10th International IFAC Symposium on Computer Applications in Biotechnology Preprints Vol.2, June 4-6, 2007, Cancún, Mexico FROM METABOLIC NETWORKS TO MINIMAL DYNAMIC BIOREACTION MODELS Agn`esProvost ∗ and Georges Bastin ∗,1 Yves-Jacques Schneider ∗∗ ∗ Center for Systems Engineering and Applied Mechanics Universit´eCatholique de Louvain 4, Avenue G. Lemaitre, 1348 Louvain-La-Neuve, Belgium ∗∗ Institut des Sciences de la Vie, Universit´eCatholique de Louvain 4-5, Croix du Sud, 1348 Louvain-La-Neuve, Belgium Abstract: The paper deals with the design of minimal dynamic bioreaction models in the situation where (a) the model is based on the knowledge of a detailed underlying metabolic network, (b) measurements of extra-cellular species in the reactor are the only available measurements. A brief but rigorous presentation of the theory is first given. Then the approach is illustrated with the example of chinese hamster ovary cells cultivated in stirred flasks. Copyright c 2007 IFAC Keywords: Bioreaction, Convex basis, Elementary flux mode, Metabolic network, Model reduction 1. INTRODUCTION (e.g. Provost and Bastin (2004), Haag et al. (2005), Provost et al. (2006), Zhou et al. (2006)). The issue of quantitative bioprocess modelling Our concern in this paper is to give a brief but from extracellular measurements is a central issue rigorous presentation of the theory that grounds in bioengineering (e.g. Nielsen et al. (2002)). In this methodology. The goal is to design mini- classical macroscopic models, the biomass is just mal dynamic bioreaction models in the situation viewed as a catalyst for the conversion of sub- where (a) the model is based on the knowledge strates into products. The process is represented of a detailed underlying metabolic network, (b) by a set of so-called bioreactions that directly measurements of extra-cellular species in the re- connect the substrates to the products, without actor are the only available measurements. We making an explicit reference to the intracellular follow a systematic “model reduction” approach metabolism. that automatically produces a family of equivalent Recently, various publications have dealt with a models involving a minimal set of bioreactions new “macro-micro” approach that aims at linking while being fully compatible with the underlying the macroscopic model design to the metabolism metabolism and consistent with the available ex- perimental data. 1 This paper presents research results of the Belgian Furthermore, as a matter of illustration and mo- Programme on Interuniversity Attraction Poles, initiated tivation to the theory, we consider the example of by the Belgian Federal Science Policy Office. The scientific responsibility rests with its author(s). 321 chinese hamster ovary (CHO) cells cultivated in consumption fluxes, weighted by their stoichio- batch mode in stirred flasks (Ballez et al. (2004)). metric coefficients, is zero. This is expressed by the algebraic relation: Nv = 0 v > 0 (2) 2. THEORY T where v = (v1, v2, . , vm) is the m-dimensional vector of fluxes and N = [n ] is the n × m The intracellular metabolism of the cells under ij stoichiometric matrix of the metabolic network consideration is supposed to be represented by (m is the number of fluxes and n the number a metabolic network. A metabolic network is a of internal nodes of the network). More precisely, directed hypergraph 2 that encodes a set of el- a flux v denotes the rate of reaction j and a ementary biochemical reactions that take place j non-zero n is the stoichiometric coefficient of within the cell. In this hypergraph, the nodes ij metabolite i in reaction j. represent the involved metabolites and the edges represent the metabolic fluxes. A typical exam- For a given metabolic network, the set S of ple of metabolic network that will be considered admissible flux distributions is the set of vectors in Section 3 is shown in Fig.1. The metabolic v that satisfy the finite set (2) of homogeneous network involves two groups of nodes: boundary linear equalities and inequalities. Each admissible nodes and internal nodes. Boundary nodes have v must necessarily be non-negative and belong to only either incoming or outgoing edges, but not the kernel of the matrix N. Hence the set S is the both together. Boundary nodes can be further pointed polyhedral cone which is the intersection separated into input (or initial) and output (or of the kernel of N and the nonnegative orthant. terminal) nodes. Input nodes correspond to sub- This implies that any flux distribution v can be strates that are supposed to be only consumed expressed as a non-negative linear combination of but not produced. Output nodes correspond to a set of vectors ei which are the edges (or extreme final products that are supposed to be only pro- rays) of the polyhedral cone and form therefore a duced but not consumed. In contrast, the internal unique convex basis (see e.g. Weyl (1950)) of the (or intermediary) nodes are the nodes that have flux space: necessarily both incoming and outgoing incident v = w e + w e + ··· + w e w 0. (3) edges. They correspond to metabolites that are 1 1 2 2 p p i > produced by some of the metabolic reactions and The m × p non-negative matrix E with column consumed by other reactions inside the cell. vectors ei obviously satisfies NE = 0 and (3) is written in matrix form as It is assumed that the cells are cultivated in batch T mode in a stirred tank reactor. The dynamics of v = Ew with w , (w1, w2, . , wp) . (4) substrates and products in the bioreactor are rep- resented by the following basic differential equa- From a metabolic viewpoint, the vectors ei of tions: the convex basis encode the simplest metabolic paths that connect the substrates (input nodes) ds(t) = −vs(t)X(t) (1a) to the products (output nodes). More precisely, dt the non-zero entries of a basis vector e enumerate dp(t) i = vp(t)X(t) (1b) the fluxes of a sequence of biochemical reactions dt starting at one or several substrates and ending at where X(t) is the biomass concentration in the one or several products. These simple pathways culture medium, s(t) is the vector of substrate between substrates and products are called ex- concentrations, p(t) the vector of product concen- treme pathways (ExPa) or elementary flux modes trations, vs(t) the vector of specific uptake rates (EFM) of the network (Schuster et al. (1999) and vp(t) the vector of specific production rates. and Nielsen et al. (2002)). Since the intermediary (From now on, the time index “t” will be omitted). reactions are assumed to be at quasi steady-state, a single macroscopic bioreaction is then readily Obviously, the specific rates v and v are s p defined from an elementary flux mode by consid- not independent. They are quantitatively related ering only the involved initial substrates and final through the intracellular metabolism represented products. by the metabolic network. In order to explicit this relation, the quasi steady-state paradigm of Let us now come back to the basic model (1) metabolic flux analysis (MFA) is adopted (e.g. in order to elucidate the relation between the Stephanopoulos et al. (1998)). This means that specific consumption and production rates vs and for each internal metabolite of the network, it vp induced by the metabolic network. Obviously is assumed that the net sum of production and vs and vp are linear combinations of some of the metabolic fluxes. This is expressed by defining appropriate matrices Ns and Np such that 2 A hypergraph is a generalization of a graph, where edges can connect any number of vertices. vs = Nsv vp = Npv. (5) 322 From (4) and (5), it follows that the model (1) is N 0 v = . (9) rewritten as: Nm vm d s −Ns Here we focus on the special case where this = EwX = KewX (6) dt p Np system is exactly determined 4 and has a single where well-defined solution which can obviously be de- −Ns composed in the convex basis as expressed by (3). Ke , E. (7) Np But even if the flux vector v satisfying equation is the stoichiometric matrix of the set of biore- (9) is unique, it must be emphasized that the actions encoded by the EFMs. Equation (6) can decomposition of v in the convex basis {ei} is not be regarded as the dynamic model of a bioprocess unique which is the algebraic expression of the fact governed by the bioreactions with stoichiometry that the set of bioreactions used in the dynamical model (6) is redundant. Using (4), system (9) is Ke and specific reaction rates w. In other terms, equivalent to the system: each weighting coefficient wi in (3) can equally be interpreted as the specific reaction rate of the NE 0 w = w > 0. (10) bioreaction encoded by the EFM ei : the flux NmE vm vector v is thus a linear combination of EFMs We observe that the first equation NEw = 0 is whose non-negative weights are the macroscopic trivially satisfied independently of w since NE = bioreaction rates wi. 0 by definition. Hence, system (10) may be re- However an important issue concerns the num- duced to the second equation: ber of distinct bioreactions that are generated NmEw = vm w > 0. when computing the EFMs. It may become very or equivalently: large because it combinatorially increases with the size of the underlying metabolic network 3 . w NmE −vm = 0 w > 0. (11) Furthermore, even when the number of EFMs is 1 rather limited, it appears that the resulting set In this form, it is clear that the set of admissible of bioreactions can be significantly redundant for reaction rate vectors w that satisfy (11) also the design of a dynamic model that fully explains constitutes a convex polyhedral cone.