I wrote in the first part that "in the the second part I will consider their extravagant periplasmic(!) flagella. There will be no "loose ends" left in either part, I promise." Spoiler: no "loose ends" but a (knowledge) gap in the middle, that is, when it comes to cell division in Borrelia. Here it goes...
Endoflagella, also called 'axial filaments' or 'periplasmic flagella', "are anchored at each end (pole) of the bacterium within the periplasmic space (between the inner and outer membranes) where they project backwards to extend the length of the cell. These cause a twisting motion which allows the spirochaete to move about" says Wikipedia in the entry for Spirochaetes. The overall architecture of endoflagella − inner‑membrane embedded motor rings, hooks, flagellar filaments − is not notably different from that of other flagella. Except that they don't possess the outer-membrane rings and do not form bundles, a common phenomenon for multiple polar flagella, but are arranged side-by-side as a belt (Figure 2.1).
This belt-like arrangement of the Borrelia endoflagella and their rotation as a belt results in a considerable torque on the 'protoplasmic cell cylinder' (the most frequently used term in research papers but I would prefer to call it a 'protoplasmic tube' rather, emphasizing its minute cross section.) This torque and the resistance of the 'protoplasmic tube' together determine the cell shape and makes the cells move in a corkscrew-like fashion. Indeed, flagella-less mutants appear flattened-out and don't move.
In case you're unhappy with the static 2D pictures in Figure 2.1 you may want to take a 3D-ride through a Borrelia burgdorferi cell here in 'The Atlas of Bacterial & Archaeal Cell Structure'. I'm tempted to warn you "fasten your belt!", the Borrelia sure did.
What happens when flagella emanating from both cell ends have wrapped as a belt around the 'protoplasmic tube' all their way to midcell? It has been found that the flagella overlap with each other as to give a belt that runs through the entire cell length (see Figure 21.1D). This makes the corkscrew-like movement of Borrelia cells through media with lower and likewise with higher viscosity (at low Reynolds numbers!) very energy saving, thus efficient, and no less efficient when crawling on minimally wetted surfaces like host tissue or agar plates. (A practical hint: If you ever get into quicksand − better you don't! − just don't start frantically rowing with your arms but recall how Borrelia does it: move corkscrew-like.) A tricky question for which there is no conclusive answer yet: how do the two flagellar motor patches "communicate" with each other to make sure that one set of patches sets the flagella into clockwise rotation while the other into counter-clockwise rotation? This is necessary to achieve a consistent direction of swimming for a cell − if both sets of flagella rotate in the same direction the cell stops moving and gets stuck with a "kink" in its middle (which is occasionally observed.) Or can the motor patches do away with "communicating" and rely instead on a subtle gradient of signaling molecules over their considerable length? We don't know.
What determines the direction of rotation of the flagella or its reversal? Just like in other flagellated bacteria: signal transduction via chemosensory arrays. These arrays "perceive environmental stimuli and collectively use them to control the phosphorylation level of receiver proteins (CheY and CheB), which mediate motor control and sensory adaptation, respectively." To break this down a bit: a signal (a sugar or an amino acid) that is temporarily bound by the periplasmic domain of its cognate (inner-)membrane-bound receptor (MCP) is "picked up" as a phosphate group by CheW, which passes it on to CheA. CheA in turn then phosphorylates CheY and CheB. CheY~P binds to the membrane-embedded flagellar motor and slows down its rotation, leading to movement reversal if enough "brakes" are applied. CheY rapidly looses its phosphate group by de-phosphorylation, thus restoring its initial state. CheB~P leads to the restoration of the receptor's "ready" state via an interplay with CheR and a methyl-group exchange on the MCP. The efficiency of this "phosphorelay" is kinetically controlled for on/off-switching, and the overall signal strength markedly amplified by the close proximity in the array of the proteins involved.
You see in Figure 2.2 a chemosensory array of Borrelia in side view, patched to and partly sticking within the inner membrane (via the MCPs), almost like a band-aid. Its location is in close proximity to the line-up of flagellar motors at the gradually narrowing cell end. There is virtually no way to make the transfer of the periplasmic signal across the cytoplasm via CheY to the flagellar motor any shorter! In the top view, the band-aid appearance of a chemosensory array is even more clearly visible, here for Treponema as I had no sample from Borrelia available but both spirochaetes cells have this narrow cross-section of <0.5 µm. The hexagonal grid of the array, the 'baseplate', is formed by the CheA and CheW proteins, and is structurally highly adaptable to varying membrane curvatures. On top of this, the hexagonal grid-structure of the array has another exquisite property: it can be expanded or shrunk on all sides at any time by self-assembly/dis-assembly, not unlike phage capsids. Keep in mind that just as the Borrelia have flagella at both cell ends, they also have the arrays at both.
These two 'cell ends' of the Borrelia as you see them in Figures 2.1 and 2.2 look more like pointed conical hats and not even remotely like the half-dome shaped 'cell poles' that you're familiar with from normal rod‑shaped bacteria like B. subtilis or E. coli, or coccoid bacteria like Streptococcus pneumoniae. Spirochaetes divide by binary fission − easily observed in the microscope − but the molecular orchestration of septum formation and cell division of Borrelia and other spirochaetes is poorly understood. Yes, Borrelia encodes the FtsZ and MreB cytoskeletal proteins but their precise roles are not known.
A recent study from Christine Jacobs-Wagner's lab came closer to an understanding of cell growth and division in Borrelia by using peptidoglycan synthesis as proxy. Peptidoglycan (PG), the thin 'support stocking' polymer that entirely covers the membrane-bounded cytoplasm, is constantly synthesized during cell growth by crosslinking sugar oligomers with short peptides in the periplasm. The short crosslinking peptides contain the unusual amino acid D-alanine and the authors used a fluorescent D-alanine derivative, 7‑hydroxycoumarin‑amino‑D‑alanine (HADA), to monitor PG synthesis in Borrelia by fluorescence microscopy. In growing cultures containing cells of all lengths, they found HADA incorporation throughout the length of the cells with slightly decreasing signal intensities at the cell ends. Stunning, however, was the concentration of strong HADA signals in the form of zones (of up to 3 µm!) rather than defined foci at midcell and, in addition, at the 1/4 and 3/4 positions relative to the cells' lengths (Figure 2.3A). Note that all 276 cells in the 'demograph' shown in Figure 2.3 were "straightend" by software prior to aligning them by size (Figure 2.3A) and for the other evaluations (Fig.2.3B,C).
It is a thoroughly confirmed observation yet without an understanding of the underlying (molecular) mechanism that the signal zones at the 1/4 and 3/4 positions appear at the time of maximum signal intensity in the center of the cell (stage II). However, it indicates that future cell division sites are already "tapped" if the intensity at these positions increases at a time when the signal intensity of the midcell zone decreases to zero, indicating cell division at this point there (stage III). Who tells the "division machinery" that the replicated chromosome plus the replicated plasmid zoo has been safely segregated to their respective cell halves? As I said above, no "loose ends" for Borrelia but still some nagging questions at midcell.