Main

Cell-free enzymatic reaction networks (ERNs) are assemblies of extracted enzymes that partially reconstitute metabolic pathways outside of living systems. This in vitro approach offers a route to scale the production of valuable compounds in a single reaction1,2,3,4,5,6,7,8,9,10 or build functional systems11,12,13,14. Compared with traditional in vivo metabolic engineering, the use of cell-free ERNs avoids difficult product purification steps, improves yields, enables compounds that are toxic to cells to be produced and negates the sequestration of input substrates by other metabolic pathways15,16,17,18,19. Although purified enzymes and added cofactors are costly, they offer greater flexibility and control over the construction of optimized multi-enzymatic cascades20.

Leveraging these advantages, several groups have recently reported major breakthroughs in the development of complex ERNs, starting from simple and cheap materials such as glucose or fixing CO2 to produce valuable compounds such as monoterpenes, polyhydroxybutyrate, cannabinoids, malate or 6-deoxyerythronolide B21,22,23,24,25,26. It should be noted that complex networks can also be constructed using crude cell lysates for production of n-butanol, mevalonate or limonene27,28,29. Despite these advances, translating such successes into consistently high-yielding, cost-effective ERNs remains challenging.

Optimizing the initial ERN reaction mixture for optimal yields can be laborious; it is often impossible to identify a single set of initial concentrations that represents a global optimum30,31,32,33,34,35,36. The underlying challenge arises because enzyme cascades are frequently partially reversible, leading to suboptimal equilibria, and because large ERNs exhibit emergent activation and inhibition processes that affect enzyme efficiency37. These emergent kinetic barriers include allosteric regulation, product inhibition and competition for shared cofactors and only become apparent once all enzymes and cofactors are added to the reactor38,39.

To overcome these barriers, we introduce a model-guided strategy to improve ERN yields by designing a time-dependent ‘recipe’ for each network component (Fig. 1a). This approach expands the accessible design space exponentially (Fig. 1b), as each reagent can be added multiple times with an independently varied timing and amount. Within this enlarged space lie trajectories that can bypass kinetic bottlenecks and drive the cascade towards higher substrate conversion.

Fig. 1: A conceptual overview of time-dependent experiments in batch.
Fig. 1: A conceptual overview of time-dependent experiments in batch.The alternative text for this image may have been generated using AI.
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a, A kinetic model is constructed utilizing an active learning cycle which maps ERN behaviour; this model is then used to design a sequence of time-dependent inputs for the batch reaction. Batch reactions with time-dependent inputs (optimized) are then compared with standard batch experiments where all reagents are added at t = 0 (control). b, A visualization of the combinatorial explosion of the potential inputs space λ if time-dependent inputs are allowed. c, An overview of the model ERN used in this study, the PPP. AMP and PRPP are the final products. The concentrations of enzymes, glucose and cofactors can be controlled and are added to the reactor. Enzymes used in the network are HK, G6PDH, 6PGDH, PRI and PRPPS; cofactors and their products are ATP, ADP, AMP, NADP and reduced NADP (NADPH). Four observed species were quantified using HPLC.

Yet, experimentally identifying such optimal trajectories is infeasible without an accurate, predictive map of ERN dynamics. Therefore, we first parameterize a model with an active learning workflow40,41,42,43,44,45,46,47 that designs maximally informative experiments about ERN behaviour48,49,50,51,52,53 (see Fig. 2 for a detailed overview). As shown previously, this workflow performs best in systems that can sustain continuous perturbations; therefore, we conduct these experiments in a flow reactor. The resulting time-series data are used to train and test a small set of mechanistic model candidates, each incorporating slightly different regulatory assumptions. The model with the highest predictive accuracy is then selected to design batch mode experiments featuring optimized, time-dependent input profiles.

Fig. 2: An overview of the active learning procedure to train models for the PPP.
Fig. 2: An overview of the active learning procedure to train models for the PPP.The alternative text for this image may have been generated using AI.
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(1) The OED algorithm designs a sequence of pulsed inputs for each species that flow into a temperature-controlled CSTR (volume ~121 µl). (2) These species include five enzymes: HK, G6PDH, 6PGDH, PRI and PRPPS and ATP, NADP and glucose. Eight individual syringes control the unique inflow profile of each species. The outflow of the CSTR reactor is collected, quenched and measured offline by HPLC. We observe AMP, ADP, ATP and NADP. (3) The first experiment is added to a database and used to train a model, and from thereon, every new experiment added to the database serves as test data as the pipeline uses the other experiments as training data. (4) When the model has sufficient predictive power to control the catalytic processes inside the reactor, the active learning cycle stops.

We applied the workflow to two model ERNs in vitro, the pentose phosphate pathway (PPP; Fig. 1c) and a branched nucleotide salvage pathway (NSP; Extended Data Fig. 1) and achieved yield improvements of more than 5- to 21-fold, respectively, compared with control reactions, where all components were added simultaneously at t = 0. This work demonstrates that an understanding of emergent ERN dynamics enables the design of non-intuitive, time-dependent input profiles, which can overcome kinetic barriers and drive cascades towards higher product formation in batch mode.

Results and discussion

Learning ERN dynamics with optimally designed training data

Figure 1c shows one of the ERN cascades used in this study. The system contains five enzymes (hexokinase (HK), glucose-6-phosphate dehydrogenase (G6PDH), 6-phosphogluconic dehydrogenase (6PGDH), phosphoriboisomerase (PRI) and phosphoribosyl pyrophosphate synthetase (PRPPS)), two cofactors (nicotinamide adenine dinucleotide phosphate oxidized (NADP) and adenosine triphosphate (ATP)) and one substrate (glucose (GLU)). At the start, glucose is converted to glucose-6-phospate (G6P) by HK using ATP as a cofactor. G6PDH using NADP as a cofactor converts G6P to 6-phosphoglucono-δ-lactone (6PGL), which undergoes spontaneous hydrolysis to 6-phosphogluconate (6PGA). Subsequently, 6PGA is converted to ribulose-5-phosphate (Ru5P) by 6PGDH using NADP as a cofactor, which is reversibly isomerized to ribose-5-phosphate (Ro5P) by PRI. In the last step, Ro5P is converted to phosphoribosyl pyrophosphate (PRPP) by PRPPS using ATP as a cofactor (PRPPS and PRI were purified according to Arthur et al.54; for more information see Supplementary Section 2.3). The final product PRPP is the precursor for the nucleotide synthesis pathway ERN40 (described in Supplementary Section 4.1), as well as a building block for DNA and RNA. The observed intermediates were quantified using high-performance liquid chromatography (HPLC). Supplementary Section 2.1 (Supplementary Table 2 and Supplementary Figs. 2 and 3) shows the concentrations used for all reagents, as well as indicative yields for the final product.

We first validated that the PPP functions as a cell-free ERN in batch mode (Supplementary Fig. 2). Under static conditions, only ~12% of glucose was converted to final products, illustrating the limitations of fixed-input operation described above. The cascade involves cofactors that induce conformational changes necessary for substrate binding through ordered sequential bi–bi mechanisms55,56,57,58,59,60. As these cofactors are consumed and regenerated, their concentrations fluctuate, rendering reaction rates inherently time-dependent. Several steps are potentially reversible within the concentration ranges tested, further constraining overall flux. In addition, literature reports extensive allosteric regulation in this pathway, notably the inhibition of PRPPS by adenosine diphosphate (ADP), a feature conserved across organisms. Together, these factors create dynamic kinetic barriers that cannot be mitigated by static dosing at t = 0. Overcoming them requires a parameterized model capable of capturing the emergent, time-dependent behaviour of the network61 (Supplementary Section 3).

To obtain such a model, we applied an active learning workflow that integrates optimal experimental design (OED) with flow chemistry to systematically probe a wide range of reaction conditions. In this approach, a continuous stirred tank reactor (CSTR; Supplementary Figs. 4 and 5) is subjected to a programmed sequence of out-of-equilibrium perturbations. Enzymes, substrates and cofactors are continuously supplied through separate inlets while the mixed reaction medium is withdrawn at the same rate, ensuring steady operation. This set-up produces kinetic data that are maximally informative for model parameterization (Supplementary Section 3.1). As shown in Fig. 2, the process is iterative. The most recent experiment serves as test data, and the previous runs are used for training. The cycle concludes once a model can predict the test data.

An in silico analysis indicated that a 6-h optimally designed flow experiment was sufficient to parameterize a model (Supplementary Section 5.1 and Supplementary Figs. 15 and 16). Thus, we implemented the workflow experimentally, performing two >6-h flow runs as the first train–test cycle. In total, eight syringe pumps, each delivering one component—HK, G6PDH, 6PGDH, PRI, PRPPS, glucose, ATP or NADP—were connected to a 121-µl CSTR with a single outlet line (Supplementary Section 2.2 and Supplementary Figs. 4 and 5). Samples were collected sequentially, quenched and analysed offline by HPLC (Supplementary Section 1.3 and Supplementary Fig. 1).

The results of two flow experiments are summarized in Fig. 3, where Fig. 3a shows the training of the model and Fig. 3b shows a flow experiment designed to test the obtained model. Each panel represents one iteration and includes both the optimally designed inflow profiles (Fig. 3a,b, left) and the corresponding experimental time-series data (Fig. 3a,b, right). The training experiment included a 250-min manually designed segment as a control (Supplementary Fig. 7). Total flow rates ranged from 97–1,670 µl h−1 in the first iteration and 297–989 µl h−1 in the second, corresponding to residence times of 4–75 min and 7–24 min, respectively. To expand the explored input space and generate sufficiently complex test data, the stock concentrations of enzymes and substrates in each syringe were varied between the two iterations (Supplementary Section 4.1 and Supplementary Fig. 8, enzyme concentrations inside the reactor).

Fig. 3: Summary of the active workflow applied to the PPP network.
Fig. 3: Summary of the active workflow applied to the PPP network.The alternative text for this image may have been generated using AI.
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a, Training the model. Left: the inflow profile of the first iteration with the flow rates of each enzyme and substrate; the dashed black line represents the total flow rate. Right: data as measured using HPLC (black triangles): AMP, ADP, ATP and NADP. The convergence of the training process to this data is given by the mean (thick coloured line); the uncertainty of the fit is visualized by shaded area (s.d. of 2) utilizing the 25 best fits. For the flow per species, the actual concentrations of enzyme species are shown in Supplementary Fig. 8. b, Testing the model. Left: the 25 best fits from iteration one are used to predict the OED experiment performed for iteration two. Right: predictions of the time series data are visualized in an identical manner to a.

Source data

Because the relevant mechanistic features of the ERN were not known a priori, we used time-series data from the first experiment to train four ordinary differential equation (ODE) models, each incorporating slightly different regulatory assumptions. These models were then tested against data from the second optimally designed experiment. The best-performing model accurately captured both training and test data (R2 = 0.892) (Supplementary Figs. 9 and 10), indicating sufficient flexibility without overparameterization. During model selection, we confirmed that ADP inhibits PRPPS, as models lacking this interaction failed to converge as well (Supplementary Fig. 11). Adding additional activation or inhibition terms, or using Hill-type kinetics, did not improve the model (Supplementary Section 5.2 and Supplementary Figs. 18 and 19).

Trained models unlock time-dependent batch inputs

With an appropriately parameterized model, we switched to batch mode and tested the central hypothesis, namely, that that time-dependent reagent additions can overcome kinetic barriers and substantially improve ERN yields. To maintain favourable conditions throughout the reaction, the optimization was unconstrained in both timing and frequency of inputs, allowing new enzyme, cofactor or substrate additions every 5 min.

Figure 4a shows three optimized input sequences, each sampling different stock concentrations to expand the explored design space (Supplementary Section 4.2 and Supplementary Table 4). The y axis indicates the added volume and the x axis the time in minutes; each colour corresponds to a specific enzyme or substrate (HK, G6PDH, 6PGDH, PRI, PRPPS, glucose, ATP or NADP). These non-intuitive dosing schedules involve repeated additions of multiple components at varying times and amounts. Figure 4b compares model predictions (orange) with control reactions (green), where all reagents were added at t = 0. Both the simulations and experiments show substantial increases in final product concentrations. The shaded regions indicate prediction uncertainty (s.d. of 2). Figure 4c shows measured adenosine monophosphate (AMP) concentrations after 120 min, demonstrating up to a 5.6-fold improvement relative to the control (experiment 3). Finally, Fig. 4d reports overall yields, expressed as the percentage of glucose converted to AMP. The largest gain occurred in experiment 2, where yield rose from ~12% in the control to ~48% under optimized, time-dependent dosing.

Fig. 4: Results of time-dependent batch experiments.
Fig. 4: Results of time-dependent batch experiments.The alternative text for this image may have been generated using AI.
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a, Three time series showing each addition of input species within the ERN: HK, G6PDH, 6PGDH, PRI, PRPPS, ATP, NADP and GLU. The y axis shows the volume of the additions to the reactor in µl and the x axis the time of additions. The algorithm allows an input to be given every 5 min, and each experiment (1–3) samples from different stock solutions. b, The time series plots show the prediction of the model for the time-dependent batch experiment (orange) and its corresponding control (green), (n = 10, 95% confidence intervals). c, A bar plot of the concentrations in μM of one of the final products AMP as measured on the HPLC. The time-dependent batch experiments (orange) have substantially higher final AMP concentrations compared with the control (green) as predicted in b and final concentration with a 5.6-fold increase compared with the control for experiment 3 (number above orange bar). Additional control experiments show that the spontaneous ATP hydrolysis does not affect outcomes (Supplementary Fig. 13). d, The yields indicate that a notably larger percentage of the initial substrate, glucose, was converted to AMP in all experiments compared with the control (yield = 100 n(AMP)final/n(glucose)added). Notably, in experiment 2, nearly half of the input substrate was transformed into the product.

Source data

Optimally timed input sequences lower costs

In Fig. 5, we demonstrate flexibility by showing we can target cost in silico by optimizing input schemes that minimize the cost per unit of the other, more valuable output molecule: PRPP (AMP serves as an experimentally observable proxy). Figure 5a shows an in silico comparison between optimized time-dependent batch runs and equivalent controls by static dosing at t = 0. Each orange circle’s size reflects the fold change between the control and the optimized reaction. The y axis is the cost per milligram produced; the x axis is the final product concentration. On average, time-dependent dosing results in substantially lower costs (~17-fold, based on 440 samples of the input space; Fig. 5b). When we take the best-performing control versus the best-performing time-dependent experiment, we obtained a ~16-fold lower cost (Fig. 5c). We used Sigma-Aldrich price listings as a reference for this investigation (Supplementary Section 5.3).

Fig. 5: In silico analysis of batch mode optimizations for cost by Monte Carlo sampling input space.
Fig. 5: In silico analysis of batch mode optimizations for cost by Monte Carlo sampling input space.The alternative text for this image may have been generated using AI.
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a, A scatterplot of 440 in silico experiments: in orange, the optimized time-dependent inputs, and in green, the equivalent static dose control at t = 0. The size of the optimized dots represents the ratio difference between the control and the time sensitive dosing. The y axis represents the log-scaled cost per milligram unit in euros; the x axis shows the final concentration of the product AMP or PRPP once the reaction is at steady state. b, The average cost per milligram for the 440 time-dependent experiments and 440 equivalent controls. Error bars signify the standard error of the mean for optimizations performed for both the optimized and control experiments (n = 440). c, The lowest-cost experiment found in the best-performing time-dependent and best-performing control experiment.

Source data

We demonstrate the generalizability of our workflow by applying it to a second ERN: a branched NSP converting PRPP with adenine or uracil to ATP and UTP via five enzymes (APRT, UPRT, AK, PK and UMPK) and ten control inputs (Extended Data Fig. 1 and Supplementary Section 4.1). We first mapped the kinetics of this network (Extended Data Fig. 2) following the method outlined above. The trained model was used to generate a time-dependent protocols for batch mode experiments, balancing substrate inputs and cofactor availability. Experimental validation achieved a ~21-fold increase in UTP yield and PRPP conversion compared with the composition-matched all-at-once control used as an internal reference (Extended Data Fig. 3 and Supplementary Section 4.2). A negative-control experiment validated the model’s predictive power. This test case demonstrates that this workflow effectively transfers to branched, competing networks and that time-dependent control can guide these ERNs through different productive states.

Conclusion

Optimizing the performance of complex ERNs remains challenging because hidden interactions and unfavourable kinetic barriers often emerge once enzymes operate together in a network. By introducing time as an explicit dimension in the optimization process, we expand the accessible design space by many orders of magnitude beyond traditional static approaches that only vary initial conditions. For example, assuming eight components each dosed between one and ten arbitrary levels yields 108 possible experiments. When time-dependent dosing is introduced by allowing 24 sequential additions (one every 5 min, as in Fig. 4a), this space grows to 10192 possible input sequences. Within this vast space, non-intuitive dosing trajectories exist that can substantially increase product yield, yet exhaustive experimental exploration is infeasible, particularly when enzymes and cofactors are costly or available in limited amounts.

To access this space efficiently, a predictive model is essential. We therefore used an active learning workflow that designs maximally informative pulse experiments, enabling rapid model parameterization within a single OED iteration. Although a flow reactor provides an ideal platform for these experiments owing to its ability to support continuous perturbations, the approach can, in principle, be adapted to other reactor formats if sufficient dynamic and multi-species time series data can be acquired. This means both the kinetic characterization and production could be done in batch mode.

The resulting model captures the key sensitivities governing ERN dynamics and allows systematic exploration of time-dependent input strategies. Using this model, we designed batch mode addition schedules that minimize inhibitory interactions and shift reversible reactions towards higher conversion. This strategy improved product yields for two ERNs between 5- and 21-fold, respectively, relative to conventional static dosing, which served as an internal benchmark. Although our experiments focused on maximizing total yield, we also demonstrated in silico that the same framework can accommodate multiple objectives by identifying a route that would yield products at 16-fold lower cost.

Overall, our results demonstrate that model-guided, time-dependent input design can overcome kinetic bottlenecks inherent to complex biochemical networks. Because our workflow is data-driven and mechanistic rather than pathway-specific, it is in principle network-agnostic. As predictive modelling and automated experimentation continue to advance, such time-dependent ‘recipes’ are poised to unlock most efficient and cost-effective routes to value-added compounds.

Methods

Microfluidic reactor and control unit

A CSTR (volume of 121 µl for the first and second iteration of PPP experiments and 158 µl for the NSP experiment) made of poly(methyl methacrylate) equipped with four or five inlets, which were doubled to eight or ten using Y connectors. The outlet tubes differed between experiments and had their own volume (109 µl and 69 µl for the first and second PPP experiments and 69 µl for NSP experiment, respectively) (Supplementary Figs. 4 and 5). Enzyme and substrate input flow rates were controlled by Cetoni neMESYS syringe pumps and Hamilton syringes and changed every 12 or 15 min. The ranges of the lowest and highest flow rates for the first and second PPP flow experiments and NSP experiment were 97–1,670, 297–988 and 97–1,670 µl h−1, respectively. The ranges of residence time for the first and second PPP flow experiments and NSP experiment were 4–75, 7–24 and 6–98 min, respectively. Reactor content was sampled by collecting droplets at the outlet (39 µl and 29 µl for the first and second PPP flow experiments and 27 µl for the NSP experiment, respectively) using a fraction collector, and the time of each drop was recorded using an infrared (IR) detector (Supplementary Fig. 5). The IR sensor was fitted to the fraction collector head to precisely monitor the time drops were collected; the control software can be found at ref. 62. PPP samples from the CSTR outlet tube were dropped directly into a guanidinium chloride solution (8 M, 117 µl and 87 µl for the first and second PPP flow experiments, respectively), where the sample-to-guanidinium-chloride ratio was 1:3, and quenched immediately. The NSP experiment samples were dropped directly into a formic acid solution (30 V%, 3 µl), where the sample-to-formic-acid ratio was 9:1, and quenched immediately. Quenched samples were centrifuged for 5 min at 6,708g, the supernatant (30 µl for PPP samples, 20 µl for NSP samples) was transferred to the micro insert of HPLC vial and measured by HPLC. PPP samples were measured by HPLC method 1, whereas NSP samples were measured by method 2. See Supplementary Section 2.2 for more information about the flow set-up, whereas stock concentrations for flow experiments can be found in Supplementary Section 4.1.

HPLC measurements

HPLC analyses were performed on Shimadzu Nexera X3 instrument. The conditions were: Inertsil ODS-4 C18 column (3-μm pore size, 150 mm × 4.6 mm; GL Science) and a guard column (3-μm pore size; 10 mm × 4.6 mm) at 40 °C. Analysis was based off of ref. 55.

Method 1 for PPP samples: we prepared elution solution A as 50 mM aqueous potassium phosphate (pH 6.0) buffer filtered over a 0.22-µm membrane and solution B as 100% MeOH. The method 1 elution gradient was as follows: 100% solution A for 2 min, 0–12.5% linear gradient of solution B for 8 min, 12.5% solution B for 2 min, 12.5–40% linear gradient of solution B for 6 min, 40% solution B for 2 min, 40–0% linear gradient of solution B for 3 min and 100% solution A for 7 min. The flow rate was maintained at 1 ml min−1; the injection volume was 10 µl, 15 µl and 20 µl for the batch, first and second flow experiments, respectively. The ultraviolet detection was at 254 nm.

Method 2 for NSP samples: we prepared elution solution A as 100 mM aqueous potassium phosphate buffer (pH 6.4) with 8 mM ion-pair reagent tetrabutylammonium bisulfate filtered over a 0.22-µm membrane and solution B as 70% solution A with 30% acetonitrile. The method 2 elution gradient was as follows: 100% solution A for 13 min, 0–77% linear gradient of solution B for 22 min, 77–100% solution B for 1 min, 100% solution B for 15 min and 100% solution A for 4 min. The flow rate was maintained at 1 ml min−1; the injection volume was 3 µl and 1 µl for the batch and flow experiments, respectively. The ultraviolet detection was at 254 nm.

See Supplementary Section 1.3 for more information on the HPLC and detected compounds.

Plasmid preparation

  1. (1)

    PRI

    The gene for PRI was amplified from Escherichia coli K12-derived XL-1 blue competent cells using primers (bold and underline are the restriction sites used for cloning into the pET28a expression vector):

    RpiA-Fw: 5′TTAGTCAGTCATATGACGCAGGATGAATTGAAAAAAGCAGTAGG3′

    RpiA-Rv: 5′ATAGATAGCCTCGAGTCATTTCACAATGGTTTTGACACCGTCAG3′

    DNA oligonucleotides were from Integrated DNA Technologies, standard desalted.

  2. (2)

    PRPPS

    The plasmid for PRPPS originates from ref. 63.

  3. (3)

    UMPK

    The gene for UMPK was amplified from Escherichia coli K12-derived XL-1 blue competent cells using primers:

    UMPK-Fw: 5′GCATGACGTAGTACATATGGCTACCAATGCAAAACCCGTCTATAAACGC3′

    UMPK-Rv: 5′ACTGGTACGCCTCGAGTTATTCCGTGATTAAAGTCCCTTCTTTTTCACCC3′

    DNA oligonucleotides were from Integrated DNA Technologies, standard desalted.

  4. (4)

    APRT and UPRT

The plasmids for enzymes UPRT and APRT were a kind gift from Arthur and Dayie’s lab and originate from ref. 54.

Materials can be obtained via contact with the corresponding authors.

See Supplementary Section 2.3 for more information on the plasmid preparation.

Preliminary batch reactions

In a batch experiment, all five PPP network enzymes (HK, G6PDH, 6PGDH, PRI and PRPPS) together with substrate glucose and cofactors ATP and NADP were mixed in a buffer (100 mM HEPES, 10 mM MgCl2 and 1 mM CaCl2, pH 7.8) and left to react for 120 min at 30 °C while shaking. At every 0, 5, 10, 20, 40, 60 and 120 min reaction timepoint a sample (20 µl) was taken from each reaction tube, quenched using guanidinium chloride (8 M, 60 µl), where the sample-to-guanidinium-chloride ratio was 1:3, and vortexed. After this, all samples were centrifuged for 5 min at 6,708g, and the supernatant (30 µl) was analysed offline by HPLC. Conditions and results of preliminary batch reactions can be found in Supplementary Section 2.1.

Time-dependent batch experiments

Optimized time-dependent input experiments and control experiments were performed in batch conditions, where all inputs of control experiments were added at t = 0, whereas in optimized experiments, inputs were added at specific times and volumes, as shown in Fig. 4a for PPP and Extended Data Fig. 3 for NSP. Enzyme and substrate stock concentrations are presented in Supplementary Table 4. At 120 and 150 min (PPP experiments 1–3), 370 min (NSP experiment 1) and 355 min (NSP experiment 2) reaction timepoints samples from control and optimized reaction tubes were taken, quenched and measured as described in Supplementary Section 1.3. Conditions for time-dependent batch experiments as well as enzyme and substrate stock concentrations can be found in Supplementary Section 1.2.

See Supplementary Section 3.1 for a comprehensive description of the optimization procedure. See Supplementary Sections 3.2 and 3.3 for more information about the algorithm and software. See Supplementary Section 5 for an additional analysis to gauge the utility of the optimal design.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.