An idea to explore: Visualization of ionization of amino acids using Mathematica.

Ionization of amino acids (AA) is very important concept in biochemistry. We integrate the mathematical concept of probability with biochemically relevant process of AA ionization. We visualize the ionization process with Mathematica software discussing intramolecular interactions between weakly acidic/basic functional groups and charge–pH variation of amino acids in water solution. The visualizations rely on the notion of probability of ionization of functional groups and demonstrate how the extent of ionization and charge varies with pH of the solution. The examples described include amino acids and weak diprotic acids and bases. The aim is to help students better appreciate the importance and consequences of AA ionization and correct some misconceptions.

Keywords: positive and negative cooperativity; teaching of ionization of amino acids; variation of amino acid charge with pH; visualization of intramolecular interactions pertaining to amino acid ionization

Understanding noncovalent interactions (both intermolecular and intramolecular) is important in learning biochemistry because these interactions underpin protein structure and function. Several approaches have been described in the education literature1,2 to study these interactions in a qualitative way. The reported approaches used specialized software which performs structure visualization (PyMol) or which calculates amino acids (AA) surfaces (NACCESS). We describe the use of general‐purpose software; computer algebra software Mathematica. Mathematica is used to visualize ionic (electrostatic) intramolecular interactions in AA quantitatively. Ionic (electrostatic) interactions, that is, noncovalent interactions between permanently charged species are important for, for example, understanding stabilization of protein structures. Ionic interactions can be visualized by calculating and plotting partial atomic charges using various programs which present graphically the structures of biomolecules. However, these visualizations do not provide dynamic portrait of events, that is, how the charges interact and influence each other as pH of solution varies. The results also depend on particular quantum chemical method (model chemistry) used. Weakly acidic or basic functional groups are ubiquitous in biomolecules, especially in AA and related peptides and proteins. Therefore, the discussion of percentage of ionization (protonation or deprotonation) of an acidic or basic functional group is found in many articles.3–6 However, the discussion often obscures the fact that AA ionization is a cooperative phenomenon (ionization of one group influences the ionization of another) and that experimental pKa values available in the literature do not in fact correspond to any particular acidic/basic functional group as Darvey had pointed out.3,4

Ionic interactions between acidic and basic functional groups can occur between different molecules or within the same molecule. They are important for understanding the behavior of proteins and enzymes.5 AA molecules incorporate several (weakly) acidic or basic functional groups and the question arises as how different groups influence each other's ionization. N.B. The word "ionization" used here implies dissociation (deprotonation) in case of a weakly acidic functional group or protonation in case of a weakly basic functional group. We shall limit ourselves to two types of weakly acidic/basic functional groups: carboxylic acid and amino group. However the algorithms devised are general and can be extended to describe ionic interactions between other functional groups of importance in biochemistry, for example, phosphates.

Mathematica software has been used previously to present pedagogically accessible models of metabolic cycles.6 The novelty of our approach is in using probabilistic concept to describe AA ionization7 and using exact general formulas which correlate concentrations of ionic species with pH of solution.8,9 We did not rely on approximation which considers ionizations of functional groups to take place independently. The fractional concentrations of ionized and nonionized species in water solution can be equated to probabilities of ionization of functional groups present in those species.7–9 We have provided a detailed guide and examples in Data S1 on how to implement our approach which will allow instructors to develop their own preferred teaching framework.

We start by giving expressions for probabilities of dissociation of monoprotic weak acid Pr(A−) and probability of protonation of monoprotic weak base Pr(BH+). These probabilities are given as functions of pH (1)–(2) and correspond Pr–pH plots are shown in Figure 1. The sample plots shown are for acetic acid (pKa = 4.76) and ethyl amine (pKa = 10.64).

1 $$\mathrm{Pr}\left({\mathrm{A}}^{-}\right)={\mathrm{Pr}}_{\text{dissoc}}=\frac{1}{1+{10}^{\left(4.76-\mathrm{pH}\right)}},$$

2 $$\mathrm{Pr}\left(\mathrm{B}{\mathrm{H}}^{+}\right)={\mathrm{Pr}}_{\text{proton}}=\frac{1}{1+{10}^{\left(\mathrm{pH}-10.64\right)}}.$$

Equations (1)–(2) were derived from Ault's8,9 fractional concentrations by converting [H+] and Ka values to pH and pKa using well‐known relationships: [H+] = 10−pHand Ka = 10−pKa. A detailed example of how to do the conversion is given for tutorial purposes in Data S1. The calculations of probabilities and plotting their dependence on pH were achieved with Mathematica throughout this work.

The interpretation of Figure 1 is straightforward. At high pH, shortage of H+ ions in water solution precludes protonation of basic group making Pr(BH+) small. Analogously, ready availability of H+ ions in solution at low pH suppresses dissociation of acidic group making Pr(A−) small. These changes are consistent with LeChatelier's principle. At pH > 6 the carboxylic group in acetic acid is almost fully dissociated hence Pr(A−) approaches unity. At pH < 10 amine group in ethylamine is fully protonated hence Pr(BH+) is also close to unity. N.B. Ionization of acidic group corresponds to dissociation (H+ loss) while ionization of basic group corresponds to protonation (H+ gain). See Data S1 for description of species and their corresponding probabilities. The ionizations of acidic and basic groups take place within specific pH ranges, for example, 3–6 and 9–12, respectively (Figure 1). Outside of these ranges, there is no change in the ionization state of the two groups.

Consider possible ionic states of diprotic weak acid: HAAH ↔ HAA− ↔ −AA− and diprotic base: BB ↔ BBH+ ↔ +HBBH+ (with ionization constants pK1 and pK2). The ionization proceeds in stepwise manner, that is, different groups are not (de)protonated simultaneously. The probabilities of double ionization, that is, deprotonation of both acidic or protonation of both basic functional groups can be derived from fractional concentrations given by Ault8,9 as shown in Data S1 where expressions (S1)–(S3) give probabilities for each possible ionic species at a given pH. From the pedagogical point of view, it is important to emphasize that probabilistic expressions in Data S1 include both ionization constants, pK1 and pK2. This fact expresses the existence of intramolecular interactions between functional groups; in other words, the two groups show "cooperativity" where (de)protonation of one group is influenced by another.

However, as Darvey has pointed out,3,4 in polyprotic acids/bases one cannot attribute pKa values to individual functional groups nor use these individual values to demonstrate "cooperative" effects between individual groups. Each pKa value already takes "cooperativity" into account. For example, two carboxylic acid groups in oxalic acid are chemically identical. Thus, they would have identical pKa values if they were truly independent of one another. However, the fact that they have different pKa values suggests that "cooperativity" between the two groups exists. In other words, as one raises the pH of a solution it is significantly "easier" to deprotonate just one of the acidic groups (smaller pKa) than to deprotonate both, since the second deprotonation causes repulsion between negative charges within the −AA− anion. This is why we measure two different pKa values despite being groups being chemically identical. Nevertheless, we can use fractional concentrations of different ionic species from Ault8,9 and argue that they represent probabilities (Pr) of existence (finding) each type of species in solution at a given pH. However, if the probability of finding a particular species which has two ionized groups is smaller than the probability of finding the same species which has one ionized group, then we can conclude that the ionization of two groups is more difficult (less probable) than single ionization. Why? Because of the intramolecular interactions, that is, cooperativity where one part of the molecule influences the behavior of another.

The algebraic expression for Pr1, Pr2, and Pr3 in Data S1 give probability of ionization of one, two, or three functional groups in the same molecule. The expressions in Data S1 are labeled as (S1)–(S3).

Since groups influence each other's ionization Pr1, Pr2, and Pr3 will all have different values. We can define ΔPr = Pri − Prref (i = 1, 2, 3) and use it as a measure of strength of interactions (cooperativity) between groups. For example, Prref in diprotic acid is the probability of ionization of reference molecule in which one of the two ionizable groups has been replaced by hydrogen. In case of oxalic acid (HOOCCOOH) the appropriate reference molecule is formic acid (HCOOH), which is obtained by removing second COOH group and replacing it by hydrogen. For all polyprotic acids/bases we have indicated which the appropriate reference molecule is in ΔPr expressions. ΔPr variable compares ionization propensity of a particular acidic/basic group in polyprotic acid/base with propensity of the same type of group in monoprotic reference analogue (within pH range where change of ionization state of that group takes place). ΔPr < 0 indicates that ionization of the first group hinders ionization of the second (negative cooperativity) while ΔPr > 0 indicates the opposite (positive cooperativity). N.B. Each Pri expression includes all pK1 − pK3 values (see Data S1) so in our reasoning we have not attributed specific pKa values to individual groups. In other words, we use Pr1 and Pr2 (actual concentrations/probabilities) instead of pK1 and pK2 to describe first and second ionization.

Plotting ΔPr versus pH, we obtain Figure 2 for diprotic acids: oxalic acid (pK1 = 1.46; pK2 = 4.40) and adipic acid (pK1 = 4.43; pK2 = 5.41). The reference molecules were formic (pKa = 3.75) and pentanoic acid (pKa = 4.84), respectively. To facilitate student learning, we have complemented each algebraic equation with corresponding symbolic equation in (3)–(14). Symbolic equations associate each term in the algebraic expression with the structure of the corresponding chemical species. We also note that probability of simultaneous occurrence of two independent events equals the product of probabilities of individual events. In our case, the probability that two groups in the same molecule are ionized at the same time (and assuming that they do not interact) equals the product of individual group probabilities. The individual probabilities can be derived from pKa value(s) of reference molecule(s).

3 $$\Delta \mathrm{Pr}\left(\text{oxalic}\right)={\mathrm{Pr}}_2-{\mathrm{Pr}}_{\text{formic}}^2=\left(\frac{1}{1+{10}^{\left(4.40-\mathrm{pH}\right)}+{10}^{\left(1.46+4.40-2\mathrm{pH}\right)}}\right)-{\left(\frac{1}{1+{10}^{\left(3.75-\mathrm{pH}\right)}}\right)}^2,$$

4 $$\Delta \mathrm{Pr}\left(\text{adipic}\right)={\mathrm{Pr}}_2-{\mathrm{Pr}}_{\text{pentanoic}}^2=\left(\frac{1}{1+{10}^{\left(5.41-\mathrm{pH}\right)}+{10}^{\left(4.43+5.41-2\mathrm{pH}\right)}}\right)-{\left(\frac{1}{1+{10}^{\left(4.84-\mathrm{pH}\right)}}\right)}^2.$$

The curves provide the following information. Sign of ΔPr tells us whether ionization of one functional group enhances (ΔPr > 0) or hinders (ΔPr < 0) ionization of another at a given pH. In other words, ΔPr > 0, ΔPr < 0 indicate whether intramolecular interactions between ionizable groups exhibit positive or negative cooperativity. The area (depth) under the curve ("dip") reveals the strength of charge interactions and the curve position indicates in which pH range the interaction is effective.

The two curves in Figure 2 show comparable negative cooperativity in both acids. Negative cooperativity is due to destabilizing intramolecular interactions between two COO− groups i.e. the ionization of first COOH group hinders ionization of the second.

It is clear from Figure 2 that intramolecular interaction in adipic is weaker than in oxalic acid. This is reflected in shallower dip in the former acid and smaller area under the curve. The difference is due to greater spatial separation of COOH groups in adipic acid which leads to smaller negative cooperativity.

Analogous analysis can be performed for diprotic bases: ethylenediamine (pK1 = 7.56; pK2 = 10.71), cadaverine (pK1 = 9.13; pK2 = 10.25) and nicotine (pK1 = 3; pK2 = 8). The reference molecules were ethylamine (pKa = 10.87) and pentylamine (pKa = 10.6) for ethylenediamine and cadaverine, respectively. For nicotine reference molecules were pyridine (pKa = 5.23) and N‐methylpyrrolidine (pKa = 10.32).

N.B. protonation of the first basic group in these bases is related to pK2 and second to pK1. Also, the expressions (S1)–(S3) refer to deprotonation of conjugate acids of these bases. The relevant symbolic and numerical equations obtained are given as (5)–(7) and ΔPr–pH curves in Figure 3.

$$\mathrm{\Delta Pr}=\mathrm{Pr}\left({}^{+}\mathrm{N}{\mathrm{H}}_3—{\left({\mathrm{CH}}_2\right)}_2—{{\mathrm{NH}}_3}^{+}\right)—{\left[\mathrm{Pr}\left({\mathrm{CH}}_3—{\mathrm{CH}}_2—{{\mathrm{NH}}_3}^{+}\right)\right]}^2\text{ethylenediamine},$$

5 $$\Delta \mathrm{Pr}\left(\text{ethylene diamine}\right)={\mathrm{Pr}}_2-{\mathrm{Pr}}_{\text{ethylamine}}^2=\left(\frac{1}{1+{10}^{\left(\mathrm{pH}-7.56\right)}+{10}^{\left(2\mathrm{pH}-7.56-10.71\right)}}\right)-{\left(\frac{1}{1+{10}^{\left(\mathrm{pH}-10.87\right)}}\right)}^2,$$

$$\mathrm{\Delta Pr}=\mathrm{Pr}\left({}^{+}\mathrm{N}{\mathrm{H}}_3—{\left({\mathrm{CH}}_2\right)}_5—{{\mathrm{NH}}_3}^{+}\right)-{\left[\mathrm{Pr}{\left({\mathrm{CH}}_3—{\mathrm{CH}}_2\right)}_4—{{\mathrm{NH}}_3}^{+}\right]}^2\text{cadaverine},$$

6 $$\Delta \mathrm{Pr}\left(\text{cadaverine}\right)={\mathrm{Pr}}_2-{\mathrm{Pr}}_{\text{pentylamine}}^2=\left(\frac{1}{1+{10}^{\left(\mathrm{pH}-9.13\right)}+{10}^{\left(2\mathrm{pH}-9.13-10.25\right)}}\right)-{\left(\frac{1}{1+{10}^{\left(\mathrm{pH}-10.6\right)}}\right)}^2,$$

7 $$\Delta \mathrm{Pr}\left(\text{nicotine}\right)={\mathrm{Pr}}_2-{\mathrm{Pr}}_{\text{pyridine}}\times {\mathrm{Pr}}_{N-\text{methylpyrrolidine}}=\left(\frac{1}{1+{10}^{\left(\mathrm{pH}-3\right)}+{10}^{\left(2\mathrm{pH}-3-8\right)}}\right)-\left(\frac{1}{1-{10}^{\left(\mathrm{pH}-5.23\right)}}\right)*\left(\frac{1}{1+{10}^{\left(\mathrm{pH}-10.32\right)}}\right).$$

The interpretation of ΔPr–pH curves for diprotic bases is similar to the interpretation for diprotic acids, that is, protonation of the second basic group is disfavored (negative cooperativity) due to positive charge repulsion.

The curves in Figure 3 resemble ΔPr–pH curves for diprotic acids (Figure 2) which also show negative cooperativity. However, the extent of negative cooperativity for the three bases varies being strongest in ethylenediamine and weakest in cadaverine. This conclusion is based on the area under the curve and depth of the minimum in each curve. The small distance between interacting groups makes negative cooperativity in ethylene‐diamine larger than in nicotine which is in turn larger than in cadaverine. The spatial separation between interacting groups once again strongly influences the extent of cooperativity.

Nevertheless, the spacing between groups is not the only factor; orientation of ionizable group dipoles, presence of hydrogen bonding or inductive effects of other groups in the molecule also influences cooperativity. This is relevant for understanding interactions and structure of proteins where all the above factors play a role.

Ionization of proteinogenic AA is of great importance in biochemistry. Furthermore, the ionization of acidic/basic functional groups in AA comprises key steps in some mechanisms of enzyme catalysis.

We begin by discussing AA without ionizable side chain groups. These acids contain zwitterionic forms which exhibit ionic intramolecular interaction. The zwitterionic probability based on Ault8,9 is given as Pr1 in Equation (S2) in Data S1.

The AAs without ionizable side chain groups are represented by glycine (pK1 = 2.34; pK2 = 9.60) and serine (pK1 = 2.13; pK2 = 9.05).10 We can distinguish two ionization events in glycine: deprotonation of COOH group (in the presence of ${\mathrm{NH}}_3^{+}$ group) and deprotonation of ${\mathrm{NH}}_3^{+}$ group in the presence of COO− group. In both cases, ionization of one group takes place in the presence of another which is already ionized. We use acetic acid (pKa = 4.76) as reference for deprotonation of COOH and methylamine (pKa = 10.54) for deprotonation of ${\mathrm{NH}}_3^{+}$ group.

Using expressions for probabilities of existence of individual ionic species, we can derive combined symbolic and algebraic ΔPr expressions for deprotonation of COOH and ${\mathrm{NH}}_3^{+}$ groups in glycine (8)–(9) followed by ΔPr–pH plot in Figure 4.

8 $$\Delta {\mathrm{Pr}}_{\text{COOH}}={\mathrm{Pr}}_1-{\mathrm{Pr}}_{\text{acetic acid}}=\left(\frac{1}{1+{10}^{\left(2.34-\mathrm{pH}\right)}+{10}^{\left(\mathrm{pH}-9.60\right)}}\right)-\left(\frac{1}{1+{10}^{\left(4.76-\mathrm{pH}\right)}}\right),$$

9 $$\Delta {\mathrm{Pr}}_{\text{amino}}={\mathrm{Pr}}_2-{\mathrm{Pr}}_{\text{methylamine}}=\left(\frac{1}{1+{10}^{\left(9.60-\mathrm{pH}\right)}+{10}^{\left(2.34+9.60-2\mathrm{pH}\right)}}\right)-\left(\frac{1}{1+{10}^{\left(10.54-\mathrm{pH}\right)}}\right).$$

ΔPr–pH curves in Figure 4 show that in pH = 2–6 range strong positive cooperativity exists for deprotonation of COOH group (blue curve), that is, it is facilitated by ${\mathrm{NH}}_3^{+}$ group. This is due to generation of stable, neutral zwitterion upon COOH deprotonation. One must bear in mind that zwitterionic stability versus stability of its neutral form strongly depends on environment, particularly on water solvation. In zwitterion, the stabilization due to intramolecular hydrogen bonding and salt‐bridges may compensate for the thermodynamic cost of (de)protonation of acidic/basic groups. At pH = 8–12 another positive (but much weaker) cooperativity exits corresponding to deprotonation of ${\mathrm{NH}}_3^{+}$ group in the presence of COO− group (yellow curve). The reason why this second cooperativity is weaker than the first is that ${\mathrm{NH}}_3^{+}$ deprotonation quenches stable zwitterion. Nevertheless, the cooperativity is still positive because the presence of COO− group facilitates solvation of NH2CH2COO− form of glycine compared to reference methylamine form CH3NH2. This results in preferred deprotonation of ${\mathrm{NH}}_3^{+}$ group and positive cooperativity. Our analysis allows us to compare relative magnitudes of zwitterionic stabilization and solvation effect.

The presence of charged side chain groups is important for stability, action and solubility of proteins11 because such groups often lie on protein surfaces. The AAs with charged (ionizable) side chains are aspartic acid, glutamic acid, lysine, histidine, and arginine. Therefore, in this section, we pay attention to ionization of side chain groups only, in particular how is their ionization affected by carboxylic and amino groups in the main AA moiety (NH2CH2COOH). The relevant expressions (symbolic and algebraic) deduced from fractional concentrations and probabilities of multiply ionized species (Pr1, Pr2, Pr3) are given in Data S1 as (S4)–(S7).

We begin by considering AAs with acidic side chains first. Using considerations described earlier, we derive symbolic and algebraic equations for ΔPr of the side chain group as a function of pH and plot them in Figure 5. The equations relating ΔPr to pH for aspartic acid (pK1 = 1.95; pK2 = 3.71; pK3 = 9.66) and glutamic acid (pK1 = 2.16; pK2=4.15; pK3 = 9.58) are given in (10)–(11). pKa values in italic fonts refer to the side chain group. The reference molecules for aspartic and glutamic acid are propanoic (pKa = 4.88) and butanoic acids (pKa = 4.82), respectively. ΔPr–pH curves based on (10)–(11) were plotted in Figure 5.

10 $$\Delta {\mathrm{Pr}}_{\text{aspartic}}={\mathrm{Pr}}_2-{\mathrm{Pr}}_{\text{propanoic}}=\left(\frac{1}{1+{10}^{\left(\mathrm{pH}-9.66\right)}+{10}^{\left(3.71-\mathrm{pH}\right)}+{10}^{\left(1.95+3.71-2\mathrm{pH}\right)}}\right)-\left(\frac{1}{1+{10}^{\left(4.88-\mathrm{pH}\right)}}\right),$$

11 $$\Delta {\mathrm{Pr}}_{\text{glutamic}}={\mathrm{Pr}}_2-{\mathrm{Pr}}_{\text{butanoic}}=\left(\frac{1}{1+{10}^{\left(\mathrm{pH}-9.58\right)}+{10}^{\left(4.15-\mathrm{pH}\right)}+{10}^{\left(2.16+4.15-2\mathrm{pH}\right)}}\right)-\left(\frac{1}{1+{10}^{\left(4.82-\mathrm{pH}\right)}}\right).$$

How should we interpret diagrams in Figure 5? The curves for the two acids (representing deprotonation of side chain COOH group) both show positive cooperativity; the weaker cooperativity found in glutamic acid is due to greater spatial separation of side chain from the AA moiety. The positive cooperativity is not due to generation of stable zwitterion which is quenched upon deprotonation of side chain COOH, but rather to the solvation effect when side chain COOH is deprotonated. Deprotonated side chain COO− group facilitates solvation and thus stabilizes the species compared to reference propanoic or butanoic acids. Aspartic and glutamic acids demonstrate again how solvation effects (and not just zwitterion) play a role in ionization of side chain groups.

Analogous considerations for AAs with basic side chain lysine (pK1 = 2.15; pK2 = 9.16; pK3 = 10.67), histidine (pK1 = 1.70; pK2 = 6.04; pK3 = 9.09) and arginine (pK1 = 2.03; pK2 = 9.00; pK3 = 12.10), lead to symbolic and algebraic equations (12)–(14) from which ΔPr–pH curves were plotted in Figure 6. The reference molecules were pentylamine (pKa = 10.6), 2‐ethyl‐imidazole (pKa = 7.99) and guanidine (pKa = 13.6), respectively. There is no available pKa value for pentylguanidine. Equations (12)–(14) and curves describe deprotonation of basic side chain group.

12 $$\Delta {\mathit{Pr}}_{\mathit{\text{lysine}}}={\mathit{Pr}}_3-P{r}_{\mathit{\text{pentylamine}}}=\left(\frac{1}{1+{10}^{\left(10.67-\mathit{pH}\right)}+{10}^{\left(9.16+10.67-2\mathit{pH}\right)}+{10}^{\left(2.15+9.16+10.67-3\mathit{pH}\right)}}\right)-\left(\frac{1}{1+{10}^{\left(10.6-\mathit{pH}\right)}}\right)$$

GRAPH

13 $$\Delta {\mathit{Pr}}_{\mathit{\text{histidine}}}={\mathit{Pr}}_2-P{r}_{\mathit{\text{ethylimidazole}}}=\left(\frac{1}{1+{10}^{\left(\mathit{pH}-9.09\right)}+{10}^{\left(6.04-\mathit{pH}\right)}+{10}^{\left(1.70+6.04-2\mathit{pH}\right)}}\right)-\left(\frac{1}{1+{10}^{\left(7.99-\mathit{pH}\right)}}\right)$$

14 $$\Delta {\mathrm{Pr}}_{\text{arginine}}={\mathrm{Pr}}_3-{\mathrm{Pr}}_{\text{guanidine}}=\left(\frac{1}{1+{10}^{\left(12.10-\mathrm{pH}\right)}+{10}^{\left(9.00+12.10-2\mathrm{pH}\right)}+{10}^{\left(2.03+9.00+12.10-3\mathrm{pH}\right)}}\right)-\left(\frac{1}{1+{10}^{\left(13.6-\mathrm{pH}\right)}}\right).$$

ΔPr–pH profiles for AAs with basic side chain (Figure 6) reveal very diverse ionization behavior. This is due to the fact that basic side chain groups are of different types: amino (lysine), imidazole (histidine), and guanine (arginine). Furthermore, side chain group in histidine is less basic than amino group in AA moiety. Histidine and arginine show similar, large positive cooperativity while lysine shows very small negative cooperativity. The lack of significant cooperativity in lysine is due to spatial isolation of its side chain amino group from AA moiety. Histidine shows strong positive cooperativity because of the formation of stable zwitterion upon deprotonation of imidazole side chain group. In arginine, deprotonation of side chain basic group quenches zwitterion, but is still facilitated by the enhancement of solvation due to the presence of COO− group. This leads to positive cooperativity when compared to reference guanidine which has no such group. Nevertheless, positive cooperativity (induced via solvation) is smaller than in histidine (induced by zwitterion formation).

We have seen that AAs often exist as charged species in water solution. On the basis of LeChatelier's principle one may expect that total charge will be strongly dependent on pH of the solution. We derive in this section the exact expressions which quantify this behavior.

The total charge of AA at given pH is important for their separation using electrophoresis. Inspection of Equation (S1)–(S7) in Data S1 allows one to derive total charges q as a function of pH simply by multiplying the Pr by the charges of appropriate functional groups. The equations for neutral, acidic and basic AAs, respectively are given below.

$$q={\mathrm{Pr}}_0-{\mathrm{Pr}}_2\left(\text{neutral}\mathrm{AA}\right);q={\mathrm{Pr}}_0-{\mathrm{Pr}}_2-{\text{2Pr}}_3\left(\text{acidic}\mathrm{AA}\right);q={\text{2Pr}}_0-{\mathrm{Pr}}_1-{\mathrm{Pr}}_3\left(\text{basic}\mathrm{AA}\right).$$

We use Mathematica to plot these functions to obtain diagrams in Figure 7. At physiological pH neutral AAs have q ≈ 0, acidic AAs q = −1 and basic AAs have q = +1 except histidine which has q ≈ 0.

q‐pH curves in Figure 7 provide an easy way to determine isoelectric point pI for individual AA. Isoelectric point is the point of intersection of the curve with pH axis. Experimentally determined values of pI available in Reference [10] match the results obtained from the curves in Figure 7.

The curves showing the variation of net charge of AA with pH have been reported earlier by the author, but the earlier results were based on the assumption that ionizations of different acidic/basic groups are independent of each other7 which is only an approximation. The approximation leads to noticeable discrepancies between earlier curves and those reported here as well discrepancies in pI values. This is more pronounced in some cases, for example, in aspartic and glutamic acids. Curves in Figure 7 can be used when discussing experimental technique of electrophoresis in biochemistry. The ionic mobilities of ionized AA species in solution are directly proportional to charge and inversely proportional to species' sizes. Based on the charge alone one would expect that AAs whose q‐curve shows distinct intersection with pH axis would be the most mobile (since they exist mostly in charged state) while those which show a plateau near pH axis would be the least mobile (they exist with near‐zero charge over a wide pH range). This is born out to some extent when comparing low mobilities of glycine and proline versus higher mobilities of arginine, histidine, and lysine.12–14 Nevertheless, one needs to bear in mind that molecular size is also important so the curves in Figure 7 provide only a qualitative guideline about electrophoretic mobility.

pKa of an ionizable group on AA residue depends on the surrounding environment, for example, whether the group is buried inside hydrophobic cavity or exposed to the solvent on protein surface or exposed to the acidic/basic residue in its neighborhood. pKa values could fluctuate even when macroscopic pKa is fixed at some value close to the corresponding value in fully solvated environment. As an illustration of the relevance of discussions above, we give examples of how the charge of one side chain affects charge and protonation state of another. For example, side chain of histidine whose standard pKa value is 6.04 can assume pKa values between 2.4 and 9.2 when it forms part of various proteins.11 One specific protein example is hemoglobin (Figure 8). The deoxyhemoglobin's T‐state is stabilized by salt bridges between proximate residues; when these interactions are broken the hemoglobin binds oxygen and reverts to R‐state (Figure 9).

Hemoglobin's biological function is regulated by changing overall protein structure which is triggered by changes in blood pH. This structure is in turn altered by binding or releasing of CO2 and H+ to the interfaces of hemoglobin subunits (Figure 9).15

Figure 9 is a schematic diagram of the interface of two protein subunits of hemoglobin protein. In the presence of CO2 and H+ (e.g., in the muscles), charged groups are formed on AA residues lining the subunit interface (Figure 9, top left). These charged groups are held together by ionic interactions, forming "salt bridges" between the two subunits and stabilizing deoxygenated T‐form of hemoglobin (Figure 9, left).

When blood passes through the alveolar capillaries of the lungs, CO2 and H+ are removed from the hemoglobin and oxygenated R‐configuration is favored (Figure 9, top right).When the concentration of protons (H+) is low (pH 9), positive charges cannot form on the residues at subunit interfaces, so the salt bridges do not form (Figure 9, right). However, at pH 7, histidine residues at subunit interfaces (not the histidine residues which bind heme groups) accept additional proton (H+) and become positively charged. When salt bridges form by the interaction of these interfacial histidine residues with nearby negatively‐charged amino‐acid residues, the spatial arrangement of subunits changes favoring deoxygenated hemoglobin structure (T‐state) and releasing oxygen (Figure 9, bottom).

The use of visualization tools in teaching and learning biochemistry is expanding. Some of these tools appear in the form of 2D and 3D representations and media presentations..1,2 The use of suitable visualization programs needs not only to be demonstrated to students by the instructor, but also needs to be assigned to students so that they can learn how to use programs independently. Some powerful and sophisticated visualization software come with high licensing fees and requires considerable effort on behalf of instructors and students to master and use properly. The use of visualization tools also needs to be linked to learning objectives. Students need to develop particular skills ("visual literacy") which will allow them to construct models (including mathematical models) and interpret the findings or representations.1,2 Noncovalent interactions and pH/pKa concepts are often taught separately in different subjects without emphasizing their mutual links or attempts at integrating them. The proposed activity seeks to remedy that students may perform mathematical manipulations without fully understanding what is that they are manipulating and why. In the guided‐inquiry described here, none of the above deficiencies applies. Mathematica software is versatile and can be used for several purposes in biochemistry teaching, for example, solving problems in enzyme kinetics or analysis of metabolic cycles.16

• Help students to identify intramolecular (electrostatic) non‐covalent and solvent interactions in AA as a template for studying such interactions within proteins.
• Guide students to discover how pH changes affect non‐covalent interactions above.
• Develop "visual literacy skills" and ability to interpret the meaning of quantitative expressions and mathematical manipulations.
• Develop conceptual understanding of how pH affects protein and AA charges and subsequently use that knowledge to explain how pH affects protein structures and intramolecular interactions.

Upper‐level undergraduates majoring in biochemistry and molecular biology.

Prior to this activity, the instructor should ascertain that students have had instruction in the concepts of non‐covalent interactions, pH and pKa. Students are provided with access to Mathematica, handed copies of papers by Ault8,9 (including the Supporting Information attached to Ault's paper and to this paper). Instructor should also give students a brief introduction to Mathematica. Mathematica introduction can be based on Data S1 provided in this paper and should include only the explanations of syntax and plotting commands. Examples of how to use of Mathematica are given in Data S1 for this work in order to provide guidance for the instructor. The instructor should not make complete Data S1 available to the students so that they will not answer their questions by "pattern recognition." One case of detailed implementation of Mathematica to, for example, diprotic acid or one of AAs could be given to students as a handout.

The students are asked to work in pairs for 1 hr in the lab which has PC workstations with printers attached and with access to Mathematica. Each pair of students is given an example of AA for which the pair needs to deduce and plot ΔPr–pH and q–pH curves. After plotting and printing their curves students are asked to interpret the curves with respect to deducing the following information:

• Which ionic species are present in water solution at pH = 1–14 and which species are not?
• What types of cooperativity are exhibited by your AA and why?
• What electrophoretic mobility do you expect from your AA (large or small)?
• Rank AA species present in terms of their lipophilicity and describe how pH affects lipophilicity of your example molecule.

The instructor may use other biologically active molecules besides AA in the activity. Nicotine given here is a possible example.

The approach outlined in this work is "an idea to explore" or "activity to explore" for teaching biochemistry by using non‐specialized software of great versatility.

The guided‐inquiry activity described here provides students with hands‐on illustration of the relationship between pH, pKa, noncovalent interactions and solvation effects. Also, students and instructors experience Mathematica as a universal tool for solving quantitative problems and mathematical relationships. This may encourage them to use it in solving other problems in biochemistry teaching and learning, for example, enzyme kinetics. Biochemistry students often shun the use of mathematical analysis due to perceived difficulty. Mathematica facilitates the task of solving and displaying quantitative relationships.

The approach described points out and corrects some student misconceptions regarding the meaning of pKa for different functional groups.

The students may be familiar with "standard" pH curves which show how the percentage of species varies with pH. However, this familiarity does not readily convey the information about intramolecular interactions or solvent effects. Also, the use of probability emphasizes the stochastic nature of ionization process. One cannot predict whether a certain functional group will ionize, only the probability of it doing so.

To be sure, many sophisticated quantum mechanical programs can calculate and display electron density maps and partial atomic charges. The snag is that this type of software may be viewed by biochemistry students as a "black box" since they usually have inadequate understanding of quantum mechanics. Besides, partial charges are not observables and their value depends on the quantum mechanical formalism used. Our approach uses only experimental data which can be verified and determined by students in laboratory (pKa values).

GRAPH: DATA S1. Detailed guide and examples of the approach