Incorporation of enzyme concentrations into FBA and identification of optimal metabolic pathways

Background In the present article, we propose a method for determining optimal metabolic pathways in terms of the level of concentration of the enzymes catalyzing various reactions in the entire metabolic network. The method, first of all, generates data on reaction fluxes in a pathway based on steady state condition. A set of constraints is formulated incorporating weighting coefficients corresponding to concentration of enzymes catalyzing reactions in the pathway. Finally, the rate of yield of the target metabolite, starting with a given substrate, is maximized in order to identify an optimal pathway through these weighting coefficients. Results The effectiveness of the present method is demonstrated on two synthetic systems existing in the literature, two pentose phosphate, two glycolytic pathways, core carbon metabolism and a large network of carotenoid biosynthesis pathway of various organisms belonging to different phylogeny. A comparative study with the existing extreme pathway analysis also forms a part of this investigation. Biological relevance and validation of the results are provided. Finally, the impact of the method on metabolic engineering is explained with a few examples. Conclusions The method may be viewed as determining an optimal set of enzymes that is required to get an optimal metabolic pathway. Although it is a simple one, it has been able to identify a carotenoid biosynthesis pathway and the optimal pathway of core carbon metabolic network that is closer to some earlier investigations than that obtained by the extreme pathway analysis. Moreover, the present method has identified correctly optimal pathways for pentose phosphate and glycolytic pathways. It has been mentioned using some examples how the method can suitably be used in the context of metabolic engineering.

The constraints on all the 7 internal fluxes are that these fluxes are all positive. As in the previous synthetic system in the manuscript (Fig. 2), flux vectors were generated. If we want to maximize the rate of yield of E for growth on substrate A, we have to maximize the quantity z = c 6 v 6 + c 7 v 7 − c 11 b 4 . Applying the present method, we have obtained the pathway R 1 → R 2 → R 3 → R 7 → R 9 as an optimal one, which is again conforming to [1]. Here R 1 : Ext → A, R 2 : A → B, R 3 : B → C, R 4 : B → D, R 5 : D → B, R 6 : C → D, R 7 : C → E, R 8 : D → E, R 9 : E → Ext. It is to be mentioned that 100 iterations were required for minimizing y. The optimal pathway is obtained at λ = 0.5. Now we intend to apply the method to a few real life pathways where there are only internal fluxes. All the internal fluxes are positive. They are more complex than the synthetic ones.
Carotenoid biosynthesis Carotenoids are organic pigments that are naturally occurring in plants and some other photosynthetic organisms like algae, some types of fungus and some photosynthetic bacteria [3,8]. Here they play a critical role in the photosynthetic process [9]. Carotenoids are known to belong to two classes. They also occur in some non-photosynthetic bacteria, yeasts and molds, where they carry out a protective function against damage by light and oxygen. Although, animals appear to be incapable of synthesizing carotenoids, many animals incorporate carotenoids from their diet. Within animals, carotenoids provide bright coloration, serve as antioxidants, and can be a source for vitamin A activity [10]. Carotenoids are natural fat-soluble pigments that play various biological roles. Structurally they are in the form of a polyene chain which is sometimes terminated by rings [4].
Carotenoid biosynthesis pathway in Fig. 9 Biosynthesis of carotenoids occurs in all photosynthetic organisms -bacteria, algae and plants, as well as in some non-photosynthetic bacteria and fungi. The intermediate steps in the carotenoid biosynthesis pathway were postulated several decades ago by standard biochemical analyses [3]. Carotenoid formation is a highly regulated process. Concentration and composition of leaf xanthophylls are affected by light intensity and the accumulation of specific carotenoids in chromoplasts of fruits and flowers is developmentally regulated. Variation in gene expression, most likely at the transcriptional level, is the key regulatory mechanism that controls carotenogenesis. The carotenoid biosynthesis genes (or cDNA) are functional when properly expressed in bacteria. Carotenogenic organisms can be found among heterotrophic bacteria and fungi (where some species possess this biosynthetic capacity) or among photosynthetic prokaryotes and eukaryotes. In the photosynthetic lower and higher plants carotenogenesis is obligatory for their photosynthetic activity.
The most universal carotenoid biosynthesis pathway is the sequence leading to the formation of pcarotene. The initial reaction yielding phytoene, the first carotene of the pathway, is the condensation of two molecules of geranylgeranyl diphosphate (GGPP) as an intermediate [4]. In green algae and higher plants α-carotene carrying a β-and an ε-ionone ring is formed simultaneously from lycopene. Xanthophylls are derived from α-and β-carotene by introduction of oxygen groups. The carotenoid biosynthesis pathway in Rhodobacter branches off at the stage of neurosporene. The entire carotenoid biosynthesis pathway is a part of the terpenoid metabolism with formation of prenyl diphosphates as a common sequence for chain elongations. From the different prenyl diphosphates formed, specific routes branch off into various terpenoid end products. Some investigations suggest that the dimerization of GGPP leads to phytoene as the first carotene [4]. However, there are several indications that carotenoid biosynthesis relies on its independent synthesis of GGPP. Furthermore, during fruit ripening, which is accompanied by massive carotenoid formation in capsicum, expression of the GGPP synthase gene is strongly enhanced [4]. Consequently, the start of the specific biosynthesis pathway of carotenoids can be considered to occur with the synthesis of GGPP [4].
Carotenoids are generally synthesized as all-trans isomers or at least as a mixture in which the all-trans form is dominant. Xanthophylls are enzymatically formed oxidation products of α-and β-carotene. The latest developments demonstrated that molecular genetic approaches were of considerable help in understanding carotenoid biosynthesis [5].
In human beings, carotenoids can serve several important functions. The most widely studied and wellunderstood nutritional role for carotenoids is their provitamin A activity [6]. Deficiency of vitamin A is a major cause of premature death. Experimental approaches that are likely to enhance our understanding of carotenoid pathway regulation are described in [7].
Pentose Phosphate and Glycolytic pathways Pentose Phosphate Pathway is an anabolic pathway that utilizes 6 carbons of glucose to generate NADPH and pentose (5-carbon) sugars. There are two distinct phases in the pathway. The first one is an oxidative phase, in which NADPH is generated, and the second phase is the non-oxidative synthesis of 5-carbon sugars. This pathway provides one of the three main ways the body creates reducing molecules to prevent oxidative stress, accounting for approximately 10% of NADPH production in humans.
Glycolytic pathway is a series of biochemical reactions by which a molecule of glucose is oxidized into two molecules of pyruvic acid. It is the initial process of many pathways of carbohydrate catabolism, and serves two principal functions: generation of high-energy molecules (ATP and NADH) and production of a variety of six or three-carbon intermediate metabolites that may be removed at various steps in the process for other intracellular purposes (such as nucleotide biosynthesis).
On Pentose Phosphate Pathway in E. coli K-12 MG1655 (Fig. 10) We use the pentose phosphate pathway (from the KEGG database [2]) in the organism E. coli K-12 MG1655. Here we are maximizing the rate of yield of D-Glyceraldehyde-3P and D-Fructose-6P, starting from the substrate α-D-Glucose-6P. There are 32 fluxes and 19 metabolites (Fig. 10). We associate the weighting factors c 1 , c 2 , . . . , c 32 corresponding to the enzymes catalyzing these reactions respectively. As in the previous systems, 32 dimensional flux vectors were generated. The objective function y is obtained by replacing z using z = c 21 v 21 −c 22 v 22 . Following the present method, we have obtained as an optimal pathway. For small values of λ, it requires more or less 50 iterations for convergence. As we increase the value of λ from 0.1 to 1.0, the optimal pathway is obtained within a few iterations (less than 50). The optimal pathway is obtained at λ = 0.5.
On Glycolytic Pathway in E. coli K-12 MG1655 (Fig. 11) The glycolytic pathway in E.coli K-12MG1655 consists of 14 metabolites and 30 fluxes (Fig. 11). The starting metabolite is α-D-Glucose-6P and the target product is pyruvate. Here we are maximizing the rate of yield z = c 30 v 30 of pyruvate, starting from the substrate α-D-Glucose-6P. We have obtained the hosphoenolpyruvate → P yruvate as an optimal one. For small values of λ, it requires more or less 46 iterations for convergence. As we increase the value of λ from 0.1 to 1.0, the optimal pathway is obtained within a very few iterations. The optimal pathway is obtained at λ = 0.5.
On Pentose Phosphate Pathway in P. falciparum (Fig. 12) The Pentose phosphate pathway in P. falciparum consists of 14 metabolites and 24 fluxes (Fig. 12). As in the previous cases, the rate of yield z = c 13 v 13 − c 14 v 14 of D-Glyceraldehyde-3P is maximized starting from the substrate α-D-Glucose-6P. Following the present method, we have obtained the same optimal pathway as α shown by bold (black) arrows, as in the previous real life example. Here for λ = 0.7, we got the optimal pathway within 70 iterations.
System boundary   . The starting metabolite is α-D-Glucose-6P and the target product is D-Glyceraldehyde-3P respectively. The bold (black) arrows represent the optimal pathway as obtained by the present method and the bold (white) arrows represent the optimal pathway as obtained by the extreme pathway analysis.