#### 3.1. Calibration of AMBR Viscometer

A linear relationship was found between the solution viscosity and the rotation period of the bead in the solution. A series of glycerol/water solutions with varying glycerol mass fraction were analyzed by the AMBR microviscometer and, in parallel, with an Ubbelohde viscometer [

16]. The microviscometer results matched both the Ubbelohde viscosity values and the theoretically predicted values for the mixtures over a viscosity range from 0.89 to 2.8 cP (

Figure 1c) [

17]. A correlation curve relating the bead rotation period with the solution viscosity was constructed and yielded excellent uniformity (

Figure 1d). The experimentally observed linear relationship between rotation period and viscosity agrees well with the theory developed for the paramagnetic AMBR system [

18]. Additionally, the linear correlation is robust to variation in the magnetic field driving frequency (

Figure A2), and for three different bead sizes tested, with the 45 µm beads giving optimal linear correlation results (

Figure A3). However, we note that the linearity does not hold as well for a frequency close to the instability threshold as shown in

Figure A2. Furthermore, the measurement of rotation period is not as reliable, because the jerky motion affects the image analysis. Therefore, we present our results at various driving frequencies in the Supplementary Materials, but chose to use only the higher driving frequency regime for the DNA measurements.

The observed linear correlation between solution viscosity and bead rotation period can be explained by the nonlinear magnetic oscillation theoretical framework [

18,

19,

20,

21,

22]. At a low driving frequency, the bead rotates at the same rate as the driving magnetic field. However, as the driving frequency increases, the bead cannot overcome the viscous drag exerted by the surrounding fluid, and thus cannot follow the rotating magnetic field. The bead then rotates slower, and asynchronously, with respect to the driving magnetic field (

Figure 1b) [

18,

20,

22,

23,

24,

25]. The nonlinear oscillation only occurs in the asynchronous regime. In a low Reynolds number environment, the force balance between the magnetic torque and the viscous drag yields the relationship between the bead rotation period and the solution viscosity. The effects of interaction between the magnetic bead and the solid surface can be neglected under the experimental conditions described in the Experimental Section. For a paramagnetic bead, the magnetic torque due to the induced magnetic dipole can be expressed as [

18],

where

χ'' is the imaginary part of the magnetic susceptibility (which is frequency dependent),

V_{m} is the volume of the bead’s magnetic content (

i.e., the magnetic nanoparticles embedded in the bead),

B is the strength of the driving magnetic field, and µ

_{0} is the permeability of free space. The torque due to the viscous drag can be expressed as,

where θ is the arc length of the rotation,

κ is the shape factor of the bead (

κ = 6 for a sphere),

η is the solution viscosity, and

V is the volume of the magnetic bead. By combining Equations (1) and (2), the equation becomes,

Therefore, in the asynchronous regime, the rotation period of a paramagnetic bead, under the rotating field of a given strength and frequency, is expected to be linearly proportional to the solution viscosity,

i.e., T ∝

η. Our experimentally observed results confirm this theoretical relationship.

To advance the practical utility of the asynchronous rotation method, we investigated the influence of the variation in bead properties on bead rotation periods. A relative standard deviation of approximately 10% is observed due to the variation in bead properties, such as size and magnetic content. As shown in

Figure 2a, the rotation periods of 20 beads in the same solution do not show a clear bead-size dependency. Thus, bead-size non-uniformity is not the primary contributor to the variation in the rotation period measurement, despite the expected correlation in Equation (3). More likely, the bead magnetic properties, such as magnetic volume and susceptibility, are more significant for the inter-bead variation than is the size variation. The scattered pattern in

Figure 2a supports the averaging over multiple beads in the construction of correlation curves and viscosity measurement experiments.

**Figure 2.**
Reproducibility of AMBR viscosity measurements at 100 Hz driving frequency. (**a**) Rotation period measurement of 20 independent beads in the same solution plotted against the optically measured bead size of each bead. (**b**) The rotation periods of two examples of 45 µm beads observed over time in the same solution. The rotation periods are calculated over a 12 s period and plotted in the graph. The average values are for 17 sequential observations

**Figure 2.**
Reproducibility of AMBR viscosity measurements at 100 Hz driving frequency. (**a**) Rotation period measurement of 20 independent beads in the same solution plotted against the optically measured bead size of each bead. (**b**) The rotation periods of two examples of 45 µm beads observed over time in the same solution. The rotation periods are calculated over a 12 s period and plotted in the graph. The average values are for 17 sequential observations

To confirm that inter-bead variation in the rotation period is primarily due to inherent bead properties, we measured the rotation period of the same bead continuously over time. The differences in rotation period over time are much smaller than the differences between two beads in the same experiment (

Figure 2b). The relative standard deviation for a single bead over time is approximately 1%, 10 times smaller than the standard deviation in the rotation period among 10 beads. Therefore, the observed measurement error is smaller than the error caused by the bead non-uniformity. A wide variation in commercial bead properties has been observed before [

26,

27]; consequently, improved uniformity of bead magnetic character and size is expected to give better sensitivity in viscosity measurement.

#### 3.2. Viscosity Measurement of DNA Aqueous Solutions

There is a linear relationship between the viscosity of common diagnostic reaction solutions and the concentration of DNA in those solutions. At a fixed temperature, the relationship between the solution viscosity, η, and the DNA concentration, C, for a very dilute solution can be expressed as η = η

_{0}(1 + C[η]), where η

_{0} is the viscosity of the solvent and [η] is the intrinsic viscosity of the DNA product. This equation gives a linear correlation between the viscosity and the macromolecule concentration. The intrinsic viscosity increases with the molecular weight of dsDNA, and this correlation has been documented [

15],

The linear relationship between the viscosity and the DNA concentration breaks down at very high molecular weight or high concentration due to the non-Newtonian property of the DNA solution [

28].

Digestion of lambda DNA with

EcoRI has a variety of uses and performs a selective cleaving of DNA at a specific site, forming DNA fragments of length 3530, 4878, 5643, 5804, 7421 and 21,226 bp from DNA of original length of 48,502 bp. With the experimental relationship given in

Figure 1d, we estimated the viscosities of the DNA

EcoRI digest solutions, at different concentrations, using the measured bead rotation periods. A linear relationship was found between the solution viscosity and the DNA concentration (

Table 1 and

Figure 3a), confirming the assumption that these solutions were in the dilute solution regime. The viscosities of the DNA solutions measured using the AMBR microviscometer are within the theoretically estimated upper and lower bounds.

**Table 1.**
Rotation periods and viscosities of lambda DNA EcoRI digest solutions at different DNA concentrations measured by AMBR microviscometer. The expected ranges of viscosities are calculated, assuming only the longest or shortest piece of DNA is present.

**Table 1.**
Rotation periods and viscosities of lambda DNA EcoRI digest solutions at different DNA concentrations measured by AMBR microviscometer. The expected ranges of viscosities are calculated, assuming only the longest or shortest piece of DNA is present.
| Experimental Results | Expected Range |
---|

DNA Conc. (g/L) | Rotation Period (s) | Viscosity (cP) | Min Viscosity (cP) | Max Viscosity (cP) |
---|

0.00 | 2.40 ± 0.24 | 0.90 ± 0.05 | 0.89 | 0.89 |

0.02 | 2.70 ± 0.64 | 0.96 ± 0.14 | 0.94 | 1.07 |

0.05 | 3.12 ± 0.62 | 1.06 ± 0.14 | 1.02 | 1.34 |

0.09 | 3.87 ± 0.21 | 1.22 ± 0.05 | 1.15 | 1.78 |

0.19 | 5.86 ± 0.49 | 1.67 ± 0.11 | 1.41 | 2.67 |

0.35 | 9.52 ± 1.53 | 2.48 ± 0.34 | 1.85 | 4.18 |

**Figure 3.**
DNA measurement using AMBR microviscometer. (**a**) Viscosities of lambda DNA EcoRI digest solutions at different concentrations, as measured by AMBR microviscometer. The green area indicates the expected range of the viscosity calculated theoretically, assuming that only the longest (top range) or only the shortest (bottom range) DNA fragment size is present. Error bars represent standard deviation among 10 beads in one measurement. (**b**) Measurement of bead rotation period of pre- and post-digestion samples of lambda DNA by AMBR microviscometer. The field driving frequency is 150 Hz. The error bars show the standard deviation among 10 beads in each measurement. (**c**) Measurement of viscosity by bead rotation period in PCR reactions sampled every 5 cycles, starting from the 6th cycle. PCR reactions with initial DNA amounts of 0 ng, 0.05 ng, 5 ng, 55 ng, and 250 ng are shown. The reaction volumes are 50 µL each. The field driving frequency is 150 Hz, and the PCR product size is 4500 bp. Each point represents the mean value, observing ten beads. (**d**) Fluorescent signal intensities of the PCR product (4500 bp band) observed on a electrophoresis gel for the same samples measured in (**c**).

**Figure 3.**
DNA measurement using AMBR microviscometer. (**a**) Viscosities of lambda DNA EcoRI digest solutions at different concentrations, as measured by AMBR microviscometer. The green area indicates the expected range of the viscosity calculated theoretically, assuming that only the longest (top range) or only the shortest (bottom range) DNA fragment size is present. Error bars represent standard deviation among 10 beads in one measurement. (**b**) Measurement of bead rotation period of pre- and post-digestion samples of lambda DNA by AMBR microviscometer. The field driving frequency is 150 Hz. The error bars show the standard deviation among 10 beads in each measurement. (**c**) Measurement of viscosity by bead rotation period in PCR reactions sampled every 5 cycles, starting from the 6th cycle. PCR reactions with initial DNA amounts of 0 ng, 0.05 ng, 5 ng, 55 ng, and 250 ng are shown. The reaction volumes are 50 µL each. The field driving frequency is 150 Hz, and the PCR product size is 4500 bp. Each point represents the mean value, observing ten beads. (**d**) Fluorescent signal intensities of the PCR product (4500 bp band) observed on a electrophoresis gel for the same samples measured in (**c**).

#### 3.3. Measurement of DNA Reaction Progression

Measurements of restriction digestion samples confirm that the AMBR microviscometer is sensitive to viscosity changes caused by the DNA size changes. As shown in

Figure 3b, a clear difference in bead rotation period can be seen between the digested and undigested lambda DNA solutions. Thus, the AMBR microviscometer can detect DNA sequence variation using a site-specific restriction endonuclease to essentially alter the solution viscosity.

Measurements of PCR reaction samples over the course of the reaction show that the AMBR microviscometer can detect the formation of PCR products in real time. As expected, the reactions with the higher initial template concentration reach the maximum product concentration sooner than those with lower template concentrations (

Figure 3c), and the plot of reaction cycle number

versus log of initial DNA concentration yields a linear correlation (

Figure A4). Comparing the AMBR measurements with the gel electrophoresis results on the same samples (

Figure 3d) confirms that the viscosity-based method is approximately 5 cycles delayed, relative to gel electrophoresis detection.

Using commercial paramagnetic beads, the AMBR microviscometer is found to be sensitive to the viscosity changes associated with DNA reactions. The results on PCR, with a product size of 4500 bp, yield a 10% relative error in the rotation period measurement. The AMBR microviscometer should be able to detect PCR product sizes as low as 1000 bp, assuming a conversion of >95% of dNTPs to its polymerized product (i.e., 0.42 g/L final product concentration). However, this sensitivity can be further improved so as to meet the need of monitoring DNA reactions with smaller viscosity changes (e.g., PCRs with shorter DNA products) by optimizing the bead size, shape, and magnetic properties. Based on the 1% relative error observed for single bead measurements, over time, we predict that the AMBR microviscometer may be able to detect PCR with product size as low as 50 bp. By measuring the changes in viscosity of DNA solutions, our technique can measure the difference in molecular length for a known concentration or the difference in concentration for a known length.