Real-time changes in the AC magnetic susceptibility of reagents during immunomagnetic reduction assays

Biofunctionalized magnetic nanoparticles show promise for applications in drug delivery,^1–4 contrast medium, hyperthermia, etc.^5–11 Researchers have used antibody-functionalized magnetic nanoparticles for ultrasensitive immunoassays with the aid of superconducting quantum interference devices (SQUIDs).^12,13 This assay technology is referred to as immunomagnetic reduction (IMR). Several groups have reported the feasibility of utilizing IMR for quantitative detection of proteins associated with cancers, prenatal conditions, neurodegenerative diseases, and so on.^14–16 Some assay kits using IMR have been approved for clinical uses in Europe and Taiwan. These results demonstrate the clinical impact of biofunctionalized magnetic nanoparticles via IMR.

In IMR, antibody-functionalized magnetic nanoparticles are homogeneously dispersed in phosphate-buffered saline (PBS) solution.¹³ Under external magnetic fields, the reagent generates an alternating current (AC) magnetic susceptibility. Magnetic nanoparticles associate with biomolecules that are to be detected. The association makes nanoparticles larger, and larger particles are less responsive to external magnetic fields. As a result, the AC magnetic susceptibility of the reagent is reduced, which is referred to as the IMR signal.¹⁷ The dependence of the IMR signal on the concentration of the biomolecule to be detected was explored. The IMR signal is enhanced by higher concentrations of target biomolecules.¹⁸ It was also found that the concentration-dependent IMR signal follows a logistic function.¹⁸ Thus, a quantitative analysis of a target biomolecule can be performed by measuring the IMR signals. Some papers have been published to investigate the characterization of concentration-dependent IMR signals.^14–16 To achieve high sensitivity, a high-temperature SQUID magnetometer is used to sense the reduction in the AC magnetic susceptibility of the reagent because SQUID is ultrasensitive to magnetic signals. The details of the SQUID IMR analyzer were reported by Chiu et al. and Chieh et al.^19,20

However, discussions on the chemical kinetics relevant to the real-time signals of the magnetic susceptibility of reagents during nanoparticle–biomolecule associations are rare. In this study, characterizations, such as associating processes, binding rates, binding entities, etc., embedded in the real-time signals of the AC magnetic susceptibility of reagents during nanoparticle–biomolecule association are explored. Here, total tau protein (T-Tau), which is a biomarker of neurodegenerative diseases, is used as an example of a biomolecule to be detected.

An antibody (T9450, Sigma-Aldrich) against T-Tau is immobilized on dextran-coated magnetic Fe₃O₄ nanoparticles (MF-TAU-0060, MagQu). The antibody-functionalized magnetic nanoparticles are dispersed in pH 7.4 PBS solution. The magnetic concentration of the IMR reagent for T-Tau is 10 mg-Fe/ml.

T-Tau solutions with various concentrations of 0.1, 1, 10, and 100 pg/ml are prepared (T7951, Sigma-Aldrich). For each concentration, duplicated measurements of the real-time signals of AC magnetic susceptibility of reagent during nanoparticle–biomolecule association are conducted. In each measurement, 60 µl T-Tau solution is mixed with 60 µl IMR T-Tau reagent. The real-time signals of the AC magnetic susceptibility, i.e., the χ_ac-t curve of the reagent, during nanoparticle–biomolecule association were recorded with an IMR analyzer (XacPro-S, MagQu).

The data points of the χ_ac-t curve for the 10-pg/ml T-Tau solution are plotted in Fig. 1. During the time interval from 0 to 100 min, χ_ac remains almost unchanged. χ_ac starts to decrease after ∼100 min. From this moment, more nanoparticles bind with T-Tau molecules. Then, χ_ac keeps decreasing until 200 min; afterward, χ_ac reaches a lower plateau. This implies that the nanoparticles start to bind with the T-Tau molecules at 100 min and finish associating at 200 min.

Although the data points in Fig. 1 are somewhat scattered, they can be well fitted by the logistic function, which is expressed as

where U, L, τ, and β are the fitting parameters. The fitting curve is plotted as the solid line in Fig. 1 with U, L, τ, and β equal to 998.5, 958.9, 141.3, and 6.44, respectively. Notably, in Eq. (1), U denotes the χ_ac value of the upper plateau before nanoparticle–biomolecule association begins, L denotes the χ_ac value of the lower plateau after nanoparticle–biomolecule association concludes, and τ denotes the time corresponding to a 50% reduction in χ_ac within the range from U to L. The mean values ± standard deviations of U, L, τ, and β for duplicated measurements of the χ_ac-t curves at various concentrations ϕ_T-Tau of T-Tau from 0.1 to 100 pg/ml are listed in Table I.

The value of (U-L)/U is the reduction percentage in the AC magnetic susceptibility of the reagent, i.e., the so-called IMR signal, for a given T-Tau solution. The IMR signal increased from 2.99% to 4.84% as the T-Tau concentration increased from 0.1 to 100 pg/ml. The increase in the IMR signal with increasing biomolecule concentration is consistent with previous studies.^21,22

Notably, τ is concentration independent (∼128 min in this case), whereas β decreases with increasing concentrations of T-Tau. In Eq. (1), -β is the slope in the χ_ac-t curve at t equals τ. Then, there is a sharper drop in χ_ac at t equals τ for lower concentrations of T-Tau.

With the values listed in Table I, the time-dependent χ_ac of the reagent is simulated. The simulated χ_ac is scaled via the following equation:

The χ_ac,s-t curves for 0.1- and 100-pg/ml T-Tau solutions are plotted with the solid line and the dashed line in Fig. 2, respectively. The time when χ_ac,s equals 95% is referred to as t_95%, and it is defined as the starting point for initiating nanoparticle–biomolecule association. The time when χ_ac,s equals 5% is referred to as t_5%, and it is defined as the endpoint for the completion of nanoparticle–biomolecule association. In Fig. 2, the t_95% and t_5% values are labeled with dots (•) and rectangular symbols (▪), respectively. Cross symbols (×) are used to label the time at τ. The time interval between t_95% and t_5% denotes the period of associating T. The value of ϕ_T-Tau/T is the reaction rate R for nanoparticle–biomolecule association. The relationship between the reaction rate R and the T-Tau concentration ϕ_T-Tau is plotted in Fig. 3. A proportionality is obtained, i.e., R ∝ ϕ_T-Tau. According to chemical kinetics,²³ the proportionality between the reaction rate and the concentration reveals that a single T-Tau molecule participates in each nanoparticle–biomolecule association.

For IMR assay, the association between magnetic nanoparticles coated with antibodies and biomarker molecules can be expressed as

where [M-Ab], [Biomarker], and [M-Ab-Biomarker] denote the molar concentrations of magnetic nanoparticles coated with antibodies, biomarker, and the nanoparticle–biomarker complex, respectively. As a magnetic nanoparticle associates with a biomarker molecule to form a complex, [M-Ab] and [Biomarker] decrease, while as [M-Ab-Biomarker] increases. The magnetic response of the complex is not so strong as the original magnetic nanoparticle to external magnetic fields. Hence, the reduction in the AC magnetic susceptibility of the mixture is reduced as a magnetic nanoparticle binds with a biomarker molecule to form a complex.

In this work, the initial concentrations of magnetic nanoparticles with antibodies are fixed in cases of assaying samples having T-Tau of different concentrations. For higher concentrations of T-Tau, more magnetic nanoparticles would bind with T-Tau molecules to form more amounts of the complex. Thus, the reduction in the AC magnetic susceptibility of the mixture would be enhanced, which is demonstrated by the observed IMR signals increasing from 2.99% to 4.84% as the T-Tau concentration increases from 0.1 to 100 pg/ml. Remarkably, at very high concentrations of biomarker, the reduction in the AC magnetic susceptibility of the mixture would not keep increasing because of the insufficient amounts of magnetic nanoparticles or the Hook effect of biomarker molecules. Hence, there exists a linear range of assaying biomarkers with IMR, as reported in Ref. 21.

According to the results in Fig. 1, the real-time signals χ_ac of the reagent show a clear lag phase, presented with t_95%, with the changes in χ_ac. This implies the binding kinetics has two steps for nanoparticle–biomolecule associations: diffusion step and binding step. Remarkably, the fact that the χ_ac-t curve in Fig. 1 follows the logistic function in Eq. (1) is not only observed for assaying T-Tau protein using IMR but also for amyloid β peptides,²⁴ α-synuclein protein,²⁵ vascular endothelial growth factor, and so on.²⁶ Hence, the consistence between the χ_ac-t curve with the logistic function in Fig. 1 would be a mathematical fact of diffusion.

The details of the discussion on the diffusion of magnetic nanoparticles suspended in water solution caused by the Brownian motion, concentrations of magnetic nanoparticles, or thermophoresis effects are available in the literature.^27,28 In the current case, the reagent has a fixed concentration of magnetic reagent at an isothermal process during the diffusion step, and the diffusion is limited by the T-Tau concentrations. As the T-Tau concentration increases, the averaged diffusion distance for the associations is reduced, resulting in lower values of t_95%. The decrease in t_95% at higher concentrations of T-Tau is evidenced in Fig. 2.

On the other hand, the binding between T-Tau molecules and nanoparticles is strongly limited by the surface tension of nanoparticles in PBS solution. In IMR, nanoparticles are oscillating at a high speed (∼20 kHz), which generates high surface tension on nanoparticles. Once T-Tau molecules bind with nanoparticles, the oscillating speed of nanoparticles is lowered and the surface tension of nanoparticles is reduced, enhancing further binding between T-Tau molecules and nanoparticles. Therefore, after some initial T-Tau molecules bind with nanoparticles, the signal χ_ac of the reagent would decrease exponentially.

In conclusion, the real-time signals of the AC magnetic susceptibility χ_ac of the reagent during nanoparticle–biomolecule association follow the logistic function. The reduction in χ_ac is enhanced at higher concentrations of the biomolecule to be detected. The association rate is proportional to the concentration of the biomolecule. It is deduced that a single biomolecule participates in each nanoparticle–biomolecule association. The binding kinetics between biomolecules and nanoparticles can be explained combining diffusion and binding steps.

This work was supported by MagQu Co., Ltd.