Brownian dynamics simulations of a DNA molecule in an extensional flow field

Transcription

1 Brownian dynamics simulations of a DNA molecule in an extensional flow field R. G. Larson a) and Hua Hu Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan D. E. Smith and S. Chu Department of Physics, Stanford University, Stanford, California (Received 19 February 1998; final revision received 11 December 1998) Synopsis The unraveling dynamics of long, isolated, molecules of DNA subjected to an extensional flow in a crossed-slot device T. T. Perkins, D. E. Smith, and S. Chu, Single polymer dynamics in an elongational flow, Science 276, ; D. E. Smith and S. Chu, Response of Flexible Polymers to a Sudden Elongational Flow, Science 281, are predicted by Brownian dynamics simulations using measured elastic and viscous properties of the DNA as the only inputs. Quantitative agreement is obtained both in the percentages of various unraveling states, such as folds, kinks, dumbbells, half-dumbbells, and coils, and in the ensemble-averaged stretch and rate of stretch. Under fast flows (De 10), unraveling is initially nearly affine, but for fractional stretch greater than 1/3, stretching is delayed to an extent that varies widely from molecule to molecule by flow-induced folded states, which are far-from-equilibrium kinetic hindrances not predicted by dumbbell models. From the computer simulations, the source of the high molecule-to-molecule heterogeneity in the experiments is traced to variability in the initial polymer configuration, which sets the unraveling path the molecule must take at De 10. Formation of folds and kinks during unraveling can be predicted fairly reliably just by examining the initial state. The high-de unraveling behavior is consistent with the predictions of one-dimensional kink dynamics simulations The Society of Rheology. S I. INTRODUCTION Traditionally, the behavior of polymer molecules in flow fields has been inferred from macroscopic measurements of birefringence, stress, and light scattering Fuller and Leal 1980 ; Keller and Odell 1985 ; Cathey and Fuller 1990 ; Tirtaatmadja and Sridhar 1993 ; Spiegelberg and McKinley 1996 ; Verhoef et al ; Cottrell et al. 1969; Menasveta and Hoagland 1991 ; Lee et al Despite progress, there is still much uncertainty and controversy regarding the true behavior at the molecular level, even in the simplest case of dilute solutions in simple homogeneous flows. Thus, there have been many debates regarding, for example, the role of internal viscosity Cerf 1969 ; Manke and Williams 1992, dissipative stress Orr and Sridhar 1996 ; Hinch 1994 ; Rallison 1997 self entanglements Armstrong et al ; King and James a Author to whom all correspondence should be addressed by The Society of Rheology, Inc. J. Rheol. 43 2, March/April /99/43 2 /267/38/$

2 268 LARSON ET AL. TABLE I. Simulation and calculation methods for polymer flows. Method Scale of resolution Hierarchy of simplifications Atomistic MD Atomic Pair-wise interatomic potentials String-of-beads MD Kuhn length No atomic vibrations; freely rotating bonds Bead-rod BD Kuhn length Solvent is a continuum Bead-spring BD Many Kuhn lengths Equilibrated local motions, phantom springs Moment equations Many Kuhn lengths Preaveraging or closures Dumbbell model Whole macromolecule Neglect of internal structure Preaveraged dumbbell Whole macromolecule Preaveraged spring constant 1983 ; James and Sridhar 1995, formation of kinked or yo-yo conformations Ryskin 1987 ; Rallison and Hinch 1988 ; Larson 1990 conformation-dependent friction de Gennes 1974 ; Hinch 1974, and straining inefficiency Hinch 1977 ; Phan Thien et al ; Dunlap and Leal In addition, adequate signal-to-noise ratios in the macroscopic experiments mentioned above require polymer concentration levels that are high enough to call into question the appropriateness of neglecting intermolecular interactions, especially when the molecules are highly extended in a flow field. Although simulations are potentially able to access molecular-level information not obtainable from experiments, serious doubts can be raised regarding the veracity of such methods, since the most accurate ones e.g., atomistic molecular dynamics are computationally too intensive to be feasible for typical polymer length and time scales, while the simpler methods make approximations that can be tested only incompletely. Table I gives a thumbnail sketch of the available molecular dynamics MD and Brownian dynamics BD simulation methods, as well as moment equations such as the Rouse Zimm type normal-mode calculations of polymer dynamics Öttinger 1987, Included in the table are some of the most important simplifications required to step down from one level to the next lower one in the table. In the case of the second-to-the bottom entry, however, preaveraging is not invoked. The accuracy of approximations invoked by the cruder methods, such as the dumbbell calculations, can be tested by using simulation methods at higher levels in the hierarchy, but at present the Brownian dynamics methods cannot be tested by molecular dynamics simulations, except when the chains are no more than a few hundred segments long Kremer and Grest 1990 ; Gao and Weiner 1992 ; Kröger Recently, however, Perkins and co-workers 1994, 1995, 1997 have introduced experimental techniques that allow direct visualization of polymer chains in simple flows. They used fluorescently stained DNA molecules long enough 20 m that their conformations could be resolved by a high-power optical microscope. Because of their long length, the DNA molecules have a flexibility, measured by the ratio of molecular contour length L to the Kuhn length b K twice the persistence length p, that is comparable to that of many typical synthetic polymers. This molecular flexibility parameter L/b K equals the ratio of the square of the molecular contour length or fully extended length to the equilibrium mean-square separation of the ends of the molecule; i.e., L/b K L 2 / R 2 0. For instance, for a DNA molecule of contour length L 20 m and Kuhn length b K 0.13 m, we obtain a flexibility parameter of L/b K 150. To compare this to values of L/b K for synthetic flexible polymers, we note that for the latter, L/b K (0.82) 2 n/c, where n M/m 0 is the number of backbone polymer bonds with M the polymer molecular weight and m 0 the molecular weight per backbone bond; C is

3 SIMULATIONS OF A DNA MOLECULE 269 the characteristic ratio. For polystyrene, m 0 52 and C 9.6 Fetters et al Thus, a 20- m-long DNA molecule has a flexibility parameter L/b K equal to that of a polystyrene molecule of molecular weight Even longer DNA molecules have been studied under flow, as long as 150 m, for which L/b K equals that of a polystyrene molecule of molecular weight Flexibility parameters of synthetic polymers used in rheological studies are often even higher than this, however. This is the case for example in Boger fluids Mackay and Boger 1987, which are viscous, highly elastic, dilute solutions. For example, for polyisobutylene of molecular weight Tirtaatmadja and Sridhar 1993, L/b K 8000, which corresponds to a DNA molecule over 1 mm long! Thus, the flow properties of DNA molecules of length m are relevant to those of synthetic polymer molecules with molecular weights at the lower to middle of the range typically studied. At least for this range of effective flexibilities, it is now possible to test by direct experimental observation whether or not the flowinduced conformational states and dynamics predicted by coarse-grained molecular simulation methods are accurate. To date, direct comparisons between computer simulations and DNA flow experiments have only been made for simple uniform-velocity flows imposed on DNA molecules of length m tethered at one end to a small particle held in a laser-optical trap Larson et al The inputs to these simulations were the experimentally determined persistence length p b K /2 of unstained DNA, the drag coefficient coil of a DNA molecule in the undisturbed, coiled state obtained from the center-of-mass diffusion coefficient via the Stokes Einstein relationship, and the drag coefficient in the fully extended state, rod, obtained from Batchelor s formula for a rigid rod. With these inputs defined a priori, and a one-dimensional self-consistent description of hydrodynamic interactions which neglects fluctuations in the hydrodynamic interactions, the steady-state average extension x of the molecule and the mass distribution downstream of the tether point agree well with the experimentally measured quantities for DNA lengths L ranging from 20 to 150 m Larson et al. 1997, The results also show that for this range of lengths, the influence of deformation-induced changes in the effective DNA drag coefficient are small, especially for the shorter chains with L near 20 m. While these experiments support molecular theory, they do not provide a very stringent test, since they are limited to steady-state behavior and involve no gradients in velocity. Recently, however, Perkins et al and Smith and Chu 1998 obtained transient experimental data for single untethered, freely suspended lambda-phage DNA contour length 21 m, when stained in transient planar extensional flows generated in a crossed-slot apparatus. In these flows, above a critical extension rate c, the DNA molecules are stretched to a highly deformed state via several different intermediate states, even at a single extension rate in repeat experiments. In this paper, we take advantage of these single-molecule DNA studies to test in detail the validity of this classical bead-spring model, and to use it to explore the origins of the heterogeneity in unraveling behavior observed at high Deborah number. The Brownian dynamics method is described in Sec. II. In Sec. III, the predictions of the simulations are described, especially the role the initial molecular configurations play in the unraveling transitions. In Sec. IV, the predictions of the method are are shown to agree well with measurements on lambda-phage DNA molecules for a planar extensional flow. Section V summarizes the results.

4 270 LARSON ET AL. II. BROWNIAN DYNAMICS SIMULATION METHOD Most of the results presented here are obtained using a Brownian dynamics implementation of a DNA bead-spring model developed earlier; this bead-spring model proved successful in predicting the behavior of tethered DNA molecules in a constant-velocity flow Larson et al. 1997, In the model, the viscous force acting along the chain is discretized by applying it to N beads, with N as large as 80, distributed uniformly along the chain backbone, and connected together sequentially by springs. The fully extended length of each spring is l L/N. This basic bead-spring model has been discussed extensively in the literature Kuhn 1934 ; Peterlin 1966 ; Tanner 1985 ; Bird et al ; Öttinger 1987 ; Magda et al ; Larson 1988 ; Kishbaugh and McHugh The lengths of DNA molecules considered in this paper will be a L 67.2 m, and b L 21 m. A DNA molecule of the former length was already considered in detail in an earlier paper Larson et al. 1997, 1998 concerned with constant-velocity stretching, and the hydrodynamic properties were well described by bead-spring chains containing N 40 beads. The more recent extensional-flow experiments were carried out using 21- m-long stained lambda-phage DNA molecules. We will show that a 20-bead representation is quantitatively accurate for these molecules by comparisons with both experiments and with 40-bead representations. The L/b K value for case a, with 67.2 m DNA molecules, is more representative of synthetic polymer molecules than is the case for the lambda-phage DNA studied in case b, and thus both lengths will be considered here. A. The spring force The elasticity of DNA molecules has been found to be well described by the wormlike chain model Yamakawa 1971 ; Smith et al ; Vologodskii 1994 ; Bustamante et al. 1994, for which the relationship between the spring force F s and the molecular extension x of a piece of the molecule of length l is given within 5% 10% by a simple analytic approximation Bustamante et al ; Marko and Siggia 1995 F s r p eff k B T 4 1 l 1 x x l. 1 The persistence length in Eq. 1 is an effective value p eff, that is somewhat larger than the true value in order to offset the flexibility artificially added to the chain by introduction of beads, which act as free hinges Larson et al. 1997, The correction to p is small or modest as long as the contour length l of the submolecule between adjacent beads remains much 10 larger than the persistence length. Thus, the chain must not be so finely subdivided that l becomes less than ten times as large as p. For a DNA chain of contour length 67.2 m and N 40, so that l L/N 1.68 m, p must be set to p eff m to obtain elastic behavior that approximates that of a true worm-like chain with p For 80 beads, l 0.83 m, p must be increased further to 0.73 m. For stained, 21- m-long lambda-phage DNA, with p m, p eff must be set to m even for only a 20-bead representation. With more than 20 beads for lambda-phage DNA, counteracting the increased flexibility merely by increasing p eff begins to become inaccurate. Thus, in simulations of lambdaphage DNA, 20 beads will be used.

5 SIMULATIONS OF A DNA MOLECULE 271 B. The drag force The drag force on each bead is given by F> d k B T(V> x> ), where x> is the velocity of a bead and V> is the imposed solvent flow velocity at the bead s location, which is assumed to be given by a uniform gradient corresponding to a planar extensional flow, as is the case in the experiments of Perkins et al ; that is, > V> k B T is an effective bead drag coefficient, which accounts for equilibrium-averaged hydrodynamic interactions among the beads. Changes in hydrodynamic drag coefficients produced by molecular deformation are here neglected. In the earlier work Larson et al. 1997, 1998, these changes were found to be modest, especially for the shorter DNA molecules. For example, for a DNA molecule of contour length 21 m, the effective hydrodynamic drag coefficient in a uniform-velocity flow increases by only 70% when the molecule is stretched from a random coil to a fully extended filament. This increase may have a modest effect on the stress produced by molecular stretching, but is expected to have very little effect on the stretching process itself. C. The Brownian force Finally, the Brownian force must be included. Using a white-noise description valid at the time scales of interest here, the Brownian force, when added to the drag and spring forces, gives the position of a bead at time step m 1 Ermak and McCammon 1978 ; Fixman 1978 ; Liu 1989 ; Zylka and Öttinger 1989 ; Grassia et al ; Grassia and Hinch 1996 ; Öttinger 1996 ; Doyle et al : x> m 1 x> m x> m > V> dt F> m s,tot /kb T dt 6dt/ 1/2 r> m, 3 where dt is the time-step size, > V> is the velocity gradient tensor, and F> s,tot is the total spring force acting on the bead. For all beads except those at either end of the molecule, F> s,tot is therefore the sum of two terms of the form given in Eq. 1. Each component of the vector r m is a random number uniformly distributed on the interval 1, 1. Because the accuracy of the time discretization in Eq. 3 is low half-order accuracy due to the Brownian term, small time steps dt must be used, even when the velocity gradient is small. For N 40, and 1s 1 (De 4), with taken to be 1 s m 2, we find that dt 10 5 s is a small enough time step to give accurate results. For 1s 1, we take dt 10 5 /. Relative to N 40, for the same value of, we double the time-step size when N 20, and halve it when N 80. D. The stress tensor The polymer contribution to the stress tensor can be calculated from the Brownian dynamics results by using the Kramers expression N 1 < s F> ir> i, i 1 where < is the stress tensor, is the number of molecules per unit volume, R> i r> i 1 r> i is the end-to-end vector of spring i, F> i s is the spring force that spring i exerts on bead i, and denotes an ensemble average over many molecules. This expression neglects 4

6 272 LARSON ET AL. any purely viscous stresses, since the springs are assumed to possess instantaneous elasticity. Also, it neglects the Brownian contribution to the stress, because this contribution is isotropic and so does not affect measurable rheological properties. An equivalent expression for the stress tensor is obtained by replacing the spring force F> i s in spring i by the drag force F> i d on bead i, replacing R> i by r> i, and summing from i 1toN, rather than N 1. The extensional viscosity can be calculated as ( )/. E. Relaxation times and steady-state stretch At low extension rates, where the molecular stretch is low and the springs are in their linear, Hookean, regime, we expect to recover the behavior of the Rouse model. For N 20, the longest relaxation time of the Rouse model is well approximated by Ferry ,R 4 2 2, 5 sp 2 where sp is related to the mean-square stretch of a spring in the absence of flow by 2 sp 2 2 Ri 0 /3, the subscript 0 denoting equilibrium; i.e., no flow. The thermal energy k B T is absent from Eq. 5 because it has been absorbed into the definition of. 2 Each spring has the same value of sp; for the whole chain, we define a corresponding quantity: N R 2 2 sp N 1, where R 2 (r> 1 r> N ) 2 is the squared end-to-end vector of the whole chain. Defining tot N, we obtain from Eqs. 5 and 6 : 1,R N N 1 6 tot R For a worm-like chain with L 67.2 m and p m, we obtain R 2 0 2L p 7.18 m 2. For N 40 and 1s m 2, tot 40 s m 2, and Eq. 7 then gives 1,R 4.97 s. Perkins et al. used an empirical method to obtain the longest relaxation time for lambda-phage DNA in an aqueous solvent thickened with added sugar to produce a solvent viscosity of 41 cp. They simply stretched the molecule to nearly full extension in a flow field, and measured the molecular retraction as a function of time following cessation of flow. The instantaneous value of the stretch x is defined as the distance between the downstream-most part of the molecule and the upstream-most part; these need not be the ends of the chain because the ends can curl up. Values of x are experimentally observable, while the precise locations of the chain ends cannot reliably be located in an optical microscope. Fitting x 2 averaged over 14 relaxing molecules versus time in the terminal-relaxation region, where x /L 0.3, to an exponential decay; i.e., x 2 A exp( t/ 1,e ) B, Perkins et al. obtained an empirical relaxation time 1,e 3.89 s for stained lambda-phage DNA in 41 cp solvent at room temperature. The same averaging procedure, applied to 100 simulated bead-spring chains with L 67.2 m, p eff m, and N 40, gives 1,e 4.1 s; see Fig. 1. This value of 1,e is 18% lower than the longest Rouse time of 1,R 4.97 s, given by Eq. 7. A similar set of calculations with a 20-bead chain representing stained lambda-phage DNA

7 SIMULATIONS OF A DNA MOLECULE 273 FIG. 1. Relaxation of stretch squared, ensemble averaged over 100 runs, for an initially fully stretched wormlike chain with L 67.2 m, N 40, 1s/ m 2, p eff m ( p m). The portion of the relaxation corresponding to x /L 0.3 is fit to an exponential decay, yielding a relaxation time of 1,e 4.1 s. with L 21 m and p eff m gives 1,e 0.90 s, lower by 10% than 1,R Although the quantity that should theoretically match the longest Rouse time is the time constant extracted from an exponential fit to the extreme tail of a plot of R 2 versus time, rather than x 2 vs time, we find from the simulations that the distinction between R 2 and x 2 is not significant. A more significant source of deviation of 1,e from the longest Rouse time 1,R is probably some sensitivity of the fitting procedure to the influence of relaxation times shorter than 1,R. Figure 2 shows the steady-state average stretch x versus dimensionless extension rate, or Deborah number De 1,e, for a chain with N 40 beads and L A coil-stretch transition occurs at 1,e near 0.5, as expected. One can define empirically the critical strain rate for this transition by following the procedure that Perkins et al. used to analyze their DNA data. That is, we fit the data of Fig. 2 to a nonbrownian dumbbell model, using Eq. 1 for the spring force. In the non-brownian dumbbell model, the drag force must balance the spring force, i.e., 1 2 k B T x F s x 0, 8 where F s (x) is obtained from Eq. 1 with p replacing p eff. Then, by adjusting the drag coefficient k B T to obtain a fit over the range of strain rates shown in Fig. 2, we obtain the critical strain rate c as the point of intersection of the fitted line with the abscissa. For N 40, we thereby obtain c 0.11, or 1,e c Perkins et al. 1997, applying these procedures to their stained lambda-phage DNA chains, obtained a slightly smaller value, 1,e c 0.4. These values of 1,e c are 10% 20% lower than the theoretical value of 0.5, presumably because 1,e is 10% 20% lower than the true longest relaxation time 1,R.

8 274 LARSON ET AL. FIG. 2. Simulated steady-state average extension x vs Deborah number De 1,e for a worm-like chain of contour length L 67.2 m, p eff m ( p m), 1s/ m 2, with N 40. The line is a fit of Eq. 8, yielding c 0.11 s 1. F. Starting configurations for simulations To generate initial conditions for simulations of start-up of planar extension, very long runs are carried out in the absence of flow, and after allowing for relaxation of the chain to a configuration typical of equilibrium, bead configurations are written to a file every interval of about one relaxation time 1,R. Typically 101 such random, relatively uncorrelated, configurations are generated and stored for use in simulations. For each such simulation, a distinct sequence of random numbers is generated by changing the seed for the random number generator from one run to the next. In this way, 101 different molecular realizations of the start-up experiment can be simulated, and averaged together, if desired. In addition, in Sec. IV, the source of heterogeneity from one run to another is identified by carrying out a series of runs in which the starting state is kept identical and the sequence of random numbers varied, or, conversely, the starting state varied and the random-number sequence held fixed. III. PREDICTIONS OF DNA UNRAVELING Here, we examine qualitatively the stretching processes for bead chains with eff L 67.2 m, p m. These chains unravel in a wide variety of different ways, depending on strain rate, initial chain configuration, and the random Brownian processes that occur during unraveling. Many of the conformations that occur during unraveling can, however, be identified as dumbbell, half-dumbbell, folded, and kinked, using the terminology of Perkins et al. Figures 3 a and 3 b shows typical examples of dumbbell and folded states produced during unraveling. These configurations compare favorably with images of such chains obtained in the DNA experiments of Perkins et al., and reproduced in Figs. 3 a and 3 b. Folded conformations have been reported by Acierno et al. 1974, Wiest et al. 1989, Liu 1989, and Doyle and Shaqfeh The dumbbell or yo-yo configurations were anticipated by Ryskin 1987 and even earlier by Frenkel Note that the folded chain stretches much more slowly than the dumbbell chain. As described in Sec. IV B, other transitional

9 SIMULATIONS OF A DNA MOLECULE 275 FIG. 3. Comparison of experimental with simulated DNA configurations during extensional flow. In a the stretching occurs via a dumbbell conformation, while in b it occurs via a folded conformation. The spacing between images is 0.1 Hencky strain units. The experimental images are from Perkins et al configurations seen in the DNA experiments also occur in the simulations, including half dumbbells, in which only one end of the chain is rolled up in a ball, and knotted or kinked, in which the rolled up portion of the chain occurs away from the chain ends. In the simulations, the chains are phantoms that can pass through themselves, and hence cannot form true topological knots; whether or not simulated knotted configurations would become real knots if noncrossability were enforced is an open question.

10 276 LARSON ET AL. FIG. 4. Fractional extension x/l vs strain for 30 chains at a (De 0.5), b 0.25 (De 1.0), c 1(De 4), and d 10 s 1 (De 40). Results for 100 s 1 are very similar to those for 10 s 1. Both the initial configurations and the sequence of random numbers varies from one run to the next. A. Stretch vs strain Figure 4 shows curves of extension x versus Hencky strain for 30 different 40-bead eff chains, again with L 67.2 m, and p m for a s 1 (De 0.5); b 0.25 s 1 (De 1); c 1s 1 (De 4); and d 10 s 1 (De 40). The results for 100 s 1 (De 400) are very similar to those for 10 s 1. In all cases there is enormous heterogeneity in the chain stretching process; some chains stretch rather quickly; some very slowly. At s 1 (De 0.5), Fig. 4 a, the strain rate is just above the critical value and the chains do not stretch monotonically. Some chains do not stretch at all in 10 strain units; others oscillate in their degree of stretch. For a single chain at s 1 one must average the stretch over a large number of strain units around 100 Hencky units to obtain a time-averaged viscosity within 10% of the true steady-state average. At 0.25 s 1 (De 1), Fig. 4 b, on the other hand, the chains stretch more-orless monotonically. Also, the curves all appear to have the same sigmoidal shape, only the inception time required for rapid extension to begin is extremely variable. The chains that are slowest to stretch generally have the smallest initial stretch while the fast, facile chains are those with a head start; i.e., their stretch x init is relatively large to begin with. A similar result is obtained in experiments with DNA Smith and Chu At 1s 1 (De 4), Fig. 4 c, there is again a set of self-similar sigmoidal curves with widely varying inception times. But, in addition, there are a few curves with an inflection; these curves may show rapid stretch early on, but they hesitate about half-way through, and then accelerate again. We will see that these curves correspond to folded configurations; the chains folded nearest the center hesitate the longest. At 10 s 1 (De 40), Fig. 4 d, the sigmoidal curves have less heterogeneity and a smaller range of inception times. However, there are more hesitating chains with

11 SIMULATIONS OF A DNA MOLECULE 277 FIG. 5. Plots of rate of stretch ẋ versus stretch x for 30 different chains for a (De 0.5), b 0.25 (De 1.0), c 1(De 4), and d 10 (De 40) s 1. Results for 100 s 1 are very similar to those for 10 s 1. Both the initial configuration and the sequence of random numbers vary from one run to the next. folded conformations than at 1s 1. The results for 100 s 1 (De 400) are very similar to those for 10. B. Rate of stretching vs stretch As described by Perkins et al., a convenient way to test the self similarity of the shapes of the stretching curves and to eliminate the role of the inception time is to plot the rate of extension ẋ versus extension x. Following Perkins et al., we smooth out Brownian fluctuations by fitting x(t) at times t corresponding to the five strains 0.2, 0.1,, 0.1, and 0.2 to a straight line and take the slope to be ẋ. This gives a single value for ẋ( ) at each strain. The curves formed by crossplotting this value of ẋ( ) against x( ) for all 101 runs are shown in Fig. 5 for 0.125, 0.25, 1, and 10 s 1. For s 1 De 0.5, Fig. 5 a, the rate of stretching is nearly random; there are no systematic trends. For 0.25 s 1 De 1, Fig. 5 b, there is a broad, but single, band, reinforcing the impression given in Fig. 4 b that all stretching curves are roughly self similar. For 1s 1 De 4, Fig. 5 c, there is no longer a single band, but rather two limiting behaviors. For the fast chains that follow the upper range of the envelope in Fig. 5 c, the rate of stretch ẋ is nearly proportional to the stretch x, up to around x 43 m, which is about 80% of the steady-state stretch. The slow chains have a wide range of behavior, but there is a lower, bounding, behavior. The slow chains near

12 278 LARSON ET AL. this lower bound follow the same stretching curve as the fast chains up to about 20 m, which is around 1/3 of the molecular contour length, L 67.2 m. Thereafter the slow chains get hung up for a while and stretch slowly until they are stretched out to half their contour length or so, and thereafter stretch more rapidly up to x 50 m. After this, their stretching behavior is identical to that of the fast chains. An examination of typical chain configurations Sec. III C reveals that the fast chains frequently unravel via dumbbell, half-dumbbell, or kink configurations, while the slowest chains on the lower bound of the envelope pass through conformations with a fold roughly half-way along the chain contour. There are also many chains that have behavior intermediate between those of the fast and the slow chains; some of these have folds that are well away from the center of the chain. For 10 s 1 De 40, Fig. 5 d, there is again a band of fast chains and a lower bound representing chains folded in half. The lower bound for 10 s 1 lies even below that for 1s 1, and the folded chains play a more important role in the average stretching behavior than they do at 1s 1. The behavior at 100 s 1 is similar to that at 10 s 1. Figure 6 a shows ẋ /, the rate of stretch averaged over all 101 chains of Fig. 5, normalized by the strain rate. This is plotted versus stretch x, for 6 strain rates between 0.25 and 1 s 1 De from 1.0 to 4. The average at each value of x is obtained by binning together all values of ẋ(x) corresponding to x values lying within 0.5 m of the given value of x. Note in Fig. 6 a that for in the range s 1 (De 2 4) there is a nearly linear relationship between ẋ and x, up to a stretch x equal to about 75% of the steady-state value for each strain rate. For lower strain rates in the range s 1 the slope is not so well approximated by a constant over this range of strains. For 2(De 10), the linear relationship between ẋ and x breaks down much more seriously; see Fig. 6 b. At Deborah numbers of 10 or greater, the onset of folded conformations leads to a sudden drop in the average stretching rate at a stretch of around 1/3 full extension; i.e., at x/l 1/3. This is the extension at which the lower bound of the envelope of slow chains breaks away from the band of fast chains in Figs. 5 c and 5 d. For 10 (De 40), the curve of ẋ / versus x in Fig. 6 b reaches a limiting behavior that is insensitive to strain rate; it contains two portions, a fast-stretching portion at low extensions that follows the same curve as that for lower extension rates, and a slow-stretching portion for x/l 1/3, where the unraveling of folds dominates the stretching. C. Influence of initial conformation We find that the conformational changes that a molecule undergoes during stretching are influenced to a large extent by the starting conformation of the molecule. To illustrate this, we consider the three typical initial, relaxed, configurations shown in Fig. 7. Figures 8 10 show the progressions of conformations that each of these starting states undergoes at the strain rates a 0.25 s 1 (De 1); b 1s 1 (De 4); and c 10 s 1 (De 40). Thus, in each figure the same starting configuration is taken for each of the three strain rates, and the same sequence of random numbers is used to generate the Brownian fluctuations during the simulations. For clarity, the vertical axes in Figs are expanded so that chain appears artificially dilated in the vertical direction relative to the horizontal.

13 SIMULATIONS OF A DNA MOLECULE 279 FIG. 6. Normalized average rate of stretch ẋ / averaged over 101 chains versus extension x for a (De 0.5), 0.25, 0.35, 0.5, 0.75, and 1 s 1 (De 4) and b 1, 2, 4, and 10 (De 40) s 1. The results for 10 s 1 were obtained for N 80 beads, for reasons given in the text. 1. Facile, dumbbell-prone, starting configuration In Fig. 8, the starting configuration is one that is relatively elongated in the stretch horizontal direction x init /L 0.06 see Fig. 7, bottom, and the molecule is therefore rapidly stretched to its final extension at each strain rate. The final, or steady-state extension is achieved after about 5 Hencky strain units at 0.25 s 1, and 3 strain units at both 1 and 10 s 1. The stretching transitions in all three cases are dominated by dumbbell conformations in which the central strand rapidly elongates by pulling chain from the slack stored near the ends. 2. Laggard, dumbbell-prone, starting configuration The starting configuration in Fig. 9 is compressed in the stretch direction x init /L 0.01 see Fig. 7, middle, and hence this laggard molecule requires 8 strain units to reach steady state at 0.25 s 1, and 6 units at both 1 and 10 s 1. In each case, there is a long inception period around 3 strain units during which little stretching

14 280 LARSON ET AL. FIG. 7. Typical configurations sampled from the equilibrium no flow ensemble of chains with L 67.2 m, N 40. These configurations are used as starting states for the runs in Figs The abscissa and ordinate are scaled so that a distance of unity corresponds to a fully extended molecule. FIG. 8. Sequence of conformations during extensional flow for an initial conformation shown in Fig. 7, bottom that predisposes the chain to rapid stretching in a dumbbell conformation, for which the chain ends are curled up and the midsection is taut. a 0.25 s 1 (De 1), b 1s 1 (De 4), c 10 s 1 (De 40). The lowest image is the initial conformation which is the same in a, b, and c, and successive images are separated by 0.8 Hencky strain units. The image dimensions are expanded vertically transverse to flow for clarity.

15 SIMULATIONS OF A DNA MOLECULE 281 FIG. 9. As in Fig. 8, except that the initial conformation Fig. 7, middle has a small projected length in the stretch direction, predisposing the chain to delayed stretching in a dumbbell conformation at 0.25 (De 1), and 1 s 1 (De 4), and to a kinked conformation at 10 s 1 (De 40). occurs; thereafter stretching occurs rather rapidly. The mode of stretching is that of a dumbbell at 0.25 and 1 s 1, but at 10 s 1, a knot or kink conformation occurs in which the rolled up portion of the chain is in the middle. 3. Fold-prone starting configuration The starting configuration in Fig. 10 is not particularly compressed in the stretch direction x init /L 0.025; see Fig. 7, top ; however, the two ends of the molecule are hardly separated at all from each other in the stretch direction, while the middle of the molecule is displaced in the stretch direction from the two ends. This starting state FIG. 10. As in Figs. 8 and 9, except that the initial conformation Fig. 7, top predisposes the chain to form a folded conformation, which is anomalously slow to unravel at high extension rates.

16 282 LARSON ET AL. FIG. 11. Unraveling of a polymer chain from three starting configurations that are identical, except for a rotation; the configurations just after the start of extensional flow ( 0.1) are shown in enlarged form at the bottom of the figure. The figures above the enlarged ones correspond to successively higher strains; i.e., 0.9, 1.7, 2.5, 3.3, 4.1, and 4.9. In b, the starting configuration is rotated by 18 relative to a, while in c the rotation is 90. requires 9 strain units to become fully stretched at 0.25 s 1. It takes even longer in units of strain to be stretched at the higher strain rate; almost 10 units at 1s 1, and at 10, the molecule is not fully stretched and is still stretching very slowly even at the final strain of 9.6. The slow stretching at 10 s 1 Fig. 10 c is obviously due to the fold that appears almost precisely in the center of this molecule at a strain of around 3. The molecule is also folded near the middle at 1s 1, but for this lower strain rate the greater relative importance of Brownian motion apparently eases the unraveling process, and the chain becomes completely unrolled by a strain of 9 or so. At 0.25 s 1, the strain rate is too weak to stretch subsections of the molecule very much, and there is in any event plenty of time for potential folds to unravel themselves by Brownian motion; so the stretching in this case is not significantly slower than that in Fig. 10 a. Thus, for this fold-prone starting configuration, a retrograde dependence of relative stretching rate on strain rate is evident: the faster the strain rate, the more strain is required for unraveling. 4. Rotation of initial configuration The influence of the initial configuration on the unraveling dynamics can be further illustrated by rotating the fold-prone configuration shown at the top of Fig. 7. Figure 11 a shows the unraveling of this configuration when it is left unrotated; i.e., Fig. 11 a is the stick-figure version of Fig. 10 c. If the initial configuration is rotated by 18, the unraveling of this chain is much faster; complete stretching is attained in around 6.2 strain units; see Fig. 11 b and Fig. 12. Even rotations as small as 1.8 have a significant effect on the unraveling process; see Fig. 12. These small rotations accelerate stretching, but do not change the mode of stretching. With a 90 rotation, however, the chain unravels via a half-dumbbell configuration, and does so very quickly in 4 strain units ; see Fig. 11 c and Fig. 12. These simulations confirm the strong effect of rotation of the initial configuration on stretching suggested by Smith and Chu 1998.

17 SIMULATIONS OF A DNA MOLECULE 283 FIG. 12. The stretch vs strain for starting configurations that are identical to that for Fig. 11 a, except for a rotation by an angle ranging from 1.8 to Molecular predestination In the above, we have highlighted the influence of the initial configuration on the unraveling process above the critical Deborah number De c 0.5. But how complete is this influence? Is the chain predestined by its initial conformation to unravel in just one particular way, or is its unraveling largely controlled by Brownian fluctuations it experiences during the stretching process? In the limit of infinite Deborah number, the unraveling must become completely predestined, but one might expect this limit to be approached only at very high Deborah number. To determine the degree of predestination as a function of Deborah number, we create bead chains with identical initial configuration, and subject each of them to the same series of extension rates, while allowing each to retain its own distinct sequence of random numbers to produce distinct Brownian-force histories during stretching. Figure 13 a shows the extension vs. strain curves for a starting state that produced a rapidly extending dumbbell at 1s 1 (De 4) in Fig. 4 c. At a strain rate of 1s 1, the strain required to reach complete stretching varies from one run to the next by about 1 Hencky unit or so; nevertheless, for all sequences of random numbers, the stretching remains rather rapid. The configurational transition in each case remains that of a facile dumbbell. Thus, in this case, the effect of the molecule s initial configuration largely wins out over the Brownian-force history during stretching. Figure 13 b compares curves of stretch versus strain at 1s 1 for different Brownian histories, but all with the same initial configuration, one that produced a roughly half-folded conformation in Fig. 4 c. This starting configuration leads to more variation in stretching rate than does that of the facile dumbbell. Although its initial stretch is very tiny contrast Figs. 13 b with 13 a at zero strain, the molecule can, under some sequences of Brownian fluctuations, escape the fate of a folded molecule, and become instead a rather rapidly stretching dumbbell. Note, however, that even in the fastest case, 4 Hencky strain units are required for unraveling; the initial configuration of Fig. 13 a usually stretches faster than this. Thus, for 1, some configurations predestine the molecule to become a rapidly unraveling dumbbell, while other initial configurations leave the molecule s unraveling behavior largely defined by Brownian fluctuations.

18 284 LARSON ET AL. FIG. 13. Fractional extension x/l vs strain at a strain rate of 1s 1 (De 4) for 10 chains with identical initial conformations but different Brownian forces during stretching for a an initial conformation favoring rapidly stretching dumbbell configurations, and b an initial conformation favoring folded conformations. We expect the influence of the initial configuration to become even more important at higher strain rates, where Brownian motion has less time to exert an influence. Figure 14 a shows that, indeed, for 10 s 1 (De 40), starting states prone to form easily unraveling dumbbells have stretching behavior that is almost independent of Brownian fluctuations. Starting states prone to form folds in the middle of the chain are predestined to do so every time at this higher strain rate; see Fig. 14 b. However, Fig. 14 b also shows that the duration of time that the fold is able to frustrate the unraveling process is very sensitive to Brownian fluctuations. Thus, Brownian motion can assist, or frustrate, the tendency of the flow to unravel the fold. Even for 100 s 1 (De 400), the Brownian motion still influence the rate of unraveling of the fold; see Fig. 15 b. For dumbbell conformations at 100 s 1, Brownian motion plays almost no role; see Fig. 15 a. The persistence of the influence of the starting state on the evolution of chain conformation in extensional flows at high Deborah number has important implications regarding the accuracy of molecular models that use preaveraging approximations to replace the detailed molecular-level description by ensembled-averaged conformation tensors, such as the second-moment tensor of the distribution function of endto-end vectors. This limitation of preaveraging is already evident in simple dumbbell models Keunings 1997 ; Herrchen and Öttinger For multiple-bead models,

19 SIMULATIONS OF A DNA MOLECULE 285 FIG. 14. Same as Fig. 13, except 10 s 1 (De 40). preaveraging over conformations will eliminate any distinction in unraveling behavior between subpopulations that form folds and those that do not. IV. COMPARISON WITH DNA EXPERIMENTS The predictions of the model are now compared qualitatively and quantitatively with the lambda-phage DNA experiments of Smith and Chu As discussed in Sec. II A eff for lambda-phage DNA we use a 20-bead model with L 21 m, and p m. The relaxation time of this chain, determined as described in Sec. II A, is 0.90 s. A. Stretching and rate of stretch Figure 16 compares the experimental and simulated molecular extension versus fluid strain t at two different Deborah numbers, De 2 and De 48 for an ensemble of individual molecules. Similar comparisons are made in Fig. 17 for the instantaneous rate of stretch ẋ versus stretch x, again for two Deborah numbers. The range of behaviors in Figs. 16 and 17 indicate good qualitative agreement between experiment and simulation. A quantitative comparison is only possible for averages over large ensembles. Figure 18 shows a direct comparison between experiment and simulation of the average stretch x versus strain at six different values of De, while Fig. 19 reports a similar comparison for the averaged rate of stretch versus stretch. These figures show nearly

20 286 LARSON ET AL. FIG. 15. Same as Figs. 13 and 14, except 100 s 1 (De 400). quantitative agreement over the entire range of Deborah numbers, even very deeply into the nonlinear regime at De 55. The only significant discrepancy between simulations and experiments is that the pronounced dip in ẋ at high De in Fig. 19 in the simulations is absent or small in the experiments. Figure 20 summarizes the agreement between simulations and experiments over the whole De range. Note that for De 10, asymptotic behavior is approached that falls short of affine stretching, in which the molecular stretching would equal the stretching of the surrounding fluid element. Thus, the highest Deborah numbers in the experiments and simulations reach into an asymptotic regime in which the behavior is no longer sensitive to De. The simulations show that in this high-de regime, the polymer molecule is compressed after a strain of unity or so into a quasi-one-dimensional object, consisting of nearly fully stretched pieces joined by sharp folds. Thereafter, stretching of the molecule occurs by way of fold-fold annihilation, producing longer straight pieces, until the chain is completely extended. Predictions of a simple one-dimensional kink dynamics model Larson 1990 ; Hinch 1994 that accounts for motion of these folds, with Brownian forces ignored since they are small compared to viscous and elastic forces are shown in Fig. 21. The term kink used in connection with the kink dynamics model means a sharp fold; in the DNA studies kink refers instead to a roll or clump of DNA in the interior of the chain, away from the ends. Note the good agreement of Figs. 21 a 21 d with the De 55 predictions and experiments in Figs These results show that the diversity in chain-

21 SIMULATIONS OF A DNA MOLECULE 287 FIG. 16. Extension x vs strain at two Deborah numbers, from DNA experiments and simulations. stretching behavior at high De does not derive from Brownian motion during the stretching, but rather from the distribution of initial chain conformations. Of course, this initial distribution results from Brownian sampling of the equilibrium distribution in the quiescent state before stretching begins. B. Configurational transitions A much more detailed comparison of experiment and simulation is possible. Smith and Chu 1998 have categorized the various unraveling transitional states as folded, kinked, dumbbell, half dumbbell, and coiled. Cartoons of these states along with the corresponding fluorescence brightness distribution observed microscopically are shown in Fig. 22. The folded states are further defined as fold2/5 and fold2/3, the latter meaning that at least 2/3 of the apparent molecular length is taken up in the fold, while in the former at least 2/5 of the length is taken up. More detailed definition of these states can be found in Smith and Chu 1998 and in the Appendix of this paper. Smith and Chu sorted over 100 DNA runs into the above six categories at six different Deborah numbers, from De 2.0 to De 55. This categorization was carried out by scrutinizing video-taped images of DNA molecules in extensional flow generated in the crossed-slot device. To reduce subjectivity, measurements were performed on the images to determine whether configurations met the criteria for fold2/3 or fold2/5. In addition, since molecules could unravel simultaneously or sequentially via more than one of the six transitional states, a priority scheme was developed in which the molecules were categorized in the following descending order of priority: fold2/3, fold2/5, kink, dumbbell, half-dumbbell, coil. Thus, a molecule that formed a fold2/3 during any portion of its unraveling history was assigned to the fold2/3 category, even if at some other time

22 288 LARSON ET AL. FIG. 17. Rate of stretch ẋ vs stretch x from experiments and simulations. FIG. 18. Stretch averaged over 100 or so runs versus strain at various Deborah numbers from experiments symbols and simulations lines.

23 SIMULATIONS OF A DNA MOLECULE 289 FIG. 19. Normalized averaged rate of stretch ẋ / vs stretch x at various Deborah numbers from experiments symbols and simulations lines. during the unraveling it became a dumbbell, or some other configuration of lower priority. A molecule was only denoted as coiled if at no time during unraveling did it fit some other category. Typically, a clear assignment of configuration could be made after the molecule had been stretched by a strain of around 2.3. Molecules which did not experience at least this much strain were therefore not analyzed. Very few of the runs extended to strains higher than 5. Despite efforts to make the method precise, some aspects of the experimental categorization were of necessity subjective or prone to error. The limits of optical resolution were around 0.5 m and the images were acquired at a rate of around 1 per 0.1 strain units. Thus, kinks or other features of the configuration that were either too small or FIG. 20. Average stretch x vs Hencky strain from experiments and simulations at De 2, 3.4, 13.8, 27, 48, and 55; the faster rising curves correspond to the higher values De. The dashed lines correspond to affine stretching, i.e., x x 0 exp( ); for the experiments, an offset of 1.3 m is added to account for blooming of the experimental DNA image which occurs when a highly intensified video camera is used.

24 290 LARSON ET AL. FIG. 21. Predictions of one-dimensional kink dynamics simulations Larson 1990 of 100-step random walks. a The stretch averaged over the ensemble of 101 kinked chains thin line, compared to affine stretching thick line. b The average normalized rate of stretch vs stretch. c The stretch vs strain for the first 30 of the chains. d The normalized rate of stretch vs stretch for all 101 chains. persisted for too short a time period to see clearly would be missed. To develop a basis of comparison of simulated transitional states with the experimental ones, we therefore produced videos of the simulated DNA unraveling dynamics at the highest Deborah number, De 55. The images were made comparable to the experimental ones by spacing the video frames at 0.1 strain units, and making each bead 1 m in size, to represent the degree of optical smearing seen in the experiments. These video images were viewed and analyzed visually by the same co-author Doug Smith who had earlier analyzed the experimental DNA images. His assignment of unraveling categories for the simulated chains at De 55 are shown in Fig. 23. Within experimental error, the percentages of different conformations obtained in the simulations agree with those of the DNA experiments. In addition, an automated scheme was developed for categorizing the simulation configurations, based on an analysis of the brightness distribution shown in Fig. 22. The scheme uses the prioritization used by Smith and Chu and accounts for experimental time and spatial resolution limits by incorporating a spatial cutoff of 0.5 m and a temporal cutoff of 0.1 strain units. Simulation runs were analyzed only over the range of strains from 2.3 to 5.0 Hencky units, in order to match the experimental limitations. The details of the automated scheme are presented in Appendix A. The scheme was tested and fine tuned using Doug Smith s visual categorization of the simulation runs at De 55, described above. Figure 23 shows that at De 55 the automated scheme gives nearly the same percentages of each unraveling category that were obtained subjectively from the videotaped images. Only 8 out of 100 molecules were categorized differently by the two methods, and the effects of this on the final distribution are within statistical error. This exercise shows 1 the validity of the automated method, 2 the robustness of the sub-