Here, we used an intermittent feeding regime (IFR) and a two‐compartment reactor (TCR) to mimic these substrate gradients at laboratory‐scale continuous cultures. The IFR was applied to simulate substrate dynamics experienced by the cells at full scale at timescales of tens of seconds to minutes , while the TCR was designed to simulate substrate gradients at an applied mean residence time (τc) of 6 min. A biological systems analysis of the response of an industrial high‐yielding P. chrysogenum strain has been performed in these continuous cultures. Compared to an undisturbed continuous feeding regime in a single reactor, the penicillin productivity (qPenG) was reduced in all scale‐down simulators. The dynamic metabolomics data indicated that in the IFRs, the cells accumulated high levels of the central metabolites during the feast phase to actively cope with external substrate deprivation during the famine phase. In contrast, in the TCR system, the storage pool (e.g. mannitol and arabitol) constituted a large contribution of carbon supply in the non‐feed compartment. Further, transcript analysis revealed that all scale‐down simulators gave different expression levels of the glucose/hexose transporter genes and the penicillin gene clusters. The results showed that qPenG did not correlate well with exposure to the substrate regimes , but there was a clear inverse relation between qPenG and the intracellular glucose level.In a 54 m Simulation results of a 54 m3 industrial penicillin fermentation case show that three substrate regimes  can be distinguished, where the regime average Cs values are Cs,starvation¯ ≈ 3.5 × 10−3 μM, Cs,limitation¯ ≈ 34.7 μM and Cs,excess¯ ≈ 294 μM, and the average substrate concentration in the reactor Cs, avg is about 34.4 μM , de Jonge et al.  at industrial scales  and regime analysis are highly advocated to identify leads for improvement of industrial fermentation processes  of the response of P. chrysogenum to the conditions in these SD simulators with glucose as the sole limiting substrate.In the present work, we compared the IFR and the TCR systems used to mimic substrate gradients as computed for the 54 m3 industrial ‐scale penicillin fermentor  obtained for the reference condition, the IFRs and the TCR systems show that all triplicate continuous cultivations  were highly reproducible  resembles that in the 3 min IFR , μ is nearly the same, and qo2 values are 10–15% higher, this implies that there are extra ATP losses. The origin of this loss is probably efficiency loss in metabolism, due to carry‐over of central metabolites from feast to famine (see below). Metabolic flux analysis, using a previously established stoichiometric model for growth and penicillin production in P. chrysogenum . However, the qo2/qs ratios in the TCR1 and TCR2 are about 0.71 and 23.01 respectively. The first ratio shows that part of the consumed glucose in the TCR1 is not metabolized, while the second ratio indicates that there is more carbon (e.g. from the storage pool) in the TCR2 metabolized than carbon from glucose uptaken. This strongly suggests that there is carry‐over of metabolites between the TCR1 and TCR2.The TCR system shows the same biomass concentration as in the chemostat, but the average μs remains at the level of 2.6 μM over 250 h, and this agrees with the glucose uptake kinetics , and the glucose consumption times in the tubing are about 9 s (TCR1→TCR2) and 30 s (TCR2→TCR1). Therefore, the cells residing in the tubing will not experience serious starvation period.In the reference chemostat cultures, the extracellular C, et al. . The extion Figs  and S2, ent Fig. . In content Fig. , and theent Fig. . The totPenG and the Cs regimes  was observed. However, Fig. PenG in the IFRs seems to be inversely related to Cs, peak in the feast periods.Combining the above results, no clear relation between qs, in) show that the Cs, in values remain . The measured dynamics of Cs, in during the IFR experiments  during a complete cycle is a hyperbolic function of Cs, according to qs = qsmaxCsKs+Cs. With the estimated parameters qsmax = 0.0417 mol CmolX−1 h−1 and Ks = 7.8 μM  is threefold higher in TCR1 (feed compartment) than in TCR2 (non‐feed compartment) (19 μM)   respectively. The value in the non‐feed compartment is very different from that calculated using the substrate balance. This suggests that the substrate uptake kinetics in the TCR system have changed to a lower affinity . This change of affinity is also supported by the reduced mRNA levels of most of the glucose/hexose transporter genes .We assume that the specific glucose uptake rate  Fig. . From thR1 Table . Howevers is still in excess  can be estimated. We observed that during the famine phase in the 3 min and 6 min IFRs, the model‐predicted carbon emission as CO2 was about 663 and 1098 μCmol respectively. The accumulated amounts of carbon in the 3 min and 6 min IFRs were about 1600 μCmol, which is far higher than the total carbon emitted as CO2, therefore, the metabolite carbon accumulated in the feast phase is enough to sustain carbon consumption in the famine phase under these feeding regimes. This makes sense as the cells need to coordinate their metabolism to deal with changing environments, failure to do so will result in growth arrest and/or imbalanced state  in the TCR1. Combining for TCR2 the glucose consumed carbon, differences in the excreted metabolites . This shows that the massive carry‐over of carbon from feed (TCR1) to non‐feed compartment (TCR2) in the TCR simulator is mostly as storage metabolites, which is in contrast to the IFRs, where carry‐over is mostly as central carbon metabolites. Surprisingly, this stored carbon pool is accumulated in the TCR1 and thereby consumed in the TCR2 at a high value of about 160 mmolC CmolX−1 h−1  than in the TCR2 (1852 μCmol) Fig. . The tottes Fig. , the int−1 Table ; it woulet al., In the TCR2, the PEP level and the pyruvate level are not much reduced compared to other glycolytic intermediates such as G6P, F6P, suggesting there might be a buffering capacity of the closely related amino acid, alanine  is still sufficiently high to avoid such triggering towards high expression, and as a result the expression of transporter genes declines. Further, the decrease of the penicillin production must be linked to the expression levels of the penicillin gene cluster  was 70 times lower at 300 h than at 200 h. While, for the 3 min IFR, the transcript level of the pcbAB gene encoding the L‐α‐(δ‐aminoadipyl)‐L‐α‐cysteinyl‐D‐α‐valine synthase (ACVS) was about six times lower in the 300 h than in the 200 h. In the 30 s IFR, from 200 to 300 h, we observed that the transcript level of the pcbAB gene was slightly increased, while at the same time the transcript levels of the pcbC gene and the penDE gene were slightly decreased. However, in the TCR system (τc = 6 min), the pcbC gene encoding the isopenicillin N synthase (IPNS) was about two times lower expressed at 300 than at 200 h. Therefore, different IFRs and the TCR system may induce largely differential expression levels of the penicillin gene clusters, and thereby differences in penicillin productions.In previous studies, it was found that the transcription of the penicillin gene cluster is repressed by extracellular glucose  substrate gradient cycles. In the feast–famine regimes of the IFRs, the cells accumulated the surplus carbon in the form of the central metabolites, rather than in carbohydrate storage pools. In the IFRs, shorter feeding cycles led to, as expected, penicillin productivities closer to that observed in the continuously fed reference chemostat. The penicillin productivity was found to be less affected in the TCR system with two different regions of substrate levels as compared with the IFRs. Further, the metabolomics data show that the storage pools (e.g. mannitol and arabitol) constitute a big contribution in the carbon supply in the non‐feed compartment. Different IFRs and the TCR system induce differential expression levels of the glucose transporter genes and the penicillin gene clusters, which result in different penicillin productions. This study shows that in order to secure that scale‐down simulators deliver results representative for large‐scale conditions, design of these simulators should be guided by a detailed systems analysis of the large‐scale and scale‐down simulators.In summary, the penicillin productivity was reduced in all scale‐down simulators, that is, the IFR and the TCR systems, relative to an undisturbed, continuous chemostat condition. However, there is no clear relation to the exposure to substrate regimes . There is a clear inverse relation between qP. chrysogenum, DS17690, was used for all experiments, and its characteristics have been reported elsewhere . All chemostat cultivations were controlled at a dilution rate of 0.05 h−1 , a gas flow rate of 2 l min−1 in each vessel, and 0.5 bar overpressure. Effluent was removed by a peristaltic pump through an overflow pipe positioned at the gas/liquid interface of the bioreactors. The temperature was kept at 25 °C, and the broth was stirred (400 RPM) by two‐six‐bladed Rushton turbine stirrers . The DOT was measured in situ with a sterilizable oxygen probe   at 0.5 bar overpressure), and the pH of the culture system was maintained at 6.5 by automatically adding 4 M NaOH and using a sterilizable pH probe (Mettler‐Toledo) mounted in the bioreactor. The off‐gas oxygen and carbon dioxide levels were real‐time monitored by off‐gas mass spectrometry . For the TCR system, the compositions of the inlet and outlet gasses of the two compartments were separately monitored. A scheme of the location of entrance and removal of circulating broth, sampling port, feed inflow and broth outflow as well as the broth recycle pump is shown in Fig. All cultivations used identical 3 l working volume fermentors, where the TCR consisted of two of these vessels connected by broth recycling through a peristaltic pump . However, mass transfer will then become a problem in laboratory‐scale reactors, and high biomass concentrations will impede fast sampling and quenching procedures. Therefore, in this study, we mainly aimed to investigate the effect of a volume‐based Cs distribution on cell metabolism and penicillin production. According to the simulation results of the 54 m3 industrial‐scale penicillin broth, the volumetric distribution of non‐starvation (excess plus limitation) and starvation is roughly the same  was supplied to one compartment while the effluent was removed from the other compartment. The broth was circulated  in between the two compartments, and therewith the total mean residence time was set at 6, with 3 min in each compartment.Reference chemostat cultivations with constant substrate feeding were performed in triplicate. Scale‐down experiments were carried out in: (i) the IFR system at three different cycle times, each was performed in triplicate. In this system, on/off feeding was applied with different cycle times, where the feed period was always 10% of the total on/off cycle. During the feeding interval, the pump speed was set 10 times higher than that under reference conditions to keep the average glucose feeding rate of the intermittently fed cultures the same as that of the control chemostats. The feed pump was precisely controlled by a timer, switching it on every first 3, 18 and 36 s of the cycle for cycle times of 30 s, 3 min and 6 min, respectively; (ii) the TCR system, in which also all experiments were performed in triplicate. Both vessels are connected by 2 m tubing (Φ = 1 cm) wherein the broth volume is 157 ml, about 2.6% of the total 6 l broth. In this system, the substrate concentrations in each ideally mixed compartment are determined by the broth recycle rate. Knowing that the full‐scale simulated mean circulation time of the 54 mRapid sampling was carried out for acquisition of snapshots of both intracellular and extracellular metabolites and as well for analysis of mRNA levels. The detailed procedures can be found in supporting information.qO2 and qCO2 in the IFRs were the average values in the chemostat cultivations of 100–200 h. The calculation of q‐rates in the TCR system can be found in Supporting information. The thus obtained q‐rates were reconciled under the constraint that the elemental conservation relations were satisfied, using the approach of  were calculated from the corresponding compound balances. The et al., et al., et al., broth) and filtrate (TOCsupernatant) was analysed by a TOC analyser  according to the manufacturers’ instructions.Metabolites of the glycolytic pathway, the TCA cycle and the PP pathway were quantified by GC‐MS using the isotope dilution mass spectrometry (IDMS) method, as described previously  profiles in the IFRs.Fig. S2. Specific glucose uptake rate (qs).Fig. S3. Extracellular Cs values as a function of time in the TCR system.Fig. S4. Average residual glucose concentrations in the TCR system are based on 10 independent data points.Fig. S6. Intracellular glucose levels as function of the culture age in the IFRs.Fig. S7. Intracellular glucose levels over a complete feed cycle in the IFRs.Fig. S8. qpenG against Cs, in in the IFRs.Fig. S9. qpenG against Cs, in in the TCR2.Fig. S10 Cs, in against Cs, peak.Fig. S11. Intracellular glucose levels as function of the culture age in the TCR.Fig. S12. The intra/extra‐cellular glucose concentration ratio in the TCR system, where 2.5 ml gDW−1 is assumed for the conversion.Fig. S13. Concentration measurements of metabolites (in μmol−1gDW−1) in the glycolysis, PPP, TCA cycle and storage pools.Fig. S14. Mass action ratios for phosphoglucose isomerase (PGI), enolase, mannose‐6‐phosphate isomerase (PMI) and fumarase.Fig. S15. The biomass specific carbon emission rates within a complete feeding cycle predicted by the 9‐pool model .Fig. S16. Extracellular metabolites in the IFRs and the TCR system.Fig. S17. Overview of the 9‐pool model for P. chrysogenum.Table S1. Average intracellular amounts (in μmol gDW−1) of glycolytic, PPP, TCA cycle intermediates measured based on three individual chemostat cultivations within the time range of 100 h to 200 h in the continuous feeding and TCR system.Table S2. Average Storage carbohydrates measured (in μmol gDW−1) based on three individual chemostat cultivations within the time range of 100 h to 200 h in the continuous feeding, the IFRs and the TCR system.Table S3. Average intracellular levels of amino acids measured (in μmol gDW−1) based on three individual chemostat cultivations within the time range of 100 h to 200 h in the IFRs and TCR system.Table S4. Complete stoichiometric matrix of the metabolically structured kinetic model .Table S5. Kinetics used in the metabolically structured kinetic model .Table S6. Optimized parameters used in the kinetics of the metabolically structured model .Click here for additional data file.


Synechocystis sp. PCC 6803 that promises to provide a further powerful resource for the cyanobacterial research community.Recent advances in synthetic biology and an emerging algal biotechnology market have spurred a prolific increase in the availability of molecular tools for cyanobacterial research. Nevertheless, work to date has focused primarily on only a small subset of model species, which arguably limits fundamental discovery and applied research towards wider commercialisation. Here, we review the requirements for uptake of new strains, including several recently characterised fast-growing species and promising non-model species. Furthermore, we discuss the potential applications of new techniques available for transformation, genetic engineering and regulation, including an up-to-date appraisal of current Clustered Regularly Interspaced Short Palindromic Repeats/CRISPR associated protein (CRISPR/Cas) and CRISPR interference (CRISPRi) research in cyanobacteria. We also provide an overview of several exciting molecular tools that could be ported to cyanobacteria for more advanced metabolic engineering approaches . Lastly, we introduce a forthcoming mutant library for the model species  Pn plants , bacterin plants , blue [1n plants ,195, gren plants ,197,198,n plants  or near-n plants . In contn plants ,201, S. revisiae ,202 and thaliana  , and demonstrated tight repression under red light and a 20-fold induction of GFP under green light [Apart from the green light-inducible CcaS/CcaR system, only one other optogenetic TCS has been characterised in PCC 6803, the near-UV activated UirS/UirR system that regulates phototaxis . The laten light . Thus, cE. coli that could be used in cyanobacterial research. For example, the near-infrared (760 nm) TCS BphP1/PpsR2 from Rhodopseudomonas palustris showed rapid response dynamics and a 2.5-fold dynamic range [Neurospora crassa was used to generate a blue-light-inducible T7 RNAP system  fused to an extein  within a larger precursor protein that can excise themselves spontaneously from flanking protein regions ) . Inteinsn extein . When thE. coli to construct a functional two input transcriptional AND gate system using a split T7 RNAP approach [N- or a C-intein half, and expressed from two different inducible promoters. Schaerli et al. [T7. Using a similar trans-splicing approach, split inteins have also been used to facilitate re-assembly of split variants of the more complex multidomain transcriptional regulator TetR [Split inteins have been used in approach . T7 RNAPi et al.  showed ttor TetR . Like T7tor TetR . Thus, ttor TetR . Split itor TetR ,214,225)Manipulating cyanobacteria for biotechnology applications is dependent not just on developing better genetic tools but also improving our understanding of cellular metabolism. Genome-scale models (GSMs) are large-scale simulation tools that comprise a global description of the metabolic reactions and pathways of an organism using stoichiometric coefficients and mass balances of the participating metabolites . GSMs arCurrently, 290 draft genomes and 85 full genomes are available for cyanobacterial species . However13C metabolic flux analysis) [2 evolution to predict metabolic reaction fluxes [−2·s−1) and high-light (600 μmol photons m−2·s−1) conditions. Similarly, the model iSynCJ816 for PCC 6803 can account for changes in energy absorption for different light qualities [E. coli (iML1515) can predict gene essentiality in minimal media with 16 different carbon sources with an average accuracy of 94% [Recent cyanobacterial GSMs have attempted to include algorithms to model components of the photosynthetic electron transport chain ,232,235.nalysis) ,237,238.nalysis) . When usnalysis) . Recentlualities , and achy of 94% . The rel13C-labeling. The flux elucidation revealed nearly complete conversion (>96%) of the assimilated carbon into biomass compared with only 86% conversion in PCC 6803. Comparison of the UTEX 2973 flux map with that of PCC 6803 revealed differences in the synthesis of key Calvin cycle metabolites, fructose-6-phosphate and sedoheptulose-7-phosphate, and production of amino acids, glycine and serine, from the photorespiratory salvage pathway [Cyanobacterial GSMs have been used to detect key metabolic differences between species  and to i pathway . in silico to maximise the production of 2,3-butanediol [n-butanol [n-butanol, resulting in a 5-fold yield increase compared to the non-repressed strain. Similarly, OptForce was used to increase the production of limonene [GSMs can be used in cyanobacteria to improve bioproduction. Recently, the model iBJ792 for PCC 7942 was used to determine an optimal solution tanediol . GSMs catanediol ,243. Opt-butanol . OptFluxlimonene . OverexpE. coli (iML1515) [Further improvements in the accuracy of GSMs in cyanobacteria are constrained by the lack of available omics data. For example, a large percentage of the predicted proteins in most cyanobacterial genomes are still annotated as unknown or hypothetical . In addiiML1515) ,246. LasiML1515) . The curChlamydomonas reinhardtii [Arabidopsis thaliana [E. coli [S. cerevisiae [Mutant generation is a key tool in bacterial research and for altering species for biotechnology applications. Individual research laboratories currently generate mutants of interest via a variety of different experimental methods, using a range of plasmid systems in sub-strains that can vary significantly at both the phenotypic and genotypic level nhardtii  and Arabthaliana , and mod[E. coli  and S. crevisiae . The exirevisiae . In addica. 1050 coding sequences (~30%) have assigned functions, compared to ~80% in E. coli and S. cerevisiae [Cyanobacteria remain underdeveloped for fundamental research and viable biotechnological exploitation. Nevertheless, cyanobacterial research is a rapidly growing field with approximately 13,000 new papers published in the last 5 years, making it the third most studied group of autotrophs behind plants and algae. In PCC 6803, only revisiae . Of thesrevisiae . The majrevisiae ,254, butGene characterisation studies remain challenging and time-consuming but recent developments in automation  can streS. cerevisiae [C. reinhardtii [In addition to automation, this library will be barcoded to allow for the tracking of individual strains within a pooled mutant library. This powerful approach has been described in revisiae  and C. rnhardtii , where bGiven the ongoing advances in complex synthetic biology tools to finely modulate metabolism in microbes, the future looks bright for progressing both fundamental and applied cyanobacterial research. The topics outlined in this review highlight just some of the current exciting methods that could be used to generate a step change for cyanobacteria researchers by improving transformation efficiencies, gene regulation and capacity for metabolic engineering. Development of the techniques and resources outlined in this review should significantly improve our knowledge of this environmentally important phylum, especially for poorly characterised species. Moreover, their application towards the development of cyanobacteria as a renewable biotechnology platform could have immense implications, not just commercially but also in replacing polluting fossil fuels usage for chemical production and reducing carbon emissions.


Escherichia coli (E. coli) is a commonly observed phenomenon, often referred to as overflow metabolism. Among various studies that have been carried over decades, a recent work  suggested and validated that it is the differential proteomic efficiencies in energy biogenesis between fermentation and respiration that lead to the production of acetate at rapid growth conditions, as the consequence of optimally allocating the limited proteomic resource. In the current work, we attempt to incorporate this newly developed proteome allocation theory into flux balance analysis (FBA) to capture quantitatively the extent of overflow metabolism in different E. coli strains.The formation of acetate by fast-growing A concise constraint was introduced into a FBA-based model with three proteomic cost parameters to represent constrained allocation of proteome over two energy (respiration and fermentation) pathways and biomass synthesis. Linear relationships were shown to exist between the three proteomic cost parameters. Tests with three different strains revealed that the proteomic cost of fermentation was consistently lower than that of respiration. A slow-growing strain appeared to have a higher proteomic cost for biomass synthesis than fast-growing strains. Different assumed levels of carbon flowing into pentose phosphate pathway affected the absolute value of model parameters, but had no qualitative impact on the comparative proteomic costs. For the prediction of biomass yield, significant errors that occurred for one of the tested strains (ML308) were rectified by adjusting the cellular energy demand according to literature data.E. coli strains. Such prediction is enabled by three linearly-correlated (as opposed to uniquely determinable) proteomic cost parameters. The linear relationships between these parameters, when determined using data from cell culturing experiments, render biologically meaningful comparative proteomic costs between fermentation and respiration pathways and between the biomass synthesis sectors of slow- and fast-growing species. Simultaneous prediction of acetate production and biomass yield in the overflow region requires the use of reliable cellular energy demand data.With the aid of a concise proteome allocation constraint, our FBA-based model is able to quantitatively predict the onset and extent of the overflow metabolism in various The online version of this article (10.1186/s12918-018-0677-4) contains supplementary material, which is available to authorized users. Escherichia coli (E. coli) grows under aerobic-glucose conditions is a commonly observed phenomenon, which has been extensively studied over decades [E. coli where acetate was accumulated in fed-batch systems [The formation of acidic by-products, predominantly acetate, when  decades –5. Lee r systems . It has  systems , represe systems , which i systems . Acetate systems  and drug systems . These c systems –12. The  systems .E. coli originates from the global physiological proteome allocation for rapid growth [E. coli.Traditionally, the aerobic formation of acetate has been referred to as overflow metabolism: the excess glucose saturates or inhibits the tricarboxylic acid (TCA) cycle, which subsequently forces the cell to modulate the redundant carbon to the acetate pathway , 14. Howd growth . In partGiven the importance of the overflow metabolism, several phenomenological models were developed to depict this effect , 17, 18,E. coli [E. coli strains, with good quantitative agreement with experimental data. However, the proteomic costs adopted in CAFBA were applied to individual metabolic reactions, without focusing on the exploration of the critical role played by specific metabolic modules such as energy biogenesis pathways.Inspired by a recent experimental work studying the proteomic cost of the core metabolic pathways of E. coli , a modelE. coli  managed E. coli strains with quantitative accuracy, i.e. predicting aerobic steady-state rates of acetate production at different growth rates and validating the model with experimental data in literature. In particular, we adopt a concise proteome allocation constraint as identified by Basan et al. [E.coli cells tend to use the more protein-efficient fermentation pathway to generate energy in order to accommodate the high proteomic demand in biosynthesis under rapid growth. The key concepts of PAT are fully embedded and realised in our model. With a parsimonious, PAT-based metabolic model capable of accurately capturing the overflow metabolism, we further analyse the interdependency between pathway-level proteomic cost parameters, the disparity in these parameters between different E. coli strains, and the impact of cellular energy demand on the accuracy of the co-prediction of the overflow metabolism and the biomass yield on substrate.In this work, we attempt to depict the overflow metabolism in various n et al. , referreϕf and ϕr are the fractions of the fermentation- and respiration-affiliated enzymes, respectively, which enable the fluxes for energy generation; ϕBM represents the fraction of the remaining part of the proteome enabling other cellular activities, broadly referred to as the sector of biomass synthesis [Following Basan et al. , the fraynthesis , 28.ϕf represents the mass abundance of the enzymes that carry fermentation fluxes involved in glycolysis (glucose to acetyl-CoA), oxidative phosphorylation and acetate synthesis pathways (phosphotransacetylase and acetate kinase). ϕr comprises all the enzymes that catalyse the respiration-associated reactions in glycolysis, tricarboxylic acid (TCA) cycle and oxidative phosphorylation system. Same as in Basan et al. [ϕf and ϕr with the fermentation and respiration fluxes respectively,vf (vr) is the fermentation (respiration) pathway flux, which in this work is represented by the enzymatic reaction “acetate kinase ACKr” (“2-oxogluterate dehydrogenase AKGDH”); wf (wr) is the pathway-level proteomic cost, denoting the proteome fraction required per unit fermentation (respiration) flux.More specifically, n et al. , in thisn et al.  to relatϕBM corresponds to the remaining part of proteome that is not covered by the fermentation and respiration sectors, including ribosomal proteins and anabolic enzymes , catabolic enzymes and cellular maintenance proteins. Motivated by the observed linear dependency between growth rate and proteome fraction for biomass synthesis [bλ is the growth rate-associated component with λ being the specific growth rate and the constant b quantifying the proteome fraction required per unit growth rate. In Basan et al. [ϕ0 was considered as a growth rate independent constant.On the biomass synthesis sector, ynthesis –32, the n et al. , ϕ0 was Combining Eqs. –3), we , we 3), ϕ0 is unlikely a growth rate independent constant: it may remain at a constant (and minimum) level in the overflow region where the proteomic resource is stretched, but become growth-rate dependent (and larger) at lower growth rates outside the overflow region, i.e. ϕ0, min ≤ ϕ0 ≤ 1, where ϕ0, min is a true constant. Defining ϕmax ≡ 1 − ϕ0, min, Eq.  minimum value. For simplicity, both sides of Eq.  , ϕ0 was s of Eq.  is dividFlux balance analysis (FBA)  is used fobj, subject tofobj is the assumed cellular objective. We specified minimizing substrate uptake as the objective function because in this study the commonly used objective ‘growth rate’ was used as the model input (with acetate production as the model output). S is the stoichiometric matrix defined by the metabolic model; v is a column vector comprising the reactions/fluxes described in the metabolic network; vL and vU represent the lower and upper limits of the reactions, respectively. The inequality constraint (iii) is same as Eq. (maxb∗ in the third constraint (PAT constraint) of Eq. (E. coli metabolic model [The prediction of the extent of overflow metabolism (rate of acetate production) requires of the parameter values of ) of Eq. . We showmentclass2pt{minimic model , referreic model  in MATLAvf is the fermentation flux (referred to as “acetate line” in [vr is the respiration flux. kf (kr) and vf, 0  are constants representing the slope and intercept of the fermentation (respiration) line. Substituting Eq. (A linear relationship was previously shown to hold between the fermentation (or respiration) flux and steady state growth rates in the overflow region :8\documeline” in ); vr is λ) in the overflow region, which requiresEquation  holds fob∗ is a linear combination of b∗ also possesses a linear relationship with Equations  and 12)12) indic2 revolution data given in [If experimental data exist that allow for both the fermentation line and the respiration line to be plotted (such as the steady state growth rate – acetate excretion and growth rate – COgiven in ), their given in .kf and vf, 0 were directly determined from the experimentally measured growth rate-acetate excretion profile  was conducted which confirmed that all the relevant fluxes used in the model were uniquely determined. After obtaining kf (kr) and vf, 0 , a set of values of b∗ can be determined by arbitrarily specifying the value for one of the parameters. In this work, we took b∗ plots. Simulation results presented in this paper were obtained with a randomly chosen value of b∗. Note that different values chosen for In this work, entclass1pt{minimaWe observed discrepancy in biomass yield between model predictions and the experimental data, especially for ML308 Fig. , and hyprATP, nominal is the overall ATP consumption rate ; subscript “nominal” indicates the default specification of the core model; vGLNS and vPFK are the fluxes of the enzymatic reactions glutamine synthetase and phosphofructokinase (one mole ATP is required per mole flux of each reaction); (ATPM + σλ) denotes the maintenance energy required for non-metabolic processes, where ATPM corresponds to the non-growth-associated maintenance (NGAM) and σ to the growth-associated maintenance (GAM) [The original cellular energy demand embedded in the core model is quantified byce (GAM) .S(λ) is the offset energy, i.e. the amount of energy over-predicted by the core model. The mathematical analysis of the growth data of ML308 suggested that in the overflow region, the offset energy is linearly related with the growth rate (R2 = 0.9998):k and c are constants. Substituting Eqs. (The adjusted cellular energy demand is formulated asing Eqs.  and 15)14\documec and N ≡ σ − k; M and N are referred to as the (adjusted) maintenance parameters. The adjustment of the cellular energy demand in our FBA is achieved by manipulating the maintenance energy, more specifically, through the maintenance parameters. Note that in the case where growth energetic data is not available (i.e. for MG1655 and NCM3722), such adjustment was not possible, therefore the default maintenance parameter values were used , the TCA cycle, the acetate pathway (PTA-ACKA) and the terminal oxidative phosphorylation system. However we noted the existence of alternative pathways in the central carbon metabolism, which include the Entner-Doudoraff (ED) pathway, the pentose phosphate (PP) pathway and the more recently explored PEP-glyoxylate cycle [The model constructed in this work considers only the central metabolism of te cycle .Sinorhizobium meliloti, Rhodobacter sphaeroides, Zymomonas mobilis, and Paracoccus versutus [E. coli, which is capable of using both the ED and the EMP pathways, tends to use the latter. Flux measurements also suggest that the usage of the ED pathway by E.coli K-12 is minimal: only about 2% of glucose catabolism proceeds by means of the ED pathway in batch cultures [E. coli has been reported under fast-growing scenarios.The ED pathway was found to be three to five-fold more protein-efficient than the EMP pathway to achieve the same glycolytic flux , which pversutus . Howevercultures  and aboucultures . Furthercultures , 41. To E. coli.As for the PEP-glyoxylate cycle, similar to the ED pathway, its usage was identified to be significant only under slow-growing conditions. Not even a trace activity of the PEP-glyoxylate cycle was found in wild-type batch cultures or more rapidly growing chemostats . FurtherE.coli with relatively sufficient substrate availability, has taken the assumption that the use of alternative ED pathway and PEP-glyoxylate cycle is negligible compared to the glycolysis (i.e. EMP pathway) and the TCA cycle.Based on above literature evidences, this work, focusing on the overflow metabolism that occurs at fast-growing cultures of E. coli. Previous studies showed that the carbon flow through the PP pathway could reach 20–35% of the total carbon intake and can vary with different growth rates [The PP pathway, on the other hand, can function as a significant alternative to the upper part of glycolysis for carbon catabolism in th rates , 41 hencPPP%) as the portion of substrate carbon directing to PP pathway to the total carbon intake:PPP% was controlled by setting the upper bound of the portion of carbon that is directed into the PP pathway, denoted as uPPP%, with the aid of an auxiliary term DM_PPP_RATIO:We define the PP pathway ratio , previous studies on the slow-growth strain MG show that the portion of carbon that goes into the PP pathway can be approximately 20% of the total carbon intake [uPPP%) to 25, 35 and 40% to investigate the potential effect of the change in PPP% on proteomic cost parameters and model prediction , the change of entclass1pt{minimaentclass1pt{minimaeters in .Fig. 2\dTo inspect the insignificant disparity in the mentclasspt{minimavf, 0 can be determined directly by the experimental measurement of acetate production. vr, 0, on the other hand, is a result of the combination of (measured) rates of acetate production and the mass and energy balance structure of the metabolic model.The slope and intercept of the vf, 0 only depends on the pattern of acetate excretion, not affected by assumed level of uPPP%. Therefore, the impact of uPPP% on the vr, 0, which turns out to be rather moderate. Between different strains, the ratio of vf, 0 and vr, 0 and the value of vr, 0 are nearly identical between MG and NCM, regardless of the level of uPPP% adopted, resulting in the very much overlapped pattern of the For a specific strain, mentclass2pt{minimuPPP% more visibly  on the vf, λ) fixed in Eq.  on the linear interdependency of the proteomic cost parameters. With an interest in the FBA solution of the flux distribution in PP pathway (at different growth rates), simulation results were recorded for three strains with uPPP%set to 35%; other uPPP% levels displayed a similar trend . Therefore, TKT2 is activated to direct the extra amount of four-carbon and five-carbon compounds back to the glycolysis.In phase III, THD2 is finally switched on and becomes significantly active in the high-growth-rates region. TKT2 increases progressively while NADTRHD remains silent. It is presumed that in this phase, as the growth rate becomes higher, more NADPH is required for biomass synthesis. NADP transhydrogenases (THD2) is activated to produce NADPH needed in rapid growth. The surplus carbon flux in the PP pathway, which might result from the high glucose uptake rate at a high growth rate, is directed back to glycolysis via TKT2.PPP% with experimental measurements, which unfortunately have not been widely reported in the literature. Nevertheless, Fig. It would be desirable to verify the theoretical prediction of the evolution of Although combining the PAT constraint with the core model succeeded in predicting the rates of acetate production, the accuracy in biomass yield varied and was especially unsatisfactory for ML strain Fig. . A similIntuitively, feature (i) suggests that in ML, the amount of energy required per unit mass of biomass formation should be less than NCM or MG. Therefore we collected the growth data of ML and remodelled the cellular energy demand see .It is worth noting that for ML, the negative value of M Table  clearly b∗ together with the adjusted maintenance energy, our model is now able to effectively capture the unique trend of biomass yield for ML, without any compromise in the accuracy of predicting acetate excretion  and the increase in glucose uptake rate (glc2 − glc1), while the growth rate rises from λ1 to λ2:It is worth noting that, our model also succeeds in matching the elevated reduction in the yield of ML in the overflow region as the growth rate increases. This captured trend appears to originate from the low energy demand of ML. Approximately, the yield reduction rate can be considered as being proportional to the ratio of the increase in the acetate excretion  for ML will be lower than NCM to achieve a given increment in the growth rate. According to Eq.  to update the linear equations describing To investigate the impact of the change in cellular energy demand on the linear relationships of f∗ (Eqs.  and 12)\documentThe switch to the adjusted energy demand makes the vr when applying the adjusted energy demand . With no change in b∗ to become higher for ML-new than ML-nom, hence the right-shifting of the Applying the adjusted energy demand also has an impact on the relationship between entclass1pt{minimauPPP% would lead to a reduction of vr, which in turn would lead to an increase in b∗ or right-shifting of the uPPP% levels is observed compared to the case of ML-nom. This implies that the effect of vr reduction due to the increase in uPPP% is more pronounced with the adjusted energy demand.In the case of ML-nom, it was shown earlier in Fig. E. coli under different growth conditions, when coupled with a simple energy balance equation. In this work, we have embedded the PAT into the core metabolic model of E. coli, taking the advantage of the latter in offering more rigorous modelling of intracellular mass and energy balances. Furthermore, the constraint-based metabolic model allows prediction of detailed metabolic fluxes as opposed to merely acetate production, which could provide more insights about metabolic pathways in connection with the overflow metabolism and pave the way for investigating acetate excretion in junction with possible manipulations of the metabolic network.In the study by Basan et al. , which hJCO2, r, whose value was deduced by subtracting the fermentation-dependent CO2 and the growth-dependent CO2 from the total CO2 production. As such, JCO2, r could not directly correspond to a specific flux in the metabolic network. In our model, the respiratory flux directly refers to a specific flux within the TCA cycle (AKGDH), which appears to be a convenient choice when the PAT is embedded into FBA. Using a constraint-based metabolic model that includes the TCA cycle with a reasonable level of detail, the respiration flux can be directly resolved via FBA, without the need for multi-step calculation along with different levels of assumptions and uncertainties.In addition, the respiratory flux in Basan’s work was associated with the carbon dioxide produced in respiration, termed Another important comparison we would like to make is with the recently developed model – CAFBA. It was mentioned in the wcvc) representing the proteome requirement for uptaking carbon source, which is not explicitly considered in this work. We have ignored this sector as the carbon overflow occurs only at the high growth rate region, where wc (proteomic cost of the C-sector) approaches zero at high substrate uptake rates, as shown in CAFBA [wc makes the C-sector negligible compared with other proteome sectors. At low growth rate region, the value of wc becomes significant, however no acetate is excreted in this region, where the equality relationship in the PAT constraint in our work becomes inactive so that the significance of the C-sector becomes irrelevant.The proteome allocation constraint in CAFBA includes a C-sector  for a genome-scale model. Similarly in FBAwMC, a large number of crowing coefficients need to be specified. In both cases, the existence of numerous cost parameters originates from associating the resource cost with individual reactions. These parameters conceptually have a clear biological meaning and in principle can be determined experimentally by e.g. proteome measurements or extensive enzyme assays. However, in practice, it has appeared to be difficult to reliably obtain precise values for all the parameters, especially for different strains growing at different growth rates or conditions. In fact, instead of pursuing the exact values for all the individual parameters, CAFBA focused on applying the average value of the proteome fraction invested per unit flux, termed as 〈w〉, to capture the key flux pattern, along with evaluating the impact of possible heterogeneous values of the proteome parameter wi on the model prediction. Similarly, FBAwMC [cat) values [The modelling approach proposed in this work can be considered as “halfway” between the coarse-grained proteome allocation model of Basan et al.  and the , FBAwMC , 22 also, FBAwMC  or the m) values .In this work, we have intended to formulate a constraint with a greatly reduced number of proteomic cost parameters, while still capturing the essence of constrained cellular resource allocation. This is achieved by formulating the proteome allocation constraint at the pathway (as opposed to reaction) level. The proposal is the concise Eq.   of the proteomic cost parameters that allow an accurate prediction of the overflow metabolism. We thus speculate that for a FBA-based model, the ability of capturing the overflow behaviour is rendered by (i) an extra constraint representing the constrained proteomic resources and (ii) certain relations or relative magnitudes of the proteomic cost parameters embedded in the proteome constraint. In reality, the proteomic efficiencies of the metabolic pathways may vary (within a certain range), at different points in time or between cells in a population which often exhibits heterogeneity . Howeveri is proportional to the growth rate-dependent component of the associated proteome fraction [ϕi linearly with the flux it carries. A similar model is also adopted in CAFBA [As indicated in the earlier section, our formulation of the proteomic cost (Eqs.  and 2b)2b)) reflfraction , we havein CAFBA , where tE. coli under glucose-limited conditions. In some other circumstances, observations not conforming to this relatively simple model have been reported. For example, Goel and his co-workers found hardly any changes in protein levels in anaerobic slow-growing Lactococcus lactis chemostats, when the cell shifted from a high yield metabolic mode to a low yield metabolic mode with an increased growth rate [L. lactis is similar to the overflow metabolism observed in E. coli, proteome allocation did not seem to accompany the changes in metabolic fluxes. In a study on yeast’s transient transcript, enzyme and metabolite responses under metabolic perturbation, it is revealed that the reaction rates are jointly regulated by enzyme capacity and metabolite concentration due to the cell’s tendency in sacrificing the local metabolite homeostasis to maintain fluxes and global metabolite homeostasis upon enzyme perturbation [Note that our model is intended specifically for predicting the steady-state overflow metabolism in wth rate . In thisurbation . Anotherurbation .E. coli (the target organism of our model) and the organisms with which those observations were made. Besides, our model has been developed to describe the relationship between (i) the observed steady-state global proteome configuration and (ii) the growth rate or the corresponding flux, drawing on evidences and hypothesises from several previous studies [The above-mentioned experimental observations suggest that the linear proteome-flux relationship modelled in this work might not be applicable to those circumstances. We hypothesise that this might be at least partially due to the differences between  studies , 30, 32.E. coli strains, we have evaluated a new model that integrates a previously proposed proteome allocation theory (PAT) into the constraint-based modelling approach – flux balance analysis (FBA), which predicts the distribution of carbon fluxes between fermentation and respiration due to the differential proteomic efficiencies of the two energy biogenesis pathways. Using a simple proteome allocation constraint, our model allows the accurate prediction of acetate production at different steady state growth rates during overflow conditions (with sufficient oxygen and glucose). The model involves three pathway-level proteomic cost parameters linearly interrelated by two equations, which is the consequence of (i) the assumed linear dependency of proteomic costs and the growth rate and (ii) the experimentally observed linear correlation between the fermentation or respiration flux and the growth rate. The non-unique optimal values of the three parameters, or the two linear relationships between them, could be obtained by fitting the model to experimentally measured acetate excretion rates at specific growth rates.With three different b∗both meet qualitatively the expectation from a biological point of view. Furthermore, the relationship between the proteomic efficiencies of fermentation and respiration, i.e. the uPPP%) varied in the modelling studies. On the other hand, the increase of uPPP% was shown to lead to a visible increase in the estimated proteomic cost of the biomass synthesis sector b∗, which mathematically results from a reduction in the predicted respiration flux.The linear relationships between the parameters were shown to be affected, in varying degrees, by (i) the acetate excretion pattern, (ii) the assumed upper limit of the substrate carbon diverting into PP pathway and (iii) the cellular energy demand. The proteomic cost of the fermentation pathway was estimated always to a lower value than that of the respiration pathway, i.e. Finally, and as a general point for constraint-based models, cellular energy demand appeared to have a major impact on the predicted biomass yield; tuning the default energy demand with strain-specific data was shown to be critical in making simultaneously accurate predictions of biomass yield and overflow metabolism.E coli; similar modelling approaches that feature this type of combination may prove suitable also for other applications.Overall, this work demonstrates the potential of combining a detailed metabolic model with a coarse-grained, pathway-level resource allocation constraint in producing quantitatively accurate predictions of the overflow phenomenon in Additional file 1: Supplementary Tables S1-S6. (XLSX 26 kb)Additional file 2: Supplementary text and Supplementary Figures S1-S17. (DOCX 229 kb)


Escherichia coli and Saccharomyces cerevisiae fed-batch cultures. The experimental investigation of both cultures presents experimental validation results since the start of the bioprocess, i.e. since the injection of inoculant solution into bioreactor. In total, four strains were analyzed, and 21 experiments were performed under varying bioprocess conditions, out of which 7 experiments were carried out with dosed substrate feeding. Development of the microorganisms’ culture invariant generic estimator of biomass concentration was the main goal of this research.The focus of this study is online estimation of biomass concentration in fed-batch cultures. It describes a bioengineering software solution, which is explored for E. coli and S. cerevisiae cultures. The cell culture stoichiometric parameters are estimated by a procedure based on the Luedeking/Piret-model and maximization of entropy. The main input signal of the approach is cumulative oxygen uptake rate at fed-batch cultivation processes. The developed noninvasive biomass estimation procedure was intentionally made to not depend on the selection of corresponding bioprocess/bioreactor parameters.The results show that stoichiometric parameters provide acceptable knowledge on the state of biomass concentrations during the whole cultivation process, including the exponential growth phase of both E. coli and S. cerevisiae. The suggested estimation procedure is identical for both cultures. This approach improves the precision achieved by other authors without compromising the simplicity of the implementation. Moreover, the suggested approach is a candidate method to be the microorganisms’ culture invariant approach. It does not depend on any numeric initial optimization conditions, it does not require any of bioreactor parameters. No numeric stability issues of convergence occurred during multiple performance tests. All this makes this approach a potential candidate for industrial tasks with adaptive feeding control or automatic inoculations when substrate feeding profile and bioreactor parameters are not provided.The precision errors, since the bioprocess start, when inoculant was injected to a bioreactor, confirmed that the approach is relevant for online biomass state estimation. This included the lag and exponential growth phases for both  Biotechnology industry development over the last years made quality assurance more stringent for pharmacy production . As a toThis study delves into biomass estimator development based on stoichiometric parameters and Luedeking–Piret model. The cell’s yields and stoichiometry both form a generic information, which is an acceptable candidate to be included in estimators when the microorganisms culture does not change from experiment to experiment. Depending on stoichiometry, the estimator of biomass concentration can be used to automatically inject the inoculant solution at a predefined level of the optical density in bioreactor medium. At this point, cumulative oxygen uptake rate signal from an off-gas analyzer is informative to determine the biomass concentration.Escherichia coli bacteria and S. cerevisiae yeast cultures. In other words, the stoichiometry optimization algorithm must be performed once for each strain to determine the necessary coefficients. These coefficients can be later used in the subsequent experiments to estimate biomass concentration online, unless the strain does not change. Such offline analysis can be considered as an estimator tuning algorithm for a specific microorganisms’ culture.The biomass estimator described in this study includes optimization algorithm, which returns the stoichiometric parameters of the controlled culture. The algorithm refers to several optimization criteria and is based on a gray box model originating from Luedeking–Piret model. Then offline maximization of entropy leads to satisfactory parameters values for estimation procedure, which is then applied to E. coli and S. cerevisiae cultures. The “The “S. cerevisiae (no DY7221) strain was used as representative of yeasts cells. The recombinant strains E. coli BL21(DE3) pET9a-IdeS, E. coli BL21 (DE3) pET21-IFN-alfa-5  [E.coli BL21(DE3) pLysS [Four types of strain cultivation were analyzed in this work to verify biomass estimation. iae no DY221 strai3) pLysS  were useSaccharomyces cerevisiae (no DY7221) strain was cultivated in the standard nutrient medium (YPD) [In order to check biomass estimator’s reliability and accuracy, data were collected from different cell strains which have been cultivated in multiple different R&D laboratories, including the laboratory of bioprocessing modeling and management in Kaunas University of Technology. um (YPD) , 10, whi2, CO2 and pressure sensors. The culture broth mass was measured online with balanced reactor vessel which contained load cell weight sensor. The initial substrate concentration in the bioreactor was equal to zero, S = 0 g/kg. Hence, after inoculation the substrate solution feeding was started. The cultivation process was performed in 5 l bioreactor.The medium temperature was maintained at 30 °C and it was monitored by using temperature sensor “Pt100”, and pH was kept constant at 4.9 by addition of NaOH(aq) . DissolvE. coli BL21 (DE3) pET21-IFN-alfa-5 was cultivated in 7 l bioreactor. Cultivation medium was based on minimal mineral medium, which was made of 46.55 g potassium dihydrogen phosphate, 14 g ammonium phosphate dibasic, 5.6 g citric acid monohydrate, 3 ml of concentrate antifoam, 35 g magnesium sulphate heptahydrate, 105 g D (+) glucose monohydrate. The initial volume of medium was 3.7 kg. At the cultivation process the environment parameters were kept constant. The temperature setpoint was 37 °C, DOT set at 20% of air saturation and pH kept at pH 6.8 by addition of NaOH(aq). The stirrer rpm range was from 800 to 1200 rpm, the air flow rage was from 1.75 to 3.75 l/min. In order to increase oxygen transfer rate during cultivation process, pure oxygen flow was provided to bioreactor at range from 0 to 7.5 l/min. The off-gas from bioreactor was measured online by BlueSens.The cell strain of E. coli BL21 (DE3) pET9a-IdeS was cultivated in 15 l bioreactor. Cultivation medium based as minimal mineral medium. At the cultivation process the environment parameters: temperature set point was 37 °C, DOT set at 30% of air saturation and pH kept at pH 6.98 by addition of NaOH(aq). The stirrer rpm range was from 300 to 750 rpm, the air flow range was from 0.3 to 15 l/min. During the cultivation process pure oxygen flow was provided to bioreactor at range from 0 to 7.5 l/min. The off-gas from bioreactor was measured online by BlueSens.The other cell strain of E. coli (BL21(DE3) pLysS) [4)2SO4, 2.46 g/l; NH4Cl, 0.5 g/l; NaH2PO4 × H2O, 3.6 g/l; Na2SO4, 2 g/l; K2HPO4, 14.6 g/l; (NH4)2-citrate, 1 g/l; 1 M MgSO4 solution, 5 ml/l; trace elements solution, 2 ml/l; and no glucose. Initial masses of all cultures were 5 kg. The glucose solution and initial substrate concentration at the bioreactor used same as at cultivation with yeasts, pH kept constant at pH 7 and temperature was regulated to 30 °C. Dissolved oxygen tension DOT was measured by an amperometric oxygen electrode (Mettler–Toledo) and the DOT set point was 30% of the saturation. The size of bioreactor was 15 l working volume  and the stirrer speed varied from 100 to 1400 rpm.For diversity of validation, the fourth cell strain was ) pLysS) . The culIn order to adaptively control and monitor chemical or biotechnological process, it is mandatory to implement a data collection system that provides desired variables at real time with acceptable precision and performance. This requires corresponding equipment, which may be unaffordable, not implementable in system or the required instrument doesn’t exist. Hence, the better alternative is to use soft or noninvasive sensors, which collect measurable variables and estimate unmeasurable parameters , 12. Esp2 and CO2 in the off-gas [Over time, the studies of both bioprocesses and industrial production perspectives have shown that a biomass estimator requires data, which is closely related to biomass growth rate and biomass concentration. It can be indirectly measured online, with well-established and validated devices and soft sensors , 13, whi off-gas . The proX growth/maintenance and oxygen uptake rate in bioreactor [The main model, dedicated to biomass concentration estimation in this work, is a Luedeking–Piret model derived from the stoichiometric equations for oxygen consumption. It represents relationship between biomass oreactor , 15:1\doα and β represent cell’s metabolisms of oxygen consumption and correspond to the yield coefficients of these biochemical conversions. In Eq. (α means specific cell’s oxygen consumption yield (β is a model parameter termed as oxygen consumption for maintenance (Stoichiometric coefficients . In Eq.  coefficiβ≡mo2/X) –20. The mentclass2pt{minimSimutis and Lübbert (2006) improved a hybrid model estimator . The maiX), oxygen uptake rate (OUR), specific biomass growth rate (µ), broth weight (w), carbon dioxide production rate (CPR), base consumption and other model coefficients. Additionally, additional equations and a fuzzy expert system are required. The latter gives an input to the combination of a dynamical mathematical model (DMM) represented by a set of nonlinear ordinary differential equations with an artificial neural network (ANN) [α and β, which both were kept static for a particular cell strain. This led to ability to calculate biomass online [When comparing stoichiometry biomass estimators’ mathematical models to the hybrid model estimator approaches, the latter contains more main state variables: biomass (rk (ANN) . The mairk (ANN) , 23, 24.s online –24, 28. s online , 30. Theα and β, which both describe oxygen consumption by a microbial culture. This is accomplished by offline analysis Fig. This paper presents a generic biomass estimation routine that is suitable for determination of biomass state in high diversity of bioreactors Fig.  with potα and β parameters at the laboratory scale bioreactors, Fig. Afterwards, industrial scale cultivation processes reuse information about strain information for corresponding biomass concentration estimation in online analysis (stage B), as shown in Fig. During the cultivation process, the real-time data collected from the devices has interference and disturbances, which may cause distortion of parameters and estimated values . Simutisβ is assumedly not a process constant. During the cultivation, parameter β—oxygen maintenance coefficient for biomass, increases due to biomass concentration growth. The phenomenon of increasing value of parameter β can be explained by the fact that the consumption of oxygen for biomass maintenance also includes the generation of the product and other factors. Such situation occurs at the end of the exponential phase of a microbial cultivation (for recombinant protein synthesis) when the induction  is performed and the synthesis of the product increases noticeably. As a result, oxygen consumption for biomass maintenance also increases [β consists of two additive termsAccording to data from bioprocesses and previous experience, the stoichiometric parameter ncreases , 32. TheThe observational data used for proposed biomass estimation was obtained from the processes that involve recombinant protein expression. As it can be seen from the Eq. , the parβ parameter is extremely small and negligible. Only after induction or specific value of biomass concentration, oxygen consumption for maintenance becomes appreciable. Hence, during a time prior to fact when the Eq.  or optical density OD value (in o.u.) multiplied by a coefficient of biomass concentration (approximately 0.4 g/l/o.u.) ;Process duration time since cells’ inoculation to bioreactor, in hours;Oxygen uptake rate (OUR) data since the inoculation;Prior to the estimation of the biomass, specific cell strain’s stoichiometric parameters must be identified during offline analysis. There are few compulsory inputs to approach this task.β is negligible:For model fitting a chosen mathematical expression is equated to gray box model since the collected experimental data is combined with fundamental knowledge about bioprocess . Consideβ. To properly describe second gray box model, the induction time or time when biomass concentration reaches specific amount must be identified. Throughout this period the maintenance term becomes significant and can’t be negligible. After applying maintenance parameter to a model, the second gray box model’s expression is generalized toIn the Eq.  the variIn summary, the Eqs.  and 11)11) both t sample is indexed by m. Discrete DCW values define variable m and In Eq.  the lastThe prediction value of the cumulative OUR model  for Eq. m-th offline sample ism has constant variance Then the posterior distribution for m-th observation value of the cumulative OUR and its unique variance is Prior distribution also has the form of Gaussian distribution  14\documE. coli and yeast cultures. It appears that the assumption of In previous work  the uncemulation  applied ous work . In otheon as in . Anothermentclasspt{minimaTo rationally prepare Eq.  for simporms Eq.  to16\docm of a sampling interval, allows to simplify Eq.  fin fin26) fE. coli, the induction time is known, i.e. it can be defined by the time moment when IPTG solution is injected into bioreactor. In the cultivation process of S. cerevisiae yeasts the IPTG solution was not used. Hence, the variable Variable mentclasspt{minima entropy , 41. TheThe knowledge of the specific time β directly depends on biomass concentrationAfter optimization of stoichiometry parameters, which had determined unknown parameters of the mathematical method, the next step is to validate those identified parameters with experimental data. Prior to comparison of theoretical and experimental data, the mathematical model, as in Eq. , must beE. coli strain Fig. S. cerevisiae oxygen consumption for maintenance is dependent linearly on biomass concentration, thus The expression of parameter β values from linear regression based on Eq. (In Eqs.  and 28)28) regred on Eq.  output29The assumed relationship of E. coli in Fig. E. coli, the induction of IPTG, which initiates product synthesis, may cause nonlinear dependence of oxygen consumption on biomass maintenance. Based on Eqs. (According to data from cultivation processes of  on Eqs.  and 28)E. coli iing Eqs.  and (28)The workflow of both stoichiometry and biomass estimations improves structure, as in Fig. The solution of Eq.  identifis in Eq. , transfomentclass2pt{minimX on In spite of the fact that Eq.  form is  all Eq.  coefficicOUR. When stoichiometric parameters are discovered in stoichiometry estimation, stage A, or it was given, only one input from bioreactor system, cumulative oxygen uptake rate, is necessary to estimate the biomass state. This procedure is depicted by stage B  in Fig. m. Prior to the specific biomass In this paper, estimation of biomass concentration is based on stoichiometric parameters and cumulative oxygen uptake rate β comes into effect as a function of biomass concentration. Equation (After biomass concentration exceeds mentclasspt{minimaThe variable  in Eqs.  and 33)\documentThis subsection initializes the online biomass estimation procedure Fig. , which cn is the number of data counts, Both mean absolute error (MAE) and mean absolute percentage error (MAPE) were used as indicators to evaluate the estimation results. MAE and MAPE methods both evaluate the errors between estimated and observed biomass values of a cultivation process. MAE approach is defined as follows :34\docum follows :35\documThe mean absolute percentage error is a statistical measure representing the accuracy of a forecast system, in percentage. Root mean square error represents the square root of residuals of the differences between predicted values and observed values. RMSE method’s formula are as follows :36\documcOUR from fed-batch experiments of E. coli and S. cerevisiae were taken from [E. coli cells in 15 l bioreactor with limited substrate feed [S. cerevisiae yeasts in 5 l bioreactor with limited substrate feed. Additionally, there was one cultivation of E. coli in 12 l bioreactor with limited substrate feed and there were 15 cultivations in 5 l bioreactor, out of which 7 cultivations were with dosed substrate feeding. As the first step, all cultivation data was analyzed in the stoichiometric parameters’ estimation (stage A). The estimation procedure ignored both metabolism pathways, occurring during dosed substrate feed cultivations, and increasing product formation due to IPTG injections. The results of offline analysis of stoichiometric parameters are present in Table Experimental biomass measurements and data of cumulative oxygen uptake rate ken from , experimate feed  and two S. cerevisiae stoichiometric results come from just two cultivation experiments. Therefore, the results might still be improved when more experimental data becomes available in the future.The tuning coefficient cOUR signal . This work’s biomass estimation method used different cultivation experiments, with different cell strains, bioreactor volumes, type of substrates feeding solution, different IPTG induction time moment and their corresponding OD levels at IPTG injection, different substrate feeding limitations and different time of starting the substrate feed. Estimation results are shown in Table In industrial processes, strain’s stoichiometric parameters are given, unless they were estimated using offline analysis, stage A. Then biomass concentration is calculated iteratively using both Eqs.  and 33)33) from Seven experiments (#5–#11) were performed with dosed substrate feeding. Meanwhile the rest of experiments had limited feeding with various combinations of control strategies described in : multiplS. cerevisiae cultivations is 0.5 g/l. RMSE value of E. coli cultivations with limited substrate feeding is 1.26 g/l and for cultivations with dosed substrate feeding is 2.44 g/l. RMSE value of E. coli cultivations before stationary phase, when DCW reaches ~ 40 g/l  with lisults in  without E. coli and S. cerevisiae. The experimental investigation of E. coli and S. cerevisiae cultures showed that the estimation procedure is identical for both cultures. The overall average MAE of biomass estimation since inoculation is 1.1 g/l and the overall average MAPE of biomass estimation since inoculation is 7.28%. RMSE value of E. coli cultivations before stationary phase, when DCW reaches ~ 40 g/l (to compare with results of other authors) with limited substrate feeding, is 1.07 g/l and for cultivations with dosed substrate feeding is 1.2 g/l. These results show that this approach improves the precision achieved by other authors without compromising the simplicity of the implementation. Moreover, the suggested approach is a candidate method to be the microorganisms’ culture invariant approach, it does not depend on any numeric initial optimization conditions, and it does not require any of bioreactor parameters. No numeric stability issues of convergence occurred during multiple performance tests. All this makes this approach a potential candidate for industrial tasks with adaptive feeding control or automatic inoculations when substrate feeding profile and bioreactor parameters are not provided.The suggested biomass estimation’s numeric approach using cumulative oxygen uptake rate signal showed no dependability on selection of the initial variable values for optimization procedures. This study assumed, by Pareto principle, that the proposed method is only dependent on stoichiometry parameters of the strain, i.e. the developed noninvasive biomass estimation procedure was made to not depend on both the manipulation with a specific growth rate variable and the selection of corresponding bioreactor parameters. The precision errors, since the bioprocess start, when inoculant was injected to a bioreactor, confirmed that the approach is relevant for online biomass state estimation. This included the lag and exponential growth phases for both Neither numeric artifacts nor abrupt worst-case scenarios were experienced during both offline and online analysis of 21 experiments, out of which 7 ones were carried out with dosed substrate feeding. The experiments executed in 5 l, 7 l, 12 l and 15 l bioreactor volumes. Feed start, inoculation, bioreactor medium, feeding limitation and other conditions varied with no manual control or adjustment. This encourages the use of such estimator in adaptive feedback control systems. Both online and offline estimations were tested on a single core CPU processing and each procedure took no more than 30 ms when overall 1-min interval data was sampled from cumulative oxygen uptake signal, which makes the approach of practical use too. Finally, this estimator does require a usage of regular industrial gas analysis equipment such as BlueSens etc.


Many biological processes are known to be based on molecular sequestration. This kind of dynamics involves two types of molecular species, namely targets and sequestrants, that bind to form a complex. In the simple framework of mass-action law, key features of these systems appear to be threshold-like profiles of the amounts of free molecules as a function of the parameters determining their possible maximum abundance. However, biochemical processes are probabilistic and take place in stochastically fluctuating environments. How these different sources of noise affect the final outcome of the network is not completely characterised yet. In this paper we specifically investigate the effects induced by a source of extrinsic noise onto a minimal stochastic model of molecular sequestration. We analytically show how bimodal distributions of the targets can appear and characterise them as a result of noise filtering mediated by the threshold response. We then address the correlations between target species induced by the sequestrant and discuss how extrinsic noise can turn the negative correlation caused by competition into a positive one. Finally, we consider the more complex scenario of competitive inhibition for enzymatic kinetics and discuss the relevance of our findings with respect to applications. The general scheme features one (or more) molecular species, the sequestrant, that bind to another molecular species, the target, to form a complex. In a simple mass-action law scenario, the system displays a threshold-like behaviour of the mean abundance of free targets as a function of the parameters determining its possible maximum amount (e.g. rate of synthesis for open systems or pool size for closed systems) as shown in Fig. e.g, refs15. This threshold-like behaviour follows from the action of the sequestrant. Intuitively, one can think about the following extreme scenarios. When the total number of the target is lower than the one of the sequestrant most of the target will tend to be bound. However, as the total number of targets begins to outnumber the sequestrant, the average of its free molecules starts to increase more markedly, often in a linear fashion. In the vicinity of the threshold, the average number of free targets moves from the mostly bound, slowly increasing regime, to the unbound, rapidly growing one. Therein, the system is defined to be ultrasensitive, meaning that, around the threshold, a small fold change in the amount of total target (input) leads to a larger fold change in the amount of free target (output), corresponding to a Hill coefficient larger than one6.Sequestration dynamics  are ubiquitous in nature. Some of the best studied examples concern protein ubiquitination18. This kind of noise may be non-specific, i.e. affects in a similar way many components of a same network and it is often homogenous on a single-cell level but varies from one to another. Sources of extrinsic fluctuations can be variations in the amount of molecular machineries , gradients of temperature or chemicals’ concentrations or, more widely, the coupling of the cell to the variability of the external environment. In the present contribution we resort to a minimal model of molecular sequestration and study the system in presence of both intrinsic and extrinsic noise19. This allows us to focus on two features of sequestration dynamics that have received relatively little attention in the past: the shape of the target probability distribution and the correlations amongst competing targets induced by the presence of the sequestrant. We analytically provide an exact solution to the intrinsic-noise case analysing both targets distributions and correlations in the case of one target and two different targets interacting with one single sequestrant. By adding an extrinsic source of noise, we show how the analytical distributions, previously found to be unimodal, can be reshaped into bimodal in a definite set of parameters. Bimodal distributions are often encountered in gene expression and other biomolecular assays data and are of particular interest as they may indicate the coexistence of two distinct phenotypes, with the modes of the distribution linked to different differentiation states or physiological conditions21. Inspired by the numerical results of ref.15, we first analyse the role of ultrasensitivity close to the threshold as a possible tool to channel extrinsic fluctuations of the parameters of the system into bimodal distributions of free target amount. We find that these bimodal distributions are originated by stochastic effects only, the underlying deterministic system being indeed always monostable. Secondly, we show that the action of the sequestrant is not only responsible for the presence of the threshold which allows to channel noise, but also plays an important role in systems with multiple targets. Indeed, targets result to be effectively correlated by their competing interaction with the same sequestrant and this competition induces a negative correlation. Here we show that, if the amount of sequestrant fluctuates due to extrinsic noise, the above mentioned correlations can, from negative, become positive.On top of the average behaviour of the systems, one should consider stochastic effects since molecular interactions are known to be probabilistic and to take place in fluctuating environments. While the first source of noise, referred to as intrinsic in the following, has been fairly studied in the past, little is known about the effect of the second one, which we shall call extrinsicS, the sequestrant, and T, the target. We shall view S as a titrating agent that binds T and sequesters it in the complex k+ and k− are respectively the binding and unbinding rates. For simplicity, we shall assume in our model the total number of molecules of each species to be conserved, this assumption defining the following conservation laws:Let us consider a simple model consisting of two species, namely T. The average behaviour for this model can be studied by considering the associated rate equation for the concentration6. We start by writing the steady state solution of the rate equation for the case of two species that bind and unbind, with constant total concentration for each species. The rate equation for the concentration of T is:Vsys being the typical volume of the system in which reactions occur. By making use of the conservation laws for concentrations that trivially follow from  as the probability of observing T free molecules at time t. The dynamics of this probability distribution is Markovian and obeys the following chemical master equation24:Very often, in biochemical systems, the numbers of individual molecules at play is low. Fluctuations around the average behaviour (described by the rate equation) become then relevant and, to correctly characterise these systems, it is mandatory to take into account their stochastic natureUsing the conservation laws, eqs  and 3),,3), the T depends both on the total number of available target molecules TT and on the total number of sequestrant molecules ST. Indeed, the minimum number of free T can either be Tmin = 0 when ST ≥ TT, or Tmin = TT − ST when ST < TT. To obtain the steady-state solution P(T) to the master equation, one can recursively use the detailed balance conditionWe are mostly interested in the long-time behaviour of the system and we will then focus on the steady-state solution Kd and depends on whether the system is in the repressed (TT < ST) or unrepressed (TT > ST) regime. For low values of Kd the distribution of free targets T is mostly concentrated around the minimum value Tmin, which, as mentioned before is 0 in the repressed regime and TT − ST in the unrepressed one. Some examples of this probability distribution for different values of the dissociation constant and TT are reported in Fig. P(T) from eq. , which completely neglected fluctuations, gives an accurate description of the average number of free target molecules.The shape of this distribution is mostly determined by the magnitude of the dissociation constant from eq. . The meaST, we here investigate the influence of extrinsic noise on the present system. Let us consider different copies of our system with different ST, randomly assigned. This mimics, for instance, the scenario in heterogeneous cell populations randomly sampled. For the sake of simplicity we draw the amount of ST in each system from a discretised Gaussian P(ST), restricted to positive values, with mean 〈ST〉 and standard deviation In the previous section we focussed on the fluctuations originated by the discrete nature of molecules and the intrinsic randomness of their interactions. By introducing fluctuations in the total number of sequestrant molecules T , P(T), is now affected by the fluctuations in ST and differs from the one given in eq.  weighed by P(ST) as follows:P(T|ST) coincides with the solution of the master equation obtained for a given ST .T. As the level of extrinsic noise is increased, the profile as a function of TT becomes smoother with a less pronounced threshold, while the regions farther away from the threshold are not heavily affected , which allow sampling between the unrepressed and repressed regimes 10. In these systems microRNA molecules bind and sequester coding messenger RNAs and inhibit their translation. The complex formation between microRNA and mRNA takes place on a scale that is much faster than the one of transcription and degradation of microRNA and mRNA. The total number of sequestrant molecules (and the one of target ones) then fluctuates on a slower scale than the sequestration dynamics .In view of these results, the combination of the threshold-like response produced by the titrative interaction and a suitable extrinsic noise on the total amount of one of the species can be considered a general mechanism to achieve bimodality. This result is also interesting from a pure biological point of view. In biological systems, bimodal distributions of gene expression are particularly interesting as they may indicate the presence of two distinct physiological states. At the level of a cell population, this effect would therefore show heterogeneity in different cell gene expression, as observed inT1 and T2) competing for binding to a third molecule (S). The reaction network that defines the minimal model is described by eqs (T1 and T2 can bind to S respectively with rate We consider here a system composed of two molecular species (d by eqs  and 15)T1 and T2T1 can be seen to act as a sponge for the common resource S, preventing the binding with the competing species T2 and vice versa. Also in this case, we assume the total amount of each species to be constant so that the following conservation laws hold:e.g. T1) is set by the number of free molecules of the sequestrant, which depends on the amount of complex between the second target and the sequestrant  and TT1, as schematically shown in Fig. T1 and T2 as independent variables, the model is described by eq. . When several molecules of a target are bound to the sequestrant, the propensity of binding for molecules of the second target is reduced. Additionally, when the total amount of molecules of the two targets globally exceeds the sequestrant, having a high number of one target molecules bound implies having fewer molecules of the other target in a complex. From this, it naturally follows that the two targets are negatively correlated via competition. To show this, we quantify such a correlation by means of the Pearson coefficient of the two targets, defined, as usual, as the ratio between the covariance and the product of the two standard deviations (30 (as done in e.g. ref.31 for a similar setup). In view of the applications of the present framework to biochemical systems, we focus on the dependence of the correlation on the targets abundances which are simpler to control in biological systems than the targets affinity for the sequestrant. Nonetheless, we report in the SI a detailed discussion of the dependence of the correlation on the dissociations constants. In Fig. TT1). We observe that the correlation starts close to zero for low target abundances and decreases close to the threshold TT2 fixed, the steepness of the sigmoidal profile as a function of TT1 is mainly governed by the value of the dissociation constant of target 1, TT1 becomes large  in the sequestrant abundance directly reflects in a change (increase) in the number of free molecules of both targets. This is no longer the case when there is an excess of one target (e.g. large TT1), for which the sequestrant is bound mostly to T1 and a change in the sequestrant affects mostly the amount of free T1 and less the amount of free T2. When the sequestrant largely outnumbers the targets, changes in its abundance will have little impact on the targets, which will mostly be bound to the sequestrant anyway. From the combination of these effects, correlation takes a non-trivial profile as a function of the abundances and can be positive for sizeable regions of the parameters space.As a result, we observe that fluctuations in the amount of the sequestrant positively correlate the competing targets. This can be understood by considering the fact that the two target species interact with the same pool of sequestrant molecules. Then, fluctuations in the amount of the sequestrant TT1, keeping TT2 fixed. As shown in Fig. TT2 we see that for larger values of TT2 the maximum correlation is reached for lower total amounts of the TT1, for which the system is still in the vicinity of the threshold. This is investigated in Fig. TT2. When the dissociation constants of the two targets are equal, such position of the maximum decreases linearly (with slope −1) with the increase of the abundance of the second target until the overall number of target molecules (TT1 + TT2) exceeds that of the sequestrant and eventually saturates at low levels. This is more evident for systems with a strong affinity for the sequestrant where the threshold is sharp. The overall trend is preserved when the two targets have different affinities for the sequestrant but the linear decrease is less pronounced when the second target (T2) has a lower affinity with the sequestrant. When the overall number of targets starts exceeding that of the sequestrant, the correlation decreases and the profile of the location of the maximum depends on the specific dissociation constants. For systems with higher levels of extrinsic noise, such profiles beyond the threshold may be non monotonic, as shown in Fig. ρmax(TT2) corresponds to the global maximum. As expected, it is located in proximity of the region of equimolarity between targets and sequestrant . Qualitatively similar behaviours are recovered by varying the average amount of sequestrant ST and the intensity of the noise. We report this in the SI.For a more quantitative discussion, we first focus on the correlation profile as a function of the total abundance of target 1, I quantifies the amount of information, measured in bits, that could be obtained on one target by measuring the other target. Its definition30 reads:P is the joint probability distribution and TT1 . Finally, it is important to note how mutual information (and correlations) are strongly damped by the presence of extrinsic noise as a consequence of the negative interference between the negative correlation due to competition and the positive one induced by fluctuations in the sequestrant.Having access to the explicit expression of the joint probability allows us to characterise the coupling between the competing targets beyond the Pearson coefficient and to study the mutual information between the competing targets. In this setting, mutual information TF), can bind to a substrate (which we assume to be at fixed concentration cs) and form the active enzyme TA from which the product is made. The free enzyme can also bind to an inhibitor (S) that prevents it from combining with the substrate and becoming active. In the language of the previous sections, the inhibitor plays the role of the sequestrant and the enzyme that of the target. The reaction network that describes this system is defined byLet us now explore how the key features we have identified in the minimal model carry over to more complex settings. To this aim, let us consider the case of competitive inhibition in an enzymatic reaction. The system is made of an enzyme that, when free (TA from which the product is formed and the inhibitor-bound enzyme TA and the inhibited one Kd = k−/k+) on bimodality are in qualitative agreement with the minimal model. Indeed, for both TA and TT becomes wider. Similarly, a low value of the dissociation constant, i.e. a steeper threshold, favours the presence of bimodality  inhibited by the same sequestrant. The corresponding reaction network is described by the following reactions:TF1 and TF2, can be activated (into TA1 and TA2) by binding to a substrate with rates proportional to the substrate concentration and to their intrinsic activation rates TA1 and TA2 can be deactivated with rates S into the inactive complexes Ta1 and Ta2) are generally lower than the ones between the sequestered, inactive enzymes , in combination with the threshold profile may result in a bimodal distribution of the other one (the target). This means that, close to the threshold, the extrinsic variability is channelled by the nonlinear sequestration mechanism into two main outputs corresponding to a repressed and an unrepressed state. This result identifies two minimal ingredients giving rise to bimodal profiles without the need of other regulatory links, i.e., (i) titrative (sequestration) interactions and (ii) extrinsic source of noise.Sequestration models are good approximations of many processes in nature, especially in biological systems. Although fairly studied, this kind of models still show novel features when their stochasticity is addressed. In this contribution we have considered how different sources of noise can affect the properties of a minimal model with sequestration dynamics. We started by analysing again the deterministic version of this system, where, as expectedWe have then extended the model to the case of two species competitively binding to a third one. Once again, taking advantage of the simplicity of the minimal model we have derived the analytical solution of the master equations to quantitatively address another key feature of the sequestration dynamics: the correlation induced by the competition for binding to the same sequestrant. We have shown that, without extrinsic fluctuations on the total abundance of the sequestrant, a significant negative correlation is present when the total amount of targets outnumbers the sequestrant. Together with this expected effect of the competition, we have identified non-trivial trends of the correlation in presence of an extrinsic source of noise. Fluctuations in the total sequestrant amount affect in a similar way the free amounts of both targets. As a result, despite the competitive dynamics of the targets, positive correlations are induced by the extrinsic fluctuations of the common sequestrant.14). A system for which our results will be of particular importance is the post-transcriptional regulation achieved by short non-coding RNAs interacting with messenger transcripts in eukaryotic cells15.These findings, obtained in a highly simplified description, have highlighted some key features that are likely to occur in more detailed models describing specific biochemical settings. To probe this, we have investigated the more complex case of competitive inhibition in enzymatic kinetics. This system features an additional reaction downstream of the sequestration interaction, which, with its stochasticity, partially dilutes the coupling between competitors, smoothening the distribution profiles and partially hindering the appearance of bimodality. However, we found that, with the due adjustments, extrinsic noise on the inhibitor can induce bimodal distributions and positive correlations between the enzymes. This suggests that the possible constructive role of extrinsic noise we have highlighted in the minimal model may be at play in more complex systems of biological relevance. Models of these systems would be more complicated, including different intermediate steps in the reactions of mediation between sequestrant and target or using saturation functions for production rates. Nonetheless, whenever sequestration dynamics are present, we expect our findings to be of relevance for such complex systems as well, where they may serve as guidelines to identify the most relevant parameters affecting correlations and the presence of bimodal distributions 15) with T1 and S only:Making use again of the conservation laws –18, the In order to find its steady-state solution, we first notice that detailed balance holds. By using detailed balance, the following relations for the steady-state probability are obtained:P(T1), we can write a recursive relation for the probability of having S free molecules conditioned on the fact that a generic number T1 of molecules are free:Dividing eq.  by P(T1)S can either be Smin = 0 if ST ≤ TT1 + TT2 − T1, or Smin = ST − TT1 − TT2 + T1 when ST > TT1 + TT2 − T1. Equation (P(S|T1) starting from the probability P(Smin|T1):In this system, as discussed in the main text  = P(S|T1)P(T1), we can insert eq. (P(T1 + 1) as follows:Writing sert eq.  in the fsert eq. . DefininP(T1):P(Tmin) is the probability of having the minimal number allowed of free T1. Tmin is defined by the total amount of S and T1. According to Table Tmin1 = 0 when ST ≥ TT1, which means that in principle all the molecules of T1 can be bound to molecules of S, or Tmin1 = TT1 − ST when ST < TT1, meaning that even if all the molecules of S are bound to molecules of T1, Tmin1 molecules are free. When Tmin1 > 0 we have that P(0|Tmin1). Conversely, if Tmin1 = 0, the conditional probability can either be P(0|Tmin1) or P can be reconstructed by using the definition of joint probability P = P(S|T1)P(T1), which leads to the following expression:Recursively applying eq. , we can PT:39\documTmin1 > 0, then Tmin1 = 0, the value of Also for the joint probability if Supplementary Information