JOURNAL OF T H E AMERICAN CHEMICAL SOCIETY 0 Copyright 1985 by the American Chemical Society
VOLUME107, NUMBER 20
OCTOBER 2, 1985
Fractal Models of Protein Structure, Dynamics, and Magnetic Relaxation Gerald C. Wagner,*+J. Trevor Colvin,$James P. Allen,$ and Harvey J. Stapleton Contribution f r o m the Departments of Physics and Biochemistry, University of Illinois at Urbana-Champaign, Urbana, Illinois 61 801. Received February 25, 1985
Abstract: A geometric model is presented to interpret the anomalous T3+*"'temperature dependence of the Raman spin-lattice relaxation rates in heme and iron-sulfur proteins. Analysis of relaxation data is based on a modified Debye relationship between the spectral exponent m and the density of vibrational states p(u) a PI,where 0 < v < v,. Magnetic relaxation measurements on cytochrome c-551 and putidaredoxin yield noninteger values of m that are influenced by changes in the ionic medium. The apparent physical significance of m is revealed, in part, by correlation to a pcotein's fractal g_eometry,which characterizes a repeating structural motif by a single parameter called the fractal dimension d . Estimates of d for 70 proteins are computed by a method that identifies geometric and statistical self-similarities of a-carbon coordinates; values range within the limits (1 4 d 4 2) of well-defined test structures and correlate principally with dominant elements of secondary structyres. In six iron proteins, the highest values of m derived from relaxation data are approximated by the estimated values of d calculated from the covalent structure. The interrelationship between the fractal models of protein structure and molecular dynamics, Le., m = d, is also evident in the good agreement between the predicted p ( u ) a 8' and the reported distribution of low-frequency normal modes ( u < 75 cm-I) calculated for bovine pancreas trypsin inhibitor. The present findings indicate d defines a fundamental parameter that is inherent to both the structural and dynamic properties of a protein.
In earlier works we reported an anomalous temperature dependence of the Raman electron spin-lattice relaxation rates in various iron-containing The well-known Raman mechanism is an inelastic, two-phonon-scattering process that balances the magnetic energy of a spin flip, gPH, of a paramagnetic ion with the change in vibrational energy h ( u l - v2) between lattice modes. At sufficiently low temperatures, the integral expression for the Raman relaxation rate, 1/ T I ,reduces to a simple 7" power law in temperature. The value of n depends strongly upon the functional form of the density of vibrational states p(u),'s2 which is defined as the number of vibrational states p(v) dv within the frequency interval u and v + dv. We asserted previously that the general form of p ( v ) is a power law relationship2 p(u)
0:
0 4u
um-'
< u,
(1)
provided the velocity of sound us is independent of u , and ~ the Debye cutoff frequency umaXis chosen to make the total number of vibrational modes equal to the degrees of freedom within the system, Le., p ( u ) = 0 for u > umax. This form of p ( u ) results in I/T,(Raman)
a
7"
T3f2m
(2)
for ions with an odd number of unpaired electrons (Kramers' ions). The spectral exponent m corresponds normally to the Euclidean dimension d of the space filled by the periodic lattice structure of a paramagnetic salt. As m = d = 3, eq 1 and 2 reduce to the 'Present address: Department of Physics, University of Illinois, Urbana. !Present address: Hughes Research Laboratories, Malibu, CA 90265. 8 Present address: Department of Physics, University of California at San Diego, La Jolla, C A 92093.
0002-7863/S5/1507-5589$01.50/0
well-known expressions in solid-state theory of a u2 density of Raman rate. In heme and ironsulfur vibrational states and a proteins,'s2 the temperature dependence of the relaxation rates also follows a 7" relationship, which is consistent with a simple power law for p ( v ) . The temperature exponents n, however, are noninteger values, well below 9, that correspond to spectral exponents between one and two. The noninteger values of m, by analogy, imply lattice structures of non-Euclidean or fractional dimension. Structures of fractional dimension are known. Mandelbrot4s5 has pioneered the theoretical concepts and physical applications of this relatively new field of geometry6,' and has popularized the term fractal for a structure characterized by a fractional dimension. By definition, any structure possessing a self-similar or repeating motif that is invariant under a transformation of scale may be represented by a fractal dimension. Self-similarity is geometric in regular structures; in random or irregular objects, self-similarity is primarily statistical in nature. The average (root mean square, RMS) end-to-end length R of an unbranched polymer chain constitutes a statistically self-similar property. The (1) Stapleton, H. J.; Allen, J . P.; Flynn, C. P.; Stinson, D. G.; Kurtz, S. R. Phys. Rev.Lett. 1980, 45, 1456-1459. (2) Allen, J. P.; Colvin, J. T.; Stinson, D. G.; Flynn, C. P.; Stapleton, H. J. Biophys. J . 1982, 38, 299-310. (3) Kittel, G. "Introduction to Solid State Physics", 5th ed.; Wiley & Sons, Inc.: New York, 1976; pp 131-136. (4) Mandelbrot, B. B. "Fractals: Form, Chance, and Dimension"; W. H. Freeman & Co.: San Francisco, 1977. (5) Mandelbrot, B. B. "The Fractal Geometry of Nature"; W. H. Freeman & Co.: San Francisco, 1982. (6) Gardner, M. Sci. Am. 1978, 238(4), 16-32. (7) McDermott, J. Smithsonian 1983, 14, 110-1 17.
0 1985 American Chemical Society
5590 J . Am. Chem. Soc.. Vol. 107, No. 20, 1985
Wagner et al.
SECOND STAGE
F O U R T H STAGE
GENERATOR
0
R/4
2R/4
3R/4
R
Figure 1. A fractal structure illustrating properties relevant to a polypeptide. Fractals with geometric self-similarity are comprised of two elements-an initiator, which in this example is a straight line over the interval [ O , R ] ,and a generator, which is the largest N-sided structure that spans the interval [ O , R ] . The most general relation characterizing a fractal is given by the uppermost expression (see page 56 of ref 5). If all lengths ri describing the generator are equal, which is a good approximation for C,-C, distances in proteins ( r N 3.8 A), then the expression simplifies to the more common form in the second line (eq 3). A fractal dimension is calculated directly from the properties of the initiator and generator ( N = 8, r = R / 4 ) . The ratio R / r also defines a scaling factor s of the generator that determines the maner of constructing the fractal in an infinite number of stages. Note that each stage of complete construction in the figure should extend over the entire interval of the initiator and that a fifth stage would be virtually imperceptible. The d of a structure, however, is independent of the number of constructions and the increasing contour length. Estimating d for an arbitrary polypeptide, without an explicit form for the initiator or generator, is based on identifying statistically the most common repeating motifs within the structure.
fundamental relationship in fractals predicts that the number of monomer segments N of length r is related to R by the fractal dimension d
N = (R/r)l/VF ( R / r ) d
(3)
where the exponent is also equal to the inverse Flory constant vF in polymer theory.* Theoretical considerations provide limits of 1 < d < 2 in eq 3 that correspond to a linear structure, R = rN, and a structure repreynted by an unrestricted random walk, Le., Brownian motion in d 3 2, where R = rN1/2.9 Our previous studies applied these basic concepts of fractals to protein structures and, as a consequence, equated the noninteger spectral exponents directly to fractal dimensions. The fractal interpretation was enhanced further by developing a method to estimate the fractal dimension of protein structure from crystallographic coordinates of the a-carbon backbone atoms. An excellent correlation was found in two hemeproteins between the d estimated from the covalent structure and m deduced from dynamic relaxation data. The implications that relaxation measurements could probe both the structural and dynamic properties of a noncrystalline polypeptide are intriguing. The present study extends the critical Jests of the proposed fractal models with complete data (m and d) for four additional proteins. New factors that alter the spectral exponents in protein dynamics are identified and interpreted within available theoretical models. Finally, fractal dimension estimates are refined from our earlier method and correlated with the structural properties of 70 proteins. Experimental Procedures Putidaredoxin from P . putida is purified according to standard procedureslo and concentrated to 8.3 m M in a buffer of 50 mM Tris-CI pH (8) Havlin, S. Ben-Avraham, D. J . Phys. A: Math. Gen. 1982, 15, L311-L328. (9) Tanford, C. "Physical Chemistry of Macromolecules"; Wiley & Sons, Inc.: New York, 1961; 152-154.
7.4 and 10 mM 8-mercaptoethanol that contained zero or 1.0 M NaC1. Samples are reduced aerobically with a minimal excess of solid Na2S204 and frozen immediately in liquid nitrogen. Cytochrome c-551 from P . aeruginosa is purified according to a modification" of the standard procedure.12 Samples are concentrated to 6.4 m M in a buffer of 50 m M KPi pH 7.0 that contained zero or 1.0 M NaC1. Electron spin-lattice relaxation rates are measured to IO4 s-I with the pulse-saturation/recvery technique. A short pulse of microwave power, 30-50 dB above a low-power monitoring level, is generated by microwave diode switches to drive the EPR absorption signal to zero. The exponential recovery of the saturated signal to the thermal equilibrium value is monitored in real time with a signal averager. Temperatures between 1.5 and 25 K are measured and controlled to within *4 mK by the output from a calibbrated germanium or carbon glass resistance thermometer that is mounted directly to the upper cavity and sample holder. The observed relaxation rates l / T l are comprised of three terms-a direct, a Raman, and an Orbach (resonant Raman) process-with each mechanism characterized by a unique temperature dependence. In lowspin ferric hemeproteins and reduced Fe2S2proteins, rate contributions from an Orbach process are negligible because of the isolated S = l / 2 ground state. Analysis of the Raman rate, therefore, requires only an explicit determination of the direct rate process. The direct relaxation mechanism is a one-phonon process that couples a narrow band of vibrational modes at the Larmor frequency (gBH/h 9 GHz) to the precessing electron spins. In comparison to the secondorder Raman mechanism, which utilizes the entire vibrational frequency spectrum (v < vmx), the direct process is dominant only at very low temperatures (
