Prediction of the Rotational Diffusion Behavior of Biopolymers on the Basis of Their Solution or Crystal Structure ~ U R C EI.NMULLER In5titut tiir Molekularbiologie

D - 0 - 1 1 15 Berlin, Robert-Rossle-Strasse 10, Federal Republic of German)

SYNOPSIS

Two low structure-resolution methods are proposed for prediction of rotational diffusion parameters. T h e indirect procedure is based on the structure of a molecule in solution or in crystal, a n d uses the structure parameters of radius of gyration, a n d low-resolution molecular surface a n d volume, determined from measured or theoretically calculated smallangle x-ray scattering intensities, t o estimate a frictional equivalent ellipsoid of revolution. T h e direct method starts mainly from the crystallographic structure of a molecule and calculates the triaxial inertia equivalent ellipsoid, experimentally calibrated by translation diffusion data, to simulate the frictional behavior. T h e predicted harmonic mean of the rotational correlation times of compact globular macromolecules with molar masses of 14,000-65,000 g/mol agree with experimental results within the error limits. T h e prediction method is recommended for expert systems in structure research and for detection of internal protein flexibility or marker mobility by nmr and electron paramagnetic resonance experiments.

INTRODUCTION T h e rotational diffusion of globular proteins in aqueous solutions is used to obtain information on the structure of the proteins or their water shell, on intramolecular motions, on the dimensions and accessibility o f hydrophobic pockets and active centers, '' on t he changed motility of proteins by matrix binding" or protein engineering. T o get reliable results ahout the mobility of any group, marker, or intrinsic chromophore, e.g., with nmr, electron paramagnetic resonance ( e p r ) , or fluorescence experiments, the rotational frictional behavior of the complete niacromolecule has to be known. A number of methods has been developed to calculate the translational and rotational frictional parameters of macromolecules. Simple model bodies, groups of spheres, ' or atomic coordinates,5 are used in the modeling procedures. While a sphere approximation is very rough. the atomic approach needs atomic co-

ordinates and suffers from the unknown water shell to be taken into account, from unknown structural changes comparing crystal and solution structure, and from the extremely high amount of computer time. At present the sensitivity of the experimental rotational diffusion methods is too low to render possible the deduction of models for glohular proteins with higher structure resolution than triaxial ellipsoids." Therefore, we have developed a fast method for calculation of the rotational diffusion parameters of one-, two-, or three-axial spheroidal models of macromolecules on the basis of their solution or crystal structure. The prediction method needs a calibration by translation diffusion or sedimentation measurements when starting Irom the crystal structure, but no additional information is necessary in general if small-angle x-ray s( attering data of the molecules are known.

METHODS T h e prediction method for rotational diffusion parameters of biopolymers is based on experimental 149

150

MULLER

small-angle x-ray scattering ( S A X S ), and/or translation diffusion and sedimentation data. In the first step of the prediction procedure, one-, two-, or three-axial spheroids are determined, which show the same translational diffusion behavior as the macromolecules, and in the second step the rotational diffusion parameters of the models are calculated. In the flow card (Figure l ) ,the two methods used are summarized for determination of the frictional adequate spheroids. The indirect semiempirical procedure for estimation of the spheroid parameters basically uses SAXS data of a molecule, and the direct method uses the inertia-equivalent ellipsoid of a particle together with a calibration by experimental translational friction data.

Indirect Semiempirical Method

T h e physical reasons of the frictional resistance of rigid macromolecules with arbitrary shape a t low Reynold's numbers are implicitly included in the low-resolution structure parameters: radius of gyration R,, surface O L R , and volume VLR, which were determined from small-angle x-ray data a t a resolution of 1.0-4.0 nm.839 R, and V L R contain information about the shape and dimensions of the mol-

indirect

method

ecule, O1,K/V1,R describes to a certain extent the surface rugosity, and the low-resolution volume VL, contains partially and unspecifically the overall solvation.8j'o Using these geometric parameters, translational friction parameters of rigid macromolecules such as the friction coefficient f T , diffusion, and sedimentation coefficients ( DT, s ) can be predicted with high accuracy by a semiempirical m e t h ~ d . ~ ' " ~ ' * The restriction to a two-parametric ellipsoid of revolution as a frictional equivalent model renders possible a fast iterative procedure for the determination of the axial ratio p of the ellipsoid from the experimental R, and ( O , , R / V I , R ) .',I2 The volume Vl,Rand p are then used for calculation of f T , DT, and s. A short summary of the algorithm is given in the Appendix. If the decisive physical processes a t the macromolecular surface are identical for translational and rotational diffusion, the translational frictional equivalent ellipsoid of revolution should describe the rotational behavior of the macromolecules too. From the low-resolution particle volume VLRand the axial ratio p , the rotational frictional coefficients f AH, f for rotation around the axes A and B can be calculated using the corrected Perrin f ~ r m u l a . " ? ~ ' ~ Therefrom rotational diffusion coefficients 0:' for rotation around the axes x and rotational correlation

direct

method

of the molecule

T experimental s.D,

p. 5. D i

n trioxial translational frictional equivalent ellipsoid ( E I EE 1 p. s*.0;

Figure 1. Flow card of two different procedures for t h e determination of rotational correlation times from molecules with known crystallographic or low-resolution solution structure.

PREDICTION OF ROTATIONAL DIFFUSION BEHAVIOR

times T ~ T~, of the axes A and B are available. The formulas are also given in the Appendix. The indirect procedure can be used too for the determination of rotational properties, if the atomic coordinates of a macromolecule are known from crystal structure analysis. Theoretical SAXS curves can be calculated by the improved cube methodI5 from the set of atomic coordinates16 and van der Waals radii with high accuracy. From these curves the low-resolution structure parameters R,, V L R , OLR have to be deduced,' and then the rotational diffusion parameters are available via the translational frictional equivalent ellipsoid of revolution. The methods discussed above suffers from a minor disadvantage. The model used is only a twoaxial ellipsoid, and the axes of this ellipsoid of revolution do not describe the real structure of the macromolecule including the surrounding water shell (Tables I and 11). For example, the frictional behavior of the myoglobin molecule lM B N described in Table I1 ( t h e four-character code follows the Protein Data Bank convention16)is equivalent to that of a n ellipsoid with the semiaxes A = 0.96 nm and B = C = 2.45 nm, although 88% of its dry volume is enclosed in a n ellipsoid with semiaxes of A = 2.28 nm, B = 2.05 nm, C = 1.31 nm, and 94% in an ellipsoid with A = 2.43 nm, B = 2.18 nm, and C = 1.40 nm, respectively. For hemoglobins, the discrepancy between real structure and model ellipsoid is evident too (Tables I and 11). Therefore a more direct structure-related method for determination of the rotational diffusion parameters was also developed. Direct Method

The direct method is mainly based on the crystallographic st,ructure of a macromolecule ( Figure 1) . Knowing the atomic coordinates, the so-called inertial ellipsoid ( I E ) can be estimated by using wellknown methods of classical mechanic^.'^^'^^'^ The surface of the IE represents the tensor of inertia of a molecule. T h e semiaxes and their directions of the spheroid are determined by solving the eigenvalue problem for this tensor. From the IE the corresponding real triaxial ellipsoid, having the same inertial properties as the IE, can be derived.I0.l8 For globular proteins this so-called inertia equivalent ellipsoid (IEE) defines a n averaged surface, like the sea level on earth. Ten percent to 15% of the atoms of a protein globule are outside of the level, but the IEE volume is about 20-30% larger than the dry volume of the molecule.'" The IEE is used for modeling the rotational diffusion behavior of the cor-

151

responding molecule. For this purpose, the IEE is to be enlarged with constant axial ratios up to a size that is representative for the translational friction of the molecule. Via such a n iterative calibration procedure using experimental s or DT values, external water is included in a n unspecific manner. The rotational diffusion parameters are then calculated for the enlarged three-axial ellipsoid ( E I E E ) as described by Perrin13 and using a computer program developed on the basis of Harding's p r ~ g r a m . ' ~ Both methods can be combined if no experimental sedimentation or diffusion coefficients are available, or if no atomic coordinates are a t one's disposal. The s and DTvalues, determined from experimental SAXS curves, then have to be used for calibration of the IEE in the first case, and the IEE calculated from atomic coordinates can be replaced by the IEE, estimated from SAXS data," in the second case. But the use of intermediate results of the indirect method without a n experimental proof should be a n exception despite the high reliability of the semiempirical procedure.

RESULTS AND DISCUSSION The structure-related semiempirical and the direct method need experimental verification of the results. Only a few molecules have been investigated by SAXS, hydrodynamic and rotation diffusion measurements under standard conditions in comparable buffers, pH, ionic strength, etc. While SAXS data and s and DT values mostly are published for standard conditions, the rotational diffusion methods very often use a n unusual, nonphysiological milieu for the macromolecules. Low ionic strength (dielectric dispersion) or high protein concentrations ( nmr ) cannot be corrected subsequently without additional measurements. Only the temperature correction of viscosity is easy to do. The data selected from literature are summarized in Table I. The experimental errors of the translational diffusion coefficients and of the sedimentation coefficients are about 2-5%, whereas the errors of the rotational diffusion parameters are significantly larger, up to 30-40%." Modeling of Rotational Diffusion by Ellipsoids of Revolution-Indirect Method

At first the results will be discussed as if the indirect method and experimental SAXS data has been used for calculation of s and DT values. Because the differences between the calculated s

2.16m 2.4

P-Lactoglobulin (bovine milk)

Hemoglobin (human,

148v

142.0t

83.0

60.25m

37.17a

29.7

25.13

24.2a

(nm:')

VLK

P

P

P

P

P

0

P

P

Shape"

1.66 1.64

6.30

1.09

1.48

1.63

2.53

1.03

0.97

H=C (nm)

6.43

5.14

4.35

3.30

0.92

2.90

2.89

A (nm)

5.58 5.76-6.2~

6.45 7.04q 5.65

7.96 7.82n

9.74 10.2k

10.3 11.3h

10.75 10.6-11.5e

10.8 10.6-11.3b

( X 1 0 ~cm'/s) '

D+ D .e,

4.06~ 4.1-4.45~

4.39 4.51r 4.11a

2.99a 2.870

2.36a 2.40k

1.92 1.9-1.971

1.86a 1.92f

1.89a 1.87-1.91~

54.8

52.8

33

20.2

10.8

10.8

8.3

8.1

Thi

(ns)

(S)

SC SexP

7;xp

* lj

47.6-54.9~

32-41s

22.313

12 f 21

10

7.6g

7-10d

(ns)

a R,: radius of gyration; (ILH:low-resolution surface; VI,R: low-resolution volume; 11%: calculated translational diffusion coefficient; Dyp: experimental translational diffusion coefficient; s": calculated sedimentation coefficient; sex&':experimental sedimentation coefficient; &: harmonic mean value of the rotational diffusion correlation times of the ellipsoid of revolution; &,: experimental harmonic mean value of the rotational correlation times. Shape of the ellipsoid of revolution; p , prolate; 0 , oblate. ' Lowercase letters indicate from what reference(s) data is from: (a) Ref. 21; (b) Refs. 22-26; (c) Refs. 23 and 27-29; (d) Refs. 22 and 30; (e) Refs. 28 a n d 31; ( f ) Ref. 32; ( 8 ) Ref. 33; ( h ) Ref. 34; ( i ) Ref. 35; (j) Ref. 36; (k)Ref. 37; (1) Ref. 38; (m) Ref. 39; ( n ) Ref. 40; ( 0 ) Ref. 41; (p) Ref. 42; (4)Ref. 43; (r) Ref. 44; (9) Refs. 45 and 46; ( t ) Ref. 47; (u) Refs. 1 and 48-51; (v) Ref. 11; (w) Refs. 25 and 52; (x) Refs. 47 a n d 53.

Serum albumin (bovine)

182.1

100.02m

58.36a

219v

1.8a

cu-Chymotrypsin (bovine pancreas)

60.02

3.ov

1.65

Myoglobin (sperm whale)

60.24a

207.3t

1.45a

w1,actalhumin (bovine milk)

60.5a

(nm')

O1.H

3.06t

1.43a'

Lysozyme (hen egg white)

OXY)

(nm)

Molecule

R,

Table I Molecular Parameters, Translational and Rotational Correlation Times, and Diffusion Coefficients of Biological Macromolecules Investigated by SAXS, and Translational and Rotational Diffusion Methods"

$

L

UI N

Y

6LYZ 1RN3 2LHR

lMBN 1LH1 1TPO

2SBT 5CHA lEST 2CGA

ZCAB 5CPA

2SOD lHHO 2HHH

Lysozyme

Ribonuclease A

Hemoglobin V

Myoglohin

Leghemoglobin

'I'rypsin

Subtilisin

tu-Chymotrypsin

Elastase

Chymotrypsinogen A

Carbonic anhydrase R

Carhoxypeptidase A

Superoxide dismutase

Hemoglobin (oxy)

Hernoglobin (deoxy) 2.459

2.396

2.074

1.908

1.805

1.760

1.739

1.722

1.784

1.692

1.632

1.607

1.583

1.502

1.477

1.325

174.89

167.31

78.39

68.92

70.33

67.76

67.26

67.25

68.44

58.74

56.63

53.73

50.42

48.51

43.68

36.82

93.53

89.93

40.56

54.77

41.48

38.56

34.61

34.75

41.59

35.24

26.13

27.16

27.1 1

17.60

20.64

15.90

P

P

P

0

P

0

P

P

0

0

0

0

0

P

P

-

5.186

5.032

4.280

1.999

3.447

1.096

3.444

3.397

1.207

1.227

0.836

0.959

1.066

3.119

2.846

2.533

1.292

1.301

1.261

2.665

1.484

2.673

1.276

1.284

2.689

2.531

2.512

2.448

2.387

0.883

1.185

1.087

-

10.7 11.3a 10.5 10.0e 9.97 9.3a 9.45 9.04a 9.74 10.2f 9.72 9.5a 9.57 9.48a 9.38 10.7a 9.04 9.2a 8.83 8.92a 6.58 7.04g 6.44

-

12.9 13.Oa' 11.8 10.6-11.3b 11.6 10.7a 10.9 1.81 1.83a 1.99 1.87-1.91~ 1.94 1.78a 1.90 1.94d 1.84 1.9-1.97a 1.96 1.8812 2.62 2.5a 2.89 2.77a 2.32 2.40s 2.81 2.6a 2.61 2.58a 3.02 3.233 3.43 3.55~1 3.11 3.35~3 4.40 4.51h 4.32 4.34h

2.4170 5.4260 1.8026 4.2051 1.3298 5.2348 2.0660 1.6472 1.8996 1.5127 1.7580 1.4082 1.5722 1.2524 1.3507 1.0771 0.9610 2.5400 0.9434 2.5605 1.3779 1.0970 0.9303 2.0790 1.2423 1.0844 0.6085 2.2603 0.2310 1.0356 0.2113 0.9998

35.15

33.40

14.38

14.01

12.69

12.98

11.24

11.21

13.23

11.37

10.15

9.41

8.65

6.33

6.40

4.87

" .'

Prediction of the rotational diffusion behavior of biopolymers on the basis of their solution or crystal structure.

Two low structure-resolution methods are proposed for prediction of rotational diffusion parameters. The indirect procedure is based on the structure ...
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