The Influence of Symmetric Internal Loops on the Flexibility of RNA

J. Mol. Biol. (1996) 257, 276–289

The Influence of Symmetric Internal Loops on the Flexibility of RNA Martin Zacharias and Paul J. Hagerman* Department of Biochemistry Biophysics, and Genetics University of Colorado Health Sciences Center, Denver, CO USA

Internal loops are structural elements, often highly conserved, that are found in many RNA molecules of biological importance. They consist of short stretches of sequence in which the bases in one strand are not able to form canonical pairs with bases in the other strand, and are bounded on either side by helical RNA. In an effort to examine the influence of internal loops on the relative angular orientations of the flanking helices, we have quantified the apparent bend angles for symmetric internal loops of the form An-An and Un-Un (n = 2, 4, and 6), located at the center of 150 to 154 bp RNA molecules, using the method of transient electric birefringence. This hydrodynamic method exploits the extreme sensitivity of the rate of rotational reorientation of the RNA molecules to the presence and magnitude of internal bends and/or points of increased flexibility. The birefringence decay behavior of the loop-containing RNA molecules was found to be much less strongly influenced by the presence of symmetric internal loops than by bulges of the same sequence and size. This general observation is mirrored by the electrophoretic behaviour of the loopcontaining molecules, which are much less strongly retarded on polyacrylamide gels than are corresponding, bulge-containing RNA molecules. The apparent bend angles for the symmetric loops range from 020° to 40° as n is increased from 2 to 6, with a marginal shift to smaller angles in the presence of Mg2+. The apparent angles were similar when represented either as fixed bends of the specified angles (static representation), or as points of increased flexibility of specified rootmean-square angle (dynamic representation). For the latter representation, the corresponding angular dispersion would correspond to a loop persistence length of 060 to 150 Å, compared to 700 Å for duplex RNA, and depending slightly on sequence and buffer conditions. 7 1996 Academic Press Limited

*Corresponding author

Keywords: RNA structure; ribozymes; transient electric birefringence; RNA bulges; hydrodynamics

Introduction In most RNA molecules of biological importance, segments of helix are interrupted by regions in which the sequence of each strand precludes classical Watson–Crick pairing. In the simplest case where two nominally non-base-paired strands are Present address: M. Zacharias, Max Delbru¨ck Centrum Berlin, Robert Ro¨ssle Str., 13122 Berlin and Humboldt Universita¨t, Institut fu¨r Biologie, Theoretische Biophysik, Invaliden Str. 42, 10115 Berlin. Abbreviations used: dsDNA, dsRNA double-stranded DNA and double-stranded RNA, respectively; TEB, transient electric birefringence; r.m.s., root-mean-square; BD, Brownian dynamics; RZ, Roitman–Zimm; nt, nucleotide. 0022–2836/96/120276–13 $18.00/0

flanked by two helices, the element is termed an internal loop (Jaeger et al., 1993). If the two strands are of equal length, the loop is termed symmetric. If the strand lengths are unequal, the loop is asymmetric, in which case the unapposed strand is termed a bulge. Knowledge of the increased flexibility and structural diversity of internal loops would be helpful for evaluating the relationship between secondary and tertiary structure in RNA. In particular, internal loops are considered to be regions within larger RNA structures where the direction of the helix axis can change, allowing larger RNAs to form more compact structures (Jaeger et al., 1993), yet there is very little information pertaining to their global conformations in solution. In this regard, such knowledge 7 1996 Academic Press Limited

277

Internal Loops and RNA Flexibility

would also be useful in considering loops as points of articulation in the function of RNAs. Finally, knowledge of the sequence dependence of loop conformation and dynamics would shed additional light on the importance of non-Watson–Crick interactions in stabilizing RNA structure, as in the case of the E loop of 5 S rRNA (Varani et al., 1989; Wimberly et al., 1993). Thermodynamic studies of double-stranded RNA (dsRNA) and double-stranded DNA (dsDNA) containing internal loops have shown that interstrand interactions within the loop region are generally less favorable than classical Watson–Crick base-pairing, thus allowing more conformational flexibility (SantaLucia et al., 1991a,b; Patel et al., 1984a,b; Aboul-ela et al., 1985; Hsieh & Griffith, 1989). However, depending on sequence, internal loops can either stabilize or destabilize adjacent dsRNA or dsDNA (SantaLucia et al., 1991a; Aboul-ela et al., 1985). In line with the above, chemical modification studies on RNAs with non-base-paired regions have demonstrated that internal loops are generally more accessible to chemical modification and nuclease cleavage than is the surrounding dsRNA (Ehresmann et al., 1987; Knapp, 1989; Chen et al., 1993). Moreover, NMR and crystallographic studies have found increased structural diversity within loop regions in both DNA and RNA (Patel et al., 1984a,b; Varani et al., 1989; SantaLucia et al., 1991a; Holbrook et al., 1991; Wimberly et al., 1993; SantaLucia & Turner, 1993). Finally, an effort has been made to quantify the increased flexibility in DNA due to the presence of a symmetric internal loop by comparing the propensity for ligase-catalyzed circle formation in a loopcontaining DNA molecule with that of a full duplex control (Kahn et al., 1994); however, the corresponding experiment has not been performed for RNA. In order to lay a foundation for an examination of the longer-range conformational properties of internal loops in RNA, the current investigation has examined symmetric internal loops of the form An -An and Un -Un , placed at the center of an otherwise dsRNA molecule, in an effort to determine the influence of the loops on the relative angular orientations of the flanking helices. The current approach is analogous to that used for the characterization of RNA bulges (Zacharias & Hagerman, 1995a), and involves the method of transient electric birefringence (TEB) to quantify the bending and/or additional flexibility introduced by the loop elements. The TEB approach, which measures the rotational diffusion of the RNA molecules of interest, is extremely sensitive to the presence of bends in the helix axis, with bent and/or flexible molecules displaying faster rotational reorientation than full-duplex control molecules of the same contour length. The TEB approach has been used by our laboratory for the study of several non-helix elements in RNA, including the central, three-helix branch of 5 S rRNA (Shen & Hagerman, 1994), a self-cleaving (hammerhead) RNA (Gast et al., 1994; Amiri &

Hagerman, 1994), the anticodon-acceptor interstem angle of yeast tRNAPhe transcripts (Friederich et al., 1995), and both simple bulges (Zacharias & Hagerman, 1995a) and ligand-induced conformational changes in the TAR bulge of HIV (Zacharias & Hagerman, 1995b). The birefringence decay curves for short (ds) RNA or DNA molecules (e.g. <150 to 200 bp) are predominantly single-exponential, reflecting principally the end-over-end motions of the molecules. However, due to the finite persistence lengths of both DNA (Hagerman, 1988) and RNA (Gast & Hagerman, 1991; Kebbekus et al., 1995), there exist populations of molecules with distinctly non-linear conformations (Hagerman, 1988). These conformers contribute to a small (05 to 10%), faster component of the birefringence decay curve; however, points of bending and/or increased flexibility lead to both a significant increase in the amplitude of this component and a decrease in the time constant of the major (slower) decay component. In fact, this behavior is observed for both uridine and adeninecontaining loops. Therefore, one aim of the current study is to represent the experimental (TEB) behavior of the internal loops both as a fixed bend and as a simple increase in flexibility. In the latter case, the apparent added flexibility is compared with the intrinsic flexibility of an equivalent stretch of RNA helix. The central result is that the static angles for each loop are remarkably similar to the r.m.s. angles for the dynamic representations; however, as will be discussed, the ramifications of these two models are quite different. Finally, for the dynamic representation of the loop (increased flexibility), the rigid-body, equilibriumensemble approach for the interpretation of the birefringence decay curves (Hagerman & Zimm, 1981) is compared with both Brownian dynamics simulations (Allison & McCammon, 1984; Lewis et al., 1988; Allison & Nambi, 1992) and a numerical solution of the diffusion equation for a flexible, three-bead (trumbell) model (Roitman & Zimm, 1984a,b). These three approaches lead to quite similar results for the apparent loop flexibilities, namely, an r.m.s. angular fluctuation (per loop base-‘‘pair’’) of 016(25)° in the absence of Mg2+, and 013(24)° in the presence of Mg2+. These quantities are significantly larger than the r.m.s. angular fluctuation of 05.1° (per Watson–Crick base-pair) for an RNA helix with a persistence ˚. length of 700 A

Results The electrophoretic mobilities of RNA molecules containing symmetric An -An or Un -Un internal loops are only slightly reduced from the corresponding mobilities of linear duplex RNA The current approach for the production of symmetric-loop heteroduplex RNA molecules is entirely analogous to the approach described earlier

278 for the study RNA heteroduplexes possessing centrally placed An or Un bulges (Zacharias & Hagerman, 1995a). In the present scheme, pairs of RNA transcripts of length (148 + n) nt are annealed to form heteroduplex molecules possessing centrally located, An -An or Un -Un internal loops. In order to provide an initial, qualitative assessment of the characteristics of these loops, heteroduplex RNAs with either the purine and pyrimidine tracts were subjected to electrophoresis on polyacrylamide gels (Figure 1). The mobilities of the loop-containing molecules are observed to decrease, relative to their fully duplex counterparts, as n increases; however, in no instance is the relative mobility of a loop-containing RNA less than 00.96 (4% reduction for the A6-A6 loop) relative to the mobility of its duplex counterpart. This behavior stands in stark contrast to the behavior of RNA molecules with centrally located bulges, for which it has been demonstrated (Bhattacharyya & Lilley, 1989; Bhattacharyya et al., 1990; Tang & Draper, 1990, 1994; Zacharias & Hagerman, 1995a,b) that the bulge-induced bends lead to substantial reductions in electrophoretic mobility (e.g. 00.32 for an A6 bulge; Zacharias & Hagerman, 1995a). Reduced mobilities have also been noted in DNA molecules possessing points of increased, isotropic flexibility, created by singlestranded ‘‘gaps’’ in otherwise duplex helix (Mills et al., 1994). Thus, the gel results indicate that the internal loops are neither substantially bent, nor as flexible as a corresponding length of single-stranded polymer. Within both the An -An and Un -Un series, the magnitude of the mobility reduction increases with increasing n. Moreover, in the absence of Mg2+, the mobilities of the uridine loops appear to be similar or only slightly smaller than the mobilities of the corresponding adenine loops (Figure 1A). In the presence of Mg2+, the U4-U4 and U6-U6 loops appear

Internal Loops and RNA Flexibility

to migrate slightly faster than their adenine counterparts (Figure 1B), reminiscent of the behavior of Un versus An bulges (Zacharias & Hagerman, 1995a). However, in the current instance, the observed difference might be due to small reductions of the relative mobilities of the adenine loops. Transient electric birefringence measurements of loop-containing RNAs indicate that loops of the form An -An or Un -Un (nE6) create only modest distortions of the helix axis In order to provide a quantitative assessment of the influence of internal loops on the angular positions of the flanking helices, TEB measurements were performed on heteroduplex RNA molecules possessing loops of the form An -An and Un -Un (n = 2, 4 and 6) along with their 152, 154, and 156 bp duplex controls. The results of those measurements are presented in Table 1, with representative birefringence decay curves displayed in Figure 2. As anticipated from the gel results and from earlier results with RNA bulges (Zacharias & Hagerman, 1995a,b), the loop-containing RNAs undergo more rapid rotational diffusion than do their linear counterparts. Furthermore, slight decreases in the relative amplitudes of the slower (terminal) decay component are observed for all loop-containing RNAs (Table 1 and Figure 2). These two features of the birefringence decay curves indicate that the loops introduce small, albeit significant angular distortions of the overall helix axis. The addition of Mg2+ to the TEB buffer reduces the difference between the terminal decay times of the loop-containing RNAs and their linear controls (Table 1). Moreover, no strong sequence-dependent effect was observed, in contrast to the observations with An versus Un bulges (Zacharias & Hagerman, 1995a). The apparent absence of such an effect

Figure 1. Electrophoretic behavior of the internal-loop-containing RNA molecules (In ) relative to bulge-containing RNAs (Bn ) and linear RNA controls, either in the absence of Mg2+ A, or in the presence of 5 mM Mg2+ B, (see Materials and Methods). Pairs of lanes labeled In or Bn represent A or U species (left to right), in the following fashion. In : An -An , Un -Un ; Bn : An , Un . For the linear species, the lengths are indicated above the lanes to the left of each gel. For each species, n represents the number of nt in a single strand; for the bulges, the extra bases are unapposed.

279

Internal Loops and RNA Flexibility

Table 1. Summary of TEB results and equilibrium ensemble analysis tslowc (ms)

tloop /thelixd

Fe (deg.)

A. In the absence of Mg 2+ 150 bp linear 0.91(20.03) A2-A2 0.87(20.03) U2-U2 0.83(20.03) 152 bp linear 0.90(20.03) A4-A4 0.85(20.02) U4-U4 0.83(20.03) 154 bp linear 0.89(20.02) 0.77(20.03) A6-A6 U6-U6 0.79(20.03)

2.79(20.03) 2.68(20.03) 2.66(20.04) 2.84(20.03) 2.60(20.03) 2.56(20.02) 2.90(20.04) 2.62(20.04) 2.59(20.04)

1.0 0.960 0.953 1.0 0.915 0.910 1.0 0.903 0.893

0 20 2 8 23 2 8 0 32 2 6 33 2 6 0 38 2 6 39 2 6

B. In the presence of Mg 2+ 150 bp linear 0.92(20.02) 0.89(20.02) A2-A2 U2-U2 0.86(20.03) 152 bp linear 0.92(20.02) A4-A4 0.89(20.03) U4-U4 0.88(20.02) 154 bp linear 0.90(20.03) A6-A6 0.79(20.03) 0.80(20.02) U6-U6

2.32(20.02) 2.28(20.03) 2.25(20.03) 2.36(20.02) 2.25(20.03) 2.28(20.04) 2.45(20.03) 2.30(20.03) 2.36(20.02)

1.0 0.983 0.970 1.0 0.953 0.966 1.0 0.938 0.963

0 12 2 10 15 2 10 0 22 2 8 19 2 8 0 29 2 6 21 2 6

RNAa

aslowb

f(F)f (deg.)

Ploopg ˚) (A

16 2 7(23) 18 2 7(25)

70 55

16 2 5(32) 17 2 5(34)

70 60

15 2 5(37) 16 2 5(39)

80 75

12 2 7(17) 14 2 7(19)

120 100

14 2 4(28) 11 2 4(22)

95 150

12 2 4(30) 11 2 4(26)

120 165

The numbers in each row represent averages of 15 independent birefringence measurements from three separate RNA preparations, and are reported as the mean values21 standard error of the mean. The TEB measurements were performed as described in Materials and Methods. a RNA duplex or loop-containing heteroduplex. Nn -Nn refers to the loop base (N = A or U) and n refers to the loop length (nt per strand). b Fractional amplitudes associated with the slow phase in the double-exponential birefringence decay curves. c Decay times for the slow phase of the birefringence decay curves. d Ratios of the slow decay times for the loop-containing heteroduplex RNAs (tloop ) to the corresponding decay time for the linear control RNA (thelix ). e Bend angles for the internal loops, under the assumption that the bend is static (i.e. possessed of no more flexibility than the surrounding helix). In the current work, upper case symbols (e.g. F) refer to the entire loop, whereas lower case symbols (e.g. f) refer to per-base-pair quantities. Symbols without brackets indicate fixed quantities; symbols within angle brackets (e.g. f) are r.m.s. quantities derived from the variance of the angle distribution. Thus, f = f21/2; F = n 1/2 × f, where n is the number of base-pairs in the loop. The corresponding r.m.s. quantities for native helix are designated by the subscript ‘‘hx’’ (e.g. fhx ). f Per-bp (r.m.s.) polar angular fluctuation required to give the observed t-ratios in column 4, under the assumption that the internal loop introduces an isotropic increase in flexibility with no static or directional component; thus: f = [f2tilt  + f2roll ]1/2 = (2 × rise/Ploop )1/2, where the rise/bp is assumed to equal that of ˚ ), and Ploop is the loop persistence length. The numbers in parentheses (F) are given helical RNA (2.8 A by n 1/2 × f. g Loop persistence lengths required to give computed t-ratios (DIFFROT) equal to the observed ratios in column 4.

might be due to the small overall difference between the loop-containing molecules and the duplex controls, which would make the detection of sequence-dependent differences more difficult than for bulges. In this regard, NMR studies of symmetric internal loops, although showing structural distortions in mismatched loops, indicate overall A-type structure (Varani et al., 1989; SantaLucia & Turner, 1993). Recent X-ray structures of two uridine-containing internal loops indicate ˚ relative changes in rise within the loop of only 0.1 A to A-form RNA (Baeyens et al., 1995; Holbrook et al., 1991). However, a change (reduction) in ˚ is necessary in the case per-base-pair rise of 0.4 A of the 6 bp loop to account for changes in the decay time that are of comparable magnitude to the experimental error of the decay times. Thus, although the current study cannot exclude the possibility of a large change in rise as a component of the observed reduction in TEB decay times, the X-ray and NMR results predict that changes in rise

would result in changes in t that would fall within our experimental error. Analysis of the decay ratios (tloop /thelix ) in terms of either static or stochastic models for the loop-induced bends using the equilibrium-ensemble approach Estimates of the magnitude of bends in RNA (and DNA) are usually based on a static-bend model in which the bend itself possesses no more flexibility than that of the surrounding helix. However, in the absence of additional information, one should always consider the possibility that an apparent bend may actually reflect an increase in the intrinsic flexibility of the element of interest (Mills et al., 1994). It is clear from previous studies that RNA bulges possess at least some ‘‘static’’ character (or strong directional asymmetry of flexibility), since the mobilities of RNA molecules with pairs of bulges display mobilities that are strongly depen-

280

Internal Loops and RNA Flexibility

Friederich et al., 1995; Zacharias & Hagerman, 1995a,b).

dent on the helical phasing of the bulges (Bhattacharyya & Lilley, 1989; Bhattacharyya et al., 1990; Tang & Draper, 1990, 1994). In the current investigation we have chosen to present both limiting cases for the loop-induced distortions (Table 1). In this section, the ratios of the terminal birefringence decay times (Table 1, column 4) have been interpreted using the equilibrium-ensemble approach (Hagerman & Zimm, 1981) in a fashion that is entirely analogous to previous TEB analyses of non-helical elements in RNA (Shen & Hagerman, 1994; Gast et al., 1994; Amiri & Hagerman, 1994;

The static bend As in the analysis of bulge-induced bends in RNA (Zacharias & Hagerman, 1995a), the experimental ratio, tloop /thelix , was compared to a plot of computed t-ratio (designated t/tc ) versus bend angle (F) for an ensemble of chains; each chain in a given ensemble possessed a central bend of F degrees, subject only to the intrinsic flexibility of the

A

B

C Figure 2. Birefringence decay curves for internal loops of the form An -An (w) and Un -Un (W), where n = 2 in A, 4 in B, and 6 in C. For comparison, in order of decreasing vertical position in each panel (·), are decay curves for the linear control (150, 152, and 154 bp in A, B, and C, respectively), the Un bulge, and the An bulge. For each decay curve, the experimental data points are displayed, with the double-exponential curves obtained from the LM analysis indicated as continuous lines.

Internal Loops and RNA Flexibility

RNA helix. The results of this analysis are presented in Table 1, column 5. The central finding of this analysis is that the apparent (static) bend angles for the symmetric internal loops are all substantially smaller than those of the corresponding bulges (same n).

The stochastic bend (increased flexibility) Gel electrophoretic studies on the phasing of internal loops in DNA suggest that there is little static (directional) character associated with symmetric internal loops (at least for DNA; Kahn et al., 1994). Accordingly, the tloop /thelix values for the symmetric internal loops listed in Table 1 (column 4) have been modeled also in terms of an isotropic (i.e without directional bias) increase in the flexibility of the base-‘‘pairs’’ within the loop. Ensembles of chains were generated in which the effective persistence length within the loop (designated Ploop ) was reduced until the computed (equilibrium-ensemble) value of t/tc was equal to each of the experimental ratios listed in Table 1, column 4 (see Materials and Methods, Figure 5). In the current, simplified model, Ploop is assigned to each base-‘‘pair’’ within the loop. Thus, the r.m.s. angle for each base-pair within the loop, f, is defined as (2 × rise/Ploop )1/2, where the rise is taken as the normal helix rise per base-pair in RNA (for angle representations, see the legend to Table 1). The values of f and Ploop are listed in Table 1, columns 6 and 7, respectively. As expected, for a given Ploop , the computed t-ratios decrease with increasing loop length (Materials and Methods, Figure 5). However, the apparent Ploop values are nearly independent of the length of the loop, with an average value of ˚ in the absence of Mg2+, and 0130 A ˚ in the 070 A 2+ presence of Mg . Moreover, there does not appear to be any significant difference in the flexibilities for the two loop sequences in the absence of Mg2+, with perhaps a slight difference appearing in the presence of Mg2+. Finally, it should be noted that the effective, total r.m.s. angles for the loops (F = n 1/2 × f) (Table 1) are remarkably close to those obtained for the static case, underscoring the need for additional experimental input to further characterize the bend. Brownian dynamics simulations are able to reproduce the birefringence decay behavior of the loop-containing RNAs The approach taken in the previous section basically computes the terminal (ensemble-averaged) decay time for a collection of rigid chains in which flexibility is introduced through the dispersion of the inter-base-pair angles. Thus, all effects due to dynamic chain rearrangements during the decay process itself are neglected. As a consequence, the equilibrium-ensemble approach is incapable of providing a description of the faster component in the birefringence decay curve. The complete birefringence decay process can, however,

281 be reproduced using either Brownian dynamics simulations of bead representations of wormlike chains (Allison & McCammon, 1984; Lewis et al., 1988; Allison & Nambi, 1992), or through a numerical solution of the diffusion equation for a three-bead (trumbell) model (Roitman & Zimm, 1984a,b). In particular, these latter methods yield relative amplitudes for the fast and slow decay components in addition to the decay times. In Figure 3, Brownian dynamics simulations are displayed for both three-bead (A) and five-bead (B) representations of RNA molecules possessing a single, central internal loop, modeled as a central point of increased flexibility (see Materials and Methods). The computed decay curves for either the three or five-bead model, using an effective ˚ , agree well with the persistence length of 0700 A experimental curves for the linear controls (Figure 2 and Table 1). Both models predict essentially two-exponential behavior in line with previous work (Roitman & Zimm, 1984a,b; see also Table 3). Moreover, both models predict that 090% of the decay amplitude is represented by the slower decay component (Table 2 and Figure 3), essentially identical to the experimental decay curves for the linear RNA controls (Figure 2 and Table 1). The slow decay times for both three and five-bead models are close to those of the experimental curves in the absence of Mg2+. Interestingly, the slow (experimental) decay times are reduced slightly in the presence of Mg2+, in accord with previous observations (Gast et al., 1994; Kebbekus et al., 1995; Zacharias & Hagerman, 1995a). This latter observation has been attributed to a slight (04%) reduction in the rise/bp of RNA in the presence of Mg2+ (Kebbekus et al., 1995). For the analysis of the birefringence decay curves of the loop-containing RNAs, only decay time ratios and relative (slow) amplitudes were used. In the case of the five-bead model, the relative amplitudes of the slow decay components agree with their corresponding experimental values to within a few percent (Tables 1 and 2). The three-bead model predicts a slightly smaller amplitude for the slow component of the linear controls, and fast decay times that differ from the experimental values by up to 50% (data not shown). Thus, it appears to be necessary to use models with more beads in order to better simulate the fast decay times, although the relative amplitudes are well-represented by the five-bead model (Allison & Nambi, 1992). The Brownian dynamics results for the three-bead model (BD; Table 3) are in good agreement with the predictions of the Roitman–Zimm model (RZ; Table 3) for a flexible trumbell. Similar agreement has been reported for Brownian dynamics on a three-bead model representing duplex DNA (Lewis et al., 1988). Notwithstanding the possibility of insufficient convergence in the Brownian dynamics simulations, the small difference between the two models, more apparent for the larger bend angle flexibilities, might be due to the inclusion of hydrodynamic interactions in the Brownian dynam-

282

Figure 3. Computed birefringence decay curves from Brownian dynamics simulations for the three-bead A, and five-bead B, representations of the RNA species currently being considered. In order to mimic a central flexible element (internal loop), the restoring force constant for the central bend angle was varied from 1.0 to 0.25 (three-bead model), and from 1.0 to 0.125 (five-bead model) relative to the force constant of a chain representing a linear control helix with a persistence ˚ (force constant = 1.0). The data points length of 700 A represent the averages of 2 × 103 to 5 × 103 trajectories; the continuous lines represent the LM fits to the simulated decay curves (Materials and Methods). The angle distribution functions for the central bend angles corresponding to each ensemble of simulations (threebead and five-bead) are displayed in the insets to each panel.

ics simulations (absent in RZ), or that bead overlap is allowed in the Roitman–Zimm model (excluded in BD).

Internal Loops and RNA Flexibility

For a given central bend flexibility, the predicted slow decay time ratios, t/tc , are slightly smaller than the corresponding values from the equilibrium-ensemble approach (Materials and Methods, Figure 6). In order to gauge the consequences of such a difference, relative slow decay times and relative slow amplitudes from the simulations (Table 2) were compared with the experimental decay curves for the loop-containing RNAs. Relative force constants for the central bend angle were chosen that provide optimal fits to the experimental decay times and amplitudes; this process was carried out by interpolation within the data set given in Table 2. The corresponding increases in flexibility of the central bend angle were translated into corresponding decreases in Ploop (Table 2). Comparing the experimental tloop / thelix ratios with the ratios, t/tc , from the Brownian dynamics simulations yields a range in Ploop values ˚ in the absence of Mg2+, and 80 to of 60 to 120 A ˚ 140 A in the presence of Mg2+, very close to the results obtained from the equilibrium-ensemble approach. In addition, the Brownian dynamics approach predicts a correlation between the flexibility of the central element and the relative amplitude of the slow decay component, aslow (M. Zacharias & P. J. Hagerman, unpublished results). Therefore, the change in aslow with increasing flexibility of the central bend was compared to the change in aslow for each internal loop. This amplitude ˚ in the analysis yields a range for Ploop of 50 to 100 A ˚ in the presence absence of Mg2+, and 90 to 120 A of Mg2+, within the error of the experimental measurements, and similar to the results of the t-ratio analysis using either equilibrium-ensemble or Brownian dynamics approaches. Using the relation, f = (2 × rise/Ploop )1/2, with ˚ /bp within the loop, the an assumed rise of 2.8 A r.m.s. average angle for each base-pair within the loop is estimated to be 16(25)° in the absence of Mg2+, and 13(24)° in the presence of Mg2+, compared to 05.1° for helical RNA with a ˚ . Due to the relatively persistence length of 700 A small differences in the decay times and amplitudes for the loop-containing RNAs, relative to their linear controls, the statistical error of the experimental results is relatively high (Table 1), with the differences among the various theoretical models being smaller than the experimental imprecision.

Discussion Knowledge of the tertiary structures and intrinsic flexibilities of simple, often phylogenetically conserved elements in RNA is of fundamental importance for an understanding of the structure and energetics of larger RNAs. To this end, the principal aim of the current study was to estimate the apparent distortion of the RNA helix axis introduced by symmetric internal loops of simple sequence; such an estimate should serve as a

283

Internal Loops and RNA Flexibility

Table 2. Results of Brownian dynamics simulations of birefringence decay for the three and five-bead model chains Model

kbenda

tfastb (ms)

tslow (ms)

aslowc

t/tcd

DFe (deg.)

P6f ˚) (A

P4 ˚) (A

P2 ˚) (A

3-bead 3-bead 3-bead 3-bead 5-bead 5-bead 5-bead 5-bead

1.0 0.76 0.53 0.31 1.0 0.50 0.25 0.15

0.39 0.53 0.61 0.68 0.21 0.34 0.38 0.40

2.84 2.72 2.42 2.10 2.77 2.59 2.39 2.15

0.87 0.77 0.71 0.59 0.92 0.85 0.76 0.69

1.0 0.958 0.852 0.745 1.0 0.935 0.861 0.753

0.0 21.4 36.4 56.7 0.0 23.7 40.2 56.5

680 178 74 33 670 152 62 33

680 129 51 22 670 111 43 23

680 72 27 11 670 63 22 11

Each row represents the birefringence decay of a model chain obtained from Brownian dynamics simulations, and averaged over 2 × 103 to 5 × 103 trajectories. Each trajectory was run for 12 to 15 ms; see Materials and Methods. a Force constants for the central bend were obtained from the angular distribution for the Brownian dynamics simulations. A force constant of 1.0 represents the linear duplex control ˚ (680 A ˚ for the three-bead model; 670 A ˚ for the chain with a persistence length of 0700 A five-bead model). b The decay curves were all well-described as a sum of two exponential terms, with the two decay constants specified in the Table. c The relative (fractional) amplitude for the slow phase. d t-ratios obtained from the Brownian dynamics simulations, where t possesses a region of increased flexibility and tc represents the duplex control. In this work, t/tc represents the computational equivalent of tloop /thelix . e The additional flexibility in the bead models relative to the linear control was computed as DF = (F2flex − F2lin )1/2, where Fflex is the r.m.s. bead bend angle for bead models with increased flexibility, and Flin is the corresponding bead bend angle (r.m.s.) for bead models representing the linear control. f The internal loop persistence lengths, Pn , are obtained from the (above) additional flexibility (DF) using the following relation: Pn = [2 × (360/2p)2 × n × rise]/[n × (5.1°)2 + DF2], which represents a translation between bead and base-pair models, and where n corresponds to the loop strand (branch) length, 5.1° is the bend angle fluctuation (per bp) ˚ and rise = 2.8 A ˚ /bp. in duplex RNA with persistence length of 700 A

baseline for additional studies of symmetric internal loops of mixed sequence and of asymmetric internal loops. The current investigation has utilized a hydrodynamic approach, TEB, which exploits the fact that molecules possessing central distortions (bends, or points of hyperflexibility) will rotationally diffuse more rapidly than their full-duplex counterparts. The difference in rotational decay times can be converted into an apparent bend angle

for the element of interest; in the current instance, represented by internal loops. In the current work, we have sought to emphasize that an apparent bend angle could reflect either a fixed bend, a completely isotropic increase in local flexibility without any static component, or some combination thereof. Accordingly, we have modeled the relatively minor distortions introduced by the symmetric internal loops as either the static or

Table 3. Comparison of the results of Brownian dynamics simulations for the three bead model with the results of the Roitman–Zimm trumbell model kbenda 1.0 (1.0) 0.76 (0.75) 0.53 (0.50) 0.31(0.25)

aslow (BD)b

aslow (RZ)

Ratio

t/tc (BD)c

t/tc (RZ)

Ratio

0.87 0.77 0.71 0.59

0.85 0.78 0.67 0.48

1.02 0.99 1.06 1.23

1.0 0.96 0.85 0.75

1.0 0.95 0.87 0.78

1.0 1.01 0.98 0.96

a Relative force constant for each three-bead Brownian dynamics run obtained from the resulting bend angle distribution function (see Materials and Methods and the legend to Table 2 for details). A value of 1.0 corresponds to a three-bead model ˚ , and was chosen to representing a 150 bp chain with a persistence length of 700 A represent the duplex reference molecule (Z = 1.0 of Roitman & Zimm, 1984). The Brownian dynamics simulations include hydrodynamic interactions, whereas the trumbell model does not. Higher-order decays, comprising less than 3% amplitude in the RZ model predictions, were not compared. b aslow values from the Brownian dynamics (BD) simulations of the birefringence decay curves, or from the numerical solution of the diffusion equation for the trumbell model (RZ; Roitman & Zimm, 1984a,b). Ratio = aslow (BD)/aslow (RZ). c Ratios of the slow decay times of the models with reduced kbend to the reference models (kbend = 1.0); in this instance, ratio = t/tc (BD)]/[t/tc (RZ)].

284 dynamic extremes. Additional experiments will usually be required to assign the properties of a given bending element more accurately; however, experiments with internal loops in DNA (Kahn et al., 1994) suggest that the latter more closely resemble points of increased flexibility than static bends. Analysis of the birefringence decay curves using either the equilibrium-ensemble approach (Hagerman & Zimm, 1981) or Brownian dynamics methods (Allison & Nambi, 1992) yield approximately equal increases in the apparent flexibilities of the internal loops, corresponding to a seven- to ninefold decrease in the persistence length of RNA ˚ ), or a five- to in the absence of Mg2+ (050 to 100 A sevenfold decrease in the presence of Mg2+ (90 to ˚ ). For 3 bp internal loops in DNA, a bend 150 A angle fluctuation of 043° has been reported using ligase-catalyzed cyclization as the assay (Kahn et al., 1994). That value is larger than the current estimate of 028° for three consecutive mismatches (31/2 × 16°). This difference could be due to the maintenance of the tighter axial packing of the ˚ for RNA, base-pairs (rise/bp of 2.7 to 2.8 A ˚ for DNA). For example, compared to 3.4 A assuming (in the absence of any direct evidence) that the fluctuations within DNA and RNA loops are scaled by the (r.m.s.) fluctuations within the respective helices, 07°(DNA)/5°(RNA), the current value of 28° for the 3 bp loop would scale to 040° for the 3 bp DNA loop, quite similar to the value reported by Kahn et al. (1994). However, there is no reason at present to suppose that loops in DNA and RNA have the same characteristics relative to their respective helices. In agreement with previous results on the electrophoresis of internal loops, we have demonstrated that symmetric internal loops affect the relative electrophoretic mobilities of RNA to a much lesser degree than do extra unmatched nucleotides in one strand (bulges) (Battacharyya & Lilley, 1989). Moreover, we observe only a small sequence-dependence in the presence of Mg2+, with slightly enhanced electrophoretic mobilities of the Un -Un loops compared to An -An loops. This observation stands in contrast to the pronounced sequence dependence observed for bulges (Zacharias & Hagerman, 1995a). Thermodynamic studies have shown that the stability of internal loops in RNA depends on the loop sequence as well as the flanking sequences (SantaLucia et al., 1991a,b). In particular, a U2-U2 internal loop was found to be more stable than an A2-A2 loop; in contrast, the T-T loop is one of the least stable loops in DNA (SantaLucia et al., 1991 a; Aboul-ela et al., 1985). Recent crystal structures of a tandem U-U pair in RNA (Baeyens et al., 1995, Holbrook et al., 1991) show hydrogen bonding within the U-U mismatch, and a perturbed, but overall A-type helical structure for a four basepair loop region. Similar NMR results on various mismatched base-pairs (Varani et al., 1989; SantaLucia & Turner, 1993) indicate that hydrogen

Internal Loops and RNA Flexibility

bonding between bases in the mismatch may modulate their stability. Those structural and thermodynamic observations suggest that the additional flexibility in dsRNA due to symmetric internal loops is, itself, sequence-dependent. Small differences in the decay profiles for the two loop sequences were observed in the current study; however, the magnitude of these differences was of the same order as the measurement error for the experimental results. Future studies using longer internal loops or shorter constructs might be helpful to further characterize the effects of loop sequence on the conformational properties of the loop. Interestingly, approximately the same per-base-pair flexibility was found for n = 2, 4, and 6, suggesting that, within the limits of uncertainty of the present study, the conformational freedom of each internal loop base-pair is relatively insensitive to its position within the loop. In considering the model in which the loop represents a point of increased, isotropic flexibility, it has been assumed that a harmonic bending potential determines the distribution of angular deformations, as is commonly assumed for the native helix. However, a formal alternative would be the co-existence of multiple, stable, slightly bent loop conformations, or a stable bend with a narrow angular dispersion (e.g. approaching that of the native helix). Finally, although the apparent angles for the fixed and dynamic cases are similar, and also result in similar end-to-end bend angle distributions of the complete RNA chain, the ramifications of the two models for the non-helical element itself are very different. For example, in the case of the A6-A6 internal loop, the apparent fixed bend angle and the total dynamic (r.m.s.) angle are approximately equal (030°; Table 1). However, the angle distributions for the two cases differ markedly (Figure 4). For the ‘‘fixed’’ bend, the actual distribution of angles will depend on the residual flexibility of the helix in and adjacent to the bend center, but is sharply peaked at the value specified by the fixed bend. By contrast, the dynamic bend with an r.m.s. angle of magnitude equal to the fixed bend angle is broadly distributed. The consequences of this difference can be more readily appreciated by considering the dynamic case, for which there is very little free energy difference for angular deflections extending out to at least 90°. By contrast, for the ‘‘fixed’’(30°) bend, a 60° angular deflection is associated with a relative free energy penalty (with fhx = 5°) of approximately 4 kcal/mol; however, this latter quantity is exquisitely sensitive to the residual flexibility of the fixed bend. It is tempting to speculate that these differences of bend character may be relevant to the selection of a bulge versus an internal loop at particular locations within a biologically active RNA molecule; for bending loci that are purely structural, bulges or other fixed elements may be utilized, whereas for positions that require conformational rearrangement, internal loops may be preferentially utilized.

285

Internal Loops and RNA Flexibility

Figure 4. Plots of the angle probability density functions, F(F), for fixed and dynamic bends. For the fixed bend: F(F) = Cfx × exp[−(F − Ffx )2/2 × f2hx,nbp ] × sin F, while for the dynamic bend: F(F) = Cdyn × exp[−F2/2 × f2] × sin F. For the fixed bend, F = 30°, fhx,1bp = 5° (curve (a)), or fhx,2bp = 21/2 × 5° (curve (d)). For the dynamic bend case, F = 30° (curve (b)) For comparison, the approximate distribution function for a section of 6 bp of native helix (dynamic) is presented (curve (c)), where Fhx = 61/2 × f. The vertical gray bar highlights the relative probabilities of finding an angle of 60° for each of the four cases (see the text for discussion).

Materials and Methods Plasmid constructs, RNA synthesis and formation of RNA heteroduplexes The construction of plasmid sets pGJ122A-NA/T and pGJ122B-NA/T, used for the production of RNA transcripts, has been described (Zacharias & Hagerman, 1995a). The two sets are derivatives of pGJ122A and pGJ122B that possess oligonucleotide inserts containing stretches of adenosine (A) or thymidine (dT) residues; the inserts are positioned exactly at the center of templates delimited by the phage T7 polymerase transcription start site and a downstream SmaI site (Zacharias & Hagerman, 1995a). The designation NA/T indicates the length (N) of the A/T stretch in base-pairs; N = 2, 4, 6 for the current study. Plasmids to be used for transcription reactions were digested with SmaI (two hours at 30°C) and extracted twice with phenol/chloroform, followed by ethanol precipitation. A standard T7 transcription reaction contained 0.05 mg/ml SmaI-digested plasmid DNA and 0.1 mg/ml purified T7 polymerase in 40 mM Tris-HCl (pH 8.1), 1 mM spermidine, 3.5 mM each of NTP, and 20 mM MgCl2 . After two hours at 37°C, the reaction mixture was brought to 20 mM NaEDTA, followed by phenol/chloroform extraction and ethanol precipitation of the RNA. Typical yields were 0.5 to 1 mg/ml transcript.

The RNA transcripts from pGJ122A-NA/T were complementary to transcripts from pGJ122B-NA/T except for the central A or U tract. To generate symmetric internal loop-containing dsRNA molecules, equimolar amounts of corresponding transcripts from pGJ122A-NA/T and pGJ122B-NA/T, with both A or both U tracts, were mixed and annealed. To obtain fully duplex linear control RNA molecules, equimolar amounts of A-tract-containing transcripts and (fully) complementary U-tract-containing transcripts were mixed and annealed. Annealing reactions were performed in TES buffer (100 mM NaCl, 0.5 mM NaEDTA, 10 mM Tris-HCl (pH 7.5)) by heating the solution to 95°C for two minutes and slow cooling to room temperature over a period of one hour. For further purification, the resulting RNA heteroduplexes were run on a 6% (w/v) non-denaturing polyacrylamide gel (acrylamide monomer/bis-acrylamide ratio, 29:1; TBE running buffer: 90 mM Tris-borate (pH 8.0), 0.5 mM NaEDTA) and excised and eluted from the macerated gel by overnight incubation in 300 mM NaCl, 10 mM NaPO4 , 0.25 mM NaEDTA (pH 7.2) at 4°C. Following ethanol precipitation, the material was dissolved in TEB buffer and further desalted using G-50 columns which had been pre-equilibrated with the appropriate buffer for TEB measurements. The sizes of the dsRNA molecules with central internal loops, and their corresponding linear control molecules, were 150 bp (2 bp loop), 152 bp (4 bp loop), and 154 bp (6 bp loop). The sequence context for all loops was 5'-CTGAGC(N)GCTCAG-3', where N represents the loop bases. Gel electrophoresis Analytical electrophoresis measurements were performed on 12% non-denaturing polyacrylamide gels (29:1, acrylamide monomer/bis-acrylamide (w/w) ratio) using TBE (90 mM Tris-borate (pH 8.0), 0.5 mM NaEDTA) or TBM (90 mM Tris-borate (pH 8.0), 5 mM MgCl2 ) as the running buffer. Gels were run at 5 to 10 V/cm at 10°C in a cold room. Duplex (150, 152, and 154 bp) RNA molecules were run as mobility standards. Transient electric birefringence measurements Transient electric birefringence (TEB) measurements were performed as described (Hagerman, 1984; Gast & Hagerman, 1991; Shen & Hagerman, 1994, Zacharias & Hagerman, 1995a,b). A recently designed TEB instrument (Ferber Schleif & P.J.H., unpublished results) was used for all measurements. The TEB cell volume was 80 ml, and was maintained at 4.0°C for all measurements. The pulse voltage was 2 kV with an electrode spacing of 2 mm (10 kV/cm field strength). The pulse duration was 1 ms with a repetition frequency of 1 Hz. The field-free decay of birefringence was observed to be independent of pulse width (1 to 2 ms) and field strength up to the highest fields used in the current study (10 kV/cm). Measurements for loop-containing molecules and corresponding linear controls were always performed under identical pulse configuration and buffer conditions. TEB buffer (5 mM NaPO4 , 0.125 mM NaEDTA (pH 7.2)) or TEBM buffer (5 mM NaPO4 , 2 mM MgCl2 pH 7.2)) were used with 10 to 15 mg RNA/80 ml sample volume (00.4 to 0.6 mM nucleotide phosphate). No significant dependence of the decay times or relative amplitudes on RNA concentration was noted. The decay curves were accumulated and averaged (256 pulses/average) using a LeCroy 9310 digitizing oscilloscope, and were stored on disks. Baseline

286

Internal Loops and RNA Flexibility

correction was performed by subtracting the buffer response curve (averaged over the same number of pulses). The solvent birefringence during the pulse was typically 5 to 10% of the RNA birefringence. Data were transferred to 80,486-based computers and were analyzed by fitting the birefringence decay curves to two-exponential functions (Zacharias & Hagerman, 1995a) using the Levenberg Marquardt method (Press et al., 1992). Gel electrophoretic analysis of the RNA after TEB measurements showed no signs of RNA degradation during the TEB experiments. Computation and analysis of rotational decay times and amplitudes

Equilibrium ensemble approach to analyse the longest terminal birefringence decay To interpret the longest decay component of the TEB curve, which corresponds largely to the end-over-end diffusion, the computer program DIFFROT (Hagerman & Zimm, 1981; Hagerman, 1981; Cooper & Hagerman, 1989) was employed. DIFFROT uses the ‘‘rigid body’’ equilibrium-ensemble approach for calculating average rotational decay times for wormlike chains, and is in substantial agreement with the analytical results of Roitman & Zimm (1984a,b) for the trumbell model. Internal-loop containing dsRNA was modeled as a string of touching beads with non-uniform persistence length but no static bends. The persistence length for the base-pairs comprising the internal loop was varied from ˚ . Note that a local decrease in persistence to 25 to 700 A length is equivalent to an increase in the dispersion of equilibrium bend angles. For small fluctuations, persistance length and per-base-pair bend angle dispersion are related through f2 = 2 × rise/P, where f2 corresponds to the per-base-pair bend angle variance (second moment of the bend angle distribution), ‘‘rise’’ is the rise/bp of RNA, and P the local persistence length of the chain. The angular fluctuations were modeled as isotropic in roll and tilt. The persistence length of the rest of ˚ . The rise/bp the chain was assigned a value of 700 A ˚ ) and hydrodynamic radius (13 A ˚ ) of the helix (02.8 A and loop regions were assumed to be those of the native RNA helix (Gast & Hagerman, 1991; Kebbekus et al., 1995). The linear control molecule was modeled as a ˚ ). worm-like chain of uniform persistence length (700 A Variation of these three parameters by up to 20% leads to a variation of the t/tc ratio of approximately 2%, smaller than the statistical error of the experimental determination of the decay times. The ensemble-averaged longer (‘‘slow’’) decay time (t) for a given molecule with locally decreased persistence length (increased flexibility) at the central 2, 4 or 6 bp internal loop was calculated and compared with the corresponding decay time constant tc of the linear control molecule with uniform persistence length (Figure 5). Calculated t/tc ratios were compared with experimental ratios (tloop /thelix ) to estimate the additional flexibility of the internal loop. For independent per-base-pair flexibilities, the variance of the bend angle distribution function f2 is additive. Throughout the paper, the r.m.s. average of the bend angle fluctuation is represented by f( = f21/2 ), the square root of the second moment (variance) of the distribution. Figure 6 presents the computed t/tc ratios plotted as a function of the square root of the difference between the bend angle variance of an internal loop (with various persistence lengths) and ˚ the variance of a duplex control RNA (with 700 A

Figure 5. Plots of the computed ratios of the slow decay times, t/tc , as a function of locally decreasing persistence length at the central 2 (R), 4 (Q), or 6 (W) base-‘‘pairs’’ representing the internal loops. The chain lengths were 150, 152, or 154 bp for internal loops of size 2, 4, or 6 bp, respectively. The persistence length of the flanking RNA ˚ , and the abscissa is expressed as helices was set to 700 A the ratio of the loop persistence length to that of the native helix. Each data point (decay ratio) was computed from the relative decay times for an ensemble of 5000 chains using DIFFROT (see Materials and Methods).

persistence length). As an alternate extreme, the calculated decay ratios are also plotted for fixed bends of corresponding f (abscissa) (thin line in Figure 6).

Brownian dynamics simulation Brownian dynamics simulation can be used to generate full birefringence decay curves, including both fast and slow components, for comparison with experimental TEB decay curves (Lewis et al., 1988; Allison & Nambi, 1992). In particular, Brownian dynamics methods have been used to simulate TEB and dynamic light-scattering of fully duplex linear DNA molecules (Allison & McCammon, 1984; Lewis et al., 1988; Allison & Nambi, 1992) and our methods follow the approach of those authors. In the current approach, RNA molecules are modeled as chains consisting of three or five beads that are connected by bonds and under the influence of a harmonic, inter-bond bending potential. Random starting bead configurations consistent with a Boltzmann ensemble were generated using a Monte Carlo approach. A trajectory was generated using the Ermak–McCammon algorithm (Ermak & McCammon, 1978). Hydrodynamic interactions between the beads were calculated from the Rotne–Prager tensor of the bead system (Rotne & Prager, 1969). Bond lengths were constrained to their preset value using SHAKE (Ryckaert et al., 1977; Allison & McCammon, 1984). The bead bond length was chosen so that the sum of all bonds equals the contour length of the RNA molecule. Gaussian random numbers with deviates weighted by the elements of the hydrodynamic interaction tensor were generated using the GGNSM-subroutine of the IMSL library (IMSL, 1980). Parametrization of

287

Internal Loops and RNA Flexibility

functions obtained from long Brownian dynamics ˚. simulations with the required persistence length of 700 A For the comparison of r.m.s. end-end distances, the standard formula: h 2/L 2 = (2P/L) × [1 − P/L + (P/L) × exp( − L/P)], was employed (e.g. see: Bloomfield et al., 1974), where h 2 is the average (squared) end-to-end distance, L is the contour length, and P is the persistence length of the chain. For the RNA molecules containing internal loops, the central angular bending potential was systematically varied and the birefringence decay compared to the simulation of the linear control molecule. The time step for each simulation varied between 0.2 ns and 2 ns. The relative (computed) birefringence signal, Dn(t)/ Dn(0), was obtained from the expression: Dn(t)/Dn(0) = C s s P2 [b i (to )·b j (to + t)], i

Figure 6. Computed decay time ratios, t/tc , as a function of the additional flexibility of the central 2 ((Q), 4 (W), or 6 (T) bp internal loop using DIFFROT. The computed ratios are the same as for Figure 5 except that the reduced central persistence length (i.e. within the loop) was translated into the excess angular dispersion DF, using the relation: DF = (F2 − f2hx,nbp )1/2, where f2hx,nbp is the variance of the bend angle distribution for a segment of normal helix of n bp, and F2 is the variance for the angle distribution for the central loop. For comparison, ratios are plotted also for a fixed bend angle (abscissa representing the fixed angle in ˚ ) persistence this case) in a chain having a uniform (700 A length (——), and for three-bead (q) and five-bead (w) BD models (Materials and Methods). It should be noted that for the bead models, in order to compute DF, the bead bend angle distribution (r.m.s.) for the linear was subtracted from a given F (similar to the procedures described in footnote e and f of Table 2). the angle force constants and bead diameters for the linear control molecule with uniform persistence length was performed by comparing ensemble averaged diffusion constants for the N-bead models (N = 3, 5) with a corresponding ‘‘full’’ touching bead model using the program DIFFROT (Hagerman & Zimm, 1981). The bead diameter for each model was scaled to result in the same ensemble averaged three principle diffusion coefficients (within 5%) and longest ensemble averaged birefringence decay time constants as the touching bead model. For a ˚ persistence length and a five-bead 150 bp RNA and 700 A model, this resulted in a Brownian dynamics bead radius ˚ . In case of a three-bead model no bead diameter of 20 A could be found which simultaneously optimized all three principle diffusion coefficients (at constant persistence length). Since the persistence length is one of the independent parameters to be established for each internal loop in the current study, we chose for the three-bead model a bead diameter resulting in the same ensemble averaged longest terminal decay time constant ˚ persistence length as the touching bead model with 700 A ˚ ). (bead radius for the three-bead model 24.5 A The force constants were refined by directly comparing the average end-to-end distances and angle distribution

j

where C is a normalization constant (to set the initial birefringence to unity at t = to ). The double sum is taken over the second Legendre polynomial (P2 ) of the dot product of the unit bond vectors b for bead bond i at time to (start of decay) and bead bond j, at time to + t (see, for example Allison & Nambi, 1992). This expression assumes that the orientation mechanism is of the induced-dipole form, and that there is no field-induced distortion of the RNA molecule. Measurements of the field-dependence of the initial birefringence signal showed essentially E 2 behavior, deviating by less than 8% at the highest fields employed in the current study. Furthermore (more importantly), the decay times and relative amplitudes were independent of field strength. The ensemble average was accumulated at every time step for ensembles of 2000 to 5000 trajectories. Each trajectory was recorded after 500 to 1000 equilibration time steps, and was extended to between 12 and 15 ms. Our implementation of the BD algorithm required 1.9/4.5 seconds CPU time for one 10,000 step trajectory for the three-bead/five-bead simulations, respectively, on an SGI-Indigo R8000 Extreme. The ensemble-average TEB decay curve was reduced to 0500 data points through window-averaging and finally fit to a double exponential decay function using the LM method (see above). Doubling of the number of trajectories led to changes in decay time ratios and relative amplitudes of less than 5%. Using a harmonic bead inter-bond length potential or varying the time step between 0.3 and 2.0 ns affected the slow component decay time and relative amplitudes also by less than 5%. The force constant for the central bend angle was systematically varied to obtain decay time constants and amplitudes as a function of the flexibility of a central element. Although it is known from numerical solutions of the diffusion equation of the three-bead trumbell model (Roitman & Zimm, 1984a,b) that the birefringence decay can contain more than two exponentials, these make up less than 3% of the decay and were not observed experimentally. Addition of a third exponential function to the least-squares fit of the birefringence relaxation from Brownian dynamics did not reduce the residual significantly.

Acknowledgements The authors thank Ferber Schleif for the design and fabrication of many of the electronic components used in

288 the current TEB instrument, Janine Mills for synthesizing the DNA oligonucleotides, Elsi Vacano for performing the hydrodynamic computations using DIFFROT, and Khaled Amiri, Peter Kebbekus, and Dr Martha Olmsted for useful discussions. This work was supported by a grant from the National Institutes of Health (P.J.H., GM 35305). Research in the laboratory is supported in part by a Molecular Biology Program grant from the Lucille P. Markey Charitable Trust.

References Aboul-ela, F., Koh, D., Tinoco, I., Jr & Martin, F. H. (1985). Base-base mismatches. Thermodynamics of double helix formation for dCA3XA3G and dCT3YT3G (X, Y = A, C, G, T). Nucl. Acids Res. 13, 4811–4824. Allison, S. A. & McCammon, J. A. (1984). Transport properties of rigid and flexible macromolecules by Brownian dynamics simulation. Biopolymers, 23, 167–187. Allison, S. A. & Nambi, P. (1992). Electric dichroism and birefringence decay of short DNA restriction fragments studied by Brownian dynamics simulation. Macromolecules, 25, 759–768. Amiri, K. M. A. & Hagerman, P. J. (1994). Global structure of a self-cleaving hammerhead RNA. Biochemistry, 33, 13172–13177. Baeyens, K. J., De Bondt, H. L. & Holbrook, S. R. (1995). Structure of an RNA double helix including uracil-uracil base-pairs in an internal loop. Nature Struct. Biol. 2, 56–62. Battacharyya, A. & Lilley, D. M. J. (1989). The contrasting structures of mismatched DNA sequences containing looped-out bases (bulges) and multiple mismatches (bubbles). Nucl. Acids Res. 17, 6821–6840. Battacharyya, A., Murchie, A. I. H. & Lilley, D. M. J. (1990). RNA bulges and the helical periodicity of double stranded RNA. Nature, 343, 484–487. Bloomfield, V. A., Crothers, D. M. & Tinoco, I. (1974). Physical Chemistry of Nucleic Acids. Harper & Row, New York. Chen, X., Woodson, S. A., Burrows, C. J. & Rokita, S. E. (1993). A highly sensitive probe for guanine N7 in folded structures of RNA: application to tRNAPhe and Tetrahymena group I intron. Biochemistry, 32, 7610– 7616. Cooper, J. P. & Hagerman, P. J. (1989). Geometry of a branched DNA structure in solution. Proc. Natl Acad. Sci. USA, 86, 7336–7340. Ehresmann, C., Baudin, F., Mougel, M., Romby, P., Ebel, J.-P. & Ehresmann, B. (1987). Probing the structure of RNA in solution. Nucl. Acid Res. 15, 9109–9128. Ermak, D. L. & McCammon, J. A. (1978). Brownian dynamics with hydrodynamic interactions. J. Chem. Phys. 69, 1352–1360. Friederich, M. W., Gast, F.-U., Vacano, E. & Hagerman, P. J. (1995). Determination of the angle between the anticodon and aminoacyl acceptor stems of yeast phenylalanyl tRNA in solution. Proc. Natl Acad. Sci. USA, 92, 4803–4807. Gast, F.-U. & Hagerman, P. J. (1991). Electrophoretic and hydrodynamic properties of duplex ribonucleic acid molecules transcribed in vitro: evidence that A-tracts do not generate curvature in RNA. Biochemistry, 30, 4268–4277. Gast, F.-U., Amiri, K. M. A. & Hagerman, P. J. (1994). Interhelix geometry of stems I and II of a self-cleaving hammerhead RNA. Biochemistry, 33, 1788–1796.

Internal Loops and RNA Flexibility

Hagerman, P. J. (1981). Investigation of the flexibility of DNA using transient electric birefringence. Biopolymers, 20, 1503–1535. Hagerman, P. J. (1984). Evidence for the existence of stable curvature of DNA in solution. Proc. Natl Acad. Sci. USA, 81, 4632–4636. Hagerman, P. J. (1988). Flexibility of DNA. Annu. Rev. Biophys. Biophys. Chem. 17, 265–286. Hagerman, P. J. & Zimm, B. (1981). Monte Carlo approach to the analysis of the rotational diffusion of wormlike chains. Biopolymers, 20, 1481–1502. Holbrook, S. R., Cheong, C., Tinoco, I. & Kim, S.-H. (1991). Crystal structure of an RNA double helix incorporating a track of non-Watson–Crick base-pairs. Nature, 353, 579–581. Hsieh, C.-H. & Griffith, J. D. (1989). Deletions of bases in one strand of duplex DNA, in contrast to single-base mismatches, produce highly kinked molecules: possible relevance to the folding of single-stranded nucleic acids. Proc. Natl Acad. Sci. USA, 86, 4833–4837. IMSL (1980). International Mathematical and Statistical Libraries, Inc., IMSL Library 3 Reference Manual, IMSL LIB3-0005, Houston, TX. Jaeger, J. A., SantaLucia, J. & Tinoco, I. (1993). Determination of RNA structure and thermodynamics. Annu. Rev. Biochem. 62, 255–287. Kahn, J. D., Yun, E. & Crothers, D. M. (1994). Detection of localized DNA flexibility. Nature, 368, 163–166. Kebbekus, P., Draper, D. E. & Hagerman, P. (1995). Persistence length of RNA. Biochemistry, 34, 4354–4357. Knapp, G. (1989). Enzymatic approaches to probing of RNA secondary and tertiary structure. Methods Enzymol. 180, 192–212. Lewis, R. J., Allison, S. A., Eden, D. & Pecora R. (1988). Brownian dynamics simulations of a three-subunit and a ten-subunit worm-like chain: comparison of results with trumbell theory and with experimental results from DNA. J. Chem. Phys. 89, 2490–2503. Mills, J. B., Cooper, J. P. & Hagerman, P. J. (1994). Electrophoretic evidence that single-stranded regions of one or more nucleotides dramatically increase the flexibility of DNA. Biochemistry, 3, 1797–1803. Patel, D. J., Kozlowski, S. A., Ikuta, S. & Itakura, K. (1984a). Deoxyguanosine-deoxyadenosine pairing in the d(CGAGAATTCGCG) duplex: conformation and dynamics at and adjacent to the dGdA mismatch site. Biochemistry, 23, 3207–3217. Patel, D. J., Kozlowski, S. A., Ikuta, S. & Itakura, K. (1984b). Deoxyadenosine-deoxycyytidine pairing in the d(CGCGAATTCACG) duplex: conformation and dynamics at and adjacent to the dAdC mismatch site. Biochemistry, 23, 3218–3226. Press, W. H., Vetterling, W. T., Teukolsky, S. A. & Flannery, B. P. (1992). Numerical Recipes in Fortran: The Art of Scientific Computing, pp. 678–683, Cambridge University Press, Cambridge, UK. Roitman, D. B. (1984). An elastic hinge model for dynamics of stiff chains. III. Viscoelastic and Kerr-effect behavior of bent molecules. J. Chem. Phys. 81, 6356–6360. Roitman, D. B. & Zimm, B. (1984a). An elastic hinge model for dynamics of stiff chains. I. Viscoelastic properties. J. Chem. Phys. 81, 6340–6347. Roitman, D. B. & Zimm, B. (1984b). An elastic hinge model for dynamics of stiff chains. II. Transient electro-optical properties. J. Chem. Phys. 81, 6348– 6354. Rotne, J. & Prager, S. (1969). Variational treatment of

Internal Loops and RNA Flexibility

hydrodynamic interaction in polymers. J. Chem. Phys. 50, 4831–4837. Ryckaert, J. P., Ciccoti, G. & Berendsen, H. J. C. (1977). Molecular dynamics of liquid alkanes. J. Comp. Phys. 23, 327–341. SantaLucia, J., Kierzek, R. & Turner, D. H. (1991a). Stabilities of consecutive AC, CC, GG, UC, and UU mismatches in RNA internal loops: evidence for stable hydrogen-bonded UU and CC + pairs. Biochemistry, 30, 8242–8251. SantaLucia, J., Kierzek, R. & Turner, D. H. (1991b). Functional group substitutions as probes of hydrogen bonding between GA mismatches in RNA internal loops. J. Am. Chem. Soc. 113, 4313–4322. SantaLucia, S. L. & Turner, D. T. (1993). Structure of rGGCGAGCC)2 in solution from NMR and restrained molecular dynamics. Biochemistry, 32, 12612–12623. Shen, Z. & Hagerman, P. J. (1994). Geometry of the central branch of the 5S ribosomal RNA of sulfolobus acidocaldarius. J. Mol. Biol. 241, 415–430.

289 Tang, R. S. & Draper, D. E. (1990). Bulge loops used to measure the helical twist of RNA in solution. Biochemistry, 29, 5232–5237. Tang, R. S. & Draper, D. E. (1994). On the use of phasing experiments to measure helical repeat and bulge loop-associated twist in RNA. Nucl. Acids Res. 22, 835–841. Varani, G., Wimberly, B. & Tinoco, I. (1989). Conformation and dynamics of an RNA internal loop. Biochemistry, 28, 7760–7772. Wimberly, B., Varani, G. & Tinoco, I. (1993). The conformation of loop E in eukaryotic 5 S ribosomal RNA. Biochemistry, 32, 1078–1087. Zacharias, M. & Hagerman, P. J. (1995a). Bulge-nduced bends in RNA: quantification by transient electric birefringence. J. Mol. Biol. 247, 486–500. Zacharias, M. & Hagerman, P. J. (1995b). The bend in RNA created by the trans-activation response element bulge of human immunodeficiency virus is straightened by arginine and by Tat-derived peptide. Proc. Natl Acad. Sci. USA, 92, 6052–6056.

Edited by D. E. Draper (Received 19 September 1995; accepted 18 December 1995)