Sensing Methodology for the Rapid Monitoring of Biomolecules at Low Concentrations over Long Time Spans.

Studies on the dynamics of biological systems and biotechnological processes require measurement techniques that can reveal time dependencies of concentrations of specific biomolecules, preferably with small time delays, short time intervals between subsequent measurements, and the possibility to record over long time spans. For low-concentration biomolecules, these requirements are very challenging since low-concentration assays are typically slow and require new reagents in every assay. Here, we present a sensing methodology that enables rapid monitoring of picomolar and sub-picomolar concentrations in a reversible affinity-based assay, studied using simulations. We demonstrate that low-concentration biomolecules can be monitored with small time delays, short time intervals, and in principle over an endless time span.

Biological systems and biotechnological processes exhibit time dependencies that are at the most basic level regulated by the dynamics of constituting biomolecules, such as small molecules, hormones, proteins, and nucleic acids. This calls for measurement technologies that allow the monitoring of biomolecular concentrations, for instance, to serve fundamental research on biological and biomedical dynamics,[1−6] to enable the development of patient monitoring strategies based on real-time biomolecular data,[7−10] and to enable the development of closed loop control strategies in biotechnological applications.[11−16] Desirable characteristics of a generic monitoring technology are (1) precise and specific measurements, (2) small time delays between sampling input and data output, (3) short time intervals between successive measurements, and (4) a long total time span over which time-dependent biomolecular concentration data can be recorded.

It is a fundamental challenge to develop a technology that can rapidly monitor low-concentration biomolecules over long time spans. Sensitive assays are available, such as ELISA and flow cytometry,[17−20] but in these assays, new reagents are needed for every sample that is taken. The repeated consumption of reagents complicates applications where biomolecular concentrations need to be monitored over long time spans. On the other hand, sensing technologies that can operate without consuming reagents, such as surface plasmon resonance,[21] redox cycling,[22] and quartz crystal microbalance,[23] have not been designed for monitoring biomolecules at low concentrations, such as in the picomolar and sub-picomolar range.

A generic principle to measure specific biomolecules at low concentrations is by using the biochemical affinity between specific binder molecules (such as antibodies and aptamers) and the biomolecule that is to be detected (the analyte). The specificity originates from molecular interactions such as charge, hydrogen bonding, van der Waals forces, and hydrophobic and steric effects.[24] To be able to measure biomolecules at low concentrations with high precision, binder molecules are needed that have strong interactions with the analyte, which corresponds to high binding energies, low equilibrium dissociation constants Kd, and low dissociation rate constants koff.[25,26] However, this conflicts with the desire to have small time delays because low dissociation rate constants would imply a need for long incubation times to reach equilibrium.[24−26] Furthermore, low dissociation rate constants result in a slow reversibility, which conflicts with the wish to enable short time intervals between successive measurements.

In this paper, a sensing methodology is presented that enables rapid monitoring of low-concentration biomolecules in the picomolar and sub-picomolar range, studied using simulations. The method is based on the use of binder molecules with a high affinity in a limited-volume assay, with a fully reversible detection principle and time-dependent sampling of the analyte of interest. The system allows optimal trade-offs between time characteristics and precision. We present the measurement concept, time dependencies of sensor signals, and a comprehensive analysis of the achievable time characteristics and precision as a function of sensor design parameters. We demonstrate that the sensing methodology enables precise quantification of low biomolecular concentrations, with time delays and interval times that are much shorter than the time dictated by the dissociation rate constant of the binder molecules. Furthermore, due to the reversible detection method, measurements can in principle be performed over an endless time span.

Basic Concept of the Sensing Methodology

The basic concepts of the sensing methodology are sketched in Figure . The sensing system features time-dependent sampling of the analyte of interest, provided by a time-controlled analyte exchange between a biological or biotechnological system of interest and a measurement chamber (see Figure a). The measurement chamber contains specific binder molecules from which signals are recorded. The data are translated into concentration–time profiles, which should resemble as close as possible the true concentration–time profile of analyte molecules in the system of interest. During the exchange of analyte molecules, various processes occur, such as mass transport by advection and diffusion and association and dissociation of analyte molecules to binder molecules (see Figure b). To illustrate the concept, a generic rectangular measurement chamber is assumed with height H, width W, and length L. The bottom surface is provided with binder molecules, where association and dissociation of analyte molecules occurs. The rates of association and dissociation depend on the association rate constant kon, the dissociation rate constant koff, the density Γb of binder molecules, and the analyte concentration Ca at the sensor surface. These processes result in a time-dependent density γab of analyte–binder complexes, also represented as a fractional occupancy f of binder molecules occupied by analyte molecules: f = γab/Γb. Variables γab and f are changing as a function of analyte concentration and time. In an affinity-based sensor, the observed sensor signal scales with f; therefore, f is used in this paper as the sensor readout parameter to determine the analyte concentration. Analyte exchange between the system of interest and the measurement chamber is facilitated by diffusion or a combination of diffusion and advection. A net diffusive molar flux Ja (orange gradient) is caused by concentration differences between the system of interest and the measurement chamber and by concentration differences within the measurement chamber. Advective mass transport of analyte molecules into the measurement chamber is facilitated by a developed laminar flow profile with flow rate Q and mean flow speed vm (black arrows). Here, it is assumed that diffusive transport occurs in both the longitudinal (x-direction) and the lateral direction (y-direction) and scales with the diffusion coefficient D, while advective transport occurs only in the longitudinal direction and scales with the mean flow speed vm.

Figure 1

Concept of the sensing methodology for the rapid monitoring of low analyte concentrations. (a) Sensing system for biomolecular monitoring. Analyte molecules are exchanged between a biological or biotechnological system of interest and a measurement chamber. The data result in a concentration–time profile which should correspond as close as possible to the true analyte concentration-time profile in the system of interest. (b) Geometry of the measurement chamber, with height H, width W, and length L. A reaction rate at the sensor surface is caused by the association and dissociation between analyte molecules (orange) and binder molecules (brown), described by the association rate constant kon, the dissociation rate constant koff, the total binder density Γb, the analyte concentration Ca near the sensor surface, and the density of analyte–binder complexes γab. Analyte exchange is facilitated by diffusion and advection, where diffusion occurs in both x- and y-direction with the diffusion coefficient D, resulting in a net molar flux Ja, and where advection occurs in the x-direction only, with a developed flow profile with flow rate Q and a mean flow speed vm. (c) Time profile of the sensor response for low analyte concentration (Ca,0 < Kd), for two conditions: infinite-volume and limited-volume assays. For the infinite-volume condition, the time-to-equilibrium τ is determined by koff, while for the limited-volume condition, τ is determined by the effective binder concentration. (d) Biomolecular monitoring using a limited-volume assay involves repeated cycles with two phases. In phase 1, analyte molecules are exchanged between the system of interest and the measurement chamber. In phase 2, analyte molecules are not exchanged between the system of interest and the measurement chamber. The time-dependent signal is recorded in the middle of the measurement chamber at distance L/2 from the entrance. The limited-volume condition gives a time dependence of the analyte concentration in the measurement chamber: a decrease over time (depletion) or an increase over time (repletion), depending on the input analyte concentration Ca,0 and the initial fractional occupancy finit of binder molecules by analyte molecules. The input analyte concentration Ca,0 is derived from the measured time-dependent fractional occupancy f.

Figure c sketches two different sensor designs, namely, an infinite-volume assay and a limited-volume assay. The graphs visualize the fractional occupancy f of binder molecules occupied by analyte molecules as a function of time, with the corresponding characteristic time-to-equilibrium τ, defined as the time needed to attain 63% of the difference between the starting level and the equilibrium level of f (see Note S2). In an infinite-volume assay, continuous analyte exchange is enabled between the system of interest and the measurement chamber, where the system of interest is assumed to be much larger than the measurement chamber. The continuous analyte exchange could, for example, be facilitated by diffusive analyte transport across a contact area between the system of interest and the measurement chamber, while another configuration may involve a continuous flow of sample fluid provided into the measurement chamber from the system of interest. When the analyte exchange is effective and gives negligible time delay, then the analyte concentration at the sensor surface (Ca) is equal to the input analyte concentration (Ca,0). In the case of low analyte concentrations (Ca,0 < Kd), the infinite-volume assay condition leads to a characteristic time-to-equilibrium τ ≅ 1/koff (see Note S2). This implies that the time-to-equilibrium is determined by the dissociation rate constant koff, so this time is long when the binder molecules strongly bind to the analyte molecules.

The sensor design with a limited-volume assay has very different properties. Here, analyte exchange between the system of interest and the measurement chamber is enabled for a limited time. After this analyte exchange, the binder molecules in the measurement chamber interact with only a limited sample volume and therefore with a limited amount of analyte molecules. Due to this limited volume, we can now define an effective volumetric concentration of binder molecules Cb,0 = Γb/H, which is based on the number of binder molecules in the measurement chamber and the volume of the measurement chamber. When Cb,0 is high, with Cb,0 > Ca,0 and Cb,0 > Kd, then the time-to-equilibrium τ of the assay becomes dominated by the high concentration of binder molecules. When diffusional transport delays can be ignored, then the time-to-equilibrium of the assay equals τ ≅ 1/(konCb,0) (see Table and Notes S1 and S2). Thus, the time-to-equilibrium of the limited-volume assay is determined by the association rate constant and the effective volumetric concentration Cb,0 of binder molecules, which leads to equilibrium timescales that are much shorter than the time-to-equilibrium of the infinite-volume assay.

Table 1

Standard Parameter Values Used in the Finite-Element Simulationsa

	parameter	value	description
input	H	200 μm	measurement chamber height
	D	10–10 m2 s–1	diffusion coefficient of the analyte molecule
	Γb	10–9 mol m–2 (600 μm–2)	binder density
	koff	10–4 s–1	dissociation rate constant
	kon	106 M–1 s–1	association rate constant
	Ca,0	0.1 pM	input analyte concentration
derived	τD = H2/D	400 s	characteristic diffusion time
		200 s	characteristic reaction time for limited-volume assay with Cb,0 ≫ Ca,0 and Cb,0 ≫ Kd
	Cb,0 = Γb/H	5 nM	effective volumetric binder concentration
	Kd = koff/kon	100 pM	equilibrium dissociation constant
	α = Γb/HKd = Cb,0/Kd	50	acceleration factor: reduction factor of the time-to-equilibrium of a limited-volume assay with τ(H, Γb), compared to an infinite-volume assay with τ(koff)
	Da = τD/τR,LV = konΓbH/D	2	Damköhler number

Details about the simulations are described in Note S4. Additional standard parameter values are given in Table S1 (see Note S1).

In monitoring applications, it is preferred to record measurements with one and the same sensor over long time spans. To realize the limited-volume assay principle in a monitoring application, the sensor needs to be switched between two different conditions: an open condition and a closed condition. In the open condition, analyte molecules are exchanged effectively between the system of interest and the measurement chamber, as sketched in Figure a,b (also see Note S5). In the closed condition, analyte molecules are not exchanged between the system of interest and the measurement chamber, causing a limited-volume incubation in the measurement chamber, as sketched in the bottom graph of Figure c. We refer to the switching concept between the open and closed condition as “time-controlled analyte exchange”. Figure d illustrates the operating principle for a sensor where time-controlled analyte exchange is realized by a modulated flow. Phase 1 is the exchange phase, where the measurement chamber is supplied with the sample fluid so that the starting concentration in the chamber equals Ca,0. Phase 2 is the incubation phase, where the exchange process is stopped so that the limited-volume assay condition is provided. During incubation under the limited-volume condition, the analyte concentration in the measurement chamber decreases over time (depletion) or increases over time (repletion), depending on the initial fractional occupancy finit of binder molecules by analyte molecules. When finit is low, the concentration of analyte molecules in the measurement chamber decreases over time, corresponding to the depletion of analytes. When finit is high, the concentration of analyte molecules in the chamber increases over time, corresponding to the repletion of analytes. For known finit, the supplied analyte concentration Ca,0 can be derived from the measured time-dependent fractional occupancy f(t) during the incubation phase. At least two measurements need to be performed to determine the input analyte concentration Ca,0, for example, a measurement at the initial value finit and a measurement at the final value fend, as indicated in the graph.

By sequentially applying cycles of open condition and closed condition, discrete samples with a limited volume are serially measured and result in time-dependent data that relate to the different samples supplied to the sensor. Each former measurement causes a varying nonzero initial fractional occupancy finit in the next measurement. The values of finit and Ca,0 determine whether depletion or repletion occurs during the incubation phase. In the case of depletion, a higher input analyte concentration Ca,0 yields a larger, positive change in fractional occupancy Δf = fend – finit since more analyte molecules are captured from solution. In the case of repletion, a higher Ca,0 yields a smaller, negative change in fractional occupancy Δf since less analyte molecules are repleted from the sensor surface into solution. An important property of the sensor is that the interactions between binder and analyte molecules are reversible. This gives the advantage that the limited-volume assay with time-controlled analyte exchange can be used over an endless time span.

In the next sections, we will study how sensor design parameters influence the time characteristics and precision of the sensing methodology. The time characteristics are quantified by finite-element simulations of mass transport in the sensor and reaction processes at the sensor surface, and the precision is quantified by calculating the stochastic variabilities in the measurements. The simulations and calculations are verified by experiments using a sensing technique with a single-molecule resolution, called biosensing by particle mobility (BPM, see Note S7).[27−29]

Experimental Section

Finite-Element Analysis

Finite-element simulations were performed by solving diffusion, advection, and reaction equations simultaneously using COMSOL (COMSOL Multiphysics 5.5) and MATLAB (MATLAB R2019a, COMSOL Multiphysics LiveLink for MATLAB) (see Note S4). From the simulations, the time-to-equilibrium τ was determined by calculating the time at which the analyte–binder complex density γab is at 63% of the difference between the starting level and the equilibrium level of γab. The time-controlled analyte exchange (see Figure ) was simulated by instantaneously increasing/decreasing the analyte concentration throughout the measurement chamber Ca to Ca,0, with which a new measurement cycle starts. The density of analyte–binder complexes γabstart at the start of a cycle was set to be equal to the density of analyte–binder complexes γabend at the end of the preceding cycle. Sensor signals are reported at distance L/2 in the measurement chamber (see Figure d). Precisions are reported at a distance L/2 in the measurement chamber, where the signal is collected over a signal collection area of 1 mm2 (Figure c–e).

Figure 3

Simulated response of the biomolecular monitoring system using time-controlled analyte exchange. (a) Analyte concentration Ca in the measurement chamber (brown line) and the fractional occupancy f of binder molecules by analyte molecules (orange line) as a function of time, for low finit and depletion of analyte in solution (left), and for high finit and repletion of analytes in solution (right). The dashed lines indicate time points where instantaneous analyte exchange occurs, where the bulk analyte concentration was set to Ca,0 = 0.1 pM after each period of approximately 50 min. The insets highlight the kinetics of the first cycle, showing a time-to-equilibrium of τ = 340 s. For many cycles (n → ∞), both curves would approach , which equals the equilibrium value when an infinite volume is supplied (see Table S1). (b) Fractional occupancy f as a function of time where cycles of analyte exchange and incubation are applied every 15 min with alternatingly Ca,0 = 0.15 pM and Ca,0 = 0.05 pM. The curve saturates at feq,IV = 10·10–4, which equals the infinite-volume equilibrium value for the average concentration value Ca,0 = 0.1 pM (see Table S1). Dashed lines: continuous supply of Ca,0 = 0.05 pM yields feq,IV = 5·10–4 and Ca,0 = 0.15 pM yields feq,IV = 15·10–4 (see Table S1). The right panel shows zoom-in images of three sections of the solid curve, each representing four cycles of instantaneous analyte exchange and subsequent incubations of 15 min. In zoom-in 1 (t = 0–1 h), all curve segments show depletion behavior. In zoom-in images 2 (t = 12–13 h) and 3 (t = 42–43 h), depletion is seen for Ca,0 = 0.15 pM, since finit < feq,IV(Ca,0 = 0.15 pM), and repletion is seen for Ca,0 = 0.05 pM, since finit > feq,IV(Ca,0 = 0.05 pM). For all curve segments, the time-to-equilibrium τ = 340 s. The vertical scale bars indicate Δf = 10–4.

Figure 5

Analytical performance of the limited-volume assay, derived from simulations of a single measurement cycle. (a) Fractional occupancy at the end of the incubation fend as a function of analyte concentration Ca,0 for different initial fractional occupancies finit. The right y-axis indicates the number of surface-bound analyte molecules at the end of the cycle γabend. (b) Absolute change in fractional occupancy Δf as a function of Ca,0 for various finit. The right y-axis indicates Δγab. A positive Δf and Δγab indicate depletion; negative values indicate repletion. Note that the lines for finit = 10–3 and lower are overlapping. The inset shows the same data on a linear–linear scale. (c) Coefficient of variation CVC with which the analyte concentration Ca,0 can be determined as a function of analyte concentration Ca,0 for various initial fractional occupancies finit. CVC scales as for low finit and high Ca,0; CVC scales as 1/Ca,0 for high finit and low Ca,0. The horizontal dashed line indicates the 10% CV level that defines the LoQ (limit of quantification). (d) CVC as a function of measurement chamber height H (top) and binder density Γb (bottom) for various initial fractional occupancies finit and Ca,0 = 0.1 pM. The arrows on the x-axes indicate the standard parameter values for H and Γb, as listed in Table . (e) CVC as a function of the observed time-to-equilibrium τ when varying the measurement chamber height H (left) or binder density Γb (right) for various initial fractional occupancies finit and Ca,0 = 0.1 pM. The sketches above the graphs visualize a measurement chamber with an increasing height or a decreasing binder density. The arrows on the x-axes indicate the obtained time-to-equilibrium using the standard parameter values for H and Γb, as listed in Table .

Fluid Cell Assembly

Glass slides (25 × 75 mm, #5, Menzel-Gläser) were cleaned by 40 min sonication in isopropanol (VWR, absolute) and twice by 10 min sonication in Milli-Q (Thermo Fisher Scientific, Pacific AFT 20). Subsequently, the glass slides were dried under nitrogen flow. A polymer mixture of PLL(20)-g[3.5]-PEG(2) (SuSoS) and PLL(15)-g[3.5]-PEG(2)-N3 (Nanosoft Polymers) was prepared at a final concentration of 0.45 and 0.05 mg/mL in Milli-Q, respectively. The glass slides were treated by oxygen plasma (Plasmatreat GmbH) for 1 min. A custom-made fluid cell sticker (Grace Biolabs), with an approximate volume of 20 μL, was attached to the glass slide and immediately filled with the polymer mixture. After 2 h of incubation, the polymer mixture was removed and the fluid cell was immediately filled with 0.5 nM dsDNA tether solution (221 bp, with DBCO at one end and biotin at the other end) in 0.5 M NaCl in phosphate-buffered saline (PBS).[30] After overnight incubation, the solution in the fluid cell was exchanged by 2 μM DBCO-functionalized dsDNA solution[30] in 0.5 M NaCl in PBS and incubated for several days until use.

Particle Functionalization

A total of 2 μL of streptavidin-functionalized particles (10 mg/mL, Dynabeads MyOne Streptavidin C1, Thermo Fisher Scientific) was incubated with 1 μL of biotinylated ssDNA binder molecules (10 μM, IDT, HPLC purification) and 4 μL of PBS for 70 min. The particles were magnetically washed in 0.05 vol % Tween-20 (Sigma-Aldrich) in PBS and resuspended in 0.5 M NaCl in PBS to a final concentration of 0.1 mg/mL and sonicated using an ultrasonic probe (Hielscher).

BPM Assay

A total of 25 μL of particle solution was added to the fluid cell and incubated for 10 min. After incubation, the fluid cell was reversed causing unbound particles to sediment. After washing with 40 μL of 0.5 M NaCl in PBS, 40 μL of mPEG-biotin (500 μM, PG1-BN-1k, Nanocs) in 0.5 M NaCl in PBS was added to the fluid cell. After 15 min of incubation, the fluid cell was washed twice with 40 μL of PBS. A mixture of ssDNA analyte molecules (IDT, standard desalting) and free binder molecules in PBS was added to the flow cell at the required concentration, immediately after preparation. The sample was observed under a white light source using a microscope (Leica DMI5000M) with a dark-field illumination setup at a total magnification of 10× (Leica objective, N plan EPI 10×/0.25 BD). A field of view of approximately 1100 × 700 μm2 with a few thousand particles was imaged using a CMOS camera (FLIR, Grasshopper3, GS3-U3-23S6M-C) with an integration time of 5 ms and a sampling frequency of 30 Hz. The particles were tracked by applying a phasor-based localization method.[31] The particle activity was determined from the x- and y-trajectories of all particles, by applying a maximum-likelihood multiple-windows change point detection algorithm.[32] The particle activity at equilibrium and the time-to-equilibrium were extracted by fitting the measured particle activity over time using the equation given in Box S1.

Results and Discussion

Timescales of the Limited-Volume Assay

Figure shows simulation results of the time-to-equilibrium of the limited-volume assay, for sensor designs with different measurement chamber heights, different binder densities, and different flow rates, assuming standard parameter values, as listed in Table . Figure a shows how the time-to-equilibrium τ depends on the measurement chamber height H, for a sensor with instantaneous analyte exchange (see Note S6 for the influence of analyte exchange on the sensor performance). The arrow on the x-axis indicates the height as listed in Table . The data show that the time-to-equilibrium increases with the measurement chamber height. At small H, this increase is caused by a decrease in the effective volumetric binder concentration, while at large H, this increase is caused by diffusive transport limitations. The inset shows the same data, plotted as a function of the Damköhler number (Da = τD/τR,LV = konΓbH/D, see Table ); low Da means that the kinetics are limited by the reaction, and high Da means that the kinetics are limited by diffusion. To achieve a fast time-to-equilibrium, the sensor should be designed with a large Cb,0, so a small H.

Figure 2

Time-to-equilibrium τ of a limited-volume assay for sensor designs with different heights, binder densities, and flow rates of analyte exchange. (a) Time-to-equilibrium τ as a function of measurement chamber height H (orange line) for instantaneous analyte exchange. For small H, the observed τ is reaction-dominated (black dotted line), while for increasing H, the observed τ becomes diffusion-dominated. The inset shows the same data, where τ is normalized to τR and plotted as a function of Damköhler number Da. The sketch above the graph visualizes a measurement chamber with an increasing H. (b) Time-to-equilibrium τ as a function of the binder density Γb (orange line) for an instantaneous analyte exchange. For low Γb, the observed τ is reaction-dominated (τ = τR, black dotted line), while for increasing Γb, the observed τ becomes diffusion-dominated. The inset shows the same data, where τ is normalized to the characteristic diffusion time τD and plotted as a function of Da. The sketch above the graph visualizes a measurement chamber with an increasing Γb. (c) Time-to-equilibrium τ as a function of flow rate Q for three aspect ratios λ = L/H, for time-controlled analyte exchange by advection where the flow duration texch equals the characteristic advection time τA. For small Q, the observed τ is limited by the advective transport of analyte molecules from the inlet toward the point of sensing at distance L/2 from the inlet. For increasing Q, this transport process becomes faster causing the observed τ to be dominated by reaction and/or diffusion at high flow rates. The inset shows the same data (Da = 2) supplemented with Da = 0.2 (reaction-limited) and Da = 20 (diffusion-limited), where τ is normalized to τR and plotted as a function of the longitudinal Péclet number PeL. The dotted lines show the τ/τR value at high Q and are equal to the values found in panel a. The sketch above the graph visualizes a measurement chamber with an increasing Q. In all panels, the black arrows on the x-axis indicate the standard parameter values for H, Γb, and Q, as listed in Table .

Figure b shows how the time-to-equilibrium depends on the binder density Γb, for a sensor with instantaneous analyte exchange. The arrow indicates the density, as listed in Table . For small Γb, the time-to-equilibrium is long and determined by the dissociation rate constant (τ ≅ 1/koff). For Γb > HKd ≅ 20 μm–2, the time-to-equilibrium decreases, until it stabilizes due to diffusive transport limitations (τ ≅ τD = H2/D). The inset shows the same data plotted as a function of Da. To achieve a fast time-to-equilibrium, the sensor should be designed with a large Cb,0, so a large Γb.

Figure c shows how analyte exchange by advection contributes to the time-to-equilibrium per measurement cycle. The exchange phase involves a temporary flow of fluid into the measurement chamber, with flow rate Q and duration texch (see Note S6 for the influence of analyte exchange on the sensor performance). In the simulations, texch was chosen to be equal to the characteristic advection time τA = HLW/Q (see Table ), which means that a total fluid volume equal to the volume of the measurement chamber is displaced. The time-to-equilibrium τ, which now includes a contribution texch related to the exchange, is shown as a function of flow rate, for several values of the chamber aspect ratio λ = L/H. The arrow indicates the flow rate, as listed in Table . For small Q, the observed τ is limited by texch, that is, the advective transport time of analyte molecules from the inlet toward the point of sensing at a distance L/2 from the inlet, as sketched in Figure d. For increasing λ, that is, increasing L with a fixed H, the time-to-equilibrium increases since τA (and thus also texch) increases. For increasing Q, the time-to-equilibrium decreases, until it stabilizes at a level where the reaction and diffusion times determine the observed τ. The inset shows the same data (Da = 2), supplemented with Da = 0.2 (reaction-limited) and Da = 20 (diffusion-limited), plotted as a function of the longitudinal Péclet number (PeL = τD/τA = Q/λDW, see Table S1); low PeL means that the analyte exchange is limited by advection, and high PeL means that the analyte exchange is limited by diffusion. A low PeL causes a long time-to-equilibrium due to slow mass transport by advection. Increasing PeL results in a decrease in the time-to-equilibrium due to rapid filling of the chamber, until it stabilizes at a τ value equal to the value indicated in Figure a. Figure c shows the flow rate required to minimize the influence of the exchange process on the time-to-equilibrium. In the following sections, exchange with a high PeL is assumed, that is, rapid filling of the measurement chamber without influence of the flow rate on the time-to-equilibrium.

Limited-Volume Assay with Time-Controlled Analyte Exchange

Figure shows simulation results for a limited-volume assay with time-controlled analyte exchange. The analyte exchange is assumed to be instantaneous and the incubation phase includes mass transport by diffusion and reaction kinetics within the measurement chamber itself but no analyte exchange between the system of interest and the measurement chamber. Figure a shows data for repeated incubations with Ca,0 = 0.1 pM. The analyte concentration Ca in the measurement chamber (brown line) and the fractional occupancy f of the binders by analyte molecules (orange line) are plotted as a function of time, for conditions of analyte depletion (left) and analyte repletion (right). The time-to-equilibrium τ of each incubation equals approximately 340 s (see Figure ), having contributions from reaction (τR = 200 s) and diffusion (τD = 400 s). The contribution from the reaction to the time-to-equilibrium is much smaller than 1/koff = 104 s, the value that would have been observed in the case of an infinite-volume assay (cf. Figure c). In the absence of diffusion limitations, the acceleration that can be achieved with a limited-volume assay compared to an infinite-volume assay equals , which clarifies how the speed of the assay is directly related to the ratio between effective volumetric binder concentration and the equilibrium dissociation constant.

Figure b shows the response of a limited-volume assay with time-controlled analyte exchange for an analyte concentration that varies in time. As an example, the sensor is incubated with a time series of analyte concentrations that alternate around 0.1 pM (orange line): the analyte concentration is alternatingly Ca,0 = 0.05 pM or Ca,0 = 0.15 pM. The infinite-volume equilibrium fractional occupancy feq,IV is given for Ca,0 = 0.15 pM and Ca,0 = 0.05 pM by the dashed black lines (see Table S1). The panels on the right show zoom-in images of the sensor response at three different time periods (starting at t = 0, 12, and 42 h). In all cases, the time-to-equilibrium is τ = 340 s = 5.7 min. Incubation with Ca,0 = 0.15 pM gives depletion behavior at all times (since finit < feq,IV, top black line); for Ca,0 = 0.05 pM, depletion behavior is seen at t < 10 h and repletion at t > 10 h (when finit > feq,IV, bottom black line).

Figure shows an experimental study on how the time-to-equilibrium in a limited-volume assay depends on the total binder concentration in the measurement chamber. Here, the total binder concentration has two contributions, namely, a contribution from surface-bound binders and a contribution from binders supplemented in solution. For detection, we made use of BPM, which is a biomolecular monitoring principle with single-molecule resolution. In the BPM sensor, the particles are transducers that record the binding of analyte molecules to specific binder molecules (see Note S7). Figure a shows a schematic representation of a measurement chamber with binder molecules present in the two forms: immobilized and nonimmobilized. Immobilized binder molecules are present with an effective volumetric concentration Cb,0. Binder molecules supplemented free in solution have concentration Cb,suppl. In the absence of supplemented binder molecules (top), the total binder concentration in the measurement chamber equals Cb,tot = Cb,0 = Γb/H. In the presence of supplemented binder molecules (bottom), the total binder concentration equals Cb,tot = Γb/H + Cb,suppl. Since the time-to-equilibrium of the reaction scales according to τR,LV ∝ 1/Cb,tot (see Table ), an increasing supplemented binder concentration Cb,suppl results in a smaller τ. Figure b shows the measured time-to-equilibrium τ (left) and the signal change ΔS (right) as a function of Cb,suppl, for an analyte concentration of 200 pM (see the Experimental Section). The data show that the time-to-equilibrium decreases for increasing Cb,suppl. The measured signal change decreases with increasing supplemented binder concentration because only surface-captured analyte molecules generate a measurable signal. The dashed lines in Figure b represent model fits (see the caption), demonstrating a good correspondence between model and experimental results. We conclude that the measurements of Figure prove the basic concept of the sensing methodology proposed in this paper, namely, that a limited-volume design with time-controlled analyte exchange allows one to control the response time by tuning the concentration of binder molecules in the measurement chamber. Once the optimal binder concentration is known, the sensor can be made with binders immobilized in the measurement chamber. This will be a topic for future research.

Figure 4

Experimental study of a limited-volume assay with varying binder concentrations using BPM. (a) Sketch of the measurement chamber in a BPM measurement (see Note S7) without (top) and with (bottom) supplemented binders with concentration Cb,suppl. For simplicity, the particles of the BPM sensor are not shown in the sketch. (b) Experimentally observed time-to-equilibrium τ (left) and normalized signal change ΔS (right) as a function of supplemented binder concentration Cb,suppl in a BPM measurement. The model assay is based on DNA–DNA interactions, with ssDNA analytes at a concentration of 200 pM (see Note S7). Left: the dashed line shows the fitted curve τ = p1/(p2 + Cb,suppl) + p3, where p1 = 1/kon (kon is assumed to be equal for all binders), p2 = Γb/H, and p3 is the delay contributed by diffusion (see τD, black line, see also Figure b) and experimental steps (see the Experimental Section). Assuming H = 200 μm (see Table ), the fit gives Γb=(3 ± 1)·10–10 mol m–2, which is comparable to the standard parameter value, as listed in Table . The fitted association rate constant is kon=(1.5 ± 0.4)·105 M–1 s–1, which is in the range of values reported in the literature for comparable DNA–DNA hybridization reactions.[33,34] Right: In the depletion condition (finit < feq,IV), the fractional occupancy scales according to f ∝ 1/Cb,tot = Ca,0/(Cb,0 + Cb,suppl). The dashed line shows the fitted curve ΔS = p1/(p2 + Cb,suppl), where p1 scales the change in fractional occupancy to signal change and p2 = Γb/H. For H = 200 μm, it was found that Γb = (7 ± 4)·10–10 mol m–2, which is comparable to the previously found value for Γb and the standard parameter value, as listed in Table . The insets show the same data on linear-logarithmic scales. The errors reported in the figure (smaller than the symbol size) and the caption are fitting errors based on a 68% confidence interval.

Analytical Performance of the Limited-Volume Assay

Figure shows how the analytical performance of the limited-volume assay depends on the sensor design. The results are based on numerical simulations with parameters, as listed in Table . The analyte exchange is assumed to be instantaneous and the incubation includes mass transport by diffusion and reaction kinetics within the measurement chamber only. All panels show curves for different values of the initial fractional occupancy finit of the binder molecules.

Figure a shows the fractional occupancy of binders by analyte molecules at the end of the incubation (fend) as a function of the input analyte concentration Ca,0. For finit = 0 (dashed black line), fend scales linearly with the analyte concentration, which makes the sensor suitable for analyte quantification. For larger values of finit, the curves start with a rather flat segment, from which one might erroneously conclude that under those conditions, low analyte concentrations cannot be determined. Interestingly, the limited-volume assay has a linear dependence on concentration by focusing not on the absolute value of fend but rather on the change in fractional occupancy Δf (see Note S2)

This equation shows that Δf depends linearly on Ca,0, independent of the value of finit. This fact is also illustrated by the simulation results in Figure b. The response scales linearly with concentration Ca,0 and is downshifted for increasing values of finit, in agreement with eq (note that the steep increase in the curves relates to the logarithmic x-axis). Positive values of Δf relate to depletion behavior and negative values to repletion. The curves cross the x-axis (Δf = 0) when finit corresponds to the equilibrium condition, that is, when there is no net association or dissociation during incubation because finit is equal to the equilibrium fractional occupancy of the infinite-volume case: . For example, the curve for finit = 10–3 crosses Δf = 0 at Ca,0 = finitKd = 0.1 pM, as is highlighted in the inset of Figure b.

Figure c shows the precision of the concentration output of the sensor, that is, the precision with which the analyte concentration in an unknown sample can be determined for a signal collection area of 1 mm2. The precision is calculated based on Poisson noise, which gives the fundamental limit of the precision that is achievable with a molecular biosensor due to stochastic fluctuations in the number of analyte molecules (see Note S3 and S8).[35] To calculate the precision, a sensor with initial fractional occupancy finit is provided with a sample with analyte concentration Ca,0, resulting in a Δf with variability σΔ, which via the slope of the calibration curve, given in Figure b, leads to a variability σC in the concentration output of the sensor (see Note S3). The precision is indicated as the concentration-based coefficient of variation CVC = σC/μC, with σC being the variability and μC being the mean of the concentration output. Figure c shows how the concentration precision depends on the analyte concentration and the initial fractional occupancy finit. For finit = 0 (dashed line), the CVC scales as , in agreement with number fluctuations in a Poisson process (see Note S3). For higher finit, a stronger dependency is observed (CVC ∝ 1/Ca,0), caused by the smaller relative change in the fractional occupancy (see Note S3). The graph indicates the 10% precision level that is used to define the limit of quantification (LoQ) of the sensor (see Figure c, dashed horizontal line). The results show that analyte concentrations in the sub-picomolar range can be measured with a precision better than 10%, even for high initial fractional occupancies.

Figure d shows the precision of the concentration output of the sensor as a function of two design parameters, namely, the measurement chamber height H (top panel) and the binder density Γb (bottom panel), at an analyte concentration Ca,0 = 0.1 pM, for an initial fractional occupancy finit between 0 and 0.01. The arrows indicate the height and density, as listed in Table . For an increasing H, a decrease in CVC is observed, caused by an increase in the number of analyte molecules present in the measurement chamber. The CVC is the smallest for finit = 0 and increases for increasing finit since the absolute change in fractional occupancy decreases. The CVC decreases for increasing Γb caused by an increase in the number of analyte molecules captured from solution. The CVC reaches a plateau for finit = 0 due to a limited number of analyte molecules in the measurement chamber. For larger finit, the absolute change in fractional occupancy decreases and causes a less precise concentration determination; this effect is in particular visible at high Γb where the absolute number of analyte–binder complexes increases due to finit.

The trade-off between precision and time-to-equilibrium is illustrated in Figure e, for sensors with different heights of the measurement chamber (left) and different binder densities (right). The arrows indicate the time-to-equilibrium that results from the height and density, as listed in Table . The left panel shows that an increase in H gives, on the one hand, a slower sensor response (due to a larger diffusion distance) but, on the other hand, a lower CVC due to a larger number of analyte molecules present in the measurement chamber. At low H, the CVC strongly depends on finit due to the low number of analyte molecules in the solution. The right panel shows again that the CVC decreases for a slower sensor response, now controlled by decreasing the binder density Γb. At high Γb, the time-to-equilibrium is diffusion-limited (resulting in τ = 130 s). At low Γb, the time-to-equilibrium is reaction-limited with τ = 1/koff = 104 (see Figure b). At high Γb, the CVC increases for increasing finit due to the larger amount of analyte molecules on the sensor surface. At low Γb, the CVC strongly increases due to the small absolute change in the number of analyte−binder complexes during the incubation phase.

Conclusions

We have presented a sensing methodology suitable for monitoring low-concentration biomolecules with high precision, with small time delays and short time intervals, over an endless time span. The sensing methodology is based on a limited-volume assay, using high-affinity binders, a fully reversible detection principle, and time-controlled analyte exchange. We studied by simulations how the kinetics of the sensor depend on mass transport and on the surface reaction in the measurement chamber, and how time-controlled analyte exchange determines the system response and enables precise measurements of analyte concentration. Experimental results show the ability to control the sensor response time by tuning the total binder concentration in the measurement chamber. Finally, simulations show that the sensing principle allows picomolar and sub-picomolar concentrations to be monitored with a high precision over long time spans.

Approaches described in the literature for measuring low-concentration biomolecules have focused primarily on assays in which every concentration determination involves consumption of reagents.[17−20,36,37] When numbers of assays become high, due to frequent measurements over long time spans, then reagent-consuming approaches are complex and costly. The sensing methodology described in this paper is based on a fully reversible assay principle, without consuming reagents with each newly recorded concentration data point, enabling measurements with high frequency over an endless time span. The described assay principle can be implemented on several sensing platforms, for example, based on optical, electrical, or acoustical transduction methods, where especially sensing platforms with single-molecule resolution seem suitable since these allow digital measurements with very high precision, limited only by the number of observed molecular interactions. The sensing method is suited for the monitoring of a wide variety of analytes, including small molecules, proteins, and viral particles (see Note S9). Furthermore, the sensing methodology can be combined with various sampling methods, including remote advection-based sampling through a sampling line or a catheter and proximal diffusion-based sampling methods for on-body and in-body monitoring devices. The presented sensing principle warrants further experimental studies, for example, to investigate trade-offs between time characteristics and precision, for various transduction methods, sampling methods, measurement chamber geometries, binder types, and for various complex biological matrices. Due to its sensing performance and generalizability, we believe that the limited-volume assay with time-controlled analyte exchange will enable research on time dependencies of low-concentration biomolecules and novel applications in the fields of dynamic biological systems, patient monitoring, and biotechnological process control.