An integrated model for bead-based immunoassays

Journal Pre-proof An integrated model for bead-based immunoassays Dan Wu, Joel Voldman PII:

S0956-5663(20)30067-1

DOI:

https://doi.org/10.1016/j.bios.2020.112070

Reference:

BIOS 112070

To appear in:

Biosensors and Bioelectronics

Received Date: 18 October 2019 Revised Date:

30 January 2020

Accepted Date: 1 February 2020

Please cite this article as: Wu, D., Voldman, J., An integrated model for bead-based immunoassays, Biosensors and Bioelectronics (2020), doi: https://doi.org/10.1016/j.bios.2020.112070. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.

Credit Author Statement Dan Wu: Conceptualization; Data curation; Investigation; Methodology; Software; Validation; Visualization; Roles/Writing - original draft; Writing - review & editing. Joel Voldman: Conceptualization; Data curation; Funding acquisition; Methodology; Project administration; Supervision; Validation; Visualization; Writing - review & editing.

An Integrated Model for Bead-based Immunoassays Dan Wua, Joel Voldmanb,* a b

*

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA To whom correspondence should be addressed at: [email protected]

Abstract Bead-based immunoassays have shown great promise for rapid and sensitive protein quantification. However, there still lacks holistic understanding of assay performance that can inform assay design and optimization. In this paper, we present an integrated mathematical model for surface coverage bead-based assays. This model examines the building blocks of surface coverage assays, including heterogeneous binding of analyte molecules on bead or sensor surfaces, attachment of bead labels to sensor surfaces, and generation of electrochemical current by bead labels. To demonstrate and validate this model, we analyze a semi-homogeneous bead-based electronic enzyme-linked immunosorbent assay and find that experimental results agree with various model predictions. We show that the model can provide design guidance for choice of various assay parameters including bead size, bead number, antibody affinity and assay time, and provide a perspective to reconcile the performance of various implementations of surface coverage assays.

Keywords Bead-based surface coverage assays; Heterogeneous binding; Bead attachment; Electrochemical readout; Integrated mathematical model

1. Introduction Immunoassays are extensively used for clinical diagnosis and scientific research (Cohen et al. 2018; Gaster et al. 2011). While enzyme-linked immunosorbent assays (ELISAs) are the gold standard, conventional ELISAs have the limitation of long turnaround time (~ hours) and large volume of sample (~hundred microliters to milliliters). Within this context, nano-/micro-particles have been incorporated to reduce the sample volume (Yu et al. 2010), expedite the assays (Wu et al. 2018), improve the limit of detection (LOD) (Rissin et al. 2010) and facilitate assay integration (Campbell et al. 2015; Wu and Voldman 2019). Bead-based assays can take various formats, and in particular, surface coverage assays have gained considerable research interest, because they can be ultra-sensitive, rapid, and readily implemented in microfluidics for point-of-care (PoC) applications (Bruls et al. 2009; Dittmer et al. 2008; Koets et al. 2009; Tekin et al. 2013; Wu et al. 2018). Various surface coverage assays are developed, and they use different parameters and manifest different performance (Bruls et al. 2009; Mulvaney et al. 2007; Otieno et al. 2014; Wu et al. 2018). For example, an optomagnetic sensor achieves sub-picomolar LOD using 500 nm nanoparticles (Bruls et al. 2009), while attomolar LOD is obtained using fluidic force discrimination assays and 2.8 µm microbeads (Mulvaney et al. 2009). However, the field lacks theories which explain differences in performance and guide design. These assays typically feature heterogeneous binding of analyte molecules to bead or sensor surfaces and attachment of beads to sensor surfaces. Many models are dedicated to those aspects. For instance, the interaction between analytes and antibodies is extensively examined on surface-based sensors (Squires et al. 2008) or bead surfaces (Chang et al. 2012). At the same time, bead attachment is essential to surface coverage assays. To this end, three-dimensional (3D) heterogeneous binding kinetics was used to model nanoparticle (~ 50 nm) attachment (Gaster et al. 2011). Similar approach was applied to larger (210 nm and 1 µm) particles, but it was found the 3D association rate constant depends on surface densities of biomolecules (Haun and Hammer 2008; Mani et al. 2012). Stochastic models were created to analyze scenarios where biomolecules (on beads or sensors) has very low surface density (Cornaglia et al. 2014). Few attempts were made to build end-to-end models, but only simple/special scenarios (e.g., irreversible heterogeneous binding on beads) were considered (Roberts et al. 2017). Therefore, a holistic model is still lacking to obtain a system-level view. In surface coverage assays, the readout is related to the number of beads attached to sensor surfaces. Electrical assays, which transduces the presence of beads to current, are promising for PoC diagnosis due to low cost and facile miniaturization of electronics (Wu et al. 2018). In contrast to optical and magnetic approaches that rely on bead-sensor physical interactions, current production involves a series of chemical reactions. A mechanistic model of electrical readout is intriguing, as it can provide insights for assay optimization and readout system design. For example, one may want to predict the current measurement to design the optimal amperometry circuits (with required sensitivity and range). Moreover, it is critical to understand how bead size affects the resulting current, or if an optimal bead diameter exists. Thus, a mathematical description of electrical readout is required. Here, we present an integrated model and experimental validation of surface coverage assays. We first develop an analytical model to describe analyte capture on bead/sensor surfaces. A model of bead attachment is then created by combining two-dimensional (2D) binding kinetics and bead transport. Additionally, a 3D finite element model (FEM) is devised for current formation in electrical assays. Finally, we demonstrate an end-to-end modeling of a semi-homogeneous electronic ELISA of interleukin 6 (IL-6). We show that our model can capture assay kinetics and inform assay optimization. Experimental validations agree with various model predictions. We also discuss model usage to design biosensors and related bioelectronics.

2. Model There are two types of surface coverage assays. In one type, analytes first bind to antibodies on a sensor surface, followed by adhesion of bead labels to the sensor, while in the other (semi-homogeneous assays), analytes are first captured by antibodies on beads, followed by attachment of the bead-analyte complex to the sensor. Both types can be dissected into three scenarios: heterogeneous analyte binding [on bead or sensor surface, Fig. 1 (a)], adhesion of bead to sensor [Fig. 1(b)] and production of readout signals [e.g., current as illustrated in Fig. 1(c)]. In this section, we formulate models for each scenario.

Figure 1. Schematic illustration of surface coverage assays. (a) Analyte capture on (i) bead or (ii) sensor surfaces. In (i), R is the half of the mean particle-to-particle distance and r denotes distance to the center of the bead of interest. In (ii) the sensor has a characteristic length of L. (b) Bead attachment to a solid surface, d is the maximum distance to form a contact and determined by the total length of the analyte and antibody molecules, and other linking molecules. z is the distance of the apex to the surface. (c) Current generation in electrical assays, where 3,3',5,5'-tetramethylbenzidine (TMB) is used as an electrochemical substrate and TMBox denotes the oxidized state of TMB.

2.1 Heterogeneous binding on bead/sensor surface Binding on bead surface: The beads experience constant diffusion and the diffusivity (e.g., ~0.49 µm2/sec for beads with diameter a = 1.0 µm) is one to two orders smaller than that of proteins (~10 to 100 µm2/sec) (Young et al. 1980). Thus, these beads are considered immobile with respect to the analyte molecules. Additionally, since the beads are uniformly dispersed, we assume the suspension of beads as a spatially periodic medium (Nadim 2009) whose unit element features a bead centered at a sphere of radius R [Fig. 1(a-i)]. If there are N beads in the solution of volume Vs, R is

π

. Therefore, in spherical coordinates, the average surface density of bound analyte per bead

is governed by:

<



=

:

= :

< :

= 〈+, 〉

∇ =

=0

=

!

!

"#$% − 〈#% 〉) −

"#$% − 〈#% 〉) −

! ** 〈#% 〉

! ** 〈#% 〉

,

(1)

. = 0: = , 〈#0 〉 = 0 where c is the analyte concentration (M), c0 is the initial (spatially uniform) concentration, r is the coordinate distance from the center of the bead, r0 = a/2 is the bead radius, Dm is the diffusivity of the analytes, NA is the Avogadro number, nab is the surface density of antibodies (molecules/µm2), and kon3D (M-1sec-1) and koff3D(sec-1) are the 3D association and disassociation rate constants, respectively. The no-flux boundary condition at r = R reflects the effects of other beads. As Eqs.1 is based on continuum assumptions, we check the limit of validity. If 500,000 beads are suspended in a 100 µL sample with 5 pM analytes, there are 600 molecules in each sphere, which approaches the limits of the continuum approximation. However, Eqs. 1 can still predict the ensemble average of bound analyte over the solution volume. In contrast to (Chang et al. 2012), where the analyte capture is assumed limited by binding reaction, Eqs. 1 can capture the diffusion-limited scenarios. In circumstances where the binding is reaction-rate limited (i.e., Damkohler number Da = nabkon3Da/Dm << 1) and the analyte concentration is low (i.e., c0 << kD3D, where kD3D = koff3D/kon3D is the equilibrium dissociation constant), becomes:

〈#0 〉 =

51 − 7 839:: ;.

1 +2, 344

4

(2)

Binding on sensor surface: We consider a scenario in Fig. 1(a-ii), where the sample is added to the sensor with characteristic length of L. Though simplified, this scenario can occur in practical assays (Gaster et al. 2011). When the total incubation time is not too large compared with the diffusion time constant L2/Dm, the analyte surface density on the sensor can be approximated using Eqs. (1), where r0 = L/2 and R =

π

(the sample is assumed

to form a hemisphere shape). This simplification permits a direct comparison with the binding on bead surface. 2.2 Bead attachment to a solid surface: When two surfaces form a contact, only the molecules in the contact area (Ac, µm2) can bind. The adhesion probability of the two surfaces Pa (at least one bond forming) is related to analyte density (nag, molecules/µm2), antibody density (nab, molecules/µm2), contact duration tc and Ac (Chesla et al. 1998; Haun and Hammer 2008; Mani et al. 2012) by, Pa = 1- exp{-AcnagnabkD2D[1-exp(-koff2Dtc)]}, where kD2D and koff2D are the 2D equilibrium dissociation constant and dissociation rate constant, respectively (Chesla et al. 1998). During bead attachment, we assume that a homogeneous bead solution is introduced into a microfluidic channel by flow and then the flow is stopped to allow beads to interact with a sensor. Because the lateral (i.e., in the width and length directions) diffusive flux of beads can be much smaller than the vertical (i.e., in the height direction) sedimentation produced by external forces (e.g., gravity), we neglect the transport in the lateral directions and reduce the problem to one-dimensional. As in Fig. 1(b), Ac = πa(d - z)[H(z) – H(z - d)], where H(z) is the Heaviside step function. Considering a bead that represents the ensemble average of all beads and following a similar approach to the attachment between two separate beads (Ju et al. 2015), we describe the probability density function (m-1), p(z, t) by: <=">, ) <

=

G = 0:

%

, )

<=">, )

G = I:

, ) <>

− C #DE #D0

=0

. = 0: F"G, .) =

!

F"G, .)

,

(3)

J

K

where Db = kBT/(3µπa) is the bead diffusivity, vF is the bead velocity under external forces and p(z, t) is the probability of finding the apex of the bead at position z and time t. Eqs. 3 are based on mass conservation, where the first term on the right is the contribution from bead diffusion, the second term captures the effects of external forces and the third term is due to analyte-antibody binding reaction. As the beads are confined in a microfluidic channel of height H, no-flux boundary conditions are exerted at z = 0 and z = H. At t = 0, the beads are homogeneously suspended, so p(z, 0) is uniform across the channel. Eqs. 3 assume that beads can attach when only one analyte-antibody bond is formed. This model does not account for assays, where external forces (e.g., shear or magnetic forces) are applied for washing and those forces are larger than the rupture strength of an analyte-antibody complex [10-100 pN (Weisel et al. 2003)]. The probability of bead attachment after duration tb is: LD = 1 − M F"G, .0 )NG . K

(4)

2.3 Bead-based amperometry: In electrical assays, horseradish peroxidase (HRP), hydrogen peroxide (HP) and 3,3’,5,5’-tetramethylbenzidine (TMB) are commonly used for amperometry(Min et al. 2018; Riahi et al. 2016). The following reactions occur to generate current(Josephy et al. 1982; Marquez and Dunford 1997): H2 O2 + HRP → H2 O + HRP_I 1

HRP_I + TMB → HRP_II + TMBZ[ 2

HRP_II + TMB → HRP + TMBZ[ 3

TMBZ[ + 7− → TMB

,

(5)

4

where k1, k2, k3 and k4 are the reaction rate constants. The first three reactions in Eqs. 5 occur on the bead surface, while the last one is on the electrode surface. Flow injection analysis is widely used in bead-based amperometry (Fanjul-Bolado et al. 2005), so we consider the situation where the substrate (mixture of HP and TMB) is injected

into a microfluidic channel. To describe the kinetics of current formation, we formulate mass transport equations for each species: ^_

Ω0 :

=

ghi

Ω0 :

^_

=

K` ∇

=−

ghi

Ω0 :

ghiop

Ωn :

=

jkl

=−

ghi

=

ghiop

+ ∇ ∙ "b

J K` #Kf`

jkl ∇

Ωn :

ghiop

K`

+ ∇ ∙ "b

jkl #Kf`_m

ghiop

jkl ∇

=

=−

K` )

jklop

−

6.1

jkl )

+ ∇ ∙ qb

jkl #Kf`_m jklop

+

6.1.1 6.2

jkl #Kf`_mm

6.2.1

jklop r

6.3

jkl #Kf`_mm

6.2.2

,

(6)

6.3.1 6.3.2

where v is the flow profile in the channel, Ωb and Ωe denotes the bead and the electrode surfaces, respectively. Eqs. 6 can be solved using 3D FEM. The current measured by amperometry is given by the surface integration over the surface electrode: s=7

∬v ∇ w

jklop Nu,

(7)

where e is the charge of electron and NA the Avogadro's number.

3. Results As a practical case, we focused on understanding the binding/attachment kinetics to minimize assay time, and the influence of bead diameter and number on assay performance. We first examined the three roles of beads and then built a complete model for bead-based electronic ELISA (Chikkaveeraiah et al. 2011; Wu et al. 2018). Analyte capture on bead/sensor surfaces: Analyte capture was examined by solving Eqs. 1 (see SI for details), where representative parameters were selected. For instance, we assumed moderate analyte diffusivity, 60 µm2/sec, given that protein diffusivity varies from 10 to 100 µm2/sec (Tyn and Gusek 1990). We first analyzed the binding kinetics [Fig. 2(a)]. For smaller beads (a = 0.5 µm), the binding kinetics are much closer to the reaction-limited case (predicted by Eqs. 2). This is as expected because the Damkohler number Da = nabkon3Da/Dm = 0.07 << 1, suggesting reaction is limiting binding. However, the binding on larger beads (2.0 and 6.0 µm) deviates from the reaction-limiting prediction. The reason is two-fold. First, indicated by larger Da (0.84 for 6.0 µm bead), diffusion becomes comparable with reaction. Second, larger beads carry more antibodies, which in turn decreases the analyte concentration and limit the binding rate. For the same reason, the binding on a 6.0 µm bead reaches equilibrium faster. We found that the approximation of a sensor as a large bead (i.e., solving using Eqs. 1) is valid (See SI), and the binding is limited by diffusion (e.g., a sensor with characteristic length of L = 200 µm, Da = 55.37) and thus less efficient than on a bead.

Figure 2. (a) Analyte binding kinetics on bead/sensor surface, and influence of (b) bead diameter and (c) bead number. The results in (a) represents binding on a single bead/sensor surface, while (b-c) show the analyte capture in practical bead assays. In

(b) the total bead surface area is fixed and equivalent to 500,000 1.0 µm beads, while in (c) the sample volume Vs = 100 µL and a = 2.0 µm. In (a-c), nab = 10000 molecules/µm2, c0 = 1 pM, kon3D = 106 M-1sec-1, koff3D = 10-3 sec-1, Dm = 60 µm2/sec. The incubation time in (b-c) is 30 min.

We further considered practical scenarios, where numerous beads are incubated with a certain volume of sample (100 µL in this case). In Fig. 2(b), we compared assays of using different bead sizes but the same total surface area (so, larger numbers of beads as the bead size decreases). One may expect the number of captured analyte molecules per bead to scale with the bead surface area (i.e., a2). However, the results indicate that such scaling breaks as bead diameter gets larger [a > 4.0 µm in Fig. 2 (b)] due to diffusion limitation. Accordingly, the total number of captured analyte molecules decreases with diameter. Fig. 2(c) presents results of using different numbers of beads (a = 2.0 µm). When the beads are sparse, the depletion zone of the analyte molecules caused by binding does not reach the boundary of the sphere in Fig. 1(a) within the time of interest. In that regime, changing bead number, or equivalently R, does not affect binding on individual beads and thus the total number of bound analyte molecules increases linearly with bead number. However, when R gets smaller, the depletion zone can reach the boundary, slowing down the binding process and reducing the number of analytes per bead. Therefore, the number of captured molecules starts to saturate. Bead attachment to sensor surfaces: We considered in Fig. 3(a) a simple scenario without external forces. Assuming kon2D of 10-5 µm2/sec and d of 30 nm (Saitakis et al. 2008), results show an interplay of analyte-antibody binding and bead diffusion. For example, for 2.0 µm bead with 200 molecules/µm2 of analytes, the attachment rate (the slope of the attachment curve) is initially large, but decreases and becomes constant after about 900 sec due to the limitation of bead diffusion. We observe that bead attachment depends on analyte surface density and bead diameter. Beads with more analytes (2.0 µm bead with 200 molecules/µm2 analytes) have higher attachment probability. For beads with few analytes (2.0 µm bead with 10 molecules/µm2 analytes), bead attachment is limited by analyte-antibody binding and attachment probability increases linearly with time. Moreover, larger beads can form larger contact area with sensor surfaces and facilitate analyte-antibody interaction. Therefore, 6.0 µm beads are more likely to attach than 2.0 µm beads with the same analyte density at the very beginning (t < 30 sec). However, larger diameter leads to slower bead diffusion, which explains why after t = 30 sec, the diffusion-controlled attachment rate for the 6.0 µm beads becomes smaller. High mass density and magnetic properties of magnetic beads enable easy applications of external forces. In Fig. 3(b), we studied bead attachment under external body forces. We observe that the external force can significantly enhance bead attachment. For example, an external force of 104 N/m3 on 2.0 µm beads (nag = 10 molecules/µm2) increases the attachment probability from ~ 0.02 [Fig. 3(a)] to ~ 1 [Fig. 3(b)], because the force can help bring the beads close to the surface. Since the velocity of beads under an external force scales with square of bead diameter, a2, varying bead diameter can dramatically affect bead attachment.

Figure 3 Kinetics of bead attachment (a) absent and (b) under external forces (f = 104 N/m3). For (a) and (b), H = 200 µm, kon2D = 10-5 µm2/sec, nab = 10000 molecules/µm2 and d = 30 nm. Note the different scale of the y-axis in (a) and (b).

Electrochemical current generated by beads: We formulated a 3D FEM in COMSOL Multiphysics 5.3 to predict the current generated by electrochemical reactions [Fig. 4(a)], where one bead is attached to the WE (see S1 for details). To solve Eqs. 6, we chose typical rate constants (Marquez and Dunford 1997). Fig. 4(b) shows the example concentration profiles of HP, TMB and TMBox in steady state, where a = 1.0 µm. We observe that very small amounts of HP and TMB are consumed and concentration changes are negligible (less than 1% decrease). TMBox accumulates around the top half surface of the bead, but it is completely depleted near the bottom surface, suggesting that the current generation is limited by the generation and transport of TMBox. It is worth noting that the flow brings TMBox generated on the surface of bead into the bulk of channel and thus away from the electrodes, potentially inhibiting current generation.

Figure 4. 3D FEM simulation of current generated by beads (a = 1.0 µm). (a) Schematic representation of the 3D model, which features a bead, an electrode (100 µm × 100 µm) and a microfluidic channel (500 µm ×100 µm × 60 µm). The mixture of HP (2 mM) and TMB (0.2 mM) is introduced from the right. In Eqs. 6, k1 = 2.6 × 107 M-1sec-1, k2 = 3.6 × 106 M-1sec-1, k3 = 1.0 × 106 M1 sec-1 and k4 = 2.0 × 107 m/s. (b) Example concentration profiles of HP, TMB and TMBox taken at the (blue dashed) cross-section in (a), where the volume flow rate is 3 µL/min. The scale bar corresponds to 2.0 µm. (c) Current produced by beads of different diameters, where the surface density of HRP is 3.0 pM/cm2 and two flow conditions are studied. The inset shows the effective turnover number of the enzymes.

Fig. 4(c) presents the current produced by beads of different diameters. The current increases as the bead diameter increases, because a larger bead carries more enzymes (constant enzyme surface density). Contrary to the argument that convection increases mass transport (Squires et al. 2008) and thus current, results show increasing the flow rate can remove TMBox from the electrode surface into the bulk of the channel [as shown by the concentration profile in Fig. 4(b)], reducing the current. This effect is more significant for larger beads. For example, the current produced by a 4.5 µm bead reduces by ~ 10% when the flow rate triples. We normalize the current by the total number of enzymes and define this as effective turnover number of the enzymes (µA/pM). The results [inset of Fig. 4(c)] show that the turnover number decreases as the bead diameter increases, because TMBox produced on larger beads must travel over a larger distance to the surface of electrode, limiting the current production. Semi-homogeneous bead-based electronic ELISA: Finally, combining the three roles, we modeled a complete semi-homogeneous bead-based electronic ELISA of human IL-6. In this assay, superparamagnetic beads first capture analytes from samples, then attach to an antibody-coated electrode, and detected by amperometry. Here, we only consider the gravitational force due to mass density difference between the beads and the solution (∆ρ = 0.8 g/cm3). The details of the electronic ELISA are included in S3. We validated the analytical capacity of the model in Fig. S5, where the calibration curve predicted by the model matches the experimental data. The effects of bead-sample incubation time (ta) are presented in Fig. 5(a), where five different concentrations were examined. Each curve follows a saturating exponential growth. Evidenced by smaller extracted time constants, the

binding kinetics for high concentration samples (50 pM, τ = 1.43 min) are faster than low concentration samples (0.4 pM, τ = 11.18 min), because more analyte molecules are captured from concentrated samples. For low concentrations, the kinetics is very close to 3D heterogeneous binding (τ = 1/koff3D = 11.49 min). Eqs. 3 suggests that when the surface density of analytes is low, bead attachment is limited by 2D binding reaction. In this case, Pa ≈ πadnab1nab2kon2D(c0/kD)tb[1-exp(-koff3Dta)]. Here, we used Eqs. 2, as the analyte capture is reaction-limited (Da = 0.083). Fig. 5(b) presents the experimental results, which compares low (2 pM) and high (50 pM) concentrations. Similar to model predictions above, for low concentration, the estimated time constant (13.56 min) is close to that of analytes binding to antibodies on beads [koff3D was estimated to be (1.45 ± 0.08) × 10-3 s-1 and 1/koff3D corresponds to 11.59 min, see S3]. In addition, the time constant for the 50 pM sample (7.22 min) is smaller, as expected. Note that the simulations used typical values of parameters (i.e., nab1 and nab2) and that these values may differ from the actual numbers in experiments, which can contribute to the discrepancies between the model-predicted and measured time constants.

Figure 5. Kinetics of bead attachment in semi-homogeneous bead-based assays. Simulation results showing (a) the influence of bead-sample incubation time and (c) the bead-surface incubation time on bead attachment, where Dm = 100 µm2/sec, kon3D = 5.0 × 105 M-1sec-1, koff 3D= 1.45 × 10-3 sec-1, a = 1.0 µm, nab1 = 10000/µm2 (antibodies on beads), nab2 = 10000/µm2 (antibodies on electrode), d = 30 nm, kon2D= 1×10-5 µm4/sec. The bead-surface interaction in (a) is 10 min and bead-sample incubation time in (c) is 30 min. (b) Measured influence of bead-sample incubation time, where the bead-surface interaction time is 10 min. (d) Measured influence of bead-surface interaction time, where the bead-sample incubation time is 30 min. For (b) and (d), human IL-6 was spiked into PBS with 0.2% BSA and detected. The dashed lines in (a) and (b) are exponential fitting, and in (d) are exponential fitting for 50 pM and linear fitting for 2 pM.

We further investigated the effects of the bead-surface interaction time (tb). Fig. 5(c) shows that the attachment probability saturates after long time incubation for high concentration cases (50 pM). For moderate concentration (2 pM), Pa does not saturate within the timescales of interest. For low concentration samples (0.4 pM), Pa increases almost linearly with tb as few analytes are captured and bead attachment is limited by 2D binding reaction. Experimental results agree with such predictions [Fig. 5(d)].

Figure 6. Optimization and prediction of a semi-homogeneous bead-based electronic ELISA using the integrated model. (a) Optimization of assay time, where the total assay time is 25 min and a = 1.0 µm, c0 = 2 pM. The inset compares the results of using various bead-electrode incubation time. The error bars represent standard error of mean (SEM) of four replicates and the maximum of relative SEM is 15.34%. (b) Prediction of the influence of sample volume, where a = 1.0 µm, c0 = 10 pM, ta = 10 min and tb = 15 min. The dashed lines represent rational fits of the data. The inset shows the experimental results, where the error bars represent SEM of three replicates and the maximum of relative SEM is 34.6%. (c) Model prediction of influence of bead size. The inset shows the experimental results and the dashed lines denotes three standard deviations above the background. The error bars represent SEM of three replicates and the maximum of SEM is 16.6 nA. In (a-c), the model assumes Dm = 100 µm2/sec, kon3D = 5.0 × 105 M-1sec-1, koff 3D= 1.45 × 10-3 sec-1, nab1 = 10000/µm2, nab2 = 10000/µm2, d = 30 nm, kon2D= 1×10-5 µm4/sec.

Understanding the kinetics allows assay optimization. For example, with a 25 min assay time, we can design for optimal performance. In Fig. 6(a), the model predicts bead attachment probability of a 25-min assay with various combination of ta and tb. At the beginning, Pa increases as ta increases, and then decreases when ta is larger than about 10 min [40% in Fig. 6(a)]. Experimental results [inset of Fig. 6(a)] are consistent with the model prediction, showing ta = 40% leads to larger Pa than ta = 60%. Sample volume can also influence bead attachment probability [Fig. 6(b)]. One may expect that the attachment probability increases with the sample volume, because larger sample volume means more analytes and thus more captured analyte on each bead. The modeling results show that rate of increase decreases as sample volume increases. For example, the attachment probability increases 65% when sample volume increases from 20 µL to 50 µL, but only 11% when sample increases from 100 µL to 200 µL. The same trend was also observed experimentally. Therefore, this model can inform assay optimization for PoC assays, where sample volume is critical. We also examined the influence of bead diameter [Fig. 6(c)] for a use case where the total surface area of beads is kept constant while bead size varies. We observed in the modeling results that bead diameter can affect LOD by orders of magnitude, and large beads (1.0 µm) tends to give an improved (lower) LOD and larger current dynamic range. This predict is supported by our experiment results: the LOD of the assays using 1.0 µm beads is 0.4 pM, while that of the assays using 500 nm beads is 10 pM.

4. Discussion Various types of beads have been incorporated into surface coverage assays. They differ in diameter, ranging from nanoparticles (Bruls et al. 2009; Jing et al. 2019) to microparticles (Mulvaney et al. 2009; Otieno et al. 2014; Wu et al. 2018). Our model is applicable to scenarios where the bead diameter is large compared with biomolecules (> 100 nm). Additionally, the beads can be made of various materials, such as latex (Kubitschko et al. 1997), noble metals [e.g., gold (Jing et al. 2019) and silver (Lee et al. 2016)], iron oxide (Gijs et al. 2010) and combinations of multiple materials (Wang et al. 2015). The material can affect surface coverage (i.e., the number of beads attached to the sensor surface) in multiple aspects. For instance, magnetic beads can be manipulated by external magnetic fields to enhance surface coverage (Bruls et al. 2009). The surface functional groups or polymer coatings on bead surfaces may cause nonspecific interactions and affect antibody surface density. Nonspecific binding is ubiquitous and has been examined theoretically and experimentally (De Palma et al. 2007). In contrast, our model is focused on specific interactions that are the core of immunoassays. Conceptualizing the beads as ideal spheres, the model incorporates

mass density and body forces, both of which are related to the bead material. It includes antibody density as a variable parameter, so it can be used to examine the influence of antibody density (Fig. S6). Together with the bio-transduction element and the readout electronics, the bio-recognition element constitutes the essential component of a biosensor system. Analyte capture using affinity reagents is the basis of the bio-recognition element, and its role in determining the overall sensor performance is critical but underappreciated (Arlett et al. 2011). Our model comprehensively examines the specific binding in bead-based assays that have been implemented into various biosensors (Bruls et al. 2009; Mulvaney et al. 2009; Otieno et al. 2014; Wu et al. 2018), and can provide insight for the design of these biosensors and related bioelectronics. PoC immunosensors are required to deliver rapid results. Fig. 6(a) demonstrated that the model can be used to optimize testing time. Additionally, the model was used to compare two implementations of surface coverage assays and found that the semi-homogeneous assays is superior to the heterogeneous compartment under similar assay settings (Fig. S7). One may consider designing integrated microfluidic devices to accommodate the workflow of semi-homogeneous assays (Wu and Voldman 2019), given that microfluidics is ideal for fluid handling. The model can provide guidelines for design of such integrated microfluidics. For example, the model predicts that a short channel height is preferred to enhance bead attachment (Fig. S8). Additionally, the model suggests that the signal increases with the sample volume [Fig. 6(b)] and thus a high channel is preferred for analyte capture. Therefore, one may need to optimize the channel height, or even consider incorporating multiple channel heights into different regions of a device. Our results also show that the increase rate of the biosensor signal actually decreases with increasing sample volume, indicating that volume increases past a certain point have minimal effect on the signal. For electrochemical biosensors, signals are measured using potentiostats, for instance, via amperometry (Wu et al. 2018). When designing a miniaturized amperometry system for PoC biosensors, one needs to know the required sensitivity and dynamic range of the current measurement. Our model can estimate the current produced by each bead. Such estimation can help determine the required sensitivity to measure down to single beads. Moreover, one can estimate the maximum current that will be encountered, if the number of beads that will be in contact with the sensor is known. Finally, since the model captures a comprehensive list of parameters (e.g., sensor size, bead diameter and number, antibody affinity and assay time), it allows comparing different implementations of surface coverage assays. In Table S1, we compared four implementations of semi-homogeneous bead-based assays (Bruls et al. 2009; Mulvaney et al. 2009; Otieno et al. 2014; Wu et al. 2018) and used the model to predict the LOD. Overall, the predictions are consistent with the reported LOD.

5. Conclusion In this paper, we developed an integrated model to develop a holistic understanding of bead-based surface coverage assays. The model predicts that analyte capture on bead surfaces and bead attachment to sensor surfaces have very different kinetics, and that bead diameter can dramatically affect the LOD. Moreover, we found that the rate of assay signal increase actually decreases with increasing sample volume. Therefore, the model can provide insight as to the design and optimization of biosensors that implement bead-based immunoassays, along with the corresponding electronics. The model is focused on specific interactions between biomolecules, and does not consider nonspecific binding. Moreover, it assumes gentle washing after bead attachment so that the washing does not break immunocomplexes. These aspects should be considered in future work to extend the model.

Acknowledgments This research was funded by Analog Devices, Inc., Maxim Integrated and the Novartis Institutes of Biomedical Research. We would like to acknowledge the Microsystems Technology Laboratory for fabrication assistance.

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Highlights:

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A holistic mathematical model to describe bead-based surface coverage assays

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Bead diameter can affect limit of detection and assay kinetics

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Model provides guidelines for design and optimization of bead-based immunoassays

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: