## Accurate detection of sample location in isotachophoresis

#### Nethanel GanOr, Shimon Rubin and Moran Bercovici

### Highlights

- A combined analytical, numerical and experimental study of counterflow ITP (isotachophoresis) in an electric field regime where body forces can be neglected. Our analysis was divided into two main aspects: prediction of the shape of the curved ITP interface, and prediction of the spatial distribution of sample focused within.
- We address the structure of the ITP interface, and construct a simple model which successfully predicts the curved shape of the ITP interface.
- Our numerical and experimental analysis of the sample spatial distribution show that while pressure driven flow is typically considered to reduce peak concentrations, certain regimes allow a net gain in analyte concentration over the non-dispersed case.

### Shape of ITP Interface surface

We formulated a simple model describing the balance between axial advection and radial diffusion, and showed it leads to accurate prediction of the shape of the ITP interface. The model clearly illustrates the simple mechanism by which steady-state counterflow is achieved; ions advected (by the flow and electromigration) away from the interface return through diffusion. Since diffusion is proportional to the surface area of the interface, at higher ITP velocities the surface is further extended to allow sufficient diffusion to achieve a steady state flux balance.

*Figure 1: Theoretical model results showing the shape of the ITP interface surface against numerical results. The colormap presents the absolute value of the difference |c _{l}-c_{t}| mM as obtained from the simulation. We identify the ITP interface, f(r) , as a geometric locus of points where c_{l=}c_{t} (white line). We present the resulting interface shape (given by equation (5.5) in [1]) (black lines) for three diffusion constants: based on the L-ions (f(r)^{L}), the T-ions (f(r)^{T}) and their average (f(r)). Results show that the model is able to predict the shape of the interface, both qualitatively (in particular the vanishing slope at the walls) and quantitatively (with the interface length well predicted), with best results obtained when providing the model with average LE-TE diffusivity. Here the mobility of the L, T and counterion was set to 80×10^{-9} , 23×10^{-9} and 30×10^{-9} m^{2}/(V·s), respectively. The radius of the channel was 20 µm, and the electric current was set to 7 µA, resulting in an ITP velocity of ~0.42 mm/s.*

### Concentration distribution of sample ions

To analyze the focusing regimes of sample at the interface we formulated a heuristic analytical model describing analyte concentration at the asymptotic regions of the interface. This analysis predicts the existence of two different sample focusing regions: one at the center of the channel and one close to its walls, and identified the two main parameters governing the sample distribution (ratio of analyte to T-ion mobilities, and ratio of the ITP velocity to analyte diffusivity). Importantly, we showed that a sufficient condition for such radial focusing to occur is the existence of a non-uniform axial velocity field, even in the absence of body forces. Using numerical simulations, we validated the existence of the different regimes and mapped the effect of these governing parameters on the analyte focusing.

Applying counterflow to an electromigrating ITP interface is a well-established and very effective way of obtaining long sample focusing times or long reaction times within the ITP interface. It is particularly attractive as it also enables the use of shorter channels and thus lower voltages. However, counterflow ITP is typically associated with significant distortion of the focused sample, reduction in peak concentration, and loss of signal. Based on analysis, we demonstrated numerically and experimentally, that when the mobility of the analyte is higher than twice the mobility of the T-ion, peak concentrations in counterflow ITP can be significantly higher than in the non-dispersed case. This result may be beneficial in the design of future ITP-based assays; For example, reactions which take place at the ITP interface between two co-focusing species can be maintained over long durations of time without loss in reaction rates or signal.

* Figure 2: Experimental and numerical results showing concentration enhancement in counterflow ITP. Each data point represents the ratio between the peak analyte concentration in counterflow ITP, c_{CF}, to its peak concentration in standard non-dispersed ITP, c_{1D}. The experiments were performed on a 20 μm deep 90 μm wide D shaped channel, and the computational results are provided for a 70 μm diameter axisymmetric channel. Results show that above a threshold current value, counterflow ITP provides peak concentrations that is higher than those obtained using standard (non-dispersed) ITP. The inset shows experimental and computational images of the sample distribution for both the counterflow and non-dispersed cases. The radial focusing and resulting higher peak concentrations are clearly evident in counterflow ITP. The effective mobilities of the L (100 mM acetic acid) and T (100 mM MES) ions in the presence of the C-ion (150 mM bistris) were calculated using SPRESSO (taking into account ionic strength effects) and implemented in the computational models as 33×10^{-9}, 14×10^{-9} and 11×10^{-9} m^{2}/(V·s), respectively. The analyte mobility was taken as 23×10^{-9} m^{2}/(V·s). In the experiment, for each current value, three independent measurements were taken of the sample distribution before and after applying the counterflow. Error bars indicate 95% confidence on the mean. *

Finally, we showed that the existence of diffusion-based focusing regimes may enable a certain degree of radial separation between co-focusing species having different diffusivities. We demonstrated this experimentally using beads and a dye, which focused close to the walls and center of the channel, respectively. This effect must be taken into account in the design of bead-based reaction and can perhaps be used favorably to separate reaction products from at least one of their components.

* Figure 3: Experimental results demonstrating radial separation of co-focused species in counterflow ITP. (a) The two species, DyLight650 and 2.8 μm magnetic beads, have similar mobilities but very different diffusivities. We chose the chemistry of the L-ion (acetic acid) and T-ion (tricine) in the presence of the counterion (bistris) such that the mobilities of the species are closer to the TE region (thus representing subfigures (e) and (f) in Figure 5 in [1]). Consequently, the dye focused at the centerline, while the beads focused closed to the walls. (b) The main figure presents the radial distribution of the two species, averaged over the area bracketed by dashed lines, and over 11 images. The bracketed lines indicate 95% confidence on the mean.*

### References

[1] GanOr N., Rubin S., and Bercovici M., Diffusion dependent focusing regimes in peak mode counterflow isotachophoresis, *Physics of Fluids, 2015, 27, 072003*