Susceptibility and Mössbauer study of temperature-dependent phase transition in biomolecules

]',,T.T-T ~ - ~ ~ Jolinial of

" " " magnetism ,M,H materials

ELSEVIER Journal of Magnetism and Magnetic Materials 150 (1995) 189-196

Susceptibility and MiSssbauer study of temperature-dependent phase transition in biomolecules

Archana Gupta, G.P. Gupta *

Physics Department, Lucknow University, Lucknow 226 007, India

Received 24 March 1994; in revised form 7 March 1995

Abstract

An analysis of temperature-induced phase transition reveals that the energy gap between the two phases is non-linear except in a very small range close to the transition temperature. Susceptibility and Mt~ssbauer data of the model heme complexes [Fe(BaldXNilm)]CB.THF (Bald = Baldwin's capped porphyrins [1], Im = imidazole, CB = carborane, THF = tetrahydrofuran) and [Fe(OEPX3-CNPy)2]C104 (OEP = octaethylporphyrinate, Py = pyridine) in the temperature range 2.1-300 K have been fitted on the basis of the theoretical model developed in this paper.

1. Introduction

Recently susceptibility and MiSssbauer data of a number of synthetic heine complexes have been successfully interpreted in terms of a quantum mechanical admixed state of S = 3 / 2 and 5 / 2 (Maltempo state) [2-5]. In an attempt to fit the data for the complexes [Fe(Bald)(NiIm)]CB.THF and [Fe(OEP)(3-CNPY)2]CIO 4 it was found necessary to consider a temperature-induced phase transition between two such states. The phenomenon of the phase transition is well known and has been studied in a number of heme proteins under the variation of parameters such as temperature, pH, solvent compo- sition, etc. [6-8]. The intermediate magnetic moments of many ferriheme compounds have been quantitatively explained by assuming a temperature- induced transition between high- and low-spin states

* Corresponding author.

[9-18]. The approximate relationship used [19] is given by

n2L + 3/3e-a/krn2 n2 = 1 + 3/3e - '~/kr ' (1)

where n L and n H are the effective Bohr magneton numbers corresponding to the spin states of the two phases, and ct and/3 are two constants related to the temperature dependence of the energy gap between the two phases, which was assumed to be linear. Assuming Eg = E 0 - AT, one may have a = E 0 and /3 = e x / k . The factor 3 has been included on account of the relative multiplicity of spin states.

The detailed analysis presented in this paper shows that the temperature dependence of the energy gap is linear only in a short range of temperatures close to the transition temperature, so that formula (1) may not be appropriate for temperatures far from the transition temperature. The theory developed below is applicable over a wide range of temperatures. The

0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved S S D 1 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 0 1 2 1 - 2

190 A. Gupta, G.P. Gupta /Journal of Magnetism and Magnetic Materials 150 (1995) 189-196

analysis also concludes that the parameter corresponding to A in the above formula corresponds to the anisotropy of the system. The susceptibility and MSssbauer data of the model heme complexes [ F e ( B a l d ) ( N i l m ) ] C B . T H F and [Fe(OEP)(3- CNPy)2]C104 have been interpreted using the revised relationship.

2. T h e o r e t i c a l a n a l y s i s

If we consider ~ as the state parameter, with o-= + 1 corresponding to the two phases, we may write the Hamiltonian as

N z

H 0 = - J Y'~ • o-i~, (2) i=l j=l

where J is a contact parameter, N is the number of active sites in the system, and z is the number of nearest neighbours in contact with a given site. Further, if we consider ( o- ) = X [20] and by impli- cation (1 :1 :X) /2 as the probabilities of the system being in phases A and B, the energy and entropy of the system may be given as

1 U = ( H o) = - - teJzX: , (3)

2

( 12X. . 1 - X l + X l n l 2 X) S = - N k . In ~- + 2 , (4)

respectively. The condition that free energy (U - TS) is minimum, provides

X X = tanh T/T~ (5)

with

T~ = Jz /k . (6)

Eq. (5), depicted in Fig. 1, shows that both X = -I- 1 are the ground states of the system at T = 0 K. As the temperature is increased X approaches zero and at T = T~ the system is completely in a disor- dered state, i.e. it has an equal probability of being in either phase A or phase B.

In order to assign a physical meaning to the parameter X, let us consider that the energies of the two phases A and B are - E g / 2 and Eg/2, respectively. Following Boltzmann's statistics the probabilities of occupation of the two phases may be related as

1 - X

2

I + X

e-Eg/2kT

= e_eg/2kr + eE,/2kr ,

e+Es/2kT

2 = e-E8/2kT _]_ eeg/2kr '

which provide

X = tanh(Eg/2kT) .

Comparing Eqs. (5) and (9) we have

X = Eg/2kT~,

(7)

(8)

(9)

(lo)

I.C

O.~

' 0

- 0 . 5

-I .O

1.0

O.~

?

-O. 5

- I .0

PHASE B P H A S E

, i

)HASE A PHASE

i i i

Fig. 1. Variation of X with temperature, based on Eq. (5). Fig. 2. Variation of the energy gap with temperature.

A. Gupta, G.P. Gupta /Journal of Magnetism and Magnetic Materials 150 (1995) 189-196 191

which reveals that the energy gap Eg between the two phases A and B is a function of temperature and the behaviour may be described as shown in Fig. 2.

Mathematically, the temperature must be zero at both ends of the temperature scale since only then can the ground state be fully occupied according to Boltzmann's statistics. However, if we consider the other extreme end as 2T c and assume that physically this is the ground state for phase B, the variation of X from - 1 to + 1 corresponds to the transition from phase A to phase B in the temperature range 0-2T~. The temperature dependence may be described as shown in Fig. 3. It may be noted that the curve shown in the figure is linear only in a short range of temperatures close to the transition temperature. Once this behaviour is known any physical property of the system, say X, as a function of temperature may be evaluated using the relation

1 - X 1 - X X=--5- -XA + - - - T X B. ( i l /

So far, according to the Hamiltonian (2), both of the states are equally probable and the contacts may be broken both ways so as to switch over from phase A to phase B. However, in a real physical system there may be some anisotropy energy involved in both phases, i.e. if the system is in one phase it prefers to remain in that phase. In order to incorpo-

t.C

O.5o 0.5 1.5 2.0 x

-0.5 ~ TITc

-I.0 i i I

Fig. 3. Variation of X over the temperature range 0-2Tc, where (I + X)/2 are the probabilities of phases A and B, respectively, at different temperatures.

1.0

o z 2 0

-O.~

°1.0 i i i

Fig. 4. Variation of X with temperature in the presence of anisotropy. X' and X" are the components of X corresponding to phases A and B, respectively.

rate this effect let us include an anisotropy term in the Hamiltonian (2) as

N

H = H o + K E or,, (12) i=1

where + K are the anisotropy constants corresponding to the two phases A and B, respectively. In this case the internal energy U will be given by

NzJX 2 U = ( H ) = 2 +NKX. (13)

The minimum free energy ( U - TS) condition gives the most probable value of X satisfying the equation

zJX T- K X = tanh k'---~ (14)

Taking Eq. (14) into consideration, the two parts of the curve X shown in Fig. 3 are modified as X' and X" shown in Fig. 4. In this situation, when the temperature corresponds to the ground state for one phase the probability of occupation for the other phase does not reduce to zero. Under these circum- stances the two values of X' and X" may be added to obtain the net value X (also shown in Fig. 4), which in turn may be used to evaluate a physical property using relation (11).

It may be noted that the anisotropy for the two phases need not necessarily be the same as has been assumed above, and consequently the curve X shown in Fig. 4 may not be symmetric about T~.

192 A. Gupta, G.P. Gupta / Journal of Magnetism and Magnetic Materials 150 (1995) 189-196

3. Analysis of susceptibility data

The experimental value of Nef f varies from 3.4 to 5.33 over the temperature range 6-300 K for the complex [Fe(OEP)(3-CNPy) 2 ]CIO 4 and from 3.52 to 5.3 in the temperature range 2.1-300 K for the complex [Fe(Bald)(Nilm)]CB.THF. The data suggest that the system may be in a quantum-mechanical admixed spin state of S = 3 / 2 and 5 /2 . The model was first suggested by Maltempo [21,22] and has been successfully used to explain the susceptibility data of several synthetic model heme complexes [2-5]. In spite of extensive efforts, the data of both complexes could not be fitted until a temperature-induced phase change in the system was taken into consideration. Accordingly, the data were fitted using Eq. (11). Nef f values for each of the two phases A and B were calculated in Maltempo approximation using the electronic Hamiltonian

= a + E ti . s , + 2t H. s (15) i

with ten IS, M~) components of t h e 6A I sextet and the 4A 2 quartet being the basis states. Here A is the energy gap between the S = 3 / 2 and 5 / 2 spin states. The second term represents the spin-orbit interaction which is responsible for mixing spin states. The summation extends over the individual spin-orbit interactions of all five 3d electrons of the Fe ion. ~" is the one-electron spin-orbit coupling constant (normally expected to be 300 cm- I ). The last term is the Zeeman interaction. Since the total orbital magnetic moment is zero in both the 6A 1 and 4A 2 states, the term does not include individual orbital magnetic moments of 3d electrons. An electrostatic energy term which is constant for the 6A l and 4A 2 states has been omitted from the expression. This Hamilto- nian can be represented by a 10 × 10 matrix. The matrix elements of spin-orbit coupling are already known [21]. The only non-zero elements are - ~'(6/5) 1/2 between 16Al + 1 /2 ) and 14Az _+ 1 /2 ) states, and -~(4/5) l/z between 16A1 + 3 / 2 ) and 14A2 + 3 / 2 ) states. The g . values in small applied fields can be related to the 6A~ - 4A z separation A by the standard method of diagonalising the electronic Hamiltonian in zero field and then perturbing

.i. 4-

' ' ~ ' ' ~ ' ' 5,2 Temperature ( K ) ' - "

Fig. 5. Experimental values ( + ) of Nat as a function of temperature for the complex [Fe(BaldXNilm)]CB.THF. Solid curve: calculated values.

the eigenstates with a small applied magnetic field. The result is:

A_---t-g" - 5 ( g ± _ 6 ) ( g ± _ 4 ) (16)

with the lower sign applying for g ± < 5. Normally g j. is proportional to the magnetic

splitting of the ground doublet and may be directly measured by EPR. It may be considered simply as a parameter indicating the percentage of spin state S = 3 / 2 in the admixed spin state, g . is linearly related to this percentage with boundary values 0% and 100% corresponding to g ± = 6 and 4, respectively. An excellent agreement between the theoretical and experimental data was achieved by optimis- ing the parameters g ±, ~" and K for both phases and their transition temperature. The values of these parameters could be estimated almost independently as they affect the shape of the susceptibility curve in different ways and the effect of one could not be compensated by that of others. The effective numbers of Bohr magnetons, N~f e are plotted versus temperature in Fig. 5 for the complex [Fe(Bald) (NiIm)]CB.THF and in Fig. 6 for the complex [Fe(OEP)(3-CNPy) 2 ]CIO 4. In general, N~ff decreases very rapidly at low temperatures when zero-field splitting effects become important. In order to see this behaviour in detail the temperature scale in Figs.


Temperature (K)-'="

Fig. 6. Experimental values ( + ) of Nefr as a function of temperature for the complex [Fe(OEPX3-CNPy)2]CIO a. Solid curve: calculated values.

Table 1 Values of g l , ~ and anisotropy constants for phases A and B, and the transition temperatures corresponding to fittings shown in Figs. 5 and 6

[Fe(OEPX3-CNPy) 2 ] [Fe(BaldXNilm)]CB. C104 THF

Transition 125 K 150 K temperature Phase A

g ± 4.5 4.8 220 c m - t 200 c m -

K 0,2 0,2 Phase B

g± 5,3 5.2 r 250 c m - i 250 c m - I K 0.1 0,1

4.1. MSssbauer spectra in zero field

5 and 6 has been chosen to be logarithmic. The magnetic susceptibility measurements are consistent with a transition from a quantum-mechanical admixed state A to another quantum-mechanical admixed state B. The values of various parameters for both phases are summarised in Table 1. A graph of the percentages of phases A and B at different temperatures is shown in Fig. 7. Interestingly, the anisotropies obtained for the two phases (0.2 and 0.1, respectively) are very small, which is why at two extreme temperatures the system is 95% in phase A or phase B. It may also be pointed out that these systems are ideal for the study of phase trans- formations, as the transition temperatures are nearly 125 and 150 K for the two compounds and the data are available from low temperatures up to 2T c where complete phase reversal is achieved.

4. Analysis of Mfissbauer data

In order to confirm the spin states investigated in the susceptibility data, MiSssbauer analyses of the two samples [Fe(OEP)(3-CNPy)2]C104 and [Fe (Bald)(Nilm)]CB.THF were undertaken [23]. The values of g j, and ~ for phase A obtained from the susceptibility data were used to fit the M6ssbauer data. The Mtissbauer spectra were recorded at several temperatures and in the presence of an external magnetic field, as discussed below.

MiSssbauer spectra of the complexes were recorded in zero field at 1.54, 4.2 and 77 K. The quadrupole doublets observed in both cases were fitted by opti- mizing the Lorentzian line width F, isomer shift 8, and quadrupole splitting A EQ. The large quadrupole splitting in both cases is very suggestive of its admixed spin state [4]. The large line width at 77 K in the case of [Fe(BaldXNilm)]CB.THF was attributed to relaxation effects, which increased to a large extent as the temperature was lowered to 1.54 K. The values of the various parameters taken for the theoretical fits of the Mossbauer spectra are given in Table 2.

IOO.C! . . . . . . . . .

I

i 50.0

I 0 .01 t I i J t i = I I

0.0 150.0 300.0 Temperature ( K )

Fig. 7. Percentages of phase A and phase B at different temperatures corresponding to anisotropy constants 0.2 and 0.1 and transition temperature 125 K.


Table 2 Values of the parameters used to simulate the M~Sssbauer spectra shown in Figs. 8 and 9

[Fe(OEPX3-CNPy) 2 ]CIO 4 [Fe(BaldXNilm)]CB. THF

Field (T) 0 0 6 0 6 Temperature (K) 77 4.2 4.2 77 4.2 g± 4.5 4.5 4.5 4.8 4.8 ~" (cm- l) 220 220 220 200 200 8 (mm/s) 0.38 0.36 0.36 0.35 0.35 A EQ (mm/s) 2.61 2.66 2.67 2.32 2.50 PJgN fin - - --2.88 - --2.98 (mm/s/unit spin) K - - 0.35 - 0.35

4.2. M 6 s s b a u e r spectra in a magnet ic f i e ld

Figs. 8 and 9 show the M/Jssbauer spectra recorded in an external magnetic field o f 6 T at 4.2 K for the two samples. Each spectrum represents an average over all possible orientations o f the molecule with reference to the mutually orthogonal gamma-ray and external field directions. For each orientation the

absorption spectrum reflects the net effect o f the interaction of the nucleus with the electric field gradient and with the sum of the applied and internal fields. The spectra were fitted by solving the electronic and nuclear Hamiltonians separately using a computer program described elsewhere [24]. This method holds good i f the applied fields are strong enough to decouple electronic and nuclear spins, i.e.

77K 0T

V V 1.54K 0T

~ -= ~ ~ .-- 4.2K 6T

-7 -5 -3 -1 I 3 5 7

Velocity (mm/s)

Fig. 8. Experimental M~ssbauer spectra (-) of [Fe(Bald)(Nilm)]CB.THF in an external field and temperatures as shown. Solid curves: calculated values with the parameters listed in Table 2. The spectrum at 0 T, 1.54 K has not been fitted due to intermediate relaxation effects.


greater than 1 mT or so. Assuming axial symmetry, the electronic Hamiltonian (15) was diagonalized for 20 equally weighted orientations of the molecule with reference to the applied field in order to carry out the simulations for a polycrystalline sample, and to determine the statistical weight, spin expectation value and effective internal magnetic field of each electronic state. The three components of the hyperfine field for each energy level are calculated as the expectation values of

H.,y = ( P / g N fiN)( a~/2 + K )Sx, r,

Hz = ( P / g N f iN)( a / 2 + K) S z, (17)

where P is the hyperfine constant and c~ is the dipolar contribution, which is zero for 6A~ states and 8/21 for 4A 2 states [25]. K, the contact constant, was taken to be 0.35 for all states [26]. The matrix elements of the orbital contribution to the hyperfine field are zero.

Once the electronic Hamiltinian was solved and the expectation value of the internal hyperfine field was known, the nuclear energy levels which deter-

mine the MiSssbauer spectra were calculated using the nuclear Hamiltonian

"~n ~- --gN ~N I" (napp '[- (H))

+ QVzz/4{IZz - I( I + 3) /3} , (18)

where I = 3 / 2 for the excited nuclear state of 57Fe and gN is the excited state nuclear gyromagnetic ratio and is equal to -0.1033. /3 N is the nuclear magneton, Q is the nuclear quadrupole moment, and we have assumed an axially symmetric quadrupole interaction. For the ground state I = 1/2, g~ is replaced by gN = 1"749gN and Q = 0. A starting assumption which is justified a posteriori by the success in simulating the experimental spectra, is that the axis of quantization of the 4A 2 quartet coincides with the z direction determined by the electric field gradient. Details of the method of determining the M~Sssbauer spectra are described elsewhere [26]. For simulations which are made in the slow relaxation rate limit, the net calculated MiSssbauer spectrum is the Boltzmann weighted sum

77K 0T

4 . 2 K 0 T

4.2K 6T

-7 -5 -3 -1 1 3 5 7

Velocity (ram/s) Fig. 9. Experimental MSssbauer spectra ( - ) of [Fc(OEP)(3-CNPy)2]CIO 4 in an external field and temperatures as shown. So]id curves: calculated values with the parameters listed in Table 2.


of spectra, one from each electronic energy level. These are then summed over a range of field directions in order to produce the powder spectrum. If the electronic spin states relax rapidly, the MiSssbauer spectrum for each molecular orientation is determined by the thermal averaged electron spin and the resulting effective magnetic field. The computer simulations suggest that the spins fluctuate in the slow relaxation limit when the magnetic field is 6 T at 4.2 K. Both the 6 T spectra shown in Figs. 8 and 9 could be fitted satisfactorily with the values of g .L and ~" obtained from the analysis of susceptibility data. The small deviations in the case of the spectrum in Fig. 8 may be attributed to the presence of some relaxation, even in the field of 6 T.

Acknowledgements

One of the authors (AG) is grateful to the Council of Scientific and Industrial Research for financial assistance. We are thankful to Professor W. R. Scheidt of University of Notre Dame, USA, and Professor C.A. Reed of the University of Southern California for providing the paramagnetic susceptibility data.

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Susceptibility and Mössbauer study of temperature-dependent phase transition in biomolecules