See text for details. Fig. 4. Reaction network of the 20E6D model. The rate expressions of the reactions shown in blue have been substituted in order to insure conservation of mass. See text for details. A case study in model reduction S. Danø et al. 4868 FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS negative. This uniquely identifies the model of lowest dimension which shows oscillations at the chosen oper- ating point. The self-consistency of this model is insured by noting that all five conservation-of-mass relations have external parts. The model structure is shown in Table 5 and Fig. 5. The elimination of varia- bles allowed us to lump a number of parameters (Table 5), but no parameter optimization was carried out. The model’s six velocity parameters and six intrin- sic parameters are listed in Table S3. As was the case with the construction of the 20E6D model, a search within the subset of reduced models defined by the ranking of the species according to their decreasing importance, successfully identifies the oscil- latory, reduced model of lowest dimension. Model properties We judge the effects of the model reductions by compar- ing the dynamic and biochemical properties of the ori- ginal 20D model to those of the three reduced models. Table 4. Rate expressions of the 20E6D model. The reaction names r refer to Table 3. The model reduction allowed us to lump some of the parameters (indicated by ~), but the underlying parameters are as in the parent 20D model. A list of the parameter values is given in Table S2. r Rate expression v r HK: ~ V 5m ~ K 5 þ ½ATP ½AMP  2 þ ~ k 22 ½ATP PFK: ~ V 5m ~ K 5 þ ½ATP ½AMP  2 ALD: V 6m ½FBPÀ ½GAP½DHAP K 6eq  K 6FBP þ½FBPþ ½FBP½GAP K 6IGAP þ ~ K 6 ½GAPK 6DHAP þ½DHAPK 6GAP þ½GAP½DHAPðÞ TIM: V 7m ½DHAPÀ ½GAP K 7eq  K 7DHAP þ½DHAPþ K 7DHAP ½GAP K 7GAP GAPDH: ~ V 8m ½GAPÀ½BPG= ~ K 8eq ðÞ 1þ ½GAP K 8GAP þ ½BPG K 8BPG lp PEP : k 9f ½BPG½ADPÀ ~ k 9r ½ATP PK: K 9f ½BPG½ADPÀ ~ k 9r ½ATP glycerol: ~ V 15m ½DHAP ~ K 15 þ½DHAP storage: ~ k 22 ½ATP ATPase: k 23 ½ATP AK: k 24f ½AMP½ATPÀk 24r ½ADP 2 Table 5. Model structure of the 20LE3D model. The stoichiometric constraint A tot ¼ [ATP] + [ADP] reduces the dimension of the model to three. The corresponding set of ODEs are constructed according to dc s dt ¼ P r v sr m r : The model reduction allowed us to lump some of the parameters (indicated by ~), but the underlying parameters are as in the parent 20L8D model. A list of the param- eter values is given in Table S3. Reaction r Rate expression v r HK–PFK: 2ATP fi 2trioseP + 2ADP ~ V 2 ½ATP 1þ ½ATP K i  q GAPDH: trioseP fi BPG ~ k 3 ½trioseP lowpart: BPG + 2ADP fi 2ATP k 4 [BPG][ADP] ATPase: ATP fi ADP k 6 [ATP] storage: 2ATP fi 2ADP ~ k 7 ½ATP glycerol: trioseP fi ~ k 8 ½trioseP Fig. 5. Reaction network of the 20LE3D model. S. Danø et al. A case study in model reduction FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS 4869 Stuart–Landau parameters and location of Hopf bifurcations The dynamic properties of the models are reflected by their Stuart–Landau parameters (Eqn 1) at the Hopf bifurcations found with [Glc x ] 0 as bifurcation param- eter. In cases where the [Glc x ] 0 parameter has been eliminated, we used the parameter that corresponds most closely to it biochemically. Table 6 shows the Stuart–Landau parameters; all the Hopf bifurcations are supercritical. The 20L8D model matches the original model quite well, but this is only because of the extensive parameter optimization of LASCO. The 20E6D model also mat- ches the original model quite well at its lower (C G6P ¼ 11 : 4 lm) Hopf bifurcation; it is remarkable that this is obtained without any parameter optimization. The existence of its upper (C G6P ¼ 5.6 mm) bifurcation is in qualitative disagreement with the original model. The 3D model was also constructed by ENVA. Again, we note that the Stuart–Landau parameters are close to those of the parent model, i.e. the 20L8D model. Biochemical components of the oscillatory plane Another measure of the models’ dynamic properties is their polar phase plane plots (Fig. 6). As described in detail elsewhere [43], such plots indicate the biochemi- cal composition of the Stuart–Landau modes, i.e. the two sets of metabolites which correspond to the two modes generating persistent oscillations. This, in turn, indicates the nature of the interactions underlying the dynamic structure of the system. The two modes are characterized by a phase difference of 90°. The leading mode is an activator, promoting the formation of the lagging mode, and the lagging mode is an inhibitor of Table 6. Stuart–Landau parameters of the models. The table lists the Stuart–Landau parameters (Eqn 1) of the Hopf bifurcations found on the borders of the oscillatory region. Re(r) and Im(r) determine the rates at which the linear instability of the stationary state and the frequency of oscillations at the stationary state, respectively, increase with the bifurcation parameter l. The addi- tional frequency change caused by amplitude changes is deter- mined by Im(g) ⁄ Re(g). The bifurcation parameter l is ([Glc x ] 0 ) [Glc x ] 0 , bif ) ⁄ [Glc x ] 0 , bif in the 20D and the 8D models, but because this parameter has been eliminated from the 20E6D and 20LE3D models, we used l ¼ (C G6P ) C G6P,bif ) ⁄ C G6P,bif and l ¼ (C Glc ) C Glc,bif ) ⁄ C Glc,bif , respectively, as proxies. (C s is the steady-state concentration of species s; see the parameter listings in the Sup- plementary material.) Because of this change in l, the Re(r) and Im(r) values of these models cannot be directly compared with those of the other models (indicated by the parentheses). All the Hopf bifurcations are supercritical. Model Location x 0 (min )1 ) Re(r) (min )1 ) Im(r) (min )1 ) Im(g) ⁄ Re(g) 20D [Glc x ] 0,bif ¼ 18.5 mM 10 1.1 )2.1 1.4 20L8D [Glc x ] 0,bif ¼ 18.5 mM 17 1.0 1.8 1.5 20E6D C G6P,bif ¼ 11.4 lM 15 (0.18) (0.028) 2.4 20E6D C G6P,bif ¼ 5.61 mM 8.9 ()27) ()6.3) 15 20LE3D C Glc,bif ¼ 6.12 mM 18 (4.1) (8.1) 1.0 FB P AT P DHAP F6P ADP GAP G6 P 20D trioseP ATP glc 20L8D DHAP ADP FBP ATP 20E6D trioseP BPG AT P 20LE3D Fig. 6. Biochemical components of the oscillatory plane: polar phase plane plots. For each of the four models, the plot is shown with and without annotations. The plots are polar plots and each dot corresponds to a species; the radius indicates its amplitude, and the angle its phase. We indicate either the phase of the maximum of the oscillation (d) or the phase of the minimum (s). All plots show the same inter- pretation of the Stuart–Landau modes. A case study in model reduction S. Danø et al. 4870 FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS the leading mode. In the framework of the Stuart– Landau equation (Eqn 1), the leading and the lagging modes can be thought of as the real and the imaginary parts of the complex variable z, respectively. Figure 6 shows that all four models have similar polar phase plane plots, indicating that the underlying dynamic structures of the models are similar. With the interpretation given in the plots, the activating mode corresponds to low energy charge, and the inhibitory mode corresponds to substrate for the lower part of glycolysis. Low energy charge promotes substrate for the lower part of glycolysis via allosteric activation of phosphofructokinase (PFK), and substrate for the lower part of glycolysis inhibits low energy charge by increasing ATP production in the phosphoglycerate kinase (PGK) and pyruvate kinase (PK) reactions [31]. In conclusion, Fig. 6 shows that the reduced models can be seen as depicting the oscillatory core of the full- scale model. Because this is one of the main objectives, this is a strong indication that the reduction proce- dures are good choices for reduction to an oscillating core. Interaction analysis The nature of the regulatory mechanisms underlying the oscillations can be assessed by ranking the species according to the importance of the feedback loops they are involved in. We do this by employing a slightly modified version of one of the methods presented in Schmidt & Jacobsen [42]. The basis of the method is the fact that the appearance of complex behaviour, such as bistability and oscillations, can be traced back to changes in the local stability properties of the sys- tem’s steady state. In the case of autonomous oscilla- tions, the underlying steady state is an unstable focus. This is reflected by the Jacobian matrix of the steady state, which has at least one pair of unstable conjugate complex eigenvalues. For each species, there exists a feedback loop conveying its effect on the other species of the system. For each of these feedback loops, the original method determines the minimal, real valued, relative perturbation required for stabilization of the linear system, which corresponds to moving the unsta- ble conjugate complex eigenvalues into the stable half plane. Another scenario leading to the disappearance of the oscillations, however, is when the unstable conju- gate complex eigenvalues are moved onto the real axis, stable or not. This possibility was not considered in the original method, which we have now modified to take this latter scenario into account. Because each feedback loop corresponds to a particular species, the importance ranking is obtained by ranking the species according to the smallness of the calculated minimal perturbations. We carried out the analysis at the operating point corresponding to [Glc x ] 0 ¼ 24 mm. The computations were performed using the Systems Biology Toolbox [44] for matlab (MathWorks, Natick, MA). The reader is referred to Schmidt & Jacobsen [42] for a detailed description of the analysis method. The importance rankings of the species are shown in Fig. 7. The most important species of the 20D model (Fig. 7, upper left) are ADP and ATP. The high importance ranking of BPG probably reflects its importance for ATP production in the lower part of glycolysis. The following six species are ranked almost equally important. This fits their localization around the central part of glycolysis. The importance ranking matches the polar phase plane plot analysis above, and the conclusions of Madsen et al. [31]. This is partic- ularly so when it is noted that the high importance ranking of BPG is due to its very low average concen- tration, which results in a high relative amplitude. In broad terms, the ranking is conserved in the model reductions. In the process of lumping and fitting which leads to the 20L8D model, the relative importance of BPG is increased, whereas the importance ranking of the pooled species Glc and trioseP is in good agree- ment with that of the corresponding species in the 20D model (Fig. 7). Further reduction of the 20L8D model to the 20LE3D model preserves the ranking. Reduc- tion of the 20D model to the 20E6D model preserves the relative ranking of ATP, ADP, BPG and FBP, whereas GAP is now ranked more important than DHAP (Fig. 7, lower left). It is interesting to note that some of the variables in the 20E6D and 20LE3D models do not have very high importance values, even though these models were constructed by the elimination of nonessential varia- bles. We suggest that such variables are essential for the connectivity of the network, rather than for the generation of the oscillations per se. Flux control The flux-control pattern is an important biochemical property. We determine this pattern using metabolic control analysis as described previously [45]. The ana- lyses are carried out at the (lower) Hopf bifurcation points of the models (Table 6). This allows the flux- control analysis to be compared with the analysis of frequency, stability and amplitude control below. The metabolic control analysis calculations were performed with the Systems Biology Toolbox [44] for matlab. S. Danø et al. A case study in model reduction FEBS Journal 273 (2006) 4862–4877 ª 2006 The Authors Journal compilation ª 2006 FEBS 4871 [...]... is available online: Doc S1 Parameter optimization of the constrained 20L8D model Table S1 Parameters of the 20L8D model Table S2 Parameters of the 20E6D model Table S3 Parameters of the 20LE3D model Table S4 Unstable stationary state of the 20L8D model provided as a check of model implementations Table S5 Unstable stationary state of the 20E6D model provided as a check of model implementations Table.. .A case study in model reduction S Danø et al Fig 7 Comparison of species’ importance rankings The heights of the bars indicate the importance of the feedback loops associated with each of the species, as determined by interaction analysis es is the smallest scalar perturbation of the linear feedback of species s which causes the unstable complex conjugate eigenvalues of the Jacobian to disappear... biochemically meaningful, models that reproduce the basic dynamic properties The strength of ENVA is that it is algorithmic, and that it does not require any input in the form of biochemical knowledge A major advantage of LASCO, however, seems to be that it results in models with more well-preserved biochemical properties We have shown these statements through extensive analysis of the resulting models Acknowledgements... steady-state approximation for each of the eliminated variables This solves the self-consistency problem in an elegant way, but results in prohibitively complicated rate expressions in our specific case In cases of large parent models with very simple rate expressions, application of the quasi steady-state approximation will probably be advantageous We evaluated the two reduction methods by comparing the properties. .. oscillations was shown several decades ago, using minimal modelling [24–27,29] The models presented here are a verification of these results, but this study has the additional strength that the models were obtained through the reduction of a realistic full-scale model Consequently, they have, for example, more realistic parameter values, fluxes, and steady-state concentrations That the biochemical properties. .. stability and amplitude The importance of the different reactions for the oscillatory dynamics can be mapped out by performing sensitivity analysis at the (lower) Hopf bifurcations of the models (Table 6) By performing the analysis in the framework of the Stuart–Landau equation (Eqn 1), it can be shown [43] that control of amplitude is equivalent to control of stability, and that reactions with a high... we want to point out that this feature might be useful when solving the general problem of model rejection LASCO can be applied at any stationary state, stable or unstable Comparison between the parent and the reduced model can be based on any property of a stationary state, e.g control coefficients as determined in metabolic control analysis [45] or elements of the Jacobian matrix [46,47] A bifurcation... present, it can be exploited for efficient comparison of dynamic properties The second method, ENVA, performs a complete search among all possible combinations of eliminated variables From these, the reduced model is picked as the most highly reduced model, which retains the basic dynamic features of the original (here, oscillations) The search among candidate models constitutes a potential combinatorial problem... properties of the three reduced models and the parent model In short, we found that the dynamic structures of the models are similar, but that their biochemical properties are different That the dynamic structures are similar can be seen from Fig 6, which shows that the oscillatory modes of the four models have the same biochemical compositions The central feedback mechanisms between these modes are the allosteric... (2004) Model reduction and ¨ analysis of robustness for the Wnt ⁄ b-catenin signal transduction pathway Genome Informatics 15, 138– 148 Strogatz SH (1994) Nonlinear Dynamics and Chaos Addison-Wesley, Reading, MA Schmidt H & Jacobsen EW (2004) Linear systems approach to analysis of complex dynamic behaviours in biochemical networks IEE Systems Biol 1, 149–158 Danø S, Madsen MF & Sørensen PG (2005) Chemical . means of mathematical models with varying degrees of detail. A primary advantage of a detailed, biochemically formulated model is that a one-to-one comparison. Reduction of a biochemical model with preservation of its basic dynamic properties Sune Danø 1 , Mads F. Madsen 1 , Henning Schmidt 2 and Gunnar Cedersund 2 1