Why a Diffusing Single‐Molecule can be Detected in Few Minutes by a Large Capturing Bioelectronic Interface

Abstract Single‐molecule detection at a nanometric interface in a femtomolar solution, can take weeks as the encounter rate between the diffusing molecule to be detected and the transducing nanodevice is negligibly small. On the other hand, several experiments prove that macroscopic label‐free sensors based on field‐effect‐transistors, engaging micrometric or millimetric detecting interfaces are capable to assay a single‐molecule in a large volume within few minutes. The present work demonstrates why at least a single molecule out of a few diffusing in a 100 µL volume has a high probability to hit a large capturing and detecting electronic interface. To this end, sensing data, measured with an electrolyte‐gated FET whose gate is functionalized with 1012 capturing anti‐immunoglobulin G, are here provided along with a Brownian diffusion‐based modeling. The EG‐FET assays solutions down to some tens of zM in concentrations with volumes ranging from 25 µL to 1 mL in which the functionalized gates are incubated for times ranging from 30 s to 20 min. The high level of accordance between the experimental data and a model based on the Einstein's diffusion‐theory proves how the single‐molecule detection process at large‐capturing interfaces is controlled by Brownian diffusion and yet is highly probable and fast.


Section 1: SiMoT I-V transfer characteristics
The drain current (I D )gate voltage (V G ) transfer curves at a fixed drain voltage V D = -0.4 V are shown in Figure S1a. The curves were measured in the forward and reverse mode to evidence the occurrence of any hysteresis. The black curve is the current measured on a sensing gate biofunctionalized with anti-IgG incubated for 10 minutes in 100 L of bare PBS after a stable current level has been accomplished. The measurement was carried out by sweeping the gate bias from 0.1 V to -0.5 V with steps of 10 mV in water. The black curve has been taken as the baseline. Subsequently, the very same sensing gate has been incubated for 10 minutes in 100 L of PBS standard solutions comprising 4 ± 2 IgG molecules, and the red curve has been measured. The same gate has been afterwards incubated in 100 L solutions encompassing N = 39 ± 6, N = 392 ± 20, N = 3.92 10 3 ± 60, N = 3.92 10 4 ± 2 10 2 , N = 3.92 10 5 ± 6 10 2 IgG molecules. Remarkably a significant current decrease has been registered already upon exposure to 4 ± 2 IgG molecules. This trend is replicated as the standard solutions with increased IgG concentration are progressively assayed, until the saturation of the response is reached.
The curves given in Figure S1b are the correspondent gate leakage currents I G , which are always about three orders of magnitude lower than I D . No faradaic activity can be evidenced by the I G curve of the anti-IgG gate, proving that mild electrochemical processes have been prevented by the fine-tuning of the inspected gate voltage window. Moreover, Figure S1b shows also no correlation between the I G current and the ligand concentration.
These are compelling evidence, proving that the field-effect induced current I D provides a capacity-coupled related sensing response. No other current flowing in the device can provide the same information. IgG comprising 4 ± 2 (red curve), 39 ± 6 (blue curve), 392 ± 20 (green curve), 3.92 10 3 ± 60 (magenta curve), 3.92 10 4 ± 2 10 2 (yellow curve) and 3.92 10 5 ± 6 10 2 (cyan curve) molecules, progressively assessed with the same anti-IgG biofunctionalized sensing gate. The gate is incubated in the samples to be assayed for 10 minutes. Here 20 transfer characteristics are measured. In the panel, only the curves measured during the last (20 th ) cycle at each concentration assayed are shown.

Section 2: The 25 l incubation volume case
For the smallest volume assayed, a 25 l, a droplet is deposited on the sensing gate. A schematic of the smallest volume assayed with the 25 l droplet deposited directly on the gate surface, is also provided ( Figure S2a). In this case, the radius of the droplet r d is measured to be 0.34 cm while that of the gate is r g = 0.25 cm as shown in Figure S2b. Equation 2 in the main text holds also in the case of the sensing gate exposed for 10 minutes to 25 l droplet of the PBS standard solutions. Figure S2a shows that the whole gate area is covered by the droplet while the incubation system is schematically depicted in Figure S2b. The volume of a liquid droplet with contact angle  and droplet radius r d is V d =  · r d 3 · (2 3cos + cos 3 ) / (3 · sin 3 ). [1] For r d ~ 3 mm a contact angle  of about 60° is computed, accounting for the relatively high wettability of the densely packed anti-IgG surface. This is in line with a measured contact angle of 65° reported for a surface covered with a protein density of 0.25 g/cm 2 , [2] comparable to the 10 12 /cm 2 anti-IgGs covering the SiMoT gate surface. Also in this case, however, the portion of the solution that is close enough to the (c) gate surface (d  r, see Figure 2c) to enable an antigen-antibody interaction within 600 s, can be approximated by a volume V r given by Equation 2, as  ~ 90° and the cylindrical disk has a radius r d with a height of r.
The plot of the P probability function (Equation 5) results in the dotted curve given in Figure S2c. As it is apparent the P function is unable to reproduce the 25 l data that are shifted towards higher N values. To fit the actual data the f function defined in the main text by the Equation 3 (main text) needs to be multiplied by a factor of  0.19. Assuming a constant  R these results suggest a reduced effective surface in the case of the incubation by drop casting. Qualitatively, these findings may be explained by the well-known spontaneous dragging of liquid from the droplet's interior towards its edges (so called coffee-ring effect) [3], [4] that may preferentially distribute the diluted IgG molecules in the outer part of the droplet and reduce the effective interaction area of the gate.

Section 3: Evaluation of the limit-of detection (LOD)
Figure S3 -Response of anti-IgG functionalized gates exposed to solutions of IgM molecules in PBS. Error bars are standard deviations over three replicates. The LOD level is computed as the average level of the signal (noise) plus three times its standard deviation.
The negative control experiment performed by exposing an anti-IgG gate to PBS standard solutions of Immunoglobulin M (IgM), defining the noise, is given in Figure S2.

Section 4: Further details on the Brownian motion theory
The translational diffusion coefficient D is given by the Stokes-Einstein equation: ionic-strength of 162 mM), we obtain a viscosity  s : [5] where  = 1·10 -2 g·cm -1 ·s -1 is the water viscosity at room temperature. In Equation S3.2 A/(mol/l) -1/2 and B/(mol/l) -1 are equal to 0.0062 and 0.0793 for NaCl; these constants are equal to 0.0052 and -0.0140 for KCl. Inserting these values and the solute concentrations of the two salts in Equation S3.2, we obtain  s = 1.013 · 10 -2 g·cm -1 ·s -1 , very close to the value for water. Taking a = 5.51  0.03 nm, the value for IgG monomers, [6] we obtain from Equation S1 a value D = 3.92  0.02·10 -7 cm 2 /s, in full agreement with photon-correlation spectroscopy data (3.89  0.02·10 -7 cm 2 /s). [6] The dynamics of an IgG moving in a PBS (water) solution is described by the Langevin equation adapted to the case of an antigen -antibody interaction, [7] with the trajectory

Section 5: Complete expression of the P probability function
The complete expression of the probability function P is provided in the following Equation S4.1: Details on the probability function P as function of the incubation volume encompassing a different number of molecules N is given in Figure S4.

Figure S4 -Probability P that in an incubation time of 600 s at least one of N IgGs, with N ranging between 4 and 36, will be close enough to the gate so as, during its random
Brownian motion, the binding with an anti-IgG will occur. The plot starts from 16 l, corresponding to the minimum volume needed to cover the gate.

Section 6: Chi-squared test
The chi-squared ( 2 ) test obtained by fitting experimental data in Figure 2 with the probability function P, by considering just the first experimental point for the saturation plateau (k = degree of freedom; p = probability >  2 ) [8] , are: The chi-squared test for the data in Figure 3 modelled also with function P gives: Since p > 0.95 in all the four sets of data at different incubation volumes, while p is larger than 0.85 when fitting the response at different incubation times, there is a high chance (95% for the data of Figure 2 and 85% for the data of Figure 3) that the null hypothesis is correct, i.e., that there is no difference between the observed and theoretical (expected) values predicted by the modeling.

Section 7: Surface Plasmon Resonance study of IgG binding to anti-IgG
The where k is the wavelength dependent sensitivity coefficient, and Δθ SPR is the experimental angular shift. For laser beams with λ = 670 nm and a thin layers (d < 100 nm), the following approximations hold true: (i) dn/dC ≈ 0.182 cm 3 g -1 , (ii) k•d ≈ 1.0•10 −7 cm deg. [9] Therefore, under these assumptions and by substitution of Thus, in Figure S5b (red squares) the dose-curve for the assayed IgG is reported as ∆θ SPR vs.
[IgG] nominal concentrations (semi-log scale). A control experiment was performed by using the BSA instead of anti-IgG as capturing bio-recognition element. Therefore, the SAM was modified with a BSA solution in PBS at concentration 100 g/mL, following the same protocol previously described for the anti-IgG anchoring. In Figure S5b