Model

To assess the effects of Nanobody conjugation on NanoLuciferase (NanoLUC), we modelled the structures of NB-NL and NB-NL-NL using AlphaFold3. The analysis focused on evaluating the stability of the core domains and the flexibility introduced by the linkers between the subunits. The structural modelling of NB-NL and NB-NL-NL shows comparable accuracy, with RMSD values of 2.03 Å and 2.00 Å, respectively, indicating reliable predictions for both models (Figure 2 and 3). In both, the main subunits, including the Nanobody and NanoLUC domains, exhibit high structural confidence and stability, particularly in their core regions. The flexibility is primarily observed in the linker regions, with the NB-NL-NL model showing increased flexibility due to the additional linker between the two NL units. This flexibility, while introducing some positional uncertainty, could provide functional benefits, such as enhanced dynamics between the NL domains. Overall, the core structures of NB and NL remain intact in both models, supporting their functional integrity.

Figure 2. The structural model of the Nanobody_Linker_NanoLUC

Figure 3. The structural model of the Nanobody_Linker_NanoLUC_Linker_NanoLUC

The structural model for the Nanobody-Chromoprotein complex, with an RMSD of 2.07 Å, indicates good overall structural alignment, though slightly less accurate compared to the NB-NL and NB-NL-NL models. The Nanobody section maintains high structural confidence, as reflected by the blue regions with pLDDT scores above 90, suggesting a reliable prediction. The Chromoprotein domain, however, exhibits areas of lower confidence, with yellow and orange regions indicating flexibility or uncertainty in these sections. The heatmap further supports this, showing low positional errors in the core regions and higher uncertainty in more flexible areas. Overall, the model demonstrates good accuracy for the Nanobody and core Chromoprotein regions but highlights some structural variability, especially in peripheral or flexible areas, consistent with the slightly higher RMSD.

Figure 4. The structural model of the Nanobody_Linker_Chromoprotein

To further investigate the impact of Nanobody conjugation on the binding affinity of NanoLUC to its substrate NanoGlo, structural dynamics was carefully analyzed using PyMol. The substrate NanoGlo (cyan) has been placed into the binding site (magenta) of NanoLUC (blue), and the analysis reveals that the binding site is located between the Nanobody (orange) and NanoLUC. This spatial arrangement suggests that the Nanobody may influence the catalytic activity by potentially obstructing NanoGlo’s access to the binding site.

Given that the Nanobody is linked to NanoLUC via a flexible linker, the dynamic movement of both components around the linker is crucial. During these motions, the Nanobody could intermittently block the entrance to the binding site, hindering NanoGlo from binding efficiently. The flexible nature of the linker introduces variability in the relative positions of the Nanobody and NanoLUC, which might cause temporary steric clashes or altered substrate access, affecting binding affinity and catalytic efficiency.

Figure 5. The structural model of the NB-NL complex bound to its substrate NanoGlo. The Nanobody (orange) and NanoLUC (magenta) are shown with the substrate NanoGlo (cyan) positioned within the binding site.

In the bioluminescence assay experiments, we tested the light production of NanoLUC and NB-NL at same molar concentrations and compared the kinetic curve to investigate the effect of Nanobody conjugation on the light production efficacy of NanoLUC.

The enzymatic reaction system contains several key molecules and parameters that define its kinetics. In a simple enzymatic reaction involving a single substrate, the enzyme NanoLuc (E) binds to the substrate NanoGlo (S) to form the enzyme-substrate complex, NanoLuc-NanoGlo (ES), as depicted in Figure 6. This complex facilitates the catalytic conversion of NanoGlo into the product P, while NanoLuc is regenerated and remains unchanged. The product molecule P is assumed to be responsible for the generation of bioluminescence. Each step in the reaction is governed by a specific rate constant (k), which quantifies the speed at which the reaction proceeds. This parameter relates rates of changes of concentrations to concentrations. The rate constants include k₁, representing the forward rate of enzyme-substrate complex formation, k_-1 describing the reverse rate for complex dissociation, and k_cat, the catalytic rate constant governing the conversion of the substrate to product.

Figure 6. Simple enzymatic reaction model

The dynamics of the enzymatic reaction system can be described using a series of ordinary differential equations (ODEs), which represent the time-dependent changes in the concentrations of the S, E, ES, and P (Figure 7).

Figure 7. ODEs for simple enzymatic reaction model

By solving these ODEs, one can simulate the time-dependent behaviour of the enzymatic reaction system, tracking the concentrations of S, E, ES, and P over time (Figure 8.). The simulation uses the proposed rate constants k₁=1, k_-1=0.5, and k_cat=0.1 as input parameters, along with various initial concentrations of enzyme NanoLuc (0.5 – 4.0 µM) and substrate NanoGlo (0.1 – 5.0 µM), to generate time-series data for these components. To mimic experimental conditions, Gaussian noise with 5% error is added to the product concentration, reflecting the variability typically encountered in laboratory measurements. Multiple replicates of the simulated data are then created to account for experimental variability across different runs, resulting in a comprehensive dataset that can be used to analyse the dynamics of the enzymatic reaction and optimise experimental conditions.

Figure 8. Simulated time-course of product concentration [P] for various initial concentrations of the enzyme NanoLuc and substrate NanoGlo.

The production of the P is influenced by both the initial concentration of the enzyme NanoLuc and the substrate NanoGlo. As the initial concentration of NanoGlo increases across the rows, the rate of product formation increases, reflecting higher substrate availability for the enzyme. Similarly, as the initial concentration of NanoLuc increases across the columns, the production rate also increases.

However, in the bioluminescence reaction, the light is generated by the NanoLuc enzyme rather than the product molecule. Therefore, to more accurately reflect the mechanism of light production in this system, we transitioned from the simple enzymatic model to a bioluminescence-specific model (Figure 9.). This revised model incorporates the role of the NanoLuc-NanoGlo complex (ES) being converted into an enzyme-product complex, NanoLuc-Product (EP), which is responsible for the production of light. The intensity of the emitted light is assumed to be proportional to the rate at which the ES is converted to the EP.

Figure 9. Bioluminescence-specific reaction model

The system of differential equations for the species in the bioluminescence system is given by:

Figure 10. ODEs for bioluminescence-specific reaction model

Data was simulated using proposed parameter values k₁ = 3, k_-1= 0.5, k_cat= 5, k₂ = 1, and k_-2 = 0.5, with various initial concentrations of NanoLuc (0.1 – 5.0 µM) and NanoGlo (2.0 – 5.0 µM). Unlike the simple model, where the product is the final output, this model emphasizes the transient formation of EP, which stabilizes quickly, reflecting the rapid onset of bioluminescence after substrate binding and conversion by NanoLuc.

Figure 11. Simulated time-course of enzyme-product complex concentration [EP] for various initial concentrations of the NanoLuc and NanoGlo in the bioluminescence model.

Nonetheless, the bioluminescence model is still not ideal because, in a real-world scenario, the bioluminescence intensity would be expected to gradually decrease and eventually disappear as the reaction progresses. This decay in light intensity is often due to substrate depletion or inhibition effects, which are not captured by the current model. To address the limitations of the previous bioluminescence model, where the light intensity remains constant after reaching a steady state, we developed a more complex model incorporating two key additional factors: inhibition by excess substrate and enzyme inactivation. In this refined model, the enzyme NanoLuc can bind to an excess of substrate NanoGlo, forming an inhibitory complex (ESS) that reduces the enzyme's ability to catalyse the reaction and, thus, decreases bioluminescence over time. Additionally, the model accounts for the inactivation of the enzyme, where NanoLuc transitions into an inactive form (E*) that is no longer able to bind the substrate or produce light (Figure 8).

Figure 12. Complex bioluminescence reaction model

Based on the reaction scheme, the system of differential equations can be constructed as follows:

Figure 13. ODEs for complex bioluminescence reaction model

These ODEs describe the inhibition and inactivation of the enzyme in the bioluminescence model. These equations incorporate various reaction rates for binding (k_on), dissociation (k_off), catalysis (k_cat), and enzyme inhibition (k_inact) by excess substrate, providing a more comprehensive representation of the enzymatic dynamics under real-world conditions. By solving the system of ODEs, the dynamics of the enzymatic reaction can be observed over time. The simulation incorporates the following rate constants: (k_{s_on}=1, k_{s_off}=0.1, k_cat=7, k_{ss_on}=1, k_{ss_off}=0.3, k_{p_off}=0.5, k_{p_on}=0. 008, and k_inact=0.1. Figure 10 illustrates the time-dependent changes in S, E, ES, ESS, EP and P concentrations.

Figure 14. The dynamics of the complex bioluminescence enzymatic reaction for fixed values of k_{s_on}=1, k_{s_off}=0.1, k_cat=7, k_{ss_on}=1, k_{ss_off}=0.3, k_{p_off}=0.5, k_{p_on}=0. 008, and k_inact=0.1.

Initially, the substrate concentration [S] decreases rapidly as it binds to the enzyme to form the ES, which then catalyses the production of the EP and P. The EP plays a crucial role in the system as it is responsible for producing bioluminescence, making it the primary point of interest in this model. Initially, the concentration of EP rises as the ES is catalytically converted into EP. This increase is sharp at the beginning of the reaction due to the high availability of substrate and active enzyme. However, as the reaction progresses, the formation of EP begins to slow down.

Two key factors contribute to this slowdown: the accumulation of the inhibitory enzyme-substrate complex and the inactivation of the enzyme. The ESS complex sequesters the enzyme, preventing it from further catalysing the formation of EP. Additionally, as the enzyme is inactivated over time, the production of EP decreases further. Consequently, the concentration of EP decreases to zero, reflecting the complete cessation of bioluminescence production. This behaviour mimics the real-world scenario, where bioluminescence intensity initially rises but gradually diminishes and ultimately disappears due to enzyme inhibition and inactivation, leading to the exhaustion of the bioluminescent signal.

Prior to designing the experimental setup for the bioluminescence assay, we conducted model fitting on various simulated datasets to evaluate the impact of experimental conditions and the number of replicates on the accuracy of point estimation for the reaction constant parameter (k) in the simple enzymatic model. This process allowed us to systematically explore how changes in enzyme and substrate concentrations, along with varying levels of experimental replication, influence the precision of parameter estimation.

Simulations were carried out using three different setups: (1) 16 enzyme+substrate conditions with 3 replicates each, (2) 16 conditions with 10 replicates each, and (3) 25 conditions with 5 replicates each. Fitted parameters were calculated with 95% confidence intervals (CI), and the quality of the estimator was assessed by evaluating its bias and consistency. The results showed that setup (3) provided the best estimator, with the lowest bias and reasonable variance. Interestingly, setup (2), despite having more replicates, resulted in a more biased estimator than setup (1). This was likely since that bias with smaller variance still leads to a larger overall error in parameter estimation.

Thus, balancing both the number of experimental conditions and replicates is essential for obtaining an accurate and reliable estimator in enzymatic kinetics modelling. In the bioluminescence assay, we tested at least 5 different enzyme concentrations for each protein, with 3 replicates for each condition. This design allowed us to capture a wide range of enzyme activities while maintaining sufficient replication to ensure data robustness.

In the bioluminescence assay experiments, we tested the light production of NanoLUC and NB-NL at the same molar concentrations to compare their kinetic profiles (Figure 15.). The experimental data, presented in the kinetic curves above, clearly demonstrate a difference in reaction rates between the two. At the start of the measurement, NB-NL showed a sharp decline in light intensity, indicating that the peak luminescence phase was reached and declining during the observation period. In contrast, NanoLUC exhibited a much more gradual decrease, suggesting that the reaction catalysed by NanoLUC had already progressed to its tail phase by the time measurements began.

Figure 15. Bioluminescence curves for [NB-NL] and [NL] at concentration of 2.5e-9M.

This kinetic difference implies that NanoLUC catalysed the reaction significantly faster than NB-NL, supporting the hypothesis that the conjugation of the Nanobody to NanoLUC may hinder the enzyme's catalytic efficiency. The slower reaction in NB-NL could be attributed to the Nanobody blocking or restricting access to the active site due to its proximity and dynamic movement around the linker, as previously suggested by the structural analysis. This experimental data strengthens the conclusion that Nanobody conjugation can affect the overall catalytic activity of NanoLUC.

Figure 16. Bioluminescence curve for [NB-NL] = 5e-9M.

Figure 16 shows the average kinetic measurement of Nanobody-NanoLuc (NB-NL) at a concentration of 5e-9M. Due to the time required for the plate reader to inject and adjust, it was challenging to capture the initial rate phase. However, the experimental bioluminescence curve was closely connected to the dynamics EP in the complex model. In the experimental curve, we observe an initial plateau in bioluminescence followed by a gradual decline. This reflects what is described by the complex enzymatic model, where EP is responsible for bioluminescence production. As the reaction progresses, however, the bioluminescence begins to decline. This decline is explained by two key factors in the complex model: the formation of the inhibitory enzyme-substrate complex and enzyme inactivation.

Ultimately, the concentration of EP approaches zero, reflecting the cessation of bioluminescence in both the model and the experimental data. This gradual decay of luminescence in the real-world experiment mimics the theoretical prediction of bioluminescence exhaustion due to enzyme inhibition and inactivation, resulting in a complete stop in light production. The alignment between the experimental and modelled data supports the validity of the complex model in describing the bioluminescence reaction dynamics.

Based on the kinetic curves measured in the bioluminescence assays, the complex model was selected to describe the enzymatic reaction dynamics, as it best captured the observed behaviour. To further investigate the properties of the enzymatic reaction, we derived the Quasi Steady-State Approximation (QSSA) for the intermediate species involved in the reaction. This approximation simplifies the system by assuming that the concentrations of ES (enzyme-substrate complex) and ESS (inhibitory enzyme-substrate complex) reach a steady state much faster than the other components of the reaction.

The QSSA allowed us to express the dynamics of the reaction in a simplified form, focusing on the slower-changing variable EP while treating the intermediates as quasi-stationary. From this, we derived a quantitative expression for the product formation rate that closely resembles the Michaelis-Menten equation, which is a standard approach to describe enzyme kinetics:

V_max represents the maximum reaction velocity that can be achieved when the enzyme is fully saturated with the substrate. K_m is the Michaelis constant, which reflects the substrate concentration at which the reaction rate is half of V_max. It provides a measure of the enzyme's affinity for the substrate: lower K_m values indicate a higher affinity, as less substrate is required to reach half of the enzyme's maximal activity, whereas higher K_m values suggest a lower affinity. k_cat, also known as the turnover number, represents the number of substrate molecules converted to product per second by a single enzyme molecule when the enzyme is fully saturated with substrate. It provides a direct measure of the enzyme's catalytic efficiency in processing the substrate.

Theoretically, these parameter values can be obtained from the Michaelis-Menten curve, which plots the initial reaction velocity (v₀′) against varying substrate concentrations ([S₀]). V_max is observed as the reaction rate approaches a plateau at high substrate concentrations, while K_m is identified as the substrate concentration at which v₀ reaches half of V_max . Once V_max is known, k_cat can be calculated by dividing V_max by the total enzyme concentration [E₀].

However, since the bioluminescence reaction does not directly measure the concentration of the EP but instead measures the light intensity, we can only observe the apparent initial reaction velocity (v₀′) rather than the true v₀. The light intensity is proportional to the amount of EP formed, but it does not provide a direct measurement of the concentration. Therefore, v₀′ needs to be converted into the true v₀ by accounting for the relationship between light intensity and product formation. This conversion typically involves calibrating the bioluminescence signal to actual product concentrations, ensuring that the measured signal accurately reflects the enzyme kinetics. Once this calibration is done, the Michaelis-Menten parameters V_max, K_m , and k_cat can be extracted in the same manner as traditional kinetic assays. Due to time constraints, we did not perform a full calibration for the bioluminescence signal in this experiment. Future work could involve conducting more accurate bioluminescence assays by recording the initial rate phase and performing a thorough calibration. This would help obtain precise kinetic parameters and ensure that the measured bioluminescence accurately reflects the underlying activity of NanoLuc.