## Sample distribution in peak mode Isotachophoresis

### Shimon Rubin, Ortal Schwartz, and Moran Bercovici

### Highlights

#### Theory

- Closed form solutions for sample distribution, electric field, as well as for leading-, trailing-, and counter-ion concentration profiles in one dimensional, peak mode ITP (isotachophoresis) problem. Our analysis reveals two major scales which govern the electric field and buffer distributions, and an additional length scale governing sample distribution.
- Experimentally verified and numerically validated model, including systems of weak electrolytes which better represent real buffer systems. The model also describes concentration profiles of multivalent analytes such as proteins and DNA.

#### Applications

- Studying reactions between multiple species having different effective mobilities which are co–focused at a single ITP (isotachophoresis) interface. We find the corresponding closed form expression for an effective-on rate which depends on reactants distributions overlap, and derive the conditions for optimizing such reactions.

### Introduction: description of a problem and theoretical model

#### Isotachophoresis

Isotachophoresis is characterized by non-uniform electric field, where the buffer ions in the steady state, are ordered according to the magnitude of their effective mobility, all moving in common velocity. The simplest example of this case is two electrolytes which dissociate into L-ion and T-ion, and common positive counterion. The mobility of the L-ion is assumed to have higher magnitude of electrophoretic mobility than the mobility of the T-ion, and without loss of generality the signs of the L-, and the T-ion are assumed to be negative while the counter-ion is positive. Consistently, the electric field in the region which is occupied with the lower mobility T-ions (TE region) is stronger than the electric field in the region occupied by higher mobility L-ions (LE region).

*Figure 1. Schematic illustration of analyte focusing in peak mode ITP. (a) An isotachophoretic front is formed between a leading and trailing electrolyte, and has a characteristic width . The analyte (typically present at a significantly lower concentration than the L or T ions) has a characteristic width typically larger than the interface width . (b) Two dimensional schematic illustration of sample distribution as would typically be obtained in fluorescence imaging.*

Importantly, sample ions, which have effective mobility bracketed between the effective mobility of the L-, and the T-ion, are focused under an electric field in the transition zone, between the LE and TE regions. In other words, when viewed in the co-moving frame of reference, the underlying velocity vector field for the analyte ions in the LE/TE region is pointing towards TE/LE and has negative divergence. Notably, as follows from the solution of Nernst-Planck equations the mutual effect of the competing electromigration and diffusion results in localized packet of ions. In contrast to the plateau mode ITP, where species achieve sufficiently high concentrations to form distinct and separate zones, peak mode ITP describes cases in which one or more species focus at the transition zone, and have sufficiently low concentrations such that their back-reaction on the electric field is negligible.

In recent years, owing to the advent in microfabrication technology, ITP has been successfully translated into microfluidic chips, giving rise to new applications such as nucleic acid purification and detection, protein analysis, bacterial detection and acceleration of reaction kinetics.

#### Our model

We follow Saville and Palusinski, [1] and solve the corresponding Nernst-Planck equations in a case of a uniform long channel of arbitrary cross-section shape and moderate pH variation between the LE and TE regions. We derive closed form expression for trailing-ion, leading-ion electric field, and sample distribution given by Eq.(9), Eq.(24), Eq.(25), Eq.(29) Eq.(32) in [2], respectively. We also derive several useful closed-form engineering approximations such as Eq.(28) and Eq.(30) in [2], which could be useful for numeric schemes.

The closed form expressions open a door towards straightforward analysis of the transition zone, as well as sample ions concentration profile with respect to different parameters in the model. In Figure 2 we bring analytical model results showing typical shape dependence of analyte concentration profile, as seen in the comoving frame, for different values of analyte effective mobility and fixed buffer properties. Particularly, Figure 2(a) shows that low/high effective mobility of the analyte results in diffusive legs towards TE/LE region.

Movie 1 shows a one-parameter family of solutions for analyte concentration profile, for different values of analyte effective mobility and fixed buffer properties. Our analysis shows analyte’s shape is much more sensitive to changes in its’ mobility for values just above the mobility of the T-ion.

*Figure 2: Analytical model results showing the effect of analyte mobility on its distributions (a) analyte concentration profiles for five different analyte mobilities, 0.26 (a) ,0.28 (b), 0.46 (c),1.14 (d), 2.04 (e), for =3.5 and =0.25. (b) The curve , relating the maximum along the channel to the analyte mobility. (c) Peak concentration as a function of the analyte mobility. Peak concentration is very sensitive to analyte mobility, when the latter approaches the TE mobility. At higher mobilities, changes in peak concentration are more moderate. The dotted line shows value , where (for N=1), i.e. that typical maximal concentration is ten times smaller than the commonly used approximation which assumes that the analyte is distributed uniformly within a region of width .*

*Movie 1: Analytical model results showing analyte concentration shape dependence as a function of analyte effective mobility for fixed buffer parameters. In this case , and in normalized units used in [2].*

#### Applications: reactions

Using our model for analyte distribution we are able to arrive a significantly more realisitc description of reactions between few co-focused analytes in the transition zone, allowing their prediction and optimization. The reaction rate in second order chemical reactions, is typically governed by the on-rate coefficient and by the product concentrations of two reactants. In our case those concentrations are non-uniform and we are naturally lead towards the effective on-rate coefficient (Eq.(50) in [2]) which takes into account the mutual distributions of the reacting analytes in the transition zone.

Our analysis in Figure 2a shows three different effective on-rates for three different cases, where in each case the mobility of one of the reactants is fixed while the mobility of the other analyte varies (between and ). Figure 2b shows that for a symmetrical concentration distribution of one of the analytes, the maximal effective on-rate is achieved when the second analyte is symmetrical as well, and bascially coincides with the concentration distribution of the first one. A somewhat counterintuitive result is that, optimal reaction rates are not always achieved when the spatial distribution of reacting species are identical. In Figure 3c,d we show analytical model results showing

Figure 3: The effect of reactants mobility on reaction rates. (a) presents the value of the form factor , defined by Eq.(51) in [2], as a function of for the three different values (corresponding to , respectively, through Eq.(52) in [2]). All quantities presented here in dimensionless units, with S=1 for convenience. The maxima points indicate the associated optimal values of (and the associated ), corresponding to maximum reaction rates. Each maxima value in (a) leads to the accompanying concentration profiles I, II, and III given in (c), (b), and (d), respectively. For a value of corresponding to a symmetric profile (shown in (b)), maximum hybridization is obtained when the two reactants’ distributions overlap exactly. However, for non-symmetric profiles of B species (as shown in (c) and (d)), optimal rates are achieved with non-overlapping A species distributions. The values of the leading-and the counter-ion are 3.5 and 0.25, respectively.

### References

[1] D. A. Saville and O. A. Palusinski, “Theory of electrophoretic separations. Part I: Formulation of a mathematical model,” AIChE Journal 32, 207-214 (1986).

[2] Rubin S., Schwartz O., and Bercovici M., Sample distribution in peak mode isotachophoresis. *Physics of Fluids 26, 012001 (2014)*.