6634 Nucleic Acids Research, 2006, Vol. 34, No. 22 Figure 4. The SAXS profiles for the two-helix model with neutral PEG loop for different ionic conditions: diamond 20 mM NaCl; plus 0.6 M MgCl2 and square 1.2 M NaCl. (A) The predicted SAXS profiles weighted by the free energy G(x, q) (see Materials and Methods); (B) The experimental SAXS profiles (49). The dashed lines, solid lines and dotted lines represent the standard SAXS profiles for the collapsed C, random R, and extended E states of the tethered helices, respectively. slightly overestimated stability in the model may be caused by the restriction of (2D) co-planar and symmetric conforma- tions for the two-helix assembly and the possible change in the hydration state of the helices and ions. With a neutral tether loop To investigate the loop effect, we consider a system of two helices tethered by a polyethylene glycol (PEG) loop with persistence length lP 3.8 s (49,85). We obtain the total free energy G(x, q) from Equation 1 with the loop-free energy Gloop(x, L) determined from Equation 2 for each given conﬁg- uration (x, q) for two helices tethered by the loop. We ﬁrst compare our prediction with the experimental measurements for a similar nucleic acid model (49). To model the short loop in the experiment (49), we assume loop length L ¼ 26 A˚ . The loop size deﬁnes the maximum helix–helix end-end distance: x < xmax ¼ 26 A˚ . We calculate the SAXS proﬁles from Equa- tion 4 for each given (x, q). The Boltzmann-weighted average by factor e G(x,q)/kBT for all the possible states (x, q) gives the mean SAXS proﬁle: where the conformational sampling covers the full conforma- tional space described by x 2 [22 A˚ , 60 A˚ ] and q 2 [0 , 180 ] with increments of 2 A˚ and 20 , respectively. Short neutral loop and comparisons with experiment. As shown in Figure 4, the predicted SAXS proﬁles for low (20 mM) and high (1.2 M) [Na+] and for high (0.6 M) [Mg2+] are close to the experimental proﬁles for a similar nucleic acid model (49). For comparison, in Figure 4A we also show the predicted proﬁle for 0.1 M Mg2+. The state in 0.1 M Mg2+ is (slightly) more compact than that the random state in 0.6 M Mg2+ but much less compact than the collapse state. Figure 4 shows close SAXS proﬁles for 1.2 M [Na+] and 0.6 M [Mg2+], while Figure 2 shows very different free energy landscape for Na+ and Mg2+ ionic conditions. How can such seemingly contrasting results be consistent with each other? The SAXS proﬁle I*(Q) is a weighted average over all the available conformations (see Equation 5). I*ðQÞ ¼ Px‚q IðQÞe Gðx‚qÞ/kBT Px‚q e Gðx‚qÞ/kBT ‚ 6 As shown in Figure 2D and H as examples, for a given loop length of 26 A˚ , the accessible conformational space on the free energy landscape is limited to the region with x < 26 A˚ . Though the full (unrestricted) landscape would certainly yield drastically different I*(Q), the restricted acces- sible region of the landscape may give similar I*(Q). Physi- cally, within the restricted conformational space, the conformational distribution for high [Na+] and high [Mg2+] are both quite inhomogeneous over a wide-range of the angle q between the helices. It is important to note that the above prediction-experiment comparisons involve several limitations of the present theory: (i) The two-helix conﬁguration is assumed to be symmetric and planar. Including the non-planar/non-symmetric conﬁgu- rations would likely weaken the predicted helix packing stability and compactness. (ii) The loop (linker) in the present model connects two helices at the helical axes instead of the helix edges in the experimental model (49). Such difference causes no signiﬁcant effects for long loops (linkers) but can play a role for shorter loops. Furthermore, the length of L ¼ 26 A˚ used in the theoretical model and the experimental system may cause additional effects. As shown in Figure 2A–H, the most stable (lowest free energy) conﬁguration occurs at inter-helix distance >26 A˚ . As a result, all the conﬁgurations considered in the model and in experiment (with x & xmax ¼ L ¼ 26 A˚ ) would result in DGE higher than the lowest free energy. Therefore, the short loop would effectively reduce the elec- trostatic force in the compaction process. Moreover, short loops would signiﬁcantly restrict the accessible conforma- tions of the extended state, resulting in a reduced conforma- tional difference between the compact and the extended states, and hence a reduced sharpness of the compaction tran- sition. For the same reason, longer loops can lead to sharper compaction transition; see Figure 3. In summary, our results agree with the experimental ﬁnd- ing that the Mg2+-mediated electrostatic interaction cannot induce the electrostatically collapsed state C for the two- helix system. Although our predicted helix–helix attraction is slightly stronger than the value derived from experiments, the predicted compact state is only slightly more compact than the random state but much less compact than the collapsed state C. A longer loop causes a sharper compaction