# Protein pKa Calculations

(Original document: http://enzyme.ucd.ie)

Table of Contents

1. Index

2. What can you do with pKa calculations?
3. Introduction
• Titratable groups and their importance in proteins
• Reaction mechanisms
• Protein stability
• The effect of the protein on an amino acid pKa value

4. Theory

5. Pitfalls and limitations

## 2. What you can do with pKa calculations

In short: pKa calculations tell you if the pKa value of a protein titratable group differs significantly from the pKa value that this group normally has in solution.

To know why this is important, it is useful to know a bit about proteins and acid-base reactions.

### 2.1 Proteins and titratable groups

One of the more frequent chemical reactions that occur in a protein-water solution is the uptake and release of protons by amino acids. The parts of amino acids in a protein that can absorb or release protons will be referred to as protein titratable groups. Thus an Aspartic acid side chain contains a single titratable group (namely the carboxy group: COO-), and an N-terminal Lysine contains two titratable groups (the N-terminal amino group, and the amino group in the side chain).
We will divide protein titratable groups into two categories: acids and bases. We define acids as Asp, Glu, Cys, Ser, Tyr, Thr and the C-terminal. Bases are the N-terminal, His, Lys and Arg.
The basic protein titratable groups are positively charged in their protonated state, and the acidic protein titratable groups are neutral in their protonated state.

Please note that this definition of acids and bases does not comply with the correct chemical definition of acids and bases, since the correct chemical definition as a base or acid is always assciated with a specific reaction. In the example below HA is an acid in the reaction with water, but in the reaction with BH, it acts as a base.

HA + H2O -> A- + H3O+

HA + BH -> H2A+ + B-

### 2.2 The pKa values in proteins

A protonation or deprotonation event is a chemical reaction. And just like any other chemical reaction, the distribution of molecules between the two states of the reaction reactions can be described by an equilibrium constant.

For acids (HA), the reaction and the associated equilibrium constant is:

HA + H2O -> H3O+ + A-

The pKa values is simply -log(Ka). Similar K values exist for bases (see section 4.1).

Therefore, if we know the pKa value for a protein titratable group, then we can predict the charge present on this group if the protein is in a solution with a given pH value, since pH=-log([H3O+]).

The pKa values of protein titratable groups in water have been estimated by comparing with the pKa values for model compounds in water (Table 2.1). One can therefore get a quite good estimate of the protonation state of a protein simply by assuming that the pKa values in the folded protein are the same as the estimated pKa values for the protein titratable groups.

In some special situations it is necessary, however, to obtain a more detailed picture of the pKa values of a protein. This is often the case when studying enzymatic mechanism and protein stability, and in these cases pKa calculation techniques provide a way of calculating the effect of the protein environment on the pKa values of the titratable groups in the protein.

Titratable group Estimated pKa value
N-term
8.0
C-term
3.0
Asp
4.0
Glu
4.3
Cys
8.7
Tyr
9.8
Ser
14.2
Thr
15.0
Arg
13
Lys
10.5
His (0 -> + transition)
6.4
Table 2.1. Estimates of acid dissociation constants for amino acid residues in a polypeptide.
From: Jack Kyte, Structure in Protein Chemistry, Garland Publishing, Inc. 1995

### 2.3 How accuarate are pKa calculations

In recent years pKa calculations have improved significantly, and the best pKa calculation algorihtms reach and rmsd between predicted and experimentally measured pKa values of 0.50 - 0.75. It is important to stress that this correlation is for a quite limited test set of approximately 120 pKa values in a limited set of proteins, and that most of the experimentally pKa values differ very little from the pKa values listed in Table 2.1.

pKa calculation algorithms are thus relatively good at predicting the pKa values that aren't that different from their model pKa value. Normally it is not that interesting to calculate essentially normal pKa values, and most applications of pKa calculation algorithms focus on calculating pKa values for "special" residues in active sites that have highly shifted pKa values. Only a handful of highly shifted pKa values have been measured experimentally, and it is therefore not straight forward to assess the accuracy of pKa calculation routines in calculating these, but as a rule of thumb one can assume that calculated pKa values are accurate within 1.25 pKa units.
It is important, however, to critically examine the structure around residues that are predicted to have highly shifted pKa values. Bumps (too short inter-atomic distances) or other structural artifacts (crystal-induced salt bridges, the presence of ion etc.) can cause pKa calculation programs to predict large shifts in pKa values that aren't observed in real life.

You need:

### 2.5 Where do I get help?

pKa calculation theory: First look in this document, then read the following articles:

Running a pKa calculation: Read section 6, and if it still doesn't work, then mail Jens.Nielsen@ucd.ie or come see me in the Conway Building Room F050.

When to trust a pKa calculation:

• Nielsen JE, McCammon JA
On the evaluation and optimization of protein X-ray structures for pKa calculations
Protein Sci 2003 Feb;12(2):313-26

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## 4. Theory

This section describes the general equations for acid-base reactions and the basic theory behind pKa calculations.

### 4.1 The basic equations for acid-base reactions

The dissociation of a proton from an acid generally takes the form

HA + H2O -> H3O+ + A- (Eq. 4.1)
(Eq. 4.2)

(Eq. 4.3)

For acid-base reactions it is customary to report the pKa values, which is simply -log(Ka). The Henderson-Hasselbalch equation is a rearrangement of equation 4.3:

(Eq. 4.4)

(Eq. 4.5)

From Eq. 4.4 it is seen that the pKa value of an acid is the pH value where the concentrations of the protonated and deprotonated forms of the acid are present at the exact same concentrations. Furthermore by rearranging Eq. 4.4:

(Eq. 4.6)

and plotting fHA as a function of pH we get the well-known sigmoid titration curve (Fig. 4.1)

Figure 4.1. The titration of an acid with a pKa value of 5.0 calculated using Eq. 4.6

For the association of a proton with a base, a similar set of equations can be produced. The major difference being that the base reaction is:

B + H3O+ -> BH+ + H2O (Eq. 4.7)

The corresponding equilibrium constant thus becomes (water excluded for simplicity)

(Eq. 4.8)

It is seen that the major difference between equations 4.8 and 4.3 is that the concentration of [H+] is in the denominator in Eq. 4.8 and in the numerator in Eq. 4.3. This means that equations 4.4 and 4.6 need to be rearranged for bases to take this into account. This is left to the enthusiastic reader.

### 4.2 Splitting up the effect of the protein environment

Every residue in a protein is, in principle, a titratable group. In the following we will limit ourselves to looking only at the titration of those groups that have pKa values in water within the range 0-14. We are thus left with the titratable groups in the side chains of Asp, Glu, Tyr, Cys, His (only the transition from His+ to His0), Lys and Arg as well as the two terminal groups. For reasons of simplicity we will refer to Asp, Glu, Cys, Tyr and the C-terminus as acids, and to His, Arg, Lys and the N-terminus as bases. Please note that the correct definition of acids and bases always is connected with a specific reaction. The definitions we use here thus represent nothing more than a convenient way of dividing the titratable residues in proteins into two groups.

The pKa value of a titratable group is a measure of the free energy difference between the neutral and charged state of the group. It is therefore possible to calculate the pKa value of a group if we can calculate the free energy difference between the charged and neutral state of that group in the protein. The calculation of this energy difference is performed in three steps:

### 4.3 Calculating the intrinsic pKa value

Figure 4.2 The thermodynamic cycle for the transfer of a titratable group from water to a protein environment. pKa(model) is the model pKa value of the group in water. pKa(protein) is the pKa value of the group in the protein, disregarding the effects from other titratable groups, and DGcharged and DGneutral are the energies associated with transferring the charged and neutral form from water into the protein.

To calculate the true pKa of the residue in the protein, we need only to account for the effect of the interaction with the rest of the titratable groups in the protein.

### 4.4 The interaction with other titratable groups

The desolvation energies and the background interaction energies can be regarded as being largely pH-independent. The interaction energy between titratable groups is obviously not pH-independent, and it is therefore not possible just to add the interaction energies with all the other titratable groups to the intrinsic pKa in order to get the true pKa value of the residue. We therefore have to use a calculation protocol that takes the pH-dependence of the interactions between titratable groups into account. This can be done if we calculate the energy for each of the possible protonation states of the protein, and use these energies to evaluate the partition function for these states at a range of pH-values.

State
Group 1
Group 2
Group 3
Energy
1
+
+
+
dGpH(1) + dGpH(2) + dGpH(3) + (1=2) + (1=3) + (2=3)
2
+
+
0
dGpH(1) + dGpH(2) + (1=2)
3
+
0
+
dGpH(1) + dGpH(3) + (1=3)
4
+
0
0
dGpH(1)
5
0
+
+
dGpH(2) + dGpH(3) + (2=3)
6
0
+
0
dGpH(2)
7
0
0
+
dGpH(3)
8
0
0
0
0

Table 4.1 Possible protonation states for a hypothetical protein consisting of three titratable group. +: charged, 0: neutral. Energy is relative to state 8. (X=Y) indicates the interaction energy between the charged forms of groups X and Y. dGpH (X) is the free energy difference between the charged and neutral forms of group X at a fixed pH value (see text for explanation).

Let us consider a protein with three titratable groups. Each of these groups can exist in two states: charged and neutral. The protein can thus occupy 23 different protonation states. These are summarised in Table 4.1.At a given pH we want to determine the free energy of all the states in Table 4.1 relative to the free energy of state 8, which we have defined to be zero. The free energy of each of the other states consists of two terms A and B:

A)    For each residue: the energy difference between the charged and neutral form of the residue disregarding the interactions between the titratable groups.

B)     The interactions between the titratable groups.

#### 4.4.1 Term A

Term A can be calculated from the intrinsic pKa for each residue by rearranging Eq. 4.10:

(Eq. 4.15)

and remembering that

(Eq. 4.16)

This gives an expression for the free energy difference between the charged and neutral state of a titratable group at a fixed pH value:

(Eq. 4.17)

#### 4.4.2 Term B

Term B is the interaction energies between the titratable groups in this particular protonation state. For state five, for example, term B should hold the following three interaction energies ([X : Y] denotes the interaction energy between X and Y):

E1: [G1:0 : G2:+] - [G1:0 : G2:0]

E2: [G1:0 : G3:+] - [G1:0 : G3:0]

E3: [G2:+ : G3:+] - [G2:0 : G3:0]

(G1 = Group 1, G2 = Group 2, G3 = Group 3, :+ = charged, :0 = neutral)

The energies E1 and E2 are already contained in the intrinsic pKa, because it is calculated by determining the energy of charging a single group in a form of the protein where all other titratable groups are in their neutral state (see section 4.2.3 and Fig. 6.2).

Thus only E3 has to be added to term A to obtain the free energy for state five. The intrinsic pKa, however, does also contain the energies E4 and E5 (in the same way that the intrinsic pKa contains E1 and E2).

E4: [G2 (+) : G3 (0)] - [G2 (0) : G3 (0)]

E5: [G2 (0) : G3 (+)] - [G2 (0) : G3 (0)]

We have to correct for this in the energy that we add to the intrinsic pKa [DGpH(2) and DGpH(3) in Table 4.1] for the interaction between the charged forms of groups two and three. A simple evaluation shows that:

E3 - (E4 + E5) = [G2(+) : G3 (+)] - [G2 (+) : G3 (0)] - [G2 (0) : G3 (+)] + [G2 (0) : G3 (0)]

and this is therefore the energy which is listed as (2<<3) in Table 4.1.

### 4.5 Calculating titration curves

We now know the energy of every possible protonation state of a protein at a given pH value, and the next step is the conversion of these energies into fractional charges at each pH value for each residue in order to get the titration curves.

A straight-forward way to find the occupancy of the different states in Table 4.1 is to evaluate the Boltzmann sum for each state.

(Eq. 4.18)

Here pi is the fraction of molecules in state i. Ei is the energy of state i, and the sum in the denominator is over all possible states of the system. k is Boltzmann's constant and T is the temperature in Kelvin.

The fractional charge of a particular group is simply the sum of the pi's for all the states where the group is charged. Thus for group 1 in Table 4.1, for example, the charge is the sum of p1, p2, p3 and p4.

It is clear from Table 4.1 that the number of states equals 2N, where N is the number of titratable groups. For values of N significantly larger than 30, it is therefore no longer possible to evaluate (Eq. 4.18). For large systems it is thus customary to use a Monte Carlo protocol [Beroza et al., 1991] to obtain pi.

From the calculated titration curves the pKa value for each group is determined as the pH where the group is half-protonated. This gives an accurate result only if the titration curve follows a Henderson-Hasselbalch shape. This is the case for most groups, but especially in active sites it is quite common to find groups that have very irregular titration curves. In these cases manual inspection of the titration curves is necessary in order to obtain meaningful results.

### 4.6 Performance of pKa calculation methods

Several pKa calculation packages are presently available. Most of these, however, have serious trouble to reach a better agreement with experimentally determined pKa values than the so-called null model. The null model assumes that the pKa values of protein side chains are not shifted at all compared to their value in water.

This poor performance of pKa calculations is not due to an incorrect theory, though, but rather to an incorrect description of the protein in the calculations. A fundamental problem with pKa calculations is that crystal structures are used as source of coordinates for the protein. The crystal symmetry induces structural changes in the protein, and thereby causes some pKa values to be shifted compared to their value in solution. It is therefore not surprising that the pKa values calculated from a crystal structure will differ from the pKa values measured in solution by NMR.

The description of the protein used in pKa calculations is, however, also often to simple. Protons are, for example, often omitted, and methods that include protons do often not model the deprotonation of a titratable group explicitly. It is our opinion that pKa calculations can improve greatly by including a more detailed description of the protein and its dynamics.

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## 5. Pitfalls and limitations

### 5.1 What pKa calculations can do and can't do

What pKa calculations can do:

What pKa calculation can't do:
So as with all other tools in computational biology pKa calculations can help you with where to start looking. Do not put too much faith in them, and always use common sense when interpreting the results. If something seems really unlikely then it probably is.

Common sense will almost always give you the best description of the situation.

### 5.5 How to prepare the 3D structure

The pKa calculations will in principle be most accurate if the structure of your protein is exactly the structure it will have at the conditions where you want to know the pKa value.
In practice this is almost never possible since X-ray structures are slightly perturbed by the crystal environment, and because of this there is a good chance that the calculated pKa values will be less accurate for residues that are involved in crystal contacts.

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