Theoretical Description

Calculation of the ion concentrations and basic characteristics of the leading and trailing ion zones are

electroneutrality:

Figure 2 The differential effects of stacking limits on the migration of DNA sequencing products on a 6% polyacrylamide gel (containing 5% bis-acrylamide cross-linker) with a 50 mmoL L~1, pH 9.0 formate leading ion, glycine trailing ion. The sequence between the two fronts, AAATTGTT, corresponds to fragment lengths of 107-114 bases; the sequence behind the front, CTGG, corresponds to fragment lengths of 167-170 bases. Fragment lengths between 115 and 166 bases are concentrated in the second front while fragment lengths <107 bases are concentrated in the first front.

electroneutrality:

allows definition of the basic relationship between the concentration of ions in the different zones and their mobilities:

This relationship can be defined for every boundary in a discontinuous electrophoresis system. The relationship is valid for strong electrolytes and weak electrolytes. For weak electrolytes, the concentration value represents the total of the ionized and unionized forms. The fraction of the trailing species that is ionized, X(n), can be determined after calculation of the amount of counterion that crosses the boundary into the trailing zone. When using a weak electrolyte as the counterion, a final expression for the counterion concentration in the trailing zone is:

Figure 2 The differential effects of stacking limits on the migration of DNA sequencing products on a 6% polyacrylamide gel (containing 5% bis-acrylamide cross-linker) with a 50 mmoL L~1, pH 9.0 formate leading ion, glycine trailing ion. The sequence between the two fronts, AAATTGTT, corresponds to fragment lengths of 107-114 bases; the sequence behind the front, CTGG, corresponds to fragment lengths of 167-170 bases. Fragment lengths between 115 and 166 bases are concentrated in the second front while fragment lengths <107 bases are concentrated in the first front.

easily accomplished through use of a few simple equations. The starting point is the Kohlrausch regulating function (R) that describes the moving boundary condition:

where the counterion concentrations are the total concentration (both ionized and un-ionized forms). The fraction ionized, X(n), can then be calculated from the ion equilibrium constants of the trailing ion and counterion. The net mobility accounts for the actual transport of the trailing acid species and is defined as:

The net mobility should be tuned closely to the sample's mobility. Sample mobilities faster than the trailing ion net mobility will be retained in the ion front. Details regarding these equations and more rigorous definitions can be found in numerous refer ences.

where C is the concentration of the ion in zone n and m is the mobility of the ion. These values are signed according to the charge of the ion. For the set-up described in Figure 1, the regulating functions characterizing the two zones would be equal under moving boundary conditions and:

where the concentration subscripts denote the zone. This relationship, along with the condition for

A few characteristics of discontinuous buffer systems are apparent from examining the above equations. First, the equilibrium concentration of the trailing ion is independent of its initial concentration. It is determined by the free mobilities of the ions in the system and the concentration of the leading ion. Therefore, only the leading ion conditions can be chosen freely. Second, since the trailing ion has a slower mobility than that of the leading ion, the concentration of the trailing ion must be lower than that of the leading ion. This bears directly on the conductance and potential gradient of the trailing c#

zone. The zone conductance, k, can be calculated from the ion concentrations, free mobilities and Faraday's constant (F). It can also be related to the field strength (volts (V) per unit length (l)) through Ohm's law, as shown (where i is the current and A the cross-sectional area of the gel):

Since the ion concentrations and ion mobilities in the trailing zone are lower than that in the preceding zone, the potential gradient in a trailing zone will be higher than the leading zone. This is what allows the trailing ions to migrate at the same rate as the leading ions, despite their lower physical mobility. Tuning the ionic strength is critical for optimizing separations. It affects the size of the migrating zone, the speed of the separation and the joule heating. The joule heating further influences resolution. Therefore, a high potential gradient for a given current, or heat output, will be preferable.

The above equations can be implemented on simple spread sheet software to determine the physical characteristics of discontinuous buffer systems. A few example buffer systems are shown in Table 1. These buffer systems have been designed to vary the trailing ion type and the trailing ion net mobility while keeping the ion speed constant. A Tris-formate buffer, at a formate concentration of 50 mmoL L_1, has been selected as a common leading ion. Tris is used as a common counterion. The ion speed and voltage gradient would be determined by the applied current. These buffer systems exemplify some of the characteristics of discontinuous buffer systems. As can be seen, the trailing ion concentration is lower than the leading ion concentration and a wide range of net mobilities can be achieved when using a common leading ion. Also notice that the net mobility, or actual transport of the trailing ion, is lower than the free mobility when using weak electrolytes. Such buffer systems can be used to assess the stacking limits, or mobility, of the analytes to be separated. Sample ions that migrate in the ion front will have a mobility intermediate between the leading and trailing ions. Those sample ions that have a mobility slower than the net mobility of the trailing ion will migrate more slowly than the ion front and will electrophorese within the trailing zone.

Other examples of calculated buffer systems are shown in Table 2. These buffer systems vary the trailing ion type while keeping the ionic strength of the trailing phase constant. This is accomplished by varying the leading ion concentration to allow for a predetermined ion concentration in the trailing zone. All the buffer systems in Table 2 use a formate leading ion of varying concentration and have a common trailing ion concentration of 30 mmol L_1. Even with a common trailing ion concentration, a 50% change in the conductance can be obtained. This allows for increased voltages to be applied without increasing the current. As with the buffer systems described in Table 1, a range of trailing ion net mobilities is obtained. Other buffer systems, that define different ranges of net mobilities, different pH values or different conductivities, could be similarly calculated to address particular separation problems or analytical characterizations.

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