Computer Simulation of Hydrodynamic Properties of Semiflexible Macromolecules: Randomly Broken Chains, Wormlike Chains, and Analysis of Properties of DNA J. J. C A R C i A M O L I N A , * M. C. LOPEZ MARTiNEZJ and J. CARCiA DE LA TORRE+.’
*Departamento de Inforrnitica y Automitica, Escuela Universitaria de Informdtica, and Departamento de Quimica Fkica, Facultad de Ciencias, Universidad de Murcia, 30100 Murcia, Spain
’
SYNOPSIS
The translational and rotational diffusion coefficients and the intrinsic viscosity of semiflexible, randomly broken, and wormlike chains have been obtained by Monte Car10 simulation in the context of the rigid-body treatment. Both approximate and rigorous rigid-body hydrodynamics are used, so that the error introduced by the approximate methods can be evaluated. A randomly broken chain and a wormlike chain having the same contour length and persistence length have the same radius of gyration but different values for any of the hydrodynamic properties. The two types of chains are compared in this regard. Considering that the cross section of the chain is represented by a cylinder better than by a string of spheres, we devise a cylindrical correction to be applied to the results simulated for chains of beads. Application is made to the analysis of experimental data for the translational and rotational coefficients of DNA fragments with up to lo3 base pairs, obtaining the persistence length for each model. The values for the wormlike chain agree well with model-independent values obtained from radii of gyration and with other literature data at varying ionic strength. The randomly broken chain is equally able to reproduce the experimental length dependence of the properties, but the resulting persistence length may be too high.
I NTRODU CTlO N Randomly Broken Chains
The solution properties of semiflexible polymers of synthetic and biological origin are usually interpreted in terms of the classical wormlike model1T2 in which flexibility takes place continuously along the contour of the chain. In an instantaneous conformation of the wormlike chain, the curvature varies smoothly. This description, however, is inadequate for polymers in which flexibility is essentially discontinuous. It is known that flexible units
c 19%) John Wiley & Sons, Inc. C ( X 000A-3525/90/050883-18 $04.00 Biopolymers, Vol. 29, 883-900 (1990) ‘ ‘ 1 ’ ~ whom correspondence should be addressed.
can be intercalated in polymer chains that otherwise would be rigid and ~ t r a i g h t Zig . ~ zag polymers having segments (of short and fixed length) connected by very flexible joints have been recently ~ynthesized.~ In completely homogeneous, helical polymers such as polyisocyanates, transient joints or breaks may form spontaneously due to fluctuations in the conformational state of the chain unit^.^,^ Among biopolymers, a case of discontinuous flexibility could be that of intermediate states in the helix-coil tran~ition.~ Another biological case of great relevance is DNA, whose partial flexibility has been customarily (and with frequent success) analyzed from solution properties as that of a wormlike coil. It has been ~uggested,~,’ however, that the flexibility of DNA could arise from spontaneous fluctuations that would cause the opening of base pairs. The places where the double-helical 883
884
G A R C ~ AMOLINA, LOPEZ MART~NEZ,AND G A R C ~ ADE LA TORRE
structure is disrupted would act as rather flexible joints. The discrimination between flexibility mechanisms of DNA is one of the main purposes of our work. We recently'0*" considered a model, the randomly broken chain, intended to represent a typical discontinuous flexibility mechanism. In an instantaneous conformation, the randomly broken chain presents some number of straight segments connected by breaks. The statistics of the number of breaks is determined by a flexibility parameter that, as for wormlike chains, can be the persistence length. Independently, Mansfield12 proposed a more general model, which he called broken wormlike chain, where the continuous segments are wormlike instead of rodlike. Thus the randomly broken chain is a particular and extreme case of Mansfield's model.12 We have restricted our study to the randomly broken chain because i t is expected to present the greatest differences with respect to the purely wormlike chain, which is the other extreme case of Mansfield's model. We are now aware of several previous studies on the randomly broken chain or closely related systems. The earliest reference is Birsthein and P t i t ~ y n , 'whose ~ work has since been applied by other authors from the USSR to biopolymer~'~, l5 and liquid-crystalline states of synthetic polymers.16,17Lattice versions of the randomly broken chain have been employed by Fujita et a1.18 and Schroll et al." we calculated equilibIn our previous rium properties, such as means of powers of the end-to-end distance, the radius of gyration, and the structure factor for radiation scattering. The results were compared with those known for the wormlike model. The study is extended in this paper to hydrodynamic properties: the translational diffusion (or sedimentation) coefficient, the rotational diffusion coefficient or the longest relaxation time, and the intrinsic viscosity Wormlike Chains and Monte Carlo Simulations
As it is apparent from the recent review by Yamakawa,2 most previous calculations of transport properties of wormlike chains involved the use of approximations in the hydrodynamics, like Kirkwood's formula for translational diffusion or sedimentation,20'21 or the preaveraging of the interaction t e n ~ o r . ~ Mor ~ -e ~recently, ~ Hagerman and Zimm25 avoided the deficiencies of these approximations using the supposedly less serious rigid-body approximation in a Monte Carlo simulation that
was restricted to rotational diffusion. In this paper we extend the calculation of these authors to translational diffusion and the intrinsic viscosity. Actually, here we report a comprehensive study of the hydrodynamic properties of the two types of semiflexible chains. The study is based on the rigid-body treatment of flexible macromolec u l e ~ , which ~ ~ - ~combines ~ Monte Carlo computer simulation of conformations and calculation of transport properties using rigid-body hydrodynamics. The computational procedure has several phases. In one of them the rigid-body treatment is used along with rigorous, rigid-body hydrodyn a m i c ~ ~ 'to - ~obtain ~ Monte Carlo averages over samples that, owing to the computational cost, are necessarily small, so that the averages have a large statistical uncertainty. In another phase, the use of approximate hydrodynamic formulas33,34 with much smaller computational requirements allows the obtention of approximate results with rather small statistical noise. The comparison of the two sets of r3sults allows an estimation of the error of the appiximate methods. Once this is done, the approximate results are corrected, thus obtaining better estimates of the rigorous values. Having obtained values of the hydrodynamic properties for wormlike and randomly broken chains as a function of length and flexibility, we make a comparison of the properties of the two types of chains, trying to determine to what extent they behave differently. Flexibility of DNA
Our computed results for the translational diffusion coefficient and the longest relaxation time of wormlike chains and randomly broken chain are applied to the interpretation of DNA data. This application is particularly relevant since it may give information about the flexibility mechanism of the DNA double helix. Although the solution properties of DNA have been successfully interpreted in terms of the wormlike model,' and the chain has a truly wormlike aspect in electron micrographs, there can be another mechanism contributing to the flexibility of DNA. Concretely spontaneous thermal fluctuations could produce local deformations in the double helix. The affected parts of the helix would act as kinks or flexible joints, and would contribute to the coiling of the molecule. This hypothesis was put forward by S ~ h e l l m a nin~ ~ a paper that is not frequently cited. Later, Manning8,9.36developed the idea in more detail, assuming the possibility of discontinuities produced by
SEMIFLEXIBLE CHAIN MODELS
transient opening of base pairs. His work has been criticized by S ~ h u r r ,who ~ ~ concluded ’~~ that such transient discontinuities should be very infrequent and should have a negligible effect on DNA flexibili t y . In this paper we tackle the DNA flexibility problem using a traditional approach in polymer physical chemistry: the goodness of a model (or the adequacy of the structural concepts that it represents) is judged in view of its ability of reproducing the observed molecular-weight dependence of several solution properties with nearly the same value of the adjustable parameter, which in the present case is the persistence length. This criterion is applied here to data for the sedimentation coeficients and relaxation times of monodisperse DNA fragments.
METHODS Chain Statistics and Simulation
The randomly broken chain model and the procedure for simulating its conformational statistics have been described in our previous paper.” Here we only summarize the notation and the main aspects. The randomly broken chain has N + 1 elements joined by N bonds of length b. The contour length of the chain is L = Nb. Each element can adopt two possible states. If the ith element is in the “rigid” state (probability p ) , the following bond b, proceeds in the same direction as the previous one, b I p l .On the other hand, if the element is flexible (probability 1 - p ) , the following bond takes a new random direction, which has no correlation with the previous one. The chain has a persistence length given by lo
u
=
b/(l - p )
(1)
885
The computer simulation of the randomly broken chain statistics is extremely simple. Bond b, is arbitrarily fixed in space, which gives the positions of elements 1 and 2. The state of element i is chosen from an uniform random number ac ( 0 , l ) ; if a 5 p the element is rigid and b, = btp,. If a 2 p the element is flexible and the polar angles 0 and + of bi (with b,-l along axis z ) are chosen as follows: cosd is a random number uniform in ( - 1 , l ) and + is a random number uniform in (0,277). Since the orientations of two adjacent segments are not correlated, the angle between them has a sine distribution, with very low probability for low angles. Thus, sharp breaks with acute angles are not likely. Following Hagerman and Zimm,25 the discrete version of the wormlike chain that is simulated is the +isotropic model of Schellman.35A wormlike chain of contour length L with persistance length a is simulated as a chain of N bonds with length b = L / N . These bonds join N + 1 chain elements that are frictional spheres with radius u = b/2. The bond angles 0,) i = 1,.. . , N - 1 are obtained as the composition of two angular displacements,
where the d,, components are normally distributed random numbers with zero mean and standard deviation equal to ( b/a)‘/2. The torsional angles, GI are isotropic and their values in the simulation are therefore random numbers with uniform distribution in (0,277). Once the pair of angles ( O , , GL)that specify the conformation of bond i have been generated, the Cartesian coordinates of element i + 1 are obtained using standard metho d from ~ ~those ~ of elements i - 1 and i and the values of the angles. This discrete version is only a true wormlike chain in the limit of high N .
so that Hydrodynamics
L/a = N(l - p ) (2) The continuous (high N ) version of the model is obtained making N + 00, b = 0, and p + 1 with constant L and constant a. We have shown” that then the mean squared end-to-end distance is given, in terms of L and a, by the same formula that holds for wormlike chains. Therefore, the mean square radius of gyration ( s 2 )is also given by ( s 2 ) = ( L a / 3 ) [ 1- ( 3 a / L )
+ (6a2/L2)
- ( 6 a 3 / L 3 ) ( 1- C L ” a )(]3 )
The Monte Carlo procedure for the calculation of hydrodynamic properties is based on the rigid-body treatment proposed by Zimm and others.25’28 Individual conformations of the chain, generated as described above, are regarded as instantaneously rigid and instantaneous values of the hydrodynamic properties are computed using the hydrodynamic theory of rigid particles. This is done for all the conformations contained in samples of adequate size, and the final values of the hydrodynamic properties are obtained as averages of the individual values. The rigid-body treatment is ac-
886
G A R C ~ AMOLINA, LOPEZ MARTINEZ, AND GARCIA DE LA TORRE
tually an approximation whose error is difficult to The chain elements are regarded as Stokes ascertain.40v41 We recall, however, that the treatspheres with hydrodynamic radius u and friction ment has given results for Gaussian chains that are coefficient 5 = 6~77~(1, where q0 is the viscosity of in excellent agreement with experimental data,28 the solvent. Although the choice of u can be someand also has been very useful in the description of what arbitrary, we have taken u = b/2, so that semiflexible chains and segmentally flexible macroneighbor elements are touching. This is a rather m o l e c u l e ~ . 44 ~~usual choice for polymers with helical structure or The most rigorous procedures for calculation of rather uniform cross section. Thus, for a given hydrodynamic properties of rigid macromolecules contour length L , the value of N fixes the hydrorequire the inversion of matrices of dimension 3( N dynamic diameter of the chain as b = L / N . The + 1 ) X 3 ( N + l ) ,or a t least the solution of linear unlikely case of acute angles in conformations of systems of 3 ( N 1) equations. As the computer randomly broken chains, for which there is overlap time required for this grows very rapidly with N , between chain elements, does not create any parone is forced to employ small sample sizes for high ticular problem, since both double-sum methods N. The obvious consequence is an increase in the and the rigorous procedure (with modified interacstatistical uncertainty of the simulation results. tion tensor) behave well in that situation. As in previous works,25we follow a strategy in For practical purposes, it is convenient to use, which simpler and faster approximate methods are instead of D,, D,, and [ q ] the following dimensionalso used. For samples of large size, the hydrodyless quantities: namic properties are evaluated using the approximate methods. The results have small statistical (7) uncertainty and can be fitted to interpolating functions. On the other hand, for small samples the properties are computed using both the approximate and the rigorous methods. This allows an estimation of the errors of the approximate methods as functions of chain length and the stiffness parameter. The final results are obtained correct(9) ing the approximate ones with the error functions. Although a variety of approximate methods are a ~ a i l a b l e , ~we ' . ~select ~ for this work those based k , is the Boltzmann constant, T is the absolute in double-sum formulas that can be evaluated with temperature, NA is the Avogadro number, and M high speed. For the translational diffusion coeffiis the molecular weight of the macromolecule. Alcient Dt, we employ the formula of K i r k ~ o o d . ~ ~ though ( s 2 ) in terms of L / a has a residual depenThe intrinsic viscosity is evaluated from the fordence on N for small N , for the sake of simplicity mula of Freire and Garcia de la T ~ r r e For . ~ ~the we always use in eqs. (7)-(9) the ( s 2 ) value obrotational diffusion tensor, D,, a double-sum procetained from Eq. (3). We stress that the X values dure that we recently proposed is used.34 thus obtained are used simply for the presentation The values of L / a covered in this work are not of results and interpolation. Of course, the simulatoo far from the rigid-rod limit. For a rigid rod, the tion results are put back in the form of observable two shortest eigenvalues of 0,are degenerate, Or,' properties in the analysis of experimental data. = D,, 2 , and the rotational relaxation time detected in electric birefringence and similar techniques is T = 1/6 D,,,. For randomly broken chains near the RESULTS rigid-rod limit we find, as Hagerman and Zimm did for weakly bending rods, that = D-,,2 (16) where the quotient q given by
depends only on L / b .
SEMIFLEXIBLE CHAIN MODELS
0
10
20
30
Lib
40
895
50
Figure 3. Ratios q, defined in Eqs. (16) and (17), for Dt, Dr, and DJ1, and best fits t o Eq. (18) with the coefficients in Table VI.
In our work, for the translational and rotational coefficients the TS values are obtained numerically for straight strings of beads, while the CYL values can be obtained from the theory of Tirado and Garcia de la T ~ r r e . ~ It '-~ is ~important to use the CYL values of these authors instead of the older theory by B r ~ e s m asince ~ ~ ,the ~ ~latter is known to be somewhat erroneous for, say, L/b < 20.51 Values of the ratio q for 0,and 0,are plotted in Figure 3. The ratio for s is the reciprocal of that for 0,. Although the rotational coefficient 0," for the long axis is not needed in the rest of this paper, its q ratio is included in Figure 3 for completeness. The values of q were fitted to equations of the form
Figure 3, Eq. (18) describes the numerical data very well, with the minor exception of 0,for L/b I 5. In the analysis of DNA data presented in the next section, we use CYL values for both wormlike and randomly broken chains. The strategy represented by Eqs. (16) and (17), and the use of the Tirado-Garcia de la Torre formula^^^^^^) for translational and rotational coefficients, assures that in the region of short fragments (say L / a I l),where DNA is practically straight, the theoretical values predicted for both wormlike and broken chains will be those of these authors for straight cylinders, which in turn have been shown to be in good agreements1 with DNA data for d = 2.6 nm.
ANALYSIS OF D N A EXPERIMENTAL D A T A The coefficients for the three properties are presented in Table IX. As it can be appreciated in Table IX Constants q, and w in Eq. (5) Property
Q,
1.037 1.089 0.96
W
0.287 0.330 0.602
The translational and rotational results calculated for each model are compared here with experimental data of solution properties of monodisperse fragments of DNA. The length of such fragments is usually measured by the number of base pairs (nbp). The commonly accepted rise per base pair is 0.34 nm. We also use the hydrodynamic diameter, b = 2.6 nm, which reproduces the properties of the shortest, strictly rodlike fragments well." Thus, we have L(nm) = 0.34 nbp and N = L/b = 0.131 nbp.
896
GARCIA MOLINA, LOPEZ MARTiNEZ, AND GARCIA DE LA TORHE
that in a log-log plot the data follow a line with quite small curvature, not far from straight. The experimental data have been fitted to the model calculations minimizing the sum of squares,
0.1 50
100
200
500
1000
nbP Figure 4. Rotational relaxation times of monodisperse DNA fragments vs length expressed as nbp. (0)Elias and Eden,55 ( 0 ) Lewis et al.,58 (H) Hagerman,56 ( 0 ) Dieckman et al.57All the values are for 20°C in water. Theoretical predictions: (-) wormlike chain with a = 73 nm and (------) randomly broken chain with a = 100 nm.
In the present case y = r and the statistical weights are taken as w, = l/yl.59 The adjustable parameter is the persistence length. The WC value a = 77 nm obtained here is slightly higher than that obtained by Hagerman, a = 64 nm. The small difference could be due to the larger set of data used. The randomly broken chains give a higher value of a , although the fit is practically as good as for the wormlike model as measured by the x 2 value. We note, however, that an inspection of Figure 4 indicates that the wormlike chain seems to describe the points for nbp > 500 somewhat better than the randomly broken chain. The fitting procedure was repeated with theoretical results calculated without the correction for cylindrical cross section. The adjusted values of a were about 15 nm, higher than those reported above. The corresponding curves were nearly coincident with those in Figure 4 for the high-nbp region, but for nbp < 300 there was a noticeable deviation and a worse fit of the experimental data. The convenience of including this correction in the analysis is thus well justified. Sedimentation Coefficient
The other solution property of DNA considered here is the sedimentation coefficient s, related to the translational diffusion coefficient through the Svedverg formula
Rotational Relaxation Times
Experimental data of the rotational relaxation times ( T = 1/6Dr) taken from the several ~ o u r c e s are ~ ~ plotted - ~ ~ in Figure 4. All of them correspond to lOP3M NaC1, which is the usual salt concentration in electro-optic studies of DNA. The range of lengths covered is 60 I nbp I 1000. The longest fragment considered is 1010 bp long. In this way, with the values of persistence length expected for D N A (say a > 400), all the data must be in the region of L / a < 10 where our calculations are valid. The data from the four sources follow a common trend, so that we assume that possible differences in experimental conditions are not relevant from the point of view of this analysis. I t is noteworthy
M(1 - up) s=
RT
Dt
The molecular weight can be written as M = MI,L where MI. is the mass per unit length, C is the partial specific volume of the macromolecule, and p is the density of the solution. The theoretical prediction of 0,requires the value for the solvent viscosity qO. As summarized by Hearst and Reese," the set of parameters for DNA in water a t 20°C is MI, = 1950 dalton/nm, 1 - u p = 0.457, and qo = 0.01 poise. Experimental data of s in the range nbp = 50-854 are taken from Kovacic and Van Holde" and plotted in Figure 5 . They were measured a t an
SEMIFLEXIBLE CHAIN MODELS
1
1
I
I
I
I
50
100
200
500
1000
897
nately, this makes it impossible to decide which model is best from the point of view of the coincidence of the values adjusted for two properties. The reason of the discrepancy is, however, rather clear: the data are measured a t an ionic strength of 10-3M while s is determined a t 10- ' M , and it is well known that the observed persistence length of DNA increases with decreasing ionic strength. Data of sedimentation coefficients of DNA fragments a t 10-3M would be desirable to obtain a value of a comparable with those from electric birefringence but to our knowledge such data have not been measured.
nbP Figure 5. Sedimentation coefficients of monodisperse DNA fragments vs length expressed as nbp. ( 0 ) Experimental data of Kovacic and Van Holde?l Theoretical predictions: () wormlike model with a = 41 nm and (------) randomly broken chain with a = 77 nm.
Table X Persistence Lengths of DNA, a in nm, Obtained for the WC and for the RBC from the Analysis of Rotational Relaxation Times ( 7 ) and Sedimentation Coefficients (s)" From For WC For RBC
From sc
T~
73 100
"'l'lie value between parenthesis is goodness of the fit. " ?ia('l = 0.001M. NaC'I = 0.1M.
41 77
x2
and indicates the
I .
ionic strength of 0.1M. The same fitting procedure described above was applied to these data, obtaining the values of the persistence length given in Table X. Again the fit is as good for the wormlike model as for the randomly broken chain. For the same set of data, Hearst and Reese reported a persistence length of 57 nm for the wormlike model. This result was obtained using either the Hearst -Stockmayer2" formula or the YamakawaFujii"2 formula (both for L / a I 5)) which is based on double-sum or preaveraging approximations. Using a more rigorous hydrodynamic treatment in this study we obtain a lower persistence length for the wormlike models, namely 41 nm. For both models we find a discrepancy between the values of a obtained from T and s. Unfortu-
Comparison with Data for the Radius of Gyration
Since the radius of gyration ( s ' ) is the only property for which both models are equivalent, i t would be convenient to have values of a obtained from the experimental dependence of ( s 2 )of short fragments on nbp and a t various ionic strengths. However, we are not aware of such an experiment. The measurement^^^,^^ of ( s ') of a single DNA molecule of high molecular weight, namely the linear form of plasmid ColE, with nbp = 6594, may be helpful. After correction for excluded volume, the (s ') data at lO-'M NaCl gives a = 44.6 nm.63 Our result of a for the wormlike model, 41 nm, agrees rather well (better than those of Hearst and Reece, 57 nm) with those data, but our value for a randomly broken chain a t lO-'M, 77 nm, is really discrepant. Manning64 has reinterpreted the (s ') data of Eisenberg et a1.,62,63finding a = 50 nm (see also Ref. 65). Our value for the randomly broken chain is still rather far from this estimate.
Comments on the Randomly Broken Chain Model
With the two values of a for the randomly broken chain at and 1O-'M we can guess that the persistence length at about lM, where the polyelectrolyte contribution to a is absent, would be a t most 60 nm. From our conformational study of the model" we know that the average number of breaks or joints is practically equal to L / a . With a = 60 nm, this implies that on the average there would be a break every 200 base pairs a t least, with a frequency of 5 x base pairs. This is of the same order as Mannings' estimate' of the fraction of base pairs open to proton exchange, but much higher than the value proposed by Wilcoxon and S c h ~ r r 38 .~~?
898
G A R C ~ AMOLINA, LOPEZ MART~NEZ,AND G A R C ~ ADE LA TORRE
Concluding in regard to the validity of the randomly broken model for DNA, we first recall that it is as able to reproduce experimental data as the wormlike model. The persistence lengths that it gives are, however, noticeably higher. As commented before, the interpretation of ( s 2 ) data is not model dependent, but no systematic data for the monodisperse short DNA are available. The randomly broken chain has not been used as yet for the calculation of other properties like linear flow dichroism and birefringence. We should also recall that our randomly broken chain is somewhat unphysical since no restriction is placed on the joints. The angle a t the joints could be restricted as in the more general model described by Mansfield,12 but at the expense of needing more model parameters which would complicate the data analysis. All these circumstances impede a detailed discussion of the values of a for the randomly broken chain for the DNA, and no clear conclusions can be drawn about the validity of this model and the underlying flexibility mechanism.
weight data with various high molecular weight results-for some of which excluded volume effects may not have been properly eliminated. Eisenberg and c o - w o r k e r ~have ~ ~ ? given ~ ~ a set of values of a a t various salt concentrations from the set of ( s 2 ) data of plasmid ColE, DNA. Another set of a values is that obtained by Rizzo and Schellman66 from flow dichroism. There are also measurements of flow birefringence of T7 DNA made by Cairney and H a r r i n g t ~ nthat ~ ~ can yield a value of a that depends on the choice of the optical anisotropy of the chain. They used three plausible choices for this quantity, of which the intermediate one is Aa = -0.124 nm3. The salt dependence of a for this Aa is the one used in our analysis. All these data are displayed in Figure 5 vs the concentration of NaC1. Although there are theories for the ionic strength dependence of the persistence length, the topic is not well established as and we have made no attempt to fit the data to any of them. Simply, we have inserted in Figure 6 our wormlike model persistence lengths, a = 41 a t 0.1M NaCl from s and a = 77 a t lOP3M from T. I t is evident that our values follow the trend of the three sets of data rather well. By the way, we note that the values of Hearst and Reese“ ( a = 57 a t 1O-’M from s) and H a g e m ~ a n(~a~= 64 nm a t 10p3M from T ) would deviate appreciably from this trend. Thus, our determinations of a are in rather good agreement with those from other properties. According to the criterion enunciated in the Introduction, this is in addition a confirmation of the valid-
Persistence Length of the Wormlike Model and Its Ionic-Strength Dependence
Our values for the wormlike model persistence and 10W’M NaCl can be length of DNA a t compared with those from other properties. We acknowledge that there could be some inconsistency in the joint analysis of our low molecular
-
1
1
1
I
-
80 -
o
x 0 1 7
E m
-
-
x t f
60-
0
0 40
x
-
h
i
x 0
I
I
I
“a
1
Cl]/M
Figure 6. Persistence length of DNA ( a ) according t o the wormlike model, for varying salt concentration. (0) Kam et al.,fi3( X ) Cairney and Harnngton,65 with a = 0.124 nm, (0) Rizzo and Schellman,6fiand ( 0 )this work.
SEMIFLEXIBLE CHAIN MODELS
ity of the wormlike model for describing the solution properties of DNA. We are grateful to Prof. Victor Bloomfield for comments on the ionic-strength dependence of the persistence length that were very helpful in the interpretation of our results. This work was supported by grant PR-0561/84 from Comisibn Asesora de Investigacibn Cientifica y TQcnica to JGT.
REFERENCES 1. Bloomfield, V. A., Crothers, D. M. & Tinoco, I. ( 1974) Physical Chemistry of Nucleic Acids, Harper
& Row, New York. 2. Yaniakawa, H. (1984) Ann. .,,>
'5.)-
Rev. Phys. Chem. 35,
I"
tI .
3. Welsh, W. J., Bahaunik, D. & Mark, J. E. (1981) J . Macromol. Sci. B20, 59-64. 4. Aharoni, S. (1987) Macromolecules 20, 877-884. 5. Tonelli, A. (1974) Macromolecules 7, 628-636. 6. Green, M. M., Gross, R. A., Cook, R. & Schilling, F. C. (1987) Macromolecules 20, 2638-2639. 7. Adelman, S. A. & Deutch, J. M. (1975) Macromolecules 8, 58-62. 8. Mannings, G. S. (1983) Biopolymers 22, 689-729. 9. Mannings, G. S. (1987) in Structure and Dynamics of lhopolymers, Nicolini, C., Ed., Martinus Nij hoff, 1)ordrecht. 10. Garcia Molina, J. J. & Garcia de la Torre, J. (1986) J . Chem. Phys. 84, 4026-4030. 11. Hammouda, B., Garcia Molina, J. J. & Garcia de la Torre, J. (1986) J . Chem. Phys. 85, 4120-4123. 12. Mansfield, M. L. (1986) Macromolecules 19, 854-859. 13. Hirsthein, T. M. & Ptitsyn, 0. B. (1969) J . Polym. Scz. Part C 15, 4617-4622. 14. Shkorbatov, A. G., Verkin, B. I. & Kulik, I. 0. (1978) Biopolymers 17, 1-10. 15. Shkorbatov, A. G. (1980) Makromol. Chem. Rapid Comrnun. 1, 171-175. 16. Khokholov, A. R. & Semenov, A. N. (1984) Macromolwules 17, 2678-2685. 17. Khokholov, A. R. & Semenov, A. N. (1986) Macromolecules 19, 373-378. 18. Fujita, S., Okamura, Y. & Chen, J. T. (1980) J . Chem. Phys. 72, 3883-3998. 19. Schroll, W. K., Walker, A. B. & Thorpe, M. F. (1982) J . Chem. Phys. 76, 6384-6392. 20. Hearst, J. E. & Stockmayer, W. H. (1962) J . Chem. I'hys. 37, 1425-1433. 21. Gray, H. B., Bloomfield, V. A. & Hearst, J. E. (1967) J . Chem. Phys. 46, 1493-1498. 22. Yamakawa, H. & Fujii, M. (1973) Macromolecules 6, 407 4 15. 23. Yamakawa, H. & Fujii, M. (1974) Macromolecules 7, -
128- 13.5.
899
24. Norisuye, T., Motowoka, M. & Fujita, H. (1979) Macromolecules 12, 320-325. 25. Hagerman, P. & Zimm, B. H. (1981) Biopolymers 20, 1481- 1502. 26. Zimm, B. H. (1980) Macromolecules 13, 592-602. 27. Garcia de la Torre, J., JimBnez, A. & Freire, J. J. (1982) Macromolecules 15, 148- 154. 28. Garcia de la Torre, J., Ihpez Martinez, M. C., Tirado, M. M. & Freire, J. J. (1984) Macromolecules 17, 2715-2722. 29. Garcia de la Torre, J. & Bloomfield, V. A. (1978) Biopolymers 17, 1605- 1627. 30. Garcia Bemal, J. M. & Garcia de la Torre, J. (1981) Biopolymers 20, 129-131. 31. Garcia de la Torre, J. & Bloomfield, V. A. (1981) Quart. Rev. Biophys. 14, 81-139. 32. Garcia de la Torre, J. (1981) in Molecular Electrooptics, Krause, S., Ed., Plenum, New York, pp. 75-103. 33. Garcia de la Torre, J., U p e z Martinez, M. C., Tirado, M. M. & Freire, J. J. (1983) Macromolecules 16, 1121- 1127. 34. Garcia de la Torre, J., Ihpez Martinez, M. C. & Garcia Molina, J. J. (1987) Macromolecules 20, 661-666. 35. Schellman, J. A. (1980) Biophys. Chem. 11, 329-337. 36. Manning, G. S. (1983) in Nucleic Acids and Proteins, Clementi, E. & Sarma, R. H., Eds., Adenine Press, New York, pp. 289-300. 37. Wilcoxon, J. & Schurr, J. M. (1983) Biopolymers 22, 2273-2321. 38. Schurr, J. M. (1984) Biopolymers 23, 191-194. 39. Flory, P. J. (1969) Stabtical Mechanics of Chain Molecules, Academic, New York, pp. 401. 40. Fixman, M. (1982) J . Chem. Phys. 76, 6124-6132. 41. Stockmayer, W. H. & Hammouda, B. (1984) Pure Appl. Chem. 56, 1373-1377. 42. Roitman, I). B. & Zimm, B. H. (1984) J . Chem. Phys. 81, 6333-6347. 43. Iniesta, A., Diaz, F. G. & Garcia de la Torre, J. (1988) Biophys. J . 54, 269-275. 44. Diaz, F. G. & Garcia de la Torre, J. (1988) J . Chem. PhyS. 88, 7698-7705. 45. Kirkwood, J . G. (1954) J . Polym. Sci. 1, 12-14. 46. Freire, J. J, & Garcia de la Torre, J. (1983) Macromolecules 16, 331-332. 47. Koyama, R. (1973) J . Phys. SOC.Jpn. 34,1029-1038. 48. Yoshizaki, T. & Yamakawa, H. (1984) J . Chem. Phys. 81, 982-996. 49. Tirado, M. M. & Garcia de la Torre, J. (1979) Chem. Phys. 71, 2581-2587. 50. Tirado, M. M. & Garcia de la Torre, J. (1980) Chem. Phys. 73, 1986-1993. 51. Tirado, M. M., Lbpez Martinez, M. C. & Garcia de la Torre, J. (1984) J . Chem. Phys. 81, 2047-2052. 52. Garcia de la Torre, J., Lbpez Martinez, M. C. & Tirado Garcia, M. M. (1984) Biopolymers 23, 611-615.
900
G A R C ~ AMOLINA, LOPEZ MART~NEZ,AND G A R C ~ ADE LA TORRE
53. Broersma, S. (1960) J. Chem. Phys. 32, 1626-1631. 54. Broersma, S. (1960) J. Chem. Phys. 32, 1632-1635. 55. Elias, J. G. & Eden, D. (1981) Macromolecules 14, 410-419. 56. Hagerman, P. (1981) Biopolymers 20, 1503-1535. 57. Dieckman, S., Hillen, W., Morgener, B., Wells, R. D. & Porschke, D. (1982) Biophys. Chem. 15, 263-270. 58. Lewis, R. J., Pecora, R. & Eden, E. (1986) Macromolecules 19, 134-139. 59. Bevington, P. R. (1969) Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York. 60. Hearst, J. E. & Reese, J. (1980) J . C h m . Phys. 73, 3007- 3009. 61. Kovacic, R. T. & Van Holde, K. E. (1977) Biochemistry 16, 1490-1498.
62. Borochov, N., Eisenberg, H. & Kam, Z. (1981) Biopolymers 20, 231-235. 63. Kam, Z., Borochov, N. & Eisenberg, H. (1981) Biopolymers 20, 2671-2690. 64. Manning, G. S. (1981) Biopolymers 20, 1751-1755. 65. Cairney, K. L. & Harrington, R. E. (1982) Biopolymers 21, 923-934. 66. Rizzo, V. & Schellman, J. A. (1981) Biopolymers 20, 2143-2163. 67. Schurr, J. M. & Allison, S. A. (1981) Biopolymers 20, 251-268.
Received November 15, 1989 Accepted April 25, 1989
*Departamento de Inforrnitica y Automitica, Escuela Universitaria de Informdtica, and Departamento de Quimica Fkica, Facultad de Ciencias, Universidad de Murcia, 30100 Murcia, Spain
’
SYNOPSIS
The translational and rotational diffusion coefficients and the intrinsic viscosity of semiflexible, randomly broken, and wormlike chains have been obtained by Monte Car10 simulation in the context of the rigid-body treatment. Both approximate and rigorous rigid-body hydrodynamics are used, so that the error introduced by the approximate methods can be evaluated. A randomly broken chain and a wormlike chain having the same contour length and persistence length have the same radius of gyration but different values for any of the hydrodynamic properties. The two types of chains are compared in this regard. Considering that the cross section of the chain is represented by a cylinder better than by a string of spheres, we devise a cylindrical correction to be applied to the results simulated for chains of beads. Application is made to the analysis of experimental data for the translational and rotational coefficients of DNA fragments with up to lo3 base pairs, obtaining the persistence length for each model. The values for the wormlike chain agree well with model-independent values obtained from radii of gyration and with other literature data at varying ionic strength. The randomly broken chain is equally able to reproduce the experimental length dependence of the properties, but the resulting persistence length may be too high.
I NTRODU CTlO N Randomly Broken Chains
The solution properties of semiflexible polymers of synthetic and biological origin are usually interpreted in terms of the classical wormlike model1T2 in which flexibility takes place continuously along the contour of the chain. In an instantaneous conformation of the wormlike chain, the curvature varies smoothly. This description, however, is inadequate for polymers in which flexibility is essentially discontinuous. It is known that flexible units
c 19%) John Wiley & Sons, Inc. C ( X 000A-3525/90/050883-18 $04.00 Biopolymers, Vol. 29, 883-900 (1990) ‘ ‘ 1 ’ ~ whom correspondence should be addressed.
can be intercalated in polymer chains that otherwise would be rigid and ~ t r a i g h t Zig . ~ zag polymers having segments (of short and fixed length) connected by very flexible joints have been recently ~ynthesized.~ In completely homogeneous, helical polymers such as polyisocyanates, transient joints or breaks may form spontaneously due to fluctuations in the conformational state of the chain unit^.^,^ Among biopolymers, a case of discontinuous flexibility could be that of intermediate states in the helix-coil tran~ition.~ Another biological case of great relevance is DNA, whose partial flexibility has been customarily (and with frequent success) analyzed from solution properties as that of a wormlike coil. It has been ~uggested,~,’ however, that the flexibility of DNA could arise from spontaneous fluctuations that would cause the opening of base pairs. The places where the double-helical 883
884
G A R C ~ AMOLINA, LOPEZ MART~NEZ,AND G A R C ~ ADE LA TORRE
structure is disrupted would act as rather flexible joints. The discrimination between flexibility mechanisms of DNA is one of the main purposes of our work. We recently'0*" considered a model, the randomly broken chain, intended to represent a typical discontinuous flexibility mechanism. In an instantaneous conformation, the randomly broken chain presents some number of straight segments connected by breaks. The statistics of the number of breaks is determined by a flexibility parameter that, as for wormlike chains, can be the persistence length. Independently, Mansfield12 proposed a more general model, which he called broken wormlike chain, where the continuous segments are wormlike instead of rodlike. Thus the randomly broken chain is a particular and extreme case of Mansfield's model.12 We have restricted our study to the randomly broken chain because i t is expected to present the greatest differences with respect to the purely wormlike chain, which is the other extreme case of Mansfield's model. We are now aware of several previous studies on the randomly broken chain or closely related systems. The earliest reference is Birsthein and P t i t ~ y n , 'whose ~ work has since been applied by other authors from the USSR to biopolymer~'~, l5 and liquid-crystalline states of synthetic polymers.16,17Lattice versions of the randomly broken chain have been employed by Fujita et a1.18 and Schroll et al." we calculated equilibIn our previous rium properties, such as means of powers of the end-to-end distance, the radius of gyration, and the structure factor for radiation scattering. The results were compared with those known for the wormlike model. The study is extended in this paper to hydrodynamic properties: the translational diffusion (or sedimentation) coefficient, the rotational diffusion coefficient or the longest relaxation time, and the intrinsic viscosity Wormlike Chains and Monte Carlo Simulations
As it is apparent from the recent review by Yamakawa,2 most previous calculations of transport properties of wormlike chains involved the use of approximations in the hydrodynamics, like Kirkwood's formula for translational diffusion or sedimentation,20'21 or the preaveraging of the interaction t e n ~ o r . ~ Mor ~ -e ~recently, ~ Hagerman and Zimm25 avoided the deficiencies of these approximations using the supposedly less serious rigid-body approximation in a Monte Carlo simulation that
was restricted to rotational diffusion. In this paper we extend the calculation of these authors to translational diffusion and the intrinsic viscosity. Actually, here we report a comprehensive study of the hydrodynamic properties of the two types of semiflexible chains. The study is based on the rigid-body treatment of flexible macromolec u l e ~ , which ~ ~ - ~combines ~ Monte Carlo computer simulation of conformations and calculation of transport properties using rigid-body hydrodynamics. The computational procedure has several phases. In one of them the rigid-body treatment is used along with rigorous, rigid-body hydrodyn a m i c ~ ~ 'to - ~obtain ~ Monte Carlo averages over samples that, owing to the computational cost, are necessarily small, so that the averages have a large statistical uncertainty. In another phase, the use of approximate hydrodynamic formulas33,34 with much smaller computational requirements allows the obtention of approximate results with rather small statistical noise. The comparison of the two sets of r3sults allows an estimation of the error of the appiximate methods. Once this is done, the approximate results are corrected, thus obtaining better estimates of the rigorous values. Having obtained values of the hydrodynamic properties for wormlike and randomly broken chains as a function of length and flexibility, we make a comparison of the properties of the two types of chains, trying to determine to what extent they behave differently. Flexibility of DNA
Our computed results for the translational diffusion coefficient and the longest relaxation time of wormlike chains and randomly broken chain are applied to the interpretation of DNA data. This application is particularly relevant since it may give information about the flexibility mechanism of the DNA double helix. Although the solution properties of DNA have been successfully interpreted in terms of the wormlike model,' and the chain has a truly wormlike aspect in electron micrographs, there can be another mechanism contributing to the flexibility of DNA. Concretely spontaneous thermal fluctuations could produce local deformations in the double helix. The affected parts of the helix would act as kinks or flexible joints, and would contribute to the coiling of the molecule. This hypothesis was put forward by S ~ h e l l m a nin~ ~ a paper that is not frequently cited. Later, Manning8,9.36developed the idea in more detail, assuming the possibility of discontinuities produced by
SEMIFLEXIBLE CHAIN MODELS
transient opening of base pairs. His work has been criticized by S ~ h u r r ,who ~ ~ concluded ’~~ that such transient discontinuities should be very infrequent and should have a negligible effect on DNA flexibili t y . In this paper we tackle the DNA flexibility problem using a traditional approach in polymer physical chemistry: the goodness of a model (or the adequacy of the structural concepts that it represents) is judged in view of its ability of reproducing the observed molecular-weight dependence of several solution properties with nearly the same value of the adjustable parameter, which in the present case is the persistence length. This criterion is applied here to data for the sedimentation coeficients and relaxation times of monodisperse DNA fragments.
METHODS Chain Statistics and Simulation
The randomly broken chain model and the procedure for simulating its conformational statistics have been described in our previous paper.” Here we only summarize the notation and the main aspects. The randomly broken chain has N + 1 elements joined by N bonds of length b. The contour length of the chain is L = Nb. Each element can adopt two possible states. If the ith element is in the “rigid” state (probability p ) , the following bond b, proceeds in the same direction as the previous one, b I p l .On the other hand, if the element is flexible (probability 1 - p ) , the following bond takes a new random direction, which has no correlation with the previous one. The chain has a persistence length given by lo
u
=
b/(l - p )
(1)
885
The computer simulation of the randomly broken chain statistics is extremely simple. Bond b, is arbitrarily fixed in space, which gives the positions of elements 1 and 2. The state of element i is chosen from an uniform random number ac ( 0 , l ) ; if a 5 p the element is rigid and b, = btp,. If a 2 p the element is flexible and the polar angles 0 and + of bi (with b,-l along axis z ) are chosen as follows: cosd is a random number uniform in ( - 1 , l ) and + is a random number uniform in (0,277). Since the orientations of two adjacent segments are not correlated, the angle between them has a sine distribution, with very low probability for low angles. Thus, sharp breaks with acute angles are not likely. Following Hagerman and Zimm,25 the discrete version of the wormlike chain that is simulated is the +isotropic model of Schellman.35A wormlike chain of contour length L with persistance length a is simulated as a chain of N bonds with length b = L / N . These bonds join N + 1 chain elements that are frictional spheres with radius u = b/2. The bond angles 0,) i = 1,.. . , N - 1 are obtained as the composition of two angular displacements,
where the d,, components are normally distributed random numbers with zero mean and standard deviation equal to ( b/a)‘/2. The torsional angles, GI are isotropic and their values in the simulation are therefore random numbers with uniform distribution in (0,277). Once the pair of angles ( O , , GL)that specify the conformation of bond i have been generated, the Cartesian coordinates of element i + 1 are obtained using standard metho d from ~ ~those ~ of elements i - 1 and i and the values of the angles. This discrete version is only a true wormlike chain in the limit of high N .
so that Hydrodynamics
L/a = N(l - p ) (2) The continuous (high N ) version of the model is obtained making N + 00, b = 0, and p + 1 with constant L and constant a. We have shown” that then the mean squared end-to-end distance is given, in terms of L and a, by the same formula that holds for wormlike chains. Therefore, the mean square radius of gyration ( s 2 )is also given by ( s 2 ) = ( L a / 3 ) [ 1- ( 3 a / L )
+ (6a2/L2)
- ( 6 a 3 / L 3 ) ( 1- C L ” a )(]3 )
The Monte Carlo procedure for the calculation of hydrodynamic properties is based on the rigid-body treatment proposed by Zimm and others.25’28 Individual conformations of the chain, generated as described above, are regarded as instantaneously rigid and instantaneous values of the hydrodynamic properties are computed using the hydrodynamic theory of rigid particles. This is done for all the conformations contained in samples of adequate size, and the final values of the hydrodynamic properties are obtained as averages of the individual values. The rigid-body treatment is ac-
886
G A R C ~ AMOLINA, LOPEZ MARTINEZ, AND GARCIA DE LA TORRE
tually an approximation whose error is difficult to The chain elements are regarded as Stokes ascertain.40v41 We recall, however, that the treatspheres with hydrodynamic radius u and friction ment has given results for Gaussian chains that are coefficient 5 = 6~77~(1, where q0 is the viscosity of in excellent agreement with experimental data,28 the solvent. Although the choice of u can be someand also has been very useful in the description of what arbitrary, we have taken u = b/2, so that semiflexible chains and segmentally flexible macroneighbor elements are touching. This is a rather m o l e c u l e ~ . 44 ~~usual choice for polymers with helical structure or The most rigorous procedures for calculation of rather uniform cross section. Thus, for a given hydrodynamic properties of rigid macromolecules contour length L , the value of N fixes the hydrorequire the inversion of matrices of dimension 3( N dynamic diameter of the chain as b = L / N . The + 1 ) X 3 ( N + l ) ,or a t least the solution of linear unlikely case of acute angles in conformations of systems of 3 ( N 1) equations. As the computer randomly broken chains, for which there is overlap time required for this grows very rapidly with N , between chain elements, does not create any parone is forced to employ small sample sizes for high ticular problem, since both double-sum methods N. The obvious consequence is an increase in the and the rigorous procedure (with modified interacstatistical uncertainty of the simulation results. tion tensor) behave well in that situation. As in previous works,25we follow a strategy in For practical purposes, it is convenient to use, which simpler and faster approximate methods are instead of D,, D,, and [ q ] the following dimensionalso used. For samples of large size, the hydrodyless quantities: namic properties are evaluated using the approximate methods. The results have small statistical (7) uncertainty and can be fitted to interpolating functions. On the other hand, for small samples the properties are computed using both the approximate and the rigorous methods. This allows an estimation of the errors of the approximate methods as functions of chain length and the stiffness parameter. The final results are obtained correct(9) ing the approximate ones with the error functions. Although a variety of approximate methods are a ~ a i l a b l e , ~we ' . ~select ~ for this work those based k , is the Boltzmann constant, T is the absolute in double-sum formulas that can be evaluated with temperature, NA is the Avogadro number, and M high speed. For the translational diffusion coeffiis the molecular weight of the macromolecule. Alcient Dt, we employ the formula of K i r k ~ o o d . ~ ~ though ( s 2 ) in terms of L / a has a residual depenThe intrinsic viscosity is evaluated from the fordence on N for small N , for the sake of simplicity mula of Freire and Garcia de la T ~ r r e For . ~ ~the we always use in eqs. (7)-(9) the ( s 2 ) value obrotational diffusion tensor, D,, a double-sum procetained from Eq. (3). We stress that the X values dure that we recently proposed is used.34 thus obtained are used simply for the presentation The values of L / a covered in this work are not of results and interpolation. Of course, the simulatoo far from the rigid-rod limit. For a rigid rod, the tion results are put back in the form of observable two shortest eigenvalues of 0,are degenerate, Or,' properties in the analysis of experimental data. = D,, 2 , and the rotational relaxation time detected in electric birefringence and similar techniques is T = 1/6 D,,,. For randomly broken chains near the RESULTS rigid-rod limit we find, as Hagerman and Zimm did for weakly bending rods, that = D-,,2 (16) where the quotient q given by
depends only on L / b .
SEMIFLEXIBLE CHAIN MODELS
0
10
20
30
Lib
40
895
50
Figure 3. Ratios q, defined in Eqs. (16) and (17), for Dt, Dr, and DJ1, and best fits t o Eq. (18) with the coefficients in Table VI.
In our work, for the translational and rotational coefficients the TS values are obtained numerically for straight strings of beads, while the CYL values can be obtained from the theory of Tirado and Garcia de la T ~ r r e . ~ It '-~ is ~important to use the CYL values of these authors instead of the older theory by B r ~ e s m asince ~ ~ ,the ~ ~latter is known to be somewhat erroneous for, say, L/b < 20.51 Values of the ratio q for 0,and 0,are plotted in Figure 3. The ratio for s is the reciprocal of that for 0,. Although the rotational coefficient 0," for the long axis is not needed in the rest of this paper, its q ratio is included in Figure 3 for completeness. The values of q were fitted to equations of the form
Figure 3, Eq. (18) describes the numerical data very well, with the minor exception of 0,for L/b I 5. In the analysis of DNA data presented in the next section, we use CYL values for both wormlike and randomly broken chains. The strategy represented by Eqs. (16) and (17), and the use of the Tirado-Garcia de la Torre formula^^^^^^) for translational and rotational coefficients, assures that in the region of short fragments (say L / a I l),where DNA is practically straight, the theoretical values predicted for both wormlike and broken chains will be those of these authors for straight cylinders, which in turn have been shown to be in good agreements1 with DNA data for d = 2.6 nm.
ANALYSIS OF D N A EXPERIMENTAL D A T A The coefficients for the three properties are presented in Table IX. As it can be appreciated in Table IX Constants q, and w in Eq. (5) Property
Q,
1.037 1.089 0.96
W
0.287 0.330 0.602
The translational and rotational results calculated for each model are compared here with experimental data of solution properties of monodisperse fragments of DNA. The length of such fragments is usually measured by the number of base pairs (nbp). The commonly accepted rise per base pair is 0.34 nm. We also use the hydrodynamic diameter, b = 2.6 nm, which reproduces the properties of the shortest, strictly rodlike fragments well." Thus, we have L(nm) = 0.34 nbp and N = L/b = 0.131 nbp.
896
GARCIA MOLINA, LOPEZ MARTiNEZ, AND GARCIA DE LA TORHE
that in a log-log plot the data follow a line with quite small curvature, not far from straight. The experimental data have been fitted to the model calculations minimizing the sum of squares,
0.1 50
100
200
500
1000
nbP Figure 4. Rotational relaxation times of monodisperse DNA fragments vs length expressed as nbp. (0)Elias and Eden,55 ( 0 ) Lewis et al.,58 (H) Hagerman,56 ( 0 ) Dieckman et al.57All the values are for 20°C in water. Theoretical predictions: (-) wormlike chain with a = 73 nm and (------) randomly broken chain with a = 100 nm.
In the present case y = r and the statistical weights are taken as w, = l/yl.59 The adjustable parameter is the persistence length. The WC value a = 77 nm obtained here is slightly higher than that obtained by Hagerman, a = 64 nm. The small difference could be due to the larger set of data used. The randomly broken chains give a higher value of a , although the fit is practically as good as for the wormlike model as measured by the x 2 value. We note, however, that an inspection of Figure 4 indicates that the wormlike chain seems to describe the points for nbp > 500 somewhat better than the randomly broken chain. The fitting procedure was repeated with theoretical results calculated without the correction for cylindrical cross section. The adjusted values of a were about 15 nm, higher than those reported above. The corresponding curves were nearly coincident with those in Figure 4 for the high-nbp region, but for nbp < 300 there was a noticeable deviation and a worse fit of the experimental data. The convenience of including this correction in the analysis is thus well justified. Sedimentation Coefficient
The other solution property of DNA considered here is the sedimentation coefficient s, related to the translational diffusion coefficient through the Svedverg formula
Rotational Relaxation Times
Experimental data of the rotational relaxation times ( T = 1/6Dr) taken from the several ~ o u r c e s are ~ ~ plotted - ~ ~ in Figure 4. All of them correspond to lOP3M NaC1, which is the usual salt concentration in electro-optic studies of DNA. The range of lengths covered is 60 I nbp I 1000. The longest fragment considered is 1010 bp long. In this way, with the values of persistence length expected for D N A (say a > 400), all the data must be in the region of L / a < 10 where our calculations are valid. The data from the four sources follow a common trend, so that we assume that possible differences in experimental conditions are not relevant from the point of view of this analysis. I t is noteworthy
M(1 - up) s=
RT
Dt
The molecular weight can be written as M = MI,L where MI. is the mass per unit length, C is the partial specific volume of the macromolecule, and p is the density of the solution. The theoretical prediction of 0,requires the value for the solvent viscosity qO. As summarized by Hearst and Reese," the set of parameters for DNA in water a t 20°C is MI, = 1950 dalton/nm, 1 - u p = 0.457, and qo = 0.01 poise. Experimental data of s in the range nbp = 50-854 are taken from Kovacic and Van Holde" and plotted in Figure 5 . They were measured a t an
SEMIFLEXIBLE CHAIN MODELS
1
1
I
I
I
I
50
100
200
500
1000
897
nately, this makes it impossible to decide which model is best from the point of view of the coincidence of the values adjusted for two properties. The reason of the discrepancy is, however, rather clear: the data are measured a t an ionic strength of 10-3M while s is determined a t 10- ' M , and it is well known that the observed persistence length of DNA increases with decreasing ionic strength. Data of sedimentation coefficients of DNA fragments a t 10-3M would be desirable to obtain a value of a comparable with those from electric birefringence but to our knowledge such data have not been measured.
nbP Figure 5. Sedimentation coefficients of monodisperse DNA fragments vs length expressed as nbp. ( 0 ) Experimental data of Kovacic and Van Holde?l Theoretical predictions: () wormlike model with a = 41 nm and (------) randomly broken chain with a = 77 nm.
Table X Persistence Lengths of DNA, a in nm, Obtained for the WC and for the RBC from the Analysis of Rotational Relaxation Times ( 7 ) and Sedimentation Coefficients (s)" From For WC For RBC
From sc
T~
73 100
"'l'lie value between parenthesis is goodness of the fit. " ?ia('l = 0.001M. NaC'I = 0.1M.
41 77
x2
and indicates the
I .
ionic strength of 0.1M. The same fitting procedure described above was applied to these data, obtaining the values of the persistence length given in Table X. Again the fit is as good for the wormlike model as for the randomly broken chain. For the same set of data, Hearst and Reese reported a persistence length of 57 nm for the wormlike model. This result was obtained using either the Hearst -Stockmayer2" formula or the YamakawaFujii"2 formula (both for L / a I 5)) which is based on double-sum or preaveraging approximations. Using a more rigorous hydrodynamic treatment in this study we obtain a lower persistence length for the wormlike models, namely 41 nm. For both models we find a discrepancy between the values of a obtained from T and s. Unfortu-
Comparison with Data for the Radius of Gyration
Since the radius of gyration ( s ' ) is the only property for which both models are equivalent, i t would be convenient to have values of a obtained from the experimental dependence of ( s 2 )of short fragments on nbp and a t various ionic strengths. However, we are not aware of such an experiment. The measurement^^^,^^ of ( s ') of a single DNA molecule of high molecular weight, namely the linear form of plasmid ColE, with nbp = 6594, may be helpful. After correction for excluded volume, the (s ') data at lO-'M NaCl gives a = 44.6 nm.63 Our result of a for the wormlike model, 41 nm, agrees rather well (better than those of Hearst and Reece, 57 nm) with those data, but our value for a randomly broken chain a t lO-'M, 77 nm, is really discrepant. Manning64 has reinterpreted the (s ') data of Eisenberg et a1.,62,63finding a = 50 nm (see also Ref. 65). Our value for the randomly broken chain is still rather far from this estimate.
Comments on the Randomly Broken Chain Model
With the two values of a for the randomly broken chain at and 1O-'M we can guess that the persistence length at about lM, where the polyelectrolyte contribution to a is absent, would be a t most 60 nm. From our conformational study of the model" we know that the average number of breaks or joints is practically equal to L / a . With a = 60 nm, this implies that on the average there would be a break every 200 base pairs a t least, with a frequency of 5 x base pairs. This is of the same order as Mannings' estimate' of the fraction of base pairs open to proton exchange, but much higher than the value proposed by Wilcoxon and S c h ~ r r 38 .~~?
898
G A R C ~ AMOLINA, LOPEZ MART~NEZ,AND G A R C ~ ADE LA TORRE
Concluding in regard to the validity of the randomly broken model for DNA, we first recall that it is as able to reproduce experimental data as the wormlike model. The persistence lengths that it gives are, however, noticeably higher. As commented before, the interpretation of ( s 2 ) data is not model dependent, but no systematic data for the monodisperse short DNA are available. The randomly broken chain has not been used as yet for the calculation of other properties like linear flow dichroism and birefringence. We should also recall that our randomly broken chain is somewhat unphysical since no restriction is placed on the joints. The angle a t the joints could be restricted as in the more general model described by Mansfield,12 but at the expense of needing more model parameters which would complicate the data analysis. All these circumstances impede a detailed discussion of the values of a for the randomly broken chain for the DNA, and no clear conclusions can be drawn about the validity of this model and the underlying flexibility mechanism.
weight data with various high molecular weight results-for some of which excluded volume effects may not have been properly eliminated. Eisenberg and c o - w o r k e r ~have ~ ~ ? given ~ ~ a set of values of a a t various salt concentrations from the set of ( s 2 ) data of plasmid ColE, DNA. Another set of a values is that obtained by Rizzo and Schellman66 from flow dichroism. There are also measurements of flow birefringence of T7 DNA made by Cairney and H a r r i n g t ~ nthat ~ ~ can yield a value of a that depends on the choice of the optical anisotropy of the chain. They used three plausible choices for this quantity, of which the intermediate one is Aa = -0.124 nm3. The salt dependence of a for this Aa is the one used in our analysis. All these data are displayed in Figure 5 vs the concentration of NaC1. Although there are theories for the ionic strength dependence of the persistence length, the topic is not well established as and we have made no attempt to fit the data to any of them. Simply, we have inserted in Figure 6 our wormlike model persistence lengths, a = 41 a t 0.1M NaCl from s and a = 77 a t lOP3M from T. I t is evident that our values follow the trend of the three sets of data rather well. By the way, we note that the values of Hearst and Reese“ ( a = 57 a t 1O-’M from s) and H a g e m ~ a n(~a~= 64 nm a t 10p3M from T ) would deviate appreciably from this trend. Thus, our determinations of a are in rather good agreement with those from other properties. According to the criterion enunciated in the Introduction, this is in addition a confirmation of the valid-
Persistence Length of the Wormlike Model and Its Ionic-Strength Dependence
Our values for the wormlike model persistence and 10W’M NaCl can be length of DNA a t compared with those from other properties. We acknowledge that there could be some inconsistency in the joint analysis of our low molecular
-
1
1
1
I
-
80 -
o
x 0 1 7
E m
-
-
x t f
60-
0
0 40
x
-
h
i
x 0
I
I
I
“a
1
Cl]/M
Figure 6. Persistence length of DNA ( a ) according t o the wormlike model, for varying salt concentration. (0) Kam et al.,fi3( X ) Cairney and Harnngton,65 with a = 0.124 nm, (0) Rizzo and Schellman,6fiand ( 0 )this work.
SEMIFLEXIBLE CHAIN MODELS
ity of the wormlike model for describing the solution properties of DNA. We are grateful to Prof. Victor Bloomfield for comments on the ionic-strength dependence of the persistence length that were very helpful in the interpretation of our results. This work was supported by grant PR-0561/84 from Comisibn Asesora de Investigacibn Cientifica y TQcnica to JGT.
REFERENCES 1. Bloomfield, V. A., Crothers, D. M. & Tinoco, I. ( 1974) Physical Chemistry of Nucleic Acids, Harper
& Row, New York. 2. Yaniakawa, H. (1984) Ann. .,,>
'5.)-
Rev. Phys. Chem. 35,
I"
tI .
3. Welsh, W. J., Bahaunik, D. & Mark, J. E. (1981) J . Macromol. Sci. B20, 59-64. 4. Aharoni, S. (1987) Macromolecules 20, 877-884. 5. Tonelli, A. (1974) Macromolecules 7, 628-636. 6. Green, M. M., Gross, R. A., Cook, R. & Schilling, F. C. (1987) Macromolecules 20, 2638-2639. 7. Adelman, S. A. & Deutch, J. M. (1975) Macromolecules 8, 58-62. 8. Mannings, G. S. (1983) Biopolymers 22, 689-729. 9. Mannings, G. S. (1987) in Structure and Dynamics of lhopolymers, Nicolini, C., Ed., Martinus Nij hoff, 1)ordrecht. 10. Garcia Molina, J. J. & Garcia de la Torre, J. (1986) J . Chem. Phys. 84, 4026-4030. 11. Hammouda, B., Garcia Molina, J. J. & Garcia de la Torre, J. (1986) J . Chem. Phys. 85, 4120-4123. 12. Mansfield, M. L. (1986) Macromolecules 19, 854-859. 13. Hirsthein, T. M. & Ptitsyn, 0. B. (1969) J . Polym. Scz. Part C 15, 4617-4622. 14. Shkorbatov, A. G., Verkin, B. I. & Kulik, I. 0. (1978) Biopolymers 17, 1-10. 15. Shkorbatov, A. G. (1980) Makromol. Chem. Rapid Comrnun. 1, 171-175. 16. Khokholov, A. R. & Semenov, A. N. (1984) Macromolwules 17, 2678-2685. 17. Khokholov, A. R. & Semenov, A. N. (1986) Macromolecules 19, 373-378. 18. Fujita, S., Okamura, Y. & Chen, J. T. (1980) J . Chem. Phys. 72, 3883-3998. 19. Schroll, W. K., Walker, A. B. & Thorpe, M. F. (1982) J . Chem. Phys. 76, 6384-6392. 20. Hearst, J. E. & Stockmayer, W. H. (1962) J . Chem. I'hys. 37, 1425-1433. 21. Gray, H. B., Bloomfield, V. A. & Hearst, J. E. (1967) J . Chem. Phys. 46, 1493-1498. 22. Yamakawa, H. & Fujii, M. (1973) Macromolecules 6, 407 4 15. 23. Yamakawa, H. & Fujii, M. (1974) Macromolecules 7, -
128- 13.5.
899
24. Norisuye, T., Motowoka, M. & Fujita, H. (1979) Macromolecules 12, 320-325. 25. Hagerman, P. & Zimm, B. H. (1981) Biopolymers 20, 1481- 1502. 26. Zimm, B. H. (1980) Macromolecules 13, 592-602. 27. Garcia de la Torre, J., JimBnez, A. & Freire, J. J. (1982) Macromolecules 15, 148- 154. 28. Garcia de la Torre, J., Ihpez Martinez, M. C., Tirado, M. M. & Freire, J. J. (1984) Macromolecules 17, 2715-2722. 29. Garcia de la Torre, J. & Bloomfield, V. A. (1978) Biopolymers 17, 1605- 1627. 30. Garcia Bemal, J. M. & Garcia de la Torre, J. (1981) Biopolymers 20, 129-131. 31. Garcia de la Torre, J. & Bloomfield, V. A. (1981) Quart. Rev. Biophys. 14, 81-139. 32. Garcia de la Torre, J. (1981) in Molecular Electrooptics, Krause, S., Ed., Plenum, New York, pp. 75-103. 33. Garcia de la Torre, J., U p e z Martinez, M. C., Tirado, M. M. & Freire, J. J. (1983) Macromolecules 16, 1121- 1127. 34. Garcia de la Torre, J., Ihpez Martinez, M. C. & Garcia Molina, J. J. (1987) Macromolecules 20, 661-666. 35. Schellman, J. A. (1980) Biophys. Chem. 11, 329-337. 36. Manning, G. S. (1983) in Nucleic Acids and Proteins, Clementi, E. & Sarma, R. H., Eds., Adenine Press, New York, pp. 289-300. 37. Wilcoxon, J. & Schurr, J. M. (1983) Biopolymers 22, 2273-2321. 38. Schurr, J. M. (1984) Biopolymers 23, 191-194. 39. Flory, P. J. (1969) Stabtical Mechanics of Chain Molecules, Academic, New York, pp. 401. 40. Fixman, M. (1982) J . Chem. Phys. 76, 6124-6132. 41. Stockmayer, W. H. & Hammouda, B. (1984) Pure Appl. Chem. 56, 1373-1377. 42. Roitman, I). B. & Zimm, B. H. (1984) J . Chem. Phys. 81, 6333-6347. 43. Iniesta, A., Diaz, F. G. & Garcia de la Torre, J. (1988) Biophys. J . 54, 269-275. 44. Diaz, F. G. & Garcia de la Torre, J. (1988) J . Chem. PhyS. 88, 7698-7705. 45. Kirkwood, J . G. (1954) J . Polym. Sci. 1, 12-14. 46. Freire, J. J, & Garcia de la Torre, J. (1983) Macromolecules 16, 331-332. 47. Koyama, R. (1973) J . Phys. SOC.Jpn. 34,1029-1038. 48. Yoshizaki, T. & Yamakawa, H. (1984) J . Chem. Phys. 81, 982-996. 49. Tirado, M. M. & Garcia de la Torre, J. (1979) Chem. Phys. 71, 2581-2587. 50. Tirado, M. M. & Garcia de la Torre, J. (1980) Chem. Phys. 73, 1986-1993. 51. Tirado, M. M., Lbpez Martinez, M. C. & Garcia de la Torre, J. (1984) J . Chem. Phys. 81, 2047-2052. 52. Garcia de la Torre, J., Lbpez Martinez, M. C. & Tirado Garcia, M. M. (1984) Biopolymers 23, 611-615.
900
G A R C ~ AMOLINA, LOPEZ MART~NEZ,AND G A R C ~ ADE LA TORRE
53. Broersma, S. (1960) J. Chem. Phys. 32, 1626-1631. 54. Broersma, S. (1960) J. Chem. Phys. 32, 1632-1635. 55. Elias, J. G. & Eden, D. (1981) Macromolecules 14, 410-419. 56. Hagerman, P. (1981) Biopolymers 20, 1503-1535. 57. Dieckman, S., Hillen, W., Morgener, B., Wells, R. D. & Porschke, D. (1982) Biophys. Chem. 15, 263-270. 58. Lewis, R. J., Pecora, R. & Eden, E. (1986) Macromolecules 19, 134-139. 59. Bevington, P. R. (1969) Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York. 60. Hearst, J. E. & Reese, J. (1980) J . C h m . Phys. 73, 3007- 3009. 61. Kovacic, R. T. & Van Holde, K. E. (1977) Biochemistry 16, 1490-1498.
62. Borochov, N., Eisenberg, H. & Kam, Z. (1981) Biopolymers 20, 231-235. 63. Kam, Z., Borochov, N. & Eisenberg, H. (1981) Biopolymers 20, 2671-2690. 64. Manning, G. S. (1981) Biopolymers 20, 1751-1755. 65. Cairney, K. L. & Harrington, R. E. (1982) Biopolymers 21, 923-934. 66. Rizzo, V. & Schellman, J. A. (1981) Biopolymers 20, 2143-2163. 67. Schurr, J. M. & Allison, S. A. (1981) Biopolymers 20, 251-268.
Received November 15, 1989 Accepted April 25, 1989
