Author: William Hancock
For electrostatic interactions of proteins, the ionic strength is a major determinant, and so we want to be able to compare buffers of different ionic strengths. The problem is that the ionic strength of a buffer depends on its pH. Adding complexity, if the buffer is divalent like PIPES, then calculating ionic strength is even more complex. It is important to understand the concept of ionic strength, both for designing experiments and making buffers.
Ionic strength is calculated as
I = ½ * Sum(CiZi2)
where Ci is the concentration of ion i and Zi is its charge. Hence for NaCl, which is fully dissociated in water and each has charge of +/-1, the ionic strength equals the concentration.
This gets more tricky when you use buffers because the essence of buffering is the compound being in equilibrium between an acid and basic form, with changes in pH being “buffered” because the acid species neutralizes OH- to become base and the base species neutralizes H+ to become the acid form. For instances ACES is monovalent:
ACES <–>ACES– + H+
So we buy ACES or K-ACES. If you make up buffer with all acid form (K-ACES), the proton will dissociate and it will start very acidic, whereas if you start with all basic form (ACES–), it will grab a proton from water and start very basic. At equilibrium, some will be charged (acidic) and some will be uncharged (basic). The fraction of ACES in its charged form can be calculated from the Henderson-Hasselbach equation, which is as follows. For a general acid HA that dissociates into H+ plus the base A–,
HA<–>H+ + A–
where pKa is the acid constant of the buffer, defined as the pH where half is in the acid form and half is in the base form. This equation can be transformed into the more useful form, which gives the fraction of the species that is in the acid form:
To use this equation, plug in the pH and the pKa for the buffer species. pKa at 25 deg C are: 6.76 for PIPES, 6.78 for ACES, 7.2 for Phosphate, and 7.48 for HEPES. The range over which buffers work is 1-1.5 pH units, centered at their pKa; beyond that either the acid or base species becomes limiting and can no longer absorb changes to maintain pH. This is one of the main reasons so many different buffer species are used (see Sigma website for tables and pKa constants and recipes).
When using PIPES, things are a bit more complex, because PIPES is either divalent or monovalent:
PIPES2-<–>PIPES– + H+
Said another way, buffer is mix of K-PIPES and K2-PIPES. If you start with all K-PIPES, then you will lose the proton to make PIPES2- and the buffer will start very acidic; it can be pH’ed with KOH to get to the proper pH. Note that if you start with all K2-PIPES and then add HCl to bring pH down, then you’re getting excess KCl in the buffer that you need to account for. The ideal way is to mix dry K-PIPES with K2-PIPES to get close to the optimal pH on your first try, and then do minor adjustments around that with KOH or HCl (there are tables for this for some buffers on Sigma website). But starting with acid form and pH’ing with KCl also works; only problem is lack of solubility at low pH.
Because of the divalent nature of PIPES and the fact that when it is properly pH’ed, part is in divalent form and part is in movalent form, it gets a bit complicated to calculate the ionic strength. As an example, let’s calculate ionic strength at the pKa of the buffer where half is divalent and half monovalent. To calculate ionic strength, you need to find the fraction in 1– form (half at the pKa), and you have single K+ here, so for BRB80 that will be 40mM. Then you take other part, which is divalent so 2K+ , and to calculate ionic strength here you use:
I = ½ ([PIPES–]*(-1)2 + [K+]*(+1)2 + [PIPES2-]*(-2)2 + 2*[K+]*(+1)2)
I = ½ (40mM*(-1)2 + 40mM*(1)2 + 40mM*(-2)2 + 2*40mM*(1)2)
Hence, at the pKa, the ionic strength of PIPES is 160mM.
In reality, the pKa of PIPES is 6.8 and we use it at pH 6.9. From handy calculator site http://www.changbioscience.com/calculator/HendersonHasselbach.html
this comes out to 45m M base (-2 charge) and 35 mM acid (-1 charge). So plugging in like above:
I = ½ (35mM*1 + 35mM*1 + 45mM*(-2)2 + 2*45mM*1) = 170mM.
Barry Grant in his electrostatics PLoS paper used BRB50 for experiments and called this 150mM I.S. for his calculations. This comes out to 28mM -2 charge and 22mM -1 charge, so I.S. is actually lower.
I = ½ (22mM*1 + 22mM*1 + 28mM*(-2)2 + 2*28mM*1) = 106mM.
Concluding thoughts
It is important to be on top of things when making buffers because it seems easy but can be complex. Check your recipes, mix dry acid and base if you can, otherwise be attentive to how you pH it because you can add excess ionic strength without realizing it. For instance, if you start with 80 mM K2-PIPES, roughly one molar equivalent (40 mM) will take up H+, making the solution very basic, and you will need to pH with about 40 mM of HCl to balance. This adds an extra 40 mM KCl to your buffer compared to starting with K-PIPES and pH’ing with KOH.
Addendum: Calculating [PIPES] at different pH to maintain Ionic Strength
We have been making BRB80 at different pH values and we realized that this changes the ionic strength. This is because the fraction of mono- and divalent PIPES changes, and this change in the divalent PIPES has an outsize effect because of the factor of 4 there. Hence, we want to figure out how to change the [PIPES] in our buffer to maintain a constant ionic strength across different pH. Here is a derivation for how to calculate [PIPES] at different pH. Recall our ionic strength calculation:
Remember, to make PIPES, we start with all divalent H2-PIPES and we pH with KOH. This involves one full equivalent for the first proton and part of the second equivalent to balance a fraction of the second proton. It is this second proton that is doing the buffering. The acid-base equilibrium is:
PIPES– <–> H+ + PIPES2-
(Acid) (Base)
From the Henderson-Hasselbalch, the fraction that are in the acid form is:
It follows that the fraction in the base form is 1 minus the fraction in the acid form:
Another fact that comes out of this is that the [K+] is equal to one PIPES equivalent (to get rid of the first free proton) plus the concentration of Divalent PIPES (which gets rid of some of the second proton)
Thus, the equation for the ionic strength sums up the monovalent PIPES, the divalent PIPES, and the K+ concentrations:
I = 0.5*[PIPES]*(1*Frac_Monovalent + 4*Frac_Divalent + 1*(1 + Frac_Divalent))
The monovalent is the acid form and the divalent is the base form. From above the fraction in the acid form is 1/(1 + 10pH-pKa). Therefore:
I = 0.5*[PIPES]*( 1/(1 + 10pH-pKa) + 4*(1 – 1/(1 + 10pH-pKa)) + (1 + (1-1/(1 + 10pH-pKa)))))
I = 0.5*[PIPES]*( 1/(1 + 10pH-pKa) + 4*1 – 4/(1 + 10pH-pKa)) + 1 + 1 – 1/(1 + 10pH-pKa)))
I = 0.5*[PIPES]*( 6 – 4/(1 + 10pH-pKa))
Solving for [PIPES] we get
So, to figure out how much PIPES to include, we plug in the desired ionic strength, I, the desired pH, and the pKa of 6.76 for PIPES.
