extfieldsofstudy: Medicine Chemistry

text:
Conformational Heterogeneity in the C-terminal Zinc Fingers of Human MTF-1

The human metalloregulatory transcription factor, metal-response element (MRE)-binding transcription factor-1 (MTF-1), contains six TFIIIA-type Cys 2 -His 2 motifs, each of which was projected to form well-structured domains upon Zn(II) binding. In this report, the structure and backbone dynamics of a fragment containing the unusual C-terminal fingers F4–F6 has been investigated. 15 N heteronuclear single quan-tum coherence (HSQC) spectra of uniformly 15 N-labeled hMTF-zf46 show that Zn(II) induces the folding of hMTF-zf46. Analysis of the secondary structure of Zn 3 hMTF-zf46 determined by 13 C (cid:2) chemical shift indexing and the magnitude of 3 J H (cid:2) -HN clearly reveal that zinc fingers F4 and F6 adopt typical (cid:1)(cid:1)(cid:2) structures. An analysis of the heteronuclear backbone 15 N relaxation dynamics behavior is consistent with this picture and further reveals independent tumbling of the finger domains in solution. Titration of apo-MTF-zf46 with Zn(II) reveals that the F4 domain binds Zn(II) significantly more tightly than do the other two finger domains. In contrast to fingers F4 and F6, the (cid:1)(cid:1)(cid:2) fold of finger F5

Zinc is the second most abundant transition metal ion present in biological systems. Zinc plays essential and ubiquitous catalytic and structural roles, present both at the active sites of many hydrolytic enzymes and as an integral tertiary structural element in many proteins, most prominently exemplified by zinc finger proteins (1,2). The intracellular concentrations of "bioavailable" essential metals ions, including zinc, appear to be tightly regulated in all cells. Although recent progress in our understanding of zinc toxicity in mammalian cell mitochondria has been made (3), the precise underlying cause for zinc toxicity is not yet known with certainty. However, too much zinc would compete for enzyme active sites requiring other metal ions or alter their homeostatic mechanisms by competing for metal transporter sites. Zinc-specific metalloregulation of gene expression has been documented to occur in essentially all organisms, including mammalian cells (4), plants (5), Saccharomyces cerevisiae (6), and bacteria (7)(8)(9)(10). Zinc metalloregulatory transcription factors, which mediate the response of cells to both zinc-deficient and zinc-excess conditions have been identified and partially characterized from many of these sources (10).
In many cell types, including mammalian (mouse, rat, and human) cells (11,12), fish (13), and Drosophila melanogaster (14), zinc-dependent transcriptional regulation of metallothionein (MT) 1 genes plays an important role in zinc homeostasis and detoxification. This regulation requires the interaction of metal-response element (MRE)-binding transcription factor-1 (MTF-1) (15) with MREs (16,17) situated in the promoters of zinc-inducible genes (17). MTF-1 is a constitutively expressed protein in mouse and human cells of ϳ80 kDa that contains six Cys 2 His 2 zinc finger domains and multiple domains for transcriptional activation (18), the latter of which likely play important roles in the signal transduction pathway that leads to a zinc response (19). MTF-1 is an essential protein (20) required for basal and heavy metal-induced expression of metallothionein I and II genes. Other toxic heavy metals, including cadmium (for a review, see ref. 21) and to a lesser extent copper, are also known to induce the expression of metallothioneins in vertebrate and Drosophila cells (22) (copper is a primary inducer of MT expression in Drosophila (14)), as do various forms of oxidative stress (23) and hypoxia (24). Some reports suggest that the zinc finger domain of MTF-1 itself is a zinc sensor that exhibits increased DNA binding activity upon zinc treatment in vivo and in vitro. It is becoming increasingly clear, however, that stimulation of MRE-binding activity of MTF-1 by Zn(II) may be necessary but not sufficient for transcriptional activation by zinc and other potent inducers, including Cd(II). Zn(II) and other inducers play a major role in modulating nuclear-cytoplasmic trafficking of MTF-1 (25,26), whereas Zn(II) and Cd(II) seem to alter the phosphorylation status of MTF-1 (19).
The zinc finger domain of MTF-1 from human, mouse, D. melanogaster, and Japanese pufferfish is remarkably conserved throughout hundreds of millions of years of evolution (13). Each zinc finger conforms to the classic (F/Y)-X-Cys-X 4 -Cys-X 3 -(Y/F)-X 5 -L-X 2 -His-X-(R/K)-X-His sequence, each of which are connected by nearly consensus TG(E/Q)(K/R)P linkers (27), with the exception of linker between fingers F3 and F4, which is one residue shorter, TGKT (28). This is likely to alter the conformation of the linker and may well alter the ability of F3 and F4 to bind to successive non-overlapping 3-base pair subsites (for recent reviews on the structures of zinc finger proteins, see Refs. 29,30). The only deviation to the absolute conservation of the (F/Y)-X-Cys-X 4 -Cys-X 3 -(Y/F)-X 5 -L-X 2 -His-X-(R/K)-X-His sequence in MTF-1 is in F4 in D. melanogaster MTF-1 where the Cys-X 4 -Cys sequence, predicted to form a ␤-type II turn, is replaced by the more common Cys-X 2 -Cys sequence, predicted to form a ␤-type I turn, with otherwise no changes in the structure of the ␤␤␣ domain (30,31).
We have begun to elucidate the biochemical and structural properties of the zinc finger domain of human MTF-1 in an effort to obtain insight into regulatory signaling by MTF-1. Our previous work suggests that the N-terminal four zinc fingers of MTF-1 (F1-F4) adopt canonical ␤␤␣ structures and make large contributions to the specific MRE-binding affinity, with C-terminal fingers F5 and F6 playing an accessory role in stabilizing the protein-DNA complex on an mMT-I MREd-containing oligonucleotide (32,33). In addition, a variety of evidence suggested that the apparent affinity of the C-terminal fingers F5 and F6 for Zn(II) was weaker than that for the major DNAbinding fingers of the molecule (33). These findings have lead to the hypothesis that the C-terminal zinc fingers of hMTF-1 play a direct role in metalloregulation via reversible Zn(II) binding to F5 and/or F6 fingers, although as yet, there is no direct evidence in support of this proposal (34,35). On the other hand, it has recently been proposed that finger F1 functions as a zinc sensor rather than F5 and/or F6 (35). Recent studies reveal that deletion of F1 from hMTF-zf destroys high affinity MREdbinding activity of the zinc finger domain (36).
Previous far-UV-circular dichroism studies of hMTF-zf were interpreted to suggest that the N-terminal four zinc fingers are folded into typical ␤␤␣ motifs (33). In contrast, one or both of the C-terminal fingers were thought to adopt one or more alternative conformations under conditions of saturating Zn(II), although the structural details remained unclear. These structural differences are maintained in the isolated threefinger domain fragments of hMTF-zf (zf13 and zf46) (33), providing justification for investigating the detailed structure of hMTF-zf46 as representative of the structure of this region within intact hMTF-zf and hMTF-1. In this study, we have used heteronuclear NMR spectroscopy to investigate the Zn(II) binding and conformational properties of the C-terminal zinc fingers of hMTF-zf46. We find that F4 and F6 do indeed adopt canonical ␤␤␣ structures, with F6 characterized by a clearly lower affinity for Zn(II). Furthermore, at saturating Zn(II), F5 does not adopt a stable ␤␤␣ structure but is instead in conformational equilibrium with one or more non-native conformations. Finally, titration of apo-hMTF-zf46 with Cd(II) induces a non-native, largely unfolded structure as revealed by 15 N HSQC spectroscopy, with a maximum Cd(II)-peptide stoichiometry of ϳ2:1, with significant S Ϫ 3 Cd(II) thiolate absorption in the complex. This finding is consistent with other results that show that, although Cd(II) is a potent inducer of MT gene expression from the MRE (37), it is unlikely to do so by inducing ␤␤␣ structure into the MRE-binding domain of MTF-1.

MATERIALS AND METHODS
Purification of hMTF-zf46 and hMTF-zf56 -pET-derived overexpression plasmids were designed to express zinc finger fragments encompassing zinc fingers F4 -F6 (residues Gly 226 -Tyr 320 ), denoted pMTF-zf46, or zinc fingers F5-F6 (Gly 253 -Tyr 320 ), denoted pMTF-zf56 (see Fig.  1A). In both cases, the C-terminal Tyr was replaced by a Trp, as in all other constructs (32,33), to facilitate optical spectroscopy studies. Basically, appropriately designed polymerase chain reaction primers were used to amplify the desired coding region using pMTF-zf as a template and subcloned between the NcoI and BamHI sites of pET3d, with overexpression in Escherichia coli BL21(DE3) essentially as described previously (33). The integrity of the coding sequence was verified by dideoxy sequencing of both DNA strands carried out by the Gene Technologies Laboratory at Texas A&M University. Protein expression was induced with 0.4 mM isopropyl-1-thio-␤-D-galactopyranoside at a cell culture density of A 600 ϭ 0.5 at 37°C, and the cells were harvested 3-4 h after induction. hMTF-zf46 and hMTF-zf56 were purified from the low speed supernatant using a chromatographic scheme on a POROS C18 column developed with a linear gradient (40 mM Mops, 5 mM dithiothreitol, pH 7.0, 0.2 M NaCl to 1 M NaCl), followed by pooling of the appropriate fractions, an anaerobic dialysis against 0.1% trifluoroacetic acid, with subsequent repeated lyophilization from doubledistilled water. Lyophilized protein was then returned to the anaerobic glovebox and resuspended in metal-free 120 mM d 18 -Hepes, pH 7.0, in the absence of exogenously added Zn(II). These samples of apo-hMTF-zf46 and apo-hMTF-zf56 typically contained Յ0.1 mol eq of Zn(II) by atomic absorption spectroscopy on a PE AAnalyst 700 and contained the expected number of reduced cysteines (ϳ6 for apo-hMTF-zf46 and ϳ4 for apo-hMTF-zf56) by 5,5Ј-dithiobis(nitrobenzoic acid) reactivity (38). In both cases, the N-terminal Met was removed in vivo to yield an N-terminal native Gly. Uniformly 15 N-and 13 C-labeled and 15 N-labeled proteins were prepared by expressing the proteins on M9 minimal media salts supplemented with 1 g/liter 15 NH 4 (SO 4 ) 2 as the sole nitrogen source, or 1 g/liter 15 NH 4 (SO 4 ) 2 and 3 g/liter [ 13 C]glucose essentially as described before (39). ␣-15 N-Lys-labeled hMTF-zf46 was prepared by growing transformed E. coli BL21(DE3) on minimal media supplemented with the other 19 unlabeled amino acids at the concentrations suggested (40).
The 1 H-15 N HSQC spectrum was recorded as a 160* ϫ 480* twodimensional matrix in the t 1 and t 2 dimensions, respectively, where n* represents n complex points. Acquisition times in each dimension were 100 ms (t 1 , 15 N) and 80 ms (t 2 , 1 H N ) as defined by sweep widths of 1600 and 6000 Hz in the t 1 and t 2 dimensions, respectively, for a total experiment time of 1.6 h. The data were apodized along t 2 by using a 90°-shifted squared sine-bell, truncated at 1% (sin 2 175). The t 1 time domain was extended to 256* using mirror-image linear prediction (eight coefficients) (43) and then apodized with an untruncated 90°shifted squared sine-bell. The data were zero-filled to a digital resolution of 3.1 (F 1 ) and 3.9 Hz (F 2 ).
The CBCA(CO)NH spectrum (44) was recorded as a 34* ϫ 55* ϫ 512* three-dimensional matrix with acquisition times of 22.4 (t 1 , 15 N), 6.5 (t 2 , 13 C ␣,␤ ), and 73.1 ms (t 3 , 1 H N ) as defined by sweep widths of 1515.2, 8444.2, and 6999.7 Hz in the t 1 , t 2 , and t 3 dimensions, respectively. The total acquisition time was 39.5 h. Acquired data were apodized with a 90°-shifted squared sine-bell in the t 3 dimension, truncated at 1% at the end of the free induction decay, and an untruncated 90°-shifted squared sine-bell in the t 2 dimension after mirror-image linear prediction was used to extend the data along t 2 to 91*. The t 1 dimension was apodized with an untruncated 90°-shifted squared sinebell after the data were extended to 58* by using mirror-image linear prediction. Data were zero-filled to digital resolutions of 15.8 (F 1 ), 44 (F 2 ), and 6.8 Hz (F 3 ).
The HNHA spectrum (48) for measurement of 3 J HNH␣ , was recorded as a 60* ϫ 78* ϫ 512* three-dimensional matrix with acquisition times of 40 (t 1 , 15 N), 15.6 (t 2 , 1 H), and 85.3 ms (t 3 , 1 H N ) and sweep widths of 1600, 5000, and 6000 Hz in the t 1 , t 2 , and t 3 dimensions, respectively. Acquired data were apodized with a 90°-shifted squared sine-bell in the t 3 dimension, truncated at 2% at the end of the free induction decay, and with a 90°-shifted sine-bell in the t 2 dimension, truncated at 16%. The t 1 dimension was apodized with an untruncated 90°-shifted squared sine-bell. Data were zero-filled such that the digital resolutions were 25 (F 1 ), 19.5 (F 2 ), and 5.86 Hz (F 3 ). Three-bond HN-H␣ coupling constants ( 3 J HN-H␣ ) were estimated from the ratio of the intensities of cross-peaks on (I diag ) and off (I cross ) the diagonal according the following expression (49), where ϭ 13.2 ms. The three-dimensional 15 N-separated NOESY spectrum (50) with a 100-ms mixing time was recorded as a 32* ϫ 128* ϫ 512* threedimensional matrix with acquisition times of 20 (t 1 , 15 N), 25.6 (t 2 , 1 H), and 85.3 ms (t 3 , 1 H N ) as defined by sweep widths of 1600, 5000, and 6000.2 Hz in the t 1 , t 2 , and t 3 dimensions, respectively. The total acquisition time was 91.3 h. Acquired data were apodized with a 90°shifted squared sine-bell in the t 3 dimension, truncated at 2%, and with a 90°-shifted sine-bell in the t 2 dimension, truncated at 16%. The t 1 dimension was apodized with a 90°-shifted sine-bell, truncated at 16%. Data were zero-filled to yield digital resolutions of 16.7 (F 1 ), 19.5 (F 2 ), and 5.86 Hz (F 3 ).
The three-dimensional 15 N-separated total correlation spectroscopy spectrum (50) with a mixing time of 120 ms was recorded as a 32* ϫ 128* ϫ 512* three-dimensional matrix with acquisition times of 20 (t 1 , 15 N), 25.6 (t 2 , 1 H), and 85.3 ms (t 3 , 1 H N ) as defined by sweep widths of 1600, 5000, and 6000.2 Hz in the t 1 , t 2 , and t 3 dimensions, respectively. Acquired data were processed similarly to the 15 N-separated NOESY experiment and were zero-filled such that the digital resolutions were 16.7 (F 1 ), 19.5 (F 2 ), and 5.86 Hz (F 3 ).
Relaxation Experiments-The pulse sequences used to determine the 15 N R 1 (1/T 1 ), R 2 (1/T 2 ) and { 1 H}- 15 N NOE values were derived from  (51), which incorporates gradient selection and were recorded at 30°C. All spectra were collected as 160* ϫ 512* data matrixes in the t 1 and t 2 dimensions, with acquisition times of 83.3 (t 1 ) and 73.1 ms (t 2 ) as defined by sweep widths of 1920 (t 1 ) and 6999.7 Hz (t 2 ). The 15 N T 1 data were collected using 15 15 N NOEs, two-dimensional spectra were recorded with (NOE experiment) and without (NONOE experiment) application of 1 H saturation applied during the delay between successive transients. In the NONOE experiment, a relaxation delay of 5 s was used between transients, whereas in the NOE experiment, a 2-s delay followed by a 3-s proton presaturation achieved with the use of 120°1H pulses applied every 5 ms. 15 N T 1 , T 2 , and 15 N{-H} NOE data sets were collected twice, several days apart. The time points for the T 1 and T 2 data sets were randomized differently for each set and interleaved such that all 10 15 N delays were sampled at each t 1 period, to minimize spectrometer drift effects. Acquisition times were 22.8, 18.0, and 29.4 h for each T 1 , T 2 , and NOE data set, respectively. All spectra were apodized in the t 2 dimension with a 90°-shifted sine-bell, truncated at 1%. The t 1 time domain was apodized with a 90°-shifted sine-bell, truncated at 6%. The spectra were zero-filled to 3.7 (t 1 ) and 4.6 Hz (t 2 ). Peak intensities were fitted to a monoexponential decay function in Kaleidagraph 3.09 (Abelbeck Software). Uncertainties in the { 1 H- 15 N} NOE values were estimated from individual data sets collected in duplicate.
Absorption Spectroscopy-Anaerobic optical titrations of apo-hMTF-zf46 or apo-hMTF-zf56 were carried out at ambient temperature with absorption spectra recorded on an HP8453 diode array spectrophotometer essentially as described previously (52,53). After each addition of Co(II) or Cd(II), the spectra were recorded after 5 min to ensure that complex formation had come to equilibrium. Fig. 1A shows the primary structure of the C-terminal zinc finger fragment of hMTF-1, hMTF-zf46 (residues 226 -320; 95 residues) used in these studies. Apo-hMTF-zf46 was purified to homogeneity in a form with a residual Zn(II) content of 0.1 g mol Ϫ1 Zn(II), with nearly all the Cys in their reduced form by 5,5Ј-dithiobis(nitrobenzoic acid) analysis (see "Materials and Methods"). Metal binding studies of metal-free apo-MTF-zf46 reveal that, as expected, this peptide fragment binds ϳ3 mol eq of Co(II) (Fig. 2). The molar absorptivity spectrum of the Co(II)saturated complex is identical to those previously published for canonical TFIIIA-type Cys 2 -His 2 zinc finger peptides and proteins (31,54). Addition of stoichiometric Zn(II) results in complete bleaching of the optical spectrum (data not shown).

RESULTS
An 15 N HSQC spectrum is shown in Fig. 3 for a sample of FIG. 1. A, the amino acid sequence of hMTF-zf46 (residues 226 -320) used in these studies. Black triangles represent residues that would be expected to make base-specific contacts with the MRE based on the structures of other zinc finger-DNA complexes (29,30). B, amino acid sequence comparison of the F5 zinc finger domain of MTF-1 from mouse (mMTF), human (hMTF), Fugu rubripes (fMTF) and D. melanogaster (dMTF).
apo-and fully reduced hMTF-zf46 to which was added Ն3 mol eq of Zn(II). As can be seen, the most intense cross-peaks in the spectrum are of a relatively uniform intensity, and of narrow linewidths, giving rise to excellent spectral dispersion, indicative of a well-folded protein. As expected, the apoprotein 15 N HSQC spectrum is essentially that expected for an unfolded protein (data not shown), revealing that Zn(II) binding folds at least a subset of the C-terminal fingers into ␤␤␣ units (55). Interestingly, only about 65 or so of 91 expected backbone 1 H-15 N correlations are actually cleanly observed in this spectrum of the zinc-saturated protein. Consistent with this, in addition to these sharp resonances, there are cross-peaks clustered in the middle of the spectrum of very weak intensity, suggestive of chemical exchange broadening in at least one region of the hMTF-zf46 molecule.
To determine which regions of MTF-zf46 are characterized by chemical exchange broadening, 1 H, 13 C, 15 N sequential resonance assignments were obtained for the regions of the zinc-saturated protein that could be connected by implemen-tation of a backbone walk obtained upon sequential application of CBCA(CO)NH (44), CBCANH (45), HBHA(CO)NH (46), and HBHANH (47) triple resonance experiments. A representative run through F4 residues Cys 231 -Leu 243 is shown in Fig. 4, with nearly complete 15 N, 13 C␣, 13 C␤, HN, H␣, and H␤ resonance assignments shown in Table IS (see  supplemental material). It was trivial to link all resonances from the residues Lys 227 to Lys 257 in F4 and Ser 293 to the C-terminal Trp 320 ; these assignments are superimposed on the 15 N HSQC spectrum shown in Fig. 3. As can be seen, these assignments identify the vast majority of intense peaks that are well-resolved in the spectrum, with the remainder of the cross-peaks poorly resolved and of low intensity. These data reveal that, under these conditions of saturating Zn(II), F5 resonances as a group experience severe chemical exchange broadening, with essentially no evidence for stably folded ␤␤␣ structure.
Secondary Structure Analysis of F4 and F6 Zinc Finger Domains in MTF-zf46 -Shown in Fig. 5 are the results of the residue-specific secondary structural analysis of F4, F6, and C-terminal tail residues in hMTF-zf46 as inferred from chemical shift indexing with the most reliable indicator, the 13 C␣ chemical shift (56), and the magnitude of the 3 J H␣-HN coupling constant. A contiguous run of at least four residues in which the 13 C␣ chemical shifts are ϩ0.8 ppm or greater than the random-coil values (i.e. ϩ 13 C␣ ⌬␦ coil ) is a strong indication of ␣-helical secondary structure (57). On the other hand, three or more residues with 13 C␣ chemical shifts Ϫ0.5 ppm or greater than the random coil value (i.e. Ϫ 13 C␣ ⌬␦ coil ) have a ␤-strand conformation. Residues 227, 228, and 239 in F4 and the corresponding residue in F6 (299) are representative of this class of residues. To obtain further support for this secondary structure assignment, the magnitude of 3 J H␣-HN was measured with a three-dimensional HNHA experiment (48). ␣-Helical regions are characterized by a continuous stretch of 3 J H␣-HN Ͻ 5 Hz, whereas residues in ␤-strand conformations have 3 J H␣-HN Ͼ 8.0 Hz. Residues in random coil or other structured regions are typically characterized by 3 J H␣-HN between these two extrema, typically 3 J H␣-HN ϳ 7.0 Hz.
Both experiments lead to essentially the same conclusion, i.e. there is excellent correspondence of secondary structure in F4 and F6 and strong evidence that the F4 and F6 domains adopt the canonical ␤␤␣ fold (Fig. 5). These structural features for hMTF-zf F4 and F6 bear a detailed similarity to the N-terminal three finger domains of TFIIIA (zf1-3) and the four zinc fingers of the Wilm's tumor suppressor protein (wt1-4) (27,58). The two anticipated short stretches of classic ␤-structure are reasonably well-defined by the data, with the N-terminal start of the ␣-helix (residue ϩ1; Thr 243 in F4 and Thr 303 in F6) in exactly the anticipated position. Furthermore, the amino acids in the Ϫ1 position relative to the start of the ␣-helix, Thr 242 and Thr 302 in F4 an F6, respectively, exhibit strongly upfieldshifted 13 C␣ and downfield-shifted 13 C␤ chemical shifts, consistent with this residue serving as the N-cap on the ␣-helix in both cases, exactly as previously found in TFIIIA zf1-3 and wt1-4 (27). Interestingly, the run of ␣-helical residues in each finger domain appears to terminate at the second residue of the His-X 3 -His motif (of residues Arg 251 and Lys 311 , respectively), as evidenced by the relatively small positive values of 13 C␣ ⌬␦ coil and 3 J H␣-HN values of ϳ8 Hz (Fig. 5). This is consistent with a distortion in the ␣-helical geometry to 3 10 -helical geometry as observed in other His-X 3 -His motifs in classic zinc fingers, particularly when not bound to DNA (27). This is required to accommodate bis-imidazole coordination of the Zn(II) in this structure. 15 Fig. 6 shows a presentation of the 15 N heteronuclear relaxation data acquired at 600-MHz 1 H frequency for F4, F6, and C-terminal tail residues. The magnitude and uniformity of T 1 /T 2 (R 2 /R 1 ) ratio when considered along with the { 1 H}-15 N heteronuclear NOE can provide insight into backbone conformational dynamics on the picosecond to nanosecond, as well as microsecond, time scales (51). The { 1 H}- 15 N NOE values reveal that, by and large, F4 residues from Phe 229 in the ␤ 1 strand to His 253 encompassing the entire ␤␤␣-structure tumble as an independently folded globular domain, with the mean (ϮS.D.) value of the NOE of 0.61 Ϯ 0.07 averaged over these 24 residues. This is comparable to the magnitude of the mean { 1 H}-15 N NOE determined in other ␤␤␣ zinc fingers under similar solution conditions (59 -61). The same is true of residues Ser 293 to His 313 in F6, with the mean (ϮS.D.) value of the { 1 H}-15 N NOE of 0.65 Ϯ 0.06 averaged over these 19 residues. As expected, the N-terminal residues Gly 226 and Lys 227 , as well as the F6 C-terminal tail residues Lys 316 to Trp 320 are characterized by weakly positive or strongly negative NOE values, consistent with these residues reorienting rapidly in the picosecond to nanosecond time scale and are otherwise unstructured.

N Heteronuclear Relaxation Behavior of F4 and F6 Zinc Finger Domains in MTF-zf46 -
The mean T 1 value for 24 F4 (229 -253) and 19 F6 residues (292-313) are essentially indistinguishable, with T 1 (F4) ϭ 0.518 Ϯ 0.020 s and T 1 (F6) ϭ 0.529 Ϯ 0.022 s. The T 2 values are significantly different, which makes the T 1 /T 2 ratio significantly larger for the C-terminal F6 domains relative to F4. For F4, T 2 ϭ 0.140 Ϯ 0.026 s to give T 1 /T 2 ϭ 3.84 Ϯ 0.09, whereas for F6 residues, T 2 ϭ 0.126 Ϯ 0.022 s to give T 1 /T 2 ϭ 4.33 Ϯ 0.09. Assuming isotropic tumbling for individual zinc finger domains, the rotational molecular correlation times, m , calculated for the F4 and F6 finger domains, are 5.3 (Ϯ0.8) ns and 5.8 (Ϯ0.8) ns, respectively. These m values are comparable to that previously measured for folded zinc finger domains within multizinc finger proteins, TFIIIA zf1-3 (60) and yeast ADR1 (59) when not bound to the DNA; thus, F4 and F6 domains are clearly characterized by essentially independent tumbling in solution. However, although the effect is small and not statistically outside of the standard deviation of the mean T 2 or T 1 /T 2 FIG. 4. Strategy for obtaining complete 1 H, 13 C, and 15 N  values observed for the residues within individual finger domains, the C-terminal F6 zinc finger domain seems to behave, relaxation-wise, as though it were a slightly larger globular particle than the F4 ␤␤␣ domain. A similar observation was made for one the zinc finger domains of the yeast ADR1 DNAbinding domain (59). The structural origin of this effect is unknown, but systematic shortening of T 2 throughout an entire globular domain is likely indicative of some other relaxation process operating in the s time scale superimposed on local short-time N-H bond vector reorientation and globular tumbling of that domain (cf. Ref. 39).
Zinc Binding Studies of Apo-MTF-zf46 -The data presented above reveal that both F4 and F6 domains fold into the canonical ␤␤␣-structural module in the presence of saturating Zn(II), with zinc finger F5 specifically characterized by unusual structural properties. In addition, previous data suggest that the C-terminal zinc finger domains form weaker zinc coordination structures than do N-terminal finger domains F1-F4 (33,36). Therefore, a series of Zn(II) binding titrations were carried out in an attempt to gain additional insight into these two processes. Fig. 7A shows selected representative 15 N HSQC spectra of

FIG. 5. Secondary structural analysis of Zn 3 hMTF-zf46 as determined by chemical shift indexing using 13 C␣ chemical shifts (57) and determination of the 3 J H␣-HN from a three-dimensional HNHA experiment (48).
Structural information on F5 remains lacking.
apo-␣-15 N-Lys-labeled hMTF-zf46 acquired over the course of a Zn(II) titration, with the normalized intensities of individual cross-peaks found in Zn 3 hMTF-zf46 plotted as a function of added Zn(II) shown in Fig. 7B. hMTF-zf46 contains 10 lysine residues, distributed relatively evenly throughout the primary structure, some in identical positions in different fingers, e.g. Lys 238 , Lys 268 , and Lys 298 in the predicted ␤2 strand in fingers F4, F5, and F6, respectively (Fig. 1A). Given the heteronuclear 15 N relaxation results described above, lysines in individual finger domains will report on Zn(II) binding to that finger only. The apoprotein spectrum (Fig. 7A, upper left) shows the expected ten resolvable cross-peaks, although one is of relatively low intensity. The integrated cross-peak intensities (Fig. 7B) derived from individual 15 N HSQC spectra are normalized to that of Lys 316 in the unstructured C-terminal tail (Fig. 6), whose chemical shift does not change as a function of bound Zn(II).
Simple inspection of the spectrum obtained upon addition of ϳ3 mol eq of Zn(II) reveals spectral dispersion indicative of a folded structure in all three zinc finger domains. However, cross-peaks assigned to F5 lysines are of very low intensity, even in this spectrum (see below). Fig. 7B reveals that the cross-peak intensities of lysines in MTF-zf46 cluster into es-sentially three groups as a function of added zinc. The crosspeaks of Lys 238 and Lys 248 reach maximum intensity at ϳ1 mol eq of Zn(II), with the other five cross-peaks not achieving maximum intensity until ϳ2 to 3 mol eq of metal is added. Finger F6 resonances (Lys 298 , Lys 307 , and Lys 311 ) remain fully developed even at excess Zn(II). A third pair of resonances, which by elimination correspond to Lys 268 and Lys 277 in F5, actually disappear completely at excess Zn(II), concomitant with the appearance of at least one new cross-peak just outside of this spectral window (data not shown). Lys 227 near the N terminus is just outside this spectral window; the intensity of Lys 257 in the F4 -F5 linker could not be easily monitored due to low intensity and spectral overlap.
These data reveal that the apo-F4 domain is preferentially loaded with Zn(II) in the presence of stoichiometric metal. After the binding of Zn(II) to apo-F4, the F5 and F6 domains compete about equally well for subsequent additions of Zn(II). Under conditions of excess Zn(II), the F6 domain is stably folded; in strong contrast, F5 resonances become exchanged broadened beyond detection (Fig. 7B).
Zinc Binding Studies of Apo-MTF-zf56 -The conformational heterogeneity in F5 was further explored within the context of C-terminal two-finger domain fragment of hMTF-1, hMTF- zf56, in an analogous Zn(II) titration. This experiment was carried out to determine if F4 influenced in any way the unusual structural properties of F5. Representative spectral regions of complete 15 N HSQC spectra (the upfield ␣-helical region is shown) for uniformly 15 N-labeled hMTF-zf56 as a function of added Zn(II) are presented in Fig. 8 with normalized peak intensities plotted in Fig. 9. Tentative resonance assignments for F5 cross-peaks were obtained by analogy with F4 and F6 cross-peaks of the corresponding residues.
These data are in qualitative accord with the Zn(II) titration of ␣-15 N-Lys-MTF-zf46 shown above (Fig. 7B). However, the surprising conclusion is that the entire F5 finger domain, in contrast to that of F6 (and F4), is only metastable under these conditions during the course of a zinc titration. In fact, at 2.0 mol eq of Zn(II), the F5 cross-peaks which derive from the ␤␤␣ fold are already significantly broadened; at ϳ3.0 mol eq of Zn(II), these peaks have effectively disappeared (Fig. 9). These data suggest that the canonical ␤␤␣ structure of F5 is reorganizing into some other structure(s), which is in intermediate conformational exchange with the canonical structure, a structural change largely complete upon the addition of approximately one stoichiometric equivalent of excess Zn(II). Cd(II) Binding to Apo-hMTF-zf46 -Although numerous reports suggest that cadmium is a potent inducer of MRE-mediated MTF-1-dependent activation of gene expression in mammalian cells (4), the mechanism of the induction remains poorly defined. Furthermore, Cd(II) does not activate the DNA-binding activity of MTF-1, like that which can be demonstrated to occur with Zn(II). Thus, if folding of ␤␤␣ zinc fingers upon addition of Zn(II) lies as the molecular basis for Zn(II)-dependent stimulation of the MRE-binding activity, the prediction can be made that Cd(II) would not be capable of inducing the folding of the finger domains F4 and F6, for example.
An anaerobic Cd(II) titration of apo-MTF-zf46 as monitored by UV spectroscopy is shown in Fig. 10. This titration indicates that Cd(II) forms metal complexes with apo-MTF-zf46 involving at least some Cd(II)-cysteine thiolate coordination bonds. The intensity of the ligand-to-metal charge transfer absorption at 250 nm is apparently maximal at about 1.5-2.0 mol eq, which falls off slightly upon addition of one additional mol eq of the metal. The molar absorptivity, depending on how many Cd(II) ions are actually bound in each case, is consistent with no more than ϳ1-2 cysteine thiolates per bound Cd(II) ion, if one assumes ϳ5000 M Ϫ1 cm Ϫ1 per Cd-S bond at 250 nm (62). Inspection of the 15 N HSQC spectrum of ␣-15 N-Lys-MTF-zf46 reveals that the addition of 2 mol eq of Cd(II) fails to properly fold the structure, and that 2.0 mol eq of Zn(II) readily displaces the bound Cd(II) to give rise to spectrum indistinguishable from that recorded when 2.0 mol eq of Zn(II) are added directly to the apoprotein (Fig. 7A). These data reveal that apo-MTF-zf46 binds Cd(II) at cysteine-thiolates to form a weak, non-native complex, which are readily displaced upon addition of Zn(II). The same spectra are obtained when greater than two equivalents of metal are added. DISCUSSION An amino acid sequence alignment of MTF-1 from human, mouse, Japanese pufferfish, and D. melanogaster reveals that the zinc finger domain is the most highly evolutionarily conserved region of the molecule (13,28). Finger F6 is the most divergent among all of the zinc fingers, but even these sequences share ϳ70% sequence similarity with most of the differences in the Cys-X 4 -Cys loop. A sequence alignment of finger F5 domains (Fig. 1B) reveals that F5 is as highly conserved as the major MRE-binding zinc fingers of the molecule (F1-F4), despite the fact that it plays an ancillary role in DNA binding as revealed by limited proteolysis and biophysical char-  acterization of finger deletion mutants (33) and has been reported to be dispensable for zinc-dependent activation of MREbinding by intact hMTF-1 and zinc-dependent transcriptional activation in transient transfection experiments in vivo in mammalian cells (34,35).
The findings presented in this paper reveal that the Cterminal zinc fingers in hMTF-1 are remarkably structurally heterogeneous. All indications are that F4 is a structural zinc finger, which exhibits high intrinsic affinity for Zn(II), and plays a critical role in maintaining high affinity binding to the MRE. Finger domains F1-F4 comprise the minimal high affinity binding domain of hMTF-zf, because deletion of either F1 or F4 results in nearly nonspecific binding by the zinc finger fragment (33,36). F4 is the only zinc finger among all six in MTF-1 that can be considered a standard zinc finger (30) and is predicted to utilize canonical recognition elements in the ϩ2, ϩ3, and ϩ6 residues of the recognition ␣-helix (28). F4 is clearly not able to mediate high affinity binding in the absence of F1-F3, however, because the fragment characterized here, hMTF-zf46, has no detectable DNA binding activity (33).
Like finger F4, F6 clearly adopts a canonical ␤␤␣ fold. However, its equilibrium affinity for Zn(II) is clearly lower, because apo-F5 and apo-F6 don't compete well with apo-F4 for limiting amounts of Zn(II). The rate of alkylation of Zn(II)-coordinating thiolates in F5 and F6 finger domains in the zinc finger fragment has also recently been shown to be far greater than those determined for F1-F4 domains, consistent with weaker Zn(II) coordination bonds in these two fingers (36). The structural origin of this difference in chelate stability is unknown, although previous studies suggest that zinc affinities can vary over a 10-to 100-fold range with substitutions outside of the zinc-coordinating residues (54,63). Because folding of the ␤␤␣motif is obligatorily coupled to formation of the Zn(II) chelate, changes in the propensities of individual residues within the short two-stranded ␤-strand and/or ␣-helix to adopt their respective secondary structures will have a measurable effect on metal affinity. In fact, Berg and coworkers (64) have used the consensus zinc finger peptide framework to estimate ␤-sheet propensities by simply measuring K Zn for ␤-sheet variants.
In contrast to F4 and F6, finger domain F5 at saturating Zn(II) is characterized by severe and uniform exchange broadening of F5 resonances. Examination of the global secondary structure of Zn(II)-saturated hMTF-zf46 and hMTF-zf56 by far-UV-circular dichroism suggests the F5 domain does not adopt the predicted ␤␤␣ fold under these conditions (32, 33) (data not shown). This is consistent with the 15 N HSQC spectrum shown in Fig. 3 and, therefore, suggests that a slight excess of Zn(II) stabilizes one or more alternate conformations of the finger domain. The structure of this alternate conformation cannot be determined using NMR methods but would appear to have little or no stable secondary structure.
Although the structural details are unclear, the sequence conservation of both F5 among different MTF-1 s is striking and unusual (Fig. 1B). A BLAST search reveals that no other Cys 2 -His 2 zinc finger domain conserves the combination of potential zinc-liganding amino acids in the Cys-X 4 -Cys loop, the His pair in the ␣-helix, and the strongly helix-promoting AFAA sequence in what is supposed to be the finger-tip region of the ␤␤␣ structure (Fig. 1B). One or more of these conserved features could significantly alter the conformational equilibria of this finger when presented with a slight excess of Zn(II). Zn(II) titration experiments reveal that the ␤␤␣ fold does become populated at low concentrations of the metal; thus, this sequence is intrinsically capable of forming a canonical zinc finger. However, as the Zn(II) concentration is raised, this structure is in equilibrium with one or more competing conformations, with chemical exchange broadening largely complete with the addition of a second equivalent of metal (Fig. 9). This situation is reversed in certain designed zinc finger peptides, which form a 2:1 peptide:metal complex at limiting metal, which then rearrange to the ␤␤␣ fold upon addition of stoichiometric or greater metal (54). The simplest interpretation of our data is that the ␤␤␣ conformation of F5 is unstable and readily binds a second mol eq of Zn(II), likely mediated by one or more the zinc-chelating residues situated in the Cys-X 4 -Cys ␤-turn and/or the solvent-exposed His pair in the ␣-helix (Fig.  1B). Puzzling, however, is the finding that Co(II) appears to stabilize the ␤␤␣ fold, as determined by Co(II) optical spectroscopy, even at excess metal (Fig. 2). Thus, the effects of Co(II) and Zn(II) on hMTF-zf structure are easily distinguished from one another. Co(II) is not a strong inducer of the MT gene expression in vivo. Unlike Co(II), the other strong inducer, Cd(II), binds to at least a subset of cysteine thiolates, but clearly induces the formation of an alternate structure(s) rather than the ␤␤␣ fold under any conditions (Figs. 6 and 9). These findings are in agreement with recent proton NMR experiments carried out with Sp1-like peptides (65).
The functional impact of this structural heterogeneity in F5 in full-length MTF-1 is unknown and currently under investigation. However, it may well be important in zinc metalloregulation in this system, particularly so if large-scale structural changes occur in MTF-1 as a result. It is tempting to speculate that this additional bound Zn(II) ion might recruit additional Zn(II)-coordinating residues in the immediately adjacent acidic activation domain or the conserved Cys-rich metallothioneinlike domain at the C terminus; such a structural change could functionally couple the DNA binding domain with the transcriptional activation domains and/or sequences responsible for nuclear export of MTF-1 (25) and therefore play an important role in signal transduction in this system.


Comments on conformal stability of brane-world models

The stability of 5-D brane-world models under conformal perturbations is investigated. The analysis is carried out in the general case and then it is applied to particular solutions. It is shown that models with the Poincare and the de Sitter branes are unstable because they have negative mass squared of gravexcitons whereas models with the Anti de Sitter branes have positive gravexciton mass squared and are stable. It is also shown that 4-D effective cosmological and gravitational constants on branes as well as gravexciton masses undergo hierarchy: they have different values on different branes.

Introduction
It is well known that part of any realistic multidimensional model should be a mechanism for extra dimension stabilization. This problem was a subject of numerous investigations. In the standard Kaluza-Klein approach cosmological models are taken in the form of warped product of Einstein spaces as internal spaces. Corresponding warp (scale) factors are assumed to be functions of external (our) space-time. If these scale factors are dynamical functions then it results in a variation of the fundamental physical constants. To be in agreement with observations, internal spaces should be compact, static (or nearly static) and less or order of electro-weak scale (the Fermi length). The stability problem of these models with respect to conformal perturbations of the internal spaces was considered in detail in our paper [1]. It was shown that stability can be achieved with the help of an effective potential of a dimensionally reduced effective 4-D theory. Small conformal excitations of the internal spaces near minima of the effective potential have the form of massive minimal scalar fields developing in the external space-time. These particles were called gravitational excitons (gravexcitons). Their physical meaning can be easily explained with the help of a simple 3-D model where 2-D spatial part has the cylindrical topology: S 1 × R 1 . Here, S 1 plays the role of the compact internal space and R 1 describes 1-D external space. Let us suppose that the size of S 1 is stabilized near some value by an effective potential. Then, conformal excitation of S 1 near its equilibrium position results in waves running along the cylinder (along R 1 ). Thus, any 1-D observer living on the cylinder (on R 1 ) will detect these oscillations as massive scalar fields. Obviously, this effect takes place for any multidimensional cosmological model with compact internal spaces. In general, it does not depend on presence or absence of branes in models. Masses of gravexcitons and equilibrium positions for the internal spaces depends on the form of the effective potential (on concrete topology and matter content of the model).
Recently [2,3,4], it was realized that it is not necessary for extra dimensions to be very small. They can be enlarged up to sub-millimeter scales in such a way that the Standard Model fields are localized on a 3-brane with thickness of the electro-weak (or less) length in the extra dimensions whereas gravitational field can propagate in all multidimensional (bulk) space. It gives possibility for lowering of the multidimensional fundamental gravitational constant down to the TeV scales (therefore this approach often call as TeV gravity approach). Cosmological models in this approach are topologically equivalent to the standard Kaluza-Klein one. Problem of their stability against conformal perturbations of additional dimensions was considered in papers [4,5]. A comparison of old and this new approaches from the point of view of conformal stability was given in paper [6]. In papers [4,5] conformal excitations of the additional dimensions near minimum position of the effective potential were called radions (to our knowledge, the first time that the term radion appeared it was in [4]). However, we prefer to call such particles gravexcitons because, first, from the point of priority and, second and it is the most important, the term radion is widely used now in the brane-world models in different content.
The brane-world models are motivated by the strongly coupled regime of E8 × E8 heterotic string theory which is interpreted as M-theory on an orbifold R 10 × S 1 /Z2 with a set of E8 gauge fields at each ten-dimensional orbifold fixed plane. After compactification on a Calabi-Yau three-fold and dimensional reduction one arrives at effective 5−dimensional solutions which describe a pair of parallel 3-branes with opposite tension, and located at the orbifold planes [7]. For these models the 5−dimensional metric contains a 4−dimensional metric component multiplied by a warp factor which is a function of the additional dimension. A cosmological solution of this type with flat 4-D branes (which we shall refer as Poincaré branes) was obtained in paper [8]. This model was generalized in numerous publications to the cases of bent branes in models with 5 and more dimensions and with single or many branes. In paper [9] it was stressed the necessity of stabilization the distance between branes to get conventional cosmology on branes. Here, the radius (scale factor) of the extra dimension was called radion. But this definition is not very precise. We should note that there is a confusion in the literature concerning the term radion: quite different forms of the metric perturbations of the brane-world models were called radions. However, in paper [10] it was clearly shown and strictly emphasized that radions describe relative distance between branes (see also papers [11,12]). It demonstrates the main difference between gravexcitons and radions: gravexcitons describe conformal excitations of geometry (more particular -conformal excitations of the additional dimensions) whereas radions describe relative motion of branes. Obviously, gravexcitons can exist in model where branes are absent at all or in models with a single brane and vice versa radions can exist in absence of gravexcitons. The latter situation can be realized for example in the TeV scale approach where branes can move relatively with respect each other due to interaction between them "sliding" on background fixed geometry (gravexcitons are absent). Branes are considered here as "probe bodies" moving in the background geometry. Nevertheless, in the brane-world models gravexcitons and radions are closely connected with each other (and this is the main reason for the confusion between them). Here, branes are 4-D surfaces along which different 5-D bulk solutions are gluing with each other. In this case positions of branes fix the shape and size of the geometry 1 and relative motion of branes results in conformal changes of the geometry. Thus, it is natural for such models to expect that solutions stable against radions are stable against gravexcitons and vice versa. Obviously, direct comparison of stability against gravexcitons and radions has only sense in models with two (and more) branes where radions exist. The radion stabilization problem was investigated in a number of papers devoted to the brane-world models. It was shown in particular, that in the case of solutions with two de Sitter branes (bent branes with 4-D effective positive cosmological constants) and two Anti de Sitter branes (bent branes with 4-D effective negative cosmological constants) radions have negative and positive mass squared respectively. Thus, the former solution is unstable but the latter one is stable against radions. In our paper we find, first, that the model with one de Sitter brane is unstable against gravexcitons and, second, that the model with one or a number of parallel Anti de Sitter branes (connected with each other via wormhole throats) is stable under conformal excitations. In the case of Randall-Sundrum solution [8] with two Poincaré branes, radions have zero mass [11,12]. From the Particle Physics point of view, such particles do not lead to instability. However, as it is well known, such ultra-light scalar fields, originated from the extra dimensions, produce a number of cosmological problems connected with flatness of their effective potential. For example, in the homogeneous case any small excitations near an equilibrium position (which can be chosen arbitrary for the flat potential) have a runaway behaviour if we do not take into account the friction term due to the cosmological expansion. However, such dynamical stabilization is very delicate problem (see e.g. [13]) and needs a separate investigation for each case. Exactly this kind instability for radions in the model with two Poincaré branes was found in [11]. It was shown here that a small departure from the equilibrium position results in either a colliding of the two branes or a runaway behaviour. In our paper we show that models with two as well as one Poincaré brane are also unstable under conformal perturbations. It is necessary to stress that in papers [11,12] the analysis was performed in the Brans-Dicke frame. In our paper the problem of stability is investigated in the Einstein frame. Obviously, if models are stable in one frame they are stable in another one because both of frames coincide in the equilibrium position. However, exact form of the dynamical behaviour (time dependence) near the equilibrium position can depend on the frame. In paper [14] it is explicitly discussed the role of conformal transformations and it is shown that some solutions of the brane-world models exist in one frame but are absent in the other one. Therefore, the equivalence between these two frames depends on the concrete discussed problem and in some cases is a matter of a delicate investigation.
In papers [11,12], mentioned above, the authors conclusions concerning the radion stability/instability were obtained for the brane-world models where matter in bulk as well as on branes is absent (correctly speaking, it is considered there in its simplest form as bulk cosmological constant and "vacuum energies" on branes). In this case, only the pair of the Anti de Sitter branes are really stable. However, it was observed that inclusion of matter can stabilize radions for different types of branes. It can be done with the help of bulk scalar field [9], [15]- [17], perfect fluid on branes [18] and the Casimir effect between branes [19,20].
Some specific forms of instability in the brane-world solutions were observed in papers [21,22]. It was shown that single Poincaré brane is unstable under small perturbations of the brane tension 2 [21] and single de Sitter brane is unstable against thermal radiation [22]. The main goal of our present comments consists in investigation of 5-D brane-world stability against conformal perturbations. First, we elaborate a method to study the stability for a large class of solutions and obtain general expressions for 4-D effective cosmological constants on branes and masses of gravitational excitons. Then we apply this method to a number of well known solutions. In particular, we find that models with the Poincaré and the de Sitter branes are unstable because they have negative mass squared of gravexcitons whereas models with the Anti de Sitter branes have positive gravexciton mass squared and are stable under conformal perturbations. We show also that 4-D effective cosmological and gravitational constants on branes as well as gravexciton masses undergo hierarchy: they have different values on different branes (if different branes have different warp factors).
The paper is organized as follows. In Section II we explain the general setup of our model, perform dimensional reduction of the brane-world models to an effective 4-D theory in general case and apply this procedure to a number of well known solutions. In Section III we elaborate a method of the investigation of brane-world solution stability against conformal perturbations and apply it to particular solutions considered in Section II. Here we show also that physical masses of gravexcitons undergo hierarchy on different branes. The brief Conclusions of the paper are followed by three appendices. In Appendix A we present useful expressions for the Ricci tensor components and scalar curvature in the case of block-diagonal metrics. Some useful formulas of the conformal transformation are summarized in Appendix B. In Appendix C we show that the results of the paper do not change if only additional dimension undergoes conformal perturbations: we arrive here to the same 4-D effective theory and the same gravexciton masses as in the case of total geometry conformal perturbations. This provides an interesting analogy between gravity and an elastic media where the eigen frequencies of an elastic body oscillations do not depend on the manner of excitation.

Model and general setup: dimensional reduction of braneworld models
We consider 5-D cosmological models on a manifold M (5) which is divided on n pieces by n − 1 branes: where r is an extra dimension. Each brane is characterized by its own tension Ti(ri) , i = 1, . . . , n − 1. We suppose that a boundary ∂M (5) also corresponds to two hypersurfaces r = const : r = r0 and r = rn and, either 4-D geometry on ∂M (5) is closed (induced 4-D metric vanishes there), or opposite points r0 and rn are identified with each other. In the first case boundary terms corresponding to ∂M (5) are equal to zero. In the second case, the boundary ∂M (5) is absent, however, if the geometry is not smoothly matched here, it results in appearance of an additional brane with a tension T0(r0)). For simplicity, bulk matter is considered in the form of a cosmological constant, in general, different for each of M (5) i . Thus, our model is described by the following action: is the standard York -Gibbons -Hawking boundary term 3 . The Einstein equation corresponding to action (2.1) reads: In equations (2.1) and (2.2) Here it was mentioned about instability of the single brane Randall-Sundrum solution under homogeneous gravitational perturbations. In our paper we show that this model is unstable also under conformal excitations. 3 In compact brane-world models it is worthy to include this term even if boundary ∂M (5) is absent because it is convenient here (as well as for all models with branes) to split manifold M (5) by branes into n submanifolds: with piecewise discontinuous functions where step functions η(r − ri) equal to zero for r < ri and unity for r > ri. Now, we suppose that a metric 4 is the solution of the Einstein equation (2.2) and has following matching conditions: ai(ri) = ai+1(ri) , i = 1, . . . , n − 1 and a1(r0) = an(rn). Scale factors ai(r) are supposed to be non-negative smooth functions in intervals [ri−1, ri]. Boundary points r0 and rn are either identified with each other: r0 ↔ rn or they are not identified and the geometry in latter case is closed: a1(r0) = an(rn) = 0 ; i.e. induced metric g (4) µν (x, r) = a 2 (r)γ (4) µν (x) vanishes in this points. Having at hands solution (2.5), we can perform dimensional reduction of action (2.1). Here, the dimensional reduction means integration over extra dimension in 5-D part of action (2.1) to get 4-D effective action. To do that, let us perform first some preliminary calculations.
Applying equation (A.3) to our case we obtain (2.7) Using properties of θ-function : , the function fi(r) can be written in the following form: in accordance with equation (B.6). As we can see, function f1 contains all information about boundary terms and for correct dimensional reduction of action (2.1) it is not necessary to include additionally boundary term SY GH because it will lead in this case to its double counting 5 . It can be easily seen also that integral where we used integration by parts. Thus, dimensional reduction of action (2.1) will result in the following effective 4-D action: S 4 Different parts of the manifold M (5) can be covered by different coordinates charts. We show an explicit example below. 5 There are two equivalent ways of the dimensional reduction. First, we can divide action integral (2.1) into n integrals in accordance with the splitting procedure described in footnote 3 and take into account the boundary terms at ∂M  In second approach, we consider full non-split action (2.1) without the brane boundary terms but take into account that scale factor a(r) is not a smooth function in points corresponding to the branes location. Thus, its second derivative has δ-function terms which completely equivalent to the brane boundary terms (see (2.8)). It can be easily checked that integration over extra dimension in both of these approaches results in the same 4-D effective action. In the present paper we applied second approach.

Examples
In this subsection we apply considered above procedure of the dimension reduction to some well known solutions (see e.g. [8,12,17,24,25]).

a) Poincaré branes
In this model 7 r0 = −L , r1 = 0 , r2 = L , b) De Sitter brane (symmetric solution) 6 In this paper we focus on the problem of stability of considered models and not discuss cosmology on branes. It is clear that from the point of an observer on a brane, physical metric is the induced metric on this brane (let it be i-th brane): g (ph)µν = a 2 i (r i )γ (4) µν . It means we should perform evident substitutions a j (r) → a j (r)/a i (r i ) in corresponding formulas. For example, for this observer physical effective 4-D cosmological and gravitational constants read as follows: Λ (ph)ef f = Λ (4) ef f /a 2 i (r i ) and κ 2 4 → κ 2 (ph)4 = κ 2 4 a 2 i (r i ). On proportionality of the effective 4-D Newton's constant on brane to a 2 (r) brane was pointed e.g. in [10] . 7 Here, we follow the original solution [8], where scale factors are dimensionless.

Stability under conformal excitations
Let us investigate now the stability of metric g (5) (X) defined in (2.5) with respect to conformal excitations. In other words, we want to investigate the dynamical behaviour of the conformal metric excitations developing on the fixed background g (5) (X). To do this, we consider a perturbed metric of the form of (B.1):ḡ (5) = Ω 2 g (5) ≡ e 2β g (5) , where Ω = 1 corresponds to the background solution and β ≪ 1 describes the small perturbation limit. Obviously, the background solution is stable against such perturbations if Ω oscillates with time near the value Ω = 1 and is unstable if Ω has a runaway behaviour from this value. According to the Perturbation Theory, full analysis should consist of two steps. The first one is the investigation of the dynamical behaviour of perturbations on the fixed background, and the second one is the study of the back reaction of perturbations on the background solution. In the present paper we are concentrating on the first problem, to find which brane-world solutions are stable against the conformal perturbations, and postpone the second problem for our future investigation.
According to the standard approach, equation of motion for perturbations (in our case for Ω or β), developing on the fixed background g (5) , can be obtained substitutingḡ (5) in equation (2.2) and taking into account the background solution g (5) (e.g. solutions from section 2.1). However, it is possible to investigate this problem by a different way: starting from action (2.1), putting there perturbed metricḡ (5) and, after that, taking into account the background solutions g (5) . Then, the resulting effective action will describe the dynamical behaviour of the perturbations on the fixed background. From this action we can obtain the energy momentum tensor of the perturbations to study the back reaction of them on the background metric.
In our paper we follow the second approach and put the perturbed metricḡ (5) into action (2.1) which yields 9 .:S With the help of equations (B.3) and (2.6) the first term in this action reads: For generality, we do not assumed the small perturbation limit β ≪ 1 keeping in action all non-linear perturbation terms. Transition to this limit can be easily performed in the final expression (see equation (3.12) below). In what follows, we shall consider a particular case when conformal prefactor is a function of 4-D space-time coordinates: Ω = Ω(x) ≡ exp(β(x)). It is well known that conformal excitations of this form behaves as scalar fields in 4-D space-time (e.g. on branes ). Because of the prefactor Ω 3 (x) in front of 4-D scalar curvature R[γ (4) ] in action, the 4-D metric γ (4) is written in the Brans-Dicke frame. However, it is more easy to investigate the conformal perturbation stability in the Einstein frame (in the introduction we mentioned the equivalence of these frames with respect to the stability analysis) : In this frame dimensionally reduced action (3.1) reads whereβ ≡ 3/2(1/κ4)β and Here, parameters Bi (i = 0, . . . , 3) are defined by equations (2.12) -(2.15). Now, the problem of the background solution (2.5) stability against the conformal excitations is reduced to existence of a minimum of the effective potential U ef f at point Ω = 1 ⇐⇒ β = 0 which corresponds to the absence of the perturbations. In Appendix B we show additionally that all other values for the minimum lead to metrics which in zero order do not satisfy the same Einstein equation as the background solution (2.5). It is clear also that the effective cosmological constant (2.11) should coincide with U ef f at Ω = 1 : Λ (4) ef f = U ef f (Ω = 1) which we explicitly obtain from (3.5). The extremum existence condition reads: Small excitations near a minimum position can be observed on branes as massive scalar fields -gravitational excitons with mass squared: Obviously, the original solution (2.5) is stable under these conformal excitations iff m 2 > 0 which prevent their runaway behaviour from the background solution. As it can be easily seen, all four models considered in previous section satisfy equation (3.6). It means that all these solutions are stationary points of U ef f if the effective potential is considered as a functional of a(r). For masses squared in the case of the Poincaré, the de Sitter (symmetric solution), the de Sitter (non-symmetric solution) and the Anti de Sitter branes we obtain respectively: Thus, three first solutions are unstable under considered conformal excitations: Ω = 1 corresponds to maximum but not to minimum of the potential (3.5). However, in the AdS brane case, mass squared is positive and decreases from 10 4 for L/l → 0 to zero for L/l → 1 (more precisely, numerical calculations show that m 2 → 0 for L/l → 0, 988). So, the AdS brane solution is stable with respect to the conformal excitations if the distance between brane and throats of wormholes is less than the AdS radius. As it was mentioned in the footnote 8, this case can be easily generalized to a number of AdS branes connected with each other via the wormhole throats. Then, in the case of n branes, for the gravexciton mass squared we obtain an expression which is simply an algebraic sum of the type (3.11) equations (with an evident substitution L → Li , i = 1, . . . , n for each member of the sum) and overall prefactor B −1 0 . Here, B0 is also a generalization of (2.35) to an evident sum. This mass squared is positive e.g. if Li/l < ∼ 1 , i = 1, . . . , n. For small fluctuations near minimum of U ef f action (3.4) reads: where the first integral corresponds to zero order theory (2.10) (background solution) and the second one describes gravitational excitons. This effective action can be used for investigation of the gravexciton back reaction on the background metric. If we put in action (3.1) conformally transformed brane tensionsT (ri) = (1/Ω)T (ri) (see (B.7) and (B.8)) instead of T (ri), the effective potential reads: In this case, Ω = 1 is not the extremum of the effective potential (3.13) for all of four considered solutions: they are not stationary points of this potential.

Conclusions
In the present paper we investigated the stability of 5-D brane-world solutions against conformal perturbations. For these models the 5-dimensional metric contains a 4-dimensional metric components multiplied by a warp (scale) factor a(r) which is a function of the additional dimension. Models contain n parallel branes "transversal" to the additional coordinate. As a matter we consider bulk cosmological constants between branes and tensions ("vacuum energies") on the branes. Scale factor is continuous piecewise function while its derivative has jumps on the branes. There are a number of well known exact solutions which belong to this class of models (e.g. [8,12,17,24,25,26,27,28]). We investigated stability of some of these models under the conformal excitations which are functions of 4-D space-time. Such excitations are of special interest because they behave as massive minimal scalar fields in 4-D space-time, for example can be observed as massive scalar particles -gravitational excitons on branes, as it takes place in the Kaluza-Klein approach [1,6].
To perform the stability analysis, we put the perturbed metric in the original 5-D action, took into account the background metric solution and integrated the action over the extra dimension (the dimension reduction). The obtained 4-D effective action describes the dynamical behaviour of the perturbations on the fixed background. The extremum of the effective potential in this action corresponds to the background solution. If this extremum is a minimum, the conformal perturbations oscillate around the background solution providing its stability. In this case, small excitations around this minimum are observed as gravitational excitons on branes. However, if this extremum is a maximum, the conformal perturbations have the runaway behaviour, and the background solution is unstable against such excitations.
We have shown that in the case of one and two Poincaré branes, one de Sitter brane (symmetric solution) and one de Sitter brane (non-symmetric solution), all these solutions are unstable with respect to these excitations because the effective potential has a maximum but not a minimum at the point corresponding to the original (background) solutions. In these models 4-D effective cosmological constant is non-negative (see (2.21), (2.27) and (2.33)). However, one AdS brane solution is stable if the distance between brane and throats of wormholes is less than the AdS radius. The effective 4-D cosmological constant is negative in this model (see (2.39)). The latter case is easily generalized to a stable model with a number of parallel AdS branes connected with each other via the wormhole throats. It is necessary to note that we found stability/instability against gravexcitons for models with the same kind of branes, which are correspondingly stable/unstable against radions considered in papers [11,12], although the direct comparison between models can be only made for cases with two and more branes.
Another remark consists in the analogy between the stability under conformal perturbations in the brane-world models considered here and the standard Kaluza-Klein models. The situation with these four solutions is similar to one we have in pure geometrical case in the standard Kaluza-Klein approach [1]. Here, the stability also takes place when 4-D effective cosmological constant is negative. If the effective cosmological constant is positive we have maximum of the effective potential instead of minimum. To shift the minimum of the effective potential to positive values, we should include matter into the model.
We found also that 4-D effective cosmological and gravitational constants on branes as well as gravexciton masses undergo hierarchy. It was shown that for observers on different branes with different warp factors these parameters have different values. Similar result with respect to the effective 4-D Newton's constant was obtained in [10].
There are a number of possible generalizations which are worth to investigate. First, it is of interest to include more rich types of matter in the model, e.g. perfect fluid in bulk as well as on branes, which simulates different forms of matter in the Universe. The presence of matter can stabilize radions in the brane-world models with non-negative 4-D effective cosmological constant on branes, as it was shown in [9], [15]- [20]. As we mentioned above, stabilization of gravexcitons in models with 4-D positive effective cosmological constant takes place also in the standard Kaluza-Klein approach if we include matter here [1]. Thus, we expect similar stabilization effect for gravexcitons in the brane-world models. Second possibility consists in generalization of the model to a multidimensional case with D > 5. It will give opportunity to include into consideration already obtained exact brane-world solutions with D = 6 and more dimensions. Additionally, as we wrote in section 3, the investigation of the background solution stability against the conformal perturbations is only the first problem. When we found the stable solutions, the second problem consists in the investigation of the perturbation back reaction on the background solution. We leave these issues for the future work.

A Block-diagonal metrics
In this appendix we present some useful formulas (see also [29]) for curvature tensors in the case of a block-diagonal metric of the form: . is a covariant derivative with respect to the metric g (D 1 ) . The scalar curvature reads correspondingly:

B Conformal transformation
For conformally transformed metric M N (X) = Ω 2 (X) g where T (ri) is the tension of the brane which causes the jump of the extrinsic curvature.

C Truncated conformal transformation
In this appendix we shall show that our results do not change if only additional dimension undergoes conformal perturbations in metric (2.5): g (5) (X) =⇒ḡ (5) (X) = Ω 2 (x)dr ⊗ dr + a 2 (r)γ (4) µν (x)dx µ ⊗ dx ν . where f1(r) is defined in (2.7). To get this expression, it is useful to go first to a new coordinate R : dR = a −1 (r)dr and then, after using conformal transformation formulas, come back to r again. It can be easily seen that after conformal transformation to the Einstein frame: and the dimensional reduction, action (3.1) is exactly reduced to effective action (3.4). Thus, gravexcitons have exactly the same masses (3.7) - (3.11). This result shows that geometry (gravitational field) under conformal transformations behaves as an elastic media. For an elastic body the eigen frequencies of its oscillations do not depend on the manner of excitation.