Practical Aspects of Measuring Intracellular Calcium Signals with Fluorescent Indicators - Methods in Cell Biology - page 146

¼

H þ ½

HA 3 A 4

ðA1:3Þ

K 4

2
K 3 K 4 A 4

¼

H þ

¼

H þ

½ ½

H 2 A 2 HA 3 ðA1:4Þ

K 3

3
K 2 K 3 K 4 A 4

H þ

¼

H þ

½ ½

H 3 A H 2 A 2 ½ ¼ ðA1:5Þ

K 2

4

H þ H þ ½ ½

H 3 A A 4 ½ H 4 A ¼ ½ ¼ ðA1:6Þ

K 1 K 1 K 2 K 3 K 4

By using these four relations, the expression for the total concentration can be
written in terms of [A 4 ]:

4 3 2

H þ

þ

H þ

þ

H þ

þ

H þ

þ

½ ½ ½ ½

A 4 A 4 A 4 A 4 A 4 C 0 ¼

K 1 K 2 K 3 K 4 K 2 K 3 K 4 K 3 K 4 K 4

ðA1:7Þ

Dividing through by [A 4 ] leads to

4
K 1 K 2 K 3 K 4 þ

3
K 2 K 3 K 4 þ

2
K 3 K 4 þ

H þ H þ H þ H ½

K 4 þ

C 0
A 4

1
a 4 ¼

½ ½ ½

¼ ðA1:8Þ

1

Writing the right side of Eq. (A1.8) as a fraction with a common denominator and
then inverting the fraction gives the desired final expression

K 1 K 2 K 3 K 4

a 4 ¼

K 1 K 2 K 3 þ K 1 K 2 K 3 K 4 :

ðA1:9Þ

4 3 2

H þ H þ H þ H þ

½ þ ½ K 1 þ ½ K 1 K 2 þ ½

An important feature of Eq. (A1.9) to notice is that each term in the expression
actually represents the contribution of a particular protonated form, thus:

A 4 K 1 K 2 K 3 K 4 ,

H þ HA 3

½ K 1 K 2 K 3 ,

2

H þ H 2 A 2

½ K 1 K 2 ,

3

H þ H 3 A

½ K 1 ,

4

H þ H 4 A

This insight makes it easy to write the fraction of the polybasic acid that is in a
particular form: the term representing the particular protonated form appears in
the numerator, while the denominator is simply the sum of all the possible terms.
For example, the fraction existing as HA 3 is

½ ,

H þ

½ K 1 K 2 K 3

a 3 ¼

K 1 K 2 K 3 þ K 1 K 2 K 3 K 4 ;

ðA1:10Þ

4 3 2

H þ H þ H þ H þ ½ þ ½ K 1 þ ½ K 1 K 2 þ ½

and the fraction existing in the doubly deprotonated H 2 A 2 form is