BiSim Tool: a binding simulation tool to aid and simplify ligand-binding assay design and development

Abstract

Ligand-binding assays (LBAs) rely on the reversible, noncovalent binding between the analyte of interest and the assay reagents, and understanding their dynamic equilibrium is key to building robust LBA methods. Although the dynamic interplay of free and bound fractions can be calculated using mathematical models, these are not routinely applied. This approach is costly in terms of both assay development time and reagents, and can result in an under-exploration of the possible parameter combinations. Therefore, we have created a user-friendly simulation tool to facilitate LBA development (the BiSim Tool). We describe the models driving the mathematical simulations and the main features of our software solution by means of case studies, illustrating the tool's value in drug development. To support drug development for all patients worldwide, the BiSim Tool is now available as an open-source code project and as a free web-based tool at https://proteinbindingsimulation.shinyapps.io/BiSim-ProteinBindingSimulation [1].

Executive summary

• Ligand-binding assays (LBAs) are essential for analyzing the interaction between an analyte and assay reagents through reversible, noncovalent binding.

• Understanding the dynamic equilibrium between free and bound fractions in LBAs is crucial for developing effective assay methods.

• Despite the importance, mathematical models to calculate this dynamic interplay are not commonly used in practice.

• We developed a new user-friendly simulation tool, BiSim, to make the application of mathematical models in LBA development more accessible.

• The tool and its underlying mathematical models are explained through case studies, such as determining assay performance requirements, predicting expected complex formation, time-to-equilibrium and assay optimization.

• BiSim is available as an open-source code project, allowing for wide accessibility and the possibility of community improvements.

• Additionally, a free web-based service hosts the BiSim Tool, ensuring easy access for users without the need for software installation: https://proteinbindingsimulation.shinyapps.io/BiSim-ProteinBindingSimulation.

• The combination of detailed use cases and the freely available tool provide a valuable resource for researchers and developers in drug development, saving both time and development costs.

Keywords: :

• assay development
• binding kinetics and interactions
• free web-based service
• ligand-binding assay
• pharmacokinetics
• simulation tool

During the drug development process for biologic drug candidates, the characterization of pharmacokinetics (PK) of the drug, the response of biomarkers to the drug and the understanding of potential immune responses to the drug through the detection of antidrug antibodies (ADAs) formed against it, are key and their measurement requires quality bioanalytical methods. In contrast to small-molecule drugs, which are commonly quantified using LC–MS methods, the bioanalytical gold standard technology for drug candidate proteins, soluble biomarkers and ADA remains ligand-binding assays (LBAs).

The establishment of a rational bioanalytical assay strategy involves the transversal contribution of multidisciplinary teams, including bioanalytical scientists, modeling and simulation scientists, pharmacologists and biomarker scientists to define the context of use and assay requirements. These, together with the drug development stage and practical bioanalytical challenges, will drive the selection of the most appropriate analytical platform, assay setup and tool reagents for LBA methods [2]. During this process and the assay development phase, a clear understanding of the capabilities and limitations of the bioanalytical assay used for analyte quantification is essential to generate fit-for-purpose and reliable bioanalytical data and enable good quality data interpretation.

To understand the characteristics of an LBA, it is imperative to realize how the binding partners interact in a reversible noncovalent manner as governed by the law of mass action. These binding interactions occur in a physiological context (between analyte and binding partner) or in LBAs (with assay reagents). Consequently, due to the dynamic equilibrium of the interactions, they are susceptible to interference from the binding of other proteins and from the effect of sample dilution.

Such interactions can be assessed by applying mathematical models based on the law of mass action that, given the expected concentrations of the binding partners and affinity constant or binding kinetics, can predict the formation of the binding complexes. The rate of complex formation is proportional to the free concentrations of each entity and the rate of bimolar association. The rate of separation of a complex back to its constitutive parts is given by a first-order dissociation rate. This yields a mathematical relationship of the free and bound concentrations of proteins interacting reversibly [3,4]. The use and benefit of mathematical approaches to simulate various assay requirements and conditions have been described previously [5,6]. However, although these calculations are not technically difficult to perform, they can be laborious and prone to error if done by hand and require coding skills to automate. In our experience, most often scientists either rely on experience with assay development or resort to a partial simulation of the assay binding conditions. As assay conditions, capabilities and limitations are experimentally explored, this manual approach is costly in terms of both assay development time and reagents. Moreover, this approach results in an under-exploration of the possible parameter combinations, possibly yielding a suboptimal assay setup. We believe that bioanalytical scientists would benefit from an easy-to-use implementation of these binding models. However, to the best of our knowledge, no such publicly available tool describing the interaction of two binding partners with competition exists.

To support bioanalytical activities during drug development, we created a user-friendly software-solution for mathematical simulations, the BiSim Tool. The solution was implemented in the software R (www.R-project.org [7]) and allows bioanalytical scientists to access the interface via an R Shiny web page. This approach hides the R code, while providing bioanalytical scientists an intuitive interface to vary parameters at will. The tool consists of two main parts: the binding tool (Figure 1, left panel) and the competition tool (Figure 1, right panel), which allow users to simulate interactions between either two or three binding partners, respectively. The graphical user interface provides contextual plots that facilitate the input of data and the translation of those input parameters to the simulated assay outputs. To use the tool, only the concentration of the binding partners and the affinity (the equilibrium dissociation constant, K_D) of the binding interaction need to be entered. A user manual is available that gives more detailed information on the different features of the tool and how to use them (see Supplementary Information 1). The tool is freely accessible via https://proteinbindingsimulation.shinyapps.io/BiSim-ProteinBindingSimulation for evaluation purposes (no installation required) [1].

Figure 1. Protein binding kinetics simulation tool.

The free and bound concentrations of proteins interacting reversibly can be estimated based on their concentrations and affinity of the interaction using the novel, web-based tool. The application has a simple and intuitive graphical user interface, consisting of a binding tool and a competition tool.

The applications of the simulation tool will be illustrated here by means of various case studies, underlining the value of the use of mathematical simulations to (1) identify appropriate bioanalytical strategies, (2) define the assay requirements, (3) support rational bioassay development and (4) support proper data generation and interpretation (Figure 2). For example, mathematical simulations offer the possibility to evaluate how much complex is formed in a bio-sample (by using the binding tool, see left panel in Figure 1), as well as the potential impact of endogenous proteins bound to the analyte of interest on the assay measurements (by using the competition tool, see right panel in Figure 1). Additionally, simulations can provide relevant information on assay requirements – for example, sensitivity – and can guide optimization of assay conditions, such as incubation times and reagent concentrations. It is worth noting that understanding binding interactions and their impact on assay performance is relevant not only for LBA but also for hybrid assays such as immunocapture LC-MS. Furthermore, the tool can also support scientists in evaluating and potentially overcoming bioanalytical challenges such as interfering factors.

Figure 2. Understanding binding interactions in a biosample and bioassay enables rational assay design and implementation.

At different stages during the assay lifecycle, it is important to understand the binding interactions of the analyte of interest, which can be the drug candidate, antidrug antibodies (ADAs) or a biomarker, with either endogenous binding partners in the biosample or with assay reagents in the bioassay. Different case studies illustrated here are covered in the main text: (1) simulate the fraction of the analyte bound to specific binding partners in the biosample; (2) simulate the assay requirements to measure specific fraction of the analyte; (3) simulate most optimal assay conditions – for example, incubation time of assay steps and concentration and affinity of assay reagents to obtain the defined assay requirements; and (4) simulate impact of sample handling (e.g., sample dilution) or presence of interfering binding partners on the binding dynamics of the analyte with endogenous binding partner.

Methods

A user-friendly tool was developed in the software R to visualize and allow adaptation of key parameters for a set of well-known (bio)chemical equations describing binding between two or three competing binding partners. The Rshiny package (https://cran.r-project.org/package=shiny [8]) provides an easy-to-use graphical user interface that allows the user to enter the anticipated concentration ranges and binding affinities of the analyte of interest and its binding partner(s) in a series of dialog boxes and slider bars. From this, a variety of informative plots and tables are generated of the resulting binding complexes formed, in both absolute and relative concentrations. The underlying computations are hidden from and requiring no involvement from the user. Information on installation and running of the application can be found in Supplementary Information 2.

The plots require the solution of a system of mathematical equations that describe the reversible binding interactions between chemical species. Those equations and the solution methods are as follows: consider the binding of an Interaction Partner 1, with concentration C1, which binds to Interaction Partner 2, with concentration C2, to form the complex of the two with a concentration that we will denote by C12. This may, for example, represent the binding of a drug to its target. This binding interaction may face competition from Interaction Partner 3, with concentration C3, which can also bind to Interaction Partner 1, forming the complex with concentration denoted by C13; the binding of Interaction Partner 3 to Interaction Partner 1 prevents the binding of Interaction Partner 2 (Figure 1, right panel). The binding interactions between the Partners are governed by the law of mass action: the formation rate of C12 is proportional to the concentrations of the Interaction Partners C1 and C2, with a proportionality constant defined to be the association rate denoted by k_on12. The complex C12 may dissociate into its constitutive parts C1 and C2 with the first-order dissociation rate k_off12. Similarly, C13 is formed at a rate proportional to C1 and C3 with proportionality constant kon13 and that complex dissociates with first-order dissociation rate k_off13. The equilibrium or dissociation binding constant K_d12 is the ratio of the dissociation and association rates – that is, K_d12 = k_off12 / k_on12; likewise K_d13 = k_off13/k_on13 is defined. That system is then described by the following set of ordinary differential equations:(Equation 1) $\begin{array}{cc}\frac{d{C}_1}{dt}=-k_{on}^{12}\times C_1\times C_2+k_{off}^{12}\times C_{12}-k_{on}^{13}\times C_1\times C_3+k_{off}^{13}\times C_{13};& C_1(0)=C_1^0\end{array}$(Equation 1) (Equation 2) $\begin{array}{cc}\frac{d{C}_2}{dt}=-k_{on}^{12}\times C_1\times C_2+k_{off}^{12}\times C_{12};& C_2(0)=C_2^0\end{array}$(Equation 2) (Equation 3) $\begin{array}{cc}\frac{d{C}_3}{dt}=-k_{on}^{13}\times C_1\times C_3+k_{off}^{13}\times C_{13};& C_3(0)=C_3^0\end{array}$(Equation 3) (Equation 4) $\begin{array}{cc}\frac{d{C}_{12}}{dt}=k_{on}^{12}\times C_1\times C_2-k_{off}^{12}\times C_{12};& C_{12}(0)=0\end{array}$(Equation 4) (Equation 5) $\begin{array}{cc}\frac{d{C}_{13}}{dt}=k_{on}^{13}\times C_1\times C_3-k_{off}^{13}\times C_{13};& C_{13}(0)=0\end{array}$(Equation 5)

The initial values for the Interaction Partners C1, C2 and C3, denoted by C10, C20 and C30, are set by the user and we assume that the initial values for the complexes C12 and C13 are zero.

The final concentrations of the Interaction Partners C1, C2 and C3, and their complexes C12 and C13 is given by the steady-state solution of the set of Equations 1–5(Equation 1) $\begin{array}{cc}\frac{d{C}_1}{dt}=-k_{on}^{12}\times C_1\times C_2+k_{off}^{12}\times C_{12}-k_{on}^{13}\times C_1\times C_3+k_{off}^{13}\times C_{13};& C_1(0)=C_1^0\end{array}$(Equation 1) and found by setting the lefthand sides to zero and assuming mass balance we can derive (in molar units):(Equation 6) $C_1^0=C_1+C_{12}+C_{13}$(Equation 6) (Equation 7) $C_2^0=C_2+C_{12}$(Equation 7) (Equation 8) $C_3^0=C_3+C_{13}$(Equation 8)

This may be substituted for the terms describing the concentrations of the complexes. The equations may then be rearranged to form three equations describing the three unknowns of the concentrations of the unbound interaction partners C₁, C₂ and C₃:(Equation 9) $0=C_1^0-C_1-\frac{C_1{C}_2}{K_d^{12}}-\frac{C_1{C}_3}{K_d^{13}}$(Equation 9) (Equation 10) $0=C_2^0-C_2-\frac{C_1{C}_2}{K_d^{12}}$(Equation 10) (Equation 11) $0=C_3^0-C_3-\frac{C_1{C}_3}{K_d^{13}}$(Equation 11)

Thus, the steady-state equations may be solved when only K_d, rather than separate K_on and K_off, parameters are given.

The set of Equations 9–11(Equation 9) $0=C_1^0-C_1-\frac{C_1{C}_2}{K_d^{12}}-\frac{C_1{C}_3}{K_d^{13}}$(Equation 9) is numerically solved in R using the R function ‘multiroot’ in the package ‘rootSolve’ (CRAN – Package rootSolve (r-project.org) https://cran.r-project.org/web/packages/rootSolve/vignettes/rootSolve.pdf [9]). Only positive roots are solved. The calculated values of C1, C2 and C3 are then used to calculate the concentrations of the complexes C12 and C13 using Equations 7(Equation 7) $C_2^0=C_2+C_{12}$(Equation 7) and 8(Equation 8) $C_3^0=C_3+C_{13}$(Equation 8) . The Rshiny tool plots related to steady-state results are then calculated by applying the solver to the set of Equations 9–11(Equation 9) $0=C_1^0-C_1-\frac{C_1{C}_2}{K_d^{12}}-\frac{C_1{C}_3}{K_d^{13}}$(Equation 9) for each set of parameters associated with each point on the plot. A reduced set of equations is not used for considering interactions between Interaction Partner 1 and Interaction Partner 2 only; the concentration of Interaction Partner 3 is instead fixed to zero. The system of Equations 9–11(Equation 9) $0=C_1^0-C_1-\frac{C_1{C}_2}{K_d^{12}}-\frac{C_1{C}_3}{K_d^{13}}$(Equation 9) has an analytical solution, which would be more computationally efficient; however, the numerical solution method has been retained here to aid code readability and possible amendments and extensions, as the chemical system being modeled is most easily written, checked and determined from the source system of ordinary differential equations.

The effects of dilution are modeled by solving the same system of Equations 9–11(Equation 9) $0=C_1^0-C_1-\frac{C_1{C}_2}{K_d^{12}}-\frac{C_1{C}_3}{K_d^{13}}$(Equation 9) , where the concentration of each chemical species is reduced by the factor of the dilution step.

Time-to-binding is determined using a numerical solution to the ordinary differential equation system (ODE) Equations 1–5(Equation 1) $\begin{array}{cc}\frac{d{C}_1}{dt}=-k_{on}^{12}\times C_1\times C_2+k_{off}^{12}\times C_{12}-k_{on}^{13}\times C_1\times C_3+k_{off}^{13}\times C_{13};& C_1(0)=C_1^0\end{array}$(Equation 1) . Time-to-binding is set as the time to reach 99% of the final C₁:C₂ complex formation, with the final concentration determined by a solution of the corresponding steady-state Equations 9-11(Equation 9) $0=C_1^0-C_1-\frac{C_1{C}_2}{K_d^{12}}-\frac{C_1{C}_3}{K_d^{13}}$(Equation 9) . The ODE system is solved in R using the ‘deSolve’ function from the package of the same name (CRAN - Package deSolve (r-project.org) [10]). Note that the calculation of time-to-binding requires the specification of the k_on and k_off parameters.

The solutions of the equations were corroborated by implementing the ODE system from Equations 1–5(Equation 1) $\begin{array}{cc}\frac{d{C}_1}{dt}=-k_{on}^{12}\times C_1\times C_2+k_{off}^{12}\times C_{12}-k_{on}^{13}\times C_1\times C_3+k_{off}^{13}\times C_{13};& C_1(0)=C_1^0\end{array}$(Equation 1) in Berkeley Madonna v8.3, using the STIFF (Rosenbrock) solver and the Equations 9–11(Equation 9) $0=C_1^0-C_1-\frac{C_1{C}_2}{K_d^{12}}-\frac{C_1{C}_3}{K_d^{13}}$(Equation 9) using the ROOTI function (www.berkeleymadonna.com [11]).

The graphical user interface was built to expose the mathematical model predictions using agile project methodology. An initial user interface was designed via a key user interview, which was then verified using mock-ups by a larger key-user group of five people. Features were built in sprints of 2 weeks, after which feedback was collected while new features were implemented. Thus, improvements were implemented in iterative feedback loops until a stable tool emerged.

All source code is publicly available at https://github.com/UCB-GitHub/BiSim [12], released under a GNU General Public License v3.0. Detailed installation instructions (local and cloud-based) can be found in Supplementary Information 2. The tool is freely accessible via https://proteinbindingsimulation.shinyapps.io/BiSim-ProteinBindingSimulation [1] for evaluation purposes (no installation required).

There are some important disclaimers when using this tool: first, the tool always assumes 1:1 stoichiometry, meaning it does not account for multivalent interactions. Second, the tool considers that the binding interactions occur in solution – that is, all binding epitopes are freely and equally available. Note that affinities are often generated using surface-based technologies, such as surface plasmon resonance, and therefore affinity values may differ from the true in-solution value [13]. Third, all interactions are assumed to be in equilibrium, which means that the number of complexes formed equals the amount that dissociates. Note that the anticipated time to equilibrium can be calculated by the tool if the binding rates (k_on and k_off) are optionally entered in the tool.

Case studies

Multiple case studies are discussed here (in the order shown in Figure 2), all of which used various features of the simulation tool, supporting different aspects of an LBA lifecycle, ranging from defining the bioanalytical strategy and needs, to assay development and data interpretation.

Context of use of the assay (bioanalytical strategy)

When an analyte of interest needs to be quantified, the following key questions should be addressed: “What do you want to measure?" and “How will the data be used?”. During drug development, measurement of free analyte (either the drug or the target) is often requested (Figure 2, step 1). The former can be used to determine the pharmacologically active drug fraction, whereas the free target fraction can inform on target occupancy. As such it is key to understand the binding interactions of the analyte in the matrix of interest, which could be bound by one or more endogenous binding partners. In the case of a protein drug candidate, this could be the target or possibly ADA, whereas the target can be bound to different binding partners (e.g., soluble receptor). Generally, measuring the free fraction is technically challenging because of the risk of complex dissociation, resulting in inaccurate measurement of the free fraction. The challenges to developing a free analyte assay are well known and have been discussed extensively [14–18], and so it is important to consider how changes in complex equilibrium can be avoided. Before beginning assay development, the anticipated drug–target complex formation in the biosample can be simulated by entering expected concentrations of the drug and target, as well as the affinity of the interaction, into the simulation tool (Figure 1, left panel). It is important to note that any mathematical calculations require accurate and relevant concentration and affinity values, although the latter is often hampered by the challenge to determine the affinity which depends on many factors (e.g., temperature, pH, buffer/matrix used) and the discrepancy between surface-based K_D values and the true in-solution K_D [18]. For example, in cases where the free drug fraction needs to be measured, the highest complex formation could be simulated by considering the lowest expected drug concentrations in the samples (lowest dose group and washout samples) while keeping in mind that target levels may increase following drug treatment, which would result in increased fraction of complexed drug. In the example given in Figure 3, when plotting the lowest drug concentration and highest target levels anticipated in the samples, the drug fraction that is bound to target is negligible. This shows that the free drug fraction approximates the total drug fraction in all PK samples, and therefore there is no concern of a disturbance of in vivo complexes when developing a free PK assay.

Figure 3. Complex formation between drug and target to inform on free drug fraction to be considered in a free pharmacokinetic assay.

The worst-case scenario is plotted here in terms of amount of drug complexed to target by entering in the simulation tool the anticipated lowest drug concentration and highest target concentration present in a pharmacokinetic sample. This simulation shows that only 0.496% of drug (depicted by the box) will be complexed, and therefore approximately all drug fractions can be considered free.

Assay requirements

As mentioned earlier, in support of target occupancy assessments, free target assays are often developed. After drug treatment, it is expected that free target levels will decrease, and thus it is relevant to understand what the maximal expected complex formation (i.e., lowest free fraction) is to define the assay sensitivity requirements (Figure 2, step 2). Therefore, the highest anticipated drug concentrations and the lowest biological target levels in the matrix of interest should be entered in the simulation tool (Figure 1, left panel). In the example given in Figure 4, at C_max, 79.2% of the target is expected to be bound by drug, so the assay should have sufficient sensitivity to measure 20.8% of the baseline target concentration (Figure 4A). If, for example, the lower end of the biological target concentration range is 100 pg/ml then the assay would need to be able to detect approximately 20 pg/ml of free target. Thus, knowing the required assay sensitivity will support the selection of the most suitable assay format and reagents. Additionally, the simulation tool plots supporting graphs, one of which is shown in Figure 4B, plotting percent target occupancy as a function of drug concentration. At the lowest expected drug concentrations, this graph shows that a limited percentage of target would expect to be occupied, indicating that a precise assay would be required to observe differences in target occupancy at low drug concentrations. In this way, the simulation tool can facilitate the determination of assay performance requirements. Note that in the example given, the curves for each of the target concentrations covering the biological range (depicted in the different colors on the bottom of Figure 4B) coincide, which indicates that the complex formation between drug and target is driven by the drug concentration and not the target concentration, due to the anticipated excess of drug present in the samples.

Figure 4. Complex formation between target and drug to inform on bound target fraction to be considered in a target occupancy assay.

To understand what the maximum anticipated complex formation will be, the highest anticipated drug concentration was used (A). At lower drug concentrations, plotted in the x-axis in the graph on the bottom, lower % target occupancy is observed, shown in the y-axis. The lines represent different target concentrations (B).

Assay optimization & establishment

Following on from the previous case study in which the required assay sensitivity for a target occupancy assay was determined (Figure 4), the simulation tool can also be used to evaluate whether a particular assay setup and assay conditions can meet these requirements (Figure 2, step 3). It is important to note that the sensitivity of an LBA depends in part on the properties of the capture tool used (affinity and concentration). In the example given, the drug is used as a capture tool to allow measurement of the free target fraction in samples. When simulating the expected complex formation using the Binding simulation tool (Figure 1, left panel), between the capture tool (drug) at a standard concentrations used in LBA (i.e., 1 μg/ml) and the target at normal biological concentrations (100–1000 pg/ml), the graph in Figure 5 shows that there is little complex formation. Furthermore, it only increases slightly when using higher capture tool concentrations within the expected target range. Only at higher target concentration (of the order of 10,000 pg/ml) can more complexes be formed but, most importantly, at the required sensitivity in the low pg/ml range, no complexes are formed. This shows that the required assay sensitivity limit will not be obtained using this assay setup and this capture tool, mainly because of the relatively low nM affinity of the interaction between the capture tool (here the drug) and target, and the low target concentrations. By simulating the binding interaction in the assay based on expected analyte concentrations and standard assay reagent concentrations as well as the affinity between the interaction partners, the expected complex formation and thus assay signal can be predicted, which can make assay development a more considered and faster process.

Figure 5. Complex formation between target and capture tool used in the free target biomarker assay.

In the proposed assay setup the drug is used as capture tool to measure the free target and thus target occupancy. Complex formation between free target in the sample and the capture tool (y-axis) is shown at different capture tool concentrations (colored lines) and at the relevant biological target range (100–1000 pg/ml, depicted on the x-axis). Limited complex formation (as representation of potential assay signal) is observed within the expected biological target range using a capture tool concentration of 1 μg/ml (depicted by the blue box in the table), and it drops to almost zero at the required sensitivity limit of 10 pg/ml target concentration (depicted by the blue circle in the table). No improvement in complex formation (i.e., no improvement in sensitivity) was observed when the capture tool concentration was increased to 10 or 100 μg/ml (depicted by the dashed blue circle in the table).

Assay incubation times can also impact assay development success (Figure 2, step 3) [15,19]. To have a robust method, it is key that all binding interactions in the assay are in equilibrium. Generally, the time it takes for a binding interaction to reach equilibrium can be quite variable and depends on the concentrations of the interaction partners and the affinity [14,20,21]. Equilibration time is inversely correlated to concentration – that is, the lower the concentration, the higher the average distance between the partners in solution and the longer it will take to rebind upon dissociation and hence reestablish equilibrium [22,23]. Translating this into bioassay conditions, this means that if lower concentrations are used, longer incubation times would be needed to reach equilibrium. Furthermore, time to equilibrium is also dependent on the affinity constant, as well as experimental conditions such as temperature, buffer (e.g., pH) and mixing steps. The faster the dissociation rate, the faster equilibrium is reached [22,23], but less complex will be formed. If a lower binding affinity is observed, it is important to consider controlling the assay conditions to minimize disruption of the equilibrium – for example, lower incubation temperatures, minimal washing and mixing steps – and to ensure consistency between assay runs. To determine the time needed to reach equilibrium the association rate (k_on) and the dissociation rate (k_off) of the binding complex must also be entered into the tool (Figure 1, left panel). As mentioned earlier, depending on the concentration, time to reach equilibration will vary; therefore, it is important to simulate the time to equilibrium at concentrations at the lower and upper limits of quantitation of the assay. In Figure 6, an example is shown where time needed to reach equilibrium was simulated for the capture step in a PK assay using an antidrug reagent as the capture tool. In the given example equilibrium is reached at 2.48 h and less than 5 min at lower and upper limits of quantitation, respectively. As well as reaching equilibrium faster, more complexes are formed at higher concentrations. Altogether, for this assay capture step, to reach equilibrium and thus to be stable, an almost 3-h incubation would be needed.

Figure 6. Time needed to reach equilibrium depends on concentration of binding interaction partners.

The equilibration time was calculated for the assay capture step based on the fixed capture tool concentration and binding affinity. Additionally, lowest and highest analyte concentrations from the assay analytical range were considered (i.e., LLOQ and ULOQ, respectively). At the time point (shown on the x-axis) where the line plateaus (depicted by the crosses) the equilibrium is reached – that is the amount of complexes (shown on the y-axis) does not change anymore.

LLOQ: Lower limit of quantification; ULOQ: Upper limit of quantification.

Data generation & interpretation

It is known that due to the noncovalent reversible nature of the interaction between binding partners, in vivo analyte complexes can be easily disrupted by sample handling steps and assay conditions [5]. To ensure that the free fraction is accurately measured, it is critical that the dynamic equilibrium between the drug–target complex in the sample is not unintentionally disturbed. Key considerations during method development are optimization of assay conditions (affinity and concentration of the capture tool and the incubation time), as well as limiting sample handling, such as dilution (Figure 2, step 3) [19,24,25]. The potential impact of such factors could be explored experimentally and can be evaluated by mathematical simulations using the simulation tool (Figure 1, left panel).

The impact of sample dilutions on the drug–target complexes depends on the binding affinity and the concentrations of the binding partners. The lower the affinity or concentration of the binding partners, the easier it is to disturb the binding. Using the target occupancy assay example mentioned earlier (Figure 4), a 1:4 sample dilution will reduce the fraction bound target in the sample from 79.2% (Figure 4A) to 48.8% (Figure 7A). At lower drug concentrations (i.e. fewer complexes) the impact of sample dilution is even greater, with 27.6% bound target reduced to 8.7% (Figure 7B). Note that the impact of sample dilution would be less pronounced with a higher binding affinity of drug to the target (pM affinity instead of nM). Thus, although sample dilutions are often needed to avoid matrix interference, often it will result in complex dissociation in a nonlinear fashion – that is, more pronounced in samples with lower amounts of complexes, resulting in an overestimation of the free fraction. Overall, these considerations should be considered during assay development and data interpretation of free/bound analyte quantifications.

Figure 7. Impact of sample dilution on complex formation.

Plots depict % target occupancy (TO; i.e., fraction of target bound to drug) on the y-axis at different sample dilutions on the x-axis, starting at different % of TO in the left and right panels. (A) Starting %TO of 79.2%: when diluting the sample (e.g., 4 times), the %TO will drop from 79.2 to 48.8% (as depicted by the boxes in the table below), which means that less target will be bound by drug due to dissociation of the complex after sample dilution. (B) Starting %TO of 27.6%: when starting with a lower %TO (i.e., lower amount of target bound to drug), the same thing happens after sample dilutions (%TO drops from 27.6 to 8.7%) but the impact is greater compared with the first example (reduction of 1.6 vs 3.2 times in (A) and (B), respectively), showing that the impact of sample dilution on complex dissociation is nonlinear.

Second, in addition to sample dilutions, assay conditions can also affect the binding equilibrium of the in vivo complexes (Figure 2, step 4). In general, the lower the affinity, the concentration and the incubation time of the capture step (thus the competing Interaction Partner), the lower the likelihood of disturbing the binding equilibrium between the drug and the target. These considerations are important when measuring the free analyte fraction and the assay setup is based on a capture tool that competes with the in vivo binding partner for binding to the analyte of interest. To avoid such competition and consequently potential complex dissociation, it helps to understand how increasing concentration and affinity of assay reagents may negatively impact the complex, which can be simulated with the competition tab of the simulation tool (Figure 1, right panel), and is illustrated in Figure 8. Note that highly sensitive assays often require high-affinity tools, which increases the possibility of competition and consequently complex dissociation (Figure 8, right graph). Also, the risk of in vivo complex dissociation increases for low affinity interactions, all of which results in an overestimation of the free fraction [17,18]. Consequently, the tool presented here can play a key role in evaluating whether the levels of free analyte measured in an assay are representative of the in vivo levels.

Figure 8. Impact of assay conditions on drug–target complexes.

The complex formation between drug and target is plotted on the y-axis as a function of target concentration. The impact of the competing capture tool concentrations (colored lines, shown in the legends below the figures) on complex formation (depicted on the y-axis) is shown: first, impact of increased concentrations – that is, a decrease in complexes is seen with increased concentrations of the capture tool (depicted by the orange boxes, arrows and text), and second, impact of increased affinity of the competing factor: increased affinity of the capture tool (shown as inhibition constant, KI), from 140,000 pM in the left panel to 140 pM in the right panel, also results in decreased complexes (depicted by the blue boxes, arrow and text).

KI: Inhibition constant.

Conclusion

The simulation tool developed (BiSim Tool) supports mathematical calculations of protein binding interactions occurring in biosamples and bioassays, thus contributing to a better understanding of the requirements, capabilities and limitations of assays. Although laboratory scientists can directly evaluate the tool on our publicly hosted website, our contribution of the source code to the public domain allows others to build on and improve the tool. The various case studies discussed here show the value of the different features of the simulation tool to understand the binding dynamics – for example, the impact of assay-related interferences on the equilibrium, which presents a large challenge for the correct quantitation of free analyte concentrations. By creating an improved understanding of what needs to be measured and what can be measured with the method, assays can be designed and developed more carefully, and data interpreted appropriately, while considering the intended use of those data. Altogether the BiSim tool allows cost savings during assay development, improved assay quality and tailored data generation with patient value in mind. It is our aspiration that the public release of the BiSim tool will contribute to faster and improved drug development at reduced costs, ultimately benefiting patients worldwide.

Author contributions

L Dejager initiated the concept and set up the tool requirements. S Banton developed the tool. M Penney developed the mathematical modeling. S Grootjans was the project manager. L Dejager, S Banton, M Penney and S Grootjans drafted the majority of the work. P Marques, G Rinikova and C Martin-Hamka provided substantial contributions to the conception and the development of the work. S Lory and ES Hickford critically reviewed the tool applications and manuscript and contributed to drafting of key aspects.

Financial disclosure

The authors have no financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript. This includes employment, consultancies, honoraria, stock ownership or options, expert testimony, grants or patents received or pending, or royalties.

Writing disclosure

No writing assistance was utilized in the production of this manuscript.

Acknowledgments

The authors thank Toon Babylon, Sucharita Shankar, Hans Ulrichts, John Smeraglia, Maarten Van Acker, Pieter Deurinck, Nassim Haddad, Julie Lescut and Dan Chapman for support and useful discussions and review during the process of tool generation. All authors were employed by UCB Biopharma when the tool was designed, created and tested & the article was drafted. Manuscript publication fees were provided by UCB Biopharma. The authors are current or former employees of UCB Biopharma and may hold stock options.

Competing interests disclosure

The authors have no competing interests or relevant affiliations with any organization or entity with the subject matter or materials discussed in the manuscript. This includes employment, consultancies, stock ownership or options and expert testimony.