# Kinetic model for protein kinase simultaneous interaction with peptide, ATP, and bi-functional inhibitor/Kineetiline mudel proteiinkinaasi interaktsiooni iseloomustamiseks peptiidi, ATP ja bifunktsionaalse inhibiitoriga.

INTRODUCTIONProtein kinases (E.C.2.7.1.37) catalyse protein phosphorylation. In this reaction the phosphorylatable fragment and the ATP--magnesium complex should be simultaneously bound in the enzyme active centre. On the basis of this fact the concept of bi-substrate inhibitors has been developed in the case of these enzymes [1]. These inhibitors consist of two structural fragments, which resemble the peptide fragment and ATP and are supposed to bind in the appropriate binding sites. For the best fit with the active site these fragments are connected by a linker group of appropriate length [2, 3]. For detailed understanding of the mechanism of the interaction of these bi-functional inhibitors with the enzyme it is important to evaluate the effectiveness of the interaction of these compounds with the free enzyme, but also separately in the binding sites for ATP and peptide and reveal the possibilities of simultaneous binding of these ligands with the protein. This kinetic analysis is, however, a rather complicated task, as the bi-substrate mechanism of the phosphorylation reaction is a complex phenomenon and most probably involves besides the non-covalent binding and catalytic steps also rate-limiting conformational transitions [4]. Moreover, as the overall reaction mechanism seems to be different for different protein kinases [5], the sophisticated methods used for kinetic analysis of the phosphorylation reaction are not handy for the screening of the bi-substrate inhibitors. Therefore we propose a simplified kinetic approach by using the second-order rate constants [k.sub.II] of the peptide phosphorylation reaction, calculated as the ratio of the apparent kinetic parameters [k.sub.cat] and [K.sub.m] [6]. In this paper the kinetic model for protein kinase interaction with peptide, ATP, and bifunctional inhibitor is analysed.

KINETIC SCHEME

The kinetic scheme involves the enzyme complexes with substrates A (ATP) and B (peptide) and with the bi-functional inhibitor I. This model includes the appropriate ternary complexes and for generalization also the multicomplex including the enzyme, inhibitor, and both substrates. The ternary complexes result from the presence of two distinct binding sites for substrates, which should allow various combinations of the enzyme bound ligands. The multicomplex formation is possible if the inhibitor molecule does not fit explicitly the active centre, for example, due to a too short or a too long linker group. Practical data analysis should reveal the feasibility of this complex formation and is one of the objectives of kinetic analysis.

As the following experimental study will be based on the use of protein kinase A, the present reaction scheme takes into consideration also the phenomenon of substrate inhibition by the excess of ATP (substrate A), assuming the formation of both EAA and EAAB complexes. On the other hand, for simplicity the full inhibition scheme is assumed for this reaction. Thus the overall reaction scheme for product (P) formation can be presented as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

It is important to stress that the following analysis will be performed for the second-order rate constants, which significantly simplifies the experimental and analytical procedures.

RATE EQUATION

The rate of the reaction (1) can be presented as follows:

v = [k.sub.cat][EAB]. (2)

Taking into consideration all the equilibriums shown in scheme 1

[K.sub.a] = [E][A]/[EA], (3)

[E] = [K.sub.a]/[A][EA]=[K.sub.a][K.sub.ab]/[A][B][EAB], (4)

[K.sub.b] = [E][B]/[EB], (5)

[EB]=[B]/[K.sub.b][E]=[K.sub.a][K.sub.ab]/[K.sub.b][A][EAB], (6)

[K.sub.ab]=[EA][B]/[EAB], (7)

[EA]=[K.sub.ab]/[B][EAB], (8)

[K.sub.aa] = [EA][A]/[EAA], (9)

[EAA] = [A]/[K.sub.aa][EA]=[K.sub.ab][A]/[K.sub.aa][B][EAB], (10)

[K.sub.aab]=[A]/[K.sub.aab][EAB], (12)

[K.sub.i]=[E][I]/[EI], (13)

[EI]=[I]/[K.sub.i][E]=[K.sub.a][K.sub.ab][I]/[K.sub.i][A][B][EAB], (14)

[K.sub.ai] = [EA][I]/[EAI], (15)

[EAI]=[I]/[K.sub.ai][EA]=[K.sub.ab][I]/[K.sub.ai][B][EAB], (16)

[K.sub.bi]=[EB][I]/[EBI], (17)

[EBI]=[I]/[K.sub.bi][EB]=[K.sub.a][K.sub.b][I]/[K.sub.bi][K.sub.b][A][EAB], (18)

[K.sub.abi]=[EAB][I]/[EABI], (19)

[EABI]=[I]/[K.sub.abi][EAB], (20)

and the distribution of the enzyme between different complexes, [E.sub.0] =

[E]+[EA]+[EB]+[EAB]+[EAA]+EAAB]+[EI][EAI]+[EBI]+[EABI], (21)

the following rate equation can be obtained for the initial rate of substrate phosphorylation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

where V = [k.sub.cat][[E.sub.0]]. It can be seen that for data processing an extensive bank of experimental data is needed, including the dependences of the initial rate upon the concentrations of all three reagents--A, B, and I.

DATA PROCESSING ALGORITHM

The procedure proposed for the analysis of the kinetic scheme (1) is based on the measurement of the initial velocity of the phosphorylation reaction at various concentrations of A, B, and I. For simplicity, these experiments can be grouped so that the concentration of two ligands is fixed and the concentration of one of the ligands is changing. This allows systematic step-by-step data processing as described in the following algorithm.

First, if the concentrations of substrate A (ATP) and inhibitor I remain constant, the plot of the initial velocity vs. substrate B concentration can be analysed by the common hyperbolic function, derived from Eq. 22:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

In this equation the steady-state kinetic parameters [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] depend on the A and I concentration and have the following meaning:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Similarly, if the concentration of substrate B (peptide) and inhibitor I are kept constant within a series of measurements, the rate will depend on the concentration of substrate A, and can be presented as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

The apparent kinetic parameters of this equation, reflecting also the phenomenon of product inhibition by ATP excess [6], have the following meaning:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

From these apparent parameters the following [k.sub.cat]/[K.sub.m] ratios, which are equivalent to the appropriate second-order rate constants and are further denoted as [k.sub.app.sub.II], can be obtained as a function of the inhibitor concentration:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

If the dependence of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is analysed at various inhibitor concentrations, the parameters [[X.sub.i], [Y.sub.i], [Z.sub.i], [Q.sub.i] and [U.sub.i] can be calculated and further used for the determination of the equilibrium constants for binary and tertiary complexes formulated in scheme 1.

ENZYME TERNARY COMPLEXES WITH SUBSTRATES AND BI-FUNCTIONAL INHIBITOR

It can be seen that all the plots obtained from equations for the second order rate constants above are hyperbolic.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

This means that the plots [X.sub.i] vs. [I], [Y.sub.i] vs. [I], [Z.sub.i] vs. [I], [Q.sub.i] vs. [I], and [U.sub.i] vs. [I] can be used and for calculation of the parameters [K.sub.ai] and [[K.sub.bi,] which were the targets of the analysis, as they characterize separately the interaction between the both enzyme-substrate complexes (EA and EB) and the inhibitor molecule, i.e. formation of the ternary complexes EAB, EAI, and EBI.

For characterization of the interaction of the free enzyme with the inhibitor the ratios [Q.sub.i]/[U.sub.i] and [X.sub.i/[Y.sub.i]] obtained at different inhibitor concentrations can be used, as illustrated for the latter case:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

It is understandable that after several transformations the errors of the obtained parameters, [K.sub.i], [K.sub.ai] and [K.sub.bi] may be rather significant. On the other hand, these error values can be kept under control if the amount of the initial data is large enough and the velocity measurements are precise and do not scatter.

FORMATION OF THE EABI COMPLEX

Although the probability of multicomplex EABI formation seems to be low, this should be demonstrated on the basis of experimental data. The necessary analysis can be made proceeding from Eqs. 24 or 27. In the latter case Eq. 27 can be transformed:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

The parameter [W.sub.i] can be calculated at different inhibitor concentrations and the constant [K.sub.abi] defined in the reaction scheme 1 can be obtained from the [W.sub.i] vs. [I] plot:

[W.sub.i]=[k.sub.cat][[E.sub.0]][k.sub.abi]/[k.sub.abi]+[I]. (39)

INFLUENCE OF PRODUCT INHIBITION

The phenomenon of product inhibition, observed in the protein kinase A catalysed phosphorylation reaction of Kemptide, and caused by excess of ATP [6], additionally complicates inhibition analysis. Therefore it was interesting to estimate the influence of the substrate inhibition effect on the data analysis, primarily on the level of the second-order rate constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For this purpose Eq. 30 and its simplified form, where the term {1/[K.sub.aa][1+([I]/[K.sub.ai])}[[A[].sup.2] was omitted, were simulated in Fig. 1. In this simulation the following conditions were used: 0, [I] = 18 [K.sub.a] = [micro]M, and [K.sub.aa] = [micro]M [6]. Substrate A concentration was raised up to 1000 [micro]M, as used in experiments [6]. It can be seen that these theoretical curves overlap at rather low substrate A (ATP) concentrations, and approximately 10% differences can be observed at concentrations above 200 [micro]M. The same conclusion is valid for experiments made in the presence of the inhibitor, as all parameters of Eq. 30 have similar dependence upon I concentration. Thus the phenomenon of product inhibition should be taken into consideration in the analysis of the inhibition data if ATP concentration exceeds 200 [micro]M.

[FIGURE 1 OMITTED]

SUMMARY

The data processing algorithm proposed in this report allows characterization of interactions taking place between separate areas of the protein kinase active centre and the bi-functional inhibitor. First, the constant [K.sub.ai] provides the possibility of characterizing the interaction of the inhibitor molecule with the binding site for peptide, as the ATP binding site is occupied by an ATP--magnesium complex. Similarly, the constant [K.sub.bi] should characterize the inhibitor interaction with the ATP binding site, measured in the presence of the bound peptide. Thirdly, the constant [K.sub.i] characterizes the interaction of the bi-functional inhibitor with the free enzyme and thus should provide information about the fit of the both inhibitor fragments within the active centre. And finally, if the fit is not very good, all the three ligands can be simultaneously bound with the enzyme, forming the multiligand complex EABI. This complex is characterized by the parameter [K.sub.abi], as defined in the reaction scheme 1.

We expect that this analysis of interactions of the enzyme with different ligands is important for understanding the inhibition mechanism and will support the targeted design of bi-functional inhibitors.

ACKNOWLEDGEMENTS

This work was supported by grant GFKOK5214 and the Ministry of Education and Science grant No. 0182592s03.

REFERENCES

[1.] Ricouart, A., Gesquiere, J. C., Tartar, A. & Sergheraert, C. Design of potent protein kinase inhibitors using the bisubstrate approach. J. Med. Chem., 1991, 34, 73-78.

[2.] Medzihradszky, D., Chen, S. L., Kenyon, G. L. & Gibson, B. W. Solid-phase synthesis of adenosine phosphopeptides as potential bisubstrate inhibitors of protein kinases. J. Amer. Chem. Soc., 1994, 116, 9413-9419.

[3.] Loog, M., Uri, A., Raidaru, G., Jarv, J. & Ek, P. Adenosine-5'-carboxylic acid peptidyl derivatives as inhibitors of protein kinases. Bioorg. Med. Chem. Lett., 1999, 9, 1447-1452.

[4.] Lew, J., Taylor, S. S. & Adams, J. A. Identification of a partially rate-determining step in the catalytic mechanism of cAMP-dependent protein kinase: a transient kinetic study using stopped-flow fluorescence spectroscopy. Biochemistry, 1997, 36, 6717-6724.

[5.] Adams, J. A. Kinetic and catalytic mechnisms of protein kinases. Chem. Rev., 2001, 101, 2271-2290.

[6.] Kuznetsov, A., Oskolkov, N., Hansen, M. & Jarv, J. Steady-state kinetic analysis of protein kinase A interaction with peptide and ATP. Proc. Estonian Acad. Sci. Chem., 2003, 52, 165-177.

Aleksei Kuznetsov, Heli Vaartnou-Jarv, and Jaak Jarv *

Institute of Organic and Bioorganic Chemistry, University of Tartu, Jakobi 2, 51014 Tartu, Estonia

Received 16 September 2003, in revised form 13 October 2003

* Corresponding author, jj@chem.ut.ee