## Correlating Migration Velocities and Migration Distances

The migration velocities may be plotted against the corresponding migration distances at the end of each time interval to correlate migration velocities and migration distances (Figure 13). The function by which v and D are interrelated is best described by the following exponential equation:

where s, Dmax and § are constants, D (mm) is the independent variable and v (mms"1) the dependent variable. Dmax represents the maximum migration distance which a protein can cover, i.e. the migration distance at which the migration velocity becomes zero. If this point is reached then Dmax = D and:

Eqn [17] can be used to relate the apparent migration velocity (v) of a protein to the PA concentration (T (%)) that corresponds to the migration distance travelled during a given period of electrophoresis. When using a linear gel gradient, the PA concentration and the gel length are interrelated by eqn [19]:

whilst Tmax (%), the stacking gel concentration, is related to the maximum distance Dmax (mm) by eqn [20]:

Substituting eqns [19] and [20] into eqn [17] yields the formula:

v (mm s"1) = s[((Tmax - P) x a"1) — ((T — p) x a"1^

which can be arranged to:

and:

This derivation shows that, indeed, the apparent migration velocity of a protein (v) is related by the same function to the distance (D) as to the PA concentration (T) it has reached in a linear pore gradient, although the constants (s and Dmax, respectively, h and Tmax) are different. The exponent § in both equations, however, is the same.

Eqn [23] predicts that zero protein mobility (v = 0) results if the apparent gel concentration (T (%)) is equal to the stacking gel concentration (Tmax (%)), i.e. if T = Tmax. The apparent free electrophoretic mobility of a protein unhindered by the PA matrix (| (mm s"1)), can be calculated by simply extrapolating its apparent mobility to zero T (%):

thus:

This expression may be used to divide eqn [23] to yield eqns [26] and [27]:

vx M"1 = (h x (Tmax — T)S) x (h x Tmax) " 1 [26]

which can be rewritten as:

The value of the quotient (Tmax — T) x Tmax ranges from one (T = 0) to zero (T = Tmax) and thus the value of v extends from the apparent free electro-phoretic mobility (M) to zero.

This means that, in a linear PA gradient, the apparent migration velocity (v) of a protein (migrating under a constant electrical field strength) is equal to its apparent free mobility (M) times a retardation factor ([1 — (Tx Tmax)"1]^ which depends on the PA concentration (T) that the protein has just reached and its exclusion limit (Tmax). This factor always takes

Figure 13 (A) Estimation of the migration velocity of a protein (OVA, ovalbumin) in a linear PA gradient gel. Tn, migration distance at a longer time of electrophoresis (fn); Tm, migration distance at a shorter time of electrophoresis (tm). (B) Plot of the resulting migration velocities (v (mm s~1) versus the corresponding gel concentrations (T(%)) at the end of each time interval.

Figure 13 (A) Estimation of the migration velocity of a protein (OVA, ovalbumin) in a linear PA gradient gel. Tn, migration distance at a longer time of electrophoresis (fn); Tm, migration distance at a shorter time of electrophoresis (tm). (B) Plot of the resulting migration velocities (v (mm s~1) versus the corresponding gel concentrations (T(%)) at the end of each time interval.

values between zero and one and increases exponentially with increasing gel concentrations.

In order to solve eqn [23] (v (mms_1) = hx (Tmax — T)s), the following sequence of calculations is recommended:

1. determination of the maximum migration distance of the protein under investigation from a plot of ln (ln D) vs. t_1/2 (eqn [5])

2. computation of the maximum gel concentration (Tmax) by use of eqn [20] (DmaX = a_1(Tmax — P));

3. calculation of the gel concentration equivalent to the migration distances with eqn [19] (D = a_1(T —^)), (the values of the constants a and P may be obtained from a gel scan at 405 nm if p-nitrophenol has been mixed into the more concentrated of the two solutions used to prepare the gradient gel);

5. finally the constants h and § in eqn [23] are calculated by plotting ln v vs. ln (Tmax — T) and performing a linear regression analysis with these data, i.e. taking the logarithmized version of eqn [23]:

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