Optimization- and explicit Runge-Kutta-based Approach (ORKA) to perform dynamic Flux Balance Analysis (dFBA), which is numerically more accurate and computationally tractable than existing approaches. ORKA is applied to a four-tissue  model of Arabidopsis thaliana, p-ath773, uniquely capturing the core-metabolism of several stages of growth from seedling to senescence at hourly intervals. Model p-ath773 has been designed to show broad agreement with published plant-scale properties such as mass, maintenance, and senescence, yet leaving reaction-level behavior unconstrainted. Hence, it serves as a framework to study the reaction-level behavior necessary for observed plant-scale behavior. Two such case studies of reaction-level behavior include the lifecycle progression of sulfur metabolism and the diurnal flow of water throughout the plant. Specifically, p-ath773 shows how transpiration drives water flow through the plant and how water produced by leaf tissue metabolism may contribute significantly to transpired water. Investigation of sulfur metabolism elucidates frequent cross-compartment exchange of a standing pool of amino acids which is used to regulate the proton flow. Overall, p-ath773 and ORKA serve as scaffolds for dFBA-based lifecycle modeling of plants and other systems to further broaden the scope of in silico metabolic investigation.In this work we introduce the generalized  Synthetic biology techniques have been applied to many plant systems such as tomatoes3, rice4, and maize3 to produce enhanced phenotypes often with application to human nutrition2, pest resistance5, and resilience to abiotic stresses6. Many of these efforts have focused on a genetic understanding and manipulation of the plant system (or plant tissue) in question, having relied on intuitive interventions such as changes in regulation, insertion of new gene(s), and deletion of gene(s) from competing pathway(s)6.The use of synthetic biology for the engineering of uni- and multi-cellular organisms to enhance desirable phenotypes in microbe, plant, and animal systems, has been well established and has been capable of affecting the lives of millions of individuals, such as in the case of artemisinin production in yeast or enhancing nutritional value of agricultural productsStoichiometric Models  links9 in an organism7. When an SM encompasses the entire chemical reaction repertoire of an organism, it is called a Genome Scale Models (abbreviated as GSM or GEM)10. Perhaps the most common analysis tool used with these models is Flux Balance Analysis (FBA)11. Assuming that the system is under pseudo-steady state, FBA can find the extremum of a given growth objective  which is defined by an objective function subject to mass balance, reaction directionality, and certain other constraints 13. The basic formulation of FBA can be found in Fig. dynamic FBA (dFBA)14 can improve the predictive abilities of SMs. dFBA can perform FBA over windows of time by solving a dynamic non-linear or a static linear problem, both of which integrate system variables over discrete time windows to solve for metabolite concentrations, reaction fluxes, and system biomass15. In general, there are two approaches to dFBA. The first approach, the Static Optimization-based Approach (SOA), has been applied to E. coli14, mammalian cells17, Saccharomyces cerevisiae (bakers’ yeast)17, Hordeum vulgare (barley)18, and Arabidopsis thaliana19 (in addition to other systems). The second, the Dynamic Optimization-based Approach (DOA), has been applied to E. coli metabolism14 and signaling networks is S. cerevisiae20 (to name a few applications). These approaches have proven useful for investigating aspects of plant-scale metabolism, such as resource partitioning in Arabidopsis19. These works have inspired the development of our new approach to perform dFBA named as Optimization- and explicit Runge-Kutta –based Approach (ORKA). ORKA significantly improves upon the SOA by utilizing the step-by-step solution approach of the SOA  with increased accuracy and solution stability. These improved characteristics are due to both the implementation of a Runge-Kutta method  to replace the first-order Taylor series approximation used by SOA and by replacing the assumption that the reaction rate is constant over each time interval with a trapezoid rule-based integral approximation.Alternatively, computation-based systems biology approaches such as Arabidopsis thaliana (hereafter Arabidopsis) has been selected as a test system for the application and demonstration of the ORKA framework, due to the fact that Arabidopsis is a model plant species with a highly characterized knowledgebase. The choice has also allowed demonstration of ORKA in a dynamic, multi-tissue system. To date, many stoichiometric models of plant metabolism, including Arabidopsis, have been developed. Some of these models including models of Arabidopsis thaliana22, Zea mayz (maize)23, Sorghum bicolor (sorghum)12, Brassica napus (rapeseed)24, and Oryza sativa (rice)25 have treated plants as single metabolic units. These models have sought to analyze metabolic maintenance, response to abiotic stimuli, enzyme regulation changes, and metabolism as a whole25. Tissue-specific models have been reconstructed for various Arabidopsis tissues26, a maize leaf27, and a barley seed28 to better understand how present metabolites, metabolic pathways, and nutrient  availability differ between tissues. Multi-tissue models have also been developed to characterize whole-plant metabolism for Arabidopsis19 and barley18 and subsequently to study whole-plant metabolic response to the diurnal cycle and the source-to-sink relationship of leaves and seeds18. These studies have considered metabolism at a single point  or a single diurnal cycle13 or have modeled only a portion of the Arabidopsis lifecycle19. The most complete dFBA work, in terms of modeling the full Arabidopsis lifecycle, models two tissues, leaf and root, across 30 days of vegetative growth (from 6 days to 36 days)19. Here, we have developed a core-carbon metabolic model of Arabidopsis, named p-ath773 , to model the full lifecycle of Arabidopsis from germination to senescence by being embedded in the ORKA framework which captures metabolic interactions between four major tissues: leaf, root, seed, and stem. These four tissues have been chosen for model reconstruction to represent core plant functions: the root for nutrient uptake and growth; the leaf for photosynthesis, carbon fixation, and as a source tissue for plant nutrition; the seed for metabolite storage and a sink tissue for metabolic investment; and the stem for metabolic transport and acting as a conduit for all metabolic interactions between other tissues. Core-metabolism pathways that are included but not limited to photosynthesis; the citrate cycle; starch and sucrose synthesis; fatty acid synthesis and degradation; and amino acid synthesis. The p-ath773 model consists of 1251 total  reactions (R), 1155 total (and 276 unique) metabolites (M), and accounts for 773 genes (G) including 42 chloroplastic and 11 mitochondrial genes. Each of the modelled tissues including leaf , root , seed , and stem  has been reconstructed individually to allow for the different tissue mass ratios found across the lifecycle of the plant. A summary of the p-ath773 model is shown in Fig. Arabidopsis as modeled by p-ath773 under 12 hour light and 12 hour darkness growth conditions accounting for changes due to diurnal metabolic differences; changes in plant mass; metabolite storage and uptake (particularly carbohydrates); changes in plant tissue mass ratios; and changes in metabolism with respect to plant growth stage. The p-ath773 model is unique among Arabidopsis models for its focus on plant-scale behavior such as focus on achieving biomass levels which correspond with in vivo data; biomass-based maintenance and senescence drains; and the logical mole-balanced exchange of nutrients between tissues. While the plant-scale behavior is well-constrained in the p-ath773 model, reaction-scale behavior is unconstrained such that the model can be used to study the reaction-scale behavior necessary to explain observed macro-scale behavior.Arabidopsis which utilized the SOA to perform dFBA19. This has been done by combining improved mass step and metabolite concentration estimates with smaller time step sizes (one hours as opposed to one day19). Further, with the inclusion of two more tissue types, stem and seed, and modeling the entire lifecycle, the p-ath773 model in the ORKA framework makes a significant improvement to current Arabodipsis dFBA-based models, despite only modelling central metabolism. It should be noted that for metabolic models with a single tissue, or single organism, O(h3) or better error order is possible depending on the Runge-Kutta method selected, compared to the O(h2) error order floor of the SOA method. This low error level has proved impossible to achieve with the p-ath773 model since the seed tissue appears and disappears over the course of the Arabidopsis lifecycle, causing difficulties due to the exponential nature of FBA-determined growth rates. The series of more accurate hourly metabolic “snapshots” produced by p-ath773 has given a framework for the investigation of the central metabolism of Arabidopsis across its lifecycle. Here, these “snapshots” have been used to investigate the diurnal patterns of water flow (from the root uptake to transpiration from the leaf), and sulfur metabolism (from root uptake to tissue biomass). Further, the p-ath773 model embedded in the ORKA framework has shown general agreement with macro-level experimental data found in the literature and is potentially useful as steppingstone for dynamic lifecycle modeling of other plant systems.When ORKA has been applied to the p-ath773 multi-tissue model, the order of error of both mass step and metabolite concentration estimates has been theoretically improved by approximately three order of magnitude as compared to that achieved in a previous model of tn is the current time, t0 is the initial time, Δt is the time step, cn are the steps in the independent variable made by the Runge-Kutta method chosen to use, Yt is the current biomass concentration, Y0 is the initial biomass concentration, ana is the weight of Runge-Kutta derivative estimate steps (kn) in the next derivative estimate, bn is the weight of the Runge-Kutta derivative estimate steps in the full Runge-Kutta derivative estimate, Yt + Δt is the biomass concentration at the next time step, zi,t is the concentration of metabolite i at time t, zi,t + Δt is the concentration of metabolite i at the next time step, Sij is the stoichiometric coefficient of metabolite i in reaction j, Γj,t is the trapezoid rule-based integral estimate of the flux of reaction j at the current timestep, vj,t is the rate of reaction j at time t, vj,tn is the rate of reaction j at Runge-Kutta time step n, set N is the number of steps in the Runge-Kutta solution method (with n as the index), and cnf is the final cn value in the Runge-Kutta method. Greater detail on the definition of each symbols used can be found in the “Symbols Used” section in Text 28 by replacing the Taylor-series approximations (details in the methods section) used to advance biomass concentration in the SOA  has been developed to make more accurate and stable estimates of the changes in biomass and metabolite concentration in a dynamic Flux Balance Analysis (dFBA). The basis of the ORKA is the same as SOA, to model a dynamic (i.e. time-dependent) metabolism across multiple time points, where each time point solution builds upon previous solutions. The pseudocode describing how the ORKA works can be found in Fig. entclass1pt{minimaArabidopsis plant, namely seed, leaf, root, and stem, corresponding tissue-level metabolic models have been reconstructed. The seed and leaf tissue have been selected to model an important source-to-sink relationship, whereas the stem and root tissues have been included to model nutrient transport and nutrient uptake in Arabidopsis, respectively. Model files for each tissue can be found in the GitHub p-ath773 repository for this work (10.5281/zenodo.3735103). Details of model reconstruction can be found in Materials and Methods, but a synopsis is as follows. The seed model has been reconstructed first using the metabolic pathways shown in the Arabidopsis seed though 13C-labeled Metabolic Flux Analysis (MFA)29. The model reactions have been distributed among extracellular space, cytosol, non-green plastid, inner mitochondria, and outer mitochondria subcellular compartments in accordance with literature evidence , photosynthesis, and gluconeogenesis as well as chloroplast and thylakoid subcellular compartments. The biomass of the leaf has been adapted slightly from that of a previously published Arabidopsis model, iRS159723, by refocusing the biomass composition on primary metabolites. Similarly, by having extracted common reactions/pathways from the seed and leaf models as a starting point and adding functionalities particular to these tissues such as nitrogen reduction in the root and the transport of metabolites through the extracellular space of the stem, the root and stem models have been reconstructed. Root and stem models have been reconstructed with metabolic differences between the two such as the presence of amino acid synthesis and the conversion of ammonium to nitrate both in the root for xylem loading. Root and stem tissues are, however, largely focus on basic carbon metabolism and metabolite uptake (root) and transport (root and stem). In the absence of Arabidopsis-specific estimates, the dry weight compositions of switchgrass (Panicum virgatum) root and stem31 have been used to define root and stem biomass compositions. Both these models contain necessary transport/exchange reactions to ensure model connectivity and to facilitate their roles in the transport processes. The stem and root models have all the subcellular compartments present in the seed model. Once initial reconstructions have been accomplished, thermodynamically infeasible cycles in addition to atom and charge imbalances have been resolved7. Figure 7. Figure 7. Once each tissue model has been reconstructed, these four models have been linked by the ORKA framework, and the lifecycle of the plant has been simulated. We have addressed the incongruities between these in silico simulation results and in vivo experimental data by adjusting their metabolism of individual tissue-specific models, tissue-tissue interactions, or by adjusting parameters  associated with the plant-scale behavior of the p-ath773 model. This portion of the workflow is illustrated in Fig. In order to track the important metabolic interactions and transactions within and between major tissues of Arabidopsis plant as it passes through various stages of growth.Once these core tissue models have been reconstructed and curated, sets of constraints have been defined to enforce logical links between tissues to facilitate the simulation of tissue metabolism. For instance, these links include ensuring that water travels from the root (source) to the leaves (sink) and that literature-supported amino acids travel from the leaf and root (sources) to the seed (sink) through the stem tissue (the link between these tissues). In addition, other constraints include environmental interactions such as with atmosphere and soil. These constraints include gas exchange in all tissues; uptake of micronutrients and water by the roots; and use of light by the leaves. These constraints are discussed in detail in the Materials and Methods. In summary, these constraints ensure that micronutrients and water are transported from the root tissue to other tissues via the stem; that sugars and amino acids travel from the leaf tissue to other tissues via the stem; that patterns of starch and sucrose storage in leaf and stem tissues are included in the model; and that the rates of tissue growth are linked in such a way that tissue mass ratios are preserved or changed in accordance with how these quantities change in an O((c2 − c1)h2). Therefore, any Runge-Kutta method with error order less than that will suffice. In this work, Heun’s third-order Runge-Kutta rule is used. This is in part because of the limitation just described such that a higher-order Runge-Kutta method is not necessary. Further, this method has the advantage over Kutta’s third-order Runge-Kutta rule in that the step size between cn values is one third (e.g. c2 − c1 = 1/3) as opposed to one half (e.g. c2 − c1 = 1/2)  are shaded from light by the soil and/or seed coat. In growth stages when plant tissue ratios are constant , the tissue mass ratio values have been taken from values typical for herbaceous plants (0.511 gDW leaf:0.0.267 gDW root:0.211 gDW stem after adjusting from fresh weight to dry weight)35 , a linear biomass “slider” is used, where a single parameter, seeding (s), is used to progress tissue mass ratios  to s = 1 (mass ratios when maximum amount of seeds have been produced and have not yet been dispersed) and is linearly incremented from the point at which the first flower is produced to when all flowers are produced then decremented to when all silique (seed pods) are shattered, thus dispersing all seeds  After Germination , 70 HAG , 90 HAG , 177 HAG , 181 HAG , 770 HAG , 810 HAG , 1155 HAG , 1170 HAG , 1190 HAG , and 1199 HAG . In summary, we incorporated the p-ath773 model in an ORKA framework to simulate Arabidopsis metabolism across the lifecycle of an individual plant.As discussed more extensively in Materials and Methods, the growth rate for an SM is an exponential growth rate. Due to this exponential nature of the growth rate, seed mass becomes problematic to model as there are points in the growth of the seed tissue where its mass is zero, is advanced from zero to a non-zero value, and is advanced from a non-zero value to zero. These changes are impossible to capture using an exponential function. Therefore, plant mass as a whole is tracked and advanced by the ORKA, and individual tissues masses are determined by multiplying total plant mass by tissue mass fraction. Since there is no whole-plant biomass function, this approach requires an approximation which defines the error floor by a second order backward difference approximation of the first derivative  with an error order of see Fig. , giving see Data  yet needSee Data . In growsee Data . A workf34. At 17, 24, and 31 Days After Germination (DAG) the total dry plant mass should be between 0.5 and 2.0 mg; 2 and 8 mg; and 10 and 30 mg, respectively34. Upon the completion of multiple iteration of design-build-test cycle, the p-ath773 model has been adequately refined, the p-ath773 model has shown a total dry plant mass of 0.676 mg at 17 days (408 hours), 4.20 mg at 24 days (576 hours), and 25.9 mg at 31 days (744 hours) after germination. Furthermore, mass-based growth targets include the peak dry weights of the leaves, the seeds, and the stems which have been reported as approximately 163.7 mg (standard deviation 52.0 mg), 127.9 mg (standard deviation 52.7 mg), and 188 mg (standard deviation 39.3 mg), respectively32. As the p-ath773 captures both plant growth and loss of seed (and other) mass in the silique ripening stage, the peak mass of each of these tissues has been comparted to this data. In the refined p-ath773 model, the peak masses of the leaves, seeds, and stems have been determined as 153 mg, 100 mg, and 151 mg, respectively, all of which are within one standard deviation of the experimental value32 (see the methods section for how tissue masses are determined). These comparisons are summarized in Fig. in silico tissue and plant mass values are similar to in vivo data, thus showing strong agreement with respect to plant- and tissue- scale growth trends. This agreement has been achieved by tuning the rate of carbon dioxide and light availability to the plant system36 which the modeled plant is allowed to utilize as well as by tuning the plant biomass yield 38. We have defined both carbon dioxide and light uptakes based on literature, with the former from the carbon assimilation rate39 and the Leaf Area Ratio (LAR) of Arabidopsis40 and the latter from the transmission spectrum of fluorescent light bulbs (used in in vivo experiments utilized in the p-ath773 model reconstruction)36, the absorption spectra of chlorophyll37, and the Leaf Area Ratio of Arabidopsis40. However, the value of biomass yield (for a given plant across its full lifecycle) has been experimentally identified as between 0.7 and 0.8537. Here, to achieve the best alignment between in silico and in vivo growth patterns, biomass yield has been defined as 0.51. There are several possible reasons for this, which are included in the Discussion section. All files necessary for p-ath773 have been included in the GitHub p-ath773 repository (10.5281/zenodo.3735103). The in silico results of the final p-ath773 model can be found in Data Once growth stages have been implemented with the p-ath773 model and the ORKA framework, the design-build-test cycle  in addition to meeting the metabolic needs (such as photosynthesis) and physiological needs (such as transpiration) of tissues. Water flow through the plant has been selected as a case study which shows tissue-level insight into the general metabolic and transport processes modeled in p-ath773. The results of this analysis are shown in Fig. 42, yet sulfur still is important to plant growth. The results of this analysis are shown in Figs. In addition to tracking the flow of water through the plant, the p-ath773 model has also been used to study and track sulfur metabolism and transport across the tissues and the lifecycle of the plant to provide an example of reaction-level window into the p-ath773 modelled plant metabolism. This has been done to provide unique insight into the core metabolism of a single micronutrient which is not as extensively studied as carbon and nitrogen metabolismAt some places in the Figs. O(h2) in the SOA to O(h4) in the ORKA, depending on the Runge-Kutta method used in the ORKA. ORKA has been developed to be general enough that several different Runge-Kutta methods could be applied to biomass step estimates  to dynamic Flux Balance Analysis (dFBA) has been developed. Inspired by the Static Optimization Approach (SOA) to perform dFBA, it seeks to achieve higher levels of model accuracy and solution stability. ORKA differs from the SOA in that it replaces first-order Taylor-series approximations for biomass and concentration steps with Runge-Kutta- and Trapezoid rule-based integration. This provides lower error floors, from tes Fig.  dependenArabidopsis thaliana has been reconstructed 19, the p-ath773 model in theory has at least a three order of magnitude lower error floor due to the smaller step sizes, increased accuracy of the dFBA approach used, and inclusion of two more tissue types. However, similar comparison with the most recent dFBA work on Arabidopsis lifecycle19 is not entirely possible since these models are quite different in structure, goals, and results. For instance, the mass of the plant for the 6 to 36 days window of time is quite different between p-ath773 and the model produced by Shaw and Cheung (2018)19 . Therefore, by these models having different biomass, different flows of nitrogen, and different biomass composition and demands, it is very difficult to make a worthwhile comparison between the two models on the basis of accuracy as the structure is so different without strongly adapting one model or the other to be more similar to the other. Even though the p-ath773 model lacks a similar model in literature for the purposes of comparison, possibly because different literature sources and goals are used in model reconstructions, it is certain that when ORKA will be applied to a modeling framework comprising of all major tissues and can recapitulate and analyze real plant phenotypes. Further, these differences do not invalidate one model or the other, but rather might consider different metabolic states due to different growth conditions, thereby representing the flexibility of biological systems. In future, OKRA can be applied either by developing more tissue models  and adding to Shaw and Cheung’s model or extending the p-ath773 model to capture the secondary metabolism, and either approach, carefully informed by literature, could greatly add to knowledge of Arabidopsis metabolism.As a test system for ORKA, a multi-tissue core metabolism stoichiometric model of ted Fig. , which ited Fig. . This moges Fig. . Once p-ges Fig.  using HeArabidopsis growth  are not modeled. This is another potentially considerable drain on plant resources which is not modeled in the p-ath773 model.Using the ORKA to perform dFBA, p-ath773 is able to simulate seven stages of wth Fig.  and showwth Fig.  and on sin vivo is unlikely to be as significant as shown by the p-ath773 model but is still likely to make some contribution. This is because not all water dynamics are accounted for in the p-ath773 model, including factors such as the amount of water necessary to keep new biomass turgid (since what is modeled is dry weight not wet weight) and the amount of water produced or consumed by the plant’s extensive secondary metabolism. The former shortcoming is common to all SMs rather than to the p-ath773 model in particular, as all such models only model dry weight.The modelled flux rates have been used to study the flow of water through the plant system, and in particular to investigate the contributions of metabolic water to that transpired Fig.  and to iFor the sulfur metabolism case study, it has been shown that part of the patterns of sulfur metabolism are as expected such as increased use of and metabolic demand for sulfur when the seed tissue is present. However, some unexpected behavior has also been observed such as higher fluxes through sulfur reactions and comparatively larger concentrations of sulfur-containing metabolites at early growth stages. It is nearly impossible to pinpoint a single cause for the unexpected metabolic behavior of the sulfur metabolism in the early growth stages. This is due to the links between sulfur and energy metabolisms, in that many steps use some type energy molecule. Sulfur metabolism is also closely linked to the proton budget of the plant, in that many transports are proton-coupled. Through links to both the energy metabolism and proton budget, sulfur metabolism is strongly connected with the rest of plant metabolism. Hypothetically, this unexpected metabolic behavior might therefore be advantageous to the plant in energy metabolism and the control of the flow of protons. Particularly in the first two growth stages when the seedling’s endosperm and cotyledons are not fully utilized and are therefore providing some amino acids (though notably not cysteine or methionine), fatty acids, and sugars. As modeled, these stores interact with the extracellular space of the leaf tissue, and often require facilitated transport  into the cytosol for use or catabolism. It is therefore possible that these unexpected behaviors aid in the transport of nutrients from the endosperm, by having standing pools of metabolites which participate in proton-coupled transport to better regulate the cell’s proton budget. This hypothesis is supported by the fact that these unexpected metabolic behaviors are reduced in magnitude as the amount of nutrients uptaken from the endosperm are reduced, and indeed the concentration of metabolites such as cysteine sharply decrease. These unexpected behaviors then appear to cease all together when the endosperm is fully utilized. While the metabolic network of p-ath773 is too convoluted to prove this theory, it does highlight the usefulness of stoichiometric modelling to identify interactions which may be too complex to deduce through non-systems approaches.in vivo data. The latter generally ensure that mass balance is maintained when metabolites are transported between tissues since each flux rate is in units of mmol/gDW tissue·h and each tissue is of a different mass. Hence, such conversions are necessary. Other constraints which fall in the category of tissue-tissue interactions ensure that nutrient flow is in a logical and well-known direction (e.g. micronutrients travel up from the roots). Few constraints, with the exception of the enforced diurnal patterns of carbon storage, apply on the reaction rate- or metabolite concentration- levels, leaving a large number of system degrees of freedom at the micro-scale. Therefore, by constraining the macro-scale behavior to what is known, the p-ath773 model can be used to determine what is, or may be, occurring in the plant system with respect to reaction rates or metabolite concentrations. From the allowed uncertainty at the micro-scale level, a study of this level allows investigation of what metabolic processes support and explain the known macro-scale behavior.While there are a number of constraints applied to the model, such as biomass yield; maintenance and senescence costs; and enforcing mass ratios between tissues, these constraints apply mostly to plant-scale behaviors. These behavioral constraints generally fall into two categories: whole-plant and tissue-tissue interactions. The former generally ensure that the pattern of modeled plant and tissue growth fits that of Oryza sativa), potatoes (Solanum tuberosum), tomatoes (Solanum lycopersicum), and soybeans (Glycine max).This work provides the basis for much future development and sophistication, both in broadening the range of approaches which can be taken to dFBA, and in the potential to use p-ath773 as a basis for modeling other plant systems. Appying ORKA to perform dFBA may provide the framework for other step-by-step dFBA approaches utilizing other ODE solving methods such as Taylor Series, Linear Multistep, or even adaptive step size methods depending on the needs of the modeled system. The current p-ath773 model could be further sophisticated by adding the secondary metabolism of the plant system, which constitutes a significant portion of metabolism in many plant systems. Further, several simplifications have been made regarding tissues, particularly related to seed tissue, at present. For instance, the model currently assumes when the plant is flowering, that flower biomass and metabolism is roughly equivalent to that of the seed. While this results in a simpler model, this model cannot be used to investigate certain metabolic hypothesis such as the cost to the plant resulting from flower pigmentation, pollen, and nectar production. Future work will include developing models for other plant tissues, such as flowers. In addition, as this is a core carbon metabolism model, it is likely quite similar to the core metabolism of other plant systems. Therefore, the p-ath773 model can serve as a basis for the development of lifecycle models for other plant systems, particularly annual eudicots which are of agricultural interest, such as rice (Static Optimization-Based dFBA Approach (SOA) was first introduced in 2002 and is a method for solving for dynamic changes to a model system on a point-by-point basis (where those points are time), as opposed to the Dynamic Optimization-based Approach (DOA) which solves all points simultaneously28. The SOA, and variations thereon, have been applied to Arabidopsis19 and barley47 plant models, making it of particular interest for the study of plant metabolism. The SOA is defined as follows28:vbiomass indicates the rate of reaction for the biomass production and is equivalent to the growth rate μt which will be used hereafter. It should also be noted that biomass concentration, Yt, is actually an element of the I′ set (the set of metabolites whose concentration is tracked). However, Eq.  around point a is defined as follows.f(x) = eμt then fn)((x) = μneμt ∀n, this Taylor series expansion may be simplified as follows.a = 0, the Taylor series expansion becomes what follows (knowing that e0 = 1).eμt, therefore:Yt gives the biomass steps which are used by the SOA, namely Eq. (O(h2). One additional assumption is made in Eq. (t) vj,t is constant.The mass step taken at each time point in the SOA method, as shown in Eq. , is deririve Eq.  that is mely Eq. . Therefoe in Eq. . The difOptimization- and explicit Runge-Kutta-based dFBA Approach (ORKA) differs in that different approximations are used in the solutions attempts to increase the accuracy of the estimation of the concentration and mass steps.j,t represents an estimate of the integral of vj,t. Equation  are represented in a triangular Butcher tableau such as shown below in Fig. The dFBA method developed in this work is similar to SOA in the sense that both solve time points in a step-by-step and cumulative fashion. The μt is calculated by performing FBA on an SM, the SM must be solved for each kn (Runge-Kutta derivative estimate for step n). Therefore, by this necessity, we would also know reaction rates at each time point where a kn is solved for, namely, vj,t, vj,t +c2Δt, …, kn occurs be equally spaced, then the multiple-application Trapezoidal Rule  might be applied to solve the integral presented in Eq. . As the exact value of zi,t + Δt may be necessary to calculate vj,t + Δt, it will be assumed that vj,t + Δt is equal to the arithmetic mean of the reaction rates in the n time points used in the Runge-Kutta method selected plus the starting point. Therefore, Eq. . From this, the ORKA method can be represented as follows.Therefore:g to Eq. , should d in Eq. . Evenly l in Eq.  is that n ∈ N, denoting the current time as tcc1 through cnf might be selected. It should be noted that the multiple application Trapezoidal rule has an error floor in the order of O((c2−c1)h3). Therefore, the integral estimate will have lower error than third-order Runge Kutta methods, which generically have a global error of O(h3) because of their use of individual steps of the Runge Kutta method rather than the full time step. The integral estimate will be the limiting accuracy factor if fourth order Runge-Kutta methods (such as the 3/8-rule fourth order method) or better are used which have a global error of O(h4).For each Runge-Kutta Step aij = 0∀j ≥ i, and implicit if this is not the case. Recall that aij defines the dependence of one derivative estimate step on another, as shown below.f is the solution of a large system of under-defined linear equations (the stoichiometric model) where the best solution is selected by optimization. Using an implicit Runge-Kutta method would require the full model to be included in an under-defined non-linear system of equations and then to be solved by non-linear programming approaches. These approaches neither guarantee that a solution would be found nor a solution found would be optimal48. The sharply increased computational costs of using an implicit method combined with the complexity of implementation and the non-guarantee of an optimal solution has made implicit Runge-Kutta methods not worthwhile or attractive for implementation.Generally, Runge-Kutta methods are defined as either implicit or explicit. A method is defined as explicit if 29 allowing an accurate reconstruction of the central carbon metabolism of the seed. Next, the leaf tissue has been reconstructed as a phototrophic tissue to supply carbon to the seed tissue. We next have reconstructed the root model to provide a mechanism for the uptake of water and micronutrients necessary for plant growth. Finally, we have reconstructed the stem model to provide a logical link between the tissues. Additional works that have been used in the reconstruction of tissue models can be found in Data The seed tissue has been modeled primarily based on a published MFA work29. We then manually have filled gaps in this model with reactions based on literature and genomic evidence49 or with reactions being necessary for ensuring model connectivity. The stoichiometric coefficients of biomass precursors have been determined using sink reactions, dry biomass weight composition, and amino acid mass ratios provided in a previous work29 .Next, we have reconstructed the leaf model by taking common reactions/pathways from the seed model and adding metabolic pathways for amino acids that are not synthesized in the seed. In addition, other leaf-specific pathways such as photosynthesis, carbon fixation, and gluconeogenesis and necessary transport reactions have also been added. We then have developed the biomass equation for the leaf tissue using that of a previously published see Text , with misee Text 29. The rArabidopsis-specific estimates, the dry weight composition of switchgrass (Panicum virgatum) root and stem31 have been assumed to be equivalent to the biomass composition of these tissues in Arabidopsis. Thus, we have found the biomass of root and stem tissues to be composed entirely of carbohydrates. The resultant root tissue model focuses on nutrient uptake, transport, and growth, consisting of 149 reactions, 324 genes, and 149 metabolites, while the stem tissue model focuses on transport and growth, consisting of 167 reactions, 291 genes, and 154 metabolites. We have included the root and stem models in the GitHub p-ath773 repository (10.5281/zenodo.3735103).We have constructed the root and stem models, similarly, by extracting common reactions/pathways from the seed model and adding necessary and root-/stem-specific transport and exchange reactions. Then exchange reactions have been added to allow the root to be linked to micronutrient uptake processes from the soil and the stem to be involved in inter-tissue transport processes. In the absence of 7. Confidence scores are integer values between 0 and 4, with higher values corresponding to higher confidence in the inclusion for a given reaction. In the scoring system used, 0 corresponds to an unevaluated reaction; 1 to a reaction included for modeling necessity; 2 to evidence from physiology or a genome annotation; 3 from knock-in knock-out in vivo experiments; and 4 for direct biochemical data giving evidence for that metabolic function. The distribution of confidence scores in the component tissue models of p-ath773 can be found in Fig. Arabidopsis is a model system. Additional information on confidence scoring of the p-ath773 model can be found in Text We have defined reaction confidence scores in a manner consistent with a previously published protocol50 for the majority of reactions using the code included in the GitHub p-ath773 repository (10.5281/zenodo.3735103). This has been followed by having manually curated the GPR links and/or inclusion rational of reactions with non-KEGG identifiers. This information can be found in Data All reactions in all four models have been balanced both in terms of elements and charge. Thermodynamically infeasible cycles have also been resolved by removing reactions, breaking composite reactions, and adding metabolic costs to transport reactions. For all four tissue models, GPR links have been established through a largely automated workflow utilizing the KEGG APIYt indicates whole-plant mass. This could require some complex hand calculations to determine the value of the RHS of Eq. . On the other hand, the biomass of each plant tissue can be tracked individually as stated in the following equation.Yθ,0 = 0 yeilds only 0 as a solution. This can be shown in that the generic solution to Eq.  no tissue can either appear in the system that is not there from the beginning, and (ii) no tissue can be removed from the system. This is particularly problematic for a plant system since certain tissues appear and are removed after the plant reaches certain levels of maturity, perhaps most notably flowers and seeds. Therefore, while more complex, determining the value of the right-hand side of Eq. (st) and several other quantities in Eq.  where h = (c2 − c1)Δt is the order of error for the estimation of the backward derivative, this limits the order of the Runge-Kutta methods which can be chosen for increased accuracy benefits. For instance, a third order Runge-Kutta method has a global error of O(h3), less than that of the backward derivative estimate. In this case, choosing any Runge-Kutta method which is higher than third order would merely add complexity with no benefits in terms of the error of the solution. Therefore, a third order Runge-Kutta method has been chosen for implementation with the p-ath773 model. Two commonly-used such methods are Heun’s and Kutta’s third order rules shown in Fig. aij of the Butcher tableau which have caused errors in earlier implementations of Kutta’s third-order rule to the p-ath773 model (but would not in the current model).Given that the steps taken for the Runge-Kutta method have equally spaced values of t), we can state that the error would be on the order of Arabidopsis models including that of Shaw and Cheung (2018)19, which by reason of using SOA and a time step of one day (as opposed to one hour) results in much higher error potential. Calculating on the basis of hours the big O error ratio between ORKA and the SOA used by Shaw and Cheung (2018)19 the is approximately 1:5184. This will provide two distinct advantages to the p-ath773 model. First, higher accuracy for calculations due to smaller step size and lower errors associated with approximations used. Second, increased solution stability, so that more solution steps may be taken without ballooning error.Given that the limiting order of error in this system is the error in parameter derivative estimates, this will be the order of error for ORKA calculations for the p-ath773 model. As the value of h is one third of that of the time step  of growth, as all plant seeds are dispersed by approximately day 61, and after which in vivo data on plant growth and mass is sparse32. More specific details can be found in the following sub-sections. The full optimization-based framework used in this work has been provided in the GitHub p-ath773 repository (10.5281/zenodo.3735103) associated with this work.The tissue models were linked using techniques similar to a well-known computational framework known for modeling microbial communitiesAs concentration is tracked using the ORKA method, the mass balance for the system need not be a strict equality, but rather metabolites should be allowed to be stored and that store should be allowed to be used up. To this end, the mass balance has been defined as follows:As has already been suggested, it is not necessarily logical for metabolites to flow between tissues without suitable constraints on that flow. For instance, water is taken up by roots and transported first to the shoot, then to the leaves and seed tissue (if present). It would not, for instance, make sense for water to travel directly from the root tissue to the seed tissue. Therefore, instead of a single metabolite pool connecting tissue, there are metabolite pools connecting each pair of tissues. These pools are shown in Figs. 54. Transpiration is only allowed during the day because it is assumed that the stomas are open during the day (or when light is available), allowing transpiration, and closed at night (or when light is not available). Transpiration is reported as approximately 2.95 mmol water per m2 per second54. This can be converted to 422.3 mmol water per gDW plant per hour based on the information such as the leaf area ratio40, which is scaled at each time point as appropriate to give the rate in mmol water per gDW leaf per hour.Mathematically, for the flow of water these logical metabolite links take the following form.Micronutrients, such as nitrates, sulfates, and phosphates, follow much of the same flow pattern through the plant as does water. This is because water transports dissolved micronutrients to the rest of the plant through the xylem. The major differences are: (i) micronutrients will be used up in each tissue so that the amount of each micronutrient leaving each tissue will be less than that entering, which is modeled by Eqs.  and 52)52) belowAs is well known, sugars in plants are synthesized in photosynthetic tissue, and are transported to the rest of the tissues through the phloem. In the p-ath773 model, it is assumed that the vast majority of photosynthesis occurs in the leaf tissue, and the photosynthetic output of other tissues is negligible. This assumption is based on two factors: i) leaves are tissues specifically designed to carry out photosynthesis, and ii) photosynthesis relies on above-ground surface area to absorb light to drive the process, and leaves have by far the most surface area. Therefore, the flow of sucrose in the modeled plant system is as being exported by the leaf tissue in Eq. , transpo29, and the root and stem models do not require amino acids for biomass production in the defined biomass composition31. Essentially, these constraints ensure that all amino acids exported by the leaf are uptaken by the stem, Eq. 60) deal πt, is defined which takes a value of 1 if light is available for growth and zero otherwise for time t. This is incorporated into the model to simplify the above equations into two equations. Note that the “in” and “out” reactions are combined such that if the model is taking up a given metabolite the reaction rate will be negative, while exporting a given reaction would correspond to a positive reaction rate.πt) term in the above equations serves as a binary switch alternating between values of −1 and 1 based on the availability of light. In addition to these constraints, some limit must be placed on the uptake of carbon dioxide and oxygen by the leaves of the plant. It has already been noted here the modeled transpiration occurs at 422.3 mmol water per gDW plant per hour54, and this is used as the basis for the exchange of other gasses as well. It is noted that the rate of carbon dioxide uptake is two order of magnitude less than the rate of water loss53, and an in vivo study identifies the rate of carbon dioxide flow into the leaf as 8 μmol/m2·s55, which is converted using the Leaf Area Ratio to 1.14 mmol per gDW plant per hour. Assuming standard atmospheric composition (0.04% Carbon Dioxide and 21% Oxygen), then there are approximately 525 oxygen molecules per carbon dioxide molecule at ground level. Here, the limit of oxygen uptake is proportional (in terms of the composition of the atmosphere) to the limit of carbon dioxide uptake, specifically 598.5 mmol per gDW plant per hour is the oxygen uptake limit used. Further, as plants lack a system to transport gasses from one organ or tissue to another  it has been assumed that each tissue is responsible for its own gas exchange. As the leaf is a tissue specifically designed for photosynthesis and gas exchange, it will be assumed that the gas exchange occurring in the leaf is at least one order of magnitude larger than that occurring in the rest of the plant. As the other tissues are modeled as heterotrophic (i.e. not significantly photosynthetic), the rate of oxygen uptake must be limited. Therefore, the limit of oxygen uptake for root, seed, and stem tissues is set at 59.85 mmol per gDW plant per hour.When light is available54. These equations are defined as follows.Al, Ast,1, Ast,2, fl, fst,1, fst,2, bl, bst,1, and bst,2 in Eqs. (Plants store carbohydrates in leaf and stem tissues in the form of starch (leaf and stem) and sucrose (stem) in a pattern where the rates of storage may be modeled by a sine wave with a period of 24 hoursArabidopsis and (ii) this prevents the problem of the model preferentially producing the “cheapest” biomass. The rate of biomass production determined by an SM is the growth rate of the biological system being modeled8; therefore, plant mass can be defined as:ωt, ηt, and λt, respectively, in Eqs. , ηt → 0 and as this occurs μleaf,t. At this point, it is worthwhile to discuss how seed biomass is lost in a non-productive way .We have discovered while building this model that tissue growth rates must have enforced links between growth rates of tissues in the system for two reasons: (i) linking tissue growth rates allows control of the tissue mass ratios so that they may be modeled as they occur in uted Eq. , linear  in Eqs.  through  in Eqs. . TherefoArabidopsis, is reproduction. The seed contain a large amount of metabolites which may be metabolized by the embryo to sustain it and allow it to grow before it can photosynthesize. These stored metabolites include fatty acids, proteins, and sugars58. Further, Arabidopsis plants produce a very large number of seeds, on the order of approximately 28,000 seeds per gram dry weight of vegetative mass34. To properly model this metabolic investment, the model must ensure that these costly metabolites from the seeds are not returned to the plant metabolism when seed biomass is lost in seed dispersal. To explain how this could happen, generally the biomass reaction consumes metabolites such as amino acids, fatty acids, sugars, and other necessary compounds in its production and in these cases, μseed,t > 0. Conversely, when seed mass is being lost from the system, μseed,t < 0, the biomass precursors are produced from the biomass pseudo-metabolite, and without careful constraints, this loss of seed biomass could cause these precursor metabolites which constitute biomass to remobilize (e.g. used in the metabolism for metabolite production elsewhere) into the metabolic model, resulting in the use of these resources which should be lost to the plant. Essentially, this would simulate a plant consuming the stores of metabolites in its own seeds, rather than releasing those seeds with its stores intact. Instead, to allow modeling of the complete and non-productive loss of seed biomass, an extra equation called “biomass loss” has been defined to be identical to the biomass equation except it does not produce the biomass pseudo-metabolite. This allows the definition of the following constraint which is in effect during the silique ripening growth stage.One of the most metabolically costly activities for many species, including for Arabidopsis growth, here named as seed germination stage and seed germination to leaf development transition, a seedling’s primary source of carbon is its reserves of stored carbohydrates, proteins, and lipids. It has been shown that seeds have stores of approximately 0.425 μg of sucrose, 6 μg of fatty acids, and 6 μg of proteins (modeled here as component amino acids) available30. As no information concerning the pattern of usage of the seed storage has been found, it has been assumed that the stores are utilized at a constant rate during the duration of the seed germination period and that all the storage is fully consumed by the end of the seed germination to leaf development transition stage, which has been defined the point at which the cotyledons are fully open and leaf development intensifies32. The rate at which the seedling should uptake the seed storage has been determined by identifying the moles (mmol) of each major component of the seed storage and dividing by the time over which the seedling consumes those. This has resulted in a mmol · h−1 quantity. See Data −1 · h−1 quantity, which is used to bound the uptake rates of stored metabolites in the seed. As the leaf has proven to be the most metabolically active tissue, it is assumed that the leaf tissue of an Arabidopsis seedling uptakes the stored fatty acids, amino acids, and carbohydrates that are provided for seedling growth during the seed germination stage when the leaves have no access to light  of 25.3 μg and have only about 7% water content30. The embryo itself is assumed equal to the seed mass less the mass of seed stores of sucrose (0.425 μg), Fatty Acids (6 μg), and proteins (6 μg)30. Having assumed that the dry matter content ratio holds for the embryo as well, this has left approximately 11.0 μg dry weight (DW) for the embryo. As information on the ratio of tissue masses in Arabidopsis has not been documented in literature, the general ratio for herbaceous plants has been used as a starting point, namely 0.46:0.24:0.3 leaf:root:stem FW59. This ratio has been converted to DW ratio for stoichiometric modeling. Experimental data has shown that the dry matter content of leaf tissue is 0.212 DW/FW, of root tissue is 0.170 DW/FW, and of the stem tissue is 0.176 DW/FW57. Having converted the FW ratios to DW ratios has given the ratio of 0.511:0.267:0.211 leaf:root:stem DW. While the dry matter content of an embryonic Arabidopsis is much higher than that of a mature plant (the source of the utilized dry matter content ratios), this DW tissue ratio has non-the-less been assumed to be accurate for the embryo due to lack of evidence to the contrary.As the model advances plant and tissue masses with respect to time, the establishment of initial mass for plant and tissues has become important in this framework. Experimental evidence has shown that 32. Stage transitions selected include the transition to stage 0.70 36 and the absorbance of Arabidopsis leaves36 and surface area to plant mass of Arabidopsis leaves40. This has been approximately estimated to be 4.00 mmol·gDW plant−1·h−1. This value has been shown to be 21.50% of the total photons output by the fluorescent light 32. Further, it appears that the plant no longer maintains current mass, but allows tissues to die and desiccate32. This has been included in the p-ath773 model in that plant senescence is increased by an order of magnitude and plant maintenance is set to zero following the end of the Flower Production stage.An important consideration in any SM is the definition of a maintenance cost, which is typically defined as ATP hydrolysiss in Eq. . A maintz has been defined as the objective variable with vgrowth,tissue being defined as the rate of biomass production, in units of h−1, of the tissue referenced. The maximization of this objective function is approximately equivalent to maximizing the growth rate (change in mass per unit time) of the plant as a whole. This objective function, in early model iterations, has led to one major issue, namely how to avoid the model producing only the metabolically “cheapest” tissue which could result in the maximum objective value but is biologically unrealistic. This is addressed by Eqs. . by Eqs.  through  by Eqs.  and will22. This objective has generally been framed as minimizing the amount of light used by the plant system, given a required growth rate22. As the purp authors gratefully acknowledge funding ose of this paper is to showcase the ORKA method, rather than the p-ath773 model, alternative objective functions have not been implemented but are possible to implement.It has been noted that the maximization of plant biomass has not been the only feasible objective function for plant SM system; for instance, one alternate objective function is the maximization of plant photonic efficiencyj, and σj has been valued at 1 for the current reaction whose maximum and minimum are to be investigated and 0 for all others and is stepped through first maximizing and then minimizing each reaction. Due to restrictions of the time allowed for model solutions, nine points has been selected at which to perform FVA. These points are 1 hour after germination , 70 HAG , 90 HAG , 177 HAG , 181 HAG , 770 HAG , 810 HAG , 1155 HAG , 1170 HAG , 1190 HAG , 1199 HAG . These results generally showed narrow ranges for allowable flux rates.Flux Variability Analysis (FVA) has also been performed on the p-ath773 model which uses all previously defined constraints and the previously defined ORKA method. All flux bounds and constraints are the same and the FVA has an objective function defined as follows:Some of the files used in this work and included in the GitHub repository are files of programming codes, and all code is not written in the same programming language. Text Supplementary Text S1Supplementary Dataset 1Supplementary Dataset 2


Propionibacterium, which has been divided into several new genres. Such data also enable a comparative genomic approach to annotation and provide a new opportunity to revisit our understanding of their metabolism. Using pan-genome analysis combined with the reconstruction of the first high-quality Propionibacterium genome-scale metabolic model and a pan-metabolic model of current and former members of the genus Propionibacterium, we demonstrate that despite sharing unique metabolic traits, these organisms have an unexpected diversity in central carbon metabolism and a hidden layer of metabolic complexity. This combined approach gave us new insights into the evolution of Propionibacterium metabolism and led us to propose a novel, putative ferredoxin-linked energy conservation strategy. The pan-genomic approach highlighted key differences in Propionibacterium metabolism that reflect adaptation to their environment. Results were mathematically captured in genome-scale metabolic reconstructions that can be used to further explore metabolism using metabolic modeling techniques. Overall, the data provide a platform to explore Propionibacterium metabolism and a tool for the rational design of strains.Propionibacteria have been studied extensively since the early 1930s due to their relevance to industry and importance as human pathogens. Still, their unique metabolism is far from fully understood. This is partly due to their signature high GC content, which has previously hampered the acquisition of quality sequence data, the accurate annotation of the available genomes, and the functional characterization of genes. The recent completion of the genome sequences for several species has led researchers to reassess the taxonomical classification of the genus  P. acidipropionici and P. freudenreichii, which are found in dairy products and the rumen and are associated with the ability to utilize a wide variety of carbon sources. They are used in various biotechnology applications: in particular, P. freudenreichii is used as a starter culture in cheese manufacture, as a probiotic, for the production of vitamin B12, and for the production of propionic acid for food applications [P. acidipropionici strains have the potential to produce propionic acid at high titers and yields nearing those required to replace the chemical production of industrial-grade propionic acid [Propionibacterium  and Acidipropionibacterium (including former P. acidipropionici) [Propionibacterium genus and have been reclassified as Cutibacterium [P. avidum inhabits moist, nutrient-scarce regions rich in sweat glands such as the axilla where they are likely dependent on proteolytic activity to supply amino acids as a major carbon and energy source, whereas P. acnes inhabits hair follicles and appears to predominantly rely on sebum degradation via lipases [Propionibacterium genus consists of Pseudopropionibacterium and includes P. propionicum, which causes actinomycosis [Propionibacterium, Acidipropionibacterium, Cutibacterium, and Pseudopropionibacterium genera, out of simplicity, we will use the terms propionibacteria and Propionibacterium to collectively refer to all of these genera and will not use the new classification to avoid confusion.Propionibacteria are characterized by an unusually efficient anaerobic energy metabolism. This is primarily due to the Wood–Werkman cycle, which is a pathway that facilitates the production of propionate while maximizing energy capture ,2. Tradiications ,6,7. Somnic acid ,9. This pionici) . The cutacterium . These sacterium ,11 and p lipases . The finThe prevalence of propionibacteria in the human microbiome has importance in human health and their industrial significance dictates a need for a thorough understanding of their metabolism. While many strains have been sequenced, accurate annotations are required to fully understand and ultimately rationally engineer metabolism in propionibacteria. The development of genome-scale metabolic models could offer new opportunities for studying propionibacteria, including the design of drugs or drug targets which could, in principle, be specifically targeted toward species involved in a secondary infection or acne vulgaris by studying species-specific metabolic capabilities and interactions with the human host . In addiPropionibacterium with the majority of metabolic activities in these organisms being inferred through homology rather than confirmed experimentally. Therefore, annotation by homology may produce erroneous substrate predictions. For example, homologues in two strains may be mapped to different functions. Finally, the genus is metabolically diverse. For example, the second major fermentation product, acetate, is produced by various pathways in different species, as illustrated when an attempt to enhance propionate production in P. acidipropionici [P. freudenreichii [P. acidipropionici [Propionibacterium models [There are several challenges to producing accurate annotations for propionibacteria. Firstly, the high GC content of actinobacteria such as propionibacteria presents unique challenges to annotation, including a higher error rate and biased read coverage in genome sequencing , althougopionici  failed dnreichii  but abseopionici . A good opionici ,26,27 arm models  that lacPropionibacterium. We extend the definition of the pan-genome to a genus basis as per several previous studies [Propionibacterium species were driven by functional cues arising from the draft metabolic models or strain and enzyme-specific information from the literature, and by genomic-context cues, which leveraged information from the pan-genomic matrix, and the local gene-context of orthologues for individual strains. P. acidipropionici ATCC 4875 was selected for deep experimental characterization, including growth on chemically defined media, proteomics, transcriptomics analysis, and phenotype arrays to drive further refinement of its annotation, which were subsequently propagated. Then, core and pan-metabolic reconstructions were generated from the individual highly curated reconstructions and utilized to compare the metabolism of five different species. Our approach demonstrates that synergistic complementation of pan-genome and pan-GEM  analyses can overcome limitations in bioinformatics tools. The analysis revealed a possible evolutionary mechanism for the Wood–Werkman cycle and suggests a putative ferredoxin-linked electron bifurcating mechanism.To address these challenges, we developed a pan-genome guided metabolic network reconstruction of  studies ,30 and i studies ,32. The Propionibacterium genomes reported in the National Center for Biotechnology Information (NCBI) database at the time of performing the analysis, 21/01/2015, we focused the analysis on the 16 closed genomes due to reduced performance of the pan-genome approach when incomplete genomes are included [Propionibacterium strains were downloaded from the NBCI FTP server as GenBank files and consisted of two strains of P. acidipropionici (now Acidipropionibacterium), P. freudenreichii CIRM-BIA1, P. avidum 44,067 (now Cutibacterium), P. propionicum F0230a (now Pseudopropionibacterium), and 11 P. acnes strains (now Cutibcterium)  . Since tici 4875  using se−5. Given there were multiple closed genomes for two species, the analysis was split into two stages. First, an intra-species analysis was performed, and representative strains for each species were selected for an inter-species analysis .A pan-genomic analysis underpinned the metabolic reconstruction effort by supplying extensive functional and sequence-based information . All cluanalysis  to reducTettelin  and WillTettelin  algorithPropionibacterium metabolism. Overall, the functional annotation collapsed the pan-genomic matrix by about two-thirds, while the number of core clusters remained roughly constant. Visualization of the orthologous and functional pan-genomic matrices was performed using Cluster 3.0 [The orthologous clusters generated from the pan-genomic matrix were found to be poor representatives of the functional capacity of the genus. Evidently with a small set of strains of differing evolutionary distance, the algorithms will cluster true homologues in separate groups based on small sequence differences. Therefore, robust analyses, such as using the intersecting clusters of the OMCL, bidirectional best hit (BDBH), and COGtriangles algorithms , tended ster 3.0  and Treester 3.0  . Then, essential reactions required for growth on any of the conditions were identified and added to the analysis. Finally, the minimal number of reactions needed to sustain growth across all conditions was simultaneously computed using a mixed-integer linear programming formulation. A final round of manual curation was performed such that all reactions incorporated into the model based on this work were either supported by genes, enzyme promiscuity information, essential, or otherwise the only reasonable candidate, and results were propagated across other reconstructions where supported by genetic or biochemical evidence. The degradation pathways of phenylalanine and two related aromatic compounds completely lacked any genetic support for either of the currently known pathways for degradation and therefore were excluded.Phenotypic growth array data  for P. aComparison of the biosynthetic and catabolic potential of amino acids in propionibacteria was performed by generating clustergrams. Reactions associated with the catabolism or anabolism of each amino acid were grouped, and their presence or absence was scored in each reconstruction. Differences between strains were quantified using the Jaccardian similarity coefficient. Details of these pathways, brief functional assessments, and Jaccard score computation can be found in Propionibacterium acidipropionici ATCC 4875 was obtained from the American Type Culture Collection. Cells stored at −80 °C were used to inoculate serum bottles containing 50 mL PAM media . Cultures were incubated statically at 32 °C for 24 h and used to inoculate Applikon Ez-Control bioreactors containing PAM and 100 g/L of glucose, sucrose, or glycerol. All experiments were performed in duplicates. Reactors were maintained at 32 °C, pH 6.5 with 5 M NaOH, agitated at 200 rpm, and the head space was degassed with nitrogen to maintain anaerobic conditions throughout the fermentation. Regular samples were taken to measure optical density using a Biochrom Libra S12 spectrophotometer and for organic acid analysis. Cells were harvested for RNA and protein extraction during the mid-exponential phase. Total RNA and mRNA enrichment, sequence alignment, and differential expression analysis were performed as described in [ribed in . For proribed in  and quanribed in , using aribed in .nifJ1 gene  was knocked out in P. freudenreichii using CRISPR/Cas9 by interruption of the gene with gfpUV. Briefly, we assembled plasmid pPAC_Cas9 using pRGO1 as the plasmid backbone and introducing the lacZ-MCS region, an erythromycin resistance gene (ermE), and a GC-optimized Cas9 under the control of promoter PermE. Then, Gibson assembly was used to create the knockout vector, pCas9_nifJ . PCR reactions were performed using Phusion Taq (NEB). Introduction of plasmid pCas9_nifJ into P. freudenreichii was performed by electroporation as previously described [The pyruvate:ferredoxin oxidoreductase (PFOR) encoded in as9_nifJ , by addiescribed . After 3Propionibacterium. Unlike previous genus level pan-genomic analyses [Propionibacterium genome and 4445 (65%) of which were strain-specific   .Propionibacterium species were generated through a novel workflow that leverages information from the pan-genomic matrix and its functional pivot (as detailed in the methods section). This included two P. acidipropionici strains, which are the parents of high propionate producing genome-shuffled mutants [The pan-genomic matrix is insufficient to compare the metabolic functionalities of different propionibacteria due to factors such as convergent evolution, lateral gene transfer, and the splitting of true orthologues due to different evolutionary distances between organisms from different genera. A functional pivot of this matrix can surpass some of these issues, but it suffers from a lack of manual curation, particularly in the context of a metabolic network. Therefore, metabolic reconstructions of five  mutants ,65 with Propionibacterium, biotin, and pantothenate , 45 in P. propionicum, 21 in P. freudenreichii, 15 in P. avidum and just 3 in P. acnes. The lower number of specific functionalities for P. avidum and P. acnes is hardly surprising given they have both been reclassified as Cutibacterium and are therefore, they are the only genus with two different species included in the analysis. Accounting for this, we found that Cutibacterium possessed 40 unique functionalities. To better understand how the metabolism of the 5 species analyzed is conserved and variable, the genus-level metabolic reconstruction was used to compare central carbon metabolic pathways and key metabolic nodes, namely redox metabolism, the pyruvate and phosphoenolpyruvate node that splits metabolism and regulates carbon flow, the characteristic trehalose and polyphosphate metabolism, and amino acid metabolism.The genus-level metabolic reconstruction consisted of 1223 metabolic reactions and 279 transporters, with the core reactome consisting of 761 metabolic reactions. Of the metabolic reactions, about 16% (124) occurred only in dairy species with 3% (23) occurring in all dairy propionibacteria analyzed. Conversely, 95 reactions (13%) were only present in cutaneous species with 16 occurring in all cutaneous species. Small differences were observed between the two P. avidum, which did not have a transaldolase (2.2.1.2). Three variations of the TCA cycle lacking a glyoxalate bypass were observed which differed in how 2-oxoglutarate was dissimilated to succinate, either with substrate-level phosphorylation via the succinyl-CoA ligase (6.2.1.5) , or without via a multifunctional menD enzyme encoding the 2-oxoglutarate decarboxylase (4.1.1.71) and succinate semialdehyde dehydrogenase (1.2.1.16) identified in all studied propionibacteria or a complete GABA shunt confined to the dairy species. The dairy species also contained alternative central carbon metabolic pathways absent in commensal species studied. P. freudenreichii can catabolize glucose through the Bifidobacterium shunt, which favors acetate production and may explain the lower ratio of propionate produced relative to acetate observed in this strain [P. acidipropionici possesses and transcribes a linear variation of the Wood–Werkman cycle that utilizes a sodium-pumping methylmalonyl-CoA decarboxylase with a similar structure to that of Propionigenium modestum [Complete pathways for glycolysis, the Wood–Werkman cycle, and the pentose phosphate pathway were found in all genomes with the exception of s strain ,66. P. amodestum  , promiscuous activity of the rhamnulose isomerase is likely able to compensate for this. Sucrose-specific degradative capabiltiies were only observed in P. acidipropionici and P. propionicum, which both contained a 6-phosphosucrose fructohydrolase, allowing the catabolism of sucrose via the PTS system, as well as a sucrose-hydrolyzing levanase. Both mechanisms are likely to be active in P. acidipropionici given upregulation of the 6-phosphosucrose fructohydrolase, fructose PTS, and phosphofructokinase on sucrose , which was potentially inherited with the pdu operon, or the ADP-dependent acetyl-CoA synthetase (6.2.1.13) in P. acidipropionici, which is co-located with the sodium-pumping methylmalonyl-CoA decarboxylase. Propionibacteria lacking the pdu operon could directly produce acetate via a pyruvate:quinone oxidoreductase (1.2.2.2), which despite its energetic inefficiency is likely the major source of acetate in P. acnes and P. avidum, suggesting these strains are more reliant on aerobic respiration. This enzyme had a higher transcription in P. acidipropionici grown on glycerol where there is minimal acetate production, suggesting that the role of this enzyme is likely for redox balancing, whereas the energetically beneficial acetyl-CoA synthase is upregulated on sugars  that consumes oxygen and produces acetyl-phosphate, which likely serves as an energy yielding oxygen-tolerance mechanism. This is consistent with the observation of higher acetate production and propionate degradation when P. freudenreichii was exposed to oxygen [Acetate metabolism is surprisingly variable between different propionibacteria , despiten sugars . Notablyo oxygen .Propionibacterium fermentations, even though all strains encoded alcohol dehydrogenases (1.1.1.1) and contained proteins that can give acetaldehyde dehydrogenase (1.2.1.10) functionality. This suggests that these enzymes either are repressed or have evolved tighter substrate specificity. Notably, P. freudenreichii and P. propionicum have acquired a propanol dehydrogenase with the pdu operon [P. propionicum appears to contain a true acetaldehyde dehydrogenase and has the hallmarks of an ethanol-producing strain including the pyruvate-formate lyase; therefore, it would be an attractive candidate to screen for propanol production. Acetaldehyde is also a precursor for acetoin production associated with the formation of eyes in cheese. All strains could produce acetoin aerobically via diacetyl (1.1.1.5), but they utilized different paths anaerobically, either via acetolactate (2.2.1.6) or acetaldehyde (2.7.2.1) or both for P. acidipropionici (P. propionicum was the only species containing a degradation pathway for acetoin (2.3.1.190).A lack of ethanol and propanol production has been observed in u operon . Only P.opionici . P. propP. propionicum or a core set of three proteins first characterized in B. subtilis [Shewanella [P. acidipropionici 4875; this finding is in line with previous observations of transient lactate production before reconsumption during sugar fermentations in this strain [Lactate is a key carbon source that is preferentially catabolized over glucose in some species , and itssubtilis  and Shewewanella , which lewanella ,78. We oDespite the extensive study of lactate fermentation by propionibacteria, no attention has been given to the energetic challenge of anaerobic lactate catabolism originally highlighted in 1973 . Three lP. acidipropionici 4875, which surprisingly suggests that ferredoxin plays a major role in central carbon metabolism. Both datasets show significant upregulation of the PFOR on sugar, which appears to be the predominant source of acetyl-CoA given that expression is 2–3 times higher than the pyruvate dehydrogenase  bind to the NfnB-like subunit and the NfnA-like subunit oxidizes quinols instead of ferredoxin. Further studies are warranted to confirm the existence of such a complex.Although no homologous genes were identified for the FixABCX complex, we postulate that a similar complex exists in propionibacteria acting in the reverse direction. The flanking genes of the PFOR were identified as candidate genes for this activity based on evidence from bioinformatics analysis, ‘omics data, and genetic knockout. Our attention was drawn to these genes because they were co-transcribed with the PFOR  and were of NADH , was oriPropionibacterium contained the pyruvate kinase (2.7.1.40) and the gluconeogenic pyrophosphate-linked pyruvate phosphate di-kinase (2.7.9.1), which is a role supported by the latter being upregulated on glycerol in P. acidipropionici , which plays a key role in gluconeogenesis in other anaerobic species.Excluding the phosphotransferase system, three separate mechanisms were identified allowing the inter-conversion of pyruvate and phosphoenolpyruvate (PEP). All opionici . P. acidP. propionicum and a recently sequenced polyphosphate-dependent variant [P. acidipropionici and P. propionicum contained an additional pyruvate carboxylase (6.4.1.1). The redundancy of carbon fixation mechanisms for PEP and pyruvate coupled with a gluconeogenic malate dehydrogenase (1.1.1.39) found in all strains suggests that there is complex regulation around the pyruvate node.The pyruvate transcarboxylase complex, which initiates the Wood–Werkman cycle through an ATP-independent carboxyl group transfer, is a defining aspect of propionibacteria metabolism. However, three separate enzymes also allow for carbon fixation to produce oxaloacetate and may promote the production of succinate. Two separate PEP carboxykinases were identified: a GTP-dependent variant (4.1.1.32) found only in  variant  found inP. acidipropionici 4875 where only a single complex was highly expressed (P. acidipropionici was grown on glycerol (All propionibacteria are able to generate ATP from the Wood–Werkman cycle via a short anaerobic electron transport chain. Electrons are transferred from NADH to intracellular fumarate via an NADH dehydrogenase (1.6.5.3) and fumarate reductase, both of which are linked by membrane potential. Then, the generated proton-motive force is converted to ATP through an ATP synthase. A second copy of the fumarate reductase is thought to act as a succinate dehydrogenase ; this thxpressed . Althougglycerol .Propionibacterium species; meanwhile, P. freudenreichii and P. propionicum had lost most of these subunits. The frameshift in the cytochrome C subunit I protein identified in P. acidipropionici 4875 [P. acidipropionici 55737, which we believe to possess full aerobic respiratory capabilities. All species contained the cytochrome bd oxidase (1.10.3.14) giving propionibacteria a degree of aerotolerance.Propionibacteria differ significantly in their capacity to use different terminal electron acceptors. Despite the genus being viewed as microaerophilic, a cytochrome C linked oxidase, oxidoreductase, and NADH dehydrogenase, allowing full aerobic respiration, were identified in three ici 4875  is not pP. acidipropionici  and phosphoadenyl-sulfate reductase (1.8.4.8) functionality was most similar to adenyl-sulfate reductase and likely encoded this function. The expanded capability of P. freudenreichii and P. acnes to utilize alternative electron acceptors was also noted. Both were previously found to contain genes allowing the use of dimethyl sulfoxide as an electron acceptor [fixABCX genes clustered in the genome and caiABC homologs in both genomes, suggesting that these species can catabolize L-carnitine to γ-butyrobetaine, which additionally acts as a terminal electron acceptor.A number of alternative electron acceptors were identified. Nitrate metabolism was highly variable between different species but most extensive in opionici . All spenci 4875  was not  strains . Althougccessful , we wereacceptor . We addiPropionibacterium metabolism and are thought to play key roles in central carbon and energy metabolism. All propionibacteria contained both a polyphosphate kinase (2.7.4.1) and polyphosphate glucokinase (2.7.1.63), allowing excess energy to be stored as polyphosphate and used to phosphorylate glucose not transported by the phosphotransferase system, as indicated by a relative upregulation on glucose relative to sucrose in P. acidipropionici. This activity is solely polyphosphate dependent in all species studied except P. propionicum, which contained an ATP-specific glucokinase. A polyphosphate kinase II was also found in P. propionicum and P. acnes, which has a tendency for the production of nucleotide triphosphates from diphosphates [P. propionicum, which lacks the ATP-dependent equivalent. Exopolyphosphatase (3.6.1.11) was also found in all genomes, which controls polyphosphate levels and likely has a regulatory function due to the many signaling roles of polyphosphate [Both pyrophosphate and polyphosphate metabolism are distinguishing features of osphates . We hypohosphate ,87.P. acidipropionici  and pyruvate phosphate dikinase; indeed omics data suggested coupled over-expression with the latter in P. acidipropionici  and relatively high expression in opionici .P. propionicum, while a final synthesis pathway linking glycogen metabolism to trehalose synthesis was identified in P. acidipropionici, P. propionicum, and P. avidum. While all strains except P. freudenreichii could degrade trehalose via trehalose phosphorylase, it has been suggested that trehalose synthase plays a degradation role in this species [P. acidipropionici and P. avidum not only contain all trehalose metabolic functionalities of Propionibacterium but also contain multiple copies of several genes including trehydrolase (3.2.1.141) and trehalose phosphorylase (2.4.1.231), while P. acidipropionici additionally contains glycine betaine synthesis capability. This expanded osmoprotectant metabolism in P. acidipropionici suggests a potential role of osmoprotectants in alleviating acid stress to enable the high acid-producing phenotype typical of the species. These findings are supported by a trehalose over-expression study in P. acidipropionici 4875 [Propionibacteria synthesize trehalose in response to stress . Three t species . It is wici 4875 .P. acidipropionici 55737 only, which apart from a methionine γ lyase was otherwise identical in amino acid metabolism to strain 4875. Branched-chain amino acids were only partially degraded to fatty acid and alcohol derivatives, consistent with experimental evidence [P. acidipropionici 4875 where they could serve as nitrogen but not carbon sources. While P. avidum is known to possess D-isomer 2-hydroxyacid dehydrogenase activity, a candidate gene encoding this functionality in P. avidum but absent in other tested strains could not be identified [Amino acid metabolism in propionibacteria is particularly relevant to industrial applications of propionibacteria given that it contributes to the formation of flavor compounds in cheese; it has been reported to play key roles in stress tolerance, and it can influence the distribution of fermentation products. The only propiogenic amino acid catabolic pathways identified were for threonine in all species and methionine in evidence  and the entified .P. freudenreichii. These findings likely reflect the greater reliance of proteolytic activity as a source of carbon and energy among commensal propionibacteria given their nutrient-scarce microenvironment [Propionibacterium genus were considered in this analysis, the analysis needs to be expanded further to encompass newly available sequenced species [P. acnes was detected as auxotrophic for some amino acids.The ability for strains to synthesize and degrade amino acids will have an influence on the redox balance of the cell and consequently, the fermentation products. Differences in the metabolic potential for amino acid synthesis or degradation between strains were quantified using the Jaccard similarity coefficient and visualized as clustergrams . Examinaironment . Althoug species ,92,93 beP. freudenreichii [P. avidum and P. acnes, while the glutamate decarboxylase and associated antiporter were found in P. freudenreichii. Both P. acidipropionici strains contained both mechanisms without antiporters, suggesting they were less specialized to a particular acid tolerance mechanism. However, these more traditional acid tolerance mechanisms associated with Gram-positive bacteria [P. acidipropionici could utilize all 20 common amino acids as a sole nitrogen source but less than half as a carbon source. This functionality is not captured in the current model of P. acidipropionici because no genes associated with these functionalities were detected, as is the suggested degradation capability of phenylalanine where no homologues could be found to any known catabolic pathways, suggesting the potential presence of a novel catabolic pathway.Acid tolerance catabolic activities were also variable between species. All species contained L-asparaginase, which has been associated with acid tolerance in nreichii . The argbacteria  are unliPropionibacterium metabolism. A pan-genomic guided metabolic reconstruction strategy was developed to help overcome bioinformatic challenges associated with these non-model, high GC-content organisms. The parallel curation of several related species leads to a significantly improved annotation for all reconstructions.We have developed six genome-scale metabolic reconstructions spanning five propionibacteria to aid in silico investigation into the poorly understood Propionibacterium species despite being the second major fermentation product. This variability was even observed at a strain level of analysis and was further exemplified in amino acid metabolism, where it was suggested that commensal organisms have acquired a broader range of degradative capabilities but a reduced set of biosynthetic capabilities compared to their dairy counterparts.Comparison of the resulting reconstructions revealed that central carbon metabolism was found to be highly variable between species across all the key nodes surveyed in this study. This is especially true for acetate production for which no conserved pathway existed across the five Propionibacterium metabolism through the metabolic reconstructions, we also identified clues for the evolution of the Wood–Werkman cycle and were able to propose a novel ferredoxin-linked energy-conserving strategy and putative gene candidates. This work lays the foundations for further advancement in the understanding of Propionibacterium metabolism, and we expect the reconstructions presented here to be instrumental in the further investigation of propionibacteria, such as designing metabolic engineering strategies to enhance the potential for propionibacteria to produce industrially relevant chemicals or to better understand the relationship between propionibacteria and human health through metabolic modeling.By exploring the functional capacity of