The availability of genome sequences for many organisms enabled the reconstruction of several genome-scale metabolic network models. Currently, significant efforts are put into the automated reconstruction of such models. For this, several computational tools have been developed that particularly assist in identifying and compiling the organism-specific lists of metabolic reactions. In contrast, the last step of the model reconstruction process, which is the definition of the thermodynamic constraints in terms of reaction directionalities, still needs to be done manually. No computational method exists that allows for an automated and systematic assignment of reaction directions in genome-scale models.E. coli. The introduced systematic direction assignment yielded 130 irreversible reactions , which corresponds to about 70% of all irreversible reactions that are required to disable thermodynamically infeasible energy production.We present an algorithm that – based on thermodynamics, network topology and heuristic rules – automatically assigns reaction directions in metabolic models such that the reaction network is thermodynamically feasible with respect to the production of energy equivalents. It first exploits all available experimentally derived Gibbs energies of formation to identify irreversible reactions. As these thermodynamic data are not available for all metabolites, in a next step, further reaction directions are assigned on the basis of network topology considerations and thermodynamics-based heuristic rules. Briefly, the algorithm identifies reaction subsets from the metabolic network that are able to convert low-energy co-substrates into their high-energy counterparts and thus net produce energy. Our algorithm aims at disabling such thermodynamically infeasible cyclic operation of reaction subnetworks by assigning reaction directions based on a set of thermodynamics-derived heuristic rules. We demonstrate our algorithm on a genome-scale metabolic model of Although not being fully comprehensive, our algorithm for systematic reaction direction assignment could define a significant number of irreversible reactions automatically with low computational effort. We envision that the presented algorithm is a valuable part of a computational framework that assists the automated reconstruction of genome-scale metabolic models. Nowadays, high-throughput experimental omics techniques are being developed and are generating large-scale data sets and information bases that can hardly be intuitively understood. Models that enable mathematical analysis and simulation are essential to benefit from the knowledge that is contained in these data sets. Consequently, the importance of models increases along with the advances in experimental technologies.One class of models that has particularily proven to be useful for the analysis of omics data is the class of stoichiometric metabolic models ,2. SeverIn the model reconstruction process – reviewed in  – typicapersonal communication). Error diagnostics in these cases is difficult as the underlying reasons for direction assignments are not provided in the currently available models.In contrast, reaction directions are often assigned manually, or are adopted from other existing models or databases on metabolic pathways (e.g. KEGG). Direction assignment is not only laborious but also error-prone due to manual execution. Indeed, it was shown that the direction assignments of published genome-scale models contain inconsistencies i.e. reaction directions that contradict each other  could be classified as irreversible [The natural approach to identify the irreversible reactions in stoichiometric models would draw on Gibbs energies of formation and physiological concentration ranges. However, experimentally determined Gibbs energies of formation are not available for all metabolites. As a workaround, a group contribution method was developed that computationally estimates Gibbs energies of formation for a large set of metabolites . Using tversible . This veab initio calculation of the reaction directions is based on an NP-complete computation [An alternative approach to assign reaction directions is to apply the nonlinear constraint that arises from the fact that there must exist a thermodynamic driving force for any mass-balanced set of reaction fluxes in a reaction network . For intputation . As a reHere, we present a computational method that is intended to close this gap. In a first step, our method draws on experimentally determined thermodynamic data, i.e. Gibbs energies of formation, and physiological intracellular metabolite concentrations to assign as many reaction directions as possible. Next, in order to assign further reaction directions, we draw on network topology and heuristic rules that exploit the knowledge of biochemical energy equivalents such as ATP or NADH. An algorithm that is capable to apply this procedure to genome-scale stoichiometric models was developed and implemented in Matlab. The respective script is available from the authors on request.E. coli's metabolic network [In the following, the algorithm . Although a computational method can roughly estimate Δolecules , we prefound and ).Furthermore, Maskow and von Stockar have shown that only with Gibbs energies of formation, that are adjusted to physiological pH and ionic strength, e.g. a flux through glycolysis is thermodynamically feasible . Thus, wμM to mM [Intracellular metabolite concentrations were also required to determine the actual Gibbs energies of reaction. These are widely unknown. As, moreover, any stoichiometric model is usually applied for a variety of growth conditions and even for mutant strains where different concentration levels can be conceived, we anyhow wanted to base our analysis on concentration ranges that cover a wide spectrum of conditions. Therefore, we here assumed broad physiological ranges for intracellular metabolite concentrations, which typically are in the order of μM to mM .mM and by using the available set of experimental values for Gibbs energies of formation, ranges of Gibbs energies of reaction could be determined for 176 (out of 920) reactions in the model. In this set of ranges, we checked for allowed operational reaction directions: A positive (negative) direction was set if the range of Gibbs energy of reaction was exclusively negative (positive). With this approach, 43 reactions were defined as irreversible in the analyzed E. coli model, while 133 where defined as reversible.Employing a respective concentration range from 0.001 to 10 As the assignment depends on the estimated Gibbs energies of formation, we performed a sensitivity analysis to assess the assignment's reliablility. We widened the allowed ranges of Gibbs energies of reaction by 1, 2, 3, 4 kJ/mol and performed assignment runs using these. Despite the broadened ranges, 40 out of 43 of our direction assignments based on thermodynamic facts were still valid. Only up to three reactions (depending on the uncertainty range used) would not had been defined as irreversible. As our earlier direction assignments are in-line with the reaction directions in the original model and also in KEGG, we believe that our irreversibility assignments are correct.The limited availability of experimental Gibbs energies of formation only allowed us to analyze a rather small subset of reactions. Thus, we expanded the direction assignment procedure by another approach. As shown in  the reacNote that it is conceivable that a direction assignment based on the topological considerations contradicts an assignment made with the thermodynamic facts. Here, this was, however, never the case. For some reactions the thermodynamics facts were only less restricting as they allowed both directions while a heuristic rule constrained the reaction into one direction. To prevent the exclusion of actually possible reaction directions, we adopted the restriction only if the heuristics-based assignment was highly reliable (see below).First, we had to identify sets of reactions (subnetworks) whose simultaneous operation is thermodynamically infeasible. A thermodynamically infeasible operation of a subnetwork is, for example, given by a cyclic operation of a reaction set that in total results in no net conversion of metabolites. The absence of such reaction cycles is a necessary condition for thermodynamically consistent operation of reaction networks . Hence, a priori determination of reaction directions – without the assumption of other reaction directions – is not possible.Consider a network that consists of three reactions with the pairwise interconversion of the reactants A, B and C cf. Fig. . Assume Next, we extent the thought experiment and assume a reaction between A and C with a different stoichiometry that is actually able to re-cycle C to A cf. Fig. . This reSeveral pairs of low/highly energetic co-substrates exist  the cycles, that do not produce or consume any metabolite, and which were already described by the null space of the original stoichiometric matrix and (ii) cycles that – when complemented with the removed co-substrates – interconvert these. In terms of the terminology introduced in network-based metabolic pathway analysis , these tpersonal communication). For this reason, we based our assignment procedure on the cycles that are described by the basis vectors of the calculated null space matrix. As we will see below this approach was not fully comprehensive but allowed excluding thermodynamically infeasible cycling to a large extent while still being computationally reasonable.Every possible energy producing cycle is a combination of the linearly independent basis vectors of the null space of the reduced stoichiometric matrix. As the running time for the computation of all linear combinations increases exponentially with system size ,26, an enull. The null space of the co-substrate reduced stoichiometric matrix was described by 227 linear independent reaction cycles with an average number of reactions of 8.85 and a median number of reactions of 4. Of all cycles within the null space matrix, 145 were energy producing cycles.Faced with the fact that we only obtained one possible set of linear independent basis vectors, the choice of the null space matrix calculation, however, was important for the assignment procedure. In preliminary tests, when we applied a null space matrix that included larger cycles, our algorithm assigned less reaction directions. Thus, one should apply a null space matrix with cycles that consist of the smallest possible number of reactions. Here, the null space matrix was calculated from the reduced echelon form of the stoichiometric matrix by the Matlab function This set of cycles was now employed to assign reaction directions by thermodynamics-based heuristic rules: In three steps that are described in the following paragraphs different kinds of cycles were analyzed and reaction directions were assigned by the heuristic rules in Fig. Cycles that consist of only two reactions occur frequently in metabolic networks. Here, the null space matrix contained 45 energy producing pair cycles. For such cycles, the direction assignment to eliminate thermodynamically infeasible energy production is straightforward due to the limited possibilities for assignment of reaction directions: There are only two reactions which can be set irreversible, and the most natural approach is to block the reaction step that produces the highly energetic co-substrate. This was the only heuristics-based assignment step that was allowed to be more restricting than the thermodynamic facts assignment step. Technically, this procedure was realized by applying heuristic rules as explained in Fig. Applying these heuristics to the identified energy producing pair cycles, 42 reactions were restricted to one direction. As none of these reactions was already previously defined as irreversible in the thermodynamic facts-based assignment, in summary 85 direction assignments were made until here.The following assignment step , we selected for those in which the analyzed reaction occurs. This subset of cycles is capable to bridge the particular reaction of the first cycle, i.e. to form bypasses that start at the reaction's educts and ends at its products. In consequence, the initial cycle, whose thermodynamically infeasible operation was already disabled, and the bypass build a new – potentially energy producing – cycle. For each identified bypass, we first checked whether it was an actually functional bypass given the previously made direction assignments. If the bypass was already blocked, there was no need for any action. Otherwise, we checked whether the co-substrate conversion of the resulting new cycle was thermodynamically infeasible by calculating the cycle's Gibbs energy of reaction. If it was infeasible, a reaction direction within the bypass was assigned by applying the heuristic rules illustrated in Fig. Analyzing the bypasses that bridge the 106 previously assigned reactions, in a first iteration step 24 additional irreversible reaction directions were defined by the heuristic rules. In a second iteration step, in which we analyzed bypasses for the reactions that were defined as irreversible in the first iteration step, no further directions could be assigned: The bypasses were either already inibited or no further reaction directions could be identified with the employed heuristics.Table Our algorithm did not completely disable thermodynamically infeasible energy production: The heuristics failed in blocking all energy producing cycles and the bypass analysis was not able to identify all possible energy producing cycles. In order to assess the completeness achieved with our approach, we estimated how many additional direction assignments had to be made to completely prohibit infeasible co-substrate conversion. For this, an iterative procedure was applied: A possible energy producing cycle was identified using flux balance analysis, and then, reaction directions were assigned manually to block this cycle (cf. Methods section). When no further energy producing cycles were found, the reactions' directionalities were assumed to reflect thermodynamic feasibility with respect to energy generation. At this point, the direction assignment was considered to be complete.With this procedure, 59 additional assignments of reaction directions were required until infeasible energy production was excluded. Simulating aerobic growth on glucose by flux balance analysis, ATP was then produced via the respiratory chain. Importantly, the production of energy equivalents such as ATP by metabolic reactions was not generally rendered impossible by our linear constraints as our algorithm only selectively disables the generation of highly energetic co-substrates. In summary, the 189 irreversible reactions  were sufficient to yield a thermodynamically reasonable model with respect to the production of energy equivalents.At this point, we checked whether the application of general biochemical rules such as defining all kinase reactions as irreversible would have been a much simpler and also valid alternative to our approach. A close inspection of the 74 kinase reactions in the model revealed that this would not had resulted in a correct model: For instance, the phosphoglycerate kinase reaction is known to operate in both directions and it is correctly defined as reversible in our assignment. This demonstrates that employing heuristic rules in combination with analyzing co-substrate converting cycles is superior to simple general biochemical rules.With the model analyzed here, the calculation time required for the assignment procedure was roughly two minutes on a Pentium 3 GHz PC, if the calculation of the null space matrix and generation of a Excel file for output documentation is included. The assignment algorithm itself required about 30 to 40 s. Such, the computational effort is small and the algorithm can be efficiently executed on a usual PC.The introduced systematic direction assignment yielded 130 reactions that were restricted in one direction. Together with the 59 manual assignments that eventually eliminated any thermodynamically infeasible cycling, we obtained 189 reactions that are irreversible in our model. In comparison to the 676 irreversible reactions in the original model from Palsson and co-workers , this isFrom a constraint-based modeling viewpoint, a direct comparison of the number of irreversible reactions, however, is misleading as one assigned reaction direction can practically render impossible the reversible operation for a set of other reactions. For example, one irreversible reaction that is part of an unbranched linear pathway restricts the operation of the whole pathway to one direction. Hence, in effect it is no difference if the direction of only one or all reactions of the pathway are defined as irreversible.de facto irreversible reactions was assessed. We found that the stoichiometric network of E. coli comprises 175 sets of correlated reactions. If one reaction in such a set is defined as irreversible, mass balance constraints rule out one particular direction for each of the other reactions in the set. In the original model, 749 reactions are practically irreversible. In comparison, our direction assignment eventually resulted in 292 reactions that practically can operate only in one direction.To allow for assessment of model flexibility due to different direction assignments, we had to identify correlated sets of reactions (cf. Methods). Using the identified correlated sets, the number of We found that only in one case – namely the UTP-glucose-1-phosphate uridylyltransferase reaction – our algorithm defines a reaction as irreversible which is reversible in the original model. Remarkably, our assignment is in agreement with the EcoCyc database  which alfrdA deletion mutant  is in sNext, we evaluated whether we could complement the employed heuristic assignment rules to increase the number of reactions that are automatically defined as irreversible. Additional or modified heuristic rules should eliminate the energy producing cycles that were not yet disabled by our algorithm.First, we closely inspected the additional manual direction assigments that were required to eliminate all the remaining energy producing cycles cf. . In thisE. coli model. When we incorporated this rule into the assignment algorithm, in total 26 more reactions were restricted in one direction .E. coli. After having exploited all available thermodynamic data to define irreversible reactions, we drew on network topology and thermodynamic heuristics to assign further reaction directions: Energy producing cycles were extracted from the reaction network and thermodynamically infeasible reaction steps that produce high-energy from low-energy co-substrates were disabled.This paper reports on a computational framework that – based on thermodynamic principles – systematically assigns reaction directionalities in genome-scale stoichiometric metabolic models. We demonstrated its application on a metabolic reconstruction of ab initio prediction of reaction directions [The proposed direction assignment procedure has several advantages over other approaches. The group contribution method to computationally estimate the Gibbs energies of formation is associated with such large uncertainties that only five reactions could be identified as irreversible in a genome-scale model . The metrections  relies oAlong with the development of mathematical methods that employ genome-scale metabolic models, these models became valuable tools in systems biology and metabolic engineering. Here, our systematic assignment procedure can be used in the reconstruction of new models or in the revision of existing ones. Currently, large efforts are put into the automated reconstruction of such models ,29 and sAll calculations were carried out employing Matlab  unless specified otherwise. Necessary input data are standard Gibbs energies of formation and physiological ranges of intracellular metabolite concentrations. As output, the algorithm generates a vector which specifies the assigned reaction directions, and in addition, creates a detailed report (in Microsoft Excel) on the respectively made assignments.E. coli data set, we employed the genome-scale model iJR904 [For the E. coli's intracellular pH of 7.6 [A prerequisite for the thermodynamic facts based direction assignment is the availability of standard Gibbs energies of formation for a large number of metabolites. With these and values for intracellular pH and ionic strength (see below), standard transformed Gibbs energies of formation specific for intracellular conditions were calculated using the software Mathematica  and a Mathematica notebook provided on . StandarH of 7.6  and ioniH of 7.6  . ShortlHaving identified these cycles, we manually defined directions for one or more reaction in the set of reactions such that the identified infeasible cycling is disabled. The employed rationales for the assignment were similar to the heuristics used in the presented algorithm. Essentially, reactions that consume low-energy or produce high-energy co-subtrates were selected. In some cases, this was not possible as also low-energy metabolites were produced or high-energy metabolites were consumed concomitantly. Then, we determined reaction directions according to the metabolic function of the respective enzyme.Two reactions are correlated if the ratio of their reaction rates is identical under any conceivable condition. To identify sets of correlated reactions, in a first step, the stoichiometric matrix was extended by exchange reactions and a reaction describing biomass formation to determine mass balancing sets of reaction rates, i.e. flux distributions. Here, exchange reactions were coupled to all extracellular metabolites and enabled their interchange with the environment. Next, we calculated the null space matrix for this extended stoichiometric matrix. Rows of this null space matrix that are linearly dependent indicate that the corresponding reactions are correlated. Sets of correlated reactions were determined by an all-against-all comparison of the rows.AK, SP and MH designed the research. AK developed the algorithm. AK and MH analyzed the results. All authors read and approved the final manuscript.Lists of assigned reaction directions. This Excel file contains three sheets with (i) the list of reactions that were assigned by the standard assignment procedure, (ii) the list of reactions in case the final electron acceptors are additionally considered, and (iii) the lists of reactions that were assigned manually to obtain a thermodynamically reasonable model. In the first two sheets, we also report in which assignment step a direction was defined by our algorithm.Click here for fileM.List of applied Gibbs energies of formation. This Excel file contains the list of the model's metabolites and – if available – the respective transformed Gibbs energy of formation at a pH of 7.6 and an ionic strength of 0.15 Click here for file


Secretion of heterologous proteins depends both on biomass concentration and on the specific product secretion rate, which in turn is not constant at varying specific growth rates. As fed batch processes usually do not maintain a steady state throughout the feed phase, it is not trivial to model and optimize such a process by mathematical means.We have developed a model for product accumulation in fed batch based on iterative calculation in Microsoft Excel spreadsheets, and used the Solver software to optimize the time course of the media feed in order to maximize the volumetric productivity. The optimum feed phase consisted of an exponential feed at maximum specific growth rate, followed by a phase with linearly increasing feed rate and consequently steadily decreasing specific growth rate. The latter phase could be modeled also by exact mathematical treatment by the calculus of variations, yielding the explicit shape of the growth function, however, with certain indeterminate parameters. To evaluate the latter, one needs a numerical optimum search algorithm. The explicit shape of the growth function provides additional evidence that the Excel model results in correct data. Experimental evaluation in two independent fed batch cultures resulted in a good correlation to the optimized model data, and a 2.2 fold improvement of the volumetric productivity.in silico quickly. Additional options like further biological and technological constraints or different functions for specific productivity and biomass yield can easily be integrated.The advantages of the procedure we describe here are the ease of use and the flexibility, applying software familiar to every scientist and engineer, and rapid calculation which makes predictions extremely easy, so that many options can be tested  Modeling of bioprocesses has been pursued since the 1970s, with the aim to rationally optimize processes. While the mathematical description of processes like growth and product formation have been fairly well achieved, it is still not routine practice to design biotechnological production processes based on model prediction. An especially difficult case in this respect is fed-batch as a dynamic system usually not reaching steady state. General attempts to model fed-batch processes have been described , it is more complex to predict the kinetics of a secreted product. A typical case for secretion systems are recombinant yeasts . As the escribed ,5. The loworkers  present QP is the most plausible target for optimization. At a given process time point t, QP is defined as:With this work we aimed at the development of an optimization tool for fed batch processes using calculation tools available for every PC. MS Excel allows the approximation of a model by numerically solving equations describing the system, and the optimization of an objective function by modifying defined fields (the decision variables), while different constraints can be defined which have to be complied with. The calculation is based on the generalized reduced gradient method described in . While tQP can be calculated from the specific growth rate μ and the specific production rate qP. μ should be one of the decision variables of the optimization (defining the feed rate profile to be developed), while qP depends on μ. The exact function, qP = f(μ), of this dependence for secreted recombinant proteins has been subject to discussion . Therefore, X'(t)/X(t) for μ(t) results in maximizing the integralby formulas (5) and (6). Inserting This integral has an extremum only if the Euler-Lagrange differential equationEvaluating the Euler Lagrange equation (9) yieldsμ(t)·X(t) for X'(t) (from eq. 4) yieldsInserting Calculating the differential and reducing to a common denominator giveskq·μ'(t) + μ3(t) + kq·μ2(t) = 0     (13)2·μ(t):We solve this differential equation and get the following equation for μ over time, with indetermined parameter c and indetermined optimal value for the total feed time T. Here the parameter c and the optimal value for the total feed period T depend on the constraints μmin, μmax and qPmax. Since we know a numerical optimal solution for the growth function, we calculate the value c by fitting the analytic curve (eq 14) to the optimal solution calculated by Excel with the method of least squares. We get as numeric value c = -1.49967812 h. Figure Eq (14) defines only a necessary condition for the optimum trajectory of μmax phase. As the optimum feed profiles obviously consist of an exponential phase followed by a phase of steadily decreasing μ, the analytic approach could only serve as evidence of the correct solution of the optimization problem obtained with Excel Solver. Both the Euler-Lagrange approach used here and Pontryagin's Maximum Principle depend on data fitting to obtain a numeric solution. Given the perfect match of the two approaches presented here, we consider it much more straight forward to apply a numeric data fitting approach directly to the equations of growth and product formation.We have developed a modeling and optimization algorithm for fed batch cultures of secreted products based on MS Excel. The validity of this iterative calculation, which is highly flexible and versatile, was proven by analytic solution of the equations forming the basis of the fed batch model. While the analytic solution fits exactly to the phase of decreasing specific growth rate of the Excel Solver solution, it is not possible to calculate the duration of the initial in silico quickly. Additional options like further biological and technological constraints or different functions for specific productivity and biomass yield can easily be integrated.The advantages of the procedure we describe here are the ease of use, applying software familiar to every scientist and engineer, and rapid calculation which makes predictions extremely easy, so that many options can be tested We could prove that the experimental data basis for the functions behind the algorithm is very important. Different to previous work this was taken into account, and especially the sensitivity at very low specific growth rates needs to be highlighted.The Excel file containing the model and optimization procedure is provided as accompanying file see .Unless stated otherwise, all chemicals were purchased from Merck Eurolab and all antisera were from Sigma.P. pastoris strain X33 (wild type strain) expressing extracellularly the Fab fragment of the anti-HIV antibody 2F5 under control of the GAP promoter was used in this study. The development of this strain has been described elsewhere  (1 ≤ n ≤ N) of length dt. Therefore,We divide the total feed period in equal intervals [tn+1 = tn + dt     (17)dt = 1 [h]. The best value for dt is determined within the optimization process.We start with an initial value tn we denote by Xn = X(tn) the amount of biomass and by Pn = P(tn) the amount of product in the bioreactor. At the beginning of the fed-batch process the initial values are X(0) = X0 and P(0) = P0, as achieved at the end of the batch phase.At every point of time First we have to describe the growth of the biomass. We use the simplest model, the exponential growth model,μ of the biomass depends on time, we calculate (eq. 18) in discrete time stepsSince the specific growth rate μn is the specific growth rate during the interval . The initial values for μn are chosen arbitrarily, for instance μn ≡ μmax. The optimal values for all of the μn's are determined within the optimization process.where tn, tn+1] by the following formulaSecond we have to describe the accumulation of the product. We simply calculate the total product yield during the interval . To do this, let Sn be the amount of substrate added to the bioreactor until the time point tn. Then the substrate consumption rate depends on the amount and on the increase of biomass, i.e.Next we have to calculate the amount of substrate mS is the maintenance coefficient and YXS is the true yield coefficient of biomass from substrate. Inserting formula (18) in (23) the amount of substrate feed in the interval  calculates aswhere YXS and mS from experimental data of chemostat cultures by the method of least squares, we use the observed biomass yield coefficient Y'XS depending on the specific growth rate μ. This is done by dX = -Y'XS·dS and inserting formula (18) and the formula for the whole substrate consumption which impliesTo calculate the parameters Formula (25) can be transformed toYXS and mS were determined by linear regression.From this double reciprocal plot V0 = 1 L. The total volume at each time interval is thenLast but not least we need the total volume for the calculation of the volumetric productivity. The model process starts with a batch volume of f and the density of the feed medium ρf. Due to the high biomass concentrations achieved in P. pastoris fermentations, the cells occupy a significant fraction of the total volume, while the product is secreted to the liquid phase, the culture supernatant. In order to calculate the product concentration, the available liquid volume Vl is calculated at each time interval with the specific volume of wet biomass, which is derived from dry biomass as the specific volume per dry biomass νYDM = 0.0033 L g-1.with the substrate concentration in the feed medium sVln = Vn - Xn·νYDM     (28)p at the time point tn is calculated asFinally, we calculate the biomass and the product concentrations. The product concentration x at the same time point isand the biomass concentration Fn at each time point isThe medium feed rate These values are used to determine the feed rate profile of the optimized fed batch process.μn and the best value for dt  such that the volumetric productivity QP calculated at the point of time tN+1 asThe goal of our optimization problem is to find the best values for the specific growth rates is maximized under the following constraints:μmin ≤ μn ≤ μmax for (1 ≤ n ≤ N)     (33)andX(tN+1) = Xmax     (34)μmax = 0.2 h-1 is the maximum specific growth rate at just below washout in chemostat cultures. Since below μ = 0.02 h-1 significant product degradation appeared, the lower boundary was set at μmin = 0.03 h-1. Also the biomass concentration needs to be limited. The upper limit is mainly defined by the cell separation step, which is practically limited with approximately 100 gL-1 dry mass.Here Additional constraints may be entered, e.g. the final product concentration may be set at a minimum level.N = 150. The values tn, μn, nX, ... are organized in columns, with each time point tn... a row. The values of Xn, Pn, Vn, ... are calculated from the respective previous row using the equations provided above. The optimization process is performed by the Excel Solver as a black box. It maximizes the final QP field by varying the μ fields within the boundaries and the dt field.In the Excel sheet we set The Excel file used for this work is provided as an additional file.To verify the Excel Solver solution, the exact solution of the optimization problem was determined with calculus of variation.The symbols, their definitions and units are summarized in table 2O double distilled waterddHFab2F5 Fab fragment of antibody 2F5GAP glyceraldehyde-3-phosphate dehydrogenaseOD optical densityYDM yeast dry massThe author(s) declare that they have no competing interests.MM supported designing the model, planned and performed the fermentations, and helped in drafting the manuscript. MK developed the model and the optimization procedure, and prepared part of the manuscript. BG developed the expression strain and supported in fermentations and analytics. DM formulated the project, supported the development of the model and drafted the manuscript. All authors read and approved the final manuscript.P. pastoris fed batch process for secreted heterologous protein production. This file contains the model as described in Material and Methods, with the optimized values for μ and dt. By adjusting the specific parameters, other fed batch processes can be modeled and optimized with the Excel Solver.Model of Click here for file


Estimation of 16 adjustable parameter values is based on fitting model simulations to nine independent data curves. The identified model is further validated by comparison to additional datasets measured from mitochondria isolated from rat heart and liver and observed at low oxygen concentration. To obtain reasonable fits to the available data, it is necessary to incorporate inorganic-phosphate-dependent activation of the dehydrogenase activity and the electron transport system. Specifically, it is shown that a model incorporating phosphate-dependent activation of complex III is able to reasonably reproduce the observed data. The resulting validated and verified model provides a foundation for building larger and more complex systems models and investigating complex physiological and pathophysiological interactions in cardiac energetics.A computational model for the mitochondrial respiratory chain that appropriately balances mass, charge, and free energy transduction is introduced and analyzed based on a previously published set of data measured on isolated cardiac mitochondria. The basic components included in the model are the reactions at complexes I, III, and IV of the electron transport system, ATP synthesis at F Cells are able to perform tasks that consume energy  by using chemical energy delivered in the form of a chemical compound called adenosine triphosphate, or ATP. Two Nobel Prizes were awarded  for the determination of how ATP is synthesized from the components adenosine diphosphate (ADP) and inorganic phosphate in a subcellular body called the mitochondrion. The operating theory, called the chemiosmotic theory, describes how a driving force called the proton motive force, which arises from the sum of contributions from the electrical potential and the hydrogen ion concentration difference across the mitochondrial inner membrane, is developed by reactions catalyzed by certain enzymes and consumed in generating ATP. Yet, to date, no computer model has successfully described the development and consumption of both the chemical and electrical components of the proton motive force in a thermodynamically balanced simulation. Beard introduces such a model, which is extensively validated based on previously published sets of data obtained on isolated mitochondria. The model is used to test hypotheses about how intracellular respiration is regulated; this model could serve as a foundation for investigating the control of mitochondrial function and for developing larger integrated simulations of cellular metabolism. As the key cellular organelle responsible for transducing free energy from primary substrates into the ATP potential that drives the majority of energy-consuming processes in a cell, the mitochondrion plays a central role in the majority of eukaryotic intracellular events. Therefore, the development of a quantitative mechanistic understanding of cellular function must rely on a reasonable quantitative description of mitochondrial function. Additionally, development of computational models of physiological systems that span multiple scales, from intracellular biochemistry to whole-organ function, requires a self-consistent integrated description of the biophysical processes observed at the molecular, cellular, tissue, and whole-organ levels of resolution.Recognizing the need for a computational model of mitochondrial energetics and the need that such a model be available for integration with other physiological systems models, a computational model of the biophysics of the respiratory system and oxidative phosphorylation was developed to meet the following requirements: 1) the model must be consistent with the available experimental data, and (2) the model must be constrained by the relevant physics/biophysics. The utility of the first requirement is self-evident. The second requirement is that models must obey applicable physical laws . The alternative to the second requirement is to use empirically derived relationships that are often useful in developing data-driven models based on specific datasets. This approach is not used here because the resulting models often fail when combined together. Physics-based models, on the other hand—i.e., models built on principles including the laws of mechanics and thermodynamics, in which assumptions and approximations are made explicit—operate with a common currency of mass, charge, energy, and momentum  the mode. Such moPrevious models of oxidative phosphorylation fail to meet either one or both of the above requirements. For example, the widely used model of Korzeniewski and Zoladz –6 invokeThe model developed by Magnus–Keizer  was inteThe goal of the current work is to introduce a quantitative model representing the chemiosmotic theory of the respiratory chain and ATP synthesis that treats the proton gradient and mitochondrial membrane potential as distinct state variables. To be of maximal utility, this biophysically based model of mitochondrial respiration must conserve charge, mass, and energy, and use a treatment of the electron transport system, ATP synthesis, and substrate transporters that does not violate the laws of thermodynamics. The model is developed based on the dataset of Bose et al. . Values This section is organized as follows. First, a thermodynamically balanced biophysical model of mitochondrial oxidative phosphorylation is developed, along with identification of model parameters based on the measured dependence of NADH and cytochrome c redox states, oxygen consumption, and matrix pH on levels of buffer phosphate. It is shown that the developed model cannot match data on the inner membrane electrostatic potential without incorporating phosphate-dependent control of oxidative phosphorylation. In the next section, the isolated mitochondrial model is modified to include phosphate-dependent activation of complex III and is fit to the data on inner membrane potential. In the final section, the behavior of the model at low oxygen level is shown to compare favorably to data reported by Gnaiger and Kuznetsov ,21 and W1F0 ATPase, are illustrated in +/H+ antiporter and passive H+ and K+ permeation. The model includes two mitochondrial compartments (matrix and intermembrane [IM] space). In the following description of the model equations, subscripts “x”, “i”, and “e” denote concentrations of reactants in the matrix, IM space, and external space, respectively; the concentration variables [fATP]i and [fATP]x, and [fADP]i and [fADP]x, denote the magnesium-unbound components of ATP in the IM space and the matrix, and unbound ADP concentrations in the IM space and matrix, respectively; variables [mATP]i and [mATP]x, and [mADP]i and [mADP]x, denote the magnesium-bound components of ATP in the IM space and the matrix, and magnesium-bound ADP concentrations in the IM space and matrix, respectively. The external space corresponds to the extra-mitochondrial buffer used in the experiments simulated below.The basic components of the mitochondrial model, which include the reactions at complexes I, III, and IV of the respiratory chain and ATP synthesis at F−1 (l mito volume)−1” and not simplified to “M s−1”, which would be ambiguous when referring to fluxes for membrane transporters.All concentrations in the following sections are expressed in molar units—specifically moles per liter of compartment volume. Fluxes are expressed in units of mass per unit time per unit mitochondrial volume. To avoid confusion, the units on flux variables are written as “mol s+ to NADH via the reaction NAD++ in the mitochondrial matrix. The following expression is used to model the dehydrogenase flux:In this model the TCA cycle and other NADH-producing reactions are not explicitly modeled. Instead, a phenomenological driving force is used to simulate the phosphate-dependent rate of reduction of NADx and [NAD]x denote the concentrations of NADH and NAD in the mitochondrial matrix; [Pi]x denotes the matrix inorganic phosphate concentration, and DH, r, kXPi,1, and kPi,2 are empirical parameters. Thus, the dehydrogenase flux drives the [NADH]x/[NAD]x ratio toward r, with a reaction rate dependent on the concentration of phosphate, which serves as a substrate for mitochondrial dehydrogenase enzymes.where [NADH]The overall reaction for electron transfer from NADH to ubiquinol at complex I is expressed as+ is used to indicate hydrogen ion transferred from the matrix to the cytosol, against the electrochemical gradient. The direction of positive flux for the reaction of equation 2 and for all reactions introduced below is left-to-right.where the notation ΔHThe flux through complex I is modeled using the following expression:XC1 is an adjustable parameter, [Q] and [QH2] denote oxidized and reduced ubiquinol, RT is the gas constant multiplied by the absolute temperature; ΔGo,C1 = −69.37 kJ mol−1 is the standard free energy for the reaction H+ + NADH + Q + + QH2 at pH = 7; and ΔGH is the proton motive energy, or the free energy change associated with pumping a proton from the matrix side to the cytosol side of the mitochondrial inner membrane. The factor of four in the exponent of equation 3 arises from the four protons pumped from the matrix space to the IM space for each pair of electrons transferred from NADH to ubiquinol. The proton motive energy is computed as follows:where F is Faraday's constant, ΔΨ is the membrane potential measured as the outer potential minus the inner potential, and [H+]e/[H+]x is the ratio of external hydrogen ion concentration to matrix concentration. Equation 3 represents a minimal thermodynamically balanced one-parameter model for the flux through the first step in respiratory system—i.e., the flux described by equations 3 and 4 drives concentrations towards thermodynamic equilibrium. Specifically, the flux is driven towards a thermochemical equilibrium defined by the effective equilibrium expressionwhere which is an explicit function of the proton gradient and electrostatic gradient across the inner membrane.For complex III, it is assumed that four protons are pumped for each pair of electrons transferred from ubiquinol to cytochrome c ,23:3+ and cytC(red)2+ denote the oxidized and reduced forms of cytochrome c, respectively. As indicated in where cytC(ox)XC3 is an adjustable parameter, ΔGo,C3 = −32.53 kJ mol−1. Below, the expression for complex III flux is modified to test the hypothesis that phosphate modulates complex III activity. It will be shown that, while much of the available experimental data can be explained by the model developed in this section, which does not consider phosphate-dependent control of the respiratory chain enzymes, the observed data are better fit by a model that incorporates phosphate-dependent control of complex III activity.where The overall reaction of complex IV involves the transfer of two protons across the membrane and a total of four charges ,25:with a flux computed asXC4 is an adjustable parameter, ΔGo,C4 = −122.94 kJ mol−1, and cytCtot = [cytC(ox)3+] + [cytC(red)2+].where 2+]/cytCtot is found to provide better fits to the observed data than are possible without it.The complex IV flux of equation 9 is expressed in a form similar to those of complexes I and III, with a few key differences. While, as for complexes I and III, the flux is formulated to drive the system toward thermodynamic equilibrium, additional multiplicative factors have been included in order to successfully reproduce the observed data. The factor ntration –21,26,271F0-ATPase reaction:ADP is phosphorylated to ATP in the matrix via the FAn ≈ 3 is the number of protons transported each time this reaction turns over. Since ATP synthesis requires magnesium as a cofactor, the flux through this complex is modeled using the thermodynamically balanced expressionwhere XF1 is an adjustable parameter, ΔGo,ATP = −36.03 kJ mol−1 and KMg-ATP and KMg-ADP >are the equilibrium dissociation constants for ATP and ADP binding with Mg2+. The factor 1 M multiplying [mATP]x is used so that the term in parenthesis is balanced in terms of units.where Binding between magnesium ion and ATP and ADP is driven via the following fluxes:XMgA is the forward binding rate constant for these reactions; the effective unbinding constant is computed to satisfy the equilibrium dissociation relations for ATP and ADP binding with Mg2+.fATP and fADP denote the concentrations of ATP and ADP that are not bound to magnesium ion in the matrix and IM space; mATP and mADP denote the magnesium-bound species. The parameter Permeation of ATP, ADP, AMP, and inorganic phosphate between the external buffer and the IM space is governed by the following fluxes:pA and pPi, respectively; γ denotes the ratio of mitochondrial outer membrane area to total cardiomyocyte cell volume.where the subscripts “e” and “i” denote external buffer and IM space, respectively. The buffer concentrations are set as constants in this study. The permeabilities of the outer membrane to adenine nucleotides and to inorganic phosphate are given by ANT flux involves the displacement of one negative charge from the matrix to the IM space, and is therefore coupled to the electrostatic membrane potential. The following empirical expression ,6 is usewhere the ANT is assumed to operate on magnesium-unbound ATP and ADP in the two compartments.+ and + and Transport of inorganic phosphate between the matrix and IM space is coupled to the hydrogen ion gradient . It is akdH is the dissociation constant for the reaction. In these expressions Pi represents the sum of species is assumed to be in equilibrium on either side of the membrane with XPiHt is an adjustable parameter, kPiHt is the Michaelis–Menten constant for where High-energy phosphates are transferred between ATP, ADP, and AMP in the IM space via the Adenylate kinase (AK) reaction:The AK flux in the IM space is computed as follows:KAK = 0.4331 is the equilibrium constant for the reaction of equation 17, and XAK is the AK enzyme activity.where + is required to buffer the matrix pH x [where  · [H+]x .1F0-ATPase fluxes include two terms, one term representing the number of protons transported across the inner membrane for a given reaction, and one term representing the number of protons consumed by the associated biochemical reaction. For example, the complex III reaction pumps four H+ out of the matrix and consumes one matrix H+ for every turnover of the reference biochemical reaction H+ NADH + Q + + QH2, resulting in a total of three matrix H+ consumed. Thus, the net stoichiometric coefficients multiplying JC1, JC3, JC4, and JF1 are −(4 − 1), −(2 + 2), −(4 − 2), and +(An − 1).The stoichiometric coefficients multiplying the complex I, III, and IV, and FCIM, which is estimated below.The membrane potential kinetics depends on the effective membrane capacitance, In addition to the 17 state variables treated in equation 22, the concentrations of oxidized matrix NAD, Q, and cytochrome c, and the matrix ADP concentration, are computed as follows:tot, Qtot, cytCtot, and Atot, are the total concentrations of NAD(H), ubiquinol, cytochrome c, and adenine nucleotide in the matrix, respectively.where NAD−1 (l mito volume)−1. The parameters have been categorized into three classes, denoted classes A, B, and C. The meaning of these categories is as follows.The values of the parameters used in this section are listed in Class A refers to free parameters with values determined by fitting model simulations to the data published by Bose et al. . In totatot, Atot, Qtot, and cytCtot are obtained from the previous models of Vendelin et al. e = 10−7.1, [ATP]e = 0, and [AMP]e = 0; [ADP]e was set at either 0 or 1.3 mM, as described below, and [Pi]e was varied from 0 to 10 mM. The total magnesium concentration in the buffer was fixed at 5.0 mM; buffer potassium concentration was fixed at 150 mM. The simulations described in this section were computed with the oxygen partial pressure in the matrix set to 20 mm Hg, or [O2] = 2.6 × 10−5 M.The extensive dataset published by Bose et al.  is used tot) as a function of external inorganic phosphate, [Pi]e. The two curves correspond to two different values of external ADP concentration, 0 and 1.3 mM, as indicated in the figure. Also shown are data from Bose et al. /cytCtot multiplying the JC4 flux expression of equation 9. It is observed that the model's fits to the cytochrome c redox data are improved by incorporating this multiplicative factor in the expression for complex IV flux. Thus the best-fit model solution assumes the flux through complex IV depends on [cytC(red)2+]2, which may be explained by the fact that two reduced cytochrome c molecules are require to donate a single electron pair one oxygen atom, generating H2O.Also shown as red curves plotted in e for the cases of [ADP]e = 0 mM and [ADP]e = 1.3 mM.In contrast to the results illustrated in e = 0), as the external (buffer) phosphate concentration is increased, the dehydrogenase flux governed by equation 1 increases, providing an increased thermodynamic driving force for electron transport and a corresponding increase in the magnitude of the membrane potential. However, the model-simulated rate of increase in ΔΨ with [Pi]e is much smaller than that observed experimentally. When the active state is simulated ([ADP]e = 1.3 mM), the model fit is even worse than for the inactive state. The addition of ADP to the external buffer, representing a sink for the free energy stored in the redox state in the matrix and the membrane potential, results in a drop in both redox and membrane potential. The simulated magnitude of the potential difference decreases with increasing [Pi]e, the opposite of the trend that is observed experimentally. With 14 adjustable parameters, it is not possible to reproduce the experimentally observed behavior of ΔΨ while maintaining reasonable fits to the curves plotted in In the inactive state , close to the reported value of 0.35 ± 0.07 μM e = 0.87 mM), corresponding to the measurements of Wilson et al. [2]e = 50 μM.The oxygen consumption curve was computed for active state-3 respiration, with [ADP]re 7A of  and Figure 7A of . The mod 0.07 μM ,21. The n et al. . The pren et al. . Wilson Thus, beyond the dataset used for parameterization, the model was further validated by comparison to additional datasets measured from mitochondria isolated from rat heart ,21 and le = 0 and 1.3 mM, respectively). In total, 16 parameters were estimated by simultaneously fitted model-simulated steady states to nine independent data curves  as putative controllers of complexes I, III, and IV and of F1F0 ATPase and the ANT system. It was found that no other choice of controlled enzyme and controller species could provide fits to the data of To obtain the model solutions illustrated in o,C3 and ΔGo,C4 to depend on phosphate concentration. However, a simple model (results not shown) using the linear relationship between the apparent cytochrome c redox potential and phosphate concentration observed by Gopal et al. /NADtot ratio, which is the driving force for electron transport and proton pumping, is approximately 0.6 for both levels of buffer ADP concentration that were studied . Likely, the observed drop in membrane potential for [ADP]e = 1.3 mM (and [Pi]e = 0) compared to at [ADP]e = 0 is due to residual phosphate in the bath and in the mitochondrial matrix, as proposed by Bose et al. e = 0 compared to when [ADP]e ≠ 0. Using the current model, one can obtain improved fits to the observed data by reducing the complex I or III activity, or by reducing the dehydrogenase activity, when [ADP]e = 1.3 mM compared to the case when [ADP]e = 0. Thus, inhibition of the activity of respiratory complexes by ADP represents a possible mechanism to explain the observations of Bose et al.  is added to the buffer at [Pi]e et al.  at [Pi]e2) redox handling at complex II of the respiratory chain. Thermodynamically balanced flux expressions for complex II flux could be developed in a manner analogous to that for complexes I and III in this study. These steps represent planned progress toward a major long-term goal of the construction of a complete energetic model of the cell x are treated as state variables. Mitochondrial ATP-dependent potassium channel flux [While it was determined that the passive potassium channel flux is effectively zero in the model parameterization developed here, modeling the Knel flux ,37,44–47http://www.cellml.org.The model was implemented, simulated, and analyzed using the MATLAB  computing environment. All calculations were performed on a desktop PC. Computer codes are available from the author upon request. In addition, the model is available in the CellML exchange format at


Organisms live and die by the amount of information they acquire about their environment. The systems analysis of complex metabolic networks allows us to ask how such information translates into fitness. A metabolic network transforms nutrients into biomass. The better it uses information on available nutrient availability, the faster it will allow a cell to divide.I here use metabolic flux balance analysis to show that the accuracy I (in bits) with which a yeast cell can sense a limiting nutrient's availability relates logarithmically to fitness as indicated by biomass yield and cell division rate. For microbes like yeast, natural selection can resolve fitness differences of genetic variants smaller than 10-6, meaning that cells would need to estimate nutrient concentrations to very high accuracy (greater than 22 bits) to ensure optimal growth. I argue that such accuracies are not achievable in practice. Natural selection may thus face fundamental limitations in maximizing the information processing capacity of cells.The analysis of metabolic networks opens a door to understanding cellular biology from a quantitative, information-theoretic perspective. Organisms need to acquire information about their environment in order to survive and reproduce. They need to respond to information about changes in temperature, soil conditions, water availability, nutrient supply, predation pressure, and many other factors. The ability to acquire and use such information arguably affects organismal fitness -6. Howev3 chemical reactions for free-living organisms. Their structure has been elucidated in several organisms by manual curation, aided by functional genomic data [Saccharomyces cerevisiae and common growth substrate compositions, such as minimal media with glucose as a sole carbon source, the predictions of flux balance about maximal biomass yield are in good agreement with experimental data, where available [In microbes, an important fitness component is a cell's growth or cell division rate. The selection pressure to grow rapidly during times of nutrient availability has left clear traces in the evolutionary record, such as strong microbial codon usage biases that allow high translation efficiency of abundant proteins ,8. It hamic data ,11,12. Fvailable ,17,18. Fvailable -21 this vailable ,19,20,22Snf3, Rgt2) as well as glycolytic intermediates form the beginning of a signaling cascade. This cascade produces an integrated cellular response that includes the expression of glucose transporter genes (HXT1-HXT7), the expression of glycolytic genes, as well as the repression of many other genes [The kind and concentration of available growth substrates influence a cell's maximal biomass yield -26. Celler genes ,25. Evener genes ,25. The k nutrients and their actual concentrations Ni (1 ≤ i ≤ k) in a cell's environment. If a cell underestimates the actual nutrient concentration in the environment for any one nutrient, then its "measurement" Ni. Now subdivide the interval  into ni equal subintervals. If the cell can place (measure) the concentration of i within the interval (Ni(ni-1)/ni, Ni) then it has Ii = log2 ni bits of information about this nutrient. If the measurement error, when viewed as a random variable, has a symmetric or a uniform distribution within this interval, then the expected sensing error is Ei = Ni/2ni. Nutrient information and sensing error thus relate to each other asInformation acquired through nutrient sensing can be represented as follows. Consider Ei and Ii are equivalent, using Ii has two advantages. First, its units (bits) are canonical measures of information [Ij bits of information on nutrient j (1 ≤ j ≤ k), then it has Although ormation ; second,N1 ,..., Nk).Flux balance analysis allows us to immediately assess the fitness value of nutrient information for a cell, because a cell's maximal biomass yield is a function of the measured nutrient concentrations 3, inorganic phosphate (Pi), sulfate, and glucose as the sole carbon source. Oxygen is available as a terminal electron acceptor. For simplicity, I first focus on a scenario where information about all substrates except glucose is perfectly accurate. I assume that the biomass yield Y per unit time is linearly proportional to a cell's division rate G, a measure of fitness. In other words, Y = cG, c being some constant. I express the effect of incomplete information on biomass yield Y as s = 1-Y/Ymax = 1-G/Gmax, where Ymax and Gmax are the maximally achievable biomass yields and cell division rates, respectively, i.e., the yields and rates for perfectly accurate glucose information. The quantity s can also be thought of as a selection coefficient, as a measure by how far a cell's fitness w = 1-s = G/Gmax is reduced by incomplete information. Figure s and information is especially simple if a binary logarithm is used to scale s, i.e., -log2(s) = IGLC+1. This simple relationship emerges numerically from flux balance analysis, but it also has a straightforward intuitive explanation. If zero bits of information are available for a growth-limiting nutrient, then under the assumptions used here, the cell's "guess" about nutrient concentrations will be randomly distributed in the interval , with an expected value of Ni/2. At this expected value, the division rate of a cell will be half the maximal growth rate, such that s = 1/2. The above relationship between s and I then holds, because -log2(1/2) = 1 = I+1. If one bit of information is available (I = 1), then the cell's measurement will be randomly distributed in the interval , with an expected value of 3Ni/4, leading to s = 1/4, and -log2(1/4) = 2 = I+1. The same line of reasoning applies to ever increasing values of I. The key assumption in this intuitive explanation is that if one nutrient is growth-limiting, then cell division rate depends linearly on the cell's ability to utilize this nutrient. This is not obvious a priori, because the nutrient's metabolic products may be fed into many different pathways that produce essential biomass components. The distribution of these products among different pathways, and the cell's final resulting division rate, might in principle depend on the concentration of the nutrient and on that of other nutrients. However, flux balance analysis shows that the dependency between nutrient concentration and biomass yield is quite simple and linear.The relationship between information and fitness is best explored for a defined environment, such as a minimal growth medium. The environment I use contains NHs decreases linearly with decreasing sensing error . Quantitatively very similar linear-log and linear relationships hold for the other four nutrients  be in order to matter to natural selection? The influence of genetic drift dominates over that of natural selection, if a reduction in growth rate is s < 1/4Ne for diploid cells, where Ne is the effective population size [Ne, in turn, can be estimated from the synonymous nucleotide diversity π in a population, and the per-generation mutation rate μ as Ne = π/4μ. Thus, if a growth rate difference is smaller than s = μ/π in a population, then the associated growth rate difference is too small to be seen by natural selection. The rate of mutations per nucleotide and generation in Saccharomyces cerevisiae has been estimated at μ = 2.2 × 10-10 [s = 7.33 × 10-8 as a "critical" growth rate difference that can still be seen by natural selection. No data on synonymous nucleotide diversity are available for S. cerevisiae itself, but a recent estimate [overall nucleotide diversity of π = 0.0046  suggests an upper bound of s = 4.78 × 10-8, rendering the critical s I use here conservative. Growth rate differences below this value are more strongly influenced by drift than by selection.How large must a growth rate difference ion size ,29. Ne,  × 10-10 . In the  × 10-10 . The aboestimate  on overas is indicated in Figure I = -log2 s-1 ≈ 22 bits are selectively neutral for yeast.The critical selection coefficient s) up to some number of bits. From that point on, additional glucose-information has no effect, because ammonium-sensing has become growth-limiting. Exactly the same considerations hold if the places of ammonium or glucose are interchanged (or for any other two nutrients), which causes the symmetry of the piecewise linear surface in Figure Thus far, I assumed limited information for only one nutrient, but what if information is limited for more than one nutrient? Consider a genotype that systematically underestimates the availability of one substrate, such as glucose, because of a poor sensing mechanism. The resulting undercapacity to metabolize this substrate renders the substrate growth-limiting. In this situation, accurate sensing of the availability of another substrate, such as ammonium, may not increase fitness. The reason is that growth is limited not by a lack of information about ammonium, but by a lack of information about glucose. As an example, Figure other nutrients varies randomly. For example, the black bars correspond to a situation where only one other substrate, oxygen, is measured at accuracies randomly and uniformly distributed between 0 and 16 bits. For increasing glucose information, the growth rate loss becomes smaller, but never as small as when only glucose information limits growth  only slightly as the amount of available information increases, until much information (>60 bits) becomes available, at which point every additional bit has a large effect. This means that the benefits of growth substrate information are limited by the poorest sensing process in a metabolic system. Only if every substrate-sensing process has high accuracy, does increasing information about any one substrate provide large benefits.Figure 2(s) = I+1, where s is a selection coefficient, the difference between the maximal growth rate with perfect sensing and the actually attained growth rate.The notion that cells process information is not new . However-6 are still visible to natural selection in microbes like yeast. Very small inaccuracies  [quantitative information about expression changes for such genes has very limited accuracy for mRNA, and is generally unavailable for proteins. Thus, although it would be highly desirable to understand both the growth cost of underestiming and overestimating nutrient concentrations [A number of caveats to this approach are in order. First, I have here emphasized sensing errors that lead to underestimation of substrate availabilities, because only such errors lead to an undercapacity to metabolize nutrients. Overestimation would lead to overcapacity of nutrient utilization systems, which in itself would still lead to maximal growth. However, if overestimation causes a systematic overexpression of signaling or utilization molecules, overestimation could carry an increased cost of gene expression. Even though few genes might be involved, these costs are not necessarily low. For example, in case of the lactose operon of  = 0.05) , which i = 0.05) . Hundred = 0.05) . Becausetrations , a quantN, and the value of its concentration sensed by a cell (possibly with some inaccuracies) is S(N). If the import system for this nutrient is far from being saturated, which is likely if the nutrient is at concentrations sufficiently low to be growth limiting, then the cell's uptake rate U of this nutrient is likely to be proportional to S(N), with some proportionality constant c, i.e., U = cS(N). In this paper, I focus on the value of improved nutrient sensing S(N). However, the proportionality constant c itself may be subject to selection pressure, thereby increasing the efficiency of uptake for a given S(N). For example, in yeast populations cultivated during several hundred generations in a chemostat, such increase in uptake efficiency occurs through gene duplication and increased expression of hexose transporter genes [A second caveat is that selection may act concurrently on multiple attributes of a metabolic system, not only on nutrient sensing. One example comes from glucose limitation experiments in chemostats, where a population's environment is held constant for hundreds of generations. Consider a nutrient whose actual concentration is er genes ,46. Note+), where in yeast the glucose sensor Snf3p is known to activate the proton pumping plasma membrane ATPase [A final caveat is that it may not always be possible to sense the availability of two compounds independently from one another. One example is the sensing of glucose and protons  is also a solution, such that one can think of v as specifying relative ratios of fluxes through a metabolic network. Further constraints, such as irreversibility of some reactions, and experimentally measured uptake fluxes of external substrates can reduce the number of allowable fluxes v in steady-state. Within the space of allowable fluxes one can then use linear programming to determine the fluxes that maximize or minimize any quantity that can be expressed as linear combinations of individual metabolic fluxes. An especially important such quantity is the biomass growth flux itself.Flux balance analysis  uses inf2: 2.4; Sulfate: 0.034; inorganic phosphate Pi: 0.09; ammonium NH3: 2.45. Among these values, the glucose uptake flux vG and oxygen uptake flux vO2 stem from experimental measurements in an aerated batch culture of S. cerevisiae [vmax given these constraints, where the stoichiometry of the growth flux is given in [max , and minimized the sulfate uptake flux for which this growth flux could be observed. The resulting sulfate uptake flux  is the smallest sulfate uptake flux that can sustain the observed vmax with the given glucose uptake flux. I subsequently carried out analogous minimization procedures for the remaining two growth substrates, thus arriving at the values listed above. This combination of values has the advantage that reduction in any one nutrient uptake flux will lead to a reduction in growth rate. In other words, no nutrient is in excess, and accurate nutrient availability estimation is critical to sustain maximal growth. All analyses were carried out with a publicly available yeast metabolic model [Ii available for a given nutrient i, I translated Ii into the appropriate number of measurement intervals ni, according to the relation Ii = log2 ni discussed in the main text. The expected measured nutrient value then calculates as Ni(2ni-1)/2ni, where Ni is the actual nutrient concentration represented through one of the uptake fluxes listed above. Flux balance analysis was carried out with this expected value to determine the growth rate achievable for the corresponding amount of information.The following substrate uptake fluxes (mmol substrate/h/g dry weight) were used here (rounded to two significant digits): Glucose: 15.3; Orevisiae . I constgiven in . I then ic model  and withlimiting nutrient changes by x%, then a cell with access to perfect nutrient sensing would also change its uptake rate of the nutrient by x%. However, it must be clarified that nutrient uptake rates are not necessarily proportional to nutrient concentrations, even if cells have perfect information. Specifically, if there is so much of a nutrient that the nutrient uptake transporters are saturated, changes in extracellular nutrient concentrations will have no effect on nutrient uptake. However, this scenario is very different from my focus here, namely an environment where the concentration of individual nutrients limits growth, and where information about such nutrients matters to the cell.As stated, nutrient concentrations are here represented through nutrient uptake rates. That is, I implicitly assume that if the concentration of a AW designed and executed the research, analyzed the data, and wrote the manuscript.


The tracers produce a huge number of metabolite forms that differ according to the number and position of labeled isotopes in the carbon skeleton (isotopomers) and such a large variety makes the analysis of isotopomer data highly complex. On the other hand, this multiplicity of forms does provide sufficient information to address cell operation in vivo. By the end of last millennium, a number of tools have been developed for estimation of metabolic flux profile from any possible isotopomer distribution data. However, although well elaborated, these tools were limited to steady state analysis, and the obtained set of fluxes remained disconnected from their biochemical context. In this review we focus on a new numerical analytical approach that integrates kinetic and metabolic flux analysis. The related computational algorithm estimates the dynamic flux based on the time-dependent distribution of all possible isotopomers of metabolic pathway intermediates that are generated from a labeled substrate. The new algorithm connects specific tracer data with enzyme kinetic characteristics, thereby extending the amount of data available for analysis: it uses enzyme kinetic data to estimate the flux profile, and vice versa, for the kinetic analysis it uses in vivo tracer data to reveal the biochemical basis of the estimated metabolic fluxes.A current trend in neuroscience research is the use of stable isotope tracers in order to address metabolic processes  Figure 13C isotopes (13C isotopomers). A given set of metabolic fluxes produces a specific distribution of isotopomer fractions, and consequently, the isotopomer distribution indicates the underlying set of fluxes.Metabolic networks of living cells produce the intricate redistribution of carbon skeleton atoms of substrates. If these substrates are artificially labeled by stable isotopes such as 3C at spein situ: nuclear magnetic resonance (NMR) and mass spectrometry (MS). The different kinds of data that can be obtained from NMR measurements were classified by Mollney et al  × [R5Pj]/([X5Ptot] × [R5Ptot])     (4)Vftot] x [R5Ptot])), which are used to get the fluxes in reactions between isotopomers (as in Equation (4)). During the small time interval dt the TK reaction (3) considered as an example consumes the amount dt × V3fij of each of the isotopomers i and j and this value is subtracted from the concentrations of respective isotopomers.Here, the indices i and j refer to the concentrations of the respective isotopomers and the index 'tot' refers to the total concentration of the metabolite, as calculated in the kinetic model. In this way, the first part, ODE solving, is linked to the second part that computes the label distribution: the kinetic model calculates global fluxes and concentrations and defines the values (as in the above example V3f/([X5Pi]t+dt = [X5Pi]t - dt × Vfij[X5Pj]t+dt = [R5Pj]t - dt × Vfij     (5)t+dt = [GAPra]t + dt × Vfij[GAPrd]t+dt = [S7Prd]t + dt × Vfij     (6)[S7P2+r5p->s7p). This demands 4 × m recalculations of r5p and s7p for one direction of the reaction. In this way the whole description of the TK reaction needs only 2×(n+4×m) recalculations of substrate-product pairs instead of n × m recalculations of four substances according to the above algorithm.The above algorithm is present in order to illustrate the main principles of TK reaction simulation between a pair of isotopomers according to the Equations 5 and 6. However this algorithm is not optimized in the best way. To simulate the reaction between n isotopomers of ketose and m isotopomers of aldose it is necessary to perform n × m cycles of such calculations. It could be optimized by the imaginary separation of the whole reaction into two steps corresponding to the aldose and ketose products formation. First scission of all ketose isotopomers could be simulated, producing respective isotopomers of aldose product according to the scheme x5p->g3p; this demands just n recalculations of x5p and g3p concentrations. Then the rest two-carbon fragment (for TK) with only 4 different possible combinations of label interacts with isotopomers of aldose substrate , when the label was added with some of the substrates.- Distribution of labeled atoms such as 2, the sum of the squares of the vertical distances of the experimental points from the calculated curve divided by the standard deviation) is defined from the numerical solution of differential equations, abrupt change of parameters demanded by the applied fitting algorithm could render the system stiff and induce failure of ODE solver. Second, if the total metabolite concentrations are not measured, the merit function does not account for them, so that the best fit sometimes corresponds to evidently unreal concentrations.Optimization of a merit function in a multidimensional space of parameters is well elaborated and usually does not represent a problem ,37. HoweThe first problem was partly solved by our approach of two-step solution, when the isotopomer analysis comes after the differential equations that describe changes in total metabolite concentrations are solved. In the first step a small number of equations for total concentrations are to be solved and at this step the solution is usually robust. A number of numerical methods for ODE solution are implemented, in particular the Bulirsch-Stoer method  could be2 change, dependent on metabolite concentrations. According to the used Powel's algorithm for the local decrease of χ2 value, the program changed parameters one by one until the minimum was reached in each direction. To avoid long descent in almost flat valley, if a change in χ2 was lower than threshold, the program switched to another direction in the space of parameters. If some metabolite concentration has reached its critical value, the program automatically increased the threshold for χ2 change if the parameter change increased the critical concentration, but decreased the threshold to 0 if the parameter change induced also a decrease of critical concentration. These rules let to control the range of concentrations, and made the solution of ODEs more stable. After termination of descent in the space of parameters, the program make a step uphill by random change of a parameter as is supposed by Simulated Annealing algorithm, and then repeats the downhill descent. All the successful steps of parameter change are saved, so that if the program comes into the area of stiffness and the solver fails, the procedure could be started again from the last successful step after the necessary correction.Another part of the stiffness problem in fact was solved when, to address the second problem , we introduced the threshold for χ2 and completes the fitting procedure, it can determine the set of parameters most relevant to the fit using singular value decomposition of the second derivative matrix of χ2 with respect the parameters (Hessian matrix). Square roots of the diagonal elements of the inverse of Hessian matrix provide the standard deviation for the essential parameters. The present in our website instruction describes the way of using this feature of the program.After the program finds the minimum of χThus, fitting algorithm accepts different kinds of data and estimates not only compatible set of metabolic fluxes but also parameters of enzyme reactions and their regulation, providing insight to the biochemical underground of the particular set of metabolic fluxes.The advantages of implementation of comprehensive enzyme kinetic mechanisms (e.g. as explored in the Figure in vivo. Fitting isotopomer distribution directly accessing the kinetic constants provided insight to the enzyme kinetic mechanisms that operate in vivo. Some of the kinetic constants obtained from in vitro study were corrected based on the analysis of in vivo isotopomer distribution. The ability to accumulate the information from different sources and to check its compatibility makes this approach scalable into analytical tool which filters and orders available data, as well as evaluates metabolic fluxes compatible with maximum of available information.On the other hand, such integrated analysis brought a new area of isotopomer distribution data to the kinetic study of enzyme operation The analysis of tracer data considered here focuses mainly on central metabolic pathways that, according to the recent review , are stoAlthough the data on the existence of different intracellular compartments not separated by membrane structures are not widely accepted, they appear in various areas of cell biology and deserve thorough analysis. Analysis of the behavior of ATP-sensitive K+ channel, located in sarcolemma revealed large differences in ATP levels at different distance from the sarcolemma ,46. Simicit, citrate; dhap, dihydroxyacetone phosphate; e4p, erythrose-4-phosphate; g6p, glucose-6-phosphate; g3p, glyceraldehyde-3-phosphate; f6p, fructose-6-phosphate; glu, glutamate; lac, lactate; oaa, oxaloacetate; pep, phosphoenolpyruvate; PPP, pentose phosphate pathway; pyr, pyruvate; r5p, ribose-5-phosphate; s7p, sedoheptulose-7-phosphate; TA, transaldolase; TK, transketolase; xu5p, xylulose-5-phosphate; NMR, nuclear magnetic resonance; MS, mass spectrometry; MFA, metabolic flux analysis.