Lo and behold, the ARDs in the TRPA1 structure are oriented in a globally distinct fashion from in TRPV1, projecting perpendicularly to the membrane in a shape that I will call a "bullet" conformation--to distinguish it from the "pinwheel"--because of the combined shape of the ARDs plus the transmembrane domains.
Their distal ends, which are not resolved in the structure, appear to form a "cane" shape. The membrane-proximal regions of the ARDs form a completely new, INTRAsubunit interface with a coiled-coil formed by C-terminal helices along the 4-fold axis. This interface has properties consistent with of a noxious cold sensor. While the two interacting surfaces have gross shape complementarity, there are few close interatomic contacts, in particular no interdigitated hydrophobic side chains. Most of the interactions are polar, and many are likely mediated by ordered waters rather than being direct. Therefore, in character, this interface does not resemble physiologically relevant binding sites, but rather is reminiscent of the artifactual contacts observed in crystal structures.
If, in the cell, proteins were to bind to each other solely when their concavity and general charge distribution match, fluidity of the cytoplasm would be reduced. Therefore, this type of interaction probably forms only at the lower end of physiological temperature range (or in crystal structures). Furthermore, based on the structure, the formation of this interface seems to be incompatible with the ordering of the beta-bridges. Therefore, electrophiles always open the channel by disrupting the beta-bridges, but small shifts in the relative stability of the beta-bridges and coiled-coil interfaces could account for the large differences in thermal behavior among TRPA1 orthologs. There also may be a third interface, between the distal halves of the ARDs that are not resolved in the structure, that further tunes the response.
The next question is, assuming this is correct, what does the warm, non-covalently-modified state of the TRPA1 ARDs look like? One could conceivably assign the difference in the ARD conformations between the TRPA1 and TRPV1 structures entirely to the opposite thermal responses, such that the TRPV1 structure represents the inactive state of both receptors, and TRPA1 represents a state, also common to both receptors, with active intracellular domains yet a non-conducting pore domain (i.e., a "desensitized-like" state). However, this model (shown in the diagram below) would require an unprecedently huge rearrangement--even the ryanodine receptor, whose activation involves large-scale motions of ankyrin repeats(2), doesn't change so dramatically. It may not even be possible for the TRPA1 ARDs to form a TRPV1-like arrangement without their long, curved ends (the "handles" of the "canes") clashing with the membrane. Also, it is unclear what would favor a TRPA1-like conformation in TRPV1 in the absence of the coiled-coils.
On the other hand, the very recently published high-resolution structure of the inositol triphosphate receptor (IP3R)(3) shows a coiled-coil very reminiscent of the TRPA1 structure, yet the membrane-proximal ARDs (there are at least TWO ARDs per subunit in IP3R, and possibly more depending on how you define domain boundaries) show a quite TRPV1-ish-looking ~45-degree angle to the membrane plane, not a 90-degree angle as in the "bullet" conformation, showing that this combination can exist in a distantly related channel. This is an inactive structure, and if an IP3-bound structure were published that showed a "bullet"-like conformation, this would make such a rearrangement for TRP channels plausible after all. This would, however, require a large (up to 50 Å) motion of the other cytoplasmic domains of the IP3R, including the ligand binding domain. Whether TRP channel (or for that matter IP3R) activation involves a "square-to-windmill", "square-to-bullet", or yet a different type of motion remains to be resolved.
The other piece of data that is very difficult to reconcile with any of these models is the fact that the ARDs were found in one study to be dispensable for both cold and electrophile sensing(4). However, this report disagrees with multiple mutation studies that trace the electrophile sensitivity to N-terminal cysteines(5,6), and show large decreases in thermosensitivity, up to reversal of the response, with small changes in the ARDs(7).
A final question now--what about TRPM8? It's cool-activated, so the beta-bridges should not be the temperature sensor there. It is unclear what they WOULD sense, since other activators like menthol seem to act directly on the pore domain(8). One would expect an alternative ARD interface, like the coiled-coil contact in TRPA1. In fact, splicing the C-terminal tail of TRPM8 onto TRPV1 generated a cool-activated channel(9), suggesting that it may form the cold-promoted (or warm-disrupted) contact(s). However, the character of this interface should be like other signaling contacts that are metastable around physiological temperature, such as for instance the switch regions of GTPases. Thus, one would expect a mixed polar/nonpolar character, with side chains that nevertheless are well-packed. Only a structure will tell if this is in fact true.
(1)Paulsen CE, Armache JP, Gao Y, Cheng Y, Julius D. Structure of the TRPA1 ion channel suggests regulatory mechanisms. Nature. 520(7548):511-7. 2015. [pubmed]
(2)Efremov RG, Leitner A, Aebersold R, Raunser S. Architecture and conformational switch mechanism of the ryanodine receptor. Nature. 517(7532):39-43. 2015. [pubmed]
(3)Fan G, Baker ML, Wang Z. Gating machinery of InsP3R channels revealed by electron cryomicroscopy. Nature. 2015. [pubmed]
(4)Moparthi L, Survery S, Kreir M. Human TRPA1 is intrinsically cold- and chemosensitive with and without its N-terminal ankyrin repeat domain. PNAS. 111(47):16901-6. 2014. [pubmed]
(5)Hinman A, Chuang H, Bautista DM, Julius D. TRP channel activation by reversible covalent modification. PNAS. 103(51):19564-19568. 2006. [article]
(6)Trevisani M, Siemens J, Materazzi S. 4-Hydroxynonenal, an endogenous aldehyde, causes pain and neurogenic inflammation through activation of the irritant receptor TRPA1. PNAS. 104(33):13519-24. 2007. [pubmed]
(7)Jabba S, Goyal R, Sosa-Pagán JO. Directionality of temperature activation in mouse TRPA1 ion channel can be inverted by single-point mutations in ankyrin repeat six. Neuron. 82(5):1017-31. 2014. [pubmed]
(8)Bandell M, Dubin AE, Petrus MJ. High-throughput random mutagenesis screen reveals TRPM8 residues specifically required for activation by menthol. Nature neuroscience. 9(4):493-500. 2006. [pubmed]
(9)Brauchi S, Orta G, Salazar M, Rosenmann E, Latorre R. A hot-sensing cold receptor: C-terminal domain determines thermosensation in transient receptor potential channels. The Journal of Neuroscience. 26(18):4835-40. 2006. [pubmed]