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(Received 30 May 1989; revised 12 September 1989) This paper presents a simple practical method for characterizing conformation of protein chains. A single number

Dr, as the fractal dimension, is assigned to each chain. Dr= L,(N)/L,(N. d/L), where N is the number of the amino acid residues in the chain, L and d are the total length and the planar diameter of the chain, respectively. In general, 1

Introduction The catalysis of enzymes has been one of the most fascinating phenomena on which studies have, over the years, been challenging problems. Various concepts and techniques have been developed in this field. As is well known, enzymes are, generally, protein molecules, and their catalysis is of relevance to protein conformations. Stapleton and coworkers I introduced a fractal model to characterize the anomalous temperature dependence of the Raman electron spin relaxation rates in proteins containing iron. The model explains the observed T" temperature dependence (5 ~

Theory and methods The protein molecules are long chain copolymers, though no bifurcation or branch is involved, usually folded through cross-linking as a consequence of the interaction * To whom all correspondenceshould be addressed. 0141-8130/90/010006~)3 © 1990 Butterworth & Co. (Publishers) Ltd 6 Int. J. Biol. Macromol., 1990, Vol 12, February

of the contiguous amino acid residues by hydrogen bonding, Van der Waals force, etc. Protein strands are neither regularly recurrent nor strictly random fabrication of the copolymers, yet they have statistical selfsimilarities and can, therefore, be characterized by an average fractal dimension. First, a protein molecular chain may be regarded as a space curve in three dimensions, and being a planar curve in two-dimensional space. Since it is a fractal object the conformation may be characterized by fractal dimension lDf). We will first look into the fractal property of a protein chain in a plane, then extend the results to three-dimensional space based on the principle of fractal geometry. Second, according to Mandelbrot's theory ¢, the general form ofa fractal dimension of a planar curve is (length) l/Dr= k(area) 1/2

(1)

where 'length' signifies the total length of the curve, 'area' is the maximum potential area the curve fills, and k is a constant. To use fractals practically, three decisions must be made from outside information. First, the appropriate size and shape of limiting planar area must be determined. In the present case, the limiting area should be what is filled by a self-avoiding random walker. A random walk chain in a plane tends to fill a circular area, so that we can choose a circle as the appropriate profile of the area. Second, the appropriate units of measurement must be chosen well and made explicit because the estimates of fractal dimensions vary with the scale of measurement. We should choose the average step size as the appropriate unit. Third, the constant k is so chosen as to ensure that the right-hand side of equation (1) yields a true onedimensional characteristic of the area. This straight-line characteristic can be the 'linear size' or 'linear scale' of the area. We can choose the diameter of the circle as the straight-line characteristic of the chosen area profile. Such choices should lead to the following general equation (L/b) 1/°, = (k/b )A 1/2

(2)

where L is the total length of the curve, b is the average step length (b = L / N , N being the total number of steps), k=2n-1/2, and A is the area of the circle potentially

Fractal analysis of protein chain conformation: H. Li et al. Table 1 Fractal dimensions for some selected proteins Protein Lysozyme

Carboxypeptidase A(ZZ,+)

Chymotrypsin (ct)

Myoglobin

Haemoglobin (fl)

Parameters

Df (calculation)

/) (simulation)

dr 2

N = 129, V=45 x 30 × 30 (A3) d t = a = 45 (A) d2=b=c=30 (,~) d--- 35 (A)

1.432 1.615 1.536

1.54 + 0.02

1.76

N = 307, V= 50 × 42 × 38 dj = a = 50 d 2 = c = 38 d = 43

1.627 1.765 1.696

1.68 _+0.02

1.56

N=245, V=51 × 40 × 40 dI =a=51 d 2 --- b = c = 40 d=44

1.416 1.668 1.625

1.63 +0.03

1.36

N = 153, V=43 × 35 × 23 dl = a = 43 d2 = c = 23 d = 34

1.493 1.834 1.605

1.62 + 0.03

1.66

N = 146, d ~ 55 (•)

1.378

1.40 + 0.03

1.64

filled by a self-avoiding random walker. Thus, the fractal dimension (D 0 is given by

De=ln(L/b)/ln[(k/b)A x/2] =ln(N)/ln(Nd/L)

(3)

For a protein molecular chain, N is the number of amino acid residues in the chain, d is the diameter of the protein, and L is the chain length. Then, L = N . b , here b is the average bond length of C - - C , C - - O and C - - N bonds, and its value is 1.48 A. To test the calculated results, computer simulations were carried out using the Monte Carlo method. The Monte Carlo method is a computational technique in which various states of a system are generated with random numbers and weighted with appropriate probabilities. As models, Monte Carlo simulations are useful in analysis of protein chain conformations. For our model, we considered a self-avoiding random walk model with massless bonds ~z and used the s-p enrichment technique on an I B M 3081 computer. Further details of this method may be found in Refs 12 to 14. In the present communication, we have utilized the Monte Carlo method to compute the fractal dimension of a protein chain. The number of monomers N(R) are counted as a function of the radial distance (R) from an arbitrary origin, and fit N(R) to R b (Refs 15, 16); such fits need to be done at several places within the structure. The average fractal dimension is obtained as the best fit of N ( R ) ~ R ' , by a least-squares linear fit ofln(N) to In(R). In the above calculations, the key procedure is to assess the appropriate planar diameter (R) of the protein chain. The simplest way to estimate this value is to find first the largest distance between two points on the curve.

Results and discussion Based on the data determined by X-ray crystallography a7 from the literature, the fractal dimensions of some protein molecular chains are calculated by equation (3) and the results are listed in Table I. It is shown that Df is the reflection of the profile of the protein molecule. Generally, a real protein molecule is an ellipsoid with three diameters, a, b, c and its volume V=a × b × c. The D r

values are calculated by the average diameter (d) and are in agreement with the results of computer simulations. The deviation of our results from those of the Stapleton group 2 is conceivable since the latter is the fracton dimension of the backbone of protein. The fracton dimension concept was first introduced by Alexander and Orbach 3, and at low frequencies (m) is defined by

or

p(m)ocoflC 1

(4)

Ntc~_td'/2

(5)

where p(og) is the density of state, Nt is the number of distinct sites in the fractal visited by a random walker up to time t. The fracton dimension (d0 was originally introduced through a consideration of the scaling properties of both the volume and the connectivity in calculating the density of states on a fractal. It differs in general from Df because df reflects the geometrical structure of the fractal and Of reflects the topological structural properties of the fractal. For example, with the Sierpinski gasket in d-dimension Euclidean space, its fractal dimension is easily found as Df = ln(d + 1)/In 2, and the fracton dimension df = 2 ln(d + 1)/ln(d + 3) ~~. In addition Alexander and Orbach pointed out 3, that the fracton dimension of percolation in any dimension d, 1

1/T l oc T 3 + 2d~f (T/O, df)

(6)

where 0 is the Debye temperature, f is a smooth analytic function of T/O, and T is the absolute temperature. The experiment indicates that the low-temperature (4 ~ 20 K) behaviour of 1/T~ is best described by a non-integer

Int. J. Biol. Macromol., 1990, Vol 12, February

7

Fractal analysis o f protein chain conformation: H. Li et al.

References

p o w e r law of the form 1 / T l oc T3+ Zd'oz T"

(7)

1

with n ~ 6 . 3 for h a e m o p r o t e i n s I a n d n ~ 5 . 6 7 for ferr e d o x i n 2. S t a p l e t o n et al. 1 o b t a i n e d for different p r o t e i n s values of dr between I a n d 2 using e q u a t i o n (7). A l e x a n d e r a n d O r b a c h 3 also established the r e l a t i o n s h i p between the fractal a n d fracton d i m e n s i o n s as follows

2

df = 2Df/dw

(8)

where dw is the e x p o n e n t c o n n e c t i n g the r o o t - m e a n square d i s p l a c e m e n t R w of a r a n d o m w a l k e r o n the fractal with the n u m b e r of steps NwOCR a~', dw is called the fractal d i m e n s i o n of the walk. Similar to dw, m a n y o t h e r fractal d i m e n s i o n s m a y be defined. I n general, we have 1 < Df ~<2 for a p r o t e i n in a plane. F o r a p r o t e i n in t h r e e - d i m e n s i o n a l space, its fractal d i m e n s i o n is Dst = (De + ADs), where ADs, the i n c r e m e n t of fractal d i m e n s i o n , can be c a l c u l a t e d by the transf o r m a t i o n of the p r o j e c t i o n 18, a n d its value A D f ~ 1 for the self-affine s t r u c t u r e 19'2°. Therefore, the Dft values of the p r o t e i n m o l e c u l a r chains with t h r e e - d i m e n s i o n a l structure m a y be estimated t h r o u g h the self-affine fractals. The fractal d i m e n s i o n s are useful in the i n t e r p r e t a t i o n of certain t h e r m o d y n a m i c p r o p e r t i e s 21'22, r e a c t i o n kinetics 23, a n d catalysis of the p r o t e i n molecules, p a r t i c u l a r l y enzymes. In o u r o t h e r p a p e r 24, we have discussed the a p p l i c a t i o n s of fractal d i m e n s i o n s to the Hill coefficients of the allosteric enzymes.

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Acknowledgements T h e a u t h o r s wish to t h a n k D r M. Xiang for c o m p u t e r simulation. This research is s u p p o r t e d by Science F u n d of the Chinese A c a d e m y of Sciences.

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Int. J. Biol. M a c r o m o l . , 1990, Vol 12, F e b r u a r y

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