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Discontinuous phase transitions in enzyme density of states
Volume 84A, number 1
PHYSICS LETTERS
DISCONTINUOUS PHASE TRANSITIONS
6 July 1981
IN ENZYME DENSITY OF STATES*
M.A.F. GOMES Departamento
de Ffsica, Universidade Federal de Pernambuco,
50000 Recife, PE, Brazil
and
R.C. FERREIRA Centro Brasileiro de Pesquisas Fisicas, 22290 Rio de Janeiro, RJ, Brazil
Received 14 April 1981
We show that the high efficiency of enzymes as biological catalysts may be due to discontinuous increase in the density of occupied states of these macromolecules as a consequence of conformational changes.
According to Koshland’s induced-fit theory a considerable change in the conformation of an enzyme is induced by the substrate molecule as it fits into the active site of the enzyme [ 11. This theoretical model is now firmly established from the evidence of various experimental techniques [2-4]. These changes in the nuclear framework of the enzyme must be accompanied by modifications in the electronic levels. In small molecules the electronic eigenvalues are altered in different degrees by a change in the molecular geometry [S] , and critical effects can be observed, such as the appearance of optical rotatory power in excited allene molecules [6]. Calculations of Kleyer and Lipscomb [7] show notable changes in the density of valence orbitals for different conformers of larger molecules such as tetraglycine. The same behaviour should occur in proteins, whose molecules have band structures [8]. In crystals [9] and linear polymers [ 10,l l] reducing the short-range order (i.e. producing a local conformational change) causes the appearance of localized states in “forbidden” energy gaps. The distribution of the electronic states of enzyme molecules plays a fundamental role in a recently proposed perturbation theory of enzymatic catalysis [ 12,131. In accordance with this model the eLUMO+ - eHOMO* energy barrier between reactants 01and * Work supported in part by CNPq. 36
fl (connected to the activation energy of the process) is lowered by a perturbation potential caused by certain enzyme valence electrons. These interactions are critically enhanced by the changed electron distribution produced in response to the distortions resulting from the substrate collision. We have shown [ 131 that changes in the density of states such as those calculated by Kleyer and Lipscomb [7] could explain the observed enzymatic rate enhancements of 108-1012 [14,15]. Typical enzyme-substrate collision times are = lo-l2 s and we have also proposed [ 131 that this effect could be detected by picosecond techniques in the UV spectra of enzymes. Enzymes act as macromolecules in solution and, less often, bound to other polymeric structures such as membranes. It seems important, therefore, to describe the enzymatic processes in terms of concepts and parameters of condensed-matter physics. We represent the enhancement ratio k,/k, by an expression of Arrhenius type: k&N
=f(&R)
=
ew-Nt,RYkBTl,
where $J(,$,R) is the catalytic perturbation @($,R) = [de
P(e, 4‘)E(e,R)
>
(1) energy (2)
P(E, E) is the density of occupied electronic states of the enzyme at the energy level E and conformational 0 03 l-9 163/g l/0000-0000/S
02.50 0 North-Holland
Publishing
Company
Volume 84A, number 1
PHYSICS LETTERS
6 July 1981
*
coordinate [.E(e,R) is the second-order energy correction term [ 131 of LUMOa due to the perturbative interaction of this orbital with the molecular orbital of the enzyme at energy e, when the intermolecular distance enzyme-substrate is R. Since E(E, R) is negative [ 131 ,f(& R) > 1. The integral in (2) runs over the valence band of the enzyme; if e. stands for the lower valence-band edge,
4-
P
...
so--
““‘“:
__??-A .........
NZ,E)=jdep(e,E) EO
is the number of occupied orbitals between eO and7. We can define a lower boundNO above whichf(t,R) can assume a value 7 such that NO = - [knT/E(e,
R)] 1nT ,
where E(e, R) is the mean value ofB(e, R). We have calculated [ 161 that E(E, R) = -0.01 eV; at physiological temperatures (kBT= 0.025 eV) and putting 7% 108-1012, the lower boundNo falls in the range of SO-70 states (orbitals). Since only the valence electrons of the enzyme are catalytically active, these numbers show that the enzyme action requires at least the cooperative action of 2-4 amino-acid residues. ’ As written in (2) f(t, R) depends on an external variable R - the enzyme-substrate intermolecular distance - and an order parameter C;(a certain conformational coordinate in the macromolecule as bond angle, bond length or similar). In terms of critical phenomena we will assume that for R > R,, [ = 0, and for R CR,, ,ij= to, where R, is a critical intermolecular distance. #Q, R) can be considered as a pseudofree energy for the enzyme-substrate system, and can be written in terms of a power series of 5: $(&R)=;a.$2+$b[3+$cC;4,
(4)
under the assumption that C#J has a extremum at E = 0 and b # 0 due to the lack of symmetry of inversion in enzyme molecules. In (4) we assume #“‘(O; R)/2! E b < 0, $““(O, R)/3! f c > 0. As in the Landau theory of phase transitions we use $“(O; R) E a = A (R - Ro) with-4 > 0, i.e. this coefficient p.asses through zero at some distance R,. For large R, a > 0 and the minimum of Q lies at [ = 0. When the intermolecular distance R is reduced, the local minimum first remains at &!= 0. Lowering the distance further we obtain a new (global) minimum of 0 at 6 = to # 0. In analogy with the case
%
R
Fig. 1. Heavy line: order parameter (conformational coordinate) versus intermolecular distance R. Dotted line: the discontinuous increase in the density of occupied orbitals.
of a first-order phase transition, the conformational coordinate (order parameter) [ changes discontinuously at the critical intermolecular separation R,, from 6 =O (disordered state) to t = to (ordered state). The pseudo-entropy S = -&@R is different in the two states r; = 0 and E = to at R = R,, with S(0, R,) S(to, R,) > 0. We can see that the discontinuous change in the order parameter leads to an exponential increase in&R) through the density of states P(E,~) using expressions (2) and (4):
=Jde
( aw)R,
Me, go) - P(C
WI> 0 *(5)
As aE/aR > 0 [ 161, expression (5) implies a discontinuous increase in the density of electronic states’ when we go from conformation E = 0 to conformation 5 = En (fig. 1). References [l] D-E. Koshland Jr., Proc. Natl. Acad. Sci. USA 44 (1958) 98. [2] D.H. Buttlaire and M. Cohn, J. Biol. Chem. 249 (1974) 5733. [3] B.A. Peters and KE. Neet, J. Biol. Chem. 253 (1978) 6828. [4] R. McDonald, T.A. Steitz and D.M. Engelman, Biochemistry 18 (1979) 338. [5] A.D. Walsh, J. Chem. Sot. (1953) 2260.
37
Volume 84A, number 1
PHYSICS LETTERS
[6] A. Rauk, A.F. Drake and SF. Mason, J. Am. Chem. Sot. 101 (1979) 2284. [7] D.A. KIeyer and W.N. Lipscomb, Int. J. Quantum Chem. Quantum Biol. Symp. 4 (1977) 73. [8] S. Suhai, J. Kaspar and J. Ladik, Int. J. Quantum Chem. 17 (1980) 995, and references therein. [9] G.F. Koster and J.C. Slater, Phys. Rev. 95 (1954) 1167; 96 (1954) 1208. [lo] D.A. Morton-Blake, Theor. Chim. Acta 56 (1980) 93; Int. J. Quantum Chem. 18 (1980) 937. [ 1 l] B. KoiIler, H.S. Brandi and R.C. Ferreira, submitted to Theor. Chim. Acta.
38
6 July 1981
[12] M.A.F. Gomes, A.A.S. Gama and R.C. Ferreira, Chem. Phys. Lett. 57 (1978) 259; M.A.F. Gomes and R.C. Ferreira, Phys. Lett. 77A (1980) 384. [ 131 M.A.F. Gomes and R.C. Ferreira, submitted to Chem. Phys. [ 141 D.E. Koshland Jr., J. Cell. Comp. Physiol. 47 Suppl. 1 (1956) 217. [ 151 W.P. Jencks, Catalysis in chemistry and enzymology (McGraw-Hill, New York, 1969). [ 161 M.A.F. Gomes, PhD Thesis, Physics Department, Universidade Federal de Pernambuco, Brazil (1980).
PHYSICS LETTERS
DISCONTINUOUS PHASE TRANSITIONS
6 July 1981
IN ENZYME DENSITY OF STATES*
M.A.F. GOMES Departamento
de Ffsica, Universidade Federal de Pernambuco,
50000 Recife, PE, Brazil
and
R.C. FERREIRA Centro Brasileiro de Pesquisas Fisicas, 22290 Rio de Janeiro, RJ, Brazil
Received 14 April 1981
We show that the high efficiency of enzymes as biological catalysts may be due to discontinuous increase in the density of occupied states of these macromolecules as a consequence of conformational changes.
According to Koshland’s induced-fit theory a considerable change in the conformation of an enzyme is induced by the substrate molecule as it fits into the active site of the enzyme [ 11. This theoretical model is now firmly established from the evidence of various experimental techniques [2-4]. These changes in the nuclear framework of the enzyme must be accompanied by modifications in the electronic levels. In small molecules the electronic eigenvalues are altered in different degrees by a change in the molecular geometry [S] , and critical effects can be observed, such as the appearance of optical rotatory power in excited allene molecules [6]. Calculations of Kleyer and Lipscomb [7] show notable changes in the density of valence orbitals for different conformers of larger molecules such as tetraglycine. The same behaviour should occur in proteins, whose molecules have band structures [8]. In crystals [9] and linear polymers [ 10,l l] reducing the short-range order (i.e. producing a local conformational change) causes the appearance of localized states in “forbidden” energy gaps. The distribution of the electronic states of enzyme molecules plays a fundamental role in a recently proposed perturbation theory of enzymatic catalysis [ 12,131. In accordance with this model the eLUMO+ - eHOMO* energy barrier between reactants 01and * Work supported in part by CNPq. 36
fl (connected to the activation energy of the process) is lowered by a perturbation potential caused by certain enzyme valence electrons. These interactions are critically enhanced by the changed electron distribution produced in response to the distortions resulting from the substrate collision. We have shown [ 131 that changes in the density of states such as those calculated by Kleyer and Lipscomb [7] could explain the observed enzymatic rate enhancements of 108-1012 [14,15]. Typical enzyme-substrate collision times are = lo-l2 s and we have also proposed [ 131 that this effect could be detected by picosecond techniques in the UV spectra of enzymes. Enzymes act as macromolecules in solution and, less often, bound to other polymeric structures such as membranes. It seems important, therefore, to describe the enzymatic processes in terms of concepts and parameters of condensed-matter physics. We represent the enhancement ratio k,/k, by an expression of Arrhenius type: k&N
=f(&R)
=
ew-Nt,RYkBTl,
where $J(,$,R) is the catalytic perturbation @($,R) = [de
P(e, 4‘)E(e,R)
>
(1) energy (2)
P(E, E) is the density of occupied electronic states of the enzyme at the energy level E and conformational 0 03 l-9 163/g l/0000-0000/S
02.50 0 North-Holland
Publishing
Company
Volume 84A, number 1
PHYSICS LETTERS
6 July 1981
*
coordinate [.E(e,R) is the second-order energy correction term [ 131 of LUMOa due to the perturbative interaction of this orbital with the molecular orbital of the enzyme at energy e, when the intermolecular distance enzyme-substrate is R. Since E(E, R) is negative [ 131 ,f(& R) > 1. The integral in (2) runs over the valence band of the enzyme; if e. stands for the lower valence-band edge,
4-
P
...
so--
““‘“:
__??-A .........
NZ,E)=jdep(e,E) EO
is the number of occupied orbitals between eO and7. We can define a lower boundNO above whichf(t,R) can assume a value 7 such that NO = - [knT/E(e,
R)] 1nT ,
where E(e, R) is the mean value ofB(e, R). We have calculated [ 161 that E(E, R) = -0.01 eV; at physiological temperatures (kBT= 0.025 eV) and putting 7% 108-1012, the lower boundNo falls in the range of SO-70 states (orbitals). Since only the valence electrons of the enzyme are catalytically active, these numbers show that the enzyme action requires at least the cooperative action of 2-4 amino-acid residues. ’ As written in (2) f(t, R) depends on an external variable R - the enzyme-substrate intermolecular distance - and an order parameter C;(a certain conformational coordinate in the macromolecule as bond angle, bond length or similar). In terms of critical phenomena we will assume that for R > R,, [ = 0, and for R CR,, ,ij= to, where R, is a critical intermolecular distance. #Q, R) can be considered as a pseudofree energy for the enzyme-substrate system, and can be written in terms of a power series of 5: $(&R)=;a.$2+$b[3+$cC;4,
(4)
under the assumption that C#J has a extremum at E = 0 and b # 0 due to the lack of symmetry of inversion in enzyme molecules. In (4) we assume #“‘(O; R)/2! E b < 0, $““(O, R)/3! f c > 0. As in the Landau theory of phase transitions we use $“(O; R) E a = A (R - Ro) with-4 > 0, i.e. this coefficient p.asses through zero at some distance R,. For large R, a > 0 and the minimum of Q lies at [ = 0. When the intermolecular distance R is reduced, the local minimum first remains at &!= 0. Lowering the distance further we obtain a new (global) minimum of 0 at 6 = to # 0. In analogy with the case
%
R
Fig. 1. Heavy line: order parameter (conformational coordinate) versus intermolecular distance R. Dotted line: the discontinuous increase in the density of occupied orbitals.
of a first-order phase transition, the conformational coordinate (order parameter) [ changes discontinuously at the critical intermolecular separation R,, from 6 =O (disordered state) to t = to (ordered state). The pseudo-entropy S = -&@R is different in the two states r; = 0 and E = to at R = R,, with S(0, R,) S(to, R,) > 0. We can see that the discontinuous change in the order parameter leads to an exponential increase in&R) through the density of states P(E,~) using expressions (2) and (4):
=Jde
( aw)R,
Me, go) - P(C
WI> 0 *(5)
As aE/aR > 0 [ 161, expression (5) implies a discontinuous increase in the density of electronic states’ when we go from conformation E = 0 to conformation 5 = En (fig. 1). References [l] D-E. Koshland Jr., Proc. Natl. Acad. Sci. USA 44 (1958) 98. [2] D.H. Buttlaire and M. Cohn, J. Biol. Chem. 249 (1974) 5733. [3] B.A. Peters and KE. Neet, J. Biol. Chem. 253 (1978) 6828. [4] R. McDonald, T.A. Steitz and D.M. Engelman, Biochemistry 18 (1979) 338. [5] A.D. Walsh, J. Chem. Sot. (1953) 2260.
37
Volume 84A, number 1
PHYSICS LETTERS
[6] A. Rauk, A.F. Drake and SF. Mason, J. Am. Chem. Sot. 101 (1979) 2284. [7] D.A. KIeyer and W.N. Lipscomb, Int. J. Quantum Chem. Quantum Biol. Symp. 4 (1977) 73. [8] S. Suhai, J. Kaspar and J. Ladik, Int. J. Quantum Chem. 17 (1980) 995, and references therein. [9] G.F. Koster and J.C. Slater, Phys. Rev. 95 (1954) 1167; 96 (1954) 1208. [lo] D.A. Morton-Blake, Theor. Chim. Acta 56 (1980) 93; Int. J. Quantum Chem. 18 (1980) 937. [ 1 l] B. KoiIler, H.S. Brandi and R.C. Ferreira, submitted to Theor. Chim. Acta.
38
6 July 1981
[12] M.A.F. Gomes, A.A.S. Gama and R.C. Ferreira, Chem. Phys. Lett. 57 (1978) 259; M.A.F. Gomes and R.C. Ferreira, Phys. Lett. 77A (1980) 384. [ 131 M.A.F. Gomes and R.C. Ferreira, submitted to Chem. Phys. [ 141 D.E. Koshland Jr., J. Cell. Comp. Physiol. 47 Suppl. 1 (1956) 217. [ 151 W.P. Jencks, Catalysis in chemistry and enzymology (McGraw-Hill, New York, 1969). [ 161 M.A.F. Gomes, PhD Thesis, Physics Department, Universidade Federal de Pernambuco, Brazil (1980).