An Iso-random Bi Bi Mechanism for Adenylate Kinase*

An iso-random Bi Bi mechanism has been proposed for adenylate kinase. In this mechanism, one of the enzyme forms can bind the substrates MgATP and AMP, whereas the other form can bind the products MgADP and ADP. In a catalytic cycle, the conformational changes of the free enzyme and the ternary complexes are the rate-limiting steps. The AP5A inhibition equations derived from this mechanism show theoretically that AP5A acts as a competitive inhibitor for the forward reaction and a mixed noncompetitive inhibitor for the backward reaction.

Preparation and Activity Assay of Adenylate Kinase-The enzyme was prepared essentially according to Zhang et al. (19) The yield was usually about 60 mg of pure enzyme per kilogram of rabbit muscle. The final preparation usually had a specific activity greater than 1,600 units/mg and showed only a single peak in SDS electrophoresis, gel filtration, and reversed-phase fast protein liquid chromatography. One unit is defined as 1 mol of ADP (MgADP) generated per minute in forward reaction and 1 mol of ATP generated per minute in backward reaction.
The rate of the backward reaction (MgADP ϩ ADP f MgATP ϩ AMP) was measured by following the reduction of NADP at 340 nm in a coupled enzyme solution with hexokinase and glucose-6-phosphate dehydrogenase. The final assay mixture was: 50 mM Tris-HCl, pH 8.1, 2 mM ␤-mercaptoethanol, 6.7 mM glucose, 0.67 mM NADP, 7.6 M bovine serum albumin, 10 units/ml hexokinase, 10 units/ml glucose-6-phosphate dehydrogenase, varying and calculated concentrations of MgAc 2 , and ADP.
Measurements of the velocity of the forward reaction (MgATP ϩ AMP f MgADP ϩ ADP) was performed by monitoring the oxidation of NADH at 340 nm coupling with pyruvate kinase and lactate dehydrogenase. The final assay mixture was: 50 mM Tris-HCl, pH 7.5, 2 mM ␤-mercaptoethanol, 75 mM KCl, 4 mM phosphoenolpyruvate, 1.0 mM of MgAc 2 as free Mg 2ϩ , 0.2 mM NADH, 10 units/ml pyruvate kinase, 20 units/ml lactate dehydrogenase, 7.6 M bovine serum albumin, plus varying and calculated amounts of MgAc 2 , ATP, and AMP.
Iso-random Bi Bi Mechanism-The catalysis and inhibition mechanism of adenylate kinase is suggested as shown Scheme 1.
Using the combined equilibrium and steady-state treatment developed by Cha (20) and described by Huang (21), Scheme 1 can be reduced to Scheme 2 and Eq. 1.
Here E 1 and E 2 denote two native forms of enzyme, one form (E 1 ) can bind with AMP (M) and MgATP (T), and the other one with ADP (D 1 ) and MgADP (D 2 ). Both interconversions of the ternary complexes and free enzymes are rate-limiting steps. This mechanism is, in principle, a random Bi Bi type, similar to that proposed by an earlier study (11), but with the major modification that two native forms of enzyme are introduced. This mechanism is based on the fact that AK undergoes large domain movements upon substrate binding (22). After aligning 17 known structures of nucleoside monophosphate kinases, Vonrhein et al. (23) gave movies to demonstrate the catalytic cycle of AK, which showed that the conformation of ternary complex of enzyme binding with AMP and ATP is different from that of enzyme binding with two ADP. That is to say, AK undergoes at least two steps of conformational changes. One step is the change of a ternary complex with bound substrates to one with bound products during the reaction, and the other is the change of the free enzyme in different forms. These conformational changes should be the rate-limiting steps in the catalytic cycle of AK, whereas the binding of substrates to and the release of products from the enzyme are quickly equilibrated. AP 5 A is composed of ATP and AMP connected by a fifth phosphate, and mimics both substrates (24). It was assumed that AP 5 A (I) can only bind to E 1 and the binding reaction of E 1 with AP 5 A is rapid with an equilibrium constant of K I . We have also assumed that the affinities for substrates of free enzyme and binary complexes, ME 1 and E 1 T, as well as D 1 E 2 and E 2 D 1 , are the same.
In the forward reaction, one binding site is specific for AMP, whereas the other less specific one is for MgATP (25), so the reaction system consists of free enzymes E 1 and E 2 , binary complexes ME 1 , E 1 M, and E 1 T and ternary complexes ME 1 T and ME 1 M. The products, D 1 and D 2 , are rapidly removed by the coupled enzymes, so that the concentration of ternary complex D 1 E 1 D 2 approaches zero.
In the absence of AP 5 Here a factor of two is introduced, because the forward reaction produces two ADP molecules.
In the presence of AP 5 Eq. 3 shows that AP 5 A acts as a competitive inhibitor for the forward reaction.
In the backward reaction, one binding site is specific for ADP, and the other less specific one for MgADP (25), so the reaction system consists of free enzymes E 1 and E 2 , binary complexes D 1 E 2 , E 2 D 1 , and E 2 D 2 , as well as ternary complexes D 1 E 2 D 2 and D 1 E 2 D 1 . The product ATP is rapidly removed by coupled enzymes, so that the concentration of ternary complex ME 1 T approaches zero. The product conversion was controlled under 10%, and the concentration of binary complex ME 1 could be considered as zero.
In the absence of AP 5 The The stability constant K Mg ϭ 2,500 M Ϫ1 was used in the calculation (26).
In the presence of AP 5 A, we have Eq. 7 .
Eq. 7 shows that AP 5 A acts as a mixed noncompetitive inhibitor for backward reaction.

RESULTS
Catalytic Patterns- Fig. 1, A and B, shows the catalytic pattern when AMP and ATP were used as substrates of the forward reaction with the concentration of magnesium ion kept constant at 1 mM. Fig. 1A shows AMP as the variable substrate, whereas Fig. 1B shows MgATP as the variable substrate. With MgATP concentration lower than 0.1 mM, the specific activity increases initially then decreases with increasing concentration of AMP, implying that AMP can bind to the MgATP site (Fig. 1A).
In the backward reaction, the concentrations of ADP and MgADP are dependent on the concentration of magnesium ion. The kinetic measurements were carried out under either varied concentrations of magnesium ion with constant ADP ( Fig. 2A) or varied concentrations of ADP with constant magnesium ion (Fig. 2B). Fig. 2A shows that when ADP concentration is lower than 0.5 mM, the specific activity increases first then decreases with the increasing concentration of magnesium ion, because the concentration of MgADP increases with the increasing of magnesium ion concentration, whereas that of ADP decreases. Fig. 2B shows that with magnesium ion concentration lower than 0.5 mM, the specific activity increases first then decreases with increasing ADP concentrations, implying that ADP can bind to the MgADP site.
The full set of data in Figs. 1 and 2 was fitted to Eqs. 2 and 4 to calculate kinetic parameters by a computer program. The best fit kinetic parameters are summarized in Table I and were used to draw the solid lines in Figs. 1 and 2, which are in good agreement with the experimental results. Because the affinities of AMP to MgATP site as well as ADP to MgADP site are relatively smaller, Eqs. 10 and 12 can be treated as linear. Fig. 3, A and B, shows the AP 5 A inhibition patterns in the forward reaction with varied AMP (Fig. 3A) and MgATP (Fig. 3B) concentrations, respectively, indicating that AP 5 A acts as a competitive inhibitor for the forward reaction. Fig. 4, A and B, shows the AP 5 A inhibition patterns in the backward reaction with varied ADP (Fig. 4A) and MgADP (Fig.  4B) concentrations calculated according to Eqs. 3 and 7, respectively, exhibiting that AP 5 A acts as a mixed noncompetitive inhibitor for the backward reaction.
A value of K I ϭ 2 Ϯ 0.5 ϫ 10 Ϫ6 mM was obtained from the full set of data in Figs. 3 and 4.

DISCUSSION
There are two nucleotide-binding sites of AK, one for the magnesium complexes and the other for uncomplexed nucleotides. X-ray and NMR studies suggested that the latter sub-   2 Ϯ 0.5 ϫ 10 Ϫ6 mM k 1 14,000 Ϯ 2,000 strate site is more specific, whereas the former can bind uncomplexed substrates to some extent (24). The above experiments demonstrate that AMP or ADP shows substrate inhibition in the forward or the backward reaction, respectively. This result, which was also observed in bakers' yeast AK by Russell and co-workers (25), is in agreement with the x-ray and NMR findings. The equilibrium concentrations of E 1 and E 2 in the absence of substrates should be different from those at steady state, so a lag (or burst) phase should occur in the beginning of the catalytic reaction. However, because the activity of AK was assayed with a coupled enzyme system, the initial process is too complicated to follow.
The fact that AP 5 A acts as a competitive inhibitor for the forward reaction and a mixed noncompetitive inhibitor for the backward reaction cannot be explained by the normal random Bi Bi mechanism. However, these inhibition patterns are consistent with an iso-random Bi Bi mechanism in which the substrates can bind to one isoform while the products bind to another isoform of the enzyme. This strongly supports the suggestion that two native forms of AK might be involved in the catalytic reactions.
The rate of enzyme conformational changes is on the order of 10 2 s Ϫ1 in the catalytic reactions, which is much higher than that determined by ANS fluorescence probe (about 10 Ϫ2 s Ϫ1 , see Ref. 15). This perhaps can be explained that substrates decrease whereas ANS increases the activation energy of en- zyme conformational changes. It was indeed observed that ANS inhibits the folding of AK, whereas AMP, ADP, and ATP can accelerate the folding (16). Another possibility is that the two native forms involved in the catalytic cycle are different from those distinguished by ANS probe, and AK might exist in more native forms in equilibrium. The current experimental results provide no information to distinguish the above possibilities.
It has been reported that some proteins may exist in more than one distinct folded form in equilibrium. Evidence for distinguishing multiple native forms of staphylococcal nuclease has come from electrophoresis and NMR studies (27)(28)(29)(30)(31)(32). For calbindin D 9K, the evidence of multiple forms has come from not only NMR studies but also from x-ray crystal structures (33,34). These results shed light on the understanding of the protein folding problem but give little information on the biological role of the multiple native forms. Is the multiple native forms are necessary for protein to perform its biological functions or only the results of protein folding? Our results suggest that the multiple native forms are necessary for AK to perform its catalytic function. AK, a small nucleoside monophosphate kinase, undergoes large domain movements during catalysis (23), because it has to shield its active center from water to avoid ATP hydrolysis. Because the folded forms of AK must be flexible, multiple native forms with small free energy difference can exist in equilibrium in the absence of substrates.
In summary, two rate-limiting steps are involved in the catalytic cycle of AK. Only one form (E 1 ) of AK can bind with AP 5 A, so AP 5 A acts as a competitive inhibitor for the forward reaction and a noncompetitive inhibitor for the backward reaction.