the fission yeast schizosaccharomyces pombe is a convenient model to study cell morphogenesis ( hayles and nurse , 2001 ) .
wild - type cells are simple elongated rods growing at the cell poles and dividing in the middle . yet
, previous studies have outlined an interesting interplay between shape , growth and cytoskeletal organization .
the first component is the rigid cell wall surrounding yeast cells that maintains cell shape independently of the cytoskeleton .
second , the actin cytoskeleton is essential for cell growth and cell wall remodeling ( la carbona et al , 2006 ) .
lastly , although microtubules ( mts ) are not required for growth per se , they control the location of growth sites by depositing specific marker proteins ( mata and nurse , 1997 ; brunner and nurse , 2000 ; sawin and snaith , 2004 ) .
abnormal deposition , occurring for example in mutants where mts are shorter , results in cells that are either bent or branched ( sawin and nurse , 1998 ; snaith and sawin , 2005 ) .
mts also position the nucleus ( tran et al , 2001 ; loiodice et al , 2005 ) and thus define the site of cytokinesis ( daga and chang , 2005 ; tolic - norrelykke et al , 2005 ) and the partitioning of the cell into daughter cells .
hence , by controlling cell growth and division , mts impact the evolution of shape in the cell lineage . as mts are constrained within the cell , the converse is also true with mt organization being dependent on cell shape . for the rigid s. pombe cells ,
the two processes occur on very different timescales ; with mt lifetimes being in the order of minutes , whereas cells typically double in size after 3 h. consequently , individual mts are enclosed in a boundary that is effectively constant during their lifetime .
this means that it is valid to first study how mts depend on cell shape , and to later include cell shape changes .
we use here computer simulation for the first step , calculating the dynamic spatial organization of mts within a fixed cell shape .
this approach complements other efforts where cell morphogenesis is modeled with reaction  diffusion equations ( csikasz - nagy et al , 2008 ) by focusing on the mt cytoskeleton .
interphase mts in fission yeast are typically forming 26 bundles , which are usually attached to the nucleus at their middle ( tran et al , 2001 ) ( figure 1 ) .
antiparallel mts overlap at their static minus ends , whereas the plus ends are dynamic and grow from the overlap zone toward the cell poles ( tran et al , 2001 ; hoog et al , 2007 ) . such bundles transmit forces produced at the cell poles by mt polymerization to the nucleus . to position the nucleus near the middle of the cell ( tran et al , 2001 ;
loiodice et al , 2005 ) , mts should efficiently target the cell poles , and have catastrophes ( the switch to depolymerization ) that are rare enough to enable mts to reach the cell cortex but not so rare as to induce bending of the mt around the polar cell wall ( tran et al , 2001 ) , a configuration that is not observed in wild - type cells .
the most popular explanation for the timing of catastrophes involves multiple molecular activities that are assembled at the cell poles ( mata and nurse , 1997 ) .
another possibility is that forces caused by mt polymerization feed back on mt dynamics ( dogterom et al , 2005 ) .
this possibility has so far not been confirmed mainly due to the lack of experimental tools in vivo . in this study , we have circumvented this problem using stochastic computer simulations to check whether the effect of force measured in vitro can explain the observed mt plus end dynamics in s. pombe .
this involved constructing three models of mt dynamics : model f ( force ) in which forces at mt tips regulate mt growth and catastrophe rates , model l ( length ) in which mt length affects catastrophe rate and model fl which combines both effects . to simulate the mt cytoskeleton in s. pombe
, we first idealized its 3d shape as a spherocylinder ( see figure 1b and c ) . a nucleus and mt bundles
were then added and confined within this volume . in vivo , mt bundles self - organize ( janson et al , 2007 ) , to form overlap zones of 0.840.29 m ( tran et al , 2001 ) . for the purpose of this study , it was sufficient to use fixed overlap zones and four bundles , each containing four antiparallel mts , which is representative of the average situation .
the nucleus was represented by a sphere , to which the overlap zones of the bundles were attached ( supplementary figure s4 ) .
the deformations of the nuclear membrane observed in vivo ( tran et al , 2001 ; daga et al , 2006 ) were incorporated into the model by attaching mt bundles to the nucleus using hookean springs of moderate stiffness .
the points to which the bundles were attached were also able to move on the nuclear surface .
this allowed elongating bundles to align with the cell axis , as in vivo ( supplementary figure s5 ) . in summary ,
the simulation comprised bundles of flexible mts and a connected spherical nucleus that were confined within a frictionless cortex . although mt minus ends were static , plus ends grew and shrank independently of each other , thus producing polymerization forces , fiber deformation and nuclear movements .
the physical equations describing the evolution of this system were solved numerically as explained before ( nedelec and foethke , 2007 ) .
the advantage of using s. pombe as a model organism is that numerous dynamical properties of mts have been measured by light microscopy , and this enables the models to be quantitatively compared to reality .
table i lists the 10 in vivo properties of mts that were used to evaluate computational models .
these 10 traits summarize most of the currently established knowledge of the wild - type cells that are relevant for mt organization .
note that it was necessary to adjust the definitions of these traits to what is available from the literature ( see supplementary information ) .
for example , it was important to distinguish  bundle catastrophes , ' that affect the longest mt in a half - bundle from other mt catastrophes .
other catastrophes are harder to observe because mts overlap in vivo , and were often not reported ( brunner and nurse , 2000 ; drummond and cross , 2000 ; tran et al , 2001 ) .
independently of the traits , characteristics of the cell geometry , mt bending elasticity and cytoplasmic viscosity were obtained from the literature or determined experimentally in the course of this study ( see supplementary information ) .
the models discussed here contain two parameters : the polymerization speed v0 and the catastrophe rate c0 .
they are evaluated by running thousands of simulations with random values of v0 and c0 , and then testing which trait was matched in each simulation .
this systematic numerical exploration of the parameter space is centered on v02.4 m / min , which is observed under standard laboratory conditions ( tran et al , 2001 ) , and extends to regions where traits start to fail .
model f corresponds to an in vitro experiment , which showed that a barrier could inhibit tubulin assembly ( dogterom and yurke , 1997 ) .
these results were described by vg = v0 exp(f / fs ) ( equation a ) , where fs1.7 pn specifies the sensitivity to force , and f is the projected force between the barrier and the mt tip ( figure 1c ) .
in addition , reduced assembly in vitro was shown to promote catastrophes according to b : c=1/(a+bvg ) ( janson et al , 2003 ; janson and dogterom , 2004 ) .
model f used equations ( a ) and ( b ) with the forces calculated for each mt .
the constants fs and a were measured experimentally with purified tubulin . the constant b was calculated as the solution of ( b ) with c = c0 and v = vo such that c0 and v0 correspond to the catastrophe and assembly rates of unconstrained mts .
the resultant value of b differs from that measured in vitro , because it represents the combined effects of mt - associated proteins ( maps ) present in the cell . a value for b corresponding to what was reported in vitro without mt regulators results in excessively long mts that curl around the cell pole .
we found that simulations with model f consistently matched t18 ( traits of the wild - type cell ) around the reference values of c0 and v0 ( supplementary figure s1 ) .
however , these simulations failed to match t9 and t10 in this region , which lead us to investigate these traits further .
experimentally , t9 was measured in cells that are made asymmetric by centrifugation , with the nucleus ending up nearer one cell pole .
it was observed that during nucleus re - centering , mts have longer contact time with the proximal than with the distal pole ( 9253 and 4636 s , respectively ; daga et al , 2006 ) .
model f showed opposite effects because , as shown by our simulation , mts experience stronger compression on the proximal side than on the distal side .
n / l ) , which indicates that the maximum force of an mt at equilibrium is inversely proportional to its length .
however , contrary to this formula , the simulation considers the rate of mts reaching the cell ends , and the motion of the nucleus and thus calculates the effective force rather than an upper limit .
the other unmatched trait t10 represents the centering precision of bundles when they are not attached to the nucleus ( carazo - salas and nurse , 2006 ) . to understand why t10 fails
, we can consider the simpler situation of an unattached bundle made of two antiparallel mts .
this bundle mechanically behaves as a single elastic beam of small viscous drag : the forces at both ends equalize very fast .
consequently , the plus tips of both mts experience similar forces no matter where the overlap zone is located : there are little centering cues .
the simulation shows that the situation with bundles of four mts is essentially similar : they are centered on average , but with a large variance of 3 m .
however , when a nucleus is present , additional averaging occurs on the positions of mt minus ends : the nucleus connects multiple independent bundles , and averages their fluctuations over time , because its mobility is much lower than that of individual bundles .
thus , t10 contains different information than t7 and t8 . in conclusion , although model f matched most measurements listed in table i , its failure to match t9 and t10 prompted us to investigate how it could be adjusted .
in the situation considered for t9 , mts are shorter on the proximal side than on the distal side , and therefore the observed asymmetry in contact times may indicate that the length of an mt influences its stability , which was not the case in model f. to test this possibility , we simulated a catastrophe rate that depends on mt length l according to c = hl/(a+bvg ) ( equation b ) .
the constant h was set to 0.2/m , so that c0 would determine the catastrophe rate at l=5 m ( the length at which mts typically undergo catastrophe in average cells ) .
this makes the resulting model fl directly comparable to model f , except that longer mts are less stable compared with short ones . with model fl , t9 was matched with , for example , contact times of 10268 s on the proximal side , and 5131 s on the distal side . the length dependence overcompensated for the effects of force in a situation where the nucleus is close to the cell pole .
in fact , model fl matched the 10 traits robustly with respect to ( v0 , c0 ) around their expected in vivo values ( figure 2a ) .
although the average mt  cortex contact time in the symmetric case was already considered in t4 , comparing the shape of their distribution would provide another test of the model . to this extent
, we measured 303 in vivo contact events ( 6636 s mean and standard deviation ) .
being approximately 20% shorter , they were comparable to what had previously been reported ( 8346 s ) ( daga et al , 2006 ) .
interestingly , model fl matched the distribution ( see figure 3a ) , showing that the contact times are explained by the memory - less ( first - order ) mt catastrophe transition , because as mts continue to grow after contact , forces building up increase the instantaneous catastrophe rate .
note that this distribution is also matched by model f ( see supplementary figure s2 ) , which was anyhow discarded for other reasons .
the simulation also predicts the location of the hidden mt catastrophes ( figure 3b ) , which are frequent but difficult to observe in vivo because these events occur on the side of a longer mt in the same bundle .
such phenomenon was observed previously in xenopus egg extracts ( dogterom et al , 1996 ) , and tischer et al have confirmed experimentally that this behavior occurs in s. pombe cells , observing that catastrophe rates increase linearly from 0.1/min at l2 m up to 0.3/min at l5 m , for mts that do not contact the cell cortex ( tischer et al , in this upload ) .
the measured magnitude of the dependence of catastrophe upon mt length is identical to what we have assumed , in the center of the region in which all traits are fulfilled ( c0=0.3/min ) .
it is tempting to speculate about the molecular mechanism that could mediate the destabilization of longer mts .
the s. cerevisiae kinesin-8 kip3p produce a length - dependent destabilizing activity in vitro , because after binding at any point along the length of the mt , it moves processively to the tips , where it has a depolymerizing activity ( gupta et al , 2006 ; varga et al , 2006 ) .
it is possible that the two homologs of kip3p in s. pombe , klp5p and klp6p ( west et al , 2001 ) have a similar activity .
consistent with this view , the double deletion of klp5 and klp6 produced elongated mts that curl at cell poles ( west et al , 2001 ) , and reduces the length dependence of catastrophe ( tischer et al ) .
simulations show that length - dependent catastrophe rates offer several potential advantages to the cell .
first , nucleus repositioning is faster by a factor 2 in model fl compared with model f ( data not shown ) , because proximal mts push for a longer time than distal mts ( t9 ) .
second , bundles that are observed to detach from the nucleus ( tran et al , 2001 ; carazo - salas and nurse , 2006 ) would keep their overlap regions better centered ( their variance was 2 m instead of 3 m for model f ) , which may allow them to reattach more rapidly ( we did not simulate this hypothesis ) . finally , a model l having the length dependence present in model fl , but without any force dependence also fails to fulfill all traits ( supplementary figure s1 ) , showing that the length dependence is not sufficient to adjust mts within the cell .
hence , in model fl , the established response to force together with the length - dependent mt - destabilizing activity of maps , leads to the accurate description of the dynamics of mts in wild - type s. pombe cells .
remarkably , in both model f and fl , it was not necessary to assume that the cell poles had any localized activity associated with them that would affect mts .
this shows that , as anticipated ( dogterom et al , 2005 ) , forces in principle are sufficient to account for the location of mt catastrophes at the cell poles . in future work
, it will be necessary to explicitly distinguish maps to be able to recapitulate their mutant phenotypes . to do so
will require knowing how the dynamic equilibrium distribution of maps near mt tips is affected by force .
however , having no free parameter , our current model is already predictive in calculating how cell morphology affects a wild - type mt cytoskeleton , and this sheds light on some mutant phenotypes . in the current model , the catastrophe rate c0 represents the combined action of all maps on mts .
this in particular includes the potential effect of tea1p , tip1p or tea2p , which are located on growing mt plus ends .
these proteins are later deposited at the cell ends , but our model does not include any influence of the deposited proteins on mts in return . in the case of tea1p ,
however , such an influence has been suggested ( brunner et al , 2000 ) , because genetic deletion affects mts .
the observed defect is that mts exhibit curling at the cell poles ( see figure 3c ) . in the simulation
, we noticed that cell diameter ( but not cell length ) affected mt curling ( see figure 3d ) .
this prompted us to measure mt curling in vivo , together with the dimensions of cells .
there also , we found a clear correlation between cell diameters and the extent of mt curling ( see figure 3e ) , and no correlation between cell lengths and curling .
thus , the curling phenotype can be explained by the fact that tea1 cells were , on average , wider than wild - type cells ( figure 3f ) .
furthermore , thin tea1 cells exhibited a comparable level of curling to wild - type cells of similar diameter , which confirmed that the mt phenotype is largely the consequence of the increased diameter , and that unlike previously speculated , tea1p may not directly influence mt catastrophes .
the variability around the average behavior is smaller in the simulation than in reality , most likely because irregularities in cell shape together with measurement errors have not been modeled .
the slopes of the best linear fit were comparable in vivo and in the computational model ( figure 3d and e ) . in summary
, it seems that unbranched tea1 cells have wild - type mts in a body that is wider than wild type
this is beyond the scope of the current study however , because it requires redefining many of the traits , which are only meaningful for cells having two ends .
the simulation available on www.cytosim.org can be extended to further investigate mt organization in s. pombe .
a successful account of interphase mt dynamics , as initiated here , is a necessary step to understand the determination of cell shape in this simple organism .
this file contains 
 a. a list of supplementary movies and files ;  
 b. a description of the simulation methods ;  
 c. a detailed description of the traits listed in table 1 ;  
 d. a description of the parameters of the simulation , with details on how their values were set from the previous literature . 

e. a description of the methods used to generate figure 3d and e. frequently asked questions   supplementary figure s2
 the cytoskeleton is essential for the maintenance of cell morphology in eukaryotes . in fission yeast , for example , polarized growth sites are organized by actin , whereas microtubules ( mts ) acting upstream control where growth occurs . 
 growth is limited to the cell poles when mts undergo catastrophes there and not elsewhere on the cortex . here , we report that the modulation of mt dynamics by forces as observed in vitro can quantitatively explain the localization of mt catastrophes in schizosaccharomyces pombe 
 . however , we found that it is necessary to add length - dependent catastrophe rates to make the model fully consistent with other previously measured traits of mts . 
 we explain the measured statistical distribution of mt  
 cortex contact times and re - examine the curling behavior of mts in unbranched straight tea1 cells . 
 importantly , the model demonstrates that mts together with associated proteins such as depolymerizing kinesins are , in principle , sufficient to mark the cell poles . 

Supplementary Material