The Cake-Cut­ting Prob­lem in the Ar­chaeal World

by Kanika Khanna

Have you ever won­dered how to slice a cake in equal pro­por­tions for a large party? Hugo Stein­haus, a fa­mous Pol­ish math­e­mati­cian in­tro­duced a math­e­mat­i­cal way to deal with the fair cake-cut­ting prob­lem af­ter the sec­ond world war. He pos­si­bly used cakes as an ab­stract rep­re­sen­ta­tion of the lim­ited re­sources avail­able dur­ing those press­ing times and in­vented games with set rules and mul­ti­ple play­ers so that each player gets a fair share of the "cake" in ques­tion. Now, what hap­pens if we in­tro­duce geo­met­ric con­straints in the cake-cut­ting prob­lem? For ex­am­ple, does the shape of the cake, whether round or rec­tan­gu­lar, de­ter­mine the place­ment of the first cut for pro­por­tional dis­tri­b­u­tion? This is pre­cisely the ques­tion that Walsh et al. (2019) try to an­swer in two pleo­mor­phic ar­chaea, Haloferax vol­canii and Haloar­cula japon­ica who both can­not eas­ily com­mit to a uni­form cell shape or size. How, then, does cell shape in­flu­ence the place­ment of di­vi­sion plane to par­ti­tion the cell in two roughly equal halves?

Fig­ure 1. A A false-col­ored over­lay of ex­per­i­men­tal Ft­sZ1-GFP dis­tribution (green) and pre­dicted di­vi­sion plane distri­bution (ma­genta). The in­ter­sec­tion of the two cre­ates white. Num­bers ad­ja­cent to each cell cor­re­spond to the cross-cor­re­la­tion value be­tween ex­per­i­men­tal and pre­dicted di­vi­sion plane distribu­tions. B Scat­ter­plot with point den­sity (color scale on right) show­ing the cross-cor­re­la­tion be­tween the pre­dicted di­vi­sion plane and the ex­per­i­men­tal Ft­sZ1-GFP dis­tri­b­u­tion as a func­tion of cell as­pect ra­tio for 22,217 an­a­lyzed cells of H. vol­canii. C An ar­ray show­ing the di­vi­sion plane ori­en­ta­tion in the cells of the CetZ1 knock­out of H. vol­canii that have formed tri­an­gu­lar shapes. The left-hand num­ber of each panel is the as­pect ra­tio of the cell out­line, while the right-hand num­ber is the re­sult­ing cross-cor­re­la­tion be­tween pre­dic­tion and ob­ser­va­tion. The left most im­age is the phase-con­trast mi­cro­graph, the mid­dle ima­ge is the ex­per­i­men­tal FtsZ1 flu­o­res­cence dis­tri­b­u­tion and the right most im­age is the pre­dicted di­vi­sion plane. Source (also for Fron­tispiece)

The au­thors demon­strate that one mecha­nism for ac­cu­rate place­ment of cell di­vi­sion ma­chin­ery in pleo­mor­phic ar­chaea in­volves Tur­ing pat­terns. Alan Tur­ing, who intro­duced the math­e­mat­i­cal con­cept in 1952 showed that in­tro­duc­ing per­tur­ba­tion in the form of dif­fu­sion to a sys­tem at steady state re­sults in spa­tiotem­po­ral con­cen­tra­tion gra­dients, giv­ing rise to pat­terns in na­ture in­cluding skin pig­men­ta­tion and an­i­mal coat, among oth­ers. Pre­vi­ously, Walsh et al. (2017) demon­strated that spa­tiotem­po­ral oscil­la­tion of Min pro­teins be­tween the two poles in E. coli cells, which re­stricts the for­ma­tion of the cell di­vi­sion site to mid-cell, fol­lows the Tur­ing pat­tern­ing mech­a­nism. How­ever, it is chal­leng­ing to study a cou­pling be­tween cell shape and di­vi­sion plane place­ment in E. coli and other rod-shaped bac­te­ria due to in­her­ent dif­fi­cul­ties in gen­er­at­ing mu­tants with aber­rant mor­pholo­gies. On the other hand, the two ar­chaea, H. vol­canii and H. ja­ponica are rel­a­tively flat and ex­hibit a va­ri­ety of shapes. Also, while in the ma­jor­ity of bac­te­ria in­clud­ing E. coli, the place­ment of the di­vi­sion plane is neg­a­tively in­flu­enced by the dis­tri­b­u­tion of DNA within the cell, DNA in H. vol­canii is more uni­formly dis­trib­uted through­out the cell cy­to­plasm, likely ex­clud­ing its par­tic­i­pa­tion in di­vi­sion plane place­ment in ar­chaea (Walsh et al., 2019). These fac­tors make ar­chaea ideal model sys­tems to study the in­flu­ence of cell shape on di­vi­sion site place­ment.

In ar­chaea, FtsZ1, a ho­molog of FtsZ, the ma­jor cell di­vi­sion pro­tein in bac­te­ria with sim­i­lar­ity to eu­kary­otic tubu­lin, is crit­i­cal for cell di­vi­sion and lo­cal­izes as a ring at mid-cell (Dug­gin et al., 2015). To test if di­vi­sion site place­ment fol­lows a Tur­ing mech­a­nism, the au­thors first the­o­ret­i­cally pre­dicted the ori­en­ta­tion of the di­vi­sion plane from cell shape as the nodal plane that is least per­turbed in re­sponse to mi­nor cell shape changes. Then, they com­pared it to GFP-tagged FtsZ1 visu­al­ized by flu­o­res­cence mi­croscopy and as­signed a cross-cor­re­la­tion score to as­sess the de­gree of agree­ment be­tween the the­o­ret­i­cal pre­dic­tion and the ex­per­i­men­tal ob­ser­va­tion. The higher the cross-cor­re­la­tion value on a scale of 0 to 1, the higher the agree­ment be­tween the two val­ues.

Us­ing this ap­proach, the au­thors made the sur­pris­ing ob­ser­va­tion that the cross-cor­re­la­tion value be­tween the­o­ret­i­cally pre­dicted di­vi­sion plane and ex­per­i­men­tally de­ter­mined Ft­sZ1-GFP plane in H. vol­canii de­pends on the as­pect ra­tio of the cells (Fig­ure 1A,B). Al­though the the­o­ret­i­cal model is able to pre­dict the Ft­sZ1-GFP plane with high ac­cu­racy for elon­gated cells with low to mod­er­ate as­pect ra­tios (~<0.85), the model fails for cir­cu­lar cells that dis­play a twofold sym­me­try with a high as­pect ra­tio (ap­proach­ing ~1). The au­thors spec­u­lated that this could be at­trib­uted to cy­toskele­tal el­e­ments known to in­flu­ence po­si­tion­ing of cell di­vi­sion ma­chin­ery in eu­kary­otes. How­ever, dele­tion of CetZ1, an ar­chaeal tubu­lin-like cy­toskele­tal pro­tein that in­hibits for­ma­tion of rod-shaped cells, did not af­fect the re­la­tion­ship be­tween mean cross-cor­re­la­tion and the as­pect ra­tio of cells, sug­gest­ing that CetZ1 does not in­flu­ence the po­si­tion­ing of the di­vi­sion plane in H. vol­canii (Walsh et al., 2019).

Fig­ure 2. Two rounds of cell di­vi­sion. Four still frames (one per row) from time-lapse imag­ing show­ing two con­sec­u­tive rounds of H. vol­canii cell di­vi­sion. The left-hand im­ages show the pha­se-con­trast mi­cro­graphs which were used to au­to­mat­i­cally ge­nerate cell out­line (shown in yel­low). The sec­ond col­umn of im­ages shows the FtsZ1 flu­o­res­cence (green) with the cell out­line su­per­posed. The third col­umn of im­ages shows the pre­dicted di­vi­sion planes (ma­genta lines) based on the cell shape. The far-right col­umn of im­ages is the over­lay of the FtsZ1 flu­o­rescence and the pre­dicted di­vi­sion plane. Source

In ad­di­tion, knock­out of CetZ1 also in­creases the frac­tion of plate-shaped cells, in­clud­ing tri­an­gu­lar cells. The au­thors com­pared the the­o­ret­i­cally pre­dicted di­vi­sion plane orien­tation to that of Ft­sZ1-GFP in 47 tri­an­gu­lar cells in H. vol­canii (Fig­ure 1C). The pre­dic­tion of the model fits well in ~77% of cells with a well-de­fined di­vi­sion plane. In ~25% of cells im­aged, the di­vi­sion plane was not sin­gu­larly de­fined by a FtsZ1 ring as "proto-di­vi­sion planes" formed in mul­ti­ple di­rec­tions (row 3,4 in Fig­ure 1C). Sim­i­lar re­sults were ob­tained for tri­an­gu­lar-shaped H. japon­ica cells. The au­thors ar­gue that the slightly less ro­bust pre­dic­tion of di­vi­sion plane in trian­gular cells as com­pared to rod-shaped bac­teria could likely be at­trib­uted to the fact that tri­an­gles are more sen­si­tive to mi­nor fluc­tu­a­tions in cell shape. More­over, even slight er­rors as­so­ci­ated with trac­ing cell out­lines from mi­croscopy im­ages can lead to ma­jor changes in how the shape of the tri­an­gle is per­ceived for the­o­ret­i­cal pre­diction (for in­stance, dis­tin­guish­ing be­tween a sub-equi­lat­eral tri­an­gle and an isosce­les tri­an­gle if the cell out­lines are mis­judged). These fac­tors make it dif­fi­cult to find a sin­gle math­e­mat­i­cal so­lu­tion with the high­est score for tri­an­gu­lar cells as com­pared to rod-shaped cells.

Fur­ther, the au­thors also in­ves­ti­gated if the ori­en­ta­tion of the di­vi­sion plane in daugh­ter cell is in­flu­enced by the ori­en­ta­tion of the di­vi­sion plane in the par­ent. In­deed, the di­vi­sion plane in daugh­ter cells is dic­tated by geom­e­try of the daugh­ter cell alone and is in­de­pen­dent of that of the par­ent cell (Fig­ure 2). In­ter­est­ingly, in one of the time lapse se­ries, the au­thors ob­served a nearly per­fect equi­lat­eral tri­an­gle, such that even small er­ror in trac­ing the cell out­line or change in its shape over time re­sults in dras­tic shifts in the­o­ret­i­cally pre­dicted di­vi­sion plane ori­en­ta­tion (Movie S2). How­ever, Ft­sZ1-GFP is sta­ble, with at least two rings that are or­thog­o­nal to each other and di­vi­sion pro­duces three daugh­ter cells which sub­se­quently es­tab­lish their own di­vi­sion plane dic­tated by the geom­e­try of the new cell (Movie S1). Based on these ob­ser­va­tions, the au­thors pro­pose that tri­an­gles can pro­vide a very strin­gent test for pre­dic­tion of di­vi­sion plane and in such de­gen­er­ate cases as well as those of spher­i­cal cells, pos­si­bly other mech­a­nisms may be able to ex­plain the ori­en­ta­tion of the di­vi­sion plane. Per­haps fu­ture stud­ies to sys­tem­at­i­cally ma­nip­u­late cell shapes by plac­ing cells in mi­cro­fab­ri­cated cham­bers of de­fined geom­e­try for both ar­chaeal and bac­te­r­ial cells will help elab­o­rate these prin­ci­ples (Minc et al., 2011). Pre­vi­ously, such de­vices have been used to ma­nip­u­late sea urchin zy­gotes to a va­ri­ety of de­fined geome­tries, in­clud­ing tri­an­gu­lar, star-shaped and el­lipses in or­der to com­pre­hend cou­pling be­tween cell shape and di­vi­sion site place­ment (Minc et al., 2011).

Based on these ob­ser­va­tions, Walsh et al. sug­gest that a Tur­ing mech­a­nism may be pri­mar­ily re­spon­sible place­ment and ori­en­ta­tion of di­vi­sion plane in pleo­mor­phic ar­chaea. In that case, the sys­tem un­der con­sid­er­a­tion should rely on dif­fu­sion of chem­i­cal sig­nals in or­der to gen­er­ate gra­dients of spa­tial con­cen­tra­tion. This process re­quires en­ergy, which may be likely pro­duced by pro­teins with ATPase/GTPase ac­tiv­ity, the func­tion of many of which are yet to be an­no­tated from genomes of H. vol­canii and other pleo­mor­phic ar­chaea. In ad­di­tion to pat­tern­ing by chem­i­cal sig­nals, it is also pos­si­ble that gra­di­ent in me­chan­i­cal forces help in the place­ment of a di­vi­sion site, such as bend­ing forces gen­er­ated by mi­cro­tubule slid­ing that lead to beat­ing of cilia and fla­gella in sperm cells (Howard, 2009). Fur­ther iden­ti­fi­ca­tion of these chem­i­cal and/or me­chan­i­cal sig­nals that reg­u­late place­ment of di­vi­sion site based on a Tur­ing mech­a­nism will help us to elu­ci­date ge­neral prin­ci­ples of di­vi­sion site place­ment based on geo­met­ric cues in the sys­tem. And also, this maybe pro­vide us a more sci­en­tific way of cut­ting a cake fairly and squarely in our next gath­er­ing, what­ever its shape may be!

Ref­er­ences

Dug­gin IG, Aylett CHS, Walsh JC, Michie KA, Wang Q, Turn­bull L, Daw­son EM, Harry EJ, Whitchurch CB, Amos LA, Löwe J. 2015. CetZ tubu­lin-like pro­teins con­trol ar­chaeal cell shape. Na­ture. doi:10.1038/nature13983

Howard J. 2009. Me­chan­i­cal Sig­nal­ing in Net­works of Mo­tor and Cy­toskele­tal Pro­teins. Annu Rev Bio­phys. doi:10.1146/annurev.biophys.050708.133732

Minc N, Burgess D, Chang F. 2011. In­flu­ence of cell geom­e­try on di­vi­sion-plane po­si­tion­ing. Cell. doi:10.1016/j.cell.2011.01.016

Walsh JC, Angst­mann CN, Bis­son-Filho AW, Gar­ner EC, Dug­gin IG, Curmi PMG. 2019. Di­vi­sion plane place­ment in pleo­mor­phic ar­chaea is dy­nam­i­cally cou­pled to cell shape. Mol Mi­cro­biol. doi:10.1111/mmi.14316
 

Leg­end to Movies

Movie S1. Four rounds of cell di­vi­sion for H. vol­canii. The time-lapse movie shows an H. vol­canii (H98 + pID­J­L40-Ft­sZ1) cell un­der­go­ing four rounds of cell di­vi­sion. Left most im­age: the phase-con­trast im­age with au­to­mat­i­cally gen­er­ated cell out­lines (yel­low); sec­ond im­age: FtsZ1 flu­o­res­cence with cell out­line; third im­age: di­vision plane pre­dic­tions (ma­genta lines) based on cell shape; and far right im­age: over­lay of FtsZ1 flu­o­res­cence and di­vi­sion plane pre­dic­tions.

Movie S2. Di­vi­sion of an H. vol­canii cell with near equi­lat­eral tri­an­gu­lar mor­phol­ogy. The time-lapse movie shows an H. vol­canii (H26 + pID­JL40-Ft­sZ1) cell with near equi­lat­eral tri­an­gu­lar mor­phol­ogy un­der­go­ing two rounds of cell di­vi­sion. Left most im­age: the phase-con­trast im­age with au­to­mat­i­cally gen­er­ated cell out­lines (yel­low); sec­ond im­age: Ft­sZ1-GFP flu­o­res­cence with cell out­line; third im­age: di­vi­sion plane pre­dic­tions (ma­genta lines) based on cell shape; and far right im­age: over­lay of FtsZ1 flu­o­res­cence and di­vi­sion plane predict­ions.
 
 

Kanika Khanna is a newly minted PhD from the labs of Prof. Eli­zabeth Villa and Prof. Kit Pogliano at Uni­ver­sity of Cal­i­for­nia, San Diego. Her the­sis project fo­cused on elu­ci­dat­ing mol­e­c­u­lar mech­a­nisms of sporu­la­tion in Bacil­lus sub­tilis and those of phage repli­ca­tion in bac­te­r­ial hosts us­ing novel modal­i­ties in the field of cryo-elec­tron mi­croscopy (yet on the photo, she holds a fold­scope.) Out­side of lab, she finds so­lace in crochet­ing, 3D origami, and sci­ence out­reach.