Microscopic constants

Note that the terms macroscopic and microscopic constants do not imply that these quantities measure macroscopic or microscopic quantities, respectively. Here, in the macroscopic view we have simply grouped the two microscopic species (0, 1) and (1, 0) into one species denoted by (1). Both of these constants can be macroscopic or microscopic, depending on whether we study the binding per molecule or per one mole of molecules. 

MACROMOLECULAR SEQUENCE ANALYSIS (Computer Methods) MACROSCOPIC CONSTANTS MICROSCOPIC CONSTANTS ZWITTERION... 

MACRQSCQPIC CQNSTANT MICROSCOPIC CONSTANT MOLECULAR CROWDING SCATCHARD PLOT... 

Bowser and Chen (10) have calculated some theoretical binding isotherms (/z - /zs = /([L]) for anticooperative, noncooperative, and cooperative complex formation at two equivalent binding sites with arbitrarily chosen microscopic constants see Table 1. 

A microscopic constant applies to a single site. Consider the dissociation of a simple carboxylic acid ... 

Either of the two protons might dissociate first as the pH is raised (Eq. 6-75). However, the two microscopic dissociation constants pK x and pKB are distinctly different. The result is that at 25°C in the neutral (monoprotonated) form 80% of the molecules carry a proton on the N, while the other 20% are protonated on the less basic - O. Notice that the subscripts a and b used in this discussion do not refer to acidic and basic but to the individual dissociation steps shown in Eq. 6-75. Microscopic constants will always be indicated with asterisks in this discussion. 

The two monoprotonated forms of pyridoxine are the tautomeric pair shown in Eq. 6-75 and whose concentrations are related by the tautomeric ratio, R = [neutral form]/[dipolar ion], a pH-independent equilibrium constant with a value of 0.204/0.796 = 0.26 at 25°C.75 Evaluation of microscopic constants for dissociation of protons from compounds containing non-identical groups depends upon measurement of the tautomeric ratio, or ratios if more than two binding sites are present. In the case of pyridoxine, a spectrophotometric method was used to estimate R. 

To calculate microscopic constants from stepwise constants and tautomeric ratios, consider Eq. 6-76 in which [HP]a and [HP]B are the concentrations of the two tautomers and kj is the first stoichiometric or macroscopic dissociation constant for the diprotonated species H2P. 

For pyridoxine p/C, and pfC2 were determined spectrophotometrically as 4.94 and 8.89. These values, together with that of R given above, were used to estimate the microscopic constants that are given in Eq. 6-74.75 Notice that the microscopic constants of Eq. 6-74 are not all independent if any three of the five equilibrium constants are known the other two can be calculated readily. In describing and measuring such equilibria it is desirable to select one pathway of dissociation, e.g., H2P —> HP(A) —> P, and to relate the species HP(B) to it via the pH-independent constant R. 

Often more complex situations arise in which additional tautomers or other forms arise via pH-independent reactions. These can all be related back to the reference ionic species by additional ratios R, which may describe equilibria for tautomerization, hydration, isomerization, etc. (Eq. 6-82).76 In the case illustrated, only one of the ratios, namely R2 or R3, is likely to be a tautomerization constant because, as a rule, H2P and P will not have tautomers. Equations analogous to Eqs. 6-76 to 6-82 can be written easily to derive Kc, K0 and any other microscopic constants desired from the stoichiometric constants plus the ratios R, to R4. While it is easy to describe tautomerism by equations such as Eqs. 6-76 and 6-82 it is often difficult... 

All of the terms in both the numerator and the denominator of Eq. 7-35 can be related back to [X], using the microscopic constants from Fig. 7-21 to give an equation (comparable to Eq. 7-8) which presents Y in terms of [X], KAX and KBX, Kt, and the interaction constants KAA, KAB, and KBB. Since the equation is too complex to grasp immediately, let us consider several specific cases in which it can be simplified. 

Each one can be formed in two ways. It is a simple matter to write down the microscopic constants for addition of the second molecule of X as the sum of three terms. Because values of the constants for aj and bk differ, it will be clear that the three ways of adding the second molecule of X are not equally probable. Thus, the oligomer will show preferred orders of "loading" with ligand X. 

The pH dependence of the action of carboxypeptidase A is determined by pKa values of 6 and 9.5 for the free enzyme422 and of 6.4 and 9 for /ccat.420 For thermolysin the values for /ccat are 5 and 8.404 Assignment of pKa values has been controversial. They may all be composites of two or more microscopic constants but probably, at least for carboxypeptidase, the low pKa is largely that of Glu 270 while the high one represents largely the dissociation of a proton from the zinc-bound H20. 

The predicted kinetics is still first-order, but the equation is simpler. Now the observed rate constant is identical with the microscopic constant kx. 

Palcic, M. M., and Klinman, J. P., 1983, Isotopic probes yield microscopic constants separation of binding energy from catalytic efficiency in the bovine plasma amine oxidase reaction. Biochemistry 22 595785966. 

The constant Mi j.i+i is composed of microscopic constants, as each O2 binding step is composed of multiple microscopic reactions, which is illustrated by the reaction arrows in Fig. 1. Thus, 4 ways exist to bind the first O2, 12 ways to bind the second O2, 12 ways to bind the third O2, and 4 ways to bind the fourth O2. Each microscopic constant is designated by the notation ij of the species formed in the binding process (Fig. 1, Table 1). For each binding step i = 1,2,3, and 4, the macroscopic constant Mi j.i4i represents the average of the microstate constants A ij (i+i)j, with accompanying statistical factors that account for the different isomeric forms of the microstate tetramers, as shown in Table 1. 

At appreciable ionic strength these microscopic constants should be written in terms of activities rather than concentrations. The constants are normally estimated by spectral means. There are three microscopic ionization constants for the loss of the first proton, six for the loss of the second, and three for the loss of the last. In Table 3-2 the subscripts 1, 2, and 3 denote the carboxyl, sulfhydryl, and ammonium protons thus 32 is the microscopic constant for the ion formed by loss of the sulfhydryl proton from the species that has already lost the ammonium proton. 

Four microscopic constants can be defined, but potentiometrically only two composite (macroscopic) acidity constants can be determined. 

Since the tautomeric ratio R equals [HP]g/ [HP], Eqs. 6-76 and 6-77 can be rearranged to Eqs. 6-78 to 6-81. These allow the evaluation of all of the microscopic constants from the two stoichiometric constants and K2 plus the tautomeric ratio R. 

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