Titration curves simulation

Using the pKa and the estimated So, the DTT procedure simulates the entire titration curve before the assay commences. Figure 6.7 shows such a titration curve of propoxyphene. The simulated curve serves as a template for the instrument to collect individual pH measurements in the course of the titration. The pH domain containing precipitation is apparent from the simulation (filled points in Fig. 6.7). Titration of the sample suspension is done in the direction of dissolution (high to low pH in Fig. 6.7), eventually well past the point of complete dissolution (pH <7.3 in Fig. 6.7). The rate of dissolution of the solid, described by the classical Noyes-Whitney expression [37], depends on a number of factors, which the instrument takes into account. For example, the instrument slows down the rate of pH data taking as the point of complete dissolution approaches, where the time needed to dissolve additional solid substantially increases (between pH 9 and 7.3 in Fig. 6.7). Only after the precipitate completely dissolves, does the instalment collect the remainder of the data rapidly (unfilled circles in Fig. 6.7). Typically, 3-10 h is required for the entire equilibrium solubility data taking. The more insoluble the... 

The accuracy of PHMD methods and their feasibility for studying pH-dependent conformational phenomena of proteins can be assessed by pKa calculations. In this case, PHMD simulations are performed with several pH values. The resulting occupancy values for deprotonated states (.S dc prot) are plotted against pH (Figure 10-3). A titration curve and pATa values (Figure 10-3) can be obtained by fitting the data to the generalized HH equation (Eq. 10-5). 

Figure 10-3. Theoretical titration curves for the model compounds of Asp and His obtained from REX-CPHMD simulations [41]. Solid curves are the obtained by fitted the computed deprotonated fraction to the generalized Henderson-Hasselbach equation. The dashed lines indicate the computed pKa values...

To use this method, the sample is dissolved in a system containing two phases (e.g., water and octanol) such that the solution is at least about 5 x 10-4 M. The solution is acidified (or basified) and titrated with base (or acid) under controlled conditions. The shape of the ensuing titration curve is compared with the shape of a simulated curve, which is created in silico. The estimated p0Ka values (together with other variables used to construct the simulated curve such as substance concentration factor, CO2 content of the solution and acidity error) are allowed to vary systematically until the simulated curve fits as closely as possible to the experimental curve. The p0Ka values required to achieve the best fit are assumed to be the correct measured p0Ka values. This computerized calculation technique is called refinement , and is described elsewhere [14, 15]. 

Scheme 3.4-1. Simulated titration curves for the catalytic model system described above. The change in the steady-state concentrations following the ligand association process is schematically depicted (the species present at relatively high concentration is underlined).

Solid lines standard titration curves, broken lines manifold systematic variations, arrows direction of the induced relative shift. F s. 1 and 2 simulate structural changes in the ligand-free complexes. Figs 3-6 inhibition and activation processes induced by the controlling ligand (kinetic control). Figs 7 and 8 simulate a variation of the catalytic concentration (see Scheme 3.3-4) or of the constants of association of L to M (thermodynamic control). 

Here we assume that [H+] from acidic or basic impurities that may influence the dissociation equilibrium. If the values of Ka, Kf(HAi), Qand Cs are known, we can get a(H+) by solving the quadratic equation. Figure 3.3 shows the simulated titration curves for various values of Kf(HAy)(Ca+Cs). If we compare the experimental titration curve with the simulated ones (curve-fitting method), we can get the value of K HA ) (see Fig. 6.6). The value of Kf(HAy) can also be obtained by the relation ... 

R. de Levie, A General Simulator for Add-Base Titrations, J. Chem. Ed. 1999, 76, 987 R. de Levie, Explicit Expressions of the General Form of the Titration Curve in Terms of Concentration, J. Chem. Ed. 1993, 70, 209 R. de Levie, General Expressions for Add-Base Titrations of Arbitrary Mixtures, Anal. Chem. 1996, 68, 585 R. de Levie, Principles ctf Quantitative Chemical Analysis (New York McGraw-Hill, 1997). 

Figure 21.5 Potential errors introduced by the assumption that Ar2+ is constant in the derivation of the linkage Eq. (21.30). The dependence of Ar2+ on [Mg2 1J was modeled by a polynomial fit to the solid gray data points in Fig. 21.4D. The polynomial was used in the integration of Eq. (21.30) to give expressions for In Kobs and 0 with [Mg2+]-dependent AT2 (in contrast to Eqs. (21.33) and (21.34), which assume AT2+ is constant). 9 is plotted for the calculated titration curve when the midpoint of the titration, [Mg2+]0, is 10 jiM (circles), 30 uA / (squares), or 100 li.M (diamonds). AT2+ (as used to calculate the displayed curves) at the titration midpoints (9 = 0.5) is 1.45, 2.30, and 2.73, respectively. The simulated data points have been fit to either a modified version of Eq. (21.34) that assumes the y-intercept of the curve has the value 6 = 0, 9 = 90 + (1 - 0o)([Mg2+]/[Mg2+]o)"/[l + ([Mg2+]/[Mg2+j0)"], or to an equation that allows a nonzero y-intercept, Eq. (21.35). The residuals of the fits are shown in the lower three panels closed symbols correspond to Eq. (21.34) and open symbols to Eq. (21.35). For the curve with a midpoint of [Mg2+]0 = 100 fiM, Eq. (21.35) could not be fit to the data because the value of Co became vanishingly small. The values of A IT. at the titration curve midpoints obtained from the modified Eq. (21.34) are 1.99, 2.39, and 2.73, in order ofincreasing [Mg2+]0. AT2+ obtained by fitting ofEq. (21.35) and application of Eq. (21.36) are 1.43 and 2.29 (10 and 30 fiM transition midpoints, respectively).

There is a caveat in using Eq. (21.34), illustrated by the simulated titration curve at 10 [iM Mg2+ (Fig. 21.5). This titration curve shows a significant fraction of folded RNA in the absence of Mg2+, where. C0bs 0.15. Spectroscopic titrations and most other methods used to assess Kohs assume a baseline value of 9 = 0 when [Mg2 = 0 and normalize the titration curve accordingly. Two of the calculated curves in Fig. 21.5 have been fit to a modified Hill equation that incorporates this assumption. The apparent value of Ar 2+ at the midpoint of one curve (1.99) is larger than the value used to calculate the titration curve (1.45 see legend to Fig. 21.5). 

A stepwise change in pH can be applied outside a protein-containing membrane applied on top of an ISFET, at the interface between the membrane and the electrolyte, using a flow-through system [ 10]. This pH step will lead to simultaneous diffusion and chemical reaction of protons and hydroxyl ions in the membrane. A theoretical description of these phenomena, elaborated in the next subsection, leads to the conclusion that the diffusion of protons in the membrane is delayed by a factor that depends linearly on the protein concentration. Consequently, the time needed to reach the end point in the obtained titration curve also depends linearly on the protein concentration. The effect of both the incubation time and the protein concentration will be simulated and experimentally verified. 

SIMULATION OF TITRATION CURVES USING A SINGLE MASTER EQUATION... 

Figure 10.2. Simulated titrations of a ligand concentration of 1 nM, demonstrating the effect of varying the conditional stability constant, / condzn - A titration curve with a condzn < 19 ( comizn [L] < 10 ) 1 indistinguishable from that of a blank solution containing no ligand. (From Bruland, 1989.)...

Calculate the volume-corrected titration curve for the titration of 0.01 N HCl with 0.01 N NaOH between pH 2 and 12 using a geochemical computer code and compare the result to the corresponding curve in Fig. 5.5. This problem can be solved using the mixing option in SOLMINEQ.88 (Kharaka et al. 1988). The titration is simulated by mixing different fractions of solution 1 (pH = 12.0, Na" = 0.01 mol/kg) with solution 2 (pH = 2.0, Cr = 0.01 mol/kg). In the output Cg = Na". Computed results are given here. 

Since the spreadsheet is eminently capable of doing tedious numerical work, exact mathematical expressions are used as much as possible in the examples involving chemical equilibria. Similarly, the treatment of titrations emphasizes the use of exact mathematical relations, which can then be fitted to experimental data. In some of the exercises, the student first computes, say, a make-believe titration curve, complete with simulated noise, and is then asked to extract from that curve the relevant parameters. The make-believe curve is clearly a stand-in for using experimental data, which can be subjected to the very same analysis. 

A. K. Covington, R. N. Goldberg, and M. Sarbar, Computer Simulation of Titration Curves with Application to Aqueous Carbonate Solutions, Ana/. Chim. Acta, 130 (1981) 103. 

Fig. 15.2-13 Examples for simulated lineshape of titrations curves, which indicate different binding mechanisms. The color of the curves changes from blue to red with increasing ligand concentration, while the protein concentration is kept constant.

Using Equation 11-3, develop a series of simulated precipitation titration curves in which Ag is used to titrate three singly charged anions, X", Y", and Z, whose log K p values are -12, -9, and -6. Repeat the exercise with other sets of log K p values. How does the log K p interval affect... 

FIGURE 1. Illustrative simulation of the titration curves for two selected heme methyl signals of D. gigas cytochrome C3 upon addition of flavodoxin. Each pair of curves represents the best possible fit of the two sets of data, three different chemical models. Curves A (dashed) binary complex (stoichiometry 1 1) curves B (dotted) cytochrome C3 / flavodoxin (2 1) with two equivalent interaction sites on the flavoprotein curves C (full line) ternary complex (2 1) with two distinc but dependent interaction sites on flavodoxin (see Figure 2 for model deffmition). 

The dynamics of lEF are illustrated in the simulations of Figure 3. This simulator is quite realistic, since it accepts 150 components and voltage gradients (300Vcm ) employed in laboratory practice. Before passage of the current, the column is at constant pH and the multitude of amphoteric buffers is randomly distributed, resulting in reciprocal neutralization. However, each individual CA species will have its own titration curve (see Figure 2, lower left... 

We modified titration curves (Figure 2.1) assuming that a certain amount of simple metal salt containing M" ions is added to the solution containing hgand L. As this takes place, M" ions (aqua complexes) and ML" " species are formed whose total concentration is equal to c,. Some examples of simulated curves are presented in Figure 4.1 where the shift of the equihbrium potential (AEg ) versus Cj is plotted. A sharp decrease in is observed at the equivalent point, the depth of which depends on the stabihty constant K) of ML" " complex. 

The first computations of ionization constants of residues in proteins for structures derived from molecular dynamics trajectories were described by Wendoloski and Matthew for tuna cytochrome c. In that study, conformers were generated using molecular dynamics simulations with a range of solvents, simulating macroscopic dielectric formalisms, and one solvent model that explicitly included solvent water molecules. The authors calculated individual pR values, overall titration curves, and electrostatic potential surfaces for average structures and structures along each simulation trajectory. However, the computational scheme for predicting electrostatic interactions in proteins used by Wendoloski and Matthew was not based on a FDPB model but on the modified Tanford-Kirkwood approach, which is not discussed in this chapter. 

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