Elucidating the Bell-Shaped Dependence of Protein Translation Activity on EF-Tu Concentration in a Reconstituted Cell-Free System Using a Mechanistic Model

Abstract

Protein synthesis in cell-free protein synthesis systems often exhibits nonintuitive input–output relationships. In the PURE system, a reconstituted cell-free system, protein production peaked at low elongation factor Tu (EF-Tu) concentrations and decreased at higher concentrations, resulting in a characteristic bell-shaped profile. Here, we investigated the origin of this behavior using a detailed mechanistic model of translation in the PURE system, designated as ePURE, which describes the reaction dynamics of hundreds of molecular species and reactions. Our computational analysis suggested that excess EF-Tu sequesters the initiator tRNA (tRNA^fMet) into nonproductive EF-Tu·GTP·Met-tRNA^fMet complexes, thereby depleting the pool of initiator tRNA available for translation initiation. This suppression arises from competition for a limited molecular resource rather than from direct inhibition. Based on this mechanism, we predicted that increasing the concentrations of tRNA^fMet and methionyl-tRNA formyltransferase would eliminate the bell-shaped dependence, and experimentally confirmed this prediction. Under these modified conditions, the bell-shaped response disappeared and protein production was enhanced. These findings demonstrate how mechanistic computational models can reveal hidden constraints underlying nonintuitive input–output relationships in complex biochemical networks and guide the rational optimization of cell-free protein synthesis systems.

1. Introduction

Cells carry out essential processes such as transcription, translation, and metabolism through intricate networks of chemical reactions. These reactions are precisely regulated and flexibly adjusted in response to environmental cues, including nutrient availability, temperature, and stress. − This regulation is inherently difficult because many molecules participate simultaneously in multiple reactions, creating competition and interdependence among pathways. Consequently, increasing the supply of a substrate or cofactor does not necessarily enhance the production of a desired molecule if the same input is required by competing reactions, or if downstream products inhibit the target pathway. , These nonintuitive input–output relationships highlight the complexity of biochemical networks, in which shared substrates and products can lead to unexpected outcomes. Understanding these dynamics is crucial for elucidating the robustness and adaptability of living systems and for advancing synthetic biology approaches that reconstruct and engineer complex cellular functions in vitro.

Experimental analysis of input–output relationships in living cells remains challenging. In vivo systems contain numerous interacting components, including unidentified ones, which obscure the specific reactions or parameters that govern the behaviors of interest. To address this limitation, we employ an Escherichia coli-based reconstituted cell-free translation system, known as the PURE system. This system consists of a set of purified factors that are sufficient for efficient protein synthesis. Experimentally, the PURE system enables quantitative exploration of input–output relationships by varying the initial concentrations of components (inputs) and measuring peptide production (outputs). Previous studies that systematically altered the initial concentrations of PURE components reported several nonintuitive production profiles, including bell-shaped dependencies in which peptide production increases with a factor’s concentration up to a certain threshold and then decreases. Such bell-shaped dependencies are characteristic signatures of competition and inhibition within complex reaction networks.

Unraveling the detailed mechanisms underlying these input–output relationships requires extensive experiments under diverse conditions along with time-resolved measurements of both final products and intermediate species. However, experimentally manipulating reaction parameters and monitoring time-series data for all components is extremely demanding. In such cases, computational modeling offers a powerful alternative. − For the PURE system, a detailed mechanistic model of the PURE system, which we refer to here as ePURE, has been developed. It comprises hundreds of components and reactions and simulates the translation process by using ordinary differential equations. This model allows the time evolution of all components in the PURE system to be calculated, making it possible to systematically investigate how variations in initial concentrations influence translation outcomes. −

In this study, we aim to elucidate the mechanisms behind complex input–output relationships in translation systems, focusing on the bell-shaped dependence of peptide production on the input concentration of the elongation factor Tu (EF-Tu). EF-Tu is essential for the peptide elongation step, yet our PURE system experiments revealed a pronounced bell-shaped dependence of protein production on the EF-Tu concentration that lacked a clear mechanistic explanation. Because the original ePURE model could not reproduce this phenomenon quantitatively, we first refined the kinetic parameters by fitting the model output to the experimental data. Analysis of the adjusted parameters allowed us to identify the reaction network responsible for the bell-shaped dependence and revealed a strategy for eliminating it. Guided by this finding, we designed an experimental setup to modulate this dependence and successfully eliminated the bell-shaped response. Additionally, the peptide production further increased owing to the elimination of the bell-shaped dependence. These results demonstrate that mechanistic computational models can not only explain counterintuitive input–output relationships in complex biochemical networks but also guide rational redesign of cell-free systems to achieve desired performance.

2. Results and Discussion

2.1. Nonmonotonic EF-Tu Dependence of Peptide Synthesis in the PURE System

We quantified how the amount of peptide production depends on the initial EF-Tu concentration in the PURE system. A reaction mixture containing 67 defined components was reconstituted, and only the initial EF-Tu concentration was varied across five levels (0, 5, 25, 50, and 65 μM), with all other components held constant (see Methods and Table S1). Green fluorescent protein (GFP) (sequence given in Table S2) was used as a reporter, and its fluorescence increase was recorded over time. To compare the experimental data with the ePURE model, which explicitly describes the synthesis of the Met-Gly-Gly tripeptide, we converted the GFP fluorescence intensity into the corresponding amount of synthesized peptide (details in Methods).

As expected, no peptide was produced in the absence of EF-Tu, which is essential for the transfer of aminoacyl-tRNAs (aa-tRNAs) to the ribosomal A site. However, when EF-Tu was supplied at concentrations above 5 μM, the peptide production decreased as EF-Tu concentration increased (Figure A). To clarify this trend, the amount of synthesized peptide at the end point (1,560 s) was plotted against the initial EF-Tu concentration (Figure B). Although increasing EF-Tu would normally be expected to enhance translation until another component becomes rate-limiting, our experiments showed the opposite: higher EF-Tu concentrations reduced the amount of peptide production. Rather than approaching saturation, the EF-Tu dependence of the peptide synthesis exhibited a bell-shaped profile. This behavior provides a counterintuitive example of a nonmonotonic input–output relationship in a biochemical network and forms the central focus of the present study.

1.

Experimental results of the effect of initial EF-Tu concentration on peptide synthesis. (A) Time course data for each initial EF-Tu concentration. (B) Peptide concentrations at the end point (1,560 s). Error bars represent the standard deviation of four replicate experiments (n = 4).

2.2. Mechanism of Bell-Shaped Behavior: Excess EF-Tu Depletes tRNA^fMet

To uncover the mechanism underlying the bell-shaped dependence observed in Figure , we first sought to reproduce the experimental results using numerical simulations of the PURE system. The ePURE model represents translation in the PURE system as a set of ordinary differential equations comprising 205 variables for molecular species and their complexes and 481 reactions, enabling the calculation of the time evolution of all component concentrations. Because the original literature-derived kinetic parameters failed to reproduce the experimental behavior shown in Figure , we optimized the kinetic parameters using a genetic algorithm (GA). The following analyses are based on a representative parameter set obtained by this fitting procedure (see the Methods section for details).

Using this fitted parameter set, we investigated the mechanism underlying the bell-shaped dependence. The PURE system comprises a complex reaction network, which makes it difficult to dissect the mechanism of the bell-shaped dependence. Figure A presents a schematic of the translation subnetwork involving EF-Tu, outlining the relevant reactions in the initiation and elongation steps.

2.

Mechanism underlying the bell-shaped dependence of the peptide production on EF-Tu concentration. (A) Schematic illustration of reactions in protein translation. For initiation, the methionine corresponding to the start codon is formylated, a reaction catalyzed by MTF, in which Met-tRNA^fMet receives a formyl group from 10-formyltetrahydrofolate (FD), a tetrahydrofolate derivative (THF). − Subsequently, it binds to IF2·GTP to form the IF2·GTP·fMet-tRNA^fMet complex, which is used for initiation. For elongation, EF-Tu plays a central role by delivering aminoacyl-tRNA (aa-tRNA) to the ribosome. − When EF-Tu·GTP binds Met-tRNA^fMet, the resulting EF-Tu·GTP·Met-tRNA^fMet complex (indicated by the dashed square) can no longer participate in elongation or initiation. (B) Numerical analysis of how increasing the initial concentration of each component affects the EF-Tu dependence of the peptide production. Each line shows the EF-Tu dependence obtained when the initial concentration of a single component was increased by 1 μM. The thick black line represents the result using the original initial concentrations. The thick blue and red lines show all the results obtained when the initial concentrations of the tRNA^fMet-containing components and MTF-containing components were increased, respectively. Kinetic parameters were taken from a representative parameter set obtained by GA fitting. This trend is not limited to the 1 μM increment shown here; consistent results were obtained across other concentration increments (see Figure S1 for results with increments ranging from 0.05 μM to 5 μM).

We hypothesized that increasing the initial EF-Tu concentration depletes specific components, and that exogenous supplementation of the depleted components would modulate the EF-Tu dependence of the peptide production. To test this, we performed simulations in which the initial concentration of each component was increased by 1 μM individually, allowing us to identify which components influence the bell-shaped response.

Figure B shows how the EF-Tu dependence of the peptide production at 1,560 s changed upon increasing the initial concentration of each component. Relative to the original bell-shaped curve shown by the thick black line, many components produced noticeable deviations, although many of the resulting curves remained bell-shaped. Among all components, we focused on cases in which the bell-shaped dependence disappeared (e.g., thick blue or red curves). Components whose elevation eliminated the bell-shaped dependence clustered within a specific region of the reaction network; when we increased the initial concentration of complexes that contain tRNA^fMet, the EF-Tu dependence changed markedly (thick blue lines). Increasing complexes that include methionyl-tRNA formyltransferase (MTF) also abolished the bell-shaped curve (thick red lines). These results indicate that initiation reactions involving tRNA^fMet, as shown in Figure A, are crucial for the observed dependence. Similar results were also obtained with other concentration increases (Figure S1).

EF-Tu delivers aa-tRNA to the ribosome during the elongation step, but EF-Tu can also bind Met-tRNA^fMet, the initiator tRNA. The EF-Tu·GTP·Met-tRNA^fMet complex indicated by the dashed square in Figure A is functionally inert in both initiation and elongation. Because no tRNA synthesis occurs in the PURE system, the total concentration of tRNA^fMet does not change. As EF-Tu increases, more Met-tRNA^fMet becomes sequestered in this nonproductive complex, reducing the Met-tRNA^fMet available for formylation and initiation.

A bell-shaped dependence often implies that increasing an input factor initially promotes the reaction but subsequently suppresses it. In this system, the suppression does not arise from direct inhibition by EF-Tu. Instead, it emerges from competition for a limited resource: tRNA^fMet. At high EF-Tu concentrations, the majority of Met-tRNA^fMet is trapped in EF-Tu·GTP·Met-tRNA^fMet, leaving insufficient fMet-tRNA^fMet for initiation. Thus, the decline in peptide synthesis reflects competition between elongation-related binding of EF-Tu and initiation-specific processing of the initiator tRNA, rather than the inhibitory activity of EF-Tu itself.

To evaluate this shortage more directly, we examined the simulated concentrations of complexes involving tRNA^fMet. Because tRNA is not synthesized de novo in this system, the total amount of tRNA^fMet across free and bound states remained constant at 0.194 μM. Because these transient complexes influence the peptide synthesis rate at any given moment, their end point concentrations do not reflect their overall effect. Therefore, we evaluated their time-averaged concentrations over the entire simulation period (0–1,560 s) across different EF-Tu initial concentrations (Figure ). As the initial EF-Tu concentration increased, EF-Tu·GTP·Met-tRNA^fMet (yellow region in Figure ) came to dominate the total pool of tRNA^fMet-related complexes, whereas IF2·GTP·fMet-tRNA^fMet (blue region) decreased to nearly zero. Because IF2·GTP·fMet-tRNA^fMet is the immediate precursor that engages the ribosome during translation initiation, its near depletion indicates that initiation becomes rate-limiting under high EF-Tu conditions (see Figure S2). Importantly, this mechanism was not specific to the representative fit shown here. Across all GA-fitted trials, excess EF-Tu consistently promoted the formation of EF-Tu·GTP·Met-tRNA^fMet complexes (Figure S3), depleted free initiator tRNA, and reduced initiation at higher EF-Tu concentrations.

3.

Time-averaged concentrations of tRNA^fMet-containing complexes as a function of initial EF-Tu concentration. The stacked area plot illustrates the shifting distribution of these complexes. As the initial EF-Tu concentration increases, EF-Tu·GTP·Met-tRNA^fMet (yellow) becomes dominant, whereas IF2·GTP·fMet-tRNA^fMet (blue), which is required for translation initiation, decreases.

We further tested whether increasing the initial tRNA^fMet concentration would suppress the bell-shaped dependence. To simulate this, we increased only the initial tRNA^fMet to 1.94 μM. Upon increasing tRNA^fMet, the decrease in the peptide production at higher EF-Tu concentrations disappeared, and the production remained comparable to that at 5 μM EF-Tu (Figure A). Consistently, the EF-Tu·GTP·Met-tRNA^fMet complex no longer dominated the total tRNA^fMet pool, and the concentration of the IF2·GTP·fMet-tRNA^fMet complex was stably maintained across the tested range of EF-Tu concentrations (Figure B). These findings indicate that, even under excess EF-Tu, a sufficient amount of fMet-tRNA^fMet can be maintained to support initiation, thereby preventing translation inhibition. Taken together, the ePURE simulations corroborate the proposed mechanism.

4.

Simulation results with an increased initial tRNA^fMet concentration. (A) Peptide concentrations at end point (t = 1,560 s) with increased tRNA^fMet concentration to 1.94 μM (from the original 0.194 μM). The inset shows the time course of peptide concentration under these conditions. (B) Time-averaged concentrations of tRNA^fMet-containing complexes as a function of initial EF-Tu concentration, with tRNA^fMet initial concentration increased 10-fold. Even with increasing initial EF-Tu concentration, the IF2·GTP·fMet-tRNA^fMet complex, which is required for translation initiation, remains stable.

2.3. Converting Bell-Shaped EF-Tu Dependence to a Monotonic Saturation Curve

Using our computational model of the PURE system, we identified a plausible mechanism underlying the bell-shaped EF-Tu dependence. This analysis further suggested a strategy to convert the bell-shaped dependence into a monotonic saturation curve, which we then tested experimentally.

We hypothesized that the bell-shaped profile arises from a shortage of Met-tRNA^fMet, caused by the sequestration of Met-tRNA^fMet in the EF-Tu·GTP·Met-tRNA^fMet complex. Increasing the initial concentration of tRNA^fMet was therefore expected to mitigate this shortage and suppress the bell-shaped dependence. In parallel, raising the concentration of MTF was expected to enhance the formylation of Met-tRNA^fMet before its interaction with EF-Tu, thereby reducing the accumulation of the nonproductive EF-Tu·GTP·Met-tRNA^fMet complex.

Based on this hypothesis, we examined EF-Tu concentration dependence experimentally under conditions with elevated initial concentrations of both tRNA and MTF. Specifically, the MTF concentration was increased from 0.039 to 0.588 μM, and tRNA from 0.194 to 0.778 μM, while all other components were kept identical to those in Figure (see Methods and Table S1). These concentrations of tRNA and MTF were chosen such that further increases no longer enhanced the amount of peptide synthesis, as determined from titration experiments.

Simulations of the ePURE model using the modified initial concentrations and the fitted parameters described above reproduced the disappearance of the bell-shaped EF-Tu dependence (Figure S4A). Further analysis of the simulations suggested that under these modified conditions, mRNA availability can limit further increases in peptide production under the high EF-Tu condition (Figure S4B). Consistently, experimental measurements using the PURE system yielded no bell-shaped profile under the modified conditions (Figure ). These results support the proposed mechanism and suggest that the origin of the bell-shaped EF-Tu dependence identified in ePURE may also operate in the PURE system.

5.

Effect of increasing the tRNA and MTF concentrations on peptide synthesis in the PURE system. Compared with the original experimental results (black; identical to those shown in Figure B), the PURE system supplemented with increased concentrations of both tRNA and MTF (green) no longer exhibits the bell-shaped dependence. Error bars represent the standard deviation of four replicate experiments.

3. Conclusion

Using the mechanistic model ePURE with kinetic parameters fitted using a GA, we reproduced the experimental time courses of the PURE system. Analysis of these simulations suggested the mechanism underlying the bell-shaped dependence of the peptide production on EF-Tu concentration: excess EF-Tu drives the formation of a nonproductive EF-Tu·GTP·Met-tRNA^fMet complex, thereby depleting the pool of initiator tRNA available for translation initiation. Guided by this insight, we experimentally altered the concentrations of tRNA^fMet and MTF, which eliminated the bell-shaped dependence as predicted.

A central challenge in interpreting complex biochemical systems is that input–output relationships often emerge from distributed and competing interactions rather than from a single rate-limiting step. Even for a well-characterized factor such as EF-Tu, increasing its concentration does not guarantee an increased translational output because numerous reactions share substrates, cofactors, and intermediate complexes. Such interdependencies make it difficult to infer causal mechanisms solely from perturbation experiments, particularly in dense reaction networks such as those of PURE or living cells.

In this context, a mechanistic framework such as ePURE is essential for disentangling the causal architecture of the system. By explicitly representing all relevant reaction pathways, the model enables a systematic evaluation of how perturbations propagate through the network. Our ability to identify tRNA^fMet sequestration as a plausible explanation for the bell-shaped response illustrates its strength. Mechanistic modeling thus provides a powerful complement to experimental approaches when interpreting nonintuitive behaviors in cell-free and cellular translation.

Beyond explaining a specific phenomenon, this study highlights a general design principle for the engineering of cell-free systems. The assumption that elevating the concentration of a single translation factor will monotonically enhance the performance does not necessarily hold in complex biochemical networks. Competition for shared molecular resources or the formation of nonproductive complexes can generate unexpected nonmonotonic outcomes. Identifying components that participate in multiple competing pathways is therefore crucial for the rational optimization of translational efficiency.

Despite the successes of the present approach, several technical limitations point toward future studies and model development. In this study, our analysis focused primarily on the decrease in peptide production observed at high EF-Tu concentrations. The model and supplementation experiment support the idea that this decline is caused, at least in part, by the sequestration of initiator tRNA into a nonproductive EF-Tu·GTP·Met-tRNA^fMet complex. However, the present experiments do not quantitatively determine how the peak position of the EF-Tu response shifts with intermediate tRNA or MTF concentrations. Such peak-position control may depend on additional kinetic and resource-allocation constraints and will require systematic titration experiments and further model refinement. In addition, simulations were performed using a short model peptide; extending the framework to longer proteins will require explicit treatment of codon-dependent elongation rates, aa-tRNA demand, and context-dependent ribosomal pausing. Moreover, the current version of ePURE does not incorporate environmental factors such as temperature, magnesium concentration, pH, or ionic strength, all of which strongly influence translation in the PURE system. These omitted factors may also contribute to the remaining discrepancies between simulations and experiments, even when the qualitative behavior is captured. While incorporating these diverse effects leads to an enormously expanded parameter space, data-driven approaches such as active learning can efficiently guide the parameter exploration and optimization. Ultimately, incorporating these factors, as well as stochastic fluctuations inherent to systems consisting of molecules with low copy numbers, will be essential for achieving a more complete and predictive description of cell-free translation dynamics, particularly in confined cellular environments.

4. Methods

4.1. Quantification of Protein Synthesis by the PURE System

The components constituting the PURE system were prepared essentially as described previously. In brief, proteins were purified via a histidine tag using affinity chromatography, and the ribosome was purified using hydrophobic interaction chromatography and ultracentrifugation. The purchased chemicals with commercial sources and their final concentrations are given in Table S1.

The sequence of plasmid DNA encoding GFP is given in Table S2. Template DNAs for mRNA synthesis were prepared by PCR using KOD Plus Neo (TOYOBO). The PCR products were purified with the NucleoSpin Gel and PCR Clean-up kit (Macherey-Nagel) according to the manufacturer’s protocols. mRNA synthesis was carried out using the ScriptMAX Thermo T7 Transcription kit (TOYOBO) at 37 °C for 2 h. After template DNA digestion with DNase I, mRNA was purified with the NucleoSpin RNA Clean-up kit (Macherey-Nagel) according to the manufacturer’s protocols. mRNA concentrations were determined by absorbance measurement at 260 nm.

GFP fluorescence was monitored in real time to quantify protein synthesis in the PURE system reactions. GFP fluorescence measurements were performed using a real-time PCR machine (Mx3005P, Agilent). Reactions were done at 20 μL per well and incubated at 37 °C throughout the measurement. 40 nM Alexa647 was incorporated to cancel the well-to-well variation. RNase inhibitor was added to a final concentration of 0.4 U/μL. GFP fluorescence was recorded at 1 min intervals using an excitation wavelength of 492 nm and an emission wavelength of 516 nm. For each reaction condition, four technical replicates were measured, and the mean fluorescence intensity was used for downstream analysis.

4.2. Conversion of GFP Fluorescence to Peptide Concentration

To compare the experimental fluorescence trajectories with the ePURE simulations, GFP fluorescence was converted into the concentration of the Met-Gly-Gly tripeptide. Fluorescence intensities were first converted to GFP concentrations using a calibration curve generated from purified GFP standards. Because the experiments were performed on two different days with two technical replicates each, interday variability was corrected by scaling the fluorescence traces so that the fluorescence values at the EF-Tu concentration of 65 μM matched across days.

The fluorescence signal corresponds to the concentration of folded GFP, whereas the ePURE model does not include folding kinetics. Therefore, we corrected for GFP maturation using a first-order folding model. The folded GFP concentration f ^model[dp/dt]­(t) was assumed to satisfy the following relationship as a functional defined by the time derivative dp/dt of the prefolded concentration p(t):

$$f^{\mathrm{model}}\left[\frac{\mathrm{d}p}{\mathrm{d}t}\right](t)={\int }_0^t\mathrm{d}{t}^{\prime }\frac{\mathrm{d}p}{\mathrm{d}t}(t^{\prime })(1-e^{-k(t-t^{\prime })})$$ 1

In this equation, the folding rate constant k was set to 1/7 s^–1 based on reported GFP maturation rates. , The prefolded GFP production rate per unit time dp/dt was estimated using Bayesian inference so that the forward model best matched the experimental folded GFP time series. A non-negative uniform prior was placed on dp/dt, while the likelihood L[dp/dt] incorporated Gaussian errors and a smoothness constraint penalizing large second derivatives of p(t):

$$\log L\left[\frac{\mathrm{d}p}{\mathrm{d}t}\right]=-\int \mathrm{d}{t}^{\prime }{(f^{\mathrm{model}}\left[\frac{\mathrm{d}p}{\mathrm{d}t}\right](t^{\prime })-f^{\exp }(t^{\prime }))}^2-\alpha \int \mathrm{d}{t}^{\prime }{\left(\frac{{\mathrm{d}}^{2}p}{\mathrm{d}{t}^2}(t^{\prime })\right)}^2$$ 2

f ^exp(t) is the time-series data obtained experimentally. A smoothness constraint and non-negativity were applied to prevent unrealistic fluctuations in p(t). The smoothness strength α was set to 1. Slice sampling generated posterior means from 1,000 MCMC samples (1,000 burn-in) per replicate/EF-Tu condition. By integrating the dp/dt, the prefolded GFP concentration p(t) was obtained.

To relate prefolded GFP production to the synthesis of the 3-amino-acid peptide simulated in ePURE, we assumed that the translation rate is proportional to peptide length. The reconstructed prefolded GFP concentration was therefore multiplied by 238/3, corresponding to the ratio of the amino acid length of GFP (238 residues) to that of the tripeptide. This conversion was applied independently to all four fluorescence trajectories, and the mean trajectory was used as the experimental reference for the kinetic parameter fitting. The experimental data presented in this paper represent the concentrations equivalent to the prefolded tripeptide converted using the method described in this subsection.

4.3. ePURE: Computational Model of the Translation System

We used a previously developed deterministic ODE model of peptide synthesis based on components of the E. coli-derived in vitro translation system (PURE system). The model describes the synthesis of the formyl-Met-Gly-Gly (MGG) tripeptide using 241 molecular species in total, including 27 species present initially, and 968 reactions. All of the kinetic parameters were obtained from the literature. The initial component concentrations and kinetic parameters used in this study are listed in Tables S3 and S4, respectively.

In the original ePURE model, 241 molecular species and 968 reactions were included. We removed reactions whose kinetic parameters were set to zero in the original implementation, resulting in a reduced network containing 481 reactions and 205 components. This modification preserved the simulation results exactly while reducing the computation time.

4.4. Parameter Fitting

As shown in Figure S5B, the original ePURE parameter set did not reproduce the experimental results shown in Figure . We reasoned that the discrepancy arose because the default kinetic parameters, previously collected from the literature, were measured in highly purified forms and under conditions different from the PURE system and may not necessarily capture the complexity of the PURE system.

To reproduce the experimental behavior, we modified the kinetic parameters of the ePURE. Because previous studies suggested that varying individual parameters has only limited effects on the translation rate, we used a GA to estimate parameter sets that reproduce the experimental time courses. GAs efficiently explore high-dimensional parameter spaces and are less prone to local optima than gradient-based methods in nonlinear dynamical systems with many variables and parameters. In the GA, each individual represents a set of kinetic parameters, which were iteratively mutated and selected according to the objective function. Details of the GA procedure are described as follows.

4.4.1. Objective Function of GA

The objective function to be minimized is defined as the sum of the squared errors between experiments and simulations and a penalty term:

$$\mathrm{\Delta }=\sum _{\mathrm{EF‐Tu}}{\mathrm{\Delta }}_{\mathrm{EF‐Tu}}+\lambda \sum _{i\in \mathrm{reaction\; group}}|x_i|$$ 3

$${\mathrm{\Delta }}_{\mathrm{EF‐Tu}}=\sum _t{(C_{\exp }^{\mathrm{EF‐Tu}}(t)-C_{\mathrm{sim}}^{\mathrm{EF‐Tu}}(t;\mathrm{k⃗}))}^2$$ 4

Here, t takes every 120 s from 120 up to 1,560 s and EF-Tu takes 5, 25, 50, and 65 μM, respectively, matching the experimental sampling excluding 0 μM where the simulation result is trivially zero. The vector k = (k ₁, k ₂, ···, k _N ) represents the set of N kinetic parameter values, and x _i represents the logarithm of the change in k _i as described in the following subsection. $C_{\exp }^{\mathrm{EF‐Tu}}(t)$ is the experimentally measured peptide concentration under each initial EF-Tu condition, and $C_{\mathrm{sim}}^{\mathrm{EF‐Tu}}(t;\mathrm{k⃗})$ is the corresponding simulated concentration computed using the parameter values k.

The first term quantifies the agreement between simulation and experiment. The second term penalizes deviations from the original parameter values to maintain biological plausibility. The penalty coefficient was set to λ = 1, chosen to be sufficiently large without substantially degrading fitting performance.

4.4.2. Mutation to Kinetic Parameter Values

The ePURE model has the original set of kinetic parameters collected from the literature. Using these values as the initial condition for GA fitting, we introduced mutations according to

$$k_i=k_i^{\mathrm{ori}}\times {10}^{x_i}$$ 5

For each reaction i, the kinetic parameter k _i is determined by a deviation x _i from the original parameter $k_i^{\mathrm{ori}}$ . Because kinetic parameters vary widely and often follow a log-normal distribution in biological systems, deviations were applied on a logarithmic scale.

To ensure biologically reasonable parameters, we imposed the following constraints: First, for reversible reactions, both forward and backward rate constants were changed by the same factor, preserving the equilibrium constant and associated chemical potentials. Second, reactions sharing the same mechanism and default kinetic parameter values were mutated identically to preserve their equivalence. Examples include the first and second glycine elongation steps in Met-Gly-Gly synthesis, ATP-binding reactions across different processes, and similar phosphate release reactions (see Table S5 for details). Third, some kinetic parameters in ePURE are assigned very large values to represent sufficiently fast reactions. These parameters were not modified during GA, and we confirmed that allowing them to vary had little effect on the simulation outcomes. Finally, release factors 1 and 2 (RF1 and RF2) were unified into a single component: all RF2-associated reactions were removed, and the initial RF1 concentration was set to the sum of RF1 and RF2. RF1 and RF2 behave symmetrically in the ePURE model; therefore, distinguishing them adds complexity without providing additional insight. After imposing these constraints, 64 reaction groups remained.

4.4.3. Procedure of GA

The GA was executed as follows:

• (1)For each individual (parameter set), the translation dynamics were simulated, and the objective function was calculated.
• (2)Among 55 individuals, 11 with the lowest values of the objective function were selected as parents. Each parent produced four offspring via random mutation. Each parameter had a 1% chance of mutation by adding a normally distributed random number to x _i in eq . If no mutation occurred in an offspring, then a random mutation was forced at one position to ensure at least one mutated parameter.
• (3)Parents and offspring were passed to the next generation, and the process was repeated from step 1.

This mutation-selection cycle was repeated for 1,000 generations to obtain the fitted parameters. The entire fitting procedure was repeated 100 times using different random seeds to examine the variability of the fitting results. Figure S5A shows the evolution of the objective function across these runs. GA fitting markedly reduced the mismatch relative to the original parameter set. In later generations, squared errors plateaued, consistent with convergence to local optima.

To assess the fidelity of simulations using the fitted parameters, we examined a representative parameter set employed throughout the main text (Figure S5C). In contrast to the simulation with the original parameter set (Figure S5B), this fitted simulation reproduced the experimental data more accurately, and the nonmonotonic dependence was consistently reproduced across all GA-fitted simulations.

The code shows the conversion from fluorescence data to simulation concentrations, the GA and its output data, and the output of each figure (https://github.com/banshunnosuke/EF-Tu-Bell-Shaped-Translation-Dependency-Analysis)

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acssynbio.6c00333.

• Figure S1: additional simulation results related to Figure B; Figure S2: evidence that IF2·GTP·fMet-tRNA^fMet is rate-limiting at high EF-Tu concentrations; Figure S3: accumulation of EF-Tu·GTP·Met-tRNA^fMet complexes across all GA-fitted trials; Figure S4: simulation results corresponding to the additional experiments; and Figure S5: parameter fitting of ePURE by GA (PDF)

• Table S1: chemicals, their commercial sources, and their final concentrations; Table S2: the sequence of the plasmid DNA pET-G5tag encoding GFP; Table S3: the initial concentrations used in the ePURE simulations, together with the initial concentrations and fluorescence data from two experiments; Table S4: kinetic parameters used in this study; and Table S5: groups of reactions sharing the same mechanism, for which kinetic parameter values were mutated identically to preserve their equivalence (XLSX)

S.B. performed the simulations and data analysis and wrote the first draft of the manuscript. Y.H., C.F., and T.M. contributed to the design and overall direction of the study. Y.H. and C.F. provided conceptual, analytical, and overall guidance throughout the study. T.M. provided guidance from the experimental perspective and contributed biological insight into the development of the study. A.K. performed all the experiments. Y.S. provided advice on the treatment of kinetic parameters in the model. All authors revised the manuscript and approved the final version.

The authors declare no competing financial interest.

Published as part of ACS Synthetic Biology special issue “Cell-Free Systems”.