Weaknesses of the original model 168

As   previously   mentioned,   the   models   that   we   developed   here   were   adapted  from  a  previously  developed  ring  model,  the  goal  of  which  was   to  reproduce  experimentally  measured  values  of  ring  tension  in  silico.  An   average  ring  tension  of  390  ±  150  pN  was  measured  from  ring  sliding  in   fission  yeast  spheroplasts,  while  the  simulation  produced  a  ring  tension   of   340   ±   57   pN   [35].   However,   we   initially   found   some   aspects   of   the   model  to  be  unrealistic,  particularly  with  the  way  that  individual  myosin   clusters/nodes  interact  with  multiple  actin  filaments,  and  this  calls  into   question  the  validity  of  the  results  previously  obtained  with  this  model.     Most   importantly,   we   found   that   the   value   of   the   parameter   maxInt,   which   controls   how   a   myosin   cluster/node   interacts   with   multiple  actin  filaments,  was  vastly  overestimated.  In  the  original  model,   myosin  clusters  were  assumed  to  contain  40  Myo2  molecules,  and  were   given  a  maxInt  value  of  10,  meaning  that  the  authors  were  claiming  that  4   Myo2  molecules  were  able  to  exert  a  time  averaged  force  of  4  pN  on  an   actin   filament   [35].   Type   II   myosins   are   known   to   be   non-­‐processive,   making   this   behaviour   unlikely   [169].   Furthermore,   when   we   implemented  an  equivalent  description  in  our  model  (20  Myo2  molecules   per   node,  maxInt   set   to   5)   using   a   cylindrical   geometry   we   found   that   large  numbers  of  actin  filaments  peeled  away  from  the  ring  as  individual   filaments  (Figure  4.2C),  and  this  was  only  resolved  by  reducing  the  value   of  maxInt   to   1   (Figure   4.2D),   so   that   the   myosin   pulling   forces   did   not   overpower  the  grabbing  forces.  

  This   would   then   suggest   that   a  maxInt   value   of   2   or   3   would   be   more  appropriate  for  the  original  simulations  performed  with  this  model,   where   there   were   40   Myo2   molecules   per   cluster.   These   maxInt   values   correspond   to   20   or   ~13   Myo2   molecules   per   cluster,   respectively,   walking   processively   along   a   single   actin   filament.   This   would   then   reduce   the   tension   generated   by   the   model   by   a   factor   of   ~5   or   ~3,   respectively,   leading   to   approximate   values   of   ~68   pN   or   ~110   pN,   respectively.   This   is   significantly   less   than   what   was   observed   experimentally,  suggesting  that  the  mechanism  of  tension  generation  in   this   model   is   not   sufficient   to   produce   the   experimentally   measured   values,  and/or  the  experimentally  measured  value  is  over  estimated.  It  is   also   possible   that   rings   in   spheroplasts   are   able   to   recruit   myosin   at   a   greater   density   than   rings   in   cells,   meaning   that   they   have   a   higher   tension.   Better   understanding   of   this   discrepancy   will   require   further   experimental   work,   including   more   measurements   of   AMR   tension   in   fission  yeast,  and  further  mathematical  modelling.  

  We  also  argue  that  the  myosin  grabbing  forces  were  implemented   in  an  unrealistic  manner.  Unlike  the  pulling  forces,  which  were  reduced   when  a  myosin  cluster/node  interacts  with  multiple  actin  filaments,  the   grabbing   forces   are   not   reduced,   meaning   that   a   single   node   or   cluster   can   hypothetically   exert   a   finite   force   on   an   infinite   number   of   actin   filaments  [35].  We  first  attempted  to  correct  this  by  limiting  the  number   of  interactions  with  actin  filaments  that  are  available  for  each  node.  We   set  this  to  2,  meaning  that  each  node  can  only  exert  pulling  and  grabbing   forces   on   2   actin   filaments   (Figure   4.3A).   However,   this   prevented   us   from  using  the  3D  cylindrical  geometry  for  our  simulations,  as  the  pulling   forces   would   now   overpower   the   grabbing   forces   and   lead   to   actin   filaments   being   pulled   away   from   the   membrane.   This   was   a   problem   that  was  present  in  subsequent  iterations  of  our  model,  and  which  we  did   not  find  a  way  to  solve.  

  One   possible,   and   somewhat   realistic,   solution   to   this   problem   would   be   to   increase   the   number   of   entities   that   are   able   to   grab   actin   filaments,   but   not   exert   pulling   forces   on   them.   For   example,   the   node  

protein  Rng2  contains  an  actin  binding  calponin  homology  domain  (CHD)   [170,171],   whilst   unpublished   research   from   our   lab   has   shown   that   Pxl1,  which  interacts  with  the  SH3  domains  of  Cdc15  and  Imp2  [75],  is   also   able   to   bind   to   tropomyosin,   which   provides   another   mechanism   whereby   actin   filaments   can   be   linked   to   the   membrane.   Therefore,   it   seems  likely  that  there  are  proteins  other  than  Myo2  that  are  able  to  bind   to  actin  filaments  and  link  them  to  the  membrane.  

  A   recent   model   of   the   fission   yeast   AMR   in   a   3D   cylindrical   geometry  also  attempted  to  model  individual  heads  of  Myo2  that  interact   with  individual  actin  filaments,  in  a  similar  way  to  the  later  iteration  of   our   model,   except   with   even   greater   detail   in   the   description   of   the   myosin   molecules   [133].   The   authors   simulated   a   number   of   potential   AMR  structures,  but  in  all  of  these  the  only  linker  between  actin  filaments   and  the  membrane  was  the  simulated  Myo2  molecules,  and  despite  this   they  did  not  see  the  peeling  away  of  actin  filaments  that  we  would  expect   to  see  from  our  simulations.  In  these  simulations,  they  use  an  actin  bead   friction   coefficient   30   times   greater   than   that   used   in   previous   simulations,   including   our   own   [133].   This   value   is   used   without   justification,  and  additionally  the  beads  in  actin  filaments  are  now  placed   only   5.5   nm   apart,   which   gives   a   friction/length   around   500   times   greater  than  that  used  in  previous  simulations  (~1000  pN  s/μm2  _{vs.  2  pN}   s/μm2_{).   This   will   significantly   slow   down   the   speed   with   which   actin}   filaments  move,  since  the  pulling  force  exerted  by  the  myosin  molecules   was  the  same  as  in  our  simulations,  which  could  mean  that  filaments  get   disassembled  by  cofilin  before  they  are  able  to  be  pulled  out  of  the  ring.     When  we  attempted  to  implement  our  own  ring  model  where  we   explicitly  include  individual  pairs  of  Myo2  heads,  we  found  that  very  few   actin   filaments   were   captured   and   pulled   on   by   the   Myo2   heads,   even   when   we   biased   the   polymerisation   direction   of   actin   filaments   to   be   parallel  to  the  ring,  and  this  lead  to  a  very  low  ring  tension  (Figure  4.6E,   Figure  4.6F).  This  is  despite  our  observation  that  this  description  of  the   myosin   clusters/nodes   worked   quite   well   in   a   toy   model   (Figure   4.6A–   Figure   4.6D).   In   the   toy   model,   the   actin   filament   was   placed   at   an  

optimum   height   above   the   cluster,   which   may   have   helped   the