Incorporating heterologous metabolic pathways into well-characterized hosts is a major strategy for obtaining these target metabolites and improving productivity. However, selecting appropriate heterologous metabolic pathways for a host microorganism remains difficult owing to the complexity of metabolic networks. Hence, metabolic network design could benefit greatly from the availability of an Escherichia coli, Corynebacterium glutamicum, and Saccharomyces cerevisiae as templates. Using this algorithm, we screened heterologous pathways for the production of all possible nonnative target metabolites contained within databases. We then assessed the feasibility of the target productions using flux balance analysis, by which we could identify target metabolites associated with maximum cellular growth rate.We developed an algorithm for finding feasible heterologous pathways by which nonnative target metabolites are produced by host microorganisms, using in silico platform, designed for targeted searching of heterologous metabolic reactions, provides essential information for cell factory improvement.This  Escherichia coli, Saccharomyces cerevisiae, and Corynebacterium glutamicum with with S. cWe developed an algorithm to identify heterologous reaction(s) producing a target metabolite within a host microorganism. The algorithm expands the host metabolic network by sequentially adding heterologous metabolic reactions from our in-house database. The heterologous pathway identification procedure is as follows:1. A set of metabolites M0 and a set of metabolic reactions R0 are defined as those present in the genome-scale metabolic network of the host microorganism.2. From the in-house database, heterologous reactions that satisfy the following conditions are collected: (i) the reaction does not exist in R0, and (ii) it can produce metabolites that do not exist in M0 from a metabolite in M0. A set of these heterologous reactions is defined as R1, and a set of metabolites produced by reactions in R1 is defined as M1.3. In the same way, Ri is the set of reactions not present in {R0, R1, … , Ri − 1} which can produce metabolites not existing in {M0, M1, … , Mi − 1} from metabolites included in those sets. This expansion procedure is iterated until no further reaction is connectable to the expanded metabolic network.Mi, we can identify a set of heterologous reactions that are necessary to produce the target metabolite. For simplicity, all metabolic reactions in the database were assumed to be reversible. Of course some reactions are known to be irreversible, such as the carboxylation and decarboxylation reactions classified by Nomenclature Committee of the international Union of Biochemistry and Molecular Biology (NC-IUBMB) [If a target metabolite is included in a nonnative metabolite set C-IUBMB) . However, where FBA is based on a genome-scale metabolic model and optimization of a specific objective flux by linear programming ,44. We umentally -41,45.2, H2O, SO4 or SO3, and NH3. All calculations were performed using MATLAB 2009b . The linear programming problem was solved using GLPK 4.34 (GNU Linear Programming Kit) [In this study, the coefficients of metabolites representing biomass production flux were extracted from earlier studies -41. We eing Kit)  via the S. cerevisiae, E. coli, and C. glutamicum, we iteratively expanded the host metabolic network by adding heterologous metabolic reactions as described in the Methods section. Figure S. cerevisiae, E. coli, and C. glutamicum respectively.7,769 metabolic reactions and 6,635 metabolites . 1,3-Propanediol was then produced, aided by 1,3-propanediol oxidoreductase (encoded by dhaT). In this study, the screened heterologous pathways for C02457 production exactly matched those of the earlier studies. In E. coli, the screened production pathways of isoprene, α-farnesene, and PHB derived by our algorithm were also identical to those of the earlier studies, while similar heterologous genes introduced to the alternative hosts (C. glutamicum and S. cerevisiae) additionally produced these targets (see Table E. coli cells can produce (R)-propane-1,2-diol when methylglyoxal reductase and lactaldehyde reductase are added to the metabolic network, which has not been reported to date. Similar alternative pathways were found for the production of itaconate, ciscis-muconate, and 2,3-dihydroxybenzoate. These results suggest that our algorithm successfully identified the metabolic reactions necessary for the target productions and could assist in screening for potential host cells.As an example, the production pathways of 1,3-propanediol C02457) by  studies ,53, are 457 by  sNext, we used glucose as a carbon source to investigate whether these nonnative metabolites are producible by FBA simulations. In this simulation, the production flux of each nonnative metabolite was treated as an objective function to be maximized under the steady-state assumption. When the maximum production flux of a nonnative metabolite is zero, this metabolite is non-producible under the given condition.E. coli could not be produced using glucose as a sole carbon source. Similarly, 33% of the connectable nonnative metabolites of S. cerevisiae and 16% of the connectable nonnative metabolites of C. glutamicum were non-producible under this condition. These non-producible metabolites were identified by their tendency to disconnect when glycolysis formed the central metabolic pathway. In E. coli, these metabolites included trans-aconitate (C02341), butyrate (C00246), acetoacetate (C00164), and l-lactaldehyde (C00424).We calculated the maximum production fluxes of all connectable nonnative metabolites. 28% of the connectable nonnative metabolites of To evaluate the feasibility of nonnative target metabolite production, we performed FBA simulations under conditions of maximizing biomass production following heterologous reaction expansion of the genome-scale metabolic model. Metabolic flux profiles calculated at maximum biomass production rates have been shown to closely represent those in real microorganisms ,59-62. SIn Figure E. coli and S. cerevisiae, whereas two heterologous reactions were necessary to produce this metabolite in C. glutamicum. Vanillin can be produced under the same mechanism by introducing 4 heterologous reactions into the E. coli and C. glutamicum metabolic networks.Table Another mechanism is associated with NADH oxidization. Under micro-aerobic conditions, the cellular growth of microorganisms can be limited by NAD regeneration, which is necessary for glycolysis activity, and which occurs through NADH oxidization. Thus, when the heterologous reactions producing the targets are associated with NADH oxidization, these heterologous reactions are activated when the biomass production is maximized This phenomenon occurs, for example, in the production of (R)-propane-1,2-diol and 2-propyn-1-al.E. coli under conditions of maximum biomass production, such as (R)-propane-1,2-diol and adipate semialdehyde. Unlike S. cerevisiae and C. glutamicum, E. coli possesses NAD transhydrogenase, which can convert NADP and NADH to NADPH and NAD  respectively (and vice versa). In E. coli cells, the excess NADH is converted to NADPH which can then enter the target production pathway.We also found that some metabolites are produced only by P < 0.001). As shown in the figure, metabolites belonging to 5 categories, namely, “tyrosine metabolism,” “dioxin degradation,” “benzoate degradation,” “chlorocyclohexane and chlorobenzene degradation,” and “xylene degradation,” tended to be producible by S. cerevisiae and C. glutamicum but were scarce in E. coli cells.While screening for heterologous pathways to produce the target metabolites discussed earlier, differences in production capacity between the three host microorganisms emerged; for example, a group of metabolites was inducible by the addition of heterologous reactions to one of the hosts, but was not produced by the other hosts. To characterize the differences in target production capacity, we categorized the producible metabolites .Similarly, the metabolites in “flavonoid biosynthesis,” “phenylpropanoid biosynthesis,” and “nicotinate and nicotinamide metabolism” were preferentially generated by p-coumaroyl-CoA . Our genome-scale metabolic model does not account for transportation capabilities between compartments, which are currently unclear for many metabolites, and which might influence the production capacities of target metabolites in real cell systems.In yeast cells, the compartments present barriers to metabolite transport. For instance, mitochondrial/cytoplasmic interfaces prohibit the production of certain target metabolites when sugar is used as a carbon source. Similarly, the production of metabolites in the “flavonoid biosynthesis” category was inhibited in yeast cells because the transportation of 4-coumarate between the mitochondria and the cytosol is not permitted; therefore, the yeast strain could not produce S. cerevisiae, E. coli, and C. glutamicum as templates. Our results are consistent with those of earlier reports and provide additional alternative heterologous pathways. Producible nonnative metabolites predicted by our platform include industrial chemical compounds such as isoprene, α-farnesene, PHB, cadaverine, 1,3-propanediol, 1,2-propanediol, and vanillin. We propose that our platform is applicable to any genome-scale models that simulate cell factories. The platform greatly reduces the time and cost of heterologous pathway searching for target metabolites. Furthermore, appropriate expansions of the proposed system , could significantly improve the scope of our system. We believe that this platform will accelerate the rational design of metabolic systems and thereby enhance microbial production of essential metabolites.In conclusion, we developed a computational platform to investigate the extent to which industrial hosts can synthesize nonnative metabolites. Biosynthetic capabilities are evaluated by pathway design and flux calculations. We tested our platform using the industrial hosts The program for our pathway search algorithm is available athttp://www-shimizu.ist.osaka-u.ac.jp/pathway_search.zip. The program is written in Python. After extracting “pathway_search.zip”, the tool can be started by double clicking “runningScript.py” or by opening “runningScript.py” in Python IDLE, followed by pressing F5. All connectable nonnative metabolites including heterologous reaction are contained in the iteration folder. The folder input contains the necessary input files for identifying heterologous reactions of nonnative metabolites induced in a specified host.No competing interests declared.SC constructed the algorithm and performed the simulations. CF participated in the design of the study and drafted the manuscript. HS conceived and supervised the study. All authors revised and approved the final manuscript.List of reactions used in this study. The sheet “kegg_reaction_information” contains the metabolic reactions from the KEGG ligand database. Click here for fileList of connectable nonnative metabolites whenCorynebacterium glutamicum  was used as the host. The sheet “C.glutamicum_connectable” contains all of the connected metabolites, including heterologous reaction(s), information about gene(s) from the KEGG database and the minimum Km value from the BRENDA database. Click here for fileList of connectable nonnative metabolites whenEscherichia coli  was used as the host. The sheet “E.coli_connectable” contains all of the connected metabolites, including heterologous reaction(s), information about gene(s) from the KEGG database and the minimum Km value from the BRENDA database. Click here for fileList of connectable nonnative metabolites whenSaccharomyces cerevisiae  was used as the host. The sheet “S.cerevisiae_connectable” contains all of the connected metabolites, including heterologous reaction(s), information about gene(s) from the KEGG database and the minimum Km value from the BRENDA database. Click here for fileList of producible nonnative metabolites whenCorynebacterium glutamicumwas used as the host.The sheet “C.glutamicum_maxTarget” contains all of the producible metabolites under the target maximization condition, including heterologous reaction(s), information about gene(s) from the KEGG database and the minimum Km value from the BRENDA database. The sheet “C.glutamicum_maxBiomass” contains the producible metabolites under the biomass maximization condition, including heterologous reaction(s), information about gene(s) from the KEGG database and the minimum Km value from the BRENDA database. Click here for fileList of producible nonnative metabolites whenEscherichia coliwas used as the host.The sheet “E.coli_maxTarget” contains all of the producible metabolites under the target maximization condition, including heterologous reaction(s), information about gene(s) from the KEGG database and the minimum Km value from the BRENDA database (nonstandard format). The sheet “E.coli_maxBiomass” contains the producible metabolites under the biomass maximization condition, including heterologous reaction(s), information about gene(s) from the KEGG database and the minimum Km value from the BRENDA database. Click here for fileList of producible nonnative metabolite whenSaccharomyces cerevisiae  was used as the host. The sheet “S.cerevisiae_maxTarget” contains all of the producible metabolites under the target maximization condition, including heterologous reaction(s), information about gene(s) from the KEGG database and the minimum Km value from the BRENDA database. The sheet “S.cerevisiae_maxBiomass” contains the producible metabolites under the biomass maximization condition, including heterologous reaction(s), information about gene(s) from the KEGG database and the minimum Km value from the BRENDA database. Click here for file


Cellulose from plant biomass is an abundant, renewable material which could be a major feedstock for low emissions transport fuels such as cellulosic ethanol. Cellulase enzymes that break down cellulose into fermentable sugars are composed of different types - cellobiohydrolases I and II, endoglucanase and β-glucosidase - with separate functions. They form a complex interacting network between themselves, soluble hydrolysis product molecules, solution and solid phase substrates and inhibitors. There have been many models proposed for enzymatic saccharification however none have yet employed a cellular automaton approach, which allows important phenomena, such as enzyme crowding on the surface of solid substrates, denaturation and substrate inhibition, to be considered in the model.de novo taking into account the size and composition of the substrate and surface-acting enzymes were ascribed behaviors based on their movements, catalytic activities and rates, affinity for, and potential for crowding of, the cellulose surface, substrates and inhibitors, and denaturation rates. A basic case modeled on literature-derived parameters obtained from Trichoderma reesei cellulases resulted in cellulose hydrolysis curves that closely matched curves obtained from published experimental data. Scenarios were tested in the model, which included variation of enzyme loadings, adsorption strengths of surface acting enzymes and reaction periods, and the effect on saccharide production over time was assessed. The model simulations indicated an optimal enzyme loading of between 0.5 and 2 of the base case concentrations where a balance was obtained between enzyme crowding on the cellulose crystal, and that the affinities of enzymes for the cellulose surface had a large effect on cellulose hydrolysis. In addition, improvements to the cellobiohydrolase I activity period substantially improved overall glucose production.The Cellulase 4D model was developed Cellulase 4D simulates the enzymatic hydrolysis of cellulose to glucose by surface and solution phase-acting enzymes and accounts for complex phenomena that have previously not been included in cellulose hydrolysis models. The model is intended as a tool for industry, researchers and educators alike to explore options for enzyme engineering and process development and to test hypotheses regarding cellulase mechanisms. As an abundant and renewable material, biomass has been investigated extensively as a precursor of biofuels that can replace current fossil oil-based transport fuels and reduce greenhouse gas emissions. Biomass occurs in a variety of forms but commonly includes cellulose, a highly stable polymer of glucose, as a major component. The saccharification of cellulose, breaking the polymer into monosaccharides, yields glucose, which can undergo microbial fermentation to biofuels such as ethanol and butanol, or be used as a feedstock for the production of lipids and chemicals. Achieving efficient saccharification from complex biomass is challenging, and one of the main factors retarding the commercial biochemical method of cellulosic fuels and chemicals production .Enzymatic saccharification of cellulose is achieved through the combined actions of 1,4-β-D-endoglucanases (EG) and 1,4-β-D-exoglucanases (or cellobiohydrolases - CBHI and CBHII), collectively known as cellulases. The products of these combined activities are the disaccharide, cellobiose, and polysaccharides such as cellotriose and cellotetraose, which are further reduced to monosaccharides by β-glucosidase (BG), and sometimes also by EG and CBH. The combined activities of these enzymes have been previously described in functionally-based cellulase models -5, and tAn approach that addresses this complexity is modeling using cellular automata (CA), that is, treating cellulose and enzymes as individual entities and assigning behaviors to each based on published kinetic values. CA methods are being applied to an ever-increasing variety of biological -11 and nOur aim was to develop a cellulose-cellulase model that provides control over many of the physical and chemical variables occurring during the enzymatic saccharification of cellulose in either the laboratory or on an industrial scale. In this report we demonstrate the model's utility in hypothesis-testing as a guide to cellulase enzyme research and improvement in rates of cellulose hydrolysis. The model is designed to be applicable across a broad range of cellulases, rather than tailored to describe enzymes from any particular source, and requires no previous knowledge of programming or scripting, or the use of specialist software packages.Trichoderma reesei. The resultant cellulose conversion curves produced by the model using the base case parameters closely mimic the shape of those found experimentally, being characterized by a fast initial rate of hydrolysis followed by a marked reduction in hydrolysis rate as cellulose degradation progresses [T. reesei system is our selection of a small enzyme 'footprint' on the cellulose surface, which has enabled simulations to be run in a reasonable timeframe by reducing the crowding effect.The base case for the model utilizes parameters selected from the literature describing the cellulases of ogresses   between the base case and doubling the loading of the enzyme mixture. The final glucose production at all other loading levels tested in the simulation  were significantly reduced (P < 0.05) compared with the base case. In the follow-up trial, where the concentration of EG alone was reduced from the base case level of 800 to 400 mM while holding the other enzymes at base case levels, no significant effect was noted on final glucose concentrations  and the rates of glucose release. Examination of the production of cellobiose with a reduction in EG concentration showed a negative effect on yield, as shown in Figure Reducing or increasing the concentration of the enzyme mixture relative to cellulose while maintaining the same concentration ratios of the enzymes generally had a negative effect on glucose release rates at the early stages of the simulations Figure . Howevers Figure . HoweverF 3618.74; P < 0.05) at the end of the simulation when adsorption strength was either increased or decreased compared to the base case  in overall glucose amounts at the end of simulation  in glucose production rate where absorption strength was set at 9800 and below  in the amounts of glucose released . Thus, the slowing effect on glucose production rates of lower enzyme absorption strengths as shown in Figure In the first iteration of the base case, the reaction period for EG was varied between 1 ms and 128 ms at an adsorption strength of 9,999 to determine the effect on amount of glucose released of varying EG activity. No noticeable effect on the production of glucose was observed between these extremes of activity period Figure . The rea Figures  and 4C. In the example where both CBHI and CBHII had their reaction periods reduced from 500 ms (base case) to 4 ms, a 125-fold improvement, a highly significant increase, in the rate of glucose and cellobiose production was observed  [The slow catalytic rates of CBH enzymes (0.1 s-1)  and theiIt has been previously reported that enzyme synergism decreases at higher enzyme loadings, and that at saturation loading, cellulose hydrolysis becomes inhibited . EarlierHowever, improvement to the CBHI activity period had a major effect on the overall glucose production rate and resulted in the greatest increase among the scenarios tested. A reduction in CBH reaction rate had no effect on the rate of glucose production, due to enzyme crowding, until after the rate of cellobiose production had slowed. At this point, BG activity was no longer the rate-limiting step in glucose production. During the initial stages of hydrolysis, the CBHs released more cellobiose to solution before crowding became too severe and slowed reactions, but this resulted in higher levels of glucose produced by the end of the simulation as BG subsequently hydrolyzed the accumulated cellobiose.EG had a strong influence on the overall rate of hydrolysis in our model. Although this may be partially the result of the much slower acting CBH enzymes, it is also possible that the longer polysaccharides released from cellulose by the EG do not dissociate from the surface as readily in the physical system compared to the model. The continued association of 4-, 5- and 6-sugar polysaccharides with the cellulose surface would likely have the effect of hindering access of EGs to appropriate surface reaction sites, and also reduce the apparent rate of glucose production by being less accessible to enzymes that further degrade them to shorter polysaccharides and, eventually, glucose. Additionally, any saccharides that are inhibitors of the surface adsorbed enzymes would have a much higher apparent concentration from the perspective of surface adsorbed EG and CBHs compared to those same inhibitors in solution.Owing to the high computational cost of such a large CA model and the need for pragmatic simulation times, certain parameters were exaggerated or undervalued in the model. The smallest plausible enzyme spatial requirement was utilized, which greatly facilitated shorter simulations; it was assumed that no enzymes were denatured on the cellulose surface and high adsorption probabilities (90%) were used. The small spatial requirement allowed EG to cleave points in the cellulose chain quite close to each other, which resulted in the rapid formation of the short, soluble polysaccharides. The time taken for each simulation could potentially be shortened by modifying the algorithms for parallel processing which would greatly reduce the time required.The model presented herein is a framework upon which further refinements can be made and currently represents only the fundamental aspects of cellulase activity. Recently, another class of cellulases has come to light that employ reactive oxygen species to attack glycosidic bonds in cellulose ,22. FutuWe have described the cellular automaton model Cellulase 4D that simulates enzymatic hydrolysis of cellulose to glucose by surface and solution phase-acting enzymes. After establishing a base case for cellulose hydrolysis by implementing kinetic values obtained from the literature, we examined the effect on rates of degradation and saccharide production of altering key enzyme parameters. Adsorption strength of EG, reaction periods for CBH and enzyme loadings used were identified as having significant effects on cellulose hydrolysis.T. reesei (Hypocrea jecorina). The main outputs of the model are the concentrations of saccharide species in solution over time and the percentage of the initial cellulose that is hydrolyzed in a given time period.Cellulase 4D describes the decomposition of a crystal of cellulose by the actions of a mixture of enzymes that work in concert on both insoluble and dissolved substrates, eventually yielding glucose and some residual polymeric material. Many parameters governing the behaviors of the enzymes can be modified to allow the input of new enzyme properties, or for hypothesized activities to be tested. A 'single-enzyme' approach was suggested by Gentry and co-workers ,24 whereIn Cellulase 4D, a single virtual cellulose crystal and enzymes exist in a three-dimensional space with periodic boundary conditions, that is, an enzyme exiting one side of the space will reappear on the opposite side of the space. This rule maintains a constant concentration of enzyme and is an effective way to simulate a representative part of a larger system. The model incorporates the activities of both the adsorbed enzymes  and those working in solution . Each enzyme and its interactions with the range of substrates and inhibitors produced are considered separately and their characteristics are described in the relevant sections. For many of the model parameters pertaining to the enzymes, such as product inhibition constants or the time taken to complete a reaction on the cellulose surface, the user can choose to either input default settings based on experimentally-determined values or they can input new hypothetical values to model the effect of those changes on cellulose hydrolysis rate and the levels of the various species in solution. Other parameters, such as the size of the enzymes' 'footprint' and the size of the three-dimensional space surrounding the cellulose, are fixed and unable to be altered. Outputs, such as saccharide species in solution and the percentages of cellulose hydrolyzed and of enzyme amount adsorbed to the surface, are plotted graphically and also saved in tabulated form as the simulation progresses.Factors that have been accounted for in the model at the level of an individual enzyme include specific activities and substrate preferences of each of the component cellulases; enzyme promiscuity, as some cellulases can hydrolyze both soluble and insoluble substrates; inhibition of enzyme activity, both in solution and on the cellulose surface, by reaction products; strength of adsorption and probability of enzymes adsorbing to the cellulose surface once in proximity; and the potential for crowding at higher concentrations of enzyme.Km, kcat and Ki of all enzymes with respect to activities, if any, on soluble saccharides, are selected, and the kinetic parameters of CBHI, CBHII and EG with respect to the cellulose are established. The option for all enzymes to be either inhibited by, or active upon, any or all soluble polysaccharides is available if the user wishes to incorporate solution activity into the model. Enzymes also have a 'lifetime' (see Denaturing below) after which they become denatured but either remain in solution or adsorb to the surface and contribute to crowding. Once adsorbed to the surface, the enzymes do not diffuse into solution unless they are adsorbed to a polysaccharide that is solubilized through the action of another enzyme.Prior to initiating simulation, kinetic parameters, such as Valonia macrophysa and imaged by Lehtiö et al. [Valonia cellulose is a very highly structured form of cellulose, with each single crystalline microfibril having an approximate cross-sectional area of 20 × 20 nm [Valonia; as small as 3 nm and containing around 30 to 36 cellulose chains [The virtual cellulose crystal used in the model is based on a highly structured cellulose crystal, such as that produced by ö et al.  in a rep × 20 nm  and cont × 20 nm . Higher e chains .In the model, the virtual cellulose consists of glucose chains running the full length of the crystal that are characterized as either 'edges', 'flatsurfaces' or 'subsurfaces' Figure  and 6B. As hydrolysis of glucose residues from cellulose progresses through the activity of EG and CBHs, shorter saccharides appear on the cellulose surface. Where the chain length is six glucoses or less, the saccharide is considered soluble in the model and contributes to the overall sugar species in solution, where it can then be further hydrolyzed by enzymes in solution. Any enzymes that were located on the dissolving saccharide are considered to be desorbed from the cellulose.Enzymes diffuse through solution according to a 'random walk' algorithm  and adsoet al. [et al. [Cellulomonas fimi cellulases on the cellulose surface ranged from 2 × 10-11 to 1.2 × 10-10 cm2/s which equates to around 200 Å2 (10 square glucose molecules) per millisecond. Our selection of 5 Å increments are supported by molecular dynamics simulations reported by Bu et al. [A random walk is used to mimic Brownian motion of enzymes in solution. At each time point, a random vector in three-dimensional space is chosen and the enzyme moves 5 Å in that direction. While it is understood that an enzyme will diffuse much further than 5 Å in 1 ms in aqueous solution, we have found our approximation gives a satisfactory representation of previously reported  adsorptiet al.  found no [et al. , the difu et al.  that indkcat for the enzyme in question, an activity period. Both the kcat and standard deviation for each enzymatic reaction occurring in solution are selected before the simulation is initiated.Once a reaction site is located on the cellulose rod and the enzyme becomes active, the time taken for the reaction to go to completion is calculated by selecting a random point under a normal distribution curve centered on 1/Cellulases display many different adsorption characteristics ,36-38 raThis is the probability of adsorption of an enzyme in solution that comes into close proximity of the cellulose surface, and is dictated by a parameter chosen for each enzyme prior to running a simulation. It is not expected that every collision of an enzyme with the cellulose would result in a successful binding event. This also provides the ability to slow down the overall rate of adsorption if users are modeling systems with slower adsorption profiles. When a desorption event occurs, determined by the adsorption strength parameter (see above), the desorbed enzyme is returned to the solution, increasing the concentration of that enzyme in solution. This sets up a dynamic equilibrium between adsorbed enzymes and those in solution.cat values for the hydrolysis of cellobiose, cellotriose, cellotetraose, cellopentaose and cellohexaose are available as default settings or user-selected for EG in the model prior to each simulation. In the cases of cellotetraose and longer polysaccharides, the hydrolysis point along the chain is chosen at random in the model. For instance, a cellopentaose can be hydrolyzed to cellobiose and cellotriose species, or glucose and cellotetraose species with equal probability.EGs act by hydrolyzing cellulose chains at random positions, creating reducing and non-reducing ends at the cleavage point. The behavior of EG in the model is described as follows. Once the EG locates an appropriate reaction position on cellulose and has become active (determined by a 'reaction probability' variable - see Equation 1), it remains stationary until the activity period has elapsed. It then performs the hydrolysis, its state is then reset to 'inactive', and it resumes a random walk pattern on the cellulose surface. EG activities are diverse, also acting on soluble substrates in solution such as cellotriose, cellotetraose, cellopentaose and cellohexaose to smaller polysaccharides -42 and aet al. [cat values for the enzyme. For crystalline bacterial cellulose with a degree of polymerization of 1700 glucose units, the processivity value for Cel7A was 176 ± 20 with a kcat of 9.5 × 10-2/s. CBHs are slow catalysts in comparison with other enzymes; one cellobiose product is released from cellulose at intervals of between approximately 1 and 10 seconds [cat values in the order of 1 per 0.01 to 1 second for β-glucosidases [CBHI and CBHII are processive cellulases that hydrolyze cellobiose units from a cellulose chain, starting from either a reducing end (CBHI) or a non-reducing end (CBHII) that occurs where the cellulose chain is cleaved. After CBHs adsorb to the cellulose surface, they move about it in a random walk in increments of 1 glucose (5 Å) until a break containing a valid reaction site for the enzyme is located. Kipper et al.  examined seconds  comparedosidases .In addition to random walk and adsorbing to the cellulose surface, CBHs (inactive) also have probabilities of desorbing from the surface or denaturing after a period of time. These are governed by the adsorption strength and 'lifetime parameters', respectively. CBH activity may also be inhibited by any or all of the saccharides in solution (see section Surface adsorbed enzyme inhibition), however, this does not affect their random walk, nor change their desorbing or denaturing probabilities in the model. If a CBH encounters another enzyme on the surface during the random walk phase it is precluded from moving to that point and will wait another program cycle (1 ms) before searching for a vacant adjacent point to move to. Likewise, if the active CBH encounters another enzyme during processive hydrolysis it is programmed to stop at that point until the cellulose chain becomes free of enzymes for a minimum distance of two glucose units ahead.Enzymes on the surface of the cellulose are either inactive  or active (complexed to the substrate and involved in a reaction). The length of time that a surface-adsorbed enzyme remains active after initiation of a reaction is user-determined prior to the start of the simulation via the activity period variable (described above), which is either set by the user or default values can be used. While active, surface adsorbed enzymes are stationary and while inactive, they are mobile.Cellulases have been shown to denature during the course of cellulose hydrolysis . DenaturSurface crowding is particularly difficult to account for using traditional Michaelis-Menten kinetics alone and most models thus far have chosen not to address this phenomenon explicitly. A coincident sharp decrease in the rate of cellulose hydrolysis in some cases with high enzyme loadings has previously been attributed to crowding ,48 and oet al. [T. viride regarding their adsorption profiles, that is, what quantity of enzyme will fit on a given amount of cellulose, with values ranging from 0.007 mg to 0.126 mg enzyme per mg Avicel. In order to estimate surface coverage by enzyme, we conducted a surface area analysis of Avicel using nitrogen adsorption and determined the surface area to be 3.74 × 10-3 m2/mg. A discussion of the various merits and drawbacks of different methods for measuring cellulose surface area is beyond the scope of this report, and these aspects have been covered in detail elsewhere [2 , then 1 mg of a T. reesei Cel7A catalytic domain (without CBM or linker) of an approximate molecular weight of 46,000 g/mol will have a footprint of 0.196 m2. So, for the lowest loading case in Beldman et al. [-3 m2 footprint on a total surface of 3.74 × 10-3 m2, or approximately one-third coverage. However, for the highest loading case of enzyme there would have been approximately six times coverage. In reality, this is likely to be significantly higher given the nitrogen probe provides an overestimate of the enzyme-accessible surface area owing to its small size compared to that of an enzyme. This could imply that EGs and CBHs can adsorb to cellulose in multiple layers [T. reesei CBM footprint is 260 Å2, approximately one-sixth that of the catalytic domain, which equates almost perfectly to total coverage of cellulose by one layer of CBMs at the highest loading described by Beldman et al. [Crowding (or jamming) of cellulases on the cellulose surface is a phenomenon that has received recent attention ,50 and het al.  showed tlsewhere ,52. If an et al.   for the enzyme is reduced by half.There are several different types of enzyme inhibition that have been experimentally observed for cellulase enzymes . In the i value, that is, an inhibitor may be highly effective because it has a high probability of a successful binding event per collision, or it may have a lower probability of a successful binding event but it may have a much longer residence time in the enzyme, making it highly effective. Therefore, in the model we have simplified the potential inhibition of cellulose-adsorbed enzymes by implementing an activity probability variable  which changes according to the concentrations of the various inhibitors in solution. If there are no inhibitors present in solution, activity probability is set to 1. Equation 1 demonstrates how the activity probability for adsorbed, inactive EG is calculated at each timestep:It is difficult to decouple the contributing effects that result in an experimentally determined KA graphical representation of how activity probability changes with glucose inhibitor concentration is shown in Figure v, in M.reactions per second) is calculated, which is then used to determine how many reactions will occur at each timestep. An example of how the solution reaction velocity between an endoglucanase and cellotriose is calculated, taking into consideration the concentrations of all other substrates and inhibitors, is shown in Equation 2:Different reaction and inhibition behaviors were programmed for the soluble enzymes to those operating on the cellulose surface; they are not considered as individual enzymes but instead mean-field solution reactions whose parameters are chosen at setup. From these, a reaction velocity  have the same definitions as understood in traditional Michaelis-Menten kinetics. The reaction velocity v is converted from M.reactions per second to reactions per millisecond  in the total volume and the appropriate polysaccharide concentrations are updated accordingly.where countOutputs such as types and concentrations of saccharide species in solution against time, the percentage of initial cellulose hydrolyzed and enzyme amounts adsorbed to the surface are plotted graphically by the program and saved in tabulated form as the simulation progresses Figure . TabularThe following variables may be user-defined prior to simulation or default values selected:• Cellulose dimensions • All enzyme concentrations (mM) kcat and kcatSD: Average number of reactions per second per enzyme and the standard deviation. All enzymes have the option to be active on all substrates, with the exception of BG on cellulose.•i: All enzymes may optionally be inhibited by all saccharide species up to celloahexaose.• K• Adsorption strength  for EG, CBHI and CBHII only• Adsorption probability: The probability that an enzyme will bind to the cellulose surface when within 20 Å.• Lifetime: All enzymes (seconds)T. reesei cellulases [Three scenarios were investigated which examined sensitivities in the model; these involved varying enzyme concentrations, adsorption strengths of EG and CBH and catalytic activities of EG and CBHI, and monitoring the rate of release of saccharides over time. The experimental designs are detailed below and each test of conditions was examined in triplicate simulation experiments. Firstly, a base scenario that employed average conditions and kinetic parameters from llulases ,34,57-60Elevated enzyme concentrations may result in an increased rate of cellulose hydrolysis but may also contribute to crowding on the cellulose surface, causing a reduction in hydrolysis; these potential outcomes were investigated using Cellulase 4D. The concentrations of all enzymes were varied by factors of 0.25, 0.5, 2, 3 and 4 in comparison to the concentrations used in the base case = 1, Table , Table 1As described earlier, the adsorption strength parameter describes the probability of an enzyme spontaneously desorbing from the cellulose surface at any particular timestep. The selection of a parameter value of one causes the enzyme to desorb from cellulose in the next timestep, and at 10,000, the enzyme never desorbs unless it is on a section of cellulose that becomes soluble. For this scenario the absorption strength parameter was varied between 10,000 and 9,700 for EG and CBH enzymes and compared with the base case where the value was set at 9,999, and the production of saccharides over time was measured. The simulations were re-conducted varying the CBHI and EG adsorption strength parameters individually while maintaining the other enzymes at base case strengths.A complementary experiment to increasing the EG concentration while holding the others constant, as described above, was to increase the catalytic rate at which EG hydrolyzed cellulose. This was assessed by decreasing the time it took for each EG enzyme to complete a hydrolysis reaction after locating an appropriate reaction point. In this scenario there was potential for the accumulation of saccharide products that may result in increased competitive inhibition of hydrolysis. Further, the adsorption strength of EG was lowered to determine any effect this may have on crowding and saccharide production at various EG reaction periods. Similarly, the CBH reaction period was incrementally reduced from that of the base case, with all other parameters held at base case values, to determine any effect on the production of glucose and cellobiose over time.P <0.05 were conducted comparing glucose production between conditions.Concentration of glucose over time for replicate experiments from the base case and all scenarios were downloaded into Microsoft Excel spreadsheets. The final glucose concentrations for each scenario were examined for differences by one-way analysis of variance; where significant differences were found within a treatment, posthoc Student's t-tests with significance level of EG: total number of EG in solution; EG: 1,4-β-D-endoglucanase; kcat: number of reactions per second of an enzyme working at vmax; kcatSD: Standard deviation of kcat: Ki: dissociation constant - the concentration of inhibitor at which the activity of an enzyme is reduced to half (v = 1/2vmax); Km: Michaelis constant - the concentration of substrate at which the activity of an enzyme is reduced to half (v = 1/2vmax); v: reaction velocity for mean-field solution reactions (Equation 2); vmax: maximum rate at which an enzyme catalyzes a reaction.BG: β-Glucosidase; CBHI: 1,4-β-D-cellobiohydrolase(exoglucanase) 1; CBHII: 1,4-β-D-cellobiohydrolase II; CBM: carbohydrate binding module; countThe authors declare that they have no competing interests.ACW conceived the idea, designed the program, wrote the initial C++ code, conducted the scenarios and drafted the manuscript. BAL wrote the GUI, modified CUDA Particles for the 3D graphics display and assisted with some of the algorithm development. VSH contributed to experimental design of the program, ideas for scenarios and drafting the manuscript. All authors suggested modifications to the draft-read preliminary versions and final version, and approved the final manuscript.


Spirulina (Arthrospira) platensis is a well-known filamentous cyanobacterium used in the production of many industrial products, including high value compounds, healthy food supplements, animal feeds, pharmaceuticals and cosmetics, for example. It has been increasingly studied around the world for scientific purposes, especially for its genome, biology, physiology, and also for the analysis of its small-scale metabolic network. However, the overall description of the metabolic and biotechnological capabilities of S. platensis requires the development of a whole cellular metabolism model. Recently, the S. platensis C1 (Arthrospira sp. PCC9438) genome sequence has become available, allowing systems-level studies of this commercial cyanobacterium.S. platensis C1, iAK692, its topological properties, and its metabolic capabilities and functions. The network was reconstructed from the S. platensis C1 annotated genomic sequence using Pathway Tools software to generate a preliminary network. Then, manual curation was performed based on a collective knowledge base and a combination of genomic, biochemical, and physiological information. The genome-scale metabolic model consists of 692 genes, 837 metabolites, and 875 reactions. We validated iAK692 by conducting fermentation experiments and simulating the model under autotrophic, heterotrophic, and mixotrophic growth conditions using COBRA toolbox. The model predictions under these growth conditions were consistent with the experimental results. The iAK692 model was further used to predict the unique active reactions and essential genes for each growth condition. Additionally, the metabolic states of iAK692 during autotrophic and mixotrophic growths were described by phenotypic phase plane (PhPP) analysis.In this work, we present the genome-scale metabolic network analysis of S. platensis C1, iAK692, which is a predictive metabolic platform for a global understanding of physiological behaviors and metabolic engineering. This platform could accelerate the integrative analysis of various “-omics” data, leading to strain improvement towards a diverse range of desired industrial products from Spirulina.This study proposes the first genome-scale model of  Spirulina (Arthrospira) platensis is a filamentous non-N2-fixing cyanobacterium that has become important as a source for commercially produced nutraceutical compounds, as this cyanobacterium utilizes sunlight and CO2 to produce chemical compounds that are essential for life. Spirulina has been consumed as a protein source for many years by North Africans and Mexicans [S. platensis is potentially one of the algae capable of producing bioenergy and renewable energy, which could help to decrease the effects of global-warming [S. platensis is capable of growing in outdoor environments at a high rate [S. platensis an attractive photobiological cell factory.Mexicans  because Mexicans . MoreoveMexicans . Further-warming . Among tigh rate . In termin silico predictions of gene functions and the metabolic capabilities of an organism [in silico models  [The growing availability of genomic sequences and software technologies has made it possible to reconstruct genome-scale metabolic networks of various organisms. Genome-scale metabolic models come from the systematic reconstruction of all cellular biochemical reactions according to the genetic information of a given organism . A vast organism . By applorganism ,12, the organism ,14 with ganisms) -17.Synechocystis sp. PCC6803 has been developed by independent research teams around the world [Synechocystis sp. PCC6803 as a photobiological cell factory. With such an impressive advantage of S. platensis, especially as a nutraceutical, as previously mentioned, S. platensis has become one of the preferred choices for a sustainable photobiological cell factory. Unfortunately, there have only been a limited number of attempts to computationally analyze the metabolism of Spirulina. A simple metabolic flux model of S. platensis consisting of 22 reactions was proposed by Meechai et al[S. platensis comprising 121 reactions and 134 metabolites was formulated by Cogne and his team [S. platensis. Recently, the genome sequence of S. platensis C1 became available [S. platensis.Various genome-scale metabolic models of many organisms are currently available . Howeverhe world -21. Eachhis team . This movailable , togetheS. platensis , representing global growth behaviors under three different growth conditions: autotrophic, heterotrophic, and mixotrophic. The metabolic network is based on the S. platensis C1 genome, a collective knowledge base, and extensive manual curation. Computational simulation was performed using COBRA toolbox [in silico predictions were further validated with experimental evidence. Various analyses of the iAK692 model were performed to identify active reactions and essential genes under each growth condition. Moreover, phenotypic phase plane (PhPP) [iAK692 during autotrophic and mixotrophic growths. The iAK692 model not only provides further physiological knowledge of the cellular system, but is also a valuable platform for integrating multilevel “-omics” data, which could provide further insight towards increasing the number of desired industrial bioproducts from Spirulina.This paper presents the first genome-scale metabolic model of  toolbox ,14. The e (PhPP)  analysisS. platensis C1, the genome-scale metabolic network was reconstructed according to a series of extensive refinements  [In order to build a genome-scale metabolic model of D 67617) . The estD 67617) ,27. UsinS. platensis C1 based on the gene name or EC number matching method. This preliminary model accounted for 102 pathway frames (including a number of pathway variants), 557 genes corresponding to 1,661 biochemical reactions, and 1,542 metabolites. A summary of the draft model is shown in Table S. platensis C1; for instance, whether or not the predicted pathways or reactions present in S. platensis C1 and the numbers of incomplete pathways (missing gene and reaction) indicate a disconnection in the network. Hence, manual-intensive curation was subsequently performed to increase the accuracy of the draft auto-generated network.The Pathway Tools software ,27 prediSpirulina physiology, such as cholesterol biosynthesis I, II, and III pathways, were eliminated from the network. Pathway Tools mapped only one gene, crtB, encoding phytoene synthase, onto two reactions out of twenty-two reactions in these pathways for the draft model. However, we found that the phytoene synthase enzyme was involved in the presence of the carotenoid biosynthesis pathway. Since carotenoid is an important molecule in photosynthesis, the synthesis pathway of this compound was retained.Biochemical information about cyanobacteria was obtained from the literature and biochemistry textbooks, and online biochemical databases were used to research several iterative methods of reconstruction and were used to refine the draft model; this is referred to as the “Refined model”. In more detail, the predicted pathways/reactions that have not been reported for Then the Blast algorithm ,29 and t+), reaction directionality, transport reaction, and the completely characterized biomass equation were curated and added to the refined model. We checked the balance of protons in a reaction and also found that different databases contain some contradictory information. In this reconstruction, we used Metacyc [+ balancing. Additionally, the stoichiometry and reversibility of the reactions were manually verified and assisted by biochemical reaction databases such as KEGG [Synchocystis sp. PCC6803 [S. platensis. Instead, a total of 88 exchange reactions were included in the model to ensure that organic nutrient metabolites , inorganic nutrient metabolites , gaseous metabolites (i.e. O2 and CO2), photons, and water could enter and leave the system in response to the physiological state. We are aware that the transport mechanisms across the cell membrane should be considered in this study because of their possible effect on the model. This is an issue that we will address in a future study.In order to make the computation-ready model, referred to as the “Final model”, the balance between each element, including metabolites and protons . A biomass equation represents all necessary precursors that form the cellular biomass. Thus, 14 reactions with no gene association were added for the synthesis of intermediate macromolecules, i.e. proteins, carbohydrates, lipids, RNA, DNA, and chlorophyll, which are considered to form the major composition of biomass. The biomass formation equations were obtained based on data from different sources of experimental efforts for nditions   [Escherichia coli (iAF1260)[iFF708) [iAK692 has a similar network topology and metabolite hub to these models  , Escheri(iAF1260), and yea[iFF708) . It was S. platensis C1 naturally grows under autotrophic conditions using carbon dioxide as a carbon source and converting light as cellular energy, like other cyanobacteria. Several strains of these microalgae, including S. platensis, were recently researched for their potential to grow under heterotrophic and mixotrophic environments [S. platensis strain C1. Thus, we investigated the cellular properties of heterotrophic and mixotrophic growth in order to gain more basic physiological knowledge of S. platensis C1. Subsequently, we validated the reconstructed metabolic network by comparing the predicted results from in silico simulations with the experimental results. The measured maximum specific growth rates of S. platensis C1 under the three growth conditions are summarized in Table ronments . HoweveriAK692 were evaluated based on constraint-based modeling using the FBA technique [The basic capabilities of echnique ,12 to quS. platensis C1 was demonstrated in batch cultivation , the in silico cell could not grow  is only slightly higher than the autotrophic growth rate (0.0255 1/h) (Table hex gene knockout that results in zero flux of this reaction. The results show that the growth rate decreased from 0.0334 1/h to 0.0257 1/h (data not shown). Therefore, the different growth behaviors found for the different physiological properties and in silico predictions might require further experimental verification in order to discover further explanations and make new discoveries. All profiles of the flux distributions of the three growth conditions are presented in Additional file In the mixotrophic mode with carbon dioxide and glucose as carbon sources and light as the energy source, the measured growth rate under this condition was found to be slightly higher than that of the autotrophic mode Table . The modhis team . They suh) Table . We also rate 0.062 1/h isin vivo growth rates of iAK692 are consistent with the experimental growth rates found under the three different modes. In order to investigate the flux distributions in terms of active reactions, this model was only used for the simulation of autotrophic and mixotrophic growths because the cells failed to grow under heterotrophic conditions. The flux results show that iAK692 had 322 (41 %) and 307 (39 %) active reactions under the autotrophic and mixotrophic conditions, respectively. These two numbers of active reactions are close to those found in previous studies of other genome-scale models such as those for Clostridium beijerinckii[E. coliStaphylococcus aureusHelicobacter pylori and yeast [S. platensis  associated with each gene was deleted, one by one, by setting both the upper and lower bounds of a reaction to zero and optimizing for the biomass formation. In this study, the iAK692 model accounting for 875 biochemical reactions and 837 metabolites was employed to identify the essential genes for survival under autotrophic and mixotrophic growth conditions using the MOMA platform [iAK692 under autotrophic and mixotrophic conditions, respectively  [iFF708) [iAK692. Fig. The capacity of d models , the iAKplatform . It was ely Fig. . A list iAF1260) , and 10 iAK692 under autotrophic condition. Within the metabolic network, inorganic carbon was transported from the medium as bicarbonate ions into cells. The intracellular bicarbonate was then dehydrated via carbonic anhydrase enzyme to become CO2. The intensity of light input was represented by absorbed photon flux (APF). The surface of a three-dimensional PhPP corresponding to the predicted maximal growth rate as a function of the photon flux (0 – 100 μEinstein/m2/s) and the bicarbonate uptake rate (0–0.4 mmol/mmol dry cell/h) was plotted , the maximal growth was found to be linearly dependent on both the absorbed light and bicarbonate availability. At the APF above 45 μEinstein/m2/s (Region III), the cell growth is limited by the bicarbonate ion availability. It is noteworthy to mention that our model fails to describe the photoinhibition at high APFs, although it has been well documented that photoinhibition is observed in autotrophic cultures of Spirulina. This is another issue that we will address in a future study.Phenotypic phase plane (PhPP) analysis is a useful approach for examining the steady-state solution space when two variables of interest are varied . We emplted Fig. A. It wasiAK692 grown under varying bicarbonate and glucose uptake rates in the presence of constant APF (46.7 μEinstein/m2/s) was performed to investigate the relationship between photosynthesis and respiration growth of iAK692 during mixotrophy. The PhPP plot shows three distinct regions describing different growth phenotypes  automatic reconstruction by Pathway Tools software [.Herein, we reconstructed a genome-scale metabolic network of the cyanobacterium software ,27; and S. platensis C1. This approach allowed an estimation of the flux distribution of the entire metabolism. The simulated results were validated by experimental evidence and showed satisfactory agreement under three different growth conditions; namely, autotrophic, heterotrophic, and mixotrophic. Analyses of the iAK692 model enabled us to gain insights into the metabolic phenotypes and essential genes of S. platensis C1 grown under these conditions. With a growing Spirulina community, the proposed model would not only be useful for studying cellular phenotypes but it could also serve as a platform for “-omics” data integration in order to achieve the beneficial stage of model-driven discovery in Spirulina systems biology [We used the COBRA toolbox ,14 to in biology .S. platensis strain C1 was used in this study. Cells were grown at 35 °C in 1,000 ml Erlenmeyer flasks with a culture volume of 500 ml and continuous stirring. Autotrophic and mixotrophic cultures were grown under fluorescent light at 100 μEinstein/m2/s. Zarrouk’s medium [s medium  was used2SO4[The maximum growth rates under autotrophic, heterotrophic, and mixotrophic growth conditions were measured by the optical density at 560 nm and compared with the standard curve. The results from these three conditions are shown in Table 2SO4. The lev2SO4. The maxS. platensis C1 was formulated using a combination of two procedures: automatic and manual reconstruction  [S. platensis C1 by comparing the data to the MetaCyc database [Spirulina from the literature, books and published databases, such as KEGG [S. platensis C1 genome via the homology search because of its physiological evidence. During the manual curation, iterative modeling was performed using FBA for checking the completeness and consistency of the model and experiment. Some reactions were added to fulfill the system connection based on reactions present in closed organisms. The process of curation is iterative until the gaps in a draft metabolic network are filled [S. platensis C1 was further used to predict growth behavior under autotrophic, heterotrophic, and mixotrophic conditions. Since no biomass composition data for S. platensis C1 are available, the stoichiometry of the biomass formation reaction used in this study was obtained from the work of Cogne et al[A metabolic network of D 67617)  were useD 67617) ,27, whicdatabase  as a key as KEGG , Brenda  as KEGG , CyanoBa as KEGG  and upda as KEGG , were us as KEGG . The Pat as KEGG ,29 was u as KEGG  , if it e filled ,53. An agne et al.S. platensis C1. The column and row represent a reaction and a metabolite, respectively; each element is a stoichiometric coefficient. For each network, metabolite connectivity is defined as the number of metabolites that participate in any given reaction. The number of occurrences of each metabolite was calculated to reveal the highly connected metabolites of the reconstructed network. We also compared the metabolite connectivity pattern between the published genome-scale metabolic networks of Synechocystis sp. PCC6803 (iSyn669) [E. coli (iAF1260)[iFF708) [In order to analyze the metabolites connected within the network, we formulated a stoichiometric matrix S, derived from reaction lists of iSyn669) , E. coli(iAF1260), and yea[iFF708) .z) that can be minimized or maximized. In the case of minimization, the linear equation can be written as min z = cTv , where c is a row vector representing the influence of individual fluxes on the objective function. The metabolic flux distributions of the network are estimated under given conditions. In addition, the constraint parameters indicate the allowable range of flux values and are needed for convex solution space. The constraints for the upper and lower boundaries of reversible and irreversible reactions were defined as -∞ ≤ vi ≤ ∞ and 0 ≤ vi ≤ ∞, respectively. More details on the FBA approach have been described elsewhere [Flux balance analysis is a modeling technique that requires a developed stoichiometric metabolic network and a list of constraint parameters of biochemical reactions ,12. A selsewhere ,55.et al[The algorithm of minimization of metabolic adjustment was introduced by Segre et al. This apet al,14 to cahttp://www.gnu.org/software/glpk/) as the linear programming solver to estimate the optimal flux distributions under the maximized biomass objective. Simulations of three different growth conditions: heterotrophic , autotrophic (without glucose in the light) and mixotrophic (with glucose in the light) were performed according to the minimal growth-dependent medium. The uptake rates of carbon, nitrogen, phosphate, and sulfate sources were set and varied under each condition as shown in Table 2/s for autotrophic and mixotrophic conditions, while it was set to zero for heterotrophic cell growth. The other external metabolites involved in transport reactions such as H2O, Na+, Mg2+, H+, Zn2+, and Fe2+ (see Addition file 1), except for the substrates, were allowed to freely enter or leave the system. The uptake rates of all metabolites absent in the medium were set to zero. The flux values were expressed in mmol/mmol dry cell/h. All the three models in SBML format are provided in Additional files 5-7.In this work, we used the COBRA toolbox ,14 with in silico constraints of the simulation were set according to the carbon source utilized, as described above.In order to investigate the system-level change in response to the growth conditions we identified modes of metabolic operation under autotrophy and mixotrophy in terms of active reaction presented the non-zero flux value of the simulations. The different z) is first computed and is used as s further constraint with multiple optimizations to minimize and maximize flux values of every reaction through FBA. The mathematical equation can be written as max(zTv). The difference between the calculated minimum and maximum values for each flux defines the flux variability of that reaction. In this study, we used the COBRA toolbox [Flux variability analysis is used to examine the full range of numerical values for each reaction flux in the metabolic system while still satisfying the given constraints set and optimizing for a particular objective . Determi toolbox ,14 to peiAK692 model were individually removed while still optimizing the growth rate. This in silico analysis was performed using the MOMA algorithm running through the COBRA toolbox [The effect of genetic changes such as gene deletion can be simulated by constraining reactions associated to gene of interest to be zero . To dete toolbox ,14 underz) changes in response to an increase availability of each metabolite. Mathematically equation used to calculate the shadow price can be written as: γi = dz/dbi where γi is the ith shadow price and bi is the ith metabolite of the metabolic network. In this study, we performed PhPP using the COBRA toolbox [2/s; and of the bicarbonate uptake rates between zero to 0.4 mmol/mmol dry cell/h. For mixotrophy, PhPP analysis was performed by varying the boundaries of the glucose uptake rates between zero to 0.034 mmol/mmol dry cell/h and the bicarbonate uptake rates between zero to 0.4 mmol/ mmol dry cell/h.A robustness analysis was performed to determine the sensitivity of the predicted growth in changing the fluxes through a pre-defined range. Phenotypic phase plane is the method which has been used to obtain the sensitivity analysis as a function of dual variables . PhPP an toolbox ,14. The The authors declare that they have no competing interests.Spirulina cell cultivation experiments. All authors read and approved the final manuscript.AK performed the reconstruction of the genome-scale metabolic model and the network analysis and wrote the manuscript; AK, AP, and AM designed the study and were involved in model simulation and evaluation. AK, AP, and CK performed the iAK692 model. Excel file containing a list of the biochemical reactions in the final version of the metabolic network reconstruction, iAK692.Additional file 1Click here for fileAdditional file 2: Fluxome distribution under three different growth simulations. This Excel file contains a list of all flux distribution (in mmol/mmol dry cell/h) profiles under the autotrophic, heterotrophic, and mixotrophic culture conditions.Click here for fileAdditional file 3: Flux variability analysis. Excel file contains flux variability of the entire metabolic network for autotrophic and mixotrophic growths. Click here for fileAdditional file 4: Gene deletion analysis using MOMA approach inS.platensisC1 metabolic model. Excel file contains the grRatio, grRateKO and grRateWT values for S. platensis C1 single gene deletion during autotrophic and mixotrophic growths. Click here for fileAdditional file 5: SBML Model for the autotrophic simulation.This file contains the stoichiometric iAK692 model in the SBML format with all of the constraints needed for simulation using the COBRA toolbox. Click here for fileAdditional file 6: SBML Model for the autotrophic simulation.This file contains the stoichiometric iAK692 model in the SBML format with all of the constraints needed for simulation using the COBRA toolbox. Click here for fileAdditional file 7: SBML Model for the autotrophic simulation.This file contains the stoichiometric iAK692 model in the SBML format with all of the constraints needed for simulation using the COBRA toolbox. Click here for file


The quantification of experimentally-induced alterations in biological pathways remains a major challenge in systems biology. One example of this is the quantitative characterization of alterations in defined, established metabolic pathways from complex metabolomic data. At present, the disruption of a given metabolic pathway is inferred from metabolomic data by observing an alteration in the level of one or more individual metabolites present within that pathway. Not only is this approach open to subjectivity, as metabolites participate in multiple pathways, but it also ignores useful information available through the pairwise correlations between metabolites. This extra information may be incorporated using a higher-level approach that looks for alterations between a pair of correlation networks. In this way experimentally-induced alterations in metabolic pathways can be quantitatively defined by characterizing group differences in metabolite clustering. Taking this approach increases the objectivity of interpreting alterations in metabolic pathways from metabolomic data.We present and justify a new technique for comparing pairs of networks--in our case these networks are based on the same set of nodes and there are two distinct types of weighted edges. The algorithm is based on the Generalized Singular Value Decomposition (GSVD), which may be regarded as an extension of Principle Components Analysis to the case of two data sets. We show how the GSVD can be interpreted as a technique for reordering the two networks in order to reveal clusters that are exclusive to only one. Here we apply this algorithm to a new set of metabolomic data from the prefrontal cortex (PFC) of a translational model relevant to schizophrenia, rats treated subchronically with the N-methyl-D-Aspartic acid (NMDA) receptor antagonist phencyclidine (PCP). This provides us with a means to quantify which predefined metabolic pathways (Kyoto Encyclopedia of Genes and Genomes (KEGG) metabolite pathway database) were altered in the PFC of PCP-treated rats. Several significant changes were discovered, notably: 1) neuroactive ligands active at glutamate and GABA receptors are disrupted in the PFC of PCP-treated animals, 2) glutamate dysfunction in these animals was not limited to compromised glutamatergic neurotransmission but also involves the disruption of metabolic pathways linked to glutamate; and 3) a specific series of purine reactions Xanthine ← Hypoxyanthine ↔ Inosine ← IMP → adenylosuccinate is also disrupted in the PFC of PCP-treated animals.Network reordering via the GSVD provides a means to discover statistically validated differences in clustering between a pair of networks. In practice this analytical approach, when applied to metabolomic data, allows us to quantify the alterations in metabolic pathways between two experimental groups. With this new computational technique we identified metabolic pathway alterations that are consistent with known results. Furthermore, we discovered disruption in a novel series of purine reactions that may contribute to the PFC dysfunction and cognitive deficits seen in schizophrenia. Schizophrenia is characterized by deficits in cognition known to be dependent upon the functional integrity of the prefrontal cortex (PFC). Furthermore, compromised PFC function in schizophrenia is supported by a multitude of neuroimaging studies reporting hypometabolism , as evidenced by decreased blood flow or glucose utilization ,2. WhileMetabolomics is the comprehensive analysis of small molecule metabolites in biological systems . It invoAdditionally, it has also been used for the detection of multiple neurotransmitters in primate cerebral cortex . HILIC cMetabolomics represents a robust approach through which alterations in diverse metabolic pathways may be determined at a biological systems level. In this way a metabolomics approach may prove useful in further elucidating the pathophysiological mechanisms contributing to PFC dysfunction in schizophrenia. Furthermore, this approach may also allow for the identification of PFC metabolic biomarkers for the cognitive deficits in this disorder. While the metabolomics approach can provide a rich and comprehensive set of data, the appropriate quantitative analysis of this data has not been adequately developed. In particular, the identification of statistical differences in metabolic pathways between experimental groups rather than the identification of statistical differences in individual metabolites alone represents a major challenge to quantitatively identifying metabolic alterations at a systems level from metabolomic data. One method through which statistical differences in metabolic pathways can be identified from metabolomic data involves the representation of this data as a large, complex network of nodes (single metabolites) connected by real-value edges (the correlation coefficient between two metabolites). This form of representation has high face validity as the relationship between two metabolites, in a given pathway, is governed by a single or series of enzymatic reactions that can be viewed as being represented by the correlation between the concentrations of the two metabolites. Another advantage is that metabolomic data consist of a range of metabolites detected in both of the experimental groups of interest meaning that these data can be expressed as two complex networks based upon the same set of nodes. This data structure is amenable to analysis through the application of the Generalized Singular Value Decomposition (GSVD) algorithm.Large, complex interaction networks arise across many applications in science and technology -36. SpecOverall, this work develops and applies a novel algorithm in network science and shows that it reveals meaningful insights when applied to cutting-edge metabolomic data.A and B in ℝN×N represent two different types of interaction between a set of N nodes. We have in mind the case where the weights play the role of correlation coefficients. Our aim is to discover clusters, in the sense of subsets of nodes that are mutually, pairwise, strongly connected through positive weights. The algorithm will also discover clusters of strong negative connectivity, although in practice this type of pattern is less likely to be present. However, we note that the arguments given below and the resulting algorithm remain valid in the case where the weights are non-negative, with zero representing the minimal level of similarity. The novelty of our approach is that in order to reveal interesting differences between the two types of connectivity data, we look for a set of nodes that form a good cluster with respect to A and a poor cluster with respect to B, or vice versa. As a starting point for a computational algorithm, we consider the identitySuppose that the square, symmetric, real-valued matrices x ∈ ℝN. Here ||·||2 denotes the Euclidean norm and x ∈ ℝN to denote such a partition, with s x= 1 if node s is placed in group 1 and s x= -1 if node s is placed in group 2.for k and l, we could argue that the existence of a third node, i, such that ik aand il aare both large and positive or both large and negative is evidence in favor of placing k and l in the same group (since they have in common a strong similarity or dissimilarity with node i). On the other hand small or oppositely signed values for ik aand il ais evidence in favor of placing k and l in different groups. In terms of the indicator vector, this translates toFixing on two nodes, 1. ikail alarge and positive ⇒ try to choose kxl x= +1,2. ikail asmall or negative ⇒ try to choose kxl x= -1.k, l and third parties i. So we could judge the quality of an indicator vector by its ability to produce a large value of Returning to the right-hand side of (1), we see that Analogously, we can argue that making A that are not present in B.is a good basis for picking out strong clusters in N, v[2], v[3],..., v[9] to denote them. From these signals, we set up two matricesMore precisely, in our computation, the value of each entry (the correlation coefficient) in a D∈ ℝ20×50: the first 5 rows are linear combinations of v[1], v[2], v[3], v[4], v[5], v[6] and v[7]. Rows 6 to 15 are combinations of v[7] and v[8]. The remaining rows (rows 16 to 20) are Gaussian pseudorandom numbers.• b D∈ ℝ20×50: the first 5 rows are linear combinations of v[1], v[2], v[3], v[4], v[5], v[6] and v[7]. Rows 6 to 14 are Gaussian pseudorandom numbers. The remaining rows (rows 15 to 20) are combinations of v[4] and v[9].• Building up the rows from the underlying signals in this manner allowed us to construct the correlation patterns seen in Figure A and B before applying our algorithm to them. Figure Although the algorithm is invariant to permutation, for visual clarity, we also shuffled the synthetic correlation data sets A and B reordered with two extreme columns of X T-, one from each of the two ends of the spectrum. The reorderings reveal the mutually exclusive cluster structures of A and B. We also applied the cluster validation method to the structures uncovered by the reorderings using random permutation. In Figure X T -picks out the continuous nodes 17, 15, 20, 18, 16, 19, which form a good cluster in B but not in A (p < 0.001). The reordering from the final column of X T -shown in Figure A but not in B was completely uncovered by the nodes 10, 14, 12, 9, 8, 7, 6, 13, 15, 11 .We present the results from our algorithm in Figures In summary, this additional synthetic test illustrates that our GSVD based algorithm can be extended to reveal the pattern difference between two relative correlation matrices in terms of clustering.SIEVE analysis (Thermo-Fisher Scientific) revealed significant PCP-induced alterations in the level of specific metabolites in the PFC of PCP-treated rats Table . These cIn the context of this study the aim of applying the GSVD algorithm to metabolomic data from control and PCP-treated animals was to quantitatively determine which predefined metabolic pathways were altered in PCP-treated animals. The inter-metabolite Pearson's correlation coefficient  was used as the metric of the functional association between each pair of metabolites and was generated from the metabolite peak intensities, as determined by Liquid Chromatography Mass Spectrometry (LC-MS), across all animals within the same experimental group (i.e. either control or PCP-treated). These correlations were Fisher transformed to give the correlation data a normal distribution. This resulted in a pair of symmetric, square, real-valued {98 × 98} partial correlation matrices  and B  with the aim of discovering a new node (metabolite) ordering that reveals clusters of nodes that exhibit strong connectivity  in one network but not the other. In the context of this data the GSVD algorithm was used to identify clusters of metabolites present in one experimental group that are not present in the other with the aim of identifying those metabolic pathways in the PFC disrupted by PCP treatment. Once the matrices had been reordered through the GSVD algorithm the significant presence of a cluster in the given network was statistically tested by comparison of the cluster quality measure in the real networks relative to that in 1000 random permutations of the initial matrices. The original metabolomic networks are shown in Figure A represents control animals and B represents PCP-treated animals. Figures -TX , respectively. The original position of each metabolite detected by LC-MS  and the bottom cluster  was confirmed, indicating that there were clusters of metabolites significantly present in control (A) animals that were not present in PCP-treated (B) animals. The identity of the metabolites, the KEGG pathways in which each metabolite is involved, and the PCP-induced alteration in the overt level of each metabolite  are shown in Tables B) that was not present in control (A) animals: (i) potential top cluster [first 10 nodes] p = 0.421; (ii) potential middle cluster [nodes 18-25] p = 0.494.Our network treats interactions between molecules as bidirectional, and so the set of interactions between molecules forms an undirected weighted network. In essence the GSVD algorithm allows the reordering of the two experimental matrices S Figure  and its S Figure  are showRigorous significance testing, involving multiple potential metabolite clusters, confirmed that there were no significant clusters of metabolites in PCP-treated animals that were not present in controls Figure . FollowiThrough its application to metabolomic data we have clearly demonstrated the added value that can be gained from applying the GSVD algorithm to two sets of complex, network data based upon the same set of nodes. In particular, we have demonstrated that the combined application of the GSVD algorithm with hypergeometric probability analysis provides an analytical framework by which statistical alterations in predefined metabolic pathways between experimental groups can be defined from complex metabolomic data. There is a great unmet need for this type of analytical approach in metabolomics, as well as in the other -omics fields (e.g. transcriptomics), which allows the quantification of alterations at the biological systems (pathways) level rather than simply identifying significant alterations of discrete measures (i.e. single metabolites).14C-2-deoxyglucose imaging technique , (2) Arginine and Proline ko00330], (3) Butanoate ko00650], 4) Nicotinate and Nicotinamide ko00760], (5) Glycine, Serine and Threonine metabolic pathways as well as an imbalance in (6) metabolites active as neurotransmitter ligands [ko04080]. The disruption of metabolic pathways involving glutamate in the PFC of PCP-treated rats seems particularly pertinent given the reported alterations in extracellular glutamate availability in the PFC following repeated PCP treatment  and the 50, (2) A30, (3) B50, (4) N Nicotinaechnique ,60., (5) GlyOur results also suggest that glutamatergic dysfunction in the PFC of PCP-treated rats is not limited to the disruption of glutamatergic neurotransmission but also involves the disruption of the metabolic pathways in which glutatmate is involved. For example, altered glutamate metabolism may directly contribute to the disruption of the Arginine-Proline metabolic pathway in the PFC of PCP-treated animals. The significant disruption of the Arginine pathway in PCP-treated animals suggests that prolonged NMDA receptor hypofunction may result in disrupted nitric oxide (NO) signalling in the PFC. There is increasing evidence that NO signalling is directly linked to NMDA receptor activity through regulation of the enzyme nitric oxide synthase (NOS)  and thatIn addition to quantitatively defining the specific metabolic pathways altered by experimental manipulation, our results suggest that the GSVD algorithm can identify discrete series of metabolic reactions altered by experimental manipulation. In this way, while we found no significant evidence to support the widespread disruption of purine metabolism, or the significant disruption of any other KEGG defined metabolic pathway in the bottom cluster as detected using the GSVD, we did find evidence in this cluster to suggest that a specific series of purine reactions were significantly disrupted in the PFC of PCP-treated animals. These disrupted purinergic reactions in the PFC of PCP-treated animals were:q13.1-q13.2 in humans ), (ii) phenylalanine, tyrosine and tryptophan metabolism (3 metabolites [ko00360]), (iii) purine metabolism (2 metabolites [ko00230]) and (iv) butanoate metabolism (2 metabolites [ko00650]). KEGG defined metabolic pathways; ko00250: Alanine, Aspartate and Glutamate metabolism; ko00627: Aminobenzoate degradation; ko00330: Arginine and Proline metabolism; ko00410: beta-Alanine metabolism; ko00780: Biotin metabolism; map00650: Butanoate metabolism; ko04973: Carbohydrate metabolism; ko00270: Cysteine and Methionine metabolism; ko00071: Fatty acid metabolism; ko00051: Fructose and Manose metabolism; ko00052: Galactose metabolism; ko00471: Glutamine and Glutamate metabolism; k00480: Glutathione metabolism; ko00561: Glycerolipid metabolism; ko00564: Glycerophospholipid metabolism; ko00260: Glycine, Serine and Threonine metabolism; ko00010: Glycolysis/Gluconeogenesis; ko00340: Histidine metabolism; ko00562: Inositol Phosphate metabolism; map00300: Lysine biosynthesis; ko00310: Lysine degradation; ko00430: Methionine metabolism; ko04080; Neuroactive ligand-receptor interaction; ko00760: Nicotinate and Nicotinamide metabolism; ko00190: Oxidative phosphorylation; ko00770: Pantoate and CoA biosynthesis; ko00550: Peptidoglycan biosynthesis; ko00360: Phenylalanine metabolism; ko00400: Phenylalanine, Tyrosine and Tryptophan biosynthesis; ko00440: Phosphonate and Phosphinate metabolism; ko00640: Propanoate metabolism; ko00230: Purine metabolism; ko00240: Pyrimidine metabolism; ko00620: Pyruvate metabolism; ko00500: Starch and Sucrose metabolism; ko00600: Sphingolipid metabolism; ko00920: Sulphur metabolism; ko00430: Taurine and Hypotaurine metabolism; ko00730: Thiamine metabolism; ko00380: Tryptophan metabolism; ko00350: Tyrosine metabolism; ko00400: Tyrosine and Tryptophan biosynthesis; ko00290; Valine, Leucine and Isoleucine biosynthesis; ko00280: Valine, Leucine and Isoleucine degradation. NA denotes a metabolite not associated with a KEGG compound ID or KEGG pathway.Click here for file98 × 98 matrix of between metabolite correlations in the PFC of control animals. The 98 × 98 matrix of the Pearson's correlation coefficients (Fisher z-transformed) between all metabolites detected in the prefrontal cortex of control  animals by LC-MS analysis is shown.Click here for file98 × 98 matrix of between metabolite correlations in the PFC of PCP-treated animals. The 98 × 98 matrix of the Pearson's correlation coefficient (Fisher z-transformed) between all metabolites detected in the prefrontal cortex of PCP-treated animals by LC-MS analysis is shown.Click here for fileTable S2 - Table showing the axes labels in Figures 10and X T -and the final column of X T-, respectively, is shown.Click here for file


Here, we propose a method that allows us to perform the required computation heuristically in a time scaling linearly with the number of reactions in the network, overcoming some limitations of similar techniques employed in recent years. As a case study, we apply it to the analysis of the human red blood cell metabolic network, whose solution space can be sampled by different exact techniques, like Hit-and-Run Monte Carlo . Remarkably accurate estimates for the true distributions of viable reaction fluxes are obtained, suggesting that, although further improvements are desirable, our method enhances our ability to analyze the space of allowed configurations for large biochemical reaction networks. Quite generally, constraint-based metabolic flux analysis describes the space of viable flux configurations for a metabolic network as a high-dimensional polytope defined by the linear constraints that enforce the balancing of production and consumption fluxes for each chemical species in the system. In some cases, the complexity of the solution space can be reduced by performing an additional optimization, while in other cases, knowing the range of variability of fluxes over the polytope provides a sufficient characterization of the allowed configurations. There are cases, however, in which the thorough information encoded in the individual distributions of viable fluxes over the polytope is required. Obtaining such distributions is known to be a highly challenging computational task when the dimensionality of the polytope is sufficiently large, and the problem of developing cost-effective  N coupled chemical reactions transforming M metabolites and let ξ = {ξiµ}  denote the stoichiometric coefficients, with the standard sign convention to distinguish substrates (ξiµ < 0) from products (ξiµ > 0) in each reaction, i. If one denotes by γµ the rate of change of the intracellular level of species µ, due to exchanges between the cell’s interior and the environment, then, under mass action kinetics, the intracellular concentration, cµ, of metabolite µ obeys the equation:γµ > 0 (resp.γµ < 0) if there is a net out-take (resp. in-take) of species µ. The development of high throughput techniques now makes available a considerable number of high quality reconstructions of the metabolism of a variety of organisms, which include the stoichiometry of the biochemical reactions in the network and the underlying enzyme-gene associations ,2,3,4. IUnluckily, addressing the above system in full generality requires knowledge about reaction mechanisms and kinetic constants (which specify how rates depend on concentrations), which is at best only partially available.  for a genome-scale reconstruction involving thousands of reactions and metabolites would be a sensible thing to do). i.e., xi  ∈  , and for exchange rates, i.e., γµ ∈ , with which conditions (1) can be written as:γµ = 0), as well as exchanged species . The solution space of Equation (2) is, in turn, given by:Computational studies of metabolic networks therefore generally assume that the cell operates at non-equilibrium steady-state (NESS) conditions, where the concentration of the metabolites is constant ,6, with S, in order to retrieve information on quantities like the distribution of the allowed values of each flux over the entire solution space. While most studies aiming at developing predictive power on metabolic phenotypes  have been thus far based on coupling (2) with a phenotypic optimization principle (the solving of which requires no sampling of S), it is being increasingly recognized that the information entailed by the structure of S might provide key insight into different aspects of metabolic network analysis, from flux-flux correlations, to robustness, to perturbations, to (more subtly) optimal experiment design . The quantity, Pl→µ(xl), is the PDF of flux l taking a value xl, when metabolite µ is removed. Those PDFs are, in turn, given by the probability, for each reaction i, to satisfy the mass balance conditions for all the metabolites they process, except µ  nicely factorize. To push mathematically forward expression (6), we need to do some type of approximation for ollowing , we assu, as see b:(9)wipt µ see c, namelyPi is a normalisation constant. Note that Equation (11) also provides the recipe to evaluate the PDFs, Pµ(γ), for the exchange rates, γµ, once the conditional marginals, Pl→µ(xl), are known. The conceptual step of removing metabolites from the system is the key that allows us to recast the problem in the set of self-consistency Equations (9) and 10), for the conditional probabilities , we could use Monte Carlo integration and draw  pairs of random variables  according to the distributions, y and z. Correspondingly, an estimate for x(x) can be written as:yi − zi), accounts for draws for which the quantity x = 1 − yi − zi must be rejected due to the condition x ≥ 0. The latter condition indeed defines a feasible triangular region in the integration plane, yz  falling outside this domain must be rejected. This method  is, hence, poised to be rather inefficient. Fortunately, we know precisely where the rejection region is, and we can rewrite Equation (12) as follows:z(z) is not normalised in the interval . Introducing the corresponding weight:z(z|y) ≡ (z)/w(y). The distributions appearing above are now properly normalized. Therefore, to evaluate Equation (16), we can simply draw  pairs  according to y(y) and z(z|y), respectively, and estimate x(x) by:i.e., by  pairs of variables and weights .As discussed in ,11, the , yz see , such thz(z|y), has a y-dependent support, such that rejection never occurs. Thus, at a price of computing a weight, w(y), we overcome the whole rejection issue, and the method becomes much more efficient. The key point of this method is that the reweighted density, , its running time goes as O(2Nk), where k is the average number of metabolites processed by each reaction. Thus, as opposed to sampling techniques that have normally super-linear mixing times [S [The great advantage of using wBP is that, at fixed ng times , wBP onltimes [S . Due to times [S , make thS and obtain exact PDFs of individual fluxes for the hRBC by a controlled method that guarantees uniformity, we have developed an optimized version of Hit-and-Run Monte Carlo, which we call the Kernel Hit-and-Run method. Let us start by re-writing constraints (2) explicitly for metabolites involved in internal reactions and the rest:| equations in (18) defines the null-space of ξ, and geometrically corresponds to a family of hyperplanes passing through the origin x = 0. Let us denote the dimension of the null space of ξ as K. Clearly, K would be at least N − || . This means, obviously, that, although the number of variables in the system is N, due to the constraints in the model, the actual dimension of the solution space S is only K. As in real metabolic networks most reactions are internal, the dimension K of the null space will be significantly smaller than the original dimension of the problem N. Additionally, it turns out that the way to implement in practice such a dimensional reduction is quite straightforward: suppose that a basis of the null-space has been found, e.g., through Gaussian elimination or singular value decomposition (SVD), and let us denote as y =  the system of coordinates with respect to such a basis, so that we can write each flux in this basis as , with Φ an N × K matrix related to the change of basis between the original space and the null subspace. Plugging this into Equations (19) and (20) allows us to write:. The set of Equations (21) defines a K-dimensional polytope in the null space mi lgorithm ,16. Finae rest:1(19)mi Hit-and-Run algorithm, this scales as the square of the dimensions times the diameter of the polytope, i.e., in practice, cubically with the number of dimensions [K, rather than N. This can yield a significant reduction in computation times if K is small compared to N, as will quite generally be the case. For the hRBC, for instance, we pass from N = 46 , to a mposed to ). This iS, with a mixing time that scales as O(K3). Finally, it is worth mentioning that the matrix Φ can be easily obtained with any standard algebra software or by standard SVD algorithms, and that no matrix inversion is required to compute the projected matrix Ψ nor to convert the obtained sample to the original, full dimensional space. Therefore, to sum up, KHR uniformly samples the solution space We have applied the wBP and KHR algorithms to the study of the metabolic network of the hRBC. As mentioned in  of size  = 500. To solve the fixed point equations (9) and (10), we performed 30 iterations of our method. We started with uniform weights  and, at each iteration t = 1,..., 30, and for each fixed value of the variable xi, we applied wBP 103 × t times to evaluate the average weight αi. Once convergence was reached, we used the variable/weight sets to compute the final 46 PDFs, Pi(x) and Pµ(γ), according to Equation (11). In this last step, we averaged the weight values over 105 wBP extractions to achieve a higher accuracy. We report the results in We run the wBP algorithm by representing the marginals, like Equation (10), with sets of variables/weights S. As the uniform measure is guaranteed to be a limiting distribution, the difficulty lies, of course, in deciding when such a limit has practically been reached in the simulations. In this regard, we decided to apply three conservative measures to ensure this: first of all, we checked that the initial conditions did not affect the results and, after that, the simulations were averaged over the initial conditions ; secondIn this work, inspired by techniques employed in the statistical mechanics of disordered systems, we have presented a novel method to estimate distributions of reaction fluxes in constraint-based models of metabolic networks. The wBP methodology has, in our view, clear advantages when compared with alternative approaches. If compared to rejection-based Monte Carlo methods  or even a priori to keep fluxes “close” to their optimal value. wBP can also be integrated with optimization-based flux balance analysis (FBA), as it easily allows us to evaluate the PDFs of the enzymatic rates close to optimality (assuming a score function is known) by just injecting N-dimensional space of solutions, one can wind up into a space that, in the best of cases, has the same number of dimensions as the number of exchange reactions. Given the considerable gains that this observation provides, we believe that this method may be worth being explored further in its own right. We have also compared the performance of wBP against the KHR method. The latter is a controlled Hit-and-Run Monte Carlo taking place in the null space defined by the set of internal reactions, where a considerable effective dimensional reduction can be achieved. Indeed, starting from the original Mycoplasma pneumoniae [Escherichia coli [The validation of the wBP algorithm in the hRBC network, which can be considered a benchmark for the sampling problem for constrained metabolic models, opens the door to future applications of the method to more relevant organisms, such as eumoniae  or Eschehia coli . In our