General model for mass transport to planar and nanowire biosensor surfaces

The detection principle of virtually all biosensors requires a reaction of analytes in a solution with receptors on the sensor surface, which is then transduced into a measurable signal. However, as analyte concentrations become increasingly low, the number of molecules in a solution decreases and the transport of the analyte to the sensor surface becomes progressively more difficult, often being the factor determining the limit of detection of biosensors. Understanding the time scales of binding is critical for sensor optimization.

Nair and Alam theoretically examined transport to biosensors providing a simple scaling law describing diffusion to different geometry biosensors analogous to electrical capacitors.¹ Their theory suggests that mass transport towards spherical and cylindrical sensors is more rapid than to planar sensors due to radial diffusion. However, in real-world scenarios, the purely diffusive case is rare, and flow is often employed to increase mass transport. This begets the question: how much enhancement of mass transport does radial diffusion provide under flow?

Solving the case of reaction and diffusion simultaneously is already quite complicated. The addition of convection complicates the situation even further, requiring complex simulations in order to generate a solution. Even in the case where a solution is found, it is only applicable under the conditions used in the simulation. The challenges and concepts involved in tackling the addition of convection are summarized concisely by Squires et al.² Herein, we use these concepts in conjunction with those utilized by Alam and Nair to develop a general model describing binding to planar and nanowire sensors that includes the effect of flow. This model provides a simple method to model binding of molecules to biosensors in a microfluidic system over a wide variety of variables. To demonstrate its effectiveness for sensor optimization, we examine mass transport to planar versus nanowire sensors with flow under multiple conditions. In contrast to convectional planar sensors, nanowire sensors have demonstrated femtomolar sensitivity.^3,4 The improved performance of nanowire sensors, in comparison with conventional planar sensors, has been attributed to increased signal-to-noise ratio,^5–7 surface-to-volume ratio,⁸ reduced screening,⁹ and faster mass transport.¹ The faster mass transport has been proposed by Nair and Alam to be caused by radial diffusion. Our results reveal that the benefits of radial diffusion are negligible given enough flow due to the compression of the depletion region. A large decrease in the settling time is observed under flow for sensors with planar diffusion. In contrast, when considering radial diffusion, the enhancement from flow is greatly reduced because mass transport under purely diffusion is already rapid. However, it is observed that decreasing the sensor length along the flow direction decreases the settling time, which is an advantage nanoscale sensors possess.

The reaction between analytes and receptors is modeled as a 1:1 binding interaction. The rate of conjugation between analytes and receptors is given by

where N is the density of conjugated receptors, N₀ is the density of receptors on the sensor surface, ρ_s is the surface concentration of the analyte, and k_a and k_d are the association and dissociation rate constants of the analyte, respectively. ρ_s is determined by the balance of mass transport toward the sensor surface and the rate of binding of the analyte to the sensor. The faster the mass transport can occur, the higher the ρ_s is and the faster the analyte can bind. If mass transport can deliver the analyte much more quickly than binding can occur, then the binding rate is determined only by the association and dissociation constants of the analyte and the number of receptors available on the sensor surface. This is the reaction-limited case and represents the upper “speed limit” for how quickly binding can occur. Independent of the rate of mass transport, the equilibrium density of bound receptors is given by $Neq=N0/(1+ka/kdρ0)$⁠, where ρ₀ is the bulk concentration of the analyte.

As the analyte adsorbs to the sensor surface, the concentration of analyte near the surface is depleted, forming a depletion zone of length, $δd=2nDt$⁠, where n is the dimensionality of the sensor (n = 1 for planar; n = 2 for nanowire). This concentration gradient drives the diffusion of analyte towards the sensor surface. ρ_s is thus the result of competition between binding depleting the concentration of analyte and diffusion replenishing it. The rate of change of ρ_s can be described by Fick's law of diffusion,

where D is the diffusion coefficient of analyte in solution. Regardless of the shape or size of the sensor, the solution to Eq. (2) at steady state is given by¹ 

where I is the integrated flux, J is the incident flux, A_D is the dimension dependent area of the sensor surface, ρ₀ is the bulk concentration, and C_D is the diffusion equivalent capacitance. This solution assumes the sensor to be a perfect sink for the analyte, i.e., $kd=0$⁠. Although no sensor is a perfect sink, this is a good approximation so long as $ka≫kd$⁠, as is commonly true. The incident flux is balanced by the rate of conjugation of the analyte to the sensor surface, giving $J=dN/dt.$ Equations (1) and (3) can then be solved simultaneously to give

where C_D(t)/A_D depends on the geometry of the sensor (Table I). In arriving at this result, we assumed quasi-steady state conditions, as described by Nair and Alam.¹ A similar assumption is made by Squires et al.² and in the two-compartment model used in surface plasmon resonance (SPR).¹⁰ In this assumption, the depletion zone evolves quasi-steadily as the analyte is continuously captured. Therefore, the time-dependent response of the biosensor can be considered to be a perturbation of the steady-state solution given by Eq. (3). This assumption is valid as long as the time scale for the binding flux to change appreciably is much greater than the time scale for the depletion zone to form,² i.e., the depletion zone must form well before the sensor surface is saturated. Figure 1(a) shows the comparison of our model with previously published numerical solutions¹ of the purely diffusive case. The model fits very well in both the nanowire and the planar cases.

In the purely diffusive case, as the analyte is continuously adsorbed to the sensor surface, the depletion region will continuously grow, while diffusive flux will likewise decrease. The effect of flow is to replenish the depleting analyte, effectively limiting the growth of the depletion region. The effect of convection on the binding rate is included by accounting for its effect on the growth of the depletion region. First, the influence of convection on the depletion region will be established.

In deriving the effects of convection, the fluid is modeled as flowing through a channel with dimensions given in Fig. 1(b). Typically, the width of the channel W_c is much greater than its height H; thus, the concentration can be assumed to be uniform across the channel (in the H direction) and the system can be treated as two-dimensional. Fluid flows through the channel with velocity

which is assumed to be parabolic the Poiseuille flow with rate Q. As can be seen, the effect of flow depends on many different dimensional parameters that may vary from system to system. In order to make meaningful comparisons, these systems are often described using dimensionless parameters. First, the Peclet number, $PeH=Q/DWc$⁠, describes the competition between diffusion and convection. When $PeH≫1$⁠, convection dominates and molecules are swept past the sensor before they can diffuse very far towards the sensor. As a result, only the analyte near the sensor surface can be collected, and the flow can be treated as a linear shear flow $u=γ˙z$ at a height z above the sensor. The depletion width in this case is limited to $δf$⁠, the distance at which the time for molecules to diffuse towards the sensor $(δf2/D)$ equals the amount of time it takes for them to be swept past the sensor $(L/γ˙δf).$ The depletion width is thus

where $Pes=6L2H2PeH$ is the shear Peclet number. The incident flux through this depletion width can be estimated as $JD∼Dρ0δfWsL$⁠. A dimensionless flux $F∼JD/Dρ0Ws$ can also be defined. From this, it is clear that

The dimensionless flux $F$ has been determined empirically for high Pe_s¹¹ and low Pe_s¹² 

Turning to the case where Pe_H ≪ 1, convection does not dominate. Flow is no longer enough to limit the growth of the depletion region due to diffusion. Instead, the depletion region continues to grow and begins to extend laterally into the channel. In this low flow limit, all molecules that flow towards the sensor are collected such that the diffusive flux, J_D= ρ₀HW_c/δ, equals the convective flux, J_c= Q ρ₀. The dimensionless flux $F$ in this case equals Pe_H and the depletion width can be defined as

This factor describes the depletion width at the very low flow rates. An effective depletion width due to diffusion and convection can now be established as

where δ_d is the depletion width of the purely diffusive case. In the first moments of binding, δ_eff is determined by δ_d until it approaches δ_f at which point it plateaus. The addition of δ_ch ensures $F$ approaches Pe_H at very low flow rates.

Figure 1(c) shows the comparison of our model to COMSOL simulation data in the literature for a planar model in the diffusion-convection case.² The model fits almost exactly with some deviation present at larger λ values, with λ being the dimensionless parameter equal to L/H. Although this comparison considers only the planar geometry, the concepts utilized to account for convection are also applicable to nanowire sensors, and the differences in mass transport to planar and nanowire sensors are assumed to be related to only the differences in diffusive transport.

We confirm the validity of the model using simulations and experimental data from the literature. Figure 2(a) shows our model compared with literature data of reaction limited binding of myoglobin in SPR.¹³ In a reaction-limited regime, diffusion and convection are negligible and the binding curve should closely match a simple 1:1 Langmuir binding model. Under conditions without mass transport limitations, our model imitates a 1:1 Langmuir binding model exactly. The model replicates the data very well at all concentrations with some minor deviation at early times. This deviation is due to divergence of the experimental data from ideal 1:1 binding at the initial stages of the experiment. Figure 2(b) shows a comparison of our model with simulations of binding in a transport limited regime.¹⁰ The model replicates the response exactly at low concentrations but slightly overestimates the response at increasing concentrations. Figures 2(c) and 2(d) show a comparison with a simulation of binding at different flow rates with two different binding site densities.¹⁴ Increasing the number of available binding sites increases the transport limitations present. In the case of a low number of binding sites, the model replicates the response almost exactly. With a higher number of binding sites, the binding becomes more mass transport limited, and the model overestimates the binding response slightly. However, the effect of changing flow rates is reproduced quite well. Figure 2(e) shows a comparison of the model with experimental data examining the effect of flow rate on binding. In this case, the diffusivity of the GST-Lcyt-YF protein is unknown. D₀= 5.59 × 10⁻⁷ cm²/s was chosen as a good fit to the data. With this diffusivity, the model replicates the data very well. Using the procedure reported by Young et al.¹⁵ to estimate diffusivity from molecular weight, an estimate of 7.2 × 10⁻⁷ cm²/s was obtained. Given the ± 20% error reported by Young when compared with experimental data, our fitted diffusivity is reasonable.

The model developed here enables simple simulation of a realistic binding response to planar and nanowire sensors over a wide range of flow rates and channel geometries. From modeled binding curves [Fig. 3(a)], relevant benchmarking metrics can be calculated, such as settling time at different concentrations. To demonstrate the usefulness of this model, settling time is used as a metric to compare the effects of radial diffusion with and without flow. To isolate this effect, all dimensions of the sensors and channels will be kept identical. The nanowire diameter will be the same as the length of the planar sensor. Settling time is defined here as the time required to bind a certain threshold number of molecules [N(τ_s) = N_eq(1-e⁻¹)]. The advantage of defining settling time in this manner is that it allows us to make consistent, meaningful comparisons over many concentration ranges. If, instead, a static threshold value was chosen, the results may change depending on where along the binding curve this threshold value appears. τ_s defined in this situation plateaus to a constant value at low concentration which is determined solely by k_d, as seen in Fig. 3(b). This constant value of τ_s will be the metric for comparison of radial vs planar diffusion. The reaction limited case will also be presented as a “speed limit” to mark when mass transport has no further effect.

Figures 4(a) and 4(b) show the effect of changing sensor length on the settling time at moderate (Pe_H = 1) and high (Pe_H= 100) flows. Considering first the purely diffusive case, the radial diffusion model has a greatly reduced settling time which decreases with the sensor length in comparison with the planar diffusion model similar to the results by Nair and Alam.¹ It should be noted that the planar diffusion model is valid until the depletion region becomes comparable to the length of the sensor. This begins to occur at time L²/D in the purely diffusive case,² after which the depletion region begins to grow radially. At the flow rates investigated here, the depletion region does not grow significantly enough to display radial diffusion, so the effects of radial diffusion do not need to be considered. In the purely diffusive case, the depletion region can grow significantly to the point where radial diffusion becomes significant. The planar diffusion model used here does not account for radial diffusion and most likely overestimates the settling time. However, the focus here is comparing the effects of radial diffusion under flow.

When moderate flow rates are included, the settling time of the planar diffusion model drops dramatically, becoming comparable to the radial diffusion model. In contrast, the radial diffusion model does not see as much enhancement from flow and the relative enhancement decreases with decreasing sensor size in agreement with the theoretical calculations by Sheehan and Whitman¹⁶ for nanobiosensors. Increasing the flow rate even further decreases the settling times in both cases, and the planar diffusion model replicates the radial diffusion model, demonstrating that the enhancement of radial diffusion is negligible under enough flow.

The decrease in the settling time due to radial diffusion can be understood by considering the growth of the depletion region due to an increase in depletion width of a planar versus a nanowire sensor. The depletion region can be thought of as the region from which analyte can be captured. Increasing the size of this region increases the effective capture area of the sensor. However, increasing the size of the depletion region also increases how far the nearest capturable analyte must travel to reach the sensor. This is the reason diffusion slowdown is observed. For sensors with radial diffusion, the increase in depletion region with depletion width is greater than that for planar sensors. In other words, the relative area from which a nanowire sensor can capture the analyte is greater than a planar sensor at the same depletion widths. The sensor response for cylindrical sensors scales with ∼t, while planar sensors scale with $∼t12$⁠.¹⁷ The effect of flow is to compress the depletion region due to a constant influx of the analyte towards the sensor. Flow has a greater effect on the planar diffusion model because it depletes the solution more rapidly in one dimension, i.e., C_D/A_D decreases with time more rapidly. Conversely, the radial diffusion model does not deplete the solution as rapidly so the constant influx of the analyte that flow provides has less of an effect.

In both models, the settling time decreases with the sensor size, eventually approaching the reaction limited regime. As the analyte flows past a sensor, it is more likely to first adsorb on the part of the sensor closest to the source. Subsequently, the parts of the sensor further downstream see a lower effective concentration of the analyte. Decreasing the sensor length (along the flow direction) limits the length that this concentration drop can occur and therefore decreases the settling time. These results suggest that, under flow, nanowires will have much lower settling times than purely due to their smaller dimensions. However, this is on a per area basis. Although a smaller sensor reaches a certain #/area of bound molecules more rapidly, a larger sensor has a larger area to capture the analyte.

Figures 4(c) and 4(d) show the effect of the changing flow rate on settling time at two different sensor lengths. In contrast to Figs. 4(a) and 4(b), where the sensor size was varied at two different, static flow rates, the effect of radial diffusion does not change within each individual plot. Similar to changing the sensor radius, increasing the flow rate decreases the settling time for both the planar diffusion and the radial diffusion models, with the planar model benefitting much more from flow. Under moderate flow rates, both models become identical regardless of the sensor size. For the 10 μm sensor, moderate flow rates are needed to see a significant decrease in the settling time for the radial diffusion model, while the planar diffusion model benefits greatly from even minute amounts of flow. As previously mentioned, the planar diffusion model most likely overestimates the settling time at very low flow rates (below ∼0.3 Pe_H in these conditions). Neither model approaches the reaction limited regime with even very large flow rates. In contrast, with a 10 nm sensor, the radial diffusion model approaches the reaction limited regime without flow, and the planar diffusion model needs minimal flow to reach the reaction limited regime, further demonstrating the significance of sensor size. These results further demonstrate that enhancement of radial diffusion becomes negligible under flow. This suggests that nanoribbon sensors should possess comparable performance to nanowire sensors under flow, from a mass transport perspective. However, nanoribbon sensors can provide advantages in terms of ease of fabrication.

Figures 4(d) and 4(e) show the effects of changing the channel height on the settling time at two different flow rates, while holding sensor dimensions constant. Changing the channel height at a constant sensor length and flow lets us examine the effect of changing Pe_s at a constant Pe_H. In other words, changing the channel height under these conditions limits the growth of the depletion region, while the characteristics of the flow with respect to the channel do not change. The effect of channel height in the purely diffusive case is not being considered here. At moderate flows, decreasing the channel height has a limited effect on the settling time, reducing the settling time by less than an order of magnitude over four orders of magnitude of change in height. The effect is much more pronounced at high flow rates, with the settling time approaching the reaction limited limit at low heights. At both flow rates considered here, the planar diffusion model is comparable to the radial diffusion model.

In the case where $Pes≪1$⁠, the depletion width is larger than the sensor and δ_f scales with $PeS−12$⁠. As the height is lowered, the depletion width becomes increasingly smaller, and the flow that the sensor experiences becomes less parabolic and more linear. For the case where $Pes≫1$⁠, the depletion width becomes comparable with the sensor size and δ_f scales with $PeS−13$⁠. When this transition point is reached, the effect of the channel height on the settling time increases noticeably. The overall effect is more pronounced at higher flows because the effect of height on Pe_S is larger at higher flows. From these results, it seems that the reaction limited regime can be reached by simply reducing the channel height regardless of whether radial or planar diffusion is considered. However, decreasing the height to the micrometer scale causes the pressure developed within the channel, at the flow rates used here, to reach extreme values beyond realistic working conditions.

A general model for binding to planar and nanowire sensors under the effects of both diffusion and convection was developed. The model approximated experimental binding data under a wide variety of conditions quite well. The model was used to draw comparisons between mass transport under planar and radial diffusion with flow and it was found that the increased mass transport to nanowires due to radial diffusion is negligible given enough flow. Moderate flow rates greatly reduced the settling time of planar sensors, whereas nanowire sensors, already possessing rapid mass transport from radial diffusion, did not benefit as significantly. However, decreasing the sensor size under flow was found to greatly reduce settling times. 10 nm sensors were able to reach the reaction limited regime regardless of the type of diffusion, whereas 10 μm sensors could not under the flow rates used here. Under flow conditions in typical experiments, radial diffusion associated with nanowire sensors is inessential for fast mass transport. Instead, the small sensor length in the flow direction limits the concentration drop that can occur as the analyte flows past the sensor. Similarly, decreasing the channel height decreases the settling time, but to a lesser extent. Although both planar and radial diffusion models were shown to be able to reach the reaction limited regime, this regime may still require incubation time on the order of hours to detect an appreciable response depending on the required number of molecules for detection and diffusion coefficient of the analyte. To further improve the limit of detection of biosensors, it is critical to reduce the number of molecules required for detection or investigate strategies involving increasing k_a, e.g., through electrostatic attraction.¹⁸ The former may be where nanobiosensors prove to be beneficial.^19,20

This work was supported in part by AVX Corporation, the Agriculture & Food Research Initiative Tri-partite Competitive Grant No. 2018-67015-28307 from the USDA National Institute of Food and Agriculture, Department of Agriculture, Environment and Rural Affairs, Northern Ireland, Department of Agriculture, Food and the Marine, Ireland, the Marcus Center for Therapeutic Cell Characterization and Manufacturing, The Georgia Tech Foundation, and the Georgia Research Alliance. This material is based on work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1650044. This work was performed in part at the Georgia Tech Institute for Electronics and Nanotechnology, a member of the National Nanotechnology Coordinated Infrastructure, which is supported by the National Science Foundation (Grant No. ECCS-1542174). The authors declare no competing financial interest.