ARTICLE Reduction of a Set of Elementary Modes Using Yield Analysis Hyun-Seob Song, Doraiswami Ramkrishna School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907; telephone: 765-494-4066; fax: 765-494-0805; e-mail:
[email protected] Received 21 February 2008; revision received 8 May 2008; accepted 17 June 2008 Published online 22 July 2008 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/bit.22062
Introduction ABSTRACT: This article proposes a new concept termed ‘‘yield analysis’’ (YA) as a method of extracting a subset of elementary modes (EMs) essential for describing metabolic behaviors. YA can be defined as the analysis of metabolic pathways in yield space where the solution space is a bounded convex hull. Two important issues arising in the analysis and modeling of a metabolic network are handled. First, from a practical sense, the minimal generating set spanning the yield space is recalculated. This refined generating set excludes all the trivial modes with negligible contribution to convex hull in yield space. Second, we revisit the problem of decomposing the measured fluxes among the EMs. A consistent way of choosing the unique, minimal active modes among a number of possible candidates is discussed and compared with two other existing methods, that is, those of Schwartz and Kanehisa (Schwartz and Kanehisa, 2005. Bioinformatics 21: 204–205) and of Provost et al. (Provost et al., 2007. Proceedings of the 10th IFAC Symposium on Computer Application in Biotechnology, 321–326). The proposed idea is tested in a case study of a metabolic network of recombinant yeasts fermenting both glucose and xylose. Due to the nature of the network with multiple substrates, the flux space is split into three independent yield spaces to each of which the two-staged reduction procedure is applied. Through a priori reduction without any experimental input, the 369 EMs in total was reduced to 35 modes, which correspond to about 91% reduction. Then, three and four modes were finally chosen among the reduced set as the smallest active sets for the cases with a single substrate of glucose and xylose, respectively. It should be noted that the refined minimal generating set obtained from a priori reduction still provides a practically complete description of all possible states in the subspace of yields, while the active set covers only a specific set of experimental data. Biotechnol. Bioeng. 2009;102: 554–568. ß 2008 Wiley Periodicals, Inc. KEYWORDS: yield analysis; elementary mode; generating mode; active mode; metabolic network; mode reduction; minimal subset; uniqueness; recombinant yeast; glucose; xylose; flux decomposition
Correspondence to: D. Ramkrishna
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Cellular metabolic routes are completely described by the set of elementary (flux) modes (EMs). The EMs are a set of nondecomposable pathways consisting of a minimal set of reactions that function in steady state (Schuster et al., 2000). Elementary mode analysis (EMA) has been successfully used in a number of biochemical applications, for example, for interpreting metabolic network functions (Carlson and Srienc, 2004a,b; Gayen and Venkatesh, 2006; Poolman et al., 2003), assessing the robustness and fragility of cellular functions (Klamt, 2006; Klamt and Gilles, 2004; Stelling et al., 2002; Wilhelm et al., 2004), predicting the gene expression patterns (Stelling et al., 2002), improving the strain performance (Carlson et al., 2002; Trinh et al., 2006), etc. In addition to steady state analysis, pathway analysis has also been combined with dynamic models such as structured cybernetic models (Young, 2005; Young et al., 2008), hybrid cybernetic models (Kim, 2005; Kim et al., 2008), macrokinetic models (Provost and Bastin, 2004; Provost et al., 2006), dynamic flux balance analysis (DFBA) (Hjersted and Henson, 2006; Mahadevan et al., 2002), etc. Application of EMA is severely hampered in the analysis of complex networks with high connectivity due to the combinatorial explosion of the number of EMs (Klamt and Stelling, 2002). To cope with this problem, in one strategy, the network is split into several subnetworks by setting metabolites with a connectivity value above a certain threshold as external (Schuster et al., 2002). Obviously, division of a network into smaller ones has merits since subnetworks would be easier to analyze. In another approach, the metabolites with a connectivity value below a threshold are set as external so that only connections between the core nodes are included (Schmidt et al., 2003; Schwarz et al., 2005). The resulting EM set is smaller in comparison to the original set, but can still be too large to be employed in metabolic modeling. Since pathway analysis takes account of only two basic conditions (i.e., the stoichiometric and thermodynamic feasibility) among many other constraints imposed by a cell, the resulting flux space is so loose that not every flux vector ß 2008 Wiley Periodicals, Inc.
is physiologically reachable (Edwards and Palsson, 2000). Thus it may not be necessary to use the full set of EMs for the design of metabolic models. FBA takes up the extreme position that the solution space converges to a single pathway maximizing a certain metabolic objective function such as growth yield (Edwards et al., 2001; Kauffman et al., 2003). Motivated by this idea, we explore a way of reducing the EMs to a smaller subset, but not necessarily a single pathway as in FBA, essential for describing and predicting the metabolic behaviors. A particular interest is placed in the subsets of EMs describing a phenotype space or a set of phenotypic data. The importance of phenotype space can be found in the fact that the internal fluxes are not independently distributed but strictly constrained by the status of external fluxes through the EMs at steady state. Thus, even if we focus on a subset covering phenotype space, it might give rise to a biologically meaningful solution space. This aspect is reflected in metabolic flux analysis, a wellknown tool for estimating internal flux distributions from a given set of known or measured fluxes which are mostly external (Pitka¨nen et al., 2003; Stephanopoulos et al., 1998). In this article, in particular, we are concerned with two important problems which can be encountered in developing metabolic models. One is to identify and calculate the minimal generating set spanning a phenotype space, which is a lower-dimensional subspace composed of external fluxes only. Wagner and Urbanczik (2005) termed the elements of the minimal generating set ‘‘generating (flux) modes’’ (GMs) and calculated them using a null-space based algorithm. In our approach, this problem is handled differently by analyzing EMs in yield space. In yield space, the solution set is a bounded convex hull, instead of an unbounded polyhedral cone as in flux space, and it becomes possible to refine the set of GMs to be smaller using the analysis of the convex hull volume. We call this new approach of analyzing EMs in yield space ‘‘yield analysis’’ (YA) in short, in contrast to ‘‘flux analysis.’’ The second issue is to select the active modes from the set of EMs. The active modes are the pathways to be turned on in metabolic models to represent an experimental data set. Since in general there are an infinite number of ways of combining pathways to represent the data, selection of an appropriate set is obscured at this stage. The problem can be restated as the determination of a particular weight vector for the decomposition of measured fluxes among EMs. YA not only provides a reasonable answer to this question, but also allows for meaningful discussions on several issues such as the theoretical minimal number of the active modes and their uniqueness. We will compare our perspective with other existing approaches (Provost et al., 2007; Schwartz and Kanehisa, 2005). In the following, the theoretical aspects of YA will be described first, followed by the idea of refining the minimal generating set. Then, we develop a systematic procedure of reducing the set of EMs for designing a metabolic model with minimal structure. While for the sake of exposition we
will confine our concern to a single substrate for the first few sections, extension to multiple substrates will be made in a subsequent section on a case study with reduction of EMs for the metabolic network of recombinant yeasts fermenting both glucose and xylose.
Yield Analysis Versus Flux Analysis: Basic Concept The concept of YA is explained by comparing with flux analysis. For an illustration, we have used the example network considered by Wagner and Urbanczik (2005), which was originally hypothesized by Edwards et al. (2002). The network (not shown) contains 13 fluxes, r1 to r13, and is decomposed into eight EMs (Table I). The admissible flux distributions in the full space of fluxes are defined as the set of vectors r which satisfies the following equality and inequality constraints: Sr ¼ 0;
r0
(1)
where S is the stoichiometric matrix where each column and row represent a particular reaction and a particular species, respectively, and r is the column vector with elements ri, i ¼ 1, 2, . . ., 13. The inequality condition of Equation (1) means that every element of r is greater than or equal to zero. Any flux vector r can be expressed as a convex (or nonnegative linear) combination of the eight EMs, that is, r ¼ Zw;
w0
(2)
where Z is the EM matrix with elements zi,j, i ¼ 1, 2, . . ., 13, j ¼ 1, 2, . . ., 8, and w is the weight vector with elements wi,
Table I. Elementary modes EM1 to EM8 considered in Figure 1, which are represented in terms of the 13 fluxes r1 to r13 or in terms of two yields y12 and y13. EM1 r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13
1 3 0 0 1 1 0 0 1 8 3 3 0
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EM2 3 9 0 0 11 11 0 0 11 0 33 33 8 3 0.73
EM3 0 0 0 1 1 1 0 0 1 7 3 3 0 3 0
EM4 0 0 0 3 10 10 0 0 10 0 30 30 7 3 0.7
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0 0 3 0 2 2 0 1 2 0 4 4 1
0 0 0 0 1 1 1 0 1 27 13 13 0
1 0
2 0.5
13 0
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The fluxes r9, r12, and r13 in gray indicate the fluxes of carbon, oxygen, and biomass, respectively, and y12 (¼r12/r9) and y13 (¼r13/r9) are the yields of oxygen and biomass from carbon, respectively.
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2
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0
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or simply,
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i ¼ 1, 2, . . ., 8. The ith column of Z corresponds to the ith EM of Table I. For the phenotype phase-plane analysis, Wagner and Urbanczik (2005) projected the space of fluxes r1 to r13 to the three-dimensional subspace with uptake rates of carbon (r9) and oxygen (r12), and biomass formation rate (r13) as the axes. The projection generates the flux cone with the five edge vectors, EM2 and EM5 to EM8 (Fig. 1a). The remaining modes, EM1, EM3, and EM4, are unseen because they lie inside. The five edge vectors are the GMs spanning the threedimensional phenotype space. Any flux vector r within the flux cone of Figure 1a can be represented by a convex combination of the five GMs, implying the weights w1, w3, and w4 in Equation (2) are equal to zero. This statement can be equivalently expressed in terms of r9, r12, and r13 as follows:
0.8
EM
8
EM
0.6 EM6
2
EM
4
0.4 0.2 EM1, EM3 0
rx ¼ Zx wx ;
wx 0
where the subscript x is used to indicate the exchange fluxes. Now, we examine how this example can be analyzed in yield space. YA begins with recasting the modes into yield vector representation. Since every EM should include at least one or more substrates and products by definition,1 yield vector can always be defined. Yield vector is obtained from the net reaction of the modes by normalizing its stoichiometric coefficients with respect to a reference substrate (e.g., carbon in this context). In this example where the uptake of a substrate and the excretion of a product are described by a single input or output flux, yields are simply defined by dividing the output fluxes by the reference input flux as shown in the last two rows of Table I where y12 and y13 denote r12/r9 and r13/r9, respectively. Note that we use the term ‘‘yield’’ even for the co-substrate (i.e., oxygen). In the general treatment, the yield of nutrients or other substrates co-consumed with the reference substrate is given the minus sign to indicate that they are input fluxes. The eight EMs are represented as the eight points in the space of y12 and y13. The solution space is the bounded convex hull in yield space (Fig. 1b), in contrast that it was the unbounded polyhedral cone in flux space
1 In principle, this statement is not exact in the sense that there can exist EMs composed of internal fluxes only. However, those modes are not considered in this study since they provide no practical meaning (Klamt and Stelling, 2003).
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Figure 1. Dual representation of the solution space in the flux and yield coordinates. a: Unbounded polyhedral cone in the space of fluxes r9, r12, and r13. b: Bounded convex hull in the space of yields y12 and y13.
(Fig. 1a). The convex hull in yield space can be considered as a dual representation of the polyhedral cone in flux space. The convex hull can be reconstructed even in flux space, in this example, by cutting the flux cone with the plane of r9 ¼ 1. The intersections between the cutting plane and the edge vectors of the flux cone correspond to the vertices of the convex hull in yield space.2 Geometrically, the edges and inside rays of the cone in flux space are mapped into the vertices and interior points of the convex hull in yield space, respectively. The convex hull is completely described by the vertices just as the flux cone by the edges, both of them being the GMs spanning the phenotype space. Any yield vector
2 If the uptake of a reference substrate or the excretion of a product is described by multiple fluxes (as in the network of Fig. 8 where the uptake of xylose is modeled by the two reactions and the production and consumption of CO2 is associated with a number of reactions), this simple cutting does not give rise to the convex hull in yield space. In general, the yield vectors and convex hull can be derived from the net reactions of EMs, but not directly from the flux vectors and flux cone.
y ¼ [y12, y13]T can be represented by a convex combination of the five EMs, that is,
let us consider another hypothetical network (not shown) which has one substrate and two products. The master set of
3 h1 6 h2 7 7 z2;12 =z8;9 6 6 h3 7 ; 7 z2;13 =z8;9 6 4 h4 5 h5 2
y12 y13
¼
z2;12 =z2;9 z2;13 =z2;9
z5;12 =z5;9 z5;13 =z5;9
z6;12 =z6;9 z6;13 =z6;9
z7;12 =z7;9 z7;13 =z7;9
or simply, y ¼ Zy h;
h 0;
jjhjj1 ¼ 1
(6)
where Zy is the normalized Zx in our example, and h indicates the weight vector, respectively. Note that the weight vector h should satisfy the additional constraint that the summation of its elements is equal to one due to the property of yield vector. While there is an analogy between the flux and yield spaces as such, yield vector representation is more informative. In yield space, we can assess the ‘‘usefulness’’ of a GM by examining the ‘‘contribution’’ of the corresponding vertex to the convex hull. The contribution of the vertex is readily quantified by comparing the volume change of the convex hull between before and after eliminating it from the set of vertices. If the contribution is insignificant, we may neglect it from the list of GMs. In Figure 1b, the contribution of EM1, EM3, and EM4 is zero because they are on the edge of or inside the convex hull, which is why they are not counted as the GMs. Meanwhile, among the vertices, the contribution of EM2 and EM6 is relatively small (Fig. 1b). By eliminating those two, the set of GMs is further reduced to the three modes from the original five. We do not consider the potential emergence of EM4 as a new vertex after deleting EM2 and EM6 because our concern is to reduce the number of the vertices by excluding unimportant ones. From this consideration, we may view that the minimal generating set is not minimal in the practical sense. The original set of vertices is contaminated with many trivial points with negligible contribution to the convex hull, which can be cleansed by using YA. This idea forms the basis of a priori reduction to be expounded in the following section. The extent of reduction varies depending on the size and structural characteristic of the network. It seems difficult to do the same analysis in flux space because the flux cone is unbounded. Although the truncated flux cone can be obtained by adding more constraints such as the maximum flux limits, in general it is difficult or practically impossible to collect such information accurately.
We present here a detailed procedure for reducing the set of EMs based on the above idea. For the purpose of illustration,
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A priori Reduction of the EM Set to Minimal Generating Modes Using the Analysis of the Convex Hull Volume
(5)
EMs (Mmas) has 30 elements which are represented as the 30 points in the space of y1 and y2 (i.e., the yield of products 1 and 2) (Fig. 2). We can draw the convex hull enclosing all the points by connecting the nine vertices which are numbered as GM1 to GM9. Hereafter, the set of GMs spanning the yield space is denoted by the notation My throughout this article. Our aim is to develop a strategy to identify the subset My from Mmas, and subsequently condense My to a smaller subset which can describe the original convex hull to a predetermined level. It should be noted that the reduction method described in this section is purely an a priori analysis in the sense that it does not require any experimental input. First, we extract My from Mmas. My can be identified using a null-space approach (Wagner and Urbanczik, 2005), or a convex hull algorithm (Barber et al., 1996). Both methods should give the same result and in this study, we calculated My using the MATLAB 7.1 built-in function ‘‘convhulln.m’’ based on the latter scheme. While it may be required to employ a more efficient way of calculating My and Mmas to handle a large scale system, this issue is not discussed here in detail because it is not the main concern with the present article. Rather, the focus is placed on the development of the reduction procedure from My to a smaller subset as described below. Next, My is refined by removing the vertices with negligible contribution. One of the simplest ways for this is to examine the impact of a vertex on the volume of the
y2
3 h1 6 h2 7 5 X 6 7 6 h3 7 0; hj ¼ 1 6 7 4 h4 5 j¼1 h5 2
0
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GM
9
GM
GM
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2
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1
Figure 2. Representation of 30 EMs in the space of yields y1 and y2 where the 9 vertices of the convex hull, GM1 to GM9, indicate the minimal generating set spanning the yield space.
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convex hull when it is eliminated (i.e., top-down approach). If the loss of the volume is negligible (significant), the vertex is removed (maintained). In this way, we can rule out all trivial vertices from My. We can do a similar calculation along the reverse direction, that is, the contribution of a vertex is assessed on its addition (i.e., bottom-up approach). If the increment of the volume is significant (negligible), the vertex is included (excluded). By this, only significant vertices will constitute the set. In this study, we take the latter approach since it performs faster especially when the number of the modes of My is large. The algorithm of the bottom-up approach is, however, a bit more complex in comparison to that of the top-down approach. In the bottom-up approach, particular care should be taken in determining an initial convex set (Mini) which means to select a triangle in our example. Once the initial triangle is given, we can sequentially add more vertices to upgrade it to a quadrangle, a pentagon, and so on. The choice of the initial three vertices should not be arbitrary to ensure that the starting convex hull has an appreciable volume (or area). For example, if GM2, GM3, and GM4 are selected as Mini by random choice in Figure 2, the algorithm of calculating the convex hull volume may fail because they are not systemically independent enough. In other words, the volume of the convex hull is almost zero. Although taking the largest triangle would be the best choice to avoid this situation, it may not be desirable in view of the computational burden with searching the whole vector space of My using nonlinear integer programming (NIP). Considering our eventual purpose, the starting set need not necessarily be the best. Thus, as a compromise between the NIP and the random choice, we suggest a rather simple but robust method for selecting Mini. Instead of the largest triangle, we try to choose a triangle with reasonable size by taking three vertices apart enough from each other. This can start with the points with the maximum y1 and y2 (Mymax) (Step 1), which are GM4 and GM6 in our example (Fig. 3a). One more point which is needed to define a triangle can be found by taking the vertex GM1 which is farthest from the line connecting GM4 and GM6 (Fig. 3b) (Step 2). These two steps complete the procedure of setting Mini. In n-dimensional yield space, the minimal number of the points for defining a convex hull is n þ 1, among which, in the example, n points (i.e., GM4 and GM6) were provided from Step 1 and the remainder (i.e., GM1) was found in Step 2. The number of the modes of Mymax is, however, not necessarily n. It can be less than n if a single EM has the maximum yield for several products, or greater than n if the maximum yield of a single product is found in more than one EM. For the former, the deficiency should be compensated in Step 2 by iteratively seeking the vertex which is most distant from the given linear subspace until the minimal requirement for the formation of a convex hull is satisfied. On the other hand, the original algorithm can be applied to the latter case by making a free choice of one EM among the candidates.
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Once Mini is given, the subsequent expansion of the convex hull is rather straightforward. As implied earlier, our strategy is to add a point at a time such that the volume increment of the convex hull is maximized at every addition (Fig. 3c). As a stopping condition, we suggest the threshold of the 99% of the volume of the original convex hull and the resulting set is denoted by M99. The change of relative volume with the addition of a vertex is shown in Table II. Due to the 99% criterion, the search will stop at M99 which is composed of the seven modes. Although the difference between My and M99 is not significant in the current example, we will see that it is substantial in the case study to be considered later. Since the determination of a stopping threshold is a rather subjective matter in nature, it might be possible to test and compare lots of different criteria such as M95, M90, etc. However, we prefer the use of M99 by which it is guaranteed that loss of information by ruling out the trivial vertices is less than 1%.
Selection of Active Modes When experimental data are available, we can further narrow down M99 to the set of active modes (Mact) to be turned on to represent them. This is not a simple task since the existence of Mact is not unique in general. A set of experimental data can be represented in numerous ways by different subsets, for example, Mmas, My (or M99), or an even smaller subset, or by just a single pathway. We have several fundamental issues regarding the choice of Mact such as: (1) what is the minimal number for the activated modes, (2) whether it exists uniquely or not, (3) if multiple choice is possible, what can be a rational criterion for the selection of a particular set, (4) how we can handle the case of multiple steady states, etc. Before attempting to directly seek the answers, we have to first figure out whether the network model can exactly represent the data set or just fit it. This basic question can be restated into whether the data set is inside or outside the convex hull in yield space. Depending on the position of the data set in yield space, the answers will be different. The case where the experimental data are positioned inside the convex hull is what we expect because all possible phenotypic states can be represented by convex combinations of elements of My (or M99). However, more often than not, the opposite case is also encountered. This can happen if the network model is somewhat simplified enough to make the solution space smaller or if the experimental data are contaminated by a measurement error. While the former is avoidable, the latter is not. We discuss the above issues separately as below.
When the Data Point Is ‘‘Inside’’ the Convex Hull Various choices of Mact are possible when the data point is inside the convex hull. In yield space, Mact is defined as any
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Figure 3. Procedure of extracting M99 from My. a: Selection of the modes (GM4 and GM6) with maximum yields (Mymax). b: Inclusion of additional vertex (GM1) farthest from the line (or generally, linear subspace) formed by Mymax. c: Addition of one vertex at a time such that the volume increment of the convex hull is maximized.
Table II. Change of relative volume of the convex hulls with the chosen subset of GMs by the mechanism depicted in Figure 3c. Chosen subset of GMs {GM1, GM4, GM6} (¼Mini) {Mini, GM2} {Mini, GM2, GM8} {Mini, GM2, GM8, GM5} {Mini, GM2, GM8, GM5, GM7} (¼M99) {M99, GM9} {M99, GM9, GM3}
V/Vy 0.4123 0.6758 0.8698 0.9358 0.9930 0.9981 1.0000
V and Vy indicate that the volume of the convex hull of the chosen subset and My, respectively.
set of points which can enclose the data. Even in the case of considering M99 only, we can find many different subsets enclosing the data which can be triangles, quadrangle, or more complex convex hulls (Fig. 4a). From this observation, we can get a quick answer to the first issue above, that is, in our example, the minimal convex hull to enclose the data is a triangle unless the data are on the line connecting two points or exactly matches one point. This statement can be generalized to that the minimal number of the modes of Mact is at most ‘‘n þ 1’’ in n-dimensional yield space. The issue on the minimal number of the modes of Mact was also handled by Provost et al. (2007) in the problem of determining minimal subsets of EMs when only external flux measurements are available. They showed that in their
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resulting triangles of Provost et al. (2007) significantly outnumbers those of YA. It is because Provost et al. (2007) searches all possible triangles among Mmas but YA focuses big triangles only among M99 (Fig. 4b). Interestingly, Provost et al. (2006) showed that all of the minimal active sets are totally equivalent in the sense that any minimal Mact chosen under a specific environmental condition provides exactly the same internal flux distribution and the same dynamical simulation results. This means that it does not make any difference whether we select the minimal active modes from M99 or from Mmas, while the selection of the minimal Mact from M99 instead of Mmas is preferred due to the reason described above. It also means even picking up a particular active set among many candidates would never hurt the model performance, justifying our effort to get a unique, minimal Mact as below. It should be pointed out, however, there exist applications where the complete set of EMs (i.e., Mmas) may be required and that the use of a reduced set (i.e., Mact, M99, or My) could have its limitations. The issue such as the robustness of metabolic network, or flux distributions by gene knockout should be handled based on Mmas while Mact still can be used for estimating flux distributions in situations where a defined set of environmental changes are envisaged as shown by Provost et al. (2006). Regarding the issue of selecting a particular set, Schwartz and Kanehisa (2005) suggested that the decomposition of fluxes into the EMs can be determined such that the sum of squared weights is minimized, that is,
0 0
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Figure 4. Determination of active modes (Mact) when a set of experimental data (&) is positioned inside the convex hull. (a) Any convex hull enclosing the data point can be Mact and the triangle is the minimal Mact. (b) Selection of the minimal Mact: In YA, the minimal Mact is determined among the big triangles which can represent the data, while Provost et al. (2007) searches all the possible triangles.
algorithm, the number of nonzero weights to EMs (i.e., the number of the active modes) as a result of decomposing the measured fluxes is given at most equal to the number of measured uptake and excretion rate. For a rigorous mathematical formulation and proof of this property, refer to Section 3.5 in Provost (2006). This statement is identical to what we already observed in YA because the ndimensional yield space implies the ‘‘n þ 1’’ measured external fluxes. Another advantage of YA is thus to interpret the difficult-to-understand mathematical property in a much easier way using the geometrical representation in yield space. Due to YA, we can see that the algorithm of Provost et al. (2007) implies to search all the triangles, lines and the point which can represent the data. The conclusions of both approaches are similar as to the size of Mact, but different as to its number. While in either case, the minimal Mact is not uniquely determined, it should be noted that the
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(7)
such that Zy h ym ¼ 0;
h 0;
jjhjj1 ¼ 1
(8)
where ym is the vector of measured yield data. The last equality constraint of Equation (8) is not considered in the original formulation of Schwartz and Kanehisa (2005), but is necessary in our case. Clearly, the resulting subset is unique but not necessarily minimal. The direct application of Equations (7) and (8) to our case is not useful since it tends to distribute the nonzero weights among as many EMs as possible as shown in Figure 5a where the numbers besides the vertices indicate the corresponding weights. The additional constraint, jjhjj1 ¼ 1 of Equation (8) is responsible for this even distribution among M99. Taking all the modes of M99 is clearly in the opposite direction of our effort for finding the minimal Mact. By limiting Mact to be the minimal convex hull (i.e., triangle in our example) which can be known a priori through YA, however, we can obtain a rather promising result. Under this constraint, the algorithm of Schwartz and Kanehisa (2005) gives rise to the triangle where the data are placed around its center of mass as close as possible (Fig. 5b), which is a desirable property for a robust metabolic model.
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GM6
1
GM5
y2
0.8
a
GM
1
0.3911
6
0.6 y1
GM6
1
0.8
1
A GM
5
GM7 0.2385
0.8
0.6
B GM4
y
y2
2
0.6
0.4
0.4
0.2
0
GM
1
0.2
0
0.4
0.6 y1
0.8
1
0
Figure 5.
Determination of a particular set of active modes (Mact) using the algorithm of Schwartz and Kanehisa (2005) when a set of experimental data (&) is positioned inside the convex hull. a: Direct application of their algorithm results in the selection of all the modes. b: By imposing the additional constraint that the number of the modes of Mact is minimal (i.e., three) on the original formulation of Schwartz and Kanehisa (2005), we can get the triangle enclosing the data around its center of mass as close as possible. The numbers besides the vertices indicate the corresponding weights in both figures.
(9)
0.6 y1
GM6
1
0.8
1
A GM
5
0.8
B GM4
C
0.6 0.4
1 min jjZy h ym jj22 h 2
0.4
c
When the Data Point Is ‘‘Outside’’ the Convex Hull We turn to the problem of selecting the Mact when the data is located outside the convex hull (Fig. 6a). In this case, the experimental data are not exactly represented by a convex combination of any set of EMs. The best available option is then to select the subset of modes best-fitting the data. The determination of Mact is not a difficult issue since the modes best-fitting the data are always uniquely found. For example, in the situation of Figure 6a, two modes GM5 and GM6 are uniquely chosen as Mact representing the data A. In general, the number of the modes of Mact would simply be the same as the dimension of yield space. Mact can be calculated by solving the following least-squares problem, that is,
0.2
y2
0
0.2
0.3703
GM8
0.2 GM
2
0 0
0.2
0.4
0.6 y1
0.8
1
Figure 6.
Determination of a particular set of active modes (Mact). a: The experimental data (&) are positioned outside the convex hull. b: Two steady states exist and both of them are positioned outside. c: Three steady states exist and one of them is inside while the others are outside.
such that h 0;
jjhjj1 ¼ 1
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(10)
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YA is also able to deal with the case of multiple steady states in a similar fashion. Suppose we obtained two different steady-state data and both are outside the convex hull (Fig. 6b). Mact is uniquely identified for the data A (GM5 and GM6) and B (GM4 and GM5), respectively. Note that one mode (GM5) is in common between the two sets and the final Mact is determined by the union of the two sets, that is, GM4, GM5, and GM6. A more complex situation is found in the mixed case that parts of the multiple steady-state data are outside (data A and B) and the remainders are inside (data C) as in Figure 6c. The selection of the Mact is achieved by a simple combination of the cases of Figures 4 and 6b. Mact are finally determined again as the union of the subsets chosen for each data.
Summary So far, we have discussed a way of selecting a reduced subset of EMs from the original master set using the concept of YA. It is composed of two stages: (I) a priori reduction without
experimental input and (II) selection of the Mact for a given set of measurements (Fig. 7). The procedure of a priori reduction can be summarized as Network ! Mmas ! My ð! Mymax ! Mini Þ ! M99
(11)
The output of the a priori reduction (i.e., M99) becomes the input to the problem of selecting the (minimal) Mact representing or fitting the experimental data. Depending on the location of the data point in yield space, the strategy and result become different, which is summarized in Table III. On comparing Provost et al. (2007) and Schwartz and Kanehisa (2005), they are complementary. The former results in the minimal but not unique subset, while the latter unique but not minimal. We hold that the combination of these two methods within YA framework is a better format. Our procedure for determining the unique, minimal Mact when the data are inside the convex hull is summarized as
Figure 7. Overall reduction procedure. In the first stage, M99 is extracted from Mmas by the analysis of convex hull volume, and in the second stage, the minimal Mact is chosen among M99 in a different fashion depending on the location of the data point in yield space.
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Table III. Uniqueness and (minimal) number of the modes of Mact depending on the position of the data in yield space. Set of active modes
Location of the data
Uniqueness
Outside the convex hull Inside the convex hull
Unique Multiple
(Minimal) number of the modes na na þ 1
the G- and X-groups which contain a single substrate, it is not clear which substrate should be taken as the reference species for the M-group. The answer is rather simple. Either one (i.e., glucose or xylose) is possible and it depends on the modeler’s choice. As a result, the flux space (V) can be decomposed into three yield spaces (Y) in two ways, that is, V ! fYG0 ; YGX ; YX0 g or fYG0 ; YXG ; YX0 g
(14)
a
n ¼ the dimension of yield space.
follows: M99 ! Provost et al: ð2007Þ
Mact |ffl{zffl}
minimal but multiple
! Schwartz and Kanehisa ð2005Þ; i:e:; Eqs: ð7Þ and ð8Þ
Mact |ffl{zffl}
minimal and unique
where the subscripts G0, X0, GX, XG indicate the G-group, the X-group, the M-group with glucose as the reference, and the M-group with xylose as the reference, respectively. We may combine the two sets of YG0 and YGX (or YX0 and YXG) into one set YG (or YX) because the yield vectors in those two sets are defined with respect to the same substrate. This means the flux space can also be decomposed into two yield spaces, that is, V ! fYG ; YX0 g or fYG0 ; YX g
(15)
(12) The reduction of EMs can be applied to any combination of yield spaces above. In this way, we are able to handle the network with the multiple substrates.
or directly, M99
The number of the modes is constrained to be minimal
! Eqs: ð7Þ and ð8Þ
Mact |ffl{zffl}
minimal and unique
(13)
Case Study: Metabolic Network of Recombinant Yeasts The proposed idea is tested in a realistic, more complex metabolic network in this section. For this purpose, metabolism of recombinant Saccharomyces cerevisiae fermenting glucose and xylose was taken as a case study. The network is composed of several fundamental pathways including glycolysis pathway, pentose phosphate pathway, citric acid cycle, pathways for pyruvate metabolism, xylose utilization pathway, and others as depicted in Figure 8. The reactions occurring within the yeast are listed up in Table IV. Whereas catabolic breakdown of the substrates were considered in somewhat detail, anabolic pathways were lumped into the biomass synthesis reaction from basic precursors as reaction 36 of Table IV. Altogether, the network contains 46 metabolites (37 internal and 9 external species) and 38 reactions. We calculated 369 EMs using the latest version of METATOOL (version 5.0) (von Kamp and Schuster, 2006). Depending on which substrate is utilized, the set of EMs can be divided into three individual groups, that is, glucose (G), xylose (X) and mixture (M) group. The EMs belonging to each group are denoted by G1 to G33, X1 to X113, and M1 to M223 for the G-, X-, and M-groups, respectively (Table V). Yield space can be defined independently for each EM group. While it is straightforward to define yield vectors for
Reduction From Mmas to M99 The first step for performing a priori reduction of EM set is to specify the coordinates of yield space. In the network decomposition into EMs, ten species were set as external, that is, two substrates (i.e., GLC, XYL) and eight products (ETH, GOLx, XOLx, CO2, ACTx, BIOM, and MAINT). If we consider the whole products, the number of coordinates of yield vectors is seven (i.e., yETH, to yMAINT) for the case of a single substrate (i.e., YG0 and YX0), but it becomes eight for the case of mixed substrates because the coordinate yGLC or yXYL is additional. It was pointed our earlier that yGLC or yXYL does not have a meaning of yield because they are being consumed not produced. However, we extend the concept of yield to include the co-substrates by assigning the minus sign to the species being co-consumed. YA is applicable even if some of coordinates are negative as long as the solution space forms a convex hull. Among the four possible combinations of the set of yield spaces as given in Equations (14) and (15), the set of three yield spaces, YG0, YGX, and YX0 were chosen for the test. We first considered the full coordinates for each yield space as in Table VI. The reduction performance varies depending on the yield space. The reduction efficiency from Mmas to M99 amounts to 45%, 55%, and 91% for the G-, X-, and Mgroup, respectively and on average, it amounts to 76%. The reduction from My to M99 is also substantial (i.e., 67%), which means there are a lot of trivial vertices in the convex hull. Reduction performance becomes much higher when we consider only partial coordinates. If it is impossible to obtain the data for certain species, or if it is the main interest to examine the yield space with a few biologically important
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Figure 8.
Metabolic network model of recombinant Saccharomyces cerevisiae fermenting both glucose and xylose.
axes as in the example of Figure 1, the analysis of the yield space with partial coordinates is meaningful. In this study, only three to five yields are considered as the axes of yield space as shown in Table VI. The selection of the coordinates was decided based on the available experimental data for our system. The resulting reduction efficiency from Mmas to M99 is 76%, 86%, and 95%, for the G-, X- and M- group, respectively and altogether, it becomes 91%. Again, the reduction from My to M99 is appreciable (i.e., 51%).
Selection of Mact From M99 The final step in the reduction process is to select Mact from M99, which requires a set of experimental data. We have used the experimental data from Krishnan et al. (1997) where the fermentation data were obtained in the batch process using the genetically engineered Saccharomyces yeast 1400
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(pLNH33). As an example, we take the two data sets for the determination of Mact. The first is the fermentation data on glucose, which provides the yields for ethanol, glycerol and biomass, and the second is the fermentation data on xylose, which provides yields for xylose, ethanol, glycerol, xylitol, and biomass. Since the full spectrum of yield data is not available in our example, we employ the partial coordinate space as M99 shown in Table VI from which Mact is selected. For the case that glucose is used as the substrate, we can build the threedimensional yield space. As can be seen in Figure 9, the data is outside the convex hull. The possible reason for this is that we have used a lumped reaction for the biomass formation in our model. In other words, by the use of reaction 36 in Table IV, we have assumed constant composition of biomass, which might cause the solution space to be smaller pushing the data outside. In this case, Mact can be determined uniquely, and three modes are selected as Mact
Table IV. List of biochemical reactions included in the metabolic network model of recombinant Saccharomyces cerevisiae.
Table V. Classification of elementary modes into three subgroups according to the substrate.
Glycolysis 1 GLC þ ATP ! G6P þ ADP 2 G6P $ F6P 3 F6P þ ATP $ DHAP þ GAP þ ADP 4 DHAP $ GAP 5 DHAP þ NADH ! GOL þ NAD 6 GOL ! GOLx 7 GAP þ NAD þ ADP $ PG3 þ NADH þ ATP 8 PG3 $ PEP 9 PEP þ ADP $ PYR þ ATP Pyruvate metabolism 10 PYR ! ACD þ CO2 11 ACD þ NADH ! ETH þ NAD 12 ACD þ NADHm ! ETH þ NADm 13 ACD þ NADP ! ACT þ NADPH 14 ACT ! ACTx 15 ACT þ CoA þ 2ATP ! AcCoA þ 2ADP 16 PYR þ ATP þ CO2 ! OAA þ ADP Pentose phosphate pathway 17 G6P þ 2NADP ! Ru5P þ CO2 þ 2NADPH 18 Ru5P $ X5P 19 Ru5P $ R5P 20 R5P þ X5P $ S7P þ GAP 21 X5P þ E4P $ F6P þ GAP 22 S7P þ GAP $ F6P þ E4P Citric acid cycle 23 PYR þ NADm þ CoAm ! AcCoAm þ CO2 þ NADHm 24 OAA þ NADm þ NADH $ OAAm þ NADHm þ NAD 25 OAAm þ AcCoAm ! ICT þ CoAm 26 ICT þ NADm ! AKG þ CO2 þ NADHm 27 ICT þ NADPm ! AKG þ CO2 þ NADPHm 28 AKG þ NADm þ ADP ! SUC þ ATP þ CO2 þ NADHm 29 SUC þ 0.5NADm $ MAL þ 0.5NADHm 30 MAL þ NADm $ OAAm þ NADHm Xylose metabolism 31 XYL þ NADH ! XOL þ NAD 32 XYL þ NADPH ! XOL þ NADP 33 XOL ! XOLx 34 XOL þ NAD ! XUL þ NADH 35 XUL þ ATP ! X5P þ ADP Biomass formation 36 1.04AKG þ 0.57E4P þ 0.11GOL þ 2.39G6P þ 1.07OAA þ 0.99PEP þ 0.57PG3 þ 1.15PYR þ 0.74R5P þ 2.36AcCoA þ 0.31AcCoAm þ 2.68NAD þ 0.53NADm þ 11.55NADPH þ 1.51NADPHm þ 30.48 ATP þ 0.43CO2 ! ‘‘1 g BIOM’’ þ 2.36CoA þ 0.31CoAm þ 2.68NADH þ 0.53NADHm þ 11.55NADP þ 1.51NADPm þ 30.48ADP Others 37 ATP ! ADP þ MAINT 38 NADH ! NAD
Group
EM
Net reaction
G
G1 G2 ... G33
X
X1 X2 ... X113
M
M1 M2 ... M223
GLC ¼ 2.73 CO2 þ 1.64 ETH þ 2.36 MAINT GLC ¼ 2 CO2 þ 2 ETH þ 2 MAINT ... 255.89 GLC ¼ 5.43 ACTx þ BIOM þ 332.89 CO2 þ 195.66 ETH þ 252.08 GOLx 2 XYL ¼ CO2 þ 1.8 XOLx 2.2 XYL ¼ CO2 þ 2 XOLx ... 84.77 XYL ¼ 7.14 ACTx þ BIOM þ 119.1 CO2 þ 64.93 ETH þ 38.86 GOLx GLC þ 12 XYL ¼ 6 CO2 þ 12 XOLx GLC þ 2 XYL ¼ 2 ACTx þ 2 CO2 þ 2 MAINT þ 2 XOLx ... 449.92 GLC þ XYL ¼ 11.53 ACTx þ 1.75 BIOM þ 585.01 CO2 þ 343.25 ETH þ 444.24 GOLx
Table VI.
as summarized in Table III. From Figure 9, we can see the modes, G2, G8, and G24 provides the facet with the minimal distance from the data point, thus are chosen as Mact. The experimental data and the fitted values of yield are compared in Table VII. For the second data set with xylose as the substrate Mact can be determined in a similar way. The only difference lies in our having to consider a fourdimensional yield space for this case. Since the data is still outside the convex hull, we can uniquely find four modes best-fitting the data (Table VII). The case that both substrates are co-consumed is not treated here due to the difficulties of extracting accurate yield information from the original fermentation data.
Concluding Remarks YA can be summarized as a method of analyzing the EMs in yield space. Using this new concept, we could describe how to define and extract a meaningful subset of EMs in yield space. The key aspect of YA is that the solution space is bounded without imposing artificial constraints, enabling a priori reduction of EMs to a core subset such as M99 which is able to describe more than 99% of all possible phenotypic states. Moreover, YA provides critical information required for the design of minimal metabolic models by revealing the relationship between the experimental data and the Mact. Due to such valuable insights obtained from YA, it was
Results of a priori reduction of Mmas in the full and partial coordinates of yield space. Full coordinates
Partial coordinates
Group
Mmas
My
M99
Axes
My
M99
Axes
G X M Total
33 113 223 369
24 90 159 273
18 51 21 90
yETH to yMAINT yETH to yMAINT, yXOLx yETH to yMAINT, yXOLx , yXYL
14 27 30 71
8 16 11 35
yETH, yGOLx , yBIOM yETH, yGOLx , yBIOM, yXOLx yETH, yGOLx , yBIOM, yXOLx , yXYL
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Table VII. Substrate Glucose Exp Fitting Xylose Exp Fitting
Experimental yield data and the best-fitted values by Mact. yETH
yGOLx
yXOLx
yCO2
yACTx
yBIOM
yMAINT
Sum of squared residuals
1.8878 1.6559
0.0556 0.0554
— —
— 1.7921
0 0
0.0169 0.0166
— 1.0050
0.0538
1.1423 1.1028
0.0474 0.0469
0.0490 0.0489
— 1.3221
0 0
0.0249 0.0247
— 0.2469
0.0016
Figure 9. Example of selecting Mact: The three modes G2, G8, and G24 are chosen as Mact in the G-group since they form the facet of the convex hull closest to the data point (&). possible to discuss the advantages and limitations of existing methods developed for the selection of the Mact and finally to suggest a rational idea of uniquely determining the minimal Mact. In the case study considered in this article, the framework for treating the network with multiple substrates was established. After splitting the flux space into several independent yield spaces, the same procedure has been applied to each yield space, resulting in substantial reduction. While YA is limited to the interpretation of phenotypic behaviors of a metabolic system, we expect that the reduced set of EMs sought in YA conveys valuable information on internal flux distributions. While the reduction technique of this article provides an effective way to all flux-based approaches using EMs, its contribution to the formulation of dynamic models is even greater. In a follow-up article we demonstrate the application of this technique to the formulation of cybernetic models that are able to portray dynamic regulatory behavior of metabolic systems on mixed substrates using a few selected modes without deteriorating the simulation performance.
M99
the set of EMs covering more than 99% of the original convex hull volume
Mact
the set of active EMs
Mini
the set of EMs chosen at the initial stage of the EM reduction procedure
Mmas
the master set of EMs
My
the set of GMs spanning the given yield space
Mymax
the set of EMs with the maximum yield for each product
r
flux vector
rx
exchange flux vector
S V
stoichiometric matrix flux space
V
volume of the convex hull of the chosen subset
Vy
volume of the convex hull of My
w
weight vector for the modes of Z
wx
weight vector for the modes of Zx
X
EM with xylose as the substrate
Y
yield space
y YG
yield vector yield space combining YG0 with YGX
YG0
yield space defined by the modes of G-group
YGX
yield space defined by the modes of M-group (normalized by glucose)
ym
vector of measured yield data
YX
yield space combining YX0 with YXG
YX0 YXG
yield space defined by the modes of X-group yield space defined by the modes of M-group (normalized by xylose)
Z
EM matrix
Zx
EM matrix
Zy
normalized EM matrix
Abbreviations DFBA
dynamic flux balance analysis
EM
elementary mode
EMA
elementary mode analysis
FBA
flux balance analysis
GM
generating mode
YA
yield analysis
Nomenclature G
EM with glucose as the substrate
h M
weight vector for the modes of Zy EM with the mixture of glucose and xylose as the substrates
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Metabolites AcCoA AcCoAm
acetyl CoA (cytosol) acetyl CoA (mitochondria)
ACD
acetaldehyde
ACT
acetate (internal)
ACTx
acetate (external)
ADP
adenosine diphosphate
AKG
alpha-ketoglutarate
ATP
adenosine triphosphate
BIOM
biomass
CO2 CoA
carbon dioxide coenzyme A
CoAm
coenzyme A (mitochondria)
DHAP
dihydroxyacetone phosphate
E4P
erythrose-4-phosphate
ETH
ethanol
F6P
fructose-6-phosphate
G6P
flucose-6-phosphate
GAP GLC
glyceraldehyde 3-phosphate glucose
GOL
glycerol (internal)
GOLx
glycerol (external)
ICT
isocitrate
MAINT
maintenance
MAL
malate
NAD
nicotinamide adenine dinucleotide, oxidized (cytosol)
NADm NADH
nicotinamide adenine dinucleotide, oxidized (mitochondria) nicotinamide adenine dinucleotide, reduced (cytosol)
NADHm
nicotinamide adenine dinucleotide, reduced (mitochondria)
NADP
nicotinamide adenine dinucleotide phosphate, oxidized (cytosol)
NADPm
nicotinamide adenine dinucleotide phosphate, oxidized (mitochondria)
NADPH
nicotinamide adenine dinucleotide phosphate, reduced (cytosol)
NADPHm
nicotinamide adenine dinucleotide phosphate, reduced (mitochondria)
OAA OAAm
oxaloacetate (cytosol) oxaloacetate (mitochondria)
PEP
phosphoenol pyruvate
PG3
3-phosphoglycerate (3PG)
PYR
pyruvate
R5P
ribose-5-phosphate
Ru5P
ribulose-5-phosphate
S7P
sedoheptulose-7-phosphate
SUC X5P
succinate xylulose-5-phosphate
XOL
xylitol (internal)
XOLx
xylitol (external)
XUL
xylulose
XYL
xylose
The authors acknowledge a special grant from the Dean’s Research Office at Purdue University that made this research possible. Acknowledgment is also made to NSF GOALI Program (BES0000961) for support of much of the earlier work which provided the basis of the work reported here.
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