Hydrodynamic properties of rodlike and disklike particles in dilute solution

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 18 8 NOVEMBER 2003 Hydrodynamic properties of rodlike and disklike particles in dilute solution A. Ortega and J. García de la Torre a) Departamento de Química Física, Facultad de Química, Universidad de Murcia, Murcia, Spain Received 22 May 2003; accepted 13 August 2003 The hydrodynamic properties of cylindrical rodlike and discoidal particles in dilute solution have been computed using the bead-shell model treatment. Previous results Tirado and García dela Torre, J. Chem. Phys. 71, ; 73, for rods with length-to-diameter ratio p 2 are now extended to short cylinders and disks down to p 0.1. The intrinsic viscosity is obtained for rods and disks, and results are presented for the three rotational relaxation times of a cylindrical particle. The hydrodynamic properties are expressed in forms that have a weak variation with p, and are therefore useful for the analysis of experimental values. We present examples of the determination of the length and diameter of the cylindrical particles, for DNA oligonucleotides and tobacco mosaic virus American Institute of Physics. DOI: / I. INTRODUCTION The rodlike, cylindrical shape is frequently found in micelles 1,2 and other colloidal particles. 3 Biological macromolecules with helical secondary structures that are short enough so that bending effects are inappreciable, can be regarded as straight, rigid rodlike particles. Such is the case of short DNA fragments: their cross-sectional structure is uniform along all their contour, and therefore they can be adequately represented as cylindrical rods. 4 6 Typically cylindrical are also certain macromolecular complexes, like the tobacco mosaic virus. 7 The hydrodynamic properties of rodlike particles in dilute solution have been the subject of much theoretical work and computational work, starting from the classical study of Burgers 8 on the translational friction coefficient of cylinders. Kirkwood and co-workers studied the translational and rotational coefficients and the intrinsic viscosity of rods modeled as straight strings of beads. 9,10 The cylindrical model seems more adequate for the practical applications than the rod of beads, and was further studied by other workers The basic aspects of the low-reynolds-number hydrodynamics and rheology of dilute suspensions of axially symmetric particles has been described in a fundamental paper by Brenner, 16 which describes the relationships obeyed by the material functions of this kind of particle, and particularly those of the long slender bodies, including cylinders of long length-to-diameter ratio. Brenner 16 also considered the properties of infinitely thin disks. This paper evidenced the need for numerical results of the material coefficients, i.e., friction coefficients and intrinsic viscosity, of cylinders and disks of arbitrary moderate or short values of the aspect ratio. From these works, it is known that the hydrodynamic properties of a rod with length L, diameter d, and aspect ratio p L/d can be expressed in the following forms: a the translational friction coefficient, D t is given by 2 N A L 3 45 M ln p C, 3 where k is the Boltzmann constant, T is the absolute temperature, 0 is the viscosity of the solvent, and M is the molecular mass of the particle. The numerical factors 1/3 and 3 in Eqs. 1 and 2 are common to all the theories. The similar numerical factor for the intrinsic viscosity reported in the early studies was affected by the so-called preaveraging approximation for hydrodynamic interaction. It was later shown, when such approximation was avoided, that the correct factor is 2/45, for rods of beads 17 and for cylinders as well In the above-mentioned works, it was usually assumed that the rod was rather long, and the so-called end-effect terms, C t, C r, and C, were obtained as numerical constants, valid in the limit of p. However, it is evident that such results are not valid for short rods with small aspect ratio; for instance, a duplex DNA dodecamer has p 2. The need of theoretical results for short cylinders was considered by Broersma, 11,12 who obtained equations for C t and C r, that were found later to give results that deviated appreciably from the experimental data for short rodlike macromolecules and macroscopic cylinders. 12,18 20 Tirado and Garcia de la Torre TG 20,21 calculated the hydrodynamic properties of short cylinders employing the shell-model methodology, first proposed by Bloomfield et al. 22,23 The surface of the cylinder were modeled as a shell of minibeads of radius. Calculations are done for seva Author to whom correspondence should be addressed. Electronic mail: jgt@um.es D t 1 kt ln p C t, L b the rotational diffusion coefficient correspond to endover-end tumbling or the rod rotation around a perpendicular axis, denoted as D r, can be formulated D r 3 kt ln p C r 0 L 3, 2 and c the intrinsic viscosity,, can be written as /2003/119(18)/9914/6/$ American Institute of Physics

2 J. Chem. Phys., Vol. 119, No. 18, 8 November 2003 Hydrodynamic properties of rodlike and disklike particles 9915 eral, decreasing values of and the results are extrapolated to the shell model limit of 0. In the calculation of the hydrodynamic properties, TG employed a bead-model procedure that accounted rigorously for the hydrodynamic interactions. 24,25 The numerical results were expressed as interpolating equations for the end-effect terms, 24,26 C t /p 0.100/p 2 and C r /p 0.050/p 2. The rotational diffusion coefficient for spinning around the axis of the cylinder, D r, was also considered by TG, finding 4kTp D r 2 A 0 0 L 3 1 C r, where A and C r 0.677/p 0.183/p 2. 7 These equations were derived, and therefore are strictly valid in the range p The TG equations were found to be in very good agreement with existing experimental data. An excellent example of their applicability is the analysis of experimental data of DNA oligonucleotides by Pecora and co-workers, 4,27 who showed that the dimensions of the DNA helix that they predict are in excellent agreement with the canonical B-DNA structure. Nowadays, the TG equations are frequently employed and regarded as the standard treatments for hydrodynamics of cylindrical particles. In the TG theory, a special hydrodynamic calculation was devised to take into account the axial symmetry of the shell-of-beads model, which made it possible to compute the properties for models with many, very small minibeads. However, the symmetry simplifications were easily applicable to the calculation of the translational and rotational coefficients but not to that of the intrinsic viscosity. Therefore, similar expressions for of short cylinders have not been available so far. Furthermore, the calculations were restricted to moderately long cylinders (2 p 20), for which most of the surface corresponds to the cylindrical wall. For very short cylinders an appreciable fraction of the surface corresponds to the caps or end-plates. In the present work, we adopt an approach that do not includes symmetry simplifications it is just a bead-shell calculation like that for any other shape but allows us a to carry on the calculation of the intrinsic viscosity and b to extend the calculation to cylinders with a rather small aspect ratio, going into the range of p 1, where the particle shape is disklike. We have evaluated numerical values of the translational and friction coefficients and intrinsic viscosity of cylinders and disks with moderate aspect ratio, p. This numerical results are employed to determine simple expressions that give the hydrodynamic properties as functions of the particle dimensions, L and d. We formulate combinations of properties that depend only on the p ratio, and therefore are particularly useful for the analysis of experimental data. For this purpose, we have developed a fitting procedure, implemented in public-domain computer programs, that allows the determination of the length and diameter of the particle from a set of FIG. 1. Bead-shell model of A a rod with p 3 B a disk with p experimental values of the solution properties. Application is made to significant bioparticles, including short DNA oligonucleotides and the rodlike tobacco mosaic virus. II. THEORY AND METHODS The bead-shell model of a cylindrical particle is constructed as follows. First, rings of minibeads are stacked to describe the cylindrical wall. A ring is staggered with respect to the ones that are immediately above and below, thus achieving a closest-packing of the surface beads. Next, the cylinder is capped with the end plates, which are planar disks constructed with concentric rings of minibeads with decreasing radius. Examples of bead models for a cylinder and a disk are shown in Fig. 1. The hydrodynamic properties for the model so constructed are evaluated as for any other shell model for a recent review of theory and computational methods, see Ref. 25. Both the model-building and the hydrodynamic calculations are carried out with the help of a computer program, HYDROSUB, 28 intended for multisubunit particles composed of cylindrical and ellipsoidal subunits, of which the present calculation is a particularly simple case, corresponding to a single cylindrical subunit. The outcome of the calculation is a set of values for the solution properties, including the translational friction coefficient f or the translational diffusion coefficient, D t k B T/ f ), the intrinsic viscosity and the rotational coefficients or relaxation times. The rotational coefficients D r and D r, for tumbling and spinning, respectively, are the primary rotational properties, the quantities that are experimentally observable are the rotational relaxation or re-orientational times. The time or frequency dependence of properties related to rotational dif-

3 9916 J. Chem. Phys., Vol. 119, No. 18, 8 November 2003 A. Ortega and J. García delatorre fusion depend on the correlation function of a first or most usually second Legendre polynomial of the cosine of the angle subtended by two successive orientations of some specific vector rigidly attached to the particle. Particularly, for the second, most habitual case, the definition is P 2 t 2 u t 3 0 u t 0 t 2 2, 1 8 where u is the unitary vector at times t 0 and t 0 t, and indicates average over all the choices of the initial time t 0. For a particle of arbitrary shape, P 2 (t) is a sum of five exponential terms, which in the case of an axially symmetric particle, as it is the case for rods and disk, reduce to three: P 2 t a a e t/ a a b e t/ b a c e t/ c. 9 In Eq. 9, the amplitudes depend on the angle,, subtended by that specific vector and the axis of the particle: a a 1 4(3 cos 2 1) 2, a b 3 sin 2 cos 2, and a c 3 4 sin 4, where is the angle subtended by u and the axis of the cylinder. The most important hydrodynamic properties which depend on the particle s size and shape but not on the vector that is monitored are the rotational times, a, b, and c, which are simply related to D r and D r as a 1 6D r, 10 1 b 5D r D r, and 11 1 c 2D r 4D r. 12 Although the rotational times can also be indexed in decreasing order as 1 2 3, so that 1 is the longest one. In the present case, this notation is somehow ambiguous, since for rods the longest time 1 is a, while for disks it is c. In some relevant cases, the observed property is associated to a vector or a tensor aligned with the symmetry axis. In such cases, P 2 (t) is a single exponential, P 2 (t) exp( t/ a ), and only the relaxation time a the longest one for rods, the shortest one for cylinders, associated to the rotational coefficient, D r, will be experimentally available. For completeness, we mention another usual solution properties, the radius of gyration, R g. This is a merely geometrical property for which a simple, exact expression is available, R g L 2 /12 d 2 /8 L 1/12 1/8p The value for R g provided by HYDROSUB is always very close to the exact ones. III. RESULTS As mentioned above, both the shell modeling, the calculation of properties for various minibead radii,, and the extrapolations to the shell model limit ( 0), are carried out by the program HYDROSUB. 28 Examples of the extrapolations are shown in Fig. 2. The variation of the properties with is not very smooth, and therefore we are not certain about FIG. 2. Shell model extrapolation for D t,, and a of a cylinder with L/d 1. whether a quadratic extrapolation would be better than a linear one. For choosing between the linear and quadratic options, we adopted a practical criterion: the chosen degree was the one which gave a result for R g closest to the exact ones. In any case the differences observed in the results for the various properties, with linear and quadratic extrapolations, are of a few percent for all the properties, and such deviations will be irrelevant when the numerical data are later smoothed by the interpolating equations. In order to complete the translational and rotational equations of TG for cylinders with 2 p 20, we have evaluated the end-effect term of the intrinsic viscosity, C from the numerical results of using Eq. 3. The results are plotted in Fig. 3. Like the similar terms for translation and rotation, these results are well described by a polynomial in powers of 1/p, whose coefficients are evaluated by leastsquares fit: C /p 8.874/p /p As we have covered in this work a wide range of p, from p 0.1 for a rather flat disk, to p 20 corresponding to a rather long cylinder, we seek common representations of the properties in the whole range. The properties vary intensely

4 J. Chem. Phys., Vol. 119, No. 18, 8 November 2003 Hydrodynamic properties of rodlike and disklike particles 9917 FIG. 3. Plot of C vs 1/p for cylinders. Data points are calculated values and the curve is the interpolating polynomial, Eq. 14. with size and shape, and therefore rather than the properties themselves, it is better to consider compound quantities having a small, smooth variation in the range of p. Following a common practice in biophysical applications, for translation we employ the ratio f / f 0, where f kt/d t is the translational friction coefficient of the cylinder or disk, and f 0 is the translational friction coefficient of a sphere having the same volume, V L 3 /4p 2 as the particle, explicitly given by f L 3/16p 2 1/3. 15 The numerical values of f / f 0 for cylinders and disks are plotted in Fig. 4, where we notice that this ratio is close to unity as if the particle were spherical for p 1, and increase to about 2 for a cylinder with p 20, and to about 1.3 for a disk with p 0.1. As for other properties, we have tried to fit the numerical values to interpolating equations, with coefficients found by least-squares fits. We find that the following expression: f / f ln p ln p ln p 3 16 gives results that are practically identical to those directly calculated; the mean absolute value of the percent difference is 0.7%. For the rotational times we proceed similarly, combining them with the rotational time of a sphere of the same volume as the rod or disk, given by 0 V 0 /kt, or more explicitly by FIG. 4. Variation of f / f 0 with p for cylinders and disks. Data points are calculated values and the curve is the interpolating Eq. 16. FIG. 5. Variation of A a / 0, B b / 0 filled circles, and C c / 0 open circles with p. Data points are calculated values and the curve is the interpolating Eqs. 18, 19, 20, and L3 0 4p 2 kt. 17 The ratios a / 0, b / 0, and c / 0 are plotted in Fig. 5. The different behavior of the three ratios for rods and disks is really noteworthy. Both a / 0, b / 0, and c / 0 have a minimum at p 1, and increase with increasing p cylinders and decreasing p disks. With the dependence on size removed after dividing by 0, the variation with p represents the dependence on shape, and here we find differences among the three ratios and between cylinders and disks. a / 0 is very sensitive to size for cylinders: a / for the highest p 20. a / 0 for disks, b / 0 for both rods and disks, and c / 0 for disks have a much less intense size dependence: they have a minimum close to unity for p 1, and their extreme values for the lowest or highest p considered here are about 4. Finally, we observe that for rods c / 0 1 over the entire range of p 1. With basis on these peculiar variations with p, we have found interpolating equations that can be summarized as follows: for p 0.75, a / ln p ln p and for p 0.75, ln p , 18 a / ln p ln p ln p while the other times are, over the whole range, 19

5 9918 J. Chem. Phys., Vol. 119, No. 18, 8 November 2003 A. Ortega and J. García delatorre IV. DISCUSSION FIG. 6. Plot of the function f (p). Data points are calculated values and the curve is the interpolating Eq. 26. b / ln p ln p ln p ln p 4, c / /p /p /p The mean absolute value of the percent difference between the numerical results and those given by Eqs is 2.0% for a, 1.3% for b and 0.8% for c. The combination of the intrinsic viscosity with the particle volume is formulated by the Einstein factor, M/VN A, where N A is Avogadro s number, which for rods and disks is related to as L3 N A 4p 2 M. 22 Values of extracted from the results computed for are well described, along with the interpolating equations ln p ln p ln p 4 for p 1, ln p ln p ln p 4 for p which produces values of whose mean absolute deviation from the calculated results is only 1.2%. A plot of versus p not shown shows a pattern very similar to that of a / 0. This fact is commented in more detail in the following section. Our Eqs. 1 and can be used to evaluate immediately the hydrodynamic properties D t, a, b, c, and of rodlike and disklike particles from their dimensions, L and d. The inverse problem, i.e., the determination of L and d from the solution properties may be more involved, but it is clearly feasible. If one of the hydrodynamic properties is experimentally available, and the molecular volume is known from the molecular mass and the partial specific volume of the particle, then the quantities f / f 0,, a / 0, etc., can be used to determine the value of p of the particle and finally, its length and diameter. The determination of the dimensions of the hydrodynamic particle can be made without any previous assumption about the possible solvation if, rather than combining one hydrodynamic property with the particle s volume, one combines two hydrodynamic properties. Several combinations which depend on the shape, i.e., on the length-to-diameter ratio, p, but not on the individual values of L and d. A wellknown case is the Scheraga Mandelkern function, 29 which combines the intrinsic viscosity with the translational friction coefficient: (p) M 1/3 1/3 0 /(100 1/3 f ). Unfortunately, it is well known in macromolecular hydrodynamics that this parameter depends very weakly on the macromolecular conformation. Our results for rods and disks are not an exception, we have (0.10) , (1) , and (20) Thus the determination of p from is very influenced by the experimental error and is not feasible in practice. Fortunately, the combination of the end-over-end rotational coefficient or equivalently the rotational time, a ) with the translational diffusion coefficient is sufficiently sensitive to the aspect ratio. The function f (p), defined as f p 9 0 kt 2/3 D t D r 1/3 9 0 kt 2/3 D t 6 a 1/3 25 was shown by Garcia de la Torre et al. 26,30 to be sufficiently sensitive to the value of p for cylinders. With our results for shorter cylinders and disks we have evaluated f (p) for the whole range of aspect ratios, p The results are plotted in Fig. 6, along with the interpolating equation that describes quite well the numerical values: f p ln p ln p ln p ln p The f (p) function can be employed to determine both dimensions, L and d of cylindrical particles. The solution properties of double-helical DNA are usually analyzed in TABLE I. Determination of length and diameter of oligonucleotides and TMV. D t,cm 2 /s a,s f (p) p L,nm d, nm r DNA 8 bp DNA 12 bp DNA 20 bp TMV

6 J. Chem. Phys., Vol. 119, No. 18, 8 November 2003 Hydrodynamic properties of rodlike and disklike particles 9919 terms of a cylindrical filament. The cross section of the double helix is not circular, but the hydration that fills the helix grooves make the cylinder an acceptable model for the hydrodynamic particle. Of course, the DNA piece has to be rather short to be represented by a rigid, straight cylinder. This condition is fulfilled by oligonucleotides such as those studied by Eimer and Pecora, 4 who determined translational diffusion coefficients and rotational times of DNA oligonucleotides with 8, 12, and 20 base pairs bp. Indeed, these authors employed our previous result for the f (p) function. 26 From the experimental D t and D r ( 1/6D r ) f (p) is calculated and the value of p is extracted from Eq. 26. Then we obtain d, L and finally the rise per base pair. Table I shows the data and results of this analysis. The agreement with the expected values, d 2 nm and r 0.44, is excellent for the 20 and 12 bp fragments. When these data were analyzed 4 in terms of the previous result for f (p),the8bp fragment produced a result for r closer to the Watson Crick 0.34 nm value. Perhaps that agreement for the shortest fragment was somehow fortituous: for L nm and d 20 nm, the length-to-diameter ratio, p 1.35, is out of the range of the previous f (p) result. Probably, the reason for the present disagreement is that the shape which has less than half-a-turn of double helix, is remarkably noncylindrical; indeed a look to a space-filling atomic model of this oligonucleotide is neither rodlike nor discoidal. A paradigmatic example of rodlike particles is tobacco mosaic virus TMV. Under the electron microscope, TMV appears as a rigid, straight rod with L 300 nm and d nm. Translational and rotational diffusion coefficients have been determined. Values corrected to water/20 C conditions are D t cm 2 /s and D r 312 s 1, which gives a s. 31,32 Following the same procedures, based on the f ( p) function as for oligonucleotides, we obtain for L and d the results reported in Table I, which are in good agreement with the electron microscopy observations. V. COMPUTER PROGRAMS HYDROSUB is available from our web site leonardo.fcu.um.es/macromol. The equations found in this work for the hydrodynamic properties of rods and disks have been implemented in FORTRAN subroutines whose source code is also available from our web site. ACKNOWLEDGMENTS This work has been supported by Grant No. BQU from the Dirección General de Investigación Científica ytécnica. A.O. is the recipient of a predoctoral fellowship from the Dirección General de Investigación. 1 P. Missel, N. Mazer, G. Benedeck, and M. Carey, J. Phys. Chem. 87, P. Neeson, B. Jennings, and G. Tiddy, Chem. Phys. Lett. 95, J. Dhont, An Introduction to Dynamics of Colloids Elsevier, Amsterdam, 1996, Chap W. Eimer and R. Pecora, J. Chem. Phys. 94, J. Lapham, J. Rife, P. B. Moore, and D. Crothers, J. Biomol. NMR 10, G. Bonifacio, T. Brown, G. Conn, and A. Lane, Biophys. J. 73, N. Santos and M. Castanho, Biophys. J. 71, J. Burgers, Second Report on Viscosity and Plasticity Amsterdam Academy of Sciences, Nordeman, Amsterdam, 1938, Chap J. Riseman and J. Kirkwood, J. Chem. Phys. 18, J. Kirkwood and P. Auer, J. Chem. Phys. 19, S. Broersma, J. Chem. Phys. 32, S. Broersma, J. Chem. Phys. 32, H. Yamakawa, Macromolecules 8, T. Yoshizaki and H. Yamakawa, J. Chem. Phys. 72, J. Bonet Avalos, J. Rubí, and D. Bedeaux, Macromolecules 26, H. Brenner, Int. J. Multiphase Flow 1, J. García de la Torre, M. López Martínez, M. Tirado, and J. Freire, Macromolecules 16, M. Record, C. Woodbury, and R. Inman, Biopolymers 14, T. Norisuye, M. Motowoka, and H. Fujita, Macromolecules 12, M. Tirado and J. García de la Torre, J. Chem. Phys. 71, M. Tirado and J. García de la Torre, J. Chem. Phys. 73, V. A. Bloomfield, W. O. Dalton, and K. E. V. Holde, Biopolymers 5, D. P. Filson and V. A. Bloomfield, Biochemistry 6, J. García de la Torre and V. Bloomfield, Q. Rev. Biophys. 14, B. Carrasco and J. García de la Torre, Biophys. J. 76, M. Tirado, M. López Martínez, and J. García de la Torre, J. Chem. Phys. 81, W. Eimer, J. Williamson, S. Boxer, and R. Pecora, Biochemistry 29, J. García de la Torre and B. Carrasco, Biopolymers 63, H. Scheraga and L. Mandelkern, J. Am. Chem. Soc. 75, J. García de la Torre, M. López Martínez, and M. Tirado, Biopolymers 23, S. J. Fujime, J. Phys. Soc. Jpn. 29, K. Kubota, H. Urabe, Y. Tominaga, and S. Fujime, Macromolecules 17,