Intergenerational continuity of cell shape dynamics in Caulobacter crescentus
Charles S. Wright, Shiladitya Banerjee, Srividya Iyer-Biswas, Sean Crosson, Aaron R. Dinner, Norbert F. Scherer
IIntergenerational continuity of cell shape dynamics in
Caulobacter crescentus
Charles S. Wright,
1, 2, † Shiladitya Banerjee, † Srividya Iyer-Biswas,
1, 2
Sean Crosson,
2, 3
Aaron R. Dinner,
1, 2, 4, ∗ and Norbert F. Scherer
1, 2, 4, ∗ James Franck Institute, The University of Chicago, Chicago IL 60637 Institute for Biophysical Dynamics, The University of Chicago, Chicago IL 60637 Department of Biochemistry and Molecular Biology, The University of Chicago, Chicago IL 60637 Department of Chemistry, The University of Chicago, Chicago IL 60637
We investigate the intergenerational shape dynamics of single
Caulobacter crescentus cells usinga novel combination of imaging techniques and theoretical modeling. We determine the dynam-ics of cell pole-to-pole lengths, cross-sectional widths, and medial curvatures from high accuracymeasurements of cell contours. Moreover, these shape parameters are determined for over 250 cellsacross approximately 10000 total generations, which affords high statistical precision. Our data andmodel show that constriction is initiated early in the cell cycle and that its dynamics are controlledby the time scale of exponential longitudinal growth. Based on our extensive and detailed growthand contour data, we develop a minimal mechanical model that quantitatively accounts for the cellshape dynamics and suggests that the asymmetric location of the division plane reflects the distinctmechanical properties of the stalked and swarmer poles. Furthermore, we find that the asymmetryin the division plane location is inherited from the previous generation. We interpret these resultsin terms of the current molecular understanding of shape, growth, and division of
C. crescentus . Cell shape both reflects [1] and regulates [2] biologicalfunction. The importance of cell shape is exemplified bybacteria, which rely on specific localization of structuralproteins for spatiotemporal organization [3]. Bacteriatake forms resembling spheres, spirals, rods, and cres-cents. These shapes are defined by cell walls [4] consist-ing of networks of glycan strands cross-linked by peptidechains to form a thin peptidoglycan meshwork [5]. Super-resolution imaging is now revealing the internal positionsof associated proteins [9]. These include cytoskeletal pro-teins such as MreB, a homolog of actin [7–10], interme-diate filament-like bundles of CreS (crescentin) [11, 12],and FtsZ, a homolog of tubulin [13]. However, due tothe inherently stochastic nature of molecular processes,understanding how these proteins act collectively to ex-ert mechanical stresses and modulate the effects of turgorpressure and other environmental factors requires com-plementary methods such as high-throughput, quantita-tive optical imaging.Multigenerational imaging data for bacterial cells cannow be obtained from microfluidic devices of various de-signs [14–18]. Still, a common limitation of most devicesis that the environmental conditions change throughoutthe course of the experiment, particularly as geometricgrowth of the population results in crowding of the ex-perimental imaging spaces. We previously addressed thisissue by engineering a
C. crescentus strain in which celladhesion is switched on and off by a small molecule (andinducible promoter) [19], allowing measurements to bemade in a simple microfluidic device [19–22]. This tech-nology allows imaging >
100 generations of growth of anidentical set of 250–500 single cells distributed over ∼ C. crescentus cells grow exponen- tially in size and divide upon reaching a critical multiple( ≈ C. crescentus and the plausible role of cell wallmechanics and dynamics in these processes. Specifically,we identify natural variables for tracking cell dynamics,and develop a minimal mechanical model that shows howlongitudinal growth can arise from an isotropic pressure.We then examine the dynamics of cell constriction andunexpectedly find that it is governed by the same timeconstant as exponential growth. This important find-ing can be understood in terms of an intuitive geometricmodel that relates the constriction dynamics to the kinet-ics of the growth of septal cell wall. We further suggestthat the site of constriction can arise from differences inmaterials properties of the poles and show that it is es-tablished in the previous generation—i.e., the location ofthe site of division can be predicted before formation ofthe divisome. We relate our results to the known dynam-ics of contributing molecular factors and existing modelsfor bacterial growth and division.
RESULTS
The length is sufficient to characterize the expo-nential growth of each cell . Various techniques havebeen put forth to analyze cell morphology gathered fromsingle cell images [24]. Recent work on image analysis a r X i v : . [ q - b i o . CB ] M a r � � � ℓ � ��� ���� � θ � FIG. 1.
Determination of cell contour and definition of shape parameters. (a)
A representative phase contrast imageof one field of view. The solution flow in the microfluidic channel is from bottom to top. (b)
Zoomed image of the yellowhighlighted cell from a and its splined contour. (c) Schematic of a contour illustrating the shape parameters. The cell medialaxis is calculated from pole to pole; it defines both cell length l ( φ ) and radius of curvature R ( φ ), which lead directly to thespanning angle θ ( φ ). The cell width w ( φ, u ) is a parametric quantity, calculated as the length of the rib perpendicular to themedial axis at a specified distance from the stalked pole, u ( φ ). The location of the global minimum of the width w min ( φ )(purple line) can be used to segment the cell into stalked ( st , red) and swarmer ( sw , blue) portions. of single cells has attempted to optimize two problems:separation of distinct (but potentially overlapping) cellsand accurate determination of the edge of each cell [25].Because crowding is not an issue in our setup, we couldfocus solely on constructing an algorithm to delineateeach cell contour accurately and precisely. As shown inFig. 1a,b and described in the Methods section, we firstsegment each cell using pixel-based edge detection simi-lar to [26], then perform spline interpolation to determinethe cell contour at sub-pixel resolution. The sequence ofsuch images for each single cell constitutes a trajectory intime t that serves as the basis for quantitative analysis.Division events are then detected in an automated fash-ion using custom Python code, and used to divide timetrajectories for each cell into individual generations.All data shown here were obtained by observing 260single C. crescentus cells perfused in complex medium(peptone-yeast extract; PYE) at 31 ◦ C over the courseof 2 days (corresponding to 9672 separate generations).Under these conditions, the mean population growth rateand division time remain constant, so we treat the tra-jectories of individual generations as members of a singleensemble. In other words, we segment each cell trajec-tory by generation and take the resulting initial frame(i.e., immediately following division) as t = 0 minutes.In order to average over the ensemble, we then bin quan-titative information according to time since division, t ,normalized by the respective division time τ . The nor-malized time, φ ≡ t/τ , serves as a cell-cycle phase vari-able.For our quantitative analysis, we focus on a set of threeintuitive and independent parameters that characterizecell shape at each stage of growth: length l , width w , and radius of curvature R (Fig. 1c). They are calculateddirectly from each splined contour as follows (see alsoSupplementary Fig. 1): • We define the length, l ( φ ), as the pole-to-pole dis-tance along the contour of the cell medial axis atthe normalized time φ (Fig. 2a). • We assign a single radius of curvature, R ( φ ), toeach cell based upon the best-fit circle to the me-dial axis (Fig. 2b). Although stalked ( R st ( φ )) andswarmer ( R sw ( φ )) portions may be described bydifferent radii of curvature toward the end of thecell cycle, the average radius obtained by averagingthe contributions of each portion yields the samevalue, i.e., (cid:104) ( R st ( φ ) + R sw ( φ )) / (cid:105) (cid:39) (cid:104) R ( φ ) (cid:105) (seeSupplementary Fig. 2c). • We define the width, w ( φ, u ) as the length of theperpendicular segment spanning from one side ofthe cell contour to the other at each position u ( φ )along the medial axis, which runs from u = 0 atthe stalked pole to u = l at the swarmer pole.Furthermore, we spatially averaged the width overpositions along the medial axis, ¯ w ( φ ), to obtain acharacteristic width at each time point (Fig. 2c).The mean division time is (cid:104) τ (cid:105) = 73 ± (cid:104) ... (cid:105) indicates a population average. We find that (cid:104) l ( φ ) (cid:105) in-creases exponentially with time constant (cid:104) κ (cid:105) − = 125 ± (cid:104) ¯ w ( φ ) (cid:105) and (cid:104) R ( φ ) (cid:105) remain approximately constant for0 < φ < . . < φ < . (cid:104) R ( φ ) (cid:105) seen for φ > . θ , using the relation l = Rθ . Mechanical model for cell shape and growth .There are many details of cell growth and shape thatrequire interpretation. For example, it is not obvious a priori that growth should be almost exclusively lon-gitudinal. Therefore, we have developed a minimal me-chanical model that can explain these observations. Weparametrize the geometry of the cell wall by a collectionof shape variables { q i ( t ) } , where q = (cid:104) R (cid:105) , q = (cid:104) ¯ w (cid:105) , and q = (cid:104) θ (cid:105) are the parameters introduced above (Fig. 1c).As the cell grows in overall size, we postulate that the rateof growth in the shape parameter q i ( t ) is proportional tothe net decrease in cell wall energy, E ( { q i ( t ) } ), per unitchange in q i ( t ) [27, 28]. Assuming linear response, theconfigurational rate of strain, q i ( t ) dq i dt , is proportional tothe corresponding driving force F i = − ∂E/∂q i , in anal-ogy with the constitutive law of Newtonian flow [29]:1 q i dq i dt = Φ i F i , (1)where the constant Φ i describes the rate of irreversibleflow corresponding to the variable q i ( t ). According toequation (1), exponential growth occurs if F i is constant,whereas q i ( t ) reaches a steady-state value if F i ( q i ) = 0along with the condition ∂F i /∂q i <
0. It thus remains tospecify the form of E .For a C. crescentus cell of total volume V and surfacearea A , our model for the total energy in the cell wall isgiven by E ( R, ¯ w, θ ) = − P V + (cid:90) dA γ + E width + E cres + E div , (2)where P is a constant pressure driving cell wall expan-sion; γ is the tension on the surface of the cell wall; E width is the energy required to maintain the cell width; E cres represents the mechanical energy required to maintainthe crescent cell shape; E div is the energy driving cell wallconstriction. Traditionally P was taken to be the turgorpressure [27]; while the importance of the turgor pressurehas recently been questioned [30], an effective pressuremust still arise from the synthesis and insertion of pep-tidoglycan strands that constitute the cell wall. We notethat a purely elastic description of cell wall mechanicswould lead to a curvature-dependent surface tension [31].However, if growth is similar to plastic deformation, thetension is uniform [5]. The effective tension in our model depends on the local surface curvatures through the en-ergy terms E width and E cres , that describe harmonic wellsaround preferred values of surface curvatures.The mechanical energy for maintaining width is givenby E width = k m (cid:90) dA (cid:18) w/ − R m (cid:19) , (3)where the constant R m is the preferred radius of cur-vature, k m is the bending rigidity and dA is a differen-tial area element [33]. Contributions to k m can comefrom the peptidoglycan cell wall as well as membrane-associated cytoskeletal proteins like MreB, MreC, RodZ,etc., which are known to control cell width [7–9].In addition to maintaining a constant average width, C. crescentus cells exhibit a characteristic crescent shape,which relies on expression of the intermediate filament-like protein crescentin [11]. Although the mechanism bywhich crescentin acts is not known, various models havebeen proposed, including modulation of elongation ratesacross the cell wall [12, 34] and bundling with a preferredcurvature [35]. We assume the latter and write the energyfor maintaining the crescent shape as E cres = k c (cid:90) l c du (cid:18) c ( u ) − R c (cid:19) , (4)where u is the arc-length parameter along the crescentinbundle attached to the cell wall, c ( u ) is the local curva-ture, R c is the preferred radius of curvature, l c is the con-tour length, and k c is the linear bending rigidity. Equa-tion (S.22) accounts for the compressive stresses gener-ated by the crescentin bundle on one side of the cell wall,leading to a reduced rate of cell growth, according toequation (1). As a result, the cell wall grows differen-tially and maintains a non-zero curvature of the center-line. In the absence of crescentin ( k c = 0), our modelpredicts an exponential decay in the cell curvature thatleads to a straight morphology, consistent with previousobservations [11].Finally, one must also account for the energy drivingcell wall constriction. Constriction proceeds via inser-tion of new peptidoglycan material at the constrictionsite. This process leads to the formation of daughterpole caps [36]. We take constriction to be governed byan energy of the form E div = − λS , where S is the surfacearea of the septal cell wall, and λ is the energy per unitarea released during peptidoglycan insertion. There exists an optimal cell geometry for agiven mechanical energy . To apply the model in-troduced above (equations (1) and (2)) to interpretingthe data in Fig. 2, we assume a minimal cell geome-try given by a toroidal segment with uniform radius ofcurvature R ( φ ), uniform cross-sectional width ¯ w ( φ ) andthe spanning angle θ ( φ ). To this end, we estimate as (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:3)(cid:2)(cid:8)(cid:9)(cid:2)(cid:1)(cid:9)(cid:2)(cid:8)(cid:4)(cid:2)(cid:1)(cid:4)(cid:2)(cid:8)(cid:8)(cid:2)(cid:1) (cid:1) (cid:1) ! (cid:2) (cid:10) " (cid:1) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:1) (cid:1) (cid:1) ! (cid:2) (cid:10) " (cid:2) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:1)(cid:2)(cid:11)(cid:1)(cid:1)(cid:2)(cid:11)(cid:3)(cid:1)(cid:2)(cid:11)(cid:4)(cid:1)(cid:2)(cid:11)(cid:5)(cid:1)(cid:2)(cid:11)(cid:6) (cid:1) (cid:2) (cid:1) ! (cid:2) (cid:10) " (cid:3) _ FIG. 2.
Dynamics of cell shape parameters. (a)
Length of the cell medial axis (data shown in black and exponential fitfrom our mechanical model in red). (b)
Radius of curvature of the cell medial axis, obtained by calculating the best-fit circleto the entire cell (black points), with a time-averaged mean (cid:104) R (cid:105) = 4 . ± . µ m (0 < φ < . (cid:104) R ( φ ) (cid:105) (red solid line), whereas by accounting for constriction dynamics, the model capturesthe dip in (cid:104) R ( φ ) (cid:105) seen for 0 . < φ < . (c) Characteristic cell width, obtained by spatially averaging thewidth at each time point (black points), with a time-averaged mean (cid:104) ¯ w (cid:105) = 0 . ± . µ m (0 < φ < . (cid:104) ¯ w ( φ ) (cid:105) (red solid line), whereas cell constriction accounts for the dip in (cid:104) ¯ w ( φ ) (cid:105) seenfor φ > . ± P = 0 . γ = 50nN/ µ m, k c = 2 nN µ m , R c = 0 . µ m, k m = 40 nN µ m, R m = 0 . µ m, τ = 73 min. many mechanical parameters as we can from the liter-ature and then determine the rest by fitting our exper-imentally measured values. Turgor pressure in Gram-negative bacteria has been measured to be in the range0 . − . P = 0 . γ = 50 nN /µ m (see Supplemen-tary model section) and multiply it by the cell surfacearea A ( φ ) = π ¯ wRθ to obtain the cell wall surface energy.First, we neglect cell constriction (setting E div = 0) andassume that the crescentin structure spans the length ofthe cell wall (excluding the endcaps) [40], with a contourlength l c ( φ ) = ( R − ¯ w/ θ . The mechanical propertiesof MreB and crescentin are likely similar to those of F-actin and intermediate filaments, respectively [11, 12].However, due to a lack of direct measurements, we ob-tain the mechanical parameters k m and k c by fitting themodel to the experimental data.As desired, we find that the total energy E has a sta-ble absolute minimum at particular values of the cross-section diameter ¯ w and the centerline radius R , given bysolution of ∂E/∂ ¯ w = ∂E/∂R = 0 (see SupplementaryFig. 4). The measured values are (cid:104) ¯ w (cid:105) = 0 . ± . µ mand (cid:104) R (cid:105) = 4 . ± . µ m (0 < φ < . k m = 40 nN µ m and k c = 2 nN µ m . Whilethe fitted value for k c is numerically close to the estimatebased on the known mechanical properties of intermedi-ate filaments ( ∼ . µ m ), the value for k m is muchhigher than the bending rigidity of MreB bundles (seeSupplementary Information). This indicates that k m isonly determined in part by MreB and can have contribu-tions from the cell wall.Given stable values for ¯ w and R , growth is completely described by the dynamics of the angle variable θ ( φ ).Consequently, we write the total energy in the scalingform E ( ¯ w, R, θ ( φ )) = θ ( φ ) U ( ¯ w, R ), with U the energydensity along the longitudinal direction. The conditionfor growth then becomes U <
0, such that the energyis minimized for increasing values of θ ( φ ). From our ex-perimental data, the angle spanned by the cell centerlineincreases by an amount ∆ θ (cid:39) .
49 during the entire cy-cle. Using our parameter estimates and fitting the datain Fig. 2, we obtain a numerical value for the energy den-sity U (cid:39) − µ m. We relate the angle dynamics to thelength by dldφ = κτ l ( φ ) = κτπ ¯ w A ( φ ) (5)where κ = − Φ θ U ( U <
0) is the rate of longitudinalgrowth, which can be interpreted as resulting from re-modeling of peptidoglycan subunits with a mean current κ/π ¯ w , across the cell surface area A ( φ ). From an expo-nential fit to the data for cell length (Fig. 2a), we ob-tain Φ θ = 1 . × − (nN µ m min) − , which gives usan estimate of the friction coefficient, 1 / ( π ¯ wR Φ θ ) (cid:39) µ m − min, associated with longitudinal growth; e.g.,MreB motion that is known to correlate strongly withthe insertion of peptidoglycan strands [8]. Our resultsare consistent with previous observations of C. crescen-tus cells with arrested division but continued growth [42].
Constriction begins early and proceeds with thesame time constant as exponential growth . Hav-ing characterized the dynamics of growth, we now turn toconstriction at the division plane. As mentioned above,we obtain the experimental width at each point alongeach cell’s medial axis. The typical width profile is non-uniform along its length, exhibiting a pronounced invagi-nation near the cell center (with width w min ( φ ); Fig. 1c).This invagination, which ultimately becomes the division (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:1)(cid:2)(cid:8)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:9)(cid:1)(cid:2)(cid:6) (cid:1) (cid:1) (cid:1) (cid:2) (cid:3) ! (cid:2) (cid:10) " (cid:1) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:1)(cid:2)(cid:9)(cid:1)(cid:1)(cid:2)(cid:9)(cid:8)(cid:1)(cid:2)(cid:6)(cid:1)(cid:1)(cid:2)(cid:6)(cid:8)(cid:1)(cid:2)(cid:11)(cid:1) (cid:2) (cid:1) (cid:1) ! (cid:1) $ (cid:1) "! (cid:2) (cid:10) " (cid:2) (cid:1) (cid:1)(cid:2) (cid:1)(cid:1) (cid:1)(cid:3) (cid:1) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:1)(cid:2)(cid:12)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:8)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:9) (cid:1) (cid:1) ! (cid:13) (cid:14) (cid:15) (cid:16) (cid:17) (cid:3) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:1)(cid:2)(cid:9)(cid:1)(cid:1)(cid:2)(cid:9)(cid:8)(cid:1)(cid:2)(cid:6)(cid:1)(cid:1)(cid:2)(cid:6)(cid:8)(cid:1)(cid:2)(cid:11)(cid:1) (cid:2) (cid:1) (cid:1) ! (cid:1) $ (cid:1) "! (cid:2) (cid:10) " (cid:4) w max st w min w max sw ℓ st ℓ sw w min w max δψ S FIG. 3.
Timing and location of the division plane. (a)
Time-dependence of (cid:104) w min ( φ ) (cid:105) , with the experimental pointsin black and the model fit (equations (S.13) and (7)) in red. Fit values: w max = 0 . µ m, κ − d = 130 .
92 min, κ = 0 . µ m /min. Inset: Minimal geometry of a constricting cell, where S (blue) is the septal cell wall synthesized during constriction, δ is longitudinal width of the constriction zone, and ψ is the tangent angle at constriction. (b) Ratios of the length of the stalked(red) and swarmer (blue) portions divided by the total length ( (cid:104) l st ( φ ) /l ( φ ) (cid:105) and (cid:104) l sw ( φ ) /l ( φ ) (cid:105) , respectively), are approximatelyconstant over the cell cycle. (c) Experimental width profile of
C. crescentus cells in the initial stage of the cycle ( φ = 0)after ensemble-averaging over all data. The width (cid:104) w ( φ = 0 , u ) (cid:105) is plotted as a function of the distance from the stalked endnormalized to the length of the cell, u/l . Inset: A representative single cell contour immediately after division ( φ = 0), showingthe location of the local minimum in the width ( w min , purple line) as well as two local maxima ( w st max , red line and w sw max , blueline). These two local maxima in the width define the pole regions (shaded in red and blue, respectively). (d) Model widthprofiles of the cell showing symmetric and asymmetric location of the invagination near the midplane for different values of theratio γ stp /γ swp =1 (blue, dashed), 1.05 (red, solid) and 1.09 (green, dotted). Parameter values are the same as in Fig. 2 with γ stp = 5 γ [27]. The shaded regions in a , b and c represent ± plane, is readily identifiable early in the cell cycle, evenbefore noticeable constriction occurs. We discuss the ki-netics of constriction in this section, and focus on its loca-tion later in the manuscript. As shown in Fig. 3a (blackpoints), (cid:104) w min ( φ ) (cid:105) progressively decreases towards zerountil pinching off at φ = 1. Due to the limited spatialresolution of our imaging (phase contrast microscopy),the pinch-off process occurring for φ > . (cid:104) w min ( φ ) (cid:105) at earlier times (i.e., φ < .
9) isprecisely determined as a function of φ .To model the dynamics of constriction, we assume as inRef. [43] (Fig. 3a, inset): (i) the shape of the zone of con-striction is given by two intersecting and partially formedhemispheres with radii w max /
2; and (ii) constriction pro-ceeds by completing the missing parts of the hemispheresuch that the newly formed cell wall surface maintainsthe curvature of the pre-formed spherical segments. Asa result, a simple geometric formula is obtained that re-lates the width of the constriction zone, w min ( φ ), to the surface area S ( φ ) of the newly formed cell wall, w min ( φ ) = w max (cid:113) − ( S ( φ ) /S max ) , (6)where S max = πw is the maximum surface areaachieved by the caps as the constriction process is com-pleted, i.e., when w min ( φ = 1) = 0. We assume that theaddition of new cell wall near the division plane initiateswith a rate, κ , and thereafter grows exponentially witha rate, κ d , according to,1 τ dSdφ = κ d S ( φ ) + κ , (7)subject to the initial condition S ( φ = 0) = 0. The firstterm on the right-hand side of equation (7) follows fromequation (1), using S ( φ ) as the shape variable, after in-corporating the constriction energy E div ( φ ). The rate ofseptal peptidoglycan synthesis, κ d , is thus directly pro-portional to the energy per unit area released during con-striction, λ . The solution, S ( φ ) = κ ( e κ d τφ − /κ d ,can then be substituted into equation (S.13) to derivethe time-dependence of w min ( φ ), whose dynamics is con-trolled by two time scales: κ − d and S max κ − .Fitting equation (S.13) with the data for (cid:104) w min ( φ ) (cid:105) , weobtain κ − d (cid:39)
131 min and S max κ − (cid:39)
118 min. The fit-ted values for the time constants controlling constrictiondynamics ( κ − d and S max κ − ) are remarkably similar tothat of exponential cell elongation ( (cid:104) κ (cid:105) − (cid:39)
125 min).This shows that septal growth proceeds at a rate com-parable to longitudinal growth. Therefore, one of themain conclusions that we draw is that cell wall constric-tion (Fig. 3a) is controlled by the same time constant asexponential longitudinal growth (Fig. 2a).Having determined the dynamics of w min ( φ ), we com-pute the average width across the entire cell ¯ w ( φ ) usingthe simplified shape of the constriction zone as shownin Fig. 3a (inset). The resultant prediction (blue solidcurve in Fig. 2c) is in excellent agreement with the ex-perimental data and captures the dip in (cid:104) ¯ w ( φ ) (cid:105) seen for φ > .
5. Constriction also leads to a drop in the aver-age radius of curvature of the centerline, as shown by theexperimental data in Fig. 2b. In the supplementary ma-terial we derive a relation between the centerline radius ofcurvature R ( φ ) and the minimum width w min ( φ ), givenby R − ( dR/dφ ) = w − ( dw min /dφ ), predicting that cellcurvature increases at the same rate as w min ( φ ) drops.Using this relation, we are able to quantitatively capturethe dip in (cid:104) R ( φ ) (cid:105) seen for φ > . Origin of the asymmetric location of the pri-mary invagination . We now consider the position ofthe division plane and its interplay with cell shape. Asshown in Fig. 3b, the distance of the width minimumfrom the stalked pole ( l st ( φ )) increases through the cellcycle at the same rate as the full length of the growing cell( l ( φ )), such that their ratio remains constant with time-averaged mean (cid:104) l st /l (cid:105) = 0 . ± .
05. The presence of theprimary invagination early in the cell cycle is reiteratedin Fig. 3c, which shows the width profile constructed byensemble-averaging over each cell at the timepoint imme-diately following division. In addition to the width mini-mum w min ( φ ), there are two characteristic maxima neareither pole, w st max ( φ ) and w sw max ( φ ), respectively (Fig. 3c,inset). As evident in Fig. 3c, the stalked pole diameter (cid:104) w st max ( φ ) (cid:105) is on average larger than its swarmer counter-part (cid:104) w sw max ( φ ) (cid:105) (also see Supplementary Fig. 2a).We show that the asymmetric location of the invagi-nation (and the asymmetric width profile) can originatefrom the distinct mechanical properties inherent to thepole caps in C. crescentus . The shapes of the cell polescan be explained by Laplace’s law that relates the pres-sure difference, P , across the cell wall to the surface ten-sions in the stalked or the swarmer pole, γ st,swp . The radiiof curvature of the poles then follow from Laplace’s law R st,swp = 2 γ st,swp P , (8) where the superscript ( st, sw ) denotes the stalked orthe swarmer pole. Thus a larger radius of curvaturein the poles has to be compensated by a higher sur-face tension to maintain a constant pressure difference P . Assuming that the poles form hemispheres, we have R st,swp = w st,sw max /
2. Our data indicate that the early timeratio for w st max ( φ ) /w sw max ( φ ) ( φ < .
1) shows a strong pos-itive correlation with the ratio l st ( φ ) /l sw ( φ ), with an av-erage value (cid:104) w st max /w sw max (cid:105) (cid:39) .
04 (see Supplementary Fig3a). Laplace’s law then requires that the stalked pole bemechanically stiffer than the swarmer pole; γ stp > γ swp .This observation suggests that the asymmetry in thelengths of the stalked and swarmer parts of the cell de-pends upon different mechanical properties of the respec-tive poles.To quantitatively support this claim, we investigate aneffective contour model for the cell shape. To this end,we assume that the fluctuations in cell shape relax morerapidly than the time scale of growth. This separationof timescales allows us to derive the equation govern-ing the cell contour by minimizing the total mechanicalenergy (equation (2)). From the solution we computethe resultant width profile for the entire cell (see Supple-mentary model section). As shown in Fig. 3d, the modelwith asymmetric surface tensions of the poles causes theprimary invagination to occur away from the cell mid-plane. The spatial location of the invagination relativeto the cell length depends linearly on the ratio γ stp /γ swp .Symmetry is restored for γ stp /γ swp = 1, as shown in Fig.3d (blue dashed curve). We note that a gradient in γ along the cell body would imply differences in longitu-dinal growth rates between the stalked and the swarmerportions of the cell (Eq. (1)). Our data exclude this pos-sibility since both l st ( φ ) and l sw ( φ ) grow at the samerate κ , as evidenced by the constancy of their ratio (Fig.3b and Supplementary Fig. 2d). Because C. crescentus does not exhibit polar growth, the polar stiffness model is consistent with the observed uniformity in longitudi-nal growth rate. In addition, the non-uniformity in cellwidth comes from the differences in mechanical responsein the cell wall due to preferential attachment of cres-centin along the concave sidewall. For a creS mutantcell (where k c = 0), our model predicts a uniform widthprofile before the onset of constriction. Cell shape evolution during wall constriction .The experimental width profiles show that the growingand constricting cells typically develop a second mini-mum in width (Fig. 4a,b). These secondary invaginationsare observed in both the stalked and swarmer portions ofsingle cells in the predivisional stage ( φ > . l st min ( φ ), defined as thedistance from the stalked pole to the secondary minimum � � � μ �� � � � � ���������������� � ( μ � ) � ( μ � ) � � � � � � ���������������� � ( μ � ) � ( μ � ) � FIG. 4.
Comparison of experimental and model cell contours and width profiles. (a)
Splined contours of a growingand constricting cell at different values of normalized time φ = 0 . .
33 (orange), 0 .
67 (green) and 1 . (b) Experimental width profiles plotted against absolute distance from the stalked pole, corresponding to contours in a . (c) Contours computed from the cell shape model at different values of w min and l corresponding to the time points in a . (d) Model width profiles corresponding to the contours in c . in the stalked part (see Fig. 5c, inset). We find that theratio (cid:104) l st min ( φ ) /l st ( φ ) (cid:105) has a mean value of (cid:39) (cid:104) l st ( φ ) /l ( φ ) (cid:105) . In fact the ky-mograph of width profiles (shown over 2 generations fora representative single cell) in Fig. 5c demonstrates thatthe predivisional secondary invaginations are inheritedas primary invaginations after division. This mechanismprovides continuity and inheritance of the invaginationsacross generations and is an intrinsic element of the mech-anism for cell division in C. crescentus .To quantitatively explain the experimental width pro-files during constriction, we use our mechanical modelto determine the instantaneous cell shape by minimiz-ing the total energy (equation (2)) at the specified timepoints (see Supplementary model section). To take con-striction into account, we impose the constraint that w ( l st , φ ) = w min ( φ ), where w min ( φ ) is determined byequations (S.13) and (7). In addition, we assume non-uniform materials properties in the cell wall by takingthe tension in the cell poles ( γ st,sw p ) and the septal re-gion to be higher than the rest of the cell. As constric-tion proceeds and w min ( φ ) decreases, we compute theshape of the cell contours (Fig. 4c) and the correspond-ing width profiles (Fig. 4d). The computed width profilesfaithfully reproduce the secondary invaginations, whichbecome more pronounced as the daughter pole caps be-come prominent. An example of the experimental widthprofiles is shown in Fig. 4b at evenly-spaced intervals in time for a single generation, and the corresponding modelwidth profiles are shown in Fig. 4d.We note that the experimental cell contours in the pre-divisional stage ( φ > .
9) bend away from the initialmidline axis and develop an alternate growth direction(Fig. 4a, blue contour). These bend deformations are in-duced by the microfluidic flow about the pinch-off plane;the cells become increasingly “floppy” as the constrictionproceeds.
DISCUSSION AND CONCLUSIONS
The consistent propagation of a specific shape throughthe processes of growth and division relies upon an in-tricate interplay between the controlled spatiotemporalexpression and localization of proteins, and cytoskeletalstructural elements. The high statistical precision of ourmeasurements allows us to gain new insights into cellmorphology. From precise determination of cell contoursover time, we observe that a typical cell width profileis non-uniform at all times with a pronounced primaryinvagination appearing during the earliest stages of thecell cycle. During cell constriction, the decrease in theminimum width is governed by the same time constantas exponential axial growth (Fig. 3a). Furthermore, thelocation of the primary invagination divides the cell con-tour into its stalked and swarmer compartments, suchthat the ratio of the length of the stalked part l st ( φ ) tothe total pole-to-pole length l ( φ ) remains constant dur- (cid:1) (cid:1)(cid:2)(cid:3)(cid:1)(cid:2) !(cid:1) (cid:1)(cid:4)(cid:5)(cid:1)(cid:2) (cid:1) (cid:1)(cid:2)(cid:3)(cid:1)(cid:3) !(cid:1) (cid:1)(cid:4)(cid:5)(cid:1)(cid:3) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:4) (cid:1)(cid:2)(cid:5) (cid:1)(cid:2)(cid:6) (cid:7)(cid:2)(cid:1)(cid:1)(cid:2)(cid:8)(cid:3)(cid:1)(cid:2)(cid:8)(cid:4)(cid:1)(cid:2)(cid:8)(cid:5)(cid:1)(cid:2)(cid:8)(cid:6)(cid:7)(cid:2)(cid:1)(cid:1)(cid:7)(cid:2)(cid:1)(cid:3) (cid:1) (cid:1) ! (cid:9) (cid:10) (cid:11) (cid:12) (cid:13) (cid:1) (cid:1) (cid:1)(cid:2)(cid:3)(cid:1)(cid:3) ! (cid:1) (cid:1)(cid:3) (cid:1) (cid:1)(cid:3) ! (cid:1) "(cid:1)(cid:2)(cid:3) "(cid:1)(cid:2)(cid:7) (cid:1)(cid:2)(cid:1) (cid:1)(cid:2)(cid:7) (cid:1)(cid:2)(cid:3)(cid:1)(cid:2)(cid:14)(cid:1)(cid:2)(cid:4)(cid:1)(cid:2)(cid:15)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:16) (cid:1) (cid:1) ! (cid:9) (cid:10) (cid:11) (cid:12) (cid:13) (cid:2) (cid:7) (cid:3) (cid:14) (cid:4) (cid:15) (cid:5)"(cid:14)(cid:1)(cid:1)(cid:14)(cid:1)(cid:5)(cid:1)(cid:8)(cid:1)(cid:7)(cid:3)(cid:1) (cid:1) (cid:2) (cid:17) (cid:12) (cid:18) $ (cid:3) (cid:1)(cid:2)(cid:15)(cid:1)(cid:2)(cid:5)(cid:1)(cid:2)(cid:16)(cid:1)(cid:2)(cid:6)(cid:1)(cid:2)(cid:8) (cid:1) (cid:400) min st l st (cid:400) (cid:1134)(cid:3) = –0.05 (cid:1134)(cid:3) = +0.05 (cid:400) st (cid:400) st FIG. 5.
Location of division plane is set in the previous generation. (a)
Value of the minimum width normalized bythe respective maximum width for the stalked ( (cid:104) w st min ( φ ) /w st max ( φ ) (cid:105) ) and the swarmer ( (cid:104) w sw min ( φ ) /w sw max ( φ ) (cid:105) ) parts. (b) Ratio oflength from stalked pole to secondary minimum normalized by length from stalked pole to primary minimum ( (cid:104) l st min ( φ ) /l st ( φ ) (cid:105) ,green) and ratio of length from stalked pole to primary minimum normalized by total length ( (cid:104) l st ( φ ) /l ( φ ) (cid:105) , red). The formerratio remains constant at (cid:39) φ from the first generation as negative (e.g., φ = − . φ < − .
25 due to increased errors in identification. (c)
Kymographof width profiles for a typical cell over two generations. The time evolution of the widths (color scale) illustrates continuity ofthe location of the minima across generations. That is, the location of the secondary minimum just before division ( l st min , whiteline) becomes the primary minimum ( l st , black line) just after division (horizontal dashed line). The schematic at right showstwo measured contours that correspond to time points immediately before ( φ = − .
05) and after ( φ = +0 .
05) the divisionevent shown in the kymograph. ing the cycle with a mean value (cid:104) l st ( φ ) /l ( φ ) (cid:105) (cid:39) .
55 (Fig.3b). These observations and our mechanical model leadto two important conclusions: first, the dynamics of cellwall constriction and septal growth occur concomitantly ,and second, the asymmetric location of the primary in-vagination can be explained by the differences in mechan-ical properties in the stalked and swarmer poles . A corol-lary of the first conclusion is that the size ratio thresholdat division occurs naturally without requiring a complextiming mechanism [19].In addition to the primary septal invagination, thecell contours exhibit a pronounced secondary invagina-tion during the predivisional stages (Fig. 4). Remark-ably, the secondary invaginations develop at a preciselocation relative to the total length of the stalked com-partments, (cid:104) l st min ( φ ) /l st ( φ ) (cid:105) (cid:39) .
55 (Fig. 5b). The datathus allow a third conclusion: these secondary invagina-tions are inherited as primary invaginations in each ofthe daughter cells, directing the formation of the divisionplane in the next generation . Thus, through consistentand controlled nucleation of invaginations across genera-tions,
C. crescentus cells maintain a constant ratio of the sizes of stalked and swarmer daughter cells.Our experimental observations and the parameters inthe cell shape model can be related to the current molec-ular understanding for Gram-negative bacteria, in par-ticular
C. crescentus . Before the onset of noticeableconstriction, cell shape is dictated by the mechanicalproperties of the peptidoglycan cell wall in addition tovarious shape-controlling proteins such as MreB, MreC,RodZ and CreS. Single molecule tracking studies haverevealed that MreB forms short filamentous bundles an-chored to the inner surface of the cell wall and movescircumferentially at a rate much faster than the rate ofcell growth [8, 44].
In vitro experiments show that MreBfilaments can induce indentation of lipid membranes, sug-gesting that they may have a preferred radius of curva-ture [46]. Thus on time scales comparable to cell growth, E width is determined in part by the energy cost of ad-hering MreB bundles to the cell wall (see Supplementarymodel section).Bacterial cell division is driven by a large complex ofproteins, commonly known as divisomes that assembleinto the Z-ring structure near the longitudinal mid-planeof the cell [13]. The Z-ring contains FtsZ protofilamentsthat are assembled in a patchy band-like structure [47].FtsZ protofilaments are anchored to the cell membranevia FtsA and ZipA, and play a crucial role in drivingcell wall constriction [48]. During constriction, the divi-some proteins also control peptidoglycan synthesis anddirect the formation of new cell wall via the activity ofpenicillin-binding proteins (PBPs) [49, 50]. Thus the di-visome plays a two-fold role by concomitantly guidingcell wall constriction and growth of the septal peptido-glycan layer. According to our model the constriction ofthe cell wall is driven by the synthesis of septal cell wallat a rate κ d ( ∼ (cid:104) κ (cid:105) ), which can be directly related tothe activity of PBPs triggered by the divisome assembly.Furthermore, in our model it is sufficient that the divi-some guide the curvature of cell wall growth in the septalregion (see Fig. 3a, inset).While the mechanism behind the precise asymmetriclocation of the division plane in C. crescentus cells is notwell understood, it is likely that the ATPase MipZ helpsdivision site placement by exhibiting an asymmetric con-centration gradient during the predivisional stage [51].MipZ activity inhibits FtsZ assembly; as a result of polarlocalization of MipZ, Z-ring assembly is promoted nearthe mid-cell [52]. Our cell shape model suggests that theearly time asymmetric location of the primary invagina-tion, which develops into the division plane, is controlledby the differences in surface tensions maintained in thepoles. The presence of this invagination at φ = 0, asinherited from the secondary invaginations in the pre-vious generation, aids in Z-ring assembly at the site ofthe invagination. The curvature-sensing capability of theZ-ring may be enabled by the minimization of the FtsZpolymer conformational energy that is determined by thedifference between cell surface curvature and FtsZ spon-taneous curvature [13, 53].A higher tension in the stalked pole can be inducedby asymmetric localization of polar proteins, such asPopZ, early in the cell cycle. Experiments have shownthat PopZ localizes to the stalked pole during the initialphase of the cell cycle and increasingly accumulates at theswarmer pole as the cell cycle proceeds [54]. Consistentwith this observation, our data show that the correlationbetween the pole sizes (determined by the ratio of sur-face tension to pressure) and the stalked and swarmercompartment lengths tend to disappear later in the cycle(Supplementary Fig. 3), as cell constriction proceeds. Arecent experimental study also demonstrates that molec-ular perturbation of Clp proteases can destroy the asym-metry of cell division in C. crescentus [55], suggesting theinterplay of subcellular protease activity with the physi-cal properties of the cell wall.Earlier theoretical models have predicted that a smallamount of pinch-off force from the Z-ring ( ∼ theearly time asymmetric invagination in the cell wall canset the direction for the insertion of new peptidoglycanstrands . Constriction results from exponential growth ofsurface area in the septum (at the same rate as longi-tudinal extension). The instantaneous cell shape is de-termined by minimizing the energy functional at givenvalues of the cell size parameters.Finally, from our estimate of the cell wall energy den-sity U ( (cid:39) − µ m), we predict that a net amount∆ θ | U | (cid:39) . µ m of mechanical energy is used by thepeptidoglycan network for cell wall growth. For a C. cres-centus cell of surface area 12 . − µ m , layered with gly-can strands of length ∼ ∼ peptidoglycan subunits. Thus on aver-age, each peptidoglycan subunit can consume mechanicalenergy of ∼ × − nN µ m, or ∼ k B T at a temper-ature T = 31 ◦ C. Cell wall remodeling and insertion ofnew peptidoglycan material can likely create defects inthe peptidoglycan network [56]. One thus expects cellu-lar materials properties to change over time, as a result ofthese molecular scale fluctuations. Although we neglectsuch variations in our mean field model, it nonethelessquantitatively captures the average trends in cell shapefeatures. In future work we plan to more closely connectthe energy terms of the continuum model with moleculardetails.
METHODS
Acquisition of experimental data . Data were ac-quired as in Ref. [19]. Briefly, the inducibly-sticky
Caulobacter crescentus strain FC1428 was introducedinto a microfluidic device and cells were incubated forone hour in the presence of the vanillate inducer. Thedevice was placed inside a homemade acrylic microscopeenclosure (39 (cid:48)(cid:48) × (cid:48)(cid:48) × (cid:48)(cid:48) ) equilibrated to 31 ◦ C (tem-perature controller: CSC32J, Omega and heater fan:HGL419, Omega). At the start of the experiment, com-plex medium (peptone-yeast extract; PYE) was infusedthrough the channel at a constant flow rate of 7 µ L/min(PHD2000, Harvard Apparatus), which flushed out non-adherent cells. A microscope (Nikon Ti Eclipse with the“perfect focus” system) and robotic XY stage (Prior Sci-entific ProScan III) under computerized control (Lab-View 8.6, National Instrument) were used to acquirephase-contrast images at a magnification of 250X (EM-CCD: Andor iXon+ DU888 1k ×
1k pixels, objective:Nikon Plan Fluor 100X oil objective plus 2.5X expander,lamp: Nikon C-HFGI) and a frame rate of 1 frame/minfor 15 unique fields of view over 48 hours. In the presentstudy we use a dataset consisting of 260 cells, correspond-ing to 9672 generations (division events).0
Analysis of single cell shape . The acquired phase-contrast images were analyzed using a novel routine wedeveloped (written in Python). Each image was pro-cessed with a pixel-based edge detection algorithm thatapplied a local smoothing filter, followed by a bottom-hatoperation. The boundary of each cell was identified bythresholding the filtered image. A smoothing B-splinewas interpolated through the boundary pixels to con-struct each cell contour. Each identified cell was thentracked over time to build a full time series. We chose toinclude only cells that divided for more than 10 genera-tions in the analysis. A minimal amount of filtering wasapplied to each growth curve to remove spurious points(e.g., resulting from cells coming together and touching,or cells twisting out of plane). The timing of every divi-sion was verified by visual inspection of the correspondingphase contrast images, so that the error in this quantityis approximately set by the image acquisition rate of 1frame/min.
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ACKNOWLEDGMENTS
We gratefully acknowledge funding from NSF Physicsof Living Systems (NSF PHY-1305542), NSF Materi-als Research Science and Engineering Center (MRSEC)at the University of Chicago (NSF DMR-1420709), theW. M. Keck Foundation and the Graduate Program inBiophysical Sciences at the University of Chicago (T32EB009412/EB/NIBIB NIH HHS/United States). N.F.S.also thanks the Office of Naval Research (ONR) for aNational Security Science and Engineering Faculty Fel-lowship (NSSEFF).
AUTHOR CONTRIBUTIONS
C.S.W., S.I.B., A.R.D., and N.F.S. designed the ex-periments; S.B., A.R.D., and N.F.S. designed the model;C.S.W. and S.I.B. performed the experiments and ob-served the phenomena reported; C.S.W. designed and im-plemented custom software to automate cell shape imageanalysis; C.S.W. and S.B. analyzed the data; S.C. con-tributed reagents and materials; C.S.W., S.B., A.R.D.,and N.F.S. wrote the manuscript; all authors discussedthe results and commented on the manuscript.
COMPETING FINANCIAL INTERESTS
The authors declare no competing financial interests.2
Supplemental Material
EXPERIMENTAL CELL SHAPE PARAMETERS
A number of quantities can be immediately calculated from each splined cell contour (Supplementary Fig. 1a),including the cross-sectional area. The cell medial axis was determined by calculating the Voronoi diagram of the cellcontour [S1], pruning the branches (Supplementary Fig. 1b), and extending this skeleton to the edges of the cell contourin a manner that preserves average curvature of the medial axis (Supplementary Fig. 1c). The intersections betweenthe cell medial axis and contour represent the stalked and swarmer poles, respectively. The cell length was calculatedby evaluating the distance along the medial axis between either pole. The cell widths were determined by creatingribs perpendicular to the medial axis along its length and determining the distances between their intersections withopposite sides of the cell contour (Supplementary Fig. 1d). To calculate time-averaged quantities, we normalizedtrajectories for each generation by the respective division time τ (thus converting each variable to a function of φ ≡ t/τ ). We then split these data into 73 bins, as (cid:104) τ (cid:105) = 73 ± (cid:43) m)4041424344 y ( (cid:43) m ) (cid:43) m)4041424344 y ( (cid:43) m ) (cid:43) m)4041424344 y ( (cid:43) m ) (cid:43) m)4041424344 y ( (cid:43) m ) ab dc Supplementary Figure Procedure for calculating shape parameters . (a) B-spline cell contour extracted fromsegmented phase contrast image. (b)
The interior part of the medial cell axis was obtained by taking the Voronoi diagram ofthe cell contour, which is equivalent to the locus of inscribed circles. (c)
The medial axis located on each side of the widthminimum (excluding the constriction zone) was extended to either pole such that average radius of curvature remained constant. (d)
The cell widths are the ribs perpendicular to the medial axis.
We can define five specific values of the width according to the local minima and maxima in the width profile ( w min , w st max , w sw max , w st min , w sw min ), not all of which may be present in any given cell. Where we identify a primary minimum3in the width profile, w min , we can also determine two local maxima, w st max and w sw max , corresponding respectively tothe stalked and swarmer portions (Supplementary Fig. 2a). These values are approximately constant throughoutthe cell cycle, with (cid:104) w st max ( φ ) (cid:105) > (cid:104) w sw max ( φ ) (cid:105) . However, at later times ( φ > .
6) the value of w sw max ( φ ) increasesuntil (cid:104) w st max ( φ ) (cid:105) ≈ (cid:104) w sw max ( φ ) (cid:105) . In some cases, additional secondary local minima are observed, w stmin and w swmin ,corresponding respectively to the stalked and swarmer portions (Supplementary Fig. 2b). Although we note the valueof these quantities for early times here (where they are approximately equal to their respective local maxima), theseminima can only be determined with certainty at later times ( φ > . mean ± S.D. φ ¯ w ± µ m [0 , . w st max ± µ m [0 , . w st min ± µ m [0 , . w sw max ± µ m [0 , . w sw min ± µ m [0 , . R ± µ m [0 , . R st ± µ m [0 , . R sw ± µ m [0 , . l st /l ± , l sw /l ± , Supplementary Table Average values of the cell shape parameters that are assumed constant over thespecified time interval of φ . The time-average value of the width (cid:104) ¯ w (cid:105) is smaller than both (cid:104) w st min (cid:105) and (cid:104) w sw min (cid:105) because itaccounts for the width at the primary minimum w min . Note the close correspondence between either width maximum and itscorresponding minimum for φ < .
6, as well as w st max and l st min . Both width and radius of curvature for the swarmer portion aresmaller than their respective values for the stalked portion. We split each cell into stalked and swarmer portions according to the location of the primary minimum in the widthprofile. The radius of curvature ( R ) was calculated as the radius of the best-fit circle to the cell medial axis. We testedtwo methods of determining the radius of curvature: (I) fitting the whole cell to a circle or (II) fitting the stalked andswarmer portions separately. At early times in the cell cycle, (II) gives poorer results because of fewer data points.At later times in the cell cycle, (I) gives poorer results because the stalked and swarmer portions can indeed havediffering radii of curvature, which we attribute to the alignment of the swarmer portion of the cell with the directionof fluid flow after the division plane has narrowed enough that it becomes mechanically decoupled from the stalkedportion, i.e., like a flexible hinge. However, in the mean the results of (II) are equal to the value calculated by (I)for earlier times. Therefore, we use only data from method (I) but exclude later time points (Supplementary Fig.2c). Because of the variation in the values of the calculated radius of curvature was large, such that we cannot definea reasonable arithmetic mean without arbitrarily filtering the dataset, we first averaged the corresponding unsignedcurvature (equivalent to (cid:104) R − (cid:105) ) and then converted to radius of curvature (i.e., what we report as (cid:104) R (cid:105) is actually theharmonic mean, calculated as (cid:104) R − (cid:105) − ).The length was also split according to the locations of the local minima, into l st (distance along cell medial axisfrom stalked pole to w min ), l sw (distance along cell medial axis from swarmer pole to w min ), l st min (distance along cellmedial axis from stalked pole to w st min ), and l sw min (distance along cell medial axis from swarmer pole to w sw min ). Thevalue of l st and l st min are compared in Supplementary Fig. 2d. Note that the length of the stalked portion l st growsexponentially, with the same time constant as the length l (as does l sw , although it is not shown here for clarity),which is a necessary condition for the addition of peptidoglycan material along the entire length of the cell when thelocation of w min is set at early times. At later times ( φ > . l st min also starts to increase.We relate the asymmetry in the length of stalked and swarmer portions at early times to asymmetries in the stalkedand swarmer poles. The model predicts a linear relationship between the ratio of lengths l st /l sw and the ratio of thetensions at either pole γ stp /γ swp . We cannot directly measure the latter quantity, but from Laplace’s law it is equalto the inverse ratio of the mean curvatures at either pole, which we approximate as the inverse of half the maximumpole width, assuming that the poles are hemispheres with diameter w st max and w sw max , respectively. Supplementary Fig.3 shows scatter plots comparing w st max /w sw max to l st /l sw for three different time intervals. The red best-fit line is shownonly for Supplementary Fig. 3a, for which R = 0 .
15 (linear fits to all other plots produced values of R < . ���� ��� ��� ��� ��� ��� ������������������� ϕ � � � � * ( μ � ) � �� + ������ ��� ��� ��� ��� ��� ��������� ϕ � * ( μ � ) � ���� ��� ��� ��� ��� ��� ������������������� ϕ � � � � * ( μ � ) � �������� ��� ��� ��� ��� ��� ��������������������� ϕ ℓ * ( μ � ) � Supplementary Figure Unnormalized values of the cell shape parameters . Time-dependence of (a) width maxima,and (b) width minima, of both stalked and swarmer portions. These data correspond to Fig. 5a in the main text. (c)
Radiusof curvature calculated for either the entire cell (black), the stalked portion only (red), or the swarmer portion only (blue). (d)
Unnormalized length from stalked pole to either the primary (red) or secondary (green) width minima. These data correspondto Fig. 5b in the main text. Superscript ∗ refers to either st or sw , respectively, as indicated in each figure legend. line runs from the point (1 . , . . , . C. crescentus cell). Note that at times φ > .
6, any correlation disappears.In order to quantify the error in our width profiles, we imaged a single field of view of 24
C. crescentus cells perfusedin complex medium at 31 ◦ C at a frame rate of 5 frames per second (300 times faster than the frame rate used toacquire all other data), and calculated the splined contours for each cell. We focused in particular on a single “dead”(non-growing and non-dividing) cell, and found the root-mean-square deviation (RMSD) of nearest points along thecell contour between subsequent frames to be 12 nm. In addition, we found the RMSD of equivalent points along thewidth profile between subsequent frames to be 28 nm, or a 3.2% pinch depth at the average value of w st max = 0.85 µ m. CELL SHAPE MODEL
Cell Wall Mechanics . The total energy E for the bacterial cell wall is given as the sum of contributions froman active internal pressure P driving cell volume ( V ) expansion, mechanical energy E wall in the cell wall, and themechanical energy of interactions with cytoskeletal proteins E proteins : E = − P V + E wall + E proteins . (S.1)The bacterial cell wall consists of a network of glycan strands cross-linked by peptide chains know as the peptidoglycannetwork. Growth occurs via the insertion of new peptidoglycan strands into the existing network along with thebreaking of existing bonds due to turgor pressure induced stretching. We assume that elastic equilibrium is reachedrapidly as compared to the rate of synthesis of new material [S2]. As a result of cell wall remodeling and irreversibleelongation, growth can be understood as resulting from plastic deformations [S3–5]. To understand the origin of thecell wall tension, γ , in the model introduced in the main text, we consider the cell wall as a thin elastic shell thatdeforms plastically when stretched beyond a maximum strain ε Y , the yield strain. A thin shell has two modes ofelastic deformations, bending and stretching [S6], such that E wall = E stretch + E bend . In the limit of small thicknessof the shell h as compared to its radii of curvature, one can neglect the bending energy (that scales as E bend ∼ h )5 ●●● ●● ● ● ●● ●● ● ●●● ● ● ● ●●●●● ● ●●●●● ● ●● ● ●● ● ●●● ● ●● ● ●●● ● ● ● ●● ●●● ● ●● ●●● ● ●● ●● ●● ● ●● ●● ● ●● ●● ●● ●● ● ●● ●● ● ●●●●● ●●●●● ●● ●●●● ● ● ● ●●● ●● ● ●●●● ●●● ●●● ●● ●● ●● ●● ●● ● ●●●●● ●● ●● ● ●●● ● ●● ● ●● ● ● ●● ●●● ● ●●●● ● ● ●● ●● ●● ●● ●●●●● ●●● ●● ●●●● ●● ● ● ●●● ●● ●●● ● ●● ●●●● ●● ●●● ●●●● ● ●●● ● ●●●●●● ●●● ● ●● ●●● ● ●● ●● ● ●● ●● ● ●● ●●●● ●● ●●●● ●● ●●● ●● ●● ●●●● ● ●●● ● ●● ●● ● ●● ●●●●●● ●●● ● ●●●●● ● ●●● ● ●● ●●●● ●● ●●● ●●● ● ●● ●● ●● ●● ● ●●●●●● ●●●● ● ●●● ● ●● ●●● ● ● ● ● ●●●● ●● ● ●●●● ●● ●●●● ● ● ●●● ●● ●● ●● ●●●● ● ●● ● ●● ● ●●●● ●● ●●● ●●●● ● ●● ●● ● ●●●● ●● ●●● ●●●● ● ●● ● ●● ●●●●●● ●●● ● ●●●● ● ●● ● ●● ●● ● ● ●●● ● ●● ● ● ● ●●● ● ●● ●● ●● ●●● ●●● ●●●● ● ●● ●● ●● ● ●●● ●● ● ●● ●●● ● ● ●●● ● ●● ●●●● ● ● ●● ●● ● ● ●●●● ●●● ● ●●● ●●● ●● ●● ●●● ●●●●● ● ●●● ● ●● ●●●● ●● ●● ●●●● ●● ● ●● ● ●●● ● ● ●● ● 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●●● ●● ●● ●● ●● ●●● ● ●●● ● ●● ●● ●● ●●●● ●●● ●●●● ● ● ●● ●● ●● ●● ● ●●● ● ● ●● ● ●●●●● ●● ●● ●● ●●● ● ● ●● ●●● ● ●●● ● ●● ● ● ●● ● ●●● ● ● ●●●● ●●● ●●● ● ●●● ●●● ●●●●● ● ●●●●● ●● ●●●●●● ●● ● ●● ●●● ● ●●● ●●● ��� ��� ��� ��� ������������������� ℓ �� / ℓ �� � � � � � � / � � � � � � ��� < ϕ < ��� � Supplementary Figure Ratios of maximum width at stalked:swarmer poles versus stalked:swarmer lengths .The three columns delineate the time period from which data were taken: (a) first 10% of the generation, (b) middle 10% ofthe generation, and (c) last 10% of the generation. Blue points show the average value of w st max /w sw max calculated by binning l st /l sw into bins of width 0.05, with size of each error bar showing the standard deviation of the values within that bin. Thebest-fit line for 1 ≤ l st /l sw ≤ . R = 0 . . , .
01) to(1 . , . whereas the stretching energy E stetch is given by, E stretch = h (cid:90) dA σ ij ε ij (S.2)where σ ij is the mechanical stress tensor and ε ij is the strain tensor. As yield strain is reached at the onset of growth,we have ε ij = ε Y δ ij (assuming isotropic stretching), where δ ij is the Kronecker delta. Furthermore, assuming aHookean constitutive relation for the stress tensor [S6] σ ij = Y − ν [(1 − ν ) ε ij + νε kk δ ij ] , (S.3)where Y is the Young’s modulus and ν is the Poisson ratio, we have E stretch = γA , where, γ = 12 σ ij ε ij = Y hε Y (1 − ν ) . (S.4)For a Gram-negative bacterial cell wall of thickness h ∼ Y ∼
40 MPa [S7] and average yieldstrain ε Y ∼ .
5, the wall tension is estimated to be γ = 50 nN /µ m. While the actual value for the turgor pressurecounteracting this tension can be contested, our choice for the numerical value for internal pressure P can be justifiedusing a simple mechanical argument. Radial force-balance dictates that in order to maintain an average cross-sectionalradius r (cid:39) . µ m, a cell wall with surface tension γ ∼
50 nm/ µ m has to balance an internal pressure of magnitude P = 2 γ/r (cid:39) P = 0 . Cytoskeletal bundle mechanics . Next, we model the mechanical energy in the cell wall due to interactions withcytoskeletal proteins. Our minimalist approach considers two crucial protein bundles that are directly responsiblefor maintaining the shape of
C. crescentus cells. MreB protein bundles contribute an effective energy E width , whichfavors a rod-like shape, and crescentin filament bundles contribute an energy E cres , which favors a crescent-like shape.We thus have E proteins = E width + E cres . MreB subunits form patchy filamentous bundles adherent to the cell walland oriented perpendicular to the long axis of the cell [S8, 9]. The elastic energy stored in an adherent MreB subunitis given by E MreB ,i = k b (cid:96) i (cid:18) r i − R m (cid:19) , (S.5)where i labels the subunit, r i is the circumferential radius of curvature of the cell wall where the subunit is attached, (cid:96) i is the length of the subunit, R m is the intrinsic radius of curvature and k b is the bending rigidity of the associated MreBbundle. The total energy imparted by a collection of N m attached MreB subunits is given by E width = (cid:80) N m i E MreB ,i .Next, employing a continuum mean field assumption, we replace individual subunit lengths by their average length (cid:104) (cid:96) (cid:105) (cid:39) ρ m of MreB subunits in the cell surface to obtain E width = k m (cid:90) dA (cid:18) r − R m (cid:19) , (S.6)6where, k m = k b ρ m (cid:104) (cid:96) (cid:105) is the effective bending modulus due to MreB induced traction forces. The bending rigidityof an MreB bundle is given by k b = k MreB n ξ , where k MreB is the flexural rigidity of MreB filaments (assumed tobe similar to F-actin), n is the number of MreB protofilaments per bundle, and ξ is an exponent in the range 1–2depending on the strength of crosslinking or bundling agents [S11]. MreB filaments in a bundle appear to have stronglateral interaction with negligible filament sliding, so we assume ξ (cid:39)
2. The diameter of an MreB protofilament is ∼ n ∼ ρ m ∼ × µ m − and using k mreb (cid:39) − nN µ m [S14], we obtain k m in the range 5.6–12.1 nN µ m.Crescentin proteins form a cohesive bundled structure anchored to the sidewall of C. crescentus cells. The energeticcontribution due to crescentin is given by E cres = k c (cid:90) ds (cid:18) c ( s ) − R c (cid:19) , (S.7)where k c is the bending rigidity, s is an arc-length parameter, c ( s ) is the longitudinal curvature of the cell wall and R c is the preferred radius of curvature of crescentin bundles. The bending rigidity of crescentin can be expressed as k c = Y c I , where Y c is the Young’s modulus and I is the area moment of inertia of the bundle (with width w c ) givenby I = πw c /
64. Since crescentin is an intermediate filament homologue, we assume that Y c is similar to the Young’smodulus of intermediate filament bundles given by Y c ∼
300 MPa [S15]. Assuming w c ∼ . µ m, we estimate thebending rigidity of a crescentin bundle to be k c ∼ . µ m . Mean field model for cell shape and size dynamics . As described in the main text, the total mechanicalenergy in the cell wall of
C. crescentus cells can be given as a sum of contributions from internal pressure, wall surfacetension, mechanical energy of interactions with bundles of cytoskeletal proteins such as crescentin and MreB, and theconstriction energy E div during cell division. In the mean field description, we neglect the contributions from E div and disregard any spatial variations in cell geometry. The mean field description is a good approximation of the cellshape dynamics for φ < .
5, where the average width and the midline radius of curvature remains constant (Fig. 2 inthe main text). We approximate the shape of
C. crescentus cells as the segment of a torus with radius of curvature R ( t ), cross-sectional width ¯ w ( t ) and spanning angle θ ( t ). We also neglect the pole caps that are mechanically rigidand do not remodel during wall growth. The total energy is then given by E [ ¯ w, R, θ ] = − P ( π ¯ w Rθ ) / γ ( πR ¯ wθ ) + E width + E cres , (S.8)where E width = k m πRwθ ) (cid:2) ( ¯ w/ − − R − m (cid:3) , (S.9) E cres = k c R − ¯ w/ θ (cid:2) ( R − ¯ w/ − − R − c (cid:3) . (S.10)From the above expressions, we see that the total energy has the scaling form E [ ¯ w, R, θ ] = θU [ ¯ w, R ]. According toEq. (1) in the main text, the dynamics of the radius of curvature R , the spanning angle θ and the width w are givenby 1 R dRdt = − Φ R ∂E∂R , (S.11a)1¯ w d ¯ wdt = − Φ w ∂E∂ ¯ w , (S.11b)1 θ dθdt = − Φ θ ∂E∂θ , (S.11c)where Φ R , Φ θ and Φ w are the rate constants. The steady-state values for the radius of curvature of the centerlineand the cross-sectional width is given by the solutions to ∂ R U = 0 and ∂ w U = 0. As shown in Supplementary Fig.4a the energy density U admits a stable absolute minimum in width, controlled by the stiffness parameter k m suchthat for lower values of k m (red curve in Supplementary Fig. 4a), U does not have a minimum and the width grows intime rendering the rod-like shape unstable. Similarly, the parameter k c controls cell curvature, such that the energydensity has a stable absolute minimum in R beyond a critical stiffness k c (Supplementary Fig. 4b). At lower valuesof k c the cell does not have a stable radius of curvature and the energy is minimized by increasing R . This leads to7straightening of the cell’s medial axis. In the parameter range where the cell maintains a stable value for ¯ w and R ,the energy density U becomes a constant during cell elongation. From the growth law introduced in the main text,the spanning angle θ evolves in time according to the equation dθdt = − (Φ θ U ) θ ( t ) , (S.12)such that the condition for exponential growth becomes U < a ! b ! Μ m U p J k m k m k m Μ m U n J k c k c k c Supplementary Figure Dependence of the mechanical energy on cell width and radius of curvature . (a) Energy density U as a function of mean cell width ¯ w at various values of bending stiffness (in units of nNµ m): k m = 30 (red), k m = 40 (green) and k m = 50 (blue), for a fixed radius of curvature R = 4 . µ m and k c = 2 nN µ m . The absolute minima (ifit exists) are indicated by solid circles. (b) U as a function of radius of curvature R at various values of crescentin stiffness (inunits of nNµ m ): k c = 1 . k c = 2 . k c = 2 . w = 0 . µ m and k m = 40 nN µ m.Other parameters: P = 0 . γ = 50 nN/ µ m, R c = 0 . µ m, R m = 0 . µ m. Constriction dynamics . As described in the main text, we assume that the shape of the constriction zone isgiven by two intersecting hemishperical segments with diameter w max . The total surface area of the septum is givenby S ( t ) = πw max l s ( t ), where l s ( t ) is the total length of the spherical segments (see Supplementary Fig. 5a). Usingelementary geometry, we obtain the following relation between l s ( t ) and the minimum width, w min ( t ): w min ( t ) = w max (cid:112) − ( l s ( t ) /w max ) . (S.13)If septal growth occurs by addition of new peptidoglycan strands while maintaining the curvature of the sphericalsegments, then l s ( t ) is the shape variable controlling the growth of septal surface area S as well as the constrictiondynamics of w min ( t ). To describe the dynamics of l s ( t ), we assume that initial phase of elongation occurs with avelocity v , and thereafter growth follows an exponential law with rate κ d , dl s dt = v + κ d l s ( t ) . (S.14)Multiplying both sides of the equation by πw max , we derive the growth dynamics of the surface area S(t), dSdt = κ + κ d S ( t ) , (S.15)where κ = πw max v . Having determined the dynamics of w min , one can evaluate the time-dependence of the averagewidth ¯ w defined as, ¯ w ( t ) = l ( t ) (cid:82) l ( t )0 w ( u, t ) du , where u is the coordinate along the centerline. Using the simplifiedgeometry of the constricting cell, given by two toroidal segments (peripheral regions) connected by two intersectinghemispheres (septal region), one gets,¯ w ( t ) = 1 l ( t ) (cid:34) ¯ w ( l ( t ) − l s ( t )) + 2 (cid:90) l s ( t ) / w s ( u, t ) du (cid:35) , (S.16)where w s ( u ) = w max (cid:112) − (2 u/w max ) , is the width in the septal region and ¯ w is the average width of the stalked andswarmer components as determined by Eq. (S.11b). As shown in Fig. 2c in the main text (blue solid line), Eq. (S.16)8is in excellent agreement with the experimental data and captures the dip in ¯ w ( t ) seen for φ > .
5. Fig. 2b in themain text also shows that the radius of curvature of the centerline ( R ( t )) drops for φ > . l s ( t ) and an angle θ s ( t ) in the septal region, as shown in Supplementary Fig. 5a.Consider a small segment δl s ( t ) along the centerline with a local radius of curvature R and spanning angle δθ s . Wethen have the relation δl s ( t ) = R ( t ) δθ s ( t ). This leads to the following identity,1 δl s d ( δl s ) dt = 1 R dRdt + 1 δθ s d ( δθ s ) dt . (S.17)Now the geometry of the septal region (Supplementary Fig. 5a) directly relates the rate of increase in the spanning angle δθ s ( t ) to the rate of drop in the tangent angle ( ψ ( t )) at the constriction site, δθ s d ( δθ s ) dt = − δψ d ( δψ ) dt . Furthermore, usingthe relations, w max sin ψ ( t ) = w min ( t ) and w max cos ψ ( t ) = l s ( t ), we get δψ ( t ) = δw min ( t ) /l s ( t ) = − δl s ( t ) /w min ( t ).The last equality follows after variations of Eq. (S.13). As a result we have the following kinetic relation, d ( δψ ) /dtδψ = dw min /dtw min − d ( δl s ) δl s = − d ( δθ s ) /dtδθ s . (S.18)Combining the identities in Eq. (S.17) and (S.18) we get, R − dR/dt = w − dw min /dt , showing that the radius ofcurvature drops at the same rate as w min ( t ) shrinks. Ψ Ψ w min w max ∆ S l s Θ s R b ! a ! p st p sw st sw st sw sep Supplementary Figure Geometry of a constricting cell contour . (a) Schematic of the constriction zone of a dividingcell where the dashed line indicates the centerline with length l s , radius of curvature R and spanning angle θ s . S (blue) denotesthe septal cell wall, ψ is the tangent angle at the constriction site and δ is the width of the division ring. (b) Compartmentalizingthe cell contour into the (1) stalked pole p st (red), (2) swarmer pole p sw (purple) (3) upper wall 1 st/sw (with crescentin bundleattached, green), (4) Lower wall 2 st/sw (black) and the (5) Septal region (blue). Contour Model for cell shape . To quantitatively capture the experimentally observed spatial variations incell shape we study an effective two-dimensional contour model for the cell shape. This approach facilitates closercomparison with the two-dimensional splined contours obtained from our experimental data. The contour descriptionof the rod-like cell shape negelects circumferential variations in cell geometry. The model incorporates non-uniformmaterials properties and mechanical constraints across the cell wall, with the poles and the septal region beingmechanically stiffer than the rest of the cell wall. By exploiting the rod-like geometry, one can use the centerlinecurve to divide the contour into two parts, the upper and the lower cell wall. As shown in Supplementary Fig. 5b wefurther subdivide the cell contour into the stalked and swarmer poles ( p st/sw ), the upper (1 st/sw ) and lower cell wall(2 st/sw ), and the septal region (sep).We parametrize the instantaneous shape of the cell contour using the two-dimensional centerline curve R ( u, t ),the length of the centerline l ( t ) and the width w ( u, t ), where u is the absolute distance along the centerline fromthe stalked pole such that u ∈ [0 , l ( t )]. If ˆ n denotes the outward unit normal vector on the centerline, the curvesdefining the upper and lower parts of the cell contour, r ± ( u, t ), are given by the relation, r ± = R ± w ± ˆ n , where w ± ( u )represent the perpendicular distances of the top and bottom curves from the centerline. The total cell width is thengiven by w ( u, t ) = w + ( u, t ) + w − ( u, t ).9It is convenient to switch to polar coordinates, where the shape of the cell contour is given by the re-parametrizedcurve r ± ( ϕ, t ), where 0 < ϕ < θ ( t ) is the angular coordinate spanning the centerline, which can be approximated asthe arc of a circle with radius R ( t ). Since the ratio w ± /R ( (cid:39) .
1) is small at all times, one can approximate the localcurvature as c ± ( ϕ, t ) = 1 R ( t ) (cid:18) − w (cid:48)± ( ϕ, t ) + w (cid:48)(cid:48)± ( ϕ, t ) R ( t ) (cid:19) + O ( w ± , w (cid:48) ± , w (cid:48)(cid:48) ± ) , (S.19)where prime denotes derivative with respect to ϕ and the subscripts ± represent the upper and the lower part ofthe cell contour respectively. Furthermore, in the linear regime, the differential arc length can be approximated as du (cid:39) Rdϕ . The dynamics of the shape parameters l ( t ), θ ( t ), and R ( t ) are determined from the kinetic law in Eq. (1)of the main text.The instantaneous width profile w ( u, t ) results from minimizing the total energy functional, which leads to thefollowing shape equation, P − γ ( u ) c ± ( u, t ) + f width ( u, t ) + f cres ( u, t ) = 0 , (S.20)where, γ ( u ) is the tension on the cell contour and f width ( u, t ) and f cres ( u, t ) are the linear force densities on the contourdue to maintenance of width and the crescent shape, respectively. In the contour model we simplify the energeticcontribution due to maintenance of width (acting on the sections 1 st/sw and 2 st/sw ) as E width = K m (cid:90) du ( w ± − R m ) , (S.21)where the elastic constant K m depends on the bending rigidity k m introduced in the main text as K m = k m /R m . Thislinear approximation holds if ( w ± − R m ) /R m (cid:28)
1. From our data and mean field model fits we get ( w ± − R m ) /R m (cid:39) .
15. The resultant force density is given by f width ( u, t ) = − K m ( w ± ( u, t ) − R m ).The bending energy induced by crescentin protein bundles anchored onto the cell wall (regions 1 st/sw ) is given by E cres = k c (cid:90) ds [ c + ( u ) − c ] , (S.22)where c = 1 /R c is the spontaneous curvature of the crescentin bundle, and s is the arc-length parameter along theupper part of the cell contour (1 st/sw ) which is related to u as ds = du − w + dϕ . To obtain the force density weconsider an infinitesimal deformation, δr , of the upper contour as r + → r + + n + δr , where n + is the outward unitnormal. Accordingly the curvature and the differential arc-length changes, c + → c + + δc + and ds → ds + δ ( ds ), where δc + = c δr + d ( δr ) /ds and δ ( ds ) = − c + ds [S16]. The resultant force density is obtained after variations of theenergy functional, δE cres = (cid:82) f cres δrds , where f cres δrds = δ [( c + − c ) ds ]. This leads to the following non-linear forcecontribution: f cres = k c (cid:18) d c + ds + c − c + c (cid:19) . (S.23)Using Eq. (S.19), f cres can be linearized and expressed using the angular coordinate ϕ as f cres ( u, t ) = k c R (cid:20) − w (cid:48)(cid:48)(cid:48)(cid:48) + ( u, t ) + 12 ( c R − w (cid:48)(cid:48) + ( u, t ) + 12 ( c R − w + ( u, t ) + R (1 − c R ) (cid:21) . (S.24) Compartmentalizing the cell contour . (1) Pole caps: The cell poles are assumed to be mechanically inert inthe sense that they do not interact with the active cytoskeletal proteins such that f width = f cres = 0. The mechanicalforces acting on the cell poles come from turgor pressure P , and the tension γ p . The shape of the cell poles are thendescribed by the two-dimensional Laplace’s law, γ st,swp c ± ( u, t ) = P , (S.25)where the superscripts st and sw denote respectively the stalked and swarmer poles.(2) Upper cell contour (1 st/sw ): The region 1 st/sw in the upper cell contour obeys the force-balance equation: P − γc + ( u, t ) + f width ( u, t ) + f cres ( u, t ) = 0 . (S.26)0At the endpoints of the segments 1 st/sw we impose the boundary condition that the curvature c must equal thelongitudinal curvature of the poles.(3) Bottom cell contour (2 st/sw ): In the absence of crescentin, the region 2 st/sw in the bottom cell contour obeysthe force-balance equation: P − γc − ( u, t ) + f width ( u, t ) = 0 . (S.27)(4) Septal region: We incorporate the effect of constriction in the cell shape equation by imposing the constraint(boundary condition) that w ( l st ( t ) , t ) = w min ( t ), where w min ( t ) evolves according to the kinetics described in Eq. (6)and (7) of the main text. Furthermore, the curvatures at the end points of the septal segments must conform to thecurvature of the newly formed poles. Subject to these boundary conditions, the width profile in the upper septalregion ( l st − l s / < u < l st + l s /
2) is given by, P − γ s c + ( u, t ) + f width ( u, t ) + f cres ( u, t ) = 0 , (S.28)with a surface tension ( γ s ) of the newly formed poles chosen to be much higher than the peripheral region, γ s (cid:39) . γ .The contour below the centerline obeys the equation, P − γ s c − ( u, t ) + f width ( u, t ) = 0 . (S.29)The governing cell shape equation given in Eq. (S.20), is then solved numerically in each part of the cell contourwith matching boundary conditions in w ± ( u, t ) and its derivatives. SUPPLEMENTARY REFERENCES † These authors contributed equally to this work. ∗ To whom correspondence may be addressed. Email: [email protected] or [email protected][S1] Brandt, J. W. Convergence and continuity criteria for discrete approximations of the continuous planar skeleton.
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