Unlocking latent kinetic information from label-free binding

Abstract

Transient affinity binding interactions are central to life, composing the fundamental elements of biological networks including cell signaling, cell metabolism and gene regulation. Assigning a defined reaction mechanism to affinity binding interactions is critical to our understanding of the associated structure-function relationship, a cornerstone of biophysical characterization. Transient kinetics are currently measured using low throughput methods such as nuclear magnetic resonance, or stop-flow spectrometry-based techniques, which are not practical in many settings. In contrast, label-free biosensors measure reaction kinetics through direct binding, and with higher throughout, impacting life sciences with thousands of publications each year. Here we have developed a methodology enabling label-free biosensors to measure transient kinetic interactions towards providing a higher throughput approach suitable for mechanistic understanding of these processes. The methodology relies on hydrodynamic dispersion modeling of a smooth analyte gradient under conditions that maintain the quasi-steady-state boundary layer assumption. A transient peptide-protein interaction of relevance to drug discovery was analyzed thermodynamically using transition state theory and numerical simulations validated the approach over a wide range of operating conditions. The data establishes the technical feasibility of this approach to transient kinetic analyses supporting further development towards higher throughput applications in life science.

Introduction

Affinity interactions occur over an extraordinarily wide range in kinetics in order to support a diversity of life-sustaining processes. Understanding the associated structure-function relationship¹ requires assigning a defined reaction mechanism², which may be accomplished through kinetic analysis. Transient protein-protein interactions are ubiquitous in cell biology³ and almost all enzymes interact transiently with substrates⁴, or each other⁵, in order to promote high catalytic turnover. Indeed the majority of early chemical matter discovered in small molecule drug discovery are transient binding compounds. Generally, kinetic interaction constants are not predictable⁶ from knowledge of structure due to their complexity^1,6,7 but are routinely measured using biophysical techniques⁸. Transient kinetics are most often measured by lower throughput methods such as NMR⁹, or stop-flow spectrometry-based techniques^10–12, thereby restricting their use and limiting our understanding and exploitation of transient binding interactions. Label-free optical biosensing has become the standard for measuring direct binding of slow-to-moderate kinetics allowing mechanistic analysis^13–16 at reasonably high throughput. Indeed the approximate number of publications mentioning surface plasmon resonance (SPR), which is the most common detection mode, now approaches 70,000¹⁷. Therefore, the use of label-free biosensors to measure transient kinetic interactions holds high potential in many fields especially drug discovery. Briefly, conventional kinetic analysis of drugs reveals that an optimal therapeutic profile may depend on achieving a particular target-specific kinetic profile^16,18,19. This suggests that kinetic and mechanistic profiling should be exploited throughout early drug discovery in order to provide more optimal starting points for lead development by favoring a given kinetic “sweet spot”. At a minimum, we envision that mechanistic characterization of transiently bound early hits through kinetic profiling may prove more effective in discriminating artifactual binding^20–22 from tractable binding modes.

We have developed an approach that adapts real-time flow-injection-based biosensors to measure transient kinetics. Briefly, microfluidic geometries employed in current flow injection analysis (FIA)-based systems are relatively large (effective hydrodynamic diameter ≥ 50 μm) in order to maintain robust low pressure sampling and instrument developers²³ have focused on minimizing microchannel “non-swept dead volumes” in order to rapidly establish a uniform analyte concentration upon injection. Despite these efforts injection rise/fall regions remain >200 ms thereby limiting the uniform concentration approximation to kinetics >500 ms. We postulated that the kinetic limit of detection could be extended by including a hydrodynamic analyte dispersion term to model changing analyte concentrations within injection rise/fall regions. We evaluate this approach using test data generated by multiphysical numerical simulation and we report an experimental proof-of-principle relevant to drug discovery. We define moderate transient kinetics to be in the order of 50–500 ms and fast transient kinetics to be in the order of 1–50 ms. For a simple 1:1 binding interaction the equilibrium dissociation affinity constant is K_D = k_d/k_a (units, M), where k_a (units, M⁻¹ s⁻¹) is the association rate constant, k_d (units, s⁻¹) is the dissociation rate constant and τ = 1/k_d (units, s) is the residence time. The equilibrium response is R_eq = R_maxc/(c + K_D), where R_max (unit, RU) is the saturation response and c is the analyte concentration (units, M). We employed unmodified commercially available technology for experimental work but instrument modifications will be required for optimal implementation including, parallel sensing spots that are short in the flow direction, optimized dispersion profiles and higher time resolution.

Results

Injection-binding process

The time evolution of analyte concentration through the injection-binding process is shown in Fig. 1a and includes three compartments as described by Quinn^24,25 previously. In previous work a coiled capillary provided a third compartment for the generation of slowly evolving (i.e. >30 s) analyte gradients. Here we have not added additional capillaries/microchannels and instead demonstrate that rise/fall regions associated with washout of residual sub-μL dead volumes generate sub-second dispersion gradients that are well suited to the analysis of transient kinetics. By convention these gradient regions are discarded but here we extend the dynamic range of optical label-free biosensing by incorporating these rise/fall regions using a three-compartment model that now allows accurate kinetic measurements into the millisecond range. Briefly, analyte at a fixed concentration c₁, is injected into a microfluidic channel and disperses into pre-existing buffer (compartment 3), because of both convective and diffusive mixing, and is quantified by a dispersion coefficient k. The analyte dispersion arrives at the flow cell as a time-dependent concentration gradient c_2(t), where a fraction of the analyte diffuses through a stagnant boundary layer (compartment 2) to reach the target-coated sensing surface (compartment 1) at concentration c_3(t). There the binding reaction of analyte with bound target is measured in real-time through label-free optical detection. Figure 1b shows the analyte concentration profile for an instant rise/fall injection (left) with a uniform concentration assumed throughout the injection. However, in reality the injected sample will disperse into a non-negligible dead-volume to produce a gradient in analyte concentration at the rise/fall regions of injections. Turbulence exists briefly upon injector actuation due to transient pressure/temperature fluctuations and the dead-volume upstream of the flow cell causes a time offset with respect to analyte arrival at the flow cell thereby allowing re-establishment of laminar flow, a pre-requisite to reproducible kinetic measurements.

Figure 1.

Compartment models in label-free biosensing. (a) Numerical simulation of a full injection-binding process with numerically computed analyte concentration gradients within the microchannel (length, l and height, h, where x is the distance along the channel length) and associated flow cell rendered as a color gradient. Concentrations increase from blue-to-red and the parabolic velocity field within the flow cell is depicted by vertical arrows with zero flow at the walls. (b) Simulated analyte concentration profiles (red) for instant rise/fall injection of a uniform concentration and a more realistic gradient rise/fall. Turbulence upon injector actuation is indicated by grey panels. (c) 1:1 binding interaction models. The associated model parameters are defined in the introduction. (d) Experimentally measured bulk RI-based dispersion curves for compounds ranging in M_r from 342 Da to 670 kDa. Equation S1 (see supplemental information) is a dispersion equation and was fit locally (average R² = 0.998) to the fall dispersion points and a subset of five curves is shown. The inset panel shows k as a function of M_r^1/3 with a linear fit given by k = 2.09 × 10⁻⁷ M_r^1/3 + 1.34 × 10⁻⁶). Here all eight M_r standards were included in duplicate.

A simple 1:1 Langmuir model (Fig. 1c) is given by equation (1) and assumes that analyte gradients do not develop. A 1:1 two-compartment model couples equations (1) and (2) and assumes that a gradient in analyte concentration can exist at the surface. This gradient results in mass transport limitation (MTL) defined by a single mass transport rate constant k_t, which limits the flux of analyte through the boundary. Similarly, the dispersion gradient has a significant impact at short injection times requiring a three-compartment model that couples equations (1) and (2) with a microchannel dispersion term, as given by equation (3). In order to experimentally assess the quality of analyte dispersions, a series of M_r standards were injected and the resulting bulk refractive index (RI) dispersions (Fig. 1d) showed relatively weak M_r-dependence in the rise-dispersion, indicating convection-limited dispersion, and the fall dispersions returned dispersion constants (k) that were proportional to M_r^1/3, indicating diffusion-limited dispersion.

Low MTL/Moderate transient regime

Transient binding response curves (Fig. 2a–d) were simulated for k_a = 1 × 10⁶ M⁻¹s⁻¹, k_d = 5 s⁻¹ and R_max = 1.5 RU (equivalent to 3 pmoles mm⁻²) producing <1.5% MTL. Typical sample contact times are >10 s but here we have reduced this time to just 1 s, a more suitable time scale for analysis of moderate transient kinetics. The time-dependent analyte concentration profiles for each injected concentration (Fig. 2a) were fit to equation (S2) (supplemental information) producing superimposable fits where the residual difference was < 1%. The associated binding curve set (Fig. 2b) was fit to a two-compartment model producing high systematic deviation (>40%). Each curve was normalized with respect its maximum response causing the curve set to superimpose (Fig. 2c) other than during the association phase. A three-compartment model was fit (Fig. 2d) returning kinetic constants with <1% error relative to “true” values and <0.05% SE associated with each parameter. The average squared residual χ² was 7.4 × 10⁻⁴ RU and the maximum systematic residual deviation was <2%.

Figure 2.

Validation of the three-compartment model using numerically simulated binding curves. The interaction constants for all simulated binding curves are given in table S1 (Supplemental information). (a) Overlay of analyte dispersion profiles (i.e. color) for six serial-doubling dilution injections fit to equation S2 (supplemental information). (b) Affinity binding curves (color) for the analyte injections in (a) binding to surface-bound target and fit to a two-compartment model (black), where R_max was 1.5 RU. (c) Binding curves in (b) normalized with respect to each respective equilibrium response, R_eq. (d) Affinity binding curves in (b) fit to a three-compartment model. (e) Affinity binding curves for the interaction of analyte at R_max settings of 0.5 RU (red), 5 RU (blue) and 50 RU (black). %MTL decreases as surface binding progresses as indicated by the broken lines color matched to each respective R_max condition. (f,g,h) Interaction parameters returned from a fit to a three-compartment model as a function of %MTL, for k_a (f), k_d (g) and k_t (h), respectively, where k_t was pre-calculated (red) from equation S3 (supplemental information), or fit globally (blue). The dotted line is the “true” parameter value and the error bars are ± SE. All χ² values were < 0.08%. (i) Parameter correlation analysis for three-compartment model (Refer to supplemental information for more details).

Variable MTL/ moderate transient regime

A time-dependent MTL factor was plotted for three simulated dispersion curves, for a serial ten-fold increase in R_max at a fixed analyte concentration, and is shown along with each respective binding curve in Fig. 2e. The simulated curve set in Fig. 1b was replicated at five R_max values corresponding to a range of 11–66%MTL. A three-compartment model was fit to each curve set and the parameter values returned from the fit were plotted with respect to %MTL. Figure 2f–h. Pre-calculating k_t reduced error in kinetic parameters to <5% when ≤ 50%MTL. The error associated with global fitting of k_t decreased with increasing %MTL and increased exponentially at <20% MTL becoming undefined at 11%MTL. The parameter correlation analysis in Fig. 2i was performed at 50% MTL and shows that kinetic constants were highly correlated and this was reduced significantly when k_t was pre-calculated.

Experimental characterization in a moderate transient regime

Binding of a ~3.8 kDa peptide to a maltose binding protein (MBP)-tagged target was evaluated over a dose-response range at seven temperatures from 5–35 °C and the fitted binding curve sets (Fig. 3a). The average SE associated with k_a and k_d was 1.7% and 0.80%, respectively, and the average χ² over all data sets was 0.75%. The interaction became transient over the temperature range with τ decreasing ~10-fold to τ ≈ 0.08 s, and is also plotted in terms of k_d in Fig. 3b. k_a was relatively insensitive to temperature, increasing by ~1.4-fold from 1.0 × 10⁶ M⁻¹ s⁻¹ over the temperature range. Greater random variation was observed in k_a estimates relative to k_d and may have been related to convection dominated dispersion during the rise dispersion (Fig. 1d) and possibly greater variation in pressure-induced microchannel compliance. Indeed, k_a was more reliably estimated from k_a = k_d/K_D which was employed for Erying analysis. The Erying plots (Fig. 3b) for both kinetic constants were linear, as was the Van’t Hoff analysis of the K_D, and the resulting thermodynamic parameters are shown in Fig. 3c as energy transitions with respect to the reaction coordinate.

Figure 3.

Experimentally measured temperature-dependence of peptide binding to MBP-tagged target molecule using analyte gradient injections. (a) Three-compartment model fit containing a bulk RI term and fit to each binding curve set with k_a, k_d and R_max fit as global parameters. k_t was pre-calculated from equation S3 (supplemental information) at each temperature and fit as a constant. Each curve set contains six curves produced from injection of a peptide over a serial doubling dilution range replicated in duplicate. High reproducibility and goodness of fit cause both the replicate curves (color) and the fitted model curves (black) to appear superimposable. (b) Thermodynamic models were fit to the data generating a thermodynamic profile. (Top left) Exponential increase in k_d with temperature k_d = 0.94^(0.0747/T). (Bottom left) Linear Erying plot for k_d. (Top right) Linear Erying plot for k_a. (Botom right) Linear Van’t Hoff plot. (c) Energy transitions at 25 °C as a function of the reaction coordinate for the thermodynamic parameters obtained from (b).

Fast transient regime

A numerical simulation of the analyte concentration across the height of the flow cell is shown at 2 ms intervals for an analyte dispersion injection in Fig. 4a. The flow cell can generally be taken as the region of the microchannel surrounding the sensing region along with all other target-coated regions upstream of the sensing region. The concentration rises to a maximum c₁ and returns to zero when the gradient falls, where the fall concentration profiles are inverted (∪) relative to the preceding rising gradient profiles (∩). At a flow rate of 0.1 m s⁻¹ the approximate washout time t_wash ≈ 3 ms (assuming t_wash ≈ 3t_tr, where t_tr is the time to traverse the flow cell) and a parabolic gradient profile was observed in the bulk flow with thin diffusion boundary layers at each wall.

Figure 4.

Simulation study of boundary layer formation and its effect on estimation of fast transient kinetics. (a) Superimposed analyte concentration profiles across the height of the flow cell at 2 ms intervals over the course of a single analyte injection. (d) Boundary layer curves (color) at a fixed concentration c₁ = K_D over a serial-doubling range in R_max, where reaction kinetics were made semi-infinite allowing k_t to drive occupancy. A three-compartment model was fit (superimposed black curves) returning k_t, where R_max and K_D were held constant. C R_max and %MTL as a function of τ_ss for the binding curves in (b). (d) Three-compartment model fit (superimposed black curves) to numerically simulated curves (color) with low R_max and with pseudo-random noise added as a normal distribution where root mean square standard deviation ≈ 0.015 RU, which is typical of current state-of-the art systems. The curve set was simulated over a serial-doubling dose response range, where R_max ≈ 1.44 RU. (e,f,g) Interaction parameters returned from fitting a three-compartment model to the curve set in (d) replicated over a range in R_max producing a wide %MTL range, for k_a (e), k_d (f) and k_t (g), respectively. k_t was pre-calculated (red) from equation S3 (see supplemental information), or fit globally (blue). The dotted lines represent the “true” parameter value from the numerical simulation and the error bars are ± SE associated with fitting the parameter. h. χ² values for the curve fits in (e), (f) and (g).

Boundary layer quasi-steady state approximation (BLQSSA)

We numerically simulated kinetic binding curves with pseudo-infinite kinetics (i.e. k_a = 10¹⁰ m⁻¹s⁻¹, k_d = 10⁴ s⁻¹, τ = 0.1 ms) allowing boundary layer mass transport to dictate the analyte binding rate. Under these conditions the observed curvature became a function of R_max, R_eq and k_t and a three-compartment model fit is shown in Fig. 4b. A quasi steady-state time, in this case associated with the development of the transport limited boundary layer, was approximated by τ_ss ≈ 1.5τ_t, with a boundary decay time approximated as τ_t ≈ R_max/k_t*R_eq. Variation in both R_max and %MTL with respect to τ_ss is shown in Fig. 4c. The influence of high MTL on transient kinetics was evaluated by fitting a three-compartment model to curves generated by numerical simulations for a fast transient interaction where k_a = 1 × 10⁷ M⁻¹s⁻¹ and k_d = 100 s⁻¹ where pseudo-random Gaussian noise typical of current instrumentation was added, as shown in Fig. 4d. %MTL was minimized by confining the surface bound target to small sensing regions specifically avoiding coating upstream of the sensing regions. These simulations were repeated over a range in R_max producing curve set exhibiting low-to-high %MTL. Kinetic constants returned from model fitting are shown (Fig. 4 e,f), where k_t was fit either globally, or held constant at a value calculated from equation (S3). The error in both k_a and k_d was ≤ 4% when k_t was held constant and χ² values were reduced significantly (Fig. 4h) supporting this approach. Indeed k_t values returned from global fitting (Fig. 4g) were highly uncertain (~250% error) when %MTL was low, again implying that low MTL destabilizes the fit as indicated by higher χ² values (Fig. 4h).

Discussion

The development and application of computational models of label-free biosensing is well understood with several insightful publications^26,27. A full numerical model is sophisticated and not suitable for fitting experimental data for estimation of reaction kinetics. It does however generate binding curves defined by absolute kinetic values, while retaining the complexities of experimental data, thereby providing an absolute standard for quantitative testing of simpler mechanistic models. The time evolution of the injection-binding process was determined by multiphysical simulation as shown in Fig. 1a. Conventionally a uniform concentration profile (Fig. 1b) can be assumed to reliably approximate experimental binding curves but this assumption does not hold for rapid injections where rise/fall zones containing analyte gradients constitute a significant fraction of the injection time. When SPR-based biosensors first became available over three decades ago, a simple 1:1 Langmuir model (equation (1) in Fig. 1c), which assumes that analyte gradients do not develop, was thought sufficient to model simple binding interactions. However, it was soon realized that the analyte flux to the sensing surface can become limiting and a 1:1 two-compartment model (coupled equations (1) and (2) in Fig. 1c) has since become the gold standard and was validated by computational modeling²⁸. In the two-compartment model the analyte concentration at the surface was approximated by adding a mass transport reaction defined by a single mass transport rate constant that drives a reversible flux of analyte through the diffusion boundary layer to the sensing surface. When the binding reaction flux coefficient k_r = k_a (R_max-R) at the sensing surface is in the same order, or higher, than the mass transport coefficient k_t, then significant mass transport limitation MTL = k_r/(k_r + k_t) will occur causing an analyte gradient to form. Another unrelated analyte gradient develops through dispersive mixing within microchannel(s) en-route to the flow cell but is invariably neglected because it is assumed fast relative to reaction kinetics. This assumption does not hold for transient reaction kinetics and a gradient dispersion term must be included as a three-compartment model, by coupling equations (1), (2) and (3) (see Fig. 1c), providing a more realistic model of the injection-binding process. These analyte dispersions (Fig. 1d) were highly reproducible and evolved in a sub-second time regime suitable for the study of transient reaction kinetics. A set of numerically generated dispersion curves (Fig. 2a) were found to fit well to equation (S2) (supplemental information) and this equation was therefore selected for all kinetic modeling.

In contrast to standard uniform injections, both dispersion and binding kinetics contribute to observed curvature of binding response curves in a transient time regime. Briefly, for a simple 1:1 interaction, an idealized uniform analyte injection should produce monotonic binding curvature towards surface saturation, as implied by the observed reaction rate k_obs = k_ac + k_d, where c = constant. However, dispersion defines time-dependent changes in analyte concentration that induce additional concentration-dependent curvature that becomes apparent by fitting a two-compartment model, which cannot account for dispersion and therefore shows high systematic deviation (Fig. 2b). More importantly, inappropriate model fitting can easily occur when τ < data acquisition rate because all response points are at quasi-steady-state²⁵ and any observed curvature must be derived from dispersion alone, yet can easily be mistaken as kinetic curvature. However, by normalizing each curve with respect to R_max, curves without kinetic curvature superimpose and curves possessing kinetic curvature exhibit non-superimposable dose-dependent curvature in the association phase (Fig. 2c). The three-compartment model accounts for dispersion and producing an ideal fit (Fig. 2d) that returned kinetic constants with <1% error.

MTL scales with the number of free target sites (R_max-R), which decrease as occupancy increases, allowing kinetic curvature to exist on approach to saturation (Fig. 2e) even at very high MTL conditions. In practice, highly MTL binding curves are often recorded before assay conditions have been optimized for a given panel of interactants and confining the fitted model to these”kinetic regions” can provide an approximate kinetic analysis. MTL delays both association, and dissociation, phases and therefore increases parameter correlation yet does not greatly impact parameter return at moderate level²⁸. In order to demonstrate that this holds for transient kinetics, we generated multiple simulated curve sets over a wide range of MTL conditions and the parameter values returned from fitting the three-compartment model were determined (Fig. 2f,g). We found that k_t estimates from equation (S3) was 1.20 × 10⁸ ms⁻¹, which was in good agreement with the value returned from a global fit (i.e. 1.21 × 10⁸ m s⁻¹) at 67% MTL. While the three-compartment model performed well for all MTL levels tested, the error in kinetic constants was at a minimum at ~30% MTL, suggesting that pre-calculation of k_t²⁹ may increase fit stability. In the case of the standard two-compartment model it has been shown that kinetic constants are highly correlated when k_t is fit as a global parameter yet the high information content of binding response curves ensures accurate parameter return. Similarly, high parameter correlations were observed for the three-compartment model (Fig. 2i) at 50% MTL but these decreased significantly upon pre-calculation of k_t improving the stability of the fitted model.

In contrast to numerical simulations, kinetic binding constants for actual bimolecular interactions are not absolute quantities thereby complicating model validation through experiment. However, kinetic binding becomes more transient with increasing temperature, providing a range of kinetic constants for a given compound that are related through a common thermodynamic relationship. This relationship can be exploited to validate our model since inaccurate transient kinetic constants would be identified as outliers from the fitted thermodynamic profile. For this analysis (Fig. 3) we employed a peptide-protein interaction exhibiting kinetic constants approaching the current kinetic limit of detection at low temperatures and unmeasurable transient kinetics at high temperatures. Binding was evaluated from 5–35 °C and each binding curve set was fit to a three-compartment model (Fig. 3a) allowing thermodynamic analysis of the resulting interaction constants (Fig. 3b,c). The absence of outliers implied that transient kinetics was well determined from application of the three-compartment model and associated methods. This transient interaction has favorable enthalpy and there is a large entropy cost associated with activation of the transition state. Such mechanistic information may be exploited to validate and/or prioritize transiently bound chemical matter in early drug discovery and this first proof-of-principle experimentally demonstrated a 6-fold extension of kinetics into the transient regime.

Kinetic analysis of fast transient binding curves depends on high resolution injection clocking and data acquisition. Briefly, the time required to attain steady-state defines the relevant time regime required to measure transient kinetic constants and, by solving equation (1), assuming a fixed concentration, the time to attain a given steady-state occupancy is t_Θ = -Ln (1-Θ)/(k_ac + k_d), where Θ is fractional progress towards steady-state. Therefore at a concentration c₁ = K_D, the 95% steady-state time Θ = 0.95 will be approximately 1.5τ. This defines a threshold time resolution for the appearance of kinetic curvature and implies that a higher time resolution (e.g. ≤ 0.5τ) is needed for resolving kinetics. The maximum data acquisition rate available in currently available systems (i.e. ≤ 40 Hz) limits the measurement of transient kinetics and >10-fold increase may be required to measure transient binding kinetics routinely in drug discovery applications. Analyte concentration gradients in the bulk flow along the length, or height, of the flow cell must be minimal in order for mechanistic binding interaction models to hold. In practice this requires that the time delay associated with sample traversing the sensing regions t_tr ≪ τ. Numerical simulations (Fig. 4a) show that at a low flow rate a parabolic analyte concentration profile across the height of the flow flow cell is produced due to the influence of flow cell washout. At higher flow rates, washout becomes rapid relative to the evolution of the analyte gradient allowing a quasi-steady state concentration across the height of the flow cell.

With the improved sensitivity of current biosensors, it has become practical to employ a low R_max (1–5 RU) such that BLQSSA will hold under most practical experimental conditions. However, higher R_max values are often unavoidable in order to offset other sources of interference and this produces thicker boundary layers that take longer to form. This allows significant analyte binding before the boundary layer equilibrates, which violates BLQSSA. We numerically simulated the formation of the boundary layer over a range in R_max, assuming pseudo-infinite reaction kinetic constants establishing approximately 100% MTL, thereby allowing pre-steady-state development of the boundary layer to be studied using the three-compartment model (Fig. 4b). The steady-state time associated with the development of the boundary layer was approximated by τ_ss ≈ 1.5τ_t, and the observed boundary decay time was found to be well approximated by τ_t ≈ R_max/k_t*R_eq. τ_ss increased linearly with increasing R_max (Fig. 4c), extending into the moderate transient kinetic regime (>50 ms). For transient kinetics we might expect highest interference when τ ≤ τ_ss and to test this we numerically simulated experimentally realistic binding curves over a range in R_max using an optimized instrument design and analyzed the data using the three-compartment model (Fig. 4d). Interestingly, global fitting overestimated k_a, and k_d (Fig. 4e,f) by ~1.2-fold at <10% MTL, which coincides with transient formation of the boundary layer at <3 ms. The error in k_t (Fig. 4g) followed an almost identical dependence on %MTL but was underestimated by as much as 2.5-fold. Interestingly, the error in k_a, k_d and k_t and associated χ² (Fig. 4h) were successfully negated over the full MTL range tested by simply pre-calculating k_t, which is well proven in SPR-based calibration-free concentration measurement^29,30.

In summary a temporal rank order where t_grad >τ> t_wash and t_grad >τ_ss> t_wash is required where flow rate, and microchannel volume are chosen to achieve appropriate parameter ranges. As mentioned earlier, the analyte concentration in the flow cell can be assumed to follow a quasi-steady-state concentration when t_grad≫ t_wash. Essentially, the gradient should evolve gradually allowing occupancy to progress relatively slowly. Indeed the maximum change in occupancy observed in this simulation over the time period t_wash was <1% assuming c₃ = K_D. Taken together the measurement of transient kinetic processes requires avoiding abrupt changes in concentration and associated target occupancy thereby maintaining the quasi-steady state boundary layer assumption. In summary, we have developed a label-free biosensing approach to measure transient kinetics reporting an experimental proof-of-principle showing mechanistic characterization through transition state theory. These data and the numerical simulation studies establish the technical feasibility of the approach and support further development towards higher throughput applications in life science. It should also be noted that transient binding requires <1 s of sample exposure and may therefore be well suited to applications in high throughput screening.

Methods

General experimental

Assays were conducted using a Biacore S200 (GE Healthcare Bio-Sciences AB, SE-751 84, Uppsala, Sweden). All reagent coupling kits and sensors were from GE Healthcare. All reagents were from Sigma-Aldrich (3300 S. Second St., St. Louis, MO 63118, USA) unless otherwise stated. The MBB-tagged target were expressed recombinantly and purified in-house using standard protocols. All experiments were performed using a buffer containing 50 mM 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid, (HEPES), pH 7.5, containing 0.15 M sodium chloride and 0.2 mM tris(2-carboxyethyl)phosphine (TCEP). The 3,879 Da peptide was synthesized by GenScript (860 Centennial Ave, Piscataway, NJ 08854, USA) and purity was >95% by HPLC.

Injection-binding process

A series S CM5 chip was employed and the analysis temperature was set to 20 °C. A series of dextran molecular weight standards were prepared from 1–10 mg/ml in standard buffer. Sucrose (M_r = 342 Da) was included as a low molecule M_r standard. Each standard was injected at 100 μL/min for 2 s in duplicate and each curve was normalized with respect to the maximum response for data analysis. Here we neglect double referencing and analyze bulk RI response curves without a binding component in order to characterize the quality of transient analyte dispersions.

Experimental characterization in a moderate transient regime

A series S CM5 chip was employed and the analysis temperature was set to 5 °C. A MBP-tagged target (Mr ~52.3 kDa) was coupled according to manufacturers recommendations with the following changes. Seven serial-doubling dilutions of the 3,879 Da peptide (synthesized by Genescript, encompasses an SH3 recognition region) were prepared from 100 μM in standard buffer. Each sample was injected at 100 μL/min for 2 s in duplicate along with blank buffer injections for referencing thereby generating a curve set containing 14 binding curves. Each curve set was referenced against an averaged blank curve and fit to a three-compartment model using Biaevaluation 3.0 (GE Healthcare Bio-Science AB). This was repeated for at analysis temperatures of 10 °C, 15 °C, 20 °C, 25 °C, 30 °C and 35 °C.

Experimental characterization- data analysis

The analysis of transient kinetic curve sets was essentially identical to conventional methods other than the inclusion of a dispersion term. The Biacore S200 supports 40 Hz data acquisition rates. The sensing regions of the Biacore S200 are in series causing a time delay to exist upon arrival of sample at the sensing region relative to its reference that overlaps entirely with the rise/fall regions needed for transient kinetic analysis. Single reference against a blank, as opposed to standard double referencing, prevented interference and the response curve contained both a bulk RI response and a binding response. The analytes bulk RI component is a good approximation of the analytes dispersion profile and a three-compartment model may be fit to this two-component response curve set by adding a bulk RI analyte dispersion defined by equation (S2). When fitting, we first determined both K_D and R_max from fitting a conventional steady-state model after double referencing. We fit a three-compartment model, holding R_max constant and global constraint of kinetic parameters. The slope coefficients (b, d) of equation (S2) were also constrained to global values while time coefficients (a, c) were fit locally.

Experimental characterization- thermodynamic analysis

The temperature dependence of the kinetic and affinity constants obtained from fitting the three compartment model was analyzed by Eyring and Van’t Hoff analysis, respectively. The linear form of the Van’t Hoff equation is ln(K_D) = ΔH°/(RT)−ΔS°/R, where ΔH° and ΔS° are the standard enthalpy and entropy of binding, respectively. T is the absolute temperature and R is the universal gas constant, 1.987 cal/ (mol K). A linearization plot of Ln(Kd) versus 1/T was plotted and fit to the Van’t Hoff equation giving a slope ΔH°/R and an intercept -ΔS°/R. The linear form of the Eyring equation for the association process is given by Ln(k_ah/k_BT) = −ΔH^ǂ/RT + ΔS^ǂ/R where ΔH^ǂ and ΔS^ǂ are the changes in transition state enthalpy and entropy, respectively, associated with the formation of the affinity complex, h is Planks constant = 6.63 × 10⁻³⁴ J s and k_B is Boltzmann constants = 1.38 × 10⁻²³ J K⁻¹, respectively. Substituting k_d for k_a gives the transition state analysis for complex dissociation. A linearization plot of Ln(k_ah/k_BT) versus 1/T returns -ΔH^ǂ/R and ΔS^ǂ/R from the slope and intercept, respectively. The free energy was calculated from ΔG = ΔH -TΔS and was plotted as a function of the reaction coordinate³¹. The associated standard error bars are the SE of the fit associated with each parameter from the linearization plots. The analysis was performed using Graphpad Prism version 6 (GraphPad Software, Inc. 7825 Fay Avenue, Suite 230, La Jolla, CA, 92037, USA).

Modeling

For details on modeling dispersion gradients, influence of mass transport limitation, general data analysis and statistics, correlation analysis and numerical simulations please refer to supplementary information that accompanies this paper.

Author contributions

J.G.Q. conceived the approach and supervised all work, conducted experiments, performed all numerical simulations, was the primary author of the manuscript and coordinated preparation of the manuscript with the secondary authors. M.S., J.M.B., A.F. and M.M.M. contributed to planning and execution of experiments and were secondary authors in writing the manuscript.

Competing interests

The authors declare no competing interests.

Supplementary information

is available for this paper at 10.1038/s41598-019-54485-4.