††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Intergenerational continuity of cell shape dynamics in Caulobacter
crescentus

Charles S. Wright  James Franck Institute, The University of Chicago, Chicago
IL 60637  Institute for Biophysical Dynamics, The University of Chicago,
Chicago IL 60637  Shiladitya Banerjee  James Franck Institute, The University
of Chicago, Chicago IL 60637  Srividya Iyer-Biswas  James Franck Institute,
The University of Chicago, Chicago IL 60637  Institute for Biophysical
Dynamics, The University of Chicago, Chicago IL 60637  Sean Crosson  Institute
for Biophysical Dynamics, The University of Chicago, Chicago IL 60637
Department of Biochemistry and Molecular Biology, The University of Chicago,
Chicago IL 60637  Aaron R. Dinner  To whom correspondence may be addressed.
Email<EMAIL_ADDRESS>or<EMAIL_ADDRESS>James Franck Institute,
The University of Chicago, Chicago IL 60637  Institute for Biophysical
Dynamics, The University of Chicago, Chicago IL 60637  Department of
Chemistry, The University of Chicago, Chicago IL 60637  Norbert F. Scherer  To
whom correspondence may be addressed. Email<EMAIL_ADDRESS>or
<EMAIL_ADDRESS>James Franck Institute, The University of Chicago,
Chicago IL 60637  Institute for Biophysical Dynamics, The University of
Chicago, Chicago IL 60637  Department of Chemistry, The University of Chicago,
Chicago IL 60637

###### Abstract

We investigate the intergenerational shape dynamics of single Caulobacter
crescentus cells using a novel combination of imaging techniques and
theoretical modeling. We determine the dynamics of cell pole-to-pole lengths,
cross-sectional widths, and medial curvatures from high accuracy measurements
of cell contours. Moreover, these shape parameters are determined for over 250
cells across approximately 10000 total generations, which affords high
statistical precision. Our data and model show that constriction is initiated
early in the cell cycle and that its dynamics are controlled by the time scale
of exponential longitudinal growth. Based on our extensive and detailed growth
and contour data, we develop a minimal mechanical model that quantitatively
accounts for the cell shape dynamics and suggests that the asymmetric location
of the division plane reflects the distinct mechanical properties of the
stalked and swarmer poles. Furthermore, we find that the asymmetry in the
division plane location is inherited from the previous generation. We
interpret these results in terms of the current molecular understanding of
shape, growth, and division of C. crescentus.

Cell shape both reflects cabeen2005  and regulates huang2000  biological
function. The importance of cell shape is exemplified by bacteria, which rely
on specific localization of structural proteins for spatiotemporal
organization laloux2014 . Bacteria take forms resembling spheres, spirals,
rods, and crescents. These shapes are defined by cell walls koch2001
consisting of networks of glycan strands cross-linked by peptide chains to
form a thin peptidoglycan meshwork gan2008 . Super-resolution imaging is now
revealing the internal positions of associated proteins gahlmann2014 . These
include cytoskeletal proteins such as MreB, a homolog of actin wachi1987 ;
iwai2002 ; jiang2011 ; figge2004 , intermediate filament-like bundles of CreS
(crescentin) ausmees2003 ; cabeen2009 , and FtsZ, a homolog of tubulin
erickson2010 . However, due to the inherently stochastic nature of molecular
processes, understanding how these proteins act collectively to exert
mechanical stresses and modulate the effects of turgor pressure and other
environmental factors requires complementary methods such as high-throughput,
quantitative optical imaging.

Multigenerational imaging data for bacterial cells can now be obtained from
microfluidic devices of various designs wang2010 ; moffitt2012 ; shaw2012 ;
long2013 ; campos2014 . Still, a common limitation of most devices is that the
environmental conditions change throughout the course of the experiment,
particularly as geometric growth of the population results in crowding of the
experimental imaging spaces. We previously addressed this issue by engineering
a C. crescentus strain in which cell adhesion is switched on and off by a
small molecule (and inducible promoter) iyer-biswas2014b , allowing
measurements to be made in a simple microfluidic device iyer-biswas2014b ;
siegal-gaskins2008 ; lin2010 ; lin2012 . This technology allows imaging $>$100
generations of growth of an identical set of 250–500 single cells distributed
over $\sim$25 fields of view. Thus cell density is low and remains constant.
These studies afforded sufficient statistical precision to show that single C.
crescentus cells grow exponentially in size and divide upon reaching a
critical multiple ($\approx$1.8) of their initial sizes iyer-biswas2014b .
Satisfaction of a series of scaling laws predicted by a simple stochastic
model for exponential growth indicates that these dynamics can be
characterized by a single time scale iyer-biswas2014a ; iyer-biswas2014b .

In this paper, we use more advanced image analysis methods to extract cell
shape contours from these data. The resulting geometric parameters, together
with mathematical models, provide insights into growth and division in C.
crescentus and the plausible role of cell wall mechanics and dynamics in these
processes. Specifically, we identify natural variables for tracking cell
dynamics, and develop a minimal mechanical model that shows how longitudinal
growth can arise from an isotropic pressure. We then examine the dynamics of
cell constriction and unexpectedly find that it is governed by the same time
constant as exponential growth. This important finding can be understood in
terms of an intuitive geometric model that relates the constriction dynamics
to the kinetics of the growth of septal cell wall. We further suggest that the
site of constriction can arise from differences in materials properties of the
poles and show that it is established in the previous generation—i.e., the
location of the site of division can be predicted before formation of the
divisome. We relate our results to the known dynamics of contributing
molecular factors and existing models for bacterial growth and division.

Figure 1: Determination of cell contour and definition of shape parameters.
(a) A representative phase contrast image of one field of view. The solution
flow in the microfluidic channel is from bottom to top. (b) Zoomed image of
the yellow highlighted cell from a and its splined contour. (c) Schematic of a
contour illustrating the shape parameters. The cell medial axis is calculated
from pole to pole; it defines both cell length $\ell(\phi)$ and radius of
curvature $R(\phi)$, which lead directly to the spanning angle $\theta(\phi)$.
The cell width $w(\phi,u)$ is a parametric quantity, calculated as the length
of the rib perpendicular to the medial axis at a specified distance from the
stalked pole, $u(\phi)$. The location of the global minimum of the width
$w_{\text{min}}(\phi)$ (purple line) can be used to segment the cell into
stalked ($st$, red) and swarmer ($sw$, blue) portions.

## Results

The length is sufficient to characterize the exponential growth of each cell.
Various techniques have been put forth to analyze cell morphology gathered
from single cell images pincus2007 . Recent work on image analysis of single
cells has attempted to optimize two problems: separation of distinct (but
potentially overlapping) cells and accurate determination of the edge of each
cell sliusarenko2011 . Because crowding is not an issue in our setup, we could
focus solely on constructing an algorithm to delineate each cell contour
accurately and precisely. As shown in Fig. 1a,b and described in the Methods
section, we first segment each cell using pixel-based edge detection similar
to guberman2008 , then perform spline interpolation to determine the cell
contour at sub-pixel resolution. The sequence of such images for each single
cell constitutes a trajectory in time $t$ that serves as the basis for
quantitative analysis. Division events are then detected in an automated
fashion using custom Python code, and used to divide time trajectories for
each cell into individual generations.

All data shown here were obtained by observing 260 single C. crescentus cells
perfused in complex medium (peptone-yeast extract; PYE) at 31∘C over the
course of 2 days (corresponding to 9672 separate generations). Under these
conditions, the mean population growth rate and division time remain constant,
so we treat the trajectories of individual generations as members of a single
ensemble. In other words, we segment each cell trajectory 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 quantitative
information according to time since division, $t$, normalized by the
respective division time $\tau$. The normalized time, $\phi\equiv t/\tau$,
serves as a cell-cycle phase variable.

For our quantitative analysis, we focus on a set of three intuitive and
independent parameters that characterize cell shape at each stage of growth:
length $\ell$, width $w$, and radius of curvature $R$ (Fig. 1c). They are
calculated directly from each splined contour as follows (see also
Supplementary Fig. 1):

  * •

We define the length, $\ell(\phi)$, as the pole-to-pole distance along the
contour of the cell medial axis at the normalized time $\phi$ (Fig. 2a).

  * •

We assign a single radius of curvature, $R(\phi)$, to each cell based upon the
best-fit circle to the medial axis (Fig. 2b). Although stalked
($R^{st}(\phi)$) and swarmer ($R^{sw}(\phi)$) portions may be described by
different radii of curvature toward the end of the cell cycle, the average
radius obtained by averaging the contributions of each portion yields the same
value, i.e., $\langle(R^{st}(\phi)+R^{sw}(\phi))/2\rangle\simeq\langle
R(\phi)\rangle$ (see Supplementary Fig. 2c).

  * •

We define the width, $w(\phi,u)$ as the length of the perpendicular segment
spanning from one side of the cell contour to the other at each position
$u(\phi)$ along the medial axis, which runs from $u=0$ at the stalked pole to
$u=\ell$ at the swarmer pole. Furthermore, we spatially averaged the width
over positions along the medial axis, $\bar{w}(\phi)$, to obtain a
characteristic width at each time point (Fig. 2c).

The mean division time is $\langle\tau\rangle=73\pm 7$ min, where
$\langle...\rangle$ indicates a population average. We find that
$\langle\ell(\phi)\rangle$ increases exponentially with time constant
$\langle\kappa\rangle^{-1}=125\pm 8$ min, essentially the same time constant
that we previously observed for the cross-sectional area iyer-biswas2014b ,
while $\langle\bar{w}(\phi)\rangle$ and $\langle R(\phi)\rangle$ remain
approximately constant for $0<\phi<0.5$ and each shows a dip for
$0.5<\phi<0.9$ when cell constriction becomes prominent. The sharp rise in
$\langle R(\phi)\rangle$ seen for $\phi>0.9$ results from independent
alignment of the stalked and swarmer portions with the microfluidic flow as
they become able to move independently (i.e., fluctuate easily about the plane
of constriction). These observations confirm the assumptions in iyer-
biswas2014b  that the length is sufficient to describe the growth of cell
size. Moreover, we can track the dynamics of the spanning angle, $\theta$,
using the relation $\ell=R\theta$.

Figure 2: Dynamics of cell shape parameters. (a) Length of the cell medial
axis (data shown in black and exponential fit from our mechanical model in
red). (b) Radius of curvature of the cell medial axis, obtained by calculating
the best-fit circle to the entire cell (black points), with a time-averaged
mean $\langle R\rangle=4.44\pm 2.12\ \mu$m ($0<\phi<0.5$). The mean-field
model predicts a constant steady-state value for $\langle R(\phi)\rangle$ (red
solid line), whereas by accounting for constriction dynamics, the model
captures the dip in $\langle R(\phi)\rangle$ seen for $0.5<\phi<0.9$ (blue
solid line). (c) Characteristic cell width, obtained by spatially averaging
the width at each time point (black points), with a time-averaged mean
$\langle\bar{w}\rangle=0.74\pm 0.02\ \mu$m ($0<\phi<0.5$). The mean field
model predicts a constant steady-state value for $\langle\bar{w}(\phi)\rangle$
(red solid line), whereas cell constriction accounts for the dip in
$\langle\bar{w}(\phi)\rangle$ seen for $\phi>0.5$ (blue solid line). The
shaded regions represent $\pm 1$ standard deviation. Model parameters: $P=0.3$
MPa, $\gamma=50$ nN/$\mu$m, $k_{c}=2$ nN$\mu$m2, $R_{c}=0.5$ $\mu$m,
$k_{m}=40$ nN$\mu$m, $R_{m}=0.31$ $\mu$m, $\tau=73$ min.

Mechanical model for cell shape and growth. There are many details of cell
growth and shape that require interpretation. For example, it is not obvious a
priori that growth should be almost exclusively longitudinal. Therefore, we
have developed a minimal mechanical model that can explain these observations.
We parametrize the geometry of the cell wall by a collection of shape
variables $\\{q_{i}(t)\\}$, where $q_{1}=\langle R\rangle$,
$q_{2}=\langle\bar{w}\rangle$, and $q_{3}=\langle\theta\rangle$ are the
parameters introduced above (Fig. 1c). As the cell grows in overall size, we
postulate that the rate of growth in the shape parameter $q_{i}(t)$ is
proportional to the net decrease in cell wall energy, $E(\\{q_{i}(t)\\})$, per
unit change in $q_{i}(t)$ lan2007 ; jiang2010 . Assuming linear response, the
configurational rate of strain, $\frac{1}{q_{i}(t)}\frac{dq_{i}}{dt}$, is
proportional to the corresponding driving force $F_{i}=-\partial E/\partial
q_{i}$, in analogy with the constitutive law of Newtonian flow batchelor2000 :

$\frac{1}{q_{i}}\frac{dq_{i}}{dt}=\Phi_{i}F_{i}\;,$ (1)

where the constant $\Phi_{i}$ describes the rate of irreversible flow
corresponding to the variable $q_{i}(t)$. According to equation (1),
exponential growth occurs if $F_{i}$ is constant, whereas $q_{i}(t)$ reaches a
steady-state value if $F_{i}(q_{i})=0$ along with the condition $\partial
F_{i}/\partial q_{i}<0$. It thus remains to specify the form of $E$.

For a C. crescentus cell of total volume $V$ and surface area $A$, our model
for the total energy in the cell wall is given by

$E(R,\bar{w},\theta)=-PV+\int dA\
\gamma+E_{\text{width}}+E_{\text{cres}}+E_{\text{div}}\;,$ (2)

where $P$ is a constant pressure driving cell wall expansion; $\gamma$ is the
tension on the surface of the cell wall; $E_{\text{width}}$ is the energy
required to maintain the cell width; $E_{\text{cres}}$ represents the
mechanical energy required to maintain the crescent cell shape;
$E_{\text{div}}$ is the energy driving cell wall constriction. Traditionally
$P$ was taken to be the turgor pressure lan2007 ; while the importance of the
turgor pressure has recently been questioned rojas2014 , an effective pressure
must still arise from the synthesis and insertion of peptidoglycan strands
that constitute the cell wall. We note that a purely elastic description of
cell wall mechanics would lead to a curvature-dependent surface tension
jiang2012b . However, if growth is similar to plastic deformation, the tension
is uniform boudaoud2003 . The effective tension in our model depends on the
local surface curvatures through the energy terms $E_{\text{width}}$ and
$E_{\text{cres}}$, that describe harmonic wells around preferred values of
surface curvatures.

The mechanical energy for maintaining width is given by

$E_{\text{width}}=\frac{k_{m}}{2}\int
dA\left(\frac{1}{\bar{w}/2}-\frac{1}{R_{m}}\right)^{2},$ (3)

where the constant $R_{m}$ is the preferred radius of curvature, $k_{m}$ is
the bending rigidity and $dA$ is a differential area element sun2011 .
Contributions to $k_{m}$ can come from the peptidoglycan cell wall as well as
membrane-associated cytoskeletal proteins like MreB, MreC, RodZ, etc., which
are known to control cell width wachi1987 ; iwai2002 ; jiang2011 .

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 ausmees2003 . Although the
mechanism by which crescentin acts is not known, various models have been
proposed, including modulation of elongation rates across the cell wall
cabeen2009 ; sliusarenko2010  and bundling with a preferred curvature
jiang2011b . We assume the latter and write the energy for maintaining the
crescent shape as

$E_{\text{cres}}=\frac{k_{c}}{2}\int_{0}^{\ell_{c}}du\left(c(u)-\frac{1}{R_{c}}\right)^{2},$
(4)

where $u$ is the arc-length parameter along the crescentin bundle attached to
the cell wall, $c(u)$ is the local curvature, $R_{c}$ is the preferred radius
of curvature, $\ell_{c}$ is the contour length, and $k_{c}$ is the linear
bending rigidity. Equation (S.22) accounts for the compressive stresses
generated by the crescentin bundle on one side of the cell wall, leading to a
reduced rate of cell growth, according to equation (1). As a result, the cell
wall grows differentially and maintains a non-zero curvature of the
centerline. In the absence of crescentin ($k_{c}=0$), our model predicts an
exponential decay in the cell curvature that leads to a straight morphology,
consistent with previous observations ausmees2003 .

Finally, one must also account for the energy driving cell wall constriction.
Constriction proceeds via insertion of new peptidoglycan material at the
constriction site. This process leads to the formation of daughter pole caps
typas2012 . We take constriction to be governed by an energy of the form
$E_{\text{div}}=-\lambda S$, where $S$ is the surface area of the septal cell
wall, and $\lambda$ is the energy per unit area released during peptidoglycan
insertion.

There exists an optimal cell geometry for a given mechanical energy. To apply
the model introduced above (equations (1) and (2)) to interpreting the data in
Fig. 2, we assume a minimal cell geometry given by a toroidal segment with
uniform radius of curvature $R(\phi)$, uniform cross-sectional width
$\bar{w}(\phi)$ and the spanning angle $\theta(\phi)$. To this end, we
estimate as many mechanical parameters as we can from the literature and then
determine the rest by fitting our experimentally measured values. Turgor
pressure in Gram-negative bacteria has been measured to be in the range
$0.03-0.5$ MPa reed1985 ; koch1987 ; deng2011 . We use a value for the
effective internal pressure close to the higher end of the measured values for
turgor pressure, $P=0.3$ MPa, in order to account for peptidoglycan insertion.
We estimate the surface tension as $\gamma=50$ nN$/\mu$m (see Supplementary
model section) and multiply it by the cell surface area
$A(\phi)=\pi\bar{w}R\theta$ to obtain the cell wall surface energy. First, we
neglect cell constriction (setting $E_{\text{div}}=0$) and assume that the
crescentin structure spans the length of the cell wall (excluding the endcaps)
charbon2009 , with a contour length $\ell_{c}(\phi)=(R-\bar{w}/2)\theta$. The
mechanical properties of MreB and crescentin are likely similar to those of
F-actin and intermediate filaments, respectively van2001 ; ausmees2003 .
However, due to a lack of direct measurements, we obtain the mechanical
parameters $k_{m}$ and $k_{c}$ by fitting the model to the experimental data.

As desired, we find that the total energy $E$ has a stable absolute minimum at
particular values of the cross-section diameter $\bar{w}$ and the centerline
radius $R$, given by solution of $\partial E/\partial\bar{w}=\partial
E/\partial R=0$ (see Supplementary Fig. 4). The measured values are
$\langle\bar{w}\rangle=0.74\pm 0.02\ \mu$m and $\langle R\rangle=4.44\pm 2.12\
\mu$m ($0<\phi<0.5$), and, as indicated by the red solid curves in Fig. 2b,c,
the model reproduces them with $k_{m}=40$ nN$\mu$m and $k_{c}=2$ nN$\mu$m2.
While the fitted value for $k_{c}$ is numerically close to the estimate based
on the known mechanical properties of intermediate filaments ($\sim 1.5$
nN$\mu$m2), the value for $k_{m}$ is much higher than the bending rigidity of
MreB bundles (see Supplementary Information). This indicates that $k_{m}$ is
only determined in part by MreB and can have contributions from the cell wall.

Given stable values for $\bar{w}$ and $R$, growth is completely described by
the dynamics of the angle variable $\theta(\phi)$. Consequently, we write the
total energy in the scaling form
$E(\bar{w},R,\theta(\phi))=\theta(\phi)U(\bar{w},R)$, with $U$ the energy
density along the longitudinal direction. The condition for growth then
becomes $U<0$, such that the energy is minimized for increasing values of
$\theta(\phi)$. From our experimental data, the angle spanned by the cell
centerline increases by an amount $\Delta\theta\simeq 0.49$ during the entire
cycle. Using our parameter estimates and fitting the data in Fig. 2, we obtain
a numerical value for the energy density $U\simeq-5$ nN$\mu$m. We relate the
angle dynamics to the length by

$\frac{d\ell}{d\phi}=\kappa\tau\ell(\phi)=\frac{\kappa\tau}{\pi\bar{w}}A(\phi)$
(5)

where $\kappa=-\Phi_{\theta}U$ ($U<0$) is the rate of longitudinal growth,
which can be interpreted as resulting from remodeling of peptidoglycan
subunits with a mean current $\kappa/\pi\bar{w}$, across the cell surface area
$A(\phi)$. From an exponential fit to the data for cell length (Fig. 2a), we
obtain $\Phi_{\theta}=1.6\times 10^{-3}$ (nN $\mu$m min)-1, which gives us an
estimate of the friction coefficient, $1/(\pi\bar{w}R\Phi_{\theta})\simeq 61$
nN $\mu$m-1 min, associated with longitudinal growth; e.g., MreB motion that
is known to correlate strongly with the insertion of peptidoglycan strands
garner2011 . Our results are consistent with previous observations of C.
crescentus cells with arrested division but continued growth fiebig2010 .

Figure 3: Timing and location of the division plane. (a) Time-dependence of
$\langle w_{\text{min}}(\phi)\rangle$, with the experimental points in black
and the model fit (equations (S.13) and (7)) in red. Fit values:
$w_{\text{max}}=0.805\ \mu$m, $\kappa_{d}^{-1}=130.92$ min, $\kappa_{0}=0.016$
$\mu$m2/min. Inset: Minimal geometry of a constricting cell, where $S$ (blue)
is the septal cell wall synthesized during constriction, $\delta$ is
longitudinal width of the constriction zone, and $\psi$ is the tangent angle
at constriction. (b) Ratios of the length of the stalked (red) and swarmer
(blue) portions divided by the total length
($\langle\ell^{st}(\phi)/\ell(\phi)\rangle$ and
$\langle\ell^{sw}(\phi)/\ell(\phi)\rangle$, respectively), are approximately
constant over the cell cycle. (c) Experimental width profile of C. crescentus
cells in the initial stage of the cycle ($\phi=0$) after ensemble-averaging
over all data. The width $\langle w(\phi=0,u)\rangle$ is plotted as a function
of the distance from the stalked end normalized to the length of the cell,
$u/\ell$. Inset: A representative single cell contour immediately after
division ($\phi=0$), showing the location of the local minimum in the width
($w_{\text{min}}$, purple line) as well as two local maxima
($w^{st}_{\text{max}}$, red line and $w^{sw}_{\text{max}}$, blue line). These
two local maxima in the width define the pole regions (shaded in red and blue,
respectively). (d) Model width profiles of the cell showing symmetric and
asymmetric location of the invagination near the midplane for different values
of the ratio $\gamma_{p}^{st}/\gamma_{p}^{sw}=$1 (blue, dashed), 1.05 (red,
solid) and 1.09 (green, dotted). Parameter values are the same as in Fig. 2
with $\gamma_{p}^{st}=5\gamma$ lan2007 . The shaded regions in a, b and c
represent $\pm 1$ standard deviation.

Constriction begins early and proceeds with the same time constant as
exponential growth. Having characterized the dynamics of growth, we now turn
to constriction at the division plane. As mentioned above, we obtain the
experimental width at each point along each cell’s medial axis. The typical
width profile is non-uniform along its length, exhibiting a pronounced
invagination near the cell center (with width $w_{\text{min}}(\phi)$; Fig.
1c). This invagination, which ultimately becomes the division plane, is
readily identifiable early in the cell cycle, even before noticeable
constriction occurs. We discuss the kinetics of constriction in this section,
and focus on its location later in the manuscript. As shown in Fig. 3a (black
points), $\langle w_{\text{min}}(\phi)\rangle$ progressively decreases towards
zero until pinching off at $\phi=1$. Due to the limited spatial resolution of
our imaging (phase contrast microscopy), the pinch-off process occurring for
$\phi>0.9$ could not be captured, but $\langle w_{\text{min}}(\phi)\rangle$ at
earlier times (i.e., $\phi<0.9$) is precisely determined as a function of
$\phi$.

To model the dynamics of constriction, we assume as in Ref. reshes2008  (Fig.
3a, inset): (i) the shape of the zone of constriction is given by two
intersecting and partially formed hemispheres with radii $w_{\text{max}}/2$;
and (ii) constriction proceeds by completing the missing parts of the
hemisphere such that the newly formed cell wall surface maintains the
curvature of the pre-formed spherical segments. As a result, a simple
geometric formula is obtained that relates the width of the constriction zone,
$w_{\text{min}}(\phi)$, to the surface area $S(\phi)$ of the newly formed cell
wall,

$w_{\text{min}}(\phi)=w_{\text{max}}\sqrt{1-\left(S(\phi)/S_{\text{max}}\right)^{2}}\;,$
(6)

where $S_{\text{max}}=\pi w_{\text{max}}^{2}$ is the maximum surface area
achieved by the caps as the constriction process is completed, i.e., when
$w_{\text{min}}(\phi=1)=0$. We assume that the addition of new cell wall near
the division plane initiates with a rate, $\kappa_{0}$, and thereafter grows
exponentially with a rate, $\kappa_{d}$, according to,

$\frac{1}{\tau}\frac{dS}{d\phi}=\kappa_{d}S(\phi)+\kappa_{0}\;,$ (7)

subject to the initial condition $S(\phi=0)=0$. The first term on the right-
hand side of equation (7) follows from equation (1), using $S(\phi)$ as the
shape variable, after incorporating the constriction energy
$E_{\text{div}}(\phi)$. The rate of septal peptidoglycan synthesis,
$\kappa_{d}$, is thus directly proportional to the energy per unit area
released during constriction, $\lambda$. The solution,
$S(\phi)=\kappa_{0}(e^{\kappa_{d}\tau\phi}-1)/\kappa_{d}$, can then be
substituted into equation (S.13) to derive the time-dependence of
$w_{\text{min}}(\phi)$, whose dynamics is controlled by two time scales:
$\kappa_{d}^{-1}$ and $S_{\text{max}}\kappa_{0}^{-1}$.

Fitting equation (S.13) with the data for $\langle
w_{\text{min}}(\phi)\rangle$, we obtain $\kappa_{d}^{-1}\simeq 131$ min and
$S_{\text{max}}\kappa_{0}^{-1}\simeq 118$ min. The fitted values for the time
constants controlling constriction dynamics ($\kappa_{d}^{-1}$ and
$S_{\text{max}}\kappa_{0}^{-1}$) are remarkably similar to that of exponential
cell elongation ($\langle\kappa\rangle^{-1}\simeq 125$ min). This shows that
septal growth proceeds at a rate comparable to longitudinal growth. Therefore,
one of the main conclusions that we draw is that cell wall constriction (Fig.
3a) is controlled by the same time constant as exponential longitudinal growth
(Fig. 2a).

Having determined the dynamics of $w_{\text{min}}(\phi)$, we compute the
average width across the entire cell $\bar{w}(\phi)$ using the simplified
shape of the constriction zone as shown in Fig. 3a (inset). The resultant
prediction (blue solid curve in Fig. 2c) is in excellent agreement with the
experimental data and captures the dip in $\langle\bar{w}(\phi)\rangle$ seen
for $\phi>0.5$. Constriction also leads to a drop in the average radius of
curvature of the centerline, as shown by the experimental data in Fig. 2b. In
the supplementary material we derive a relation between the centerline radius
of curvature $R(\phi)$ and the minimum width $w_{\text{min}}(\phi)$, given by
$R^{-1}(dR/d\phi)=w_{\text{min}}^{-1}(dw_{\text{min}}/d\phi)$, predicting that
cell curvature increases at the same rate as $w_{\text{min}}(\phi)$ drops.
Using this relation, we are able to quantitatively capture the dip in $\langle
R(\phi)\rangle$ seen for $\phi>0.5$ (solid blue curve in Fig. 2b) without
invoking any additional fitting parameters.

Origin of the asymmetric location of the primary invagination. We now consider
the position of the division plane and its interplay with cell shape. As shown
in Fig. 3b, the distance of the width minimum from the stalked pole
($\ell^{st}(\phi)$) increases through the cell cycle at the same rate as the
full length of the growing cell ($\ell(\phi)$), such that their ratio remains
constant with time-averaged mean $\langle\ell^{st}/\ell\rangle=0.54\pm 0.05$.
The presence of the primary invagination early in the cell cycle is reiterated
in Fig. 3c, which shows the width profile constructed by ensemble-averaging
over each cell at the timepoint immediately following division. In addition to
the width minimum $w_{\text{min}}(\phi)$, there are two characteristic maxima
near either pole, $w^{st}_{\text{max}}(\phi)$ and $w^{sw}_{\text{max}}(\phi)$,
respectively (Fig. 3c, inset). As evident in Fig. 3c, the stalked pole
diameter $\langle w^{st}_{\text{max}}(\phi)\rangle$ is on average larger than
its swarmer counterpart $\langle w^{sw}_{\text{max}}(\phi)\rangle$ (also see
Supplementary Fig. 2a).

We show that the asymmetric location of the invagination (and the asymmetric
width profile) can originate from the distinct mechanical properties inherent
to the pole caps in C. crescentus. The shapes of the cell poles can be
explained by Laplace’s law that relates the pressure difference, $P$, across
the cell wall to the surface tensions in the stalked or the swarmer pole,
$\gamma_{p}^{st,sw}$. The radii of curvature of the poles then follow from
Laplace’s law

$R_{p}^{st,sw}=\frac{2\gamma_{p}^{st,sw}}{P}\;,$ (8)

where the superscript ($st,sw$) denotes the stalked or the swarmer pole. Thus
a larger radius of curvature in the poles has to be compensated by a higher
surface tension to maintain a constant pressure difference $P$. Assuming that
the poles form hemispheres, we have $R_{p}^{st,sw}=w_{\text{max}}^{st,sw}/2$.
Our data indicate that the early time ratio for
$w^{st}_{\text{max}}(\phi)/w^{sw}_{\text{max}}(\phi)$ ($\phi<0.1$) shows a
strong positive correlation with the ratio $\ell^{st}(\phi)/\ell^{sw}(\phi)$,
with an average value $\langle
w^{st}_{\text{max}}/w^{sw}_{\text{max}}\rangle\simeq 1.04$ (see Supplementary
Fig 3a). Laplace’s law then requires that the stalked pole be mechanically
stiffer than the swarmer pole; $\gamma_{p}^{st}>\gamma_{p}^{sw}$. This
observation suggests that the asymmetry in the lengths of the stalked and
swarmer parts of the cell depends upon different mechanical properties of the
respective poles.

To quantitatively support this claim, we investigate an effective contour
model for the cell shape. To this end, we assume that the fluctuations in cell
shape relax more rapidly than the time scale of growth. This separation of
timescales allows us to derive the equation governing the cell contour by
minimizing the total mechanical energy (equation (2)). From the solution we
compute the resultant width profile for the entire cell (see Supplementary
model section). As shown in Fig. 3d, the model with asymmetric surface
tensions of the poles causes the primary invagination to occur away from the
cell mid-plane. The spatial location of the invagination relative to the cell
length depends linearly on the ratio $\gamma_{p}^{st}/\gamma_{p}^{sw}$.
Symmetry is restored for $\gamma_{p}^{st}/\gamma_{p}^{sw}=1$, as shown in Fig.
3d (blue dashed curve). We note that a gradient in $\gamma$ along the cell
body would imply differences in longitudinal growth rates between the stalked
and the swarmer portions of the cell (Eq. (1)). Our data exclude this
possibility since both $\ell^{st}(\phi)$ and $\ell^{sw}(\phi)$ grow at the
same rate $\kappa$, 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
longitudinal growth rate. In addition, the non-uniformity in cell width comes
from the differences in mechanical response in the cell wall due to
preferential attachment of crescentin along the concave sidewall. For a creS
mutant cell (where $k_{c}=0$), our model predicts a uniform width profile
before the onset of constriction.

Figure 4: Comparison of experimental and model cell contours and width
profiles. (a) Splined contours of a growing and constricting cell at different
values of normalized time $\phi=0.0$ (red), $0.33$ (orange), $0.67$ (green)
and $1.0$ (blue). (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 $w_{\text{min}}$ and
$\ell$ corresponding to the time points in a. (d) Model width profiles
corresponding to the contours in c. Figure 5: Location of division plane is
set in the previous generation. (a) Value of the minimum width normalized by
the respective maximum width for the stalked ($\langle
w^{st}_{\text{min}}(\phi)/w^{st}_{\text{max}}(\phi)\rangle$) and the swarmer
($\langle w^{sw}_{\text{min}}(\phi)/w^{sw}_{\text{max}}(\phi)\rangle$) parts.
(b) Ratio of length from stalked pole to secondary minimum normalized by
length from stalked pole to primary minimum
($\langle\ell^{st}_{\text{min}}(\phi)/\ell^{st}(\phi)\rangle$, green) and
ratio of length from stalked pole to primary minimum normalized by total
length ($\langle\ell^{st}(\phi)/\ell(\phi)\rangle$, red). The former ratio
remains constant at $\simeq$ 0.55, while the latter obtains this value at the
end of the cell cycle. In comparing averages between two generations here, we
indicate values of $\phi$ from the first generation as negative (e.g.,
$\phi=-0.2$ is 20% of the way from the subsequent division). The green points
are not shown for $\phi<-0.25$ due to increased errors in identification. (c)
Kymograph of width profiles for a typical cell over two generations. The time
evolution of the widths (color scale) illustrates continuity of the location
of the minima across generations. That is, the location of the secondary
minimum just before division ($\ell^{st}_{\text{min}}$, white line) becomes
the primary minimum ($\ell^{st}$, black line) just after division (horizontal
dashed line). The schematic at right shows two measured contours that
correspond to time points immediately before ($\phi=-0.05$) and after
($\phi=+0.05$) the division event shown in the kymograph.

Cell shape evolution during wall constriction. The experimental width profiles
show that the growing and constricting cells typically develop a second
minimum in width (Fig. 4a,b). These secondary invaginations are observed in
both the stalked and swarmer portions of single cells in the predivisional
stage ($\phi>0.6$), although they are more common in the stalked portions
(Fig. 5a). We show here that these secondary minima become the primary minima
in each of the daughter cells. To study the dynamics of the development of the
secondary minimum we introduce a new quantity, $\ell^{st}_{\text{min}}(\phi)$,
defined as the distance from the stalked pole to the secondary minimum in the
stalked part (see Fig. 5c, inset). We find that the ratio
$\langle\ell^{st}_{\text{min}}(\phi)/\ell^{st}(\phi)\rangle$ has a mean value
of $\simeq$ 0.55 at later points in the cell cycle (Fig. 5b), equal to the
constant ratio maintained by the distance from the stalked pole to the primary
minimum, $\langle\ell^{st}(\phi)/\ell(\phi)\rangle$. In fact the kymograph of
width profiles (shown over 2 generations for a representative single cell) in
Fig. 5c demonstrates that the predivisional secondary invaginations are
inherited as primary invaginations after division. This mechanism provides
continuity and inheritance of the invaginations across generations and is an
intrinsic element of the mechanism for cell division in C. crescentus.

To quantitatively explain the experimental width profiles during constriction,
we use our mechanical model to determine the instantaneous cell shape by
minimizing the total energy (equation (2)) at the specified time points (see
Supplementary model section). To take constriction into account, we impose the
constraint that $w(\ell^{st},\phi)=w_{\text{min}}(\phi)$, where
$w_{\text{min}}(\phi)$ is determined by equations (S.13) and (7). In addition,
we assume non-uniform materials properties in the cell wall by taking the
tension in the cell poles ($\gamma_{\text{p}}^{st,sw}$) and the septal region
to be higher than the rest of the cell. As constriction proceeds and
$w_{\text{min}}(\phi)$ decreases, we compute the shape of the cell contours
(Fig. 4c) and the corresponding width profiles (Fig. 4d). The computed width
profiles faithfully reproduce the secondary invaginations, which become more
pronounced as the daughter pole caps become prominent. An example of the
experimental width profiles is shown in Fig. 4b at evenly-spaced intervals in
time for a single generation, and the corresponding model width profiles are
shown in Fig. 4d.

We note that the experimental cell contours in the predivisional stage
($\phi>0.9$) bend away from the initial midline axis and develop an alternate
growth direction (Fig. 4a, blue contour). These bend deformations are induced
by the microfluidic flow about the pinch-off plane; the cells become
increasingly “floppy” as the constriction proceeds.

## Discussion and Conclusions

The consistent propagation of a specific shape through the processes of growth
and division relies upon an intricate interplay between the controlled
spatiotemporal expression and localization of proteins, and cytoskeletal
structural elements. The high statistical precision of our measurements allows
us to gain new insights into cell morphology. From precise determination of
cell contours over time, we observe that a typical cell width profile is non-
uniform at all times with a pronounced primary invagination appearing during
the earliest stages of the cell cycle. During cell constriction, the decrease
in the minimum width is governed by the same time constant as exponential
axial growth (Fig. 3a). Furthermore, the location of the primary invagination
divides the cell contour into its stalked and swarmer compartments, such that
the ratio of the length of the stalked part $\ell^{st}(\phi)$ to the total
pole-to-pole length $\ell(\phi)$ remains constant during the cycle with a mean
value $\langle\ell^{st}(\phi)/\ell(\phi)\rangle\simeq 0.55$ (Fig. 3b). These
observations and our mechanical model lead to two important conclusions:
first, the dynamics of cell wall constriction and septal growth occur
concomitantly, and second, the asymmetric location of the primary invagination
can be explained by the differences in mechanical properties in the stalked
and swarmer poles. A corollary of the first conclusion is that the size ratio
threshold at division occurs naturally without requiring a complex timing
mechanism iyer-biswas2014b .

In addition to the primary septal invagination, the cell contours exhibit a
pronounced secondary invagination during the predivisional stages (Fig. 4).
Remarkably, the secondary invaginations develop at a precise location relative
to the total length of the stalked compartments,
$\langle\ell^{st}_{\text{min}}(\phi)/\ell^{st}(\phi)\rangle\simeq 0.55$ (Fig.
5b). The data thus allow a third conclusion: these secondary invaginations are
inherited as primary invaginations in each of the daughter cells, directing
the formation of the division plane in the next generation. Thus, through
consistent and controlled nucleation of invaginations across generations, C.
crescentus cells maintain a constant ratio of the sizes of stalked and swarmer
daughter cells.

Our experimental observations and the parameters in the cell shape model can
be related to the current molecular understanding for Gram-negative bacteria,
in particular C. crescentus. Before the onset of noticeable constriction, cell
shape is dictated by the mechanical properties of the peptidoglycan cell wall
in addition to various shape-controlling proteins such as MreB, MreC, RodZ and
CreS. Single molecule tracking studies have revealed that MreB forms short
filamentous bundles anchored to the inner surface of the cell wall and moves
circumferentially at a rate much faster than the rate of cell growth kim2006 ;
garner2011 . In vitro experiments show that MreB filaments can induce
indentation of lipid membranes, suggesting that they may have a preferred
radius of curvature salje2011 . Thus on time scales comparable to cell growth,
$E_{\text{width}}$ is determined in part by the energy cost of adhering MreB
bundles to the cell wall (see Supplementary model section).

Bacterial cell division is driven by a large complex of proteins, commonly
known as divisomes that assemble into the Z-ring structure near the
longitudinal mid-plane of the cell erickson2010 . The Z-ring contains FtsZ
protofilaments that are assembled in a patchy band-like structure holden2014 .
FtsZ protofilaments are anchored to the cell membrane via FtsA and ZipA, and
play a crucial role in driving cell wall constriction erickson1996 . During
constriction, the divisome proteins also control peptidoglycan synthesis and
direct the formation of new cell wall via the activity of penicillin-binding
proteins (PBPs) scheffers2004 ; adams2009 . Thus the divisome plays a two-fold
role by concomitantly guiding cell wall constriction and growth of the septal
peptidoglycan layer. According to our model the constriction of the cell wall
is driven by the synthesis of septal cell wall at a rate $\kappa_{d}$
($\sim\langle\kappa\rangle$), which can be directly related to the activity of
PBPs triggered by the divisome assembly. Furthermore, in our model it is
sufficient that the divisome guide the curvature of cell wall growth in the
septal region (see Fig. 3a, inset).

While the mechanism behind the precise asymmetric location of the division
plane in C. crescentus cells is not well understood, it is likely that the
ATPase MipZ helps division site placement by exhibiting an asymmetric
concentration gradient during the predivisional stage kiekebusch2012 . MipZ
activity inhibits FtsZ assembly; as a result of polar localization of MipZ,
Z-ring assembly is promoted near the mid-cell thanbichler2006 . Our cell shape
model suggests that the early time asymmetric location of the primary
invagination, which develops into the division plane, is controlled by the
differences in surface tensions maintained in the poles. The presence of this
invagination at $\phi=0$, as inherited from the secondary invaginations in the
previous generation, aids in Z-ring assembly at the site of the invagination.
The curvature-sensing capability of the Z-ring may be enabled by the
minimization of the FtsZ polymer conformational energy that is determined by
the difference between cell surface curvature and FtsZ spontaneous curvature
erickson2010 ; arumugam2012 .

A higher tension in the stalked pole can be induced by asymmetric localization
of polar proteins, such as PopZ, early in the cell cycle. Experiments have
shown that PopZ localizes to the stalked pole during the initial phase of the
cell cycle and increasingly accumulates at the swarmer pole as the cell cycle
proceeds bowman2010 . Consistent with this observation, our data show that the
correlation between the pole sizes (determined by the ratio of surface tension
to pressure) and the stalked and swarmer compartment lengths tend to disappear
later in the cycle (Supplementary Fig. 3), as cell constriction proceeds. A
recent experimental study also demonstrates that molecular perturbation of Clp
proteases can destroy the asymmetry of cell division in C. crescentus
williams2014 , suggesting the interplay of subcellular protease activity with
the physical properties of the cell wall.

Earlier theoretical models have predicted that a small amount of pinch-off
force from the Z-ring ($\sim 8$ pN) is sufficient to accomplish division by
establishing a direction along which new peptidoglycan strands can be inserted
lan2007 . In contrast, our data combined with the mathematical model allows
the interpretation that the early time asymmetric invagination in the cell
wall can set the direction for the insertion of new peptidoglycan strands.
Constriction results from exponential growth of surface area in the septum (at
the same rate as longitudinal extension). The instantaneous cell shape is
determined by minimizing the energy functional at given values of the cell
size parameters.

Finally, from our estimate of the cell wall energy density $U$ ($\simeq-5$
nN$\mu$m), we predict that a net amount $\Delta\theta|U|\simeq 2.5$ nN$\mu$m
of mechanical energy is used by the peptidoglycan network for cell wall
growth. For a C. crescentus cell of surface area $12.5-25\mu$m2, layered with
glycan strands of length $\sim$5 nm and cross-linked by peptide chains with
maximally stretched length $\sim$4 nm gan2008 , there are roughly $10^{6}$
peptidoglycan subunits. Thus on average, each peptidoglycan subunit can
consume mechanical energy of $\sim$2.4$\times 10^{-6}$ nN$\mu$m, or $\sim$0.6
$k_{B}T$ at a temperature $T=31^{\circ}$C. Cell wall remodeling and insertion
of new peptidoglycan material can likely create defects in the peptidoglycan
network amir2012 . One thus expects cellular materials properties to change
over time, as a result of these molecular scale fluctuations. Although we
neglect such variations in our mean field model, it nonetheless quantitatively
captures the average trends in cell shape features. In future work we plan to
more closely connect the energy terms of the continuum model with molecular
details.

## Methods

Acquisition of experimental data. Data were acquired as in Ref. iyer-
biswas2014b . Briefly, the inducibly-sticky Caulobacter crescentus strain
FC1428 was introduced into a microfluidic device and cells were incubated for
one hour in the presence of the vanillate inducer. The device was placed
inside a homemade acrylic microscope enclosure ($39^{\prime\prime}\times
28^{\prime\prime}\times 27^{\prime\prime}$) equilibrated to 31∘C (temperature
controller: CSC32J, Omega and heater fan: HGL419, Omega). At the start of the
experiment, complex medium (peptone-yeast extract; PYE) was infused through
the channel at a constant flow rate of 7 $\mu$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
Scientific ProScan III) under computerized control (LabView 8.6, National
Instrument) were used to acquire phase-contrast images at a magnification of
250X (EMCCD: Andor iXon+ DU888 1k $\times$ 1k pixels, objective: Nikon Plan
Fluor 100X oil objective plus 2.5X expander, lamp: Nikon C-HFGI) and a frame
rate of 1 frame/min for 15 unique fields of view over 48 hours. In the present
study we use a dataset consisting of 260 cells, corresponding to 9672
generations (division events).

Analysis of single cell shape. The acquired phase-contrast images were
analyzed using a novel routine we developed (written in Python). Each image
was processed with a pixel-based edge detection algorithm that applied a local
smoothing filter, followed by a bottom-hat operation. The boundary of each
cell was identified by thresholding the filtered image. A smoothing B-spline
was interpolated through the boundary pixels to construct each cell contour.
Each identified cell was then tracked over time to build a full time series.
We chose to include only cells that divided for more than 10 generations in
the analysis. A minimal amount of filtering was applied 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 division was
verified by visual inspection of the corresponding phase contrast images, so
that the error in this quantity is approximately set by the image acquisition
rate of 1 frame/min.

## References

## References

  * (1) Cabeen, M. T. & Jacobs-Wagner, C.  Bacterial cell shape.  _Nat Rev Microbiol_ 3, 601–610 (2005).
  * (2) Huang, S. & Ingber, D. E.  Shape-dependent control of cell growth, differentiation, and apoptosis: switching between attractors in cell regulatory networks.  _Exp Cell Res_ 261, 91–103 (2000).
  * (3) Laloux, G. & Jacobs-Wagner, C.  How do bacteria localize proteins to the cell pole?  _J Cell Sci_ 127, 11–19 (2014).
  * (4) Koch, A. L.  _Bacterial Growth and Form_ (Springer Science+Business Media, Dordrecht, 2001).
  * (5) Gan, L., Chen, S. & Jensen, G. J.  Molecular organization of Gram-negative peptidoglycan.  _Proc Natl Acad Sci USA_ 105, 18953–18957 (2008).
  * (6) Gahlmann, A. & Moerner, W.  Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging.  _Nat Rev Microbiol_ 12, 9–22 (2014).
  * (7) Wachi, M. _et al._ Mutant isolation and molecular cloning of mre genes, which determine cell shape, sensitivity to mecillinam, and amount of penicillin-binding proteins in Escherichia coli.  _J Bacteriol_ 169, 4935–4940 (1987).
  * (8) Iwai, N., Nagai, K. & Wachi, M.  Novel S-benzylisothiourea compound that induces spherical cells in Escherichia coli probably by acting on a rod-shape-determining protein(s) other than penicillin-binding protein 2.  _Biosci Biotech Bioch_ 66, 2658–2662 (2002).
  * (9) Jiang, H., Si, F., Margolin, W. & Sun, S. X.  Mechanical control of bacterial cell shape.  _Biophys J_ 101, 327–335 (2011).
  * (10) Figge, R. M., Divakaruni, A. V. & Gober, J. W.  MreB, the cell shape-determining bacterial actin homologue, co-ordinates cell wall morphogenesis in Caulobacter crescentus.  _Mol Microbiol_ 51, 1321–1332 (2004).
  * (11) Ausmees, N., Kuhn, J. R. & Jacobs-Wagner, C.  The bacterial cytoskeleton: an intermediate filament-like function in cell shape.  _Cell_ 115, 705–713 (2003).
  * (12) Cabeen, M. T. _et al._ Bacterial cell curvature through mechanical control of cell growth.  _EMBO J_ 28, 1208–1219 (2009).
  * (13) Erickson, H. P., Anderson, D. E. & Osawa, M.  FtsZ in bacterial cytokinesis: cytoskeleton and force generator all in one.  _Microbiol Mol Biol R_ 74, 504–528 (2010).
  * (14) Wang, P. _et al._ Robust growth of Escherichia coli.  _Curr Biol_ 20, 1099–1103 (2010).
  * (15) Moffitt, J. R., Lee, J. B. & Cluzel, P.  The single-cell chemostat: an agarose-based, microfluidic device for high-throughput, single-cell studies of bacteria and bacterial communities.  _Lab Chip_ 12, 1487–1494 (2012).
  * (16) Shaw, J., Payer, K., Son, S., Grover, W. H. & Manalis, S. R.  A microfluidic “baby machine” for cell synchronization.  _Lab Chip_ 12, 2656–2663 (2012).
  * (17) Long, Z. _et al._ Microfluidic chemostat for measuring single cell dynamics in bacteria.  _Lab Chip_ 13, 947–954 (2013).
  * (18) Campos, M. _et al._ A constant size extension drives bacterial cell size homeostasis.  _Cell_ 159, 1433–1446 (2014).
  * (19) Iyer-Biswas, S. _et al._ Scaling laws governing stochastic growth and division of single bacterial cells.  _Proc Natl Acad Sci USA_ 111, 15912–15917 (2014).
  * (20) Siegal-Gaskins, D. & Crosson, S.  Tightly regulated and heritable division control in single bacterial cells.  _Biophys J_ 95, 2063–2072 (2009).
  * (21) Lin, Y., Crosson, S. & Scherer, N. F.  Single-gene tuning of Caulobacter cell cycle period and noise, swarming motility, and surface adhesion.  _Mol Syst Biol_ 6, 1–13 (2010).
  * (22) Lin, Y., Li, Y., Crosson, S., Dinner, A. R. & Scherer, N. F.  Phase resetting reveals network dynamics underlying a bacterial cell cycle.  _PLoS Comput Biol_ 8, e1002778 (2012).
  * (23) Iyer-Biswas, S., Crooks, G. E., Scherer, N. F. & Dinner, A. R.  Universality in stochastic exponential growth.  _Phys Rev Lett_ 113, 028101 (2014).
  * (24) Pincus, Z. & Theriot, J. A.  Comparison of quantitative methods for cell-shape analysis.  _J Microsc_ 227, 140–156 (2007).
  * (25) Sliusarenko, O., Heinritz, J., Emonet, T. & Jacobs-Wagner, C.  High-throughput, subpixel precision analysis of bacterial morphogenesis and intracellular spatio-temporal dynamics.  _Mol Microbiol_ 80, 612–627 (2011).
  * (26) Guberman, J. M., Fay, A., Dworkin, J., Wingreen, N. S. & Gitai, Z.  PSICIC: noise and asymmetry in bacterial division revealed by computational image analysis at sub-pixel resolution.  _PLoS Comput Biol_ 4, e1000233 (2008).
  * (27) Lan, G., Wolgemuth, C. W. & Sun, S. X.  Z-ring force and cell shape during division in rod-like bacteria.  _Proc Natl Acad Sci USA_ 104, 16110–16115 (2007).
  * (28) Jiang, H. & Sun, S. X.  Morphology, growth, and size limit of bacterial cells.  _Phys Rev Lett_ 105, 028101 (2010).
  * (29) Batchelor, G. K.  _An Introduction to Fluid Dynamics_ (Cambridge University Press, Cambridge, 2000).
  * (30) Rojas, E., Theriot, J. A. & Huang, K. C.  Response of Escherichia coli growth rate to osmotic shock.  _Proc Natl Acad Sci USA_ 111, 7807–7812 (2014).
  * (31) Jiang, H. & Sun, S. X.  Growth of curved and helical bacterial cells.  _Soft Matter_ 8, 7446–7451 (2012).
  * (32) Boudaoud, A.  Growth of walled cells: from shells to vesicles.  _Phys Rev Lett_ 91, 018104 (2003).
  * (33) Sun, S. X. & Jiang, H.  Physics of bacterial morphogenesis.  _Microbiol Mol Biol R_ 75, 543–565 (2011).
  * (34) Sliusarenko, O., Cabeen, M. T., Wolgemuth, C. W., Jacobs-Wagner, C. & Emonet, T.  Processivity of peptidoglycan synthesis provides a built-in mechanism for the robustness of straight-rod cell morphology.  _Proc Natl Acad Sci USA_ 107, 10086–10091 (2010).
  * (35) Jiang, H., Si, F., Margolin, W. & Sun, S. X.  Mechanical control of bacterial cell shape.  _Biophys J_ 101, 327–335 (2011).
  * (36) Typas, A., Banzhaf, M., Gross, C. A. & Vollmer, W.  From the regulation of peptidoglycan synthesis to bacterial growth and morphology.  _Nat Rev Microbiol_ 10, 123–136 (2012).
  * (37) Reed, R. & Walsby, A.  Changes in turgor pressure in response to increases in external NaCl concentration in the gas-vacuolate cyanobacterium Microcystis sp.  _Arch Microbiol_ 143, 290–296 (1985).
  * (38) Koch, A. L. & Pinette, M. F.  Nephelometric determination of turgor pressure in growing gram-negative bacteria.  _J Bacteriol_ 169, 3654–3663 (1987).
  * (39) Deng, Y., Sun, M. & Shaevitz, J. W.  Direct measurement of cell wall stress stiffening and turgor pressure in live bacterial cells.  _Phys Rev Lett_ 107, 158101 (2011).
  * (40) Charbon, G., Cabeen, M. T. & Jacobs-Wagner, C.  Bacterial intermediate filaments: in vivo assembly, organization, and dynamics of crescentin.  _Gene Dev_ 23, 1131–1144 (2009).
  * (41) van den Ent, F., Amos, L. A. & Löwe, J.  Prokaryotic origin of the actin cytoskeleton.  _Nature_ 413, 39–44 (2001).
  * (42) Fiebig, A., Castro Rojas, C. M., Siegal-Gaskins, D. & Crosson, S.  Interaction specificity, toxicity and regulation of a paralogous set of ParE/RelE-family toxin–antitoxin systems.  _Mol Microbiol_ 77, 236–251 (2010).
  * (43) Reshes, G., Vanounou, S., Fishov, I. & Feingold, M.  Cell shape dynamics in Escherichia coli.  _Biophys J_ 94, 251–264 (2008).
  * (44) Kim, S. Y., Gitai, Z., Kinkhabwala, A., Shapiro, L. & Moerner, W.  Single molecules of the bacterial actin MreB undergo directed treadmilling motion in Caulobacter crescentus.  _Proc Natl Acad Sci USA_ 103, 10929–10934 (2006).
  * (45) Garner, E. C. _et al._ Coupled, circumferential motions of the cell wall synthesis machinery and MreB filaments in B. subtilis.  _Science_ 333, 222–225 (2011).
  * (46) Salje, J., van den Ent, F., de Boer, P. & Löwe, J.  Direct membrane binding by bacterial actin MreB.  _Mol Cell_ 43, 478–487 (2011).
  * (47) Holden, S. J. _et al._ High throughput 3D super-resolution microscopy reveals Caulobacter crescentus in vivo Z-ring organization.  _Proc Natl Acad Sci USA_ 201313368 (2014).
  * (48) Erickson, H. P., Taylor, D. W., Taylor, K. A. & Bramhill, D.  Bacterial cell division protein FtsZ assembles into protofilament sheets and minirings, structural homologs of tubulin polymers.  _Proc Natl Acad Sci USA_ 93, 519–523 (1996).
  * (49) Scheffers, D.-J., Jones, L. J. & Errington, J.  Several distinct localization patterns for penicillin-binding proteins in Bacillus subtilis.  _Mol Microbiol_ 51, 749–764 (2004).
  * (50) Adams, D. W. & Errington, J.  Bacterial cell division: assembly, maintenance and disassembly of the Z ring.  _Nat Rev Microbiol_ 7, 642–653 (2009).
  * (51) Kiekebusch, D., Michie, K. A., Essen, L.-O., Löwe, J. & Thanbichler, M.  Localized dimerization and nucleoid binding drive gradient formation by the bacterial cell division inhibitor MipZ.  _Mol Cell_ 46, 245–259 (2012).
  * (52) Thanbichler, M. & Shapiro, L.  MipZ, a spatial regulator coordinating chromosome segregation with cell division in Caulobacter.  _Cell_ 126, 147–162 (2006).
  * (53) Arumugam, S. _et al._ Surface topology engineering of membranes for the mechanical investigation of the tubulin homologue FtsZ.  _Angew Chem Int Ed_ 51, 11858–11862 (2012).
  * (54) Bowman, G. R. _et al._ Caulobacter PopZ forms a polar subdomain dictating sequential changes in pole composition and function.  _Mol Microbiol_ 76, 173–189 (2010).
  * (55) Williams, B., Bhat, N., Chien, P. & Shapiro, L.  ClpXP and ClpAP proteolytic activity on divisome substrates is differentially regulated following the Caulobacter asymmetric cell division.  _Mol Microbiol_ 93, 853–866 (2014).
  * (56) Amir, A. & Nelson, D. R.  Dislocation-mediated growth of bacterial cell walls.  _Proc Natl Acad Sci USA_ 109, 9833–9838 (2012).

## Acknowledgments

We gratefully acknowledge funding from NSF Physics of Living Systems (NSF
PHY-1305542), NSF Materials Research Science and Engineering Center (MRSEC) at
the University of Chicago (NSF DMR-1420709), the W. M. Keck Foundation and the
Graduate Program in Biophysical Sciences at the University of Chicago (T32
EB009412/EB/NIBIB NIH HHS/United States). N.F.S. also thanks the Office of
Naval Research (ONR) for a National Security Science and Engineering Faculty
Fellowship (NSSEFF).

## Author contributions

C.S.W., S.I.B., A.R.D., and N.F.S. designed the experiments; S.B., A.R.D., and
N.F.S. designed the model; C.S.W. and S.I.B. performed the experiments and
observed the phenomena reported; C.S.W. designed and implemented custom
software to automate cell shape image analysis; C.S.W. and S.B. analyzed the
data; S.C. contributed reagents and materials; C.S.W., S.B., A.R.D., and
N.F.S. wrote the manuscript; all authors discussed the results and commented
on the manuscript.

## Competing financial interests

The authors declare no competing financial interests.

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 cell
contour [Sbrandt1994, ], pruning the branches (Supplementary Fig. 1b), and
extending this skeleton to the edges of the cell contour in a manner that
preserves average curvature of the medial axis (Supplementary Fig. 1c). The
intersections between the cell medial axis and contour represent the stalked
and swarmer poles, respectively. The cell length was calculated by evaluating
the distance along the medial axis between either pole. The cell widths were
determined by creating ribs perpendicular to the medial axis along its length
and determining the distances between their intersections with opposite sides
of the cell contour (Supplementary Fig. 1d). To calculate time-averaged
quantities, we normalized trajectories for each generation by the respective
division time $\tau$ (thus converting each variable to a function of
$\phi\equiv t/\tau$). We then split these data into 73 bins, as
$\langle\tau\rangle=73\pm 7$ min under these conditions, and ensemble-averaged
each of these bins over every generation.

Supplementary Figure 1: Procedure for calculating shape parameters. (a)
B-spline cell contour extracted from segmented phase contrast image. (b) The
interior part of the medial cell axis was obtained by taking the Voronoi
diagram of the cell contour, which is equivalent to the locus of inscribed
circles. (c) The medial axis located on each side of the width minimum
(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 ($w_{\text{min}}$, $w^{st}_{\text{max}}$,
$w^{sw}_{\text{max}}$, $w^{st}_{\text{min}}$, $w^{sw}_{\text{min}}$), not all
of which may be present in any given cell. Where we identify a primary minimum
in the width profile, $w_{\text{min}}$, we can also determine two local
maxima, $w^{st}_{\text{max}}$ and $w^{sw}_{\text{max}}$, corresponding
respectively to the stalked and swarmer portions (Supplementary Fig. 2a).
These values are approximately constant throughout the cell cycle, with
$\langle w^{st}_{\text{max}}(\phi)\rangle>\langle
w^{sw}_{\text{max}}(\phi)\rangle$. However, at later times ($\phi>0.6$) the
value of $w^{sw}_{\text{max}}(\phi)$ increases until $\langle
w^{st}_{\text{max}}(\phi)\rangle\approx\langle
w^{sw}_{\text{max}}(\phi)\rangle$. In some cases, additional secondary local
minima are observed, $w^{st}_{min}$ and $w^{sw}_{min}$, corresponding
respectively to the stalked and swarmer portions (Supplementary Fig. 2b).
Although we note the value of these quantities for early times here (where
they are approximately equal to their respective local maxima), these minima
can only be determined with certainty at later times ($\phi>0.6$). There, we
observe the presence of a secondary minimum in the stalked portions of most
cells.

|  mean $\pm$ S.D. |  | $\phi$
---|---|---|---
$\bar{w}$ |  0.74 $\pm$ 0.02 |  $\mu$m | $\left[0,0.5\right]$
$w^{st}_{\text{max}}$ |  0.85 $\pm$ 0.02 |  $\mu$m | $\left[0,0.5\right]$
$w^{st}_{\text{min}}$ |  0.84 $\pm$ 0.03 |  $\mu$m | $\left[0,0.5\right]$
$w^{sw}_{\text{max}}$ |  0.83 $\pm$ 0.02 |  $\mu$m | $\left[0,0.5\right]$
$w^{sw}_{\text{min}}$ |  0.82 $\pm$ 0.02 |  $\mu$m | $\left[0,0.5\right]$
$R$ |  4.44 $\pm$ 2.12 |  $\mu$m | $\left[0,0.5\right]$
$R^{st}$ |  4.34 $\pm$ 2.50 |  $\mu$m | $\left[0,0.5\right]$
$R^{sw}$ |  3.94 $\pm$ 2.42 |  $\mu$m | $\left[0,0.5\right]$
$\ell^{st}/\ell$ |  54.3 $\pm$ 5.2 |  % | $\left[0,1\right]$
$\ell^{sw}/\ell$ |  45.7 $\pm$ 5.2 |  % | $\left[0,1\right]$
Supplementary Table 1: Average values of the cell shape parameters that are
assumed constant over the specified time interval of $\phi$. The time-average
value of the width $\langle\bar{w}\rangle$ is smaller than both $\langle
w^{st}_{\text{min}}\rangle$ and $\langle w^{sw}_{\text{min}}\rangle$ because
it accounts for the width at the primary minimum $w_{\text{min}}$. Note the
close correspondence between either width maximum and its corresponding
minimum for $\phi<0.6$, as well as $w^{st}_{\text{max}}$ and
$\ell^{st}_{\text{min}}$. Both width and radius of curvature for the swarmer
portion are smaller 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 width profile. The radius of curvature ($R$) was
calculated as the radius of the best-fit circle to the cell medial axis. We
tested two methods of determining the radius of curvature: (I) fitting the
whole cell to a circle or (II) fitting the stalked and swarmer 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 have
differing radii of curvature, which we attribute to the alignment of the
swarmer portion of the cell with the direction of fluid flow after the
division plane has narrowed enough that it becomes mechanically decoupled from
the stalked portion, 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 define a
reasonable arithmetic mean without arbitrarily filtering the dataset, we first
averaged the corresponding unsigned curvature (equivalent to $\langle
R^{-1}\rangle$) and then converted to radius of curvature (i.e., what we
report as $\langle R\rangle$ is actually the harmonic mean, calculated as
$\langle R^{-1}\rangle^{-1}$).

Supplementary Figure 2: 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)
Radius of 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 correspond to Fig. 5b in the main text. Superscript
$*$ refers to either $st$ or $sw$, respectively, as indicated in each figure
legend.

The length was also split according to the locations of the local minima, into
$\ell^{st}$ (distance along cell medial axis from stalked pole to
$w_{\text{min}}$), $\ell^{sw}$ (distance along cell medial axis from swarmer
pole to $w_{\text{min}}$), $\ell^{st}_{\text{min}}$ (distance along cell
medial axis from stalked pole to $w^{st}_{\text{min}}$), and
$\ell^{sw}_{\text{min}}$ (distance along cell medial axis from swarmer pole to
$w^{sw}_{\text{min}}$). The value of $\ell^{st}$ and $\ell^{st}_{\text{min}}$
are compared in Supplementary Fig. 2d. Note that the length of the stalked
portion $\ell^{st}$ grows exponentially, with the same time constant as the
length $\ell$ (as does $\ell^{sw}$, 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 the location of
$w_{\text{min}}$ is set at early times. At later times ($\phi>0.6$), the
length from stalked pole to secondary minimum $\ell^{st}_{\text{min}}$ also
starts to increase.

We relate the asymmetry in the length of stalked and swarmer portions at early
times to asymmetries in the stalked and swarmer poles. The model predicts a
linear relationship between the ratio of lengths $\ell^{st}/\ell^{sw}$ and the
ratio of the tensions at either pole $\gamma^{st}_{p}/\gamma^{sw}_{p}$. We
cannot directly measure the latter quantity, but from Laplace’s law it is
equal to the inverse ratio of the mean curvatures at either pole, which we
approximate as the inverse of half the maximum pole width, assuming that the
poles are hemispheres with diameter $w^{st}_{\text{max}}$ and
$w^{sw}_{\text{max}}$, respectively. Supplementary Fig. 3 shows scatter plots
comparing $w^{st}_{\text{max}}/w^{sw}_{\text{max}}$ to $\ell^{st}/\ell^{sw}$
for three different time intervals. The red best-fit line is shown only for
Supplementary Fig. 3a, for which $R^{2}=0.15$ (linear fits to all other plots
produced values of $R^{2}<0.05$). This line runs from the point $(1.0,1.01)$,
corresponding to the dashed blue curve in Fig. 3c of the main text (which is
the case of a symmetric cell) to $(1.2,1.04)$, corresponding to the red solid
curve in Fig. 3d of the main text (which is the case of the average asymmetric
C. crescentus cell). Note that at times $\phi>0.6$, 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 perfused in complex medium at 31∘C at a
frame rate of 5 frames per second (300 times faster than the frame rate used
to acquire 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
the cell contour between subsequent frames to be 12 nm. In addition, we found
the RMSD of equivalent points along the width profile between subsequent
frames to be 28 nm, or a 3.2% pinch depth at the average value of
$w^{st}_{\text{max}}$ = 0.85 $\mu$m.

## Cell Shape Model

Supplementary Figure 3: 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% of
the generation, and (c) last 10% of the generation. Blue points show the
average value of $w^{st}_{\text{max}}/w^{sw}_{\text{max}}$ calculated by
binning $\ell^{st}/\ell^{sw}$ into bins of width 0.05, with size of each error
bar showing the standard deviation of the values within that bin. The best-fit
line for $1\leq\ell^{st}/\ell^{sw}\leq 1.2$ is indicated in red (coefficient
of determination $R^{2}=0.15$). This line runs from $(1.0,1.01)$ to
$(1.2,1.04)$.

Cell Wall Mechanics. The total energy $E$ for the bacterial cell wall is given
as the sum of contributions from an active internal pressure $P$ driving cell
volume ($V$) expansion, mechanical energy $E_{\text{wall}}$ in the cell wall,
and the mechanical energy of interactions with cytoskeletal proteins
$E_{\text{proteins}}$:

$E=-PV+E_{\text{wall}}+E_{\text{proteins}}\;.$ (S.1)

The bacterial cell wall consists of a network of glycan strands cross-linked
by peptide chains know as the peptidoglycan network. Growth occurs via the
insertion of new peptidoglycan strands into the existing network along with
the breaking of existing bonds due to turgor pressure induced stretching. We
assume that elastic equilibrium is reached rapidly as compared to the rate of
synthesis of new material [Sgoriely2008, ]. As a result of cell wall
remodeling and irreversible elongation, growth can be understood as resulting
from plastic deformations [Scosgrove1985, ; koch2001, ; boudaoud2003, ]. To
understand the origin of the cell wall tension, $\gamma$, in the model
introduced in the main text, we consider the cell wall as a thin elastic shell
that deforms plastically when stretched beyond a maximum strain
$\varepsilon_{Y}$, the yield strain. A thin shell has two modes of elastic
deformations, bending and stretching [Slandau1959, ], such that
$E_{\text{wall}}=E_{\text{stretch}}+E_{\text{bend}}$. In the limit of small
thickness of the shell $h$ as compared to its radii of curvature, one can
neglect the bending energy (that scales as $E_{\text{bend}}\sim h^{3}$)
whereas the stretching energy $E_{\text{stetch}}$ is given by,

$E_{\text{stretch}}=\frac{h}{2}\int dA\ \sigma_{ij}\varepsilon_{ij}$ (S.2)

where $\sigma_{ij}$ is the mechanical stress tensor and $\varepsilon_{ij}$ is
the strain tensor. As yield strain is reached at the onset of growth, we have
$\varepsilon_{ij}=\varepsilon_{Y}\delta_{ij}$ (assuming isotropic stretching),
where $\delta_{ij}$ is the Kronecker delta. Furthermore, assuming a Hookean
constitutive relation for the stress tensor [Slandau1959, ]

$\sigma_{ij}=\frac{Y}{1-\nu^{2}}\left[(1-\nu)\varepsilon_{ij}+\nu\varepsilon_{kk}\delta_{ij}\right]\;,$
(S.3)

where $Y$ is the Young’s modulus and $\nu$ is the Poisson ratio, we have
$E_{\text{stretch}}=\gamma A$, where,

$\gamma=\frac{1}{2}\sigma_{ij}\varepsilon_{ij}=\frac{Yh\varepsilon_{Y}^{2}}{(1-\nu)}\;.$
(S.4)

For a Gram-negative bacterial cell wall of thickness $h\sim 3$ nm, elastic
modulus $Y\sim 40$ MPa [Sboulbitch2000, ] and average yield strain
$\varepsilon_{Y}\sim 0.5$, the wall tension is estimated to be $\gamma=50$
nN$/\mu$m. While the actual value for the turgor pressure counteracting this
tension can be contested, our choice for the numerical value for internal
pressure $P$ can be justified using a simple mechanical argument. Radial
force-balance dictates that in order to maintain an average cross-sectional
radius $r\simeq 0.4$ $\mu$m, a cell wall with surface tension $\gamma\sim$ 50
nm/$\mu$m has to balance an internal pressure of magnitude $P=2\gamma/r\simeq$
0.25 MPa, which is numerically very close to our choice for $P=0.3$ MPa.

Cytoskeletal bundle mechanics. Next, we model the mechanical energy in the
cell wall due to interactions with cytoskeletal proteins. Our minimalist
approach considers two crucial protein bundles that are directly responsible
for maintaining the shape of C. crescentus cells. MreB protein bundles
contribute an effective energy $E_{\text{width}}$, which favors a rod-like
shape, and crescentin filament bundles contribute an energy $E_{\text{cres}}$,
which favors a crescent-like shape. We thus have
$E_{\text{proteins}}=E_{\text{width}}+E_{\text{cres}}$. MreB subunits form
patchy filamentous bundles adherent to the cell wall and oriented
perpendicular to the long axis of the cell [Sgarner2011, ; gahlmann2014, ].
The elastic energy stored in an adherent MreB subunit is given by

$E_{\text{MreB},i}=\frac{k_{\text{b}}}{2}\ell_{i}\left(\frac{1}{r_{i}}-\frac{1}{R_{m}}\right)^{2}\;,$
(S.5)

where $i$ labels the subunit, $r_{i}$ is the circumferential radius of
curvature of the cell wall where the subunit is attached, $\ell_{i}$ is the
length of the subunit, $R_{m}$ is the intrinsic radius of curvature and
$k_{b}$ is the bending rigidity of the associated MreB bundle. The total
energy imparted by a collection of $N_{m}$ attached MreB subunits is given by
$E_{\text{width}}=\sum_{i}^{N_{m}}E_{\text{MreB},i}$. Next, employing a
continuum mean field assumption, we replace individual subunit lengths by
their average length $\langle\ell\rangle\simeq 5$ nm [Scolavin2014, ] and
assume a uniform number density $\rho_{m}$ of MreB subunits in the cell
surface to obtain

$E_{\text{width}}=\frac{k_{\text{m}}}{2}\int
dA\left(\frac{1}{r}-\frac{1}{R_{m}}\right)^{2}\;,$ (S.6)

where, $k_{m}=k_{b}\rho_{m}\langle\ell\rangle$ is the effective bending
modulus due to MreB induced traction forces. The bending rigidity of an MreB
bundle is given by $k_{b}=k_{\text{MreB}}n^{\xi}$, where $k_{\text{MreB}}$ is
the flexural rigidity of MreB filaments (assumed to be similar to F-actin),
$n$ is the number of MreB protofilaments per bundle, and $\xi$ is an exponent
in the range 1–2 depending on the strength of crosslinking or bundling agents
[Sbathe2008, ]. MreB filaments in a bundle appear to have strong lateral
interaction with negligible filament sliding, so we assume $\xi\simeq 2$. The
diameter of an MreB protofilament is $\sim 4$ nm [Svan2001, ] and the average
width of an MreB bundle has been determined from super-resolution imaging to
be in the range 60-90 nm [Sreimold2013, ], giving the estimate $n\sim$ 15-22.
Estimating the surface number density of MreB subunits as $\rho_{m}\sim
5\times 10^{4}$ $\mu$m-2 and using $k_{\text{mreb}}\simeq 10^{-4}$ nN$\mu$m2
[Sgittes1993, ], we obtain $k_{m}$ in the range 5.6–12.1 nN$\mu$m.

Crescentin proteins form a cohesive bundled structure anchored to the sidewall
of C. crescentus cells. The energetic contribution due to crescentin is given
by

$E_{\text{cres}}=\frac{k_{c}}{2}\int
ds\left(c(s)-\frac{1}{R_{c}}\right)^{2}\;,$ (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 $w_{c}$) given by
$I=\pi w_{c}^{4}/64$. Since crescentin is an intermediate filament homologue,
we assume that $Y_{c}$ is similar to the Young’s modulus of intermediate
filament bundles given by $Y_{c}\sim 300$ MPa [Sguzman2006, ]. Assuming
$w_{c}\sim 0.1$ $\mu$m, we estimate the bending rigidity of a crescentin
bundle to be $k_{c}\sim 1.5$ nN$\mu$m2.

Mean field model for cell shape and size dynamics. As described in the main
text, the total mechanical energy in the cell wall of C. crescentus cells can
be given as a sum of contributions from internal pressure, wall surface
tension, mechanical energy of interactions with bundles of cytoskeletal
proteins such as crescentin and MreB, and the constriction energy
$E_{\text{div}}$ during cell division. In the mean field description, we
neglect the contributions from $E_{\text{div}}$ and disregard any spatial
variations in cell geometry. The mean field description is a good
approximation of the cell shape dynamics for $\phi<0.5$, where the average
width and the midline radius of curvature remains constant (Fig. 2 in the 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 $\bar{w}(t)$ and
spanning angle $\theta(t)$. We also neglect the pole caps that are
mechanically rigid and do not remodel during wall growth. The total energy is
then given by

$E[\bar{w},R,\theta]=-P(\pi\bar{w}^{2}R\theta)/4+\gamma(\pi
R\bar{w}\theta)+E_{\text{width}}+E_{\text{cres}}\;,$ (S.8)

where

$E_{\text{width}}=\frac{k_{m}}{2}(\pi
Rw\theta)\left[(\bar{w}/2)^{-1}-R_{m}^{-1}\right]^{2}\;,$ (S.9)
$E_{\text{cres}}=\frac{k_{c}}{2}(R-\bar{w}/2)\theta\left[(R-\bar{w}/2)^{-1}-R_{c}^{-1}\right]^{2}\;.$
(S.10)

From the above expressions, we see that the total energy has the scaling form
$E[\bar{w},R,\theta]=\theta U[\bar{w},R]$. According to Eq. (1) in the main
text, the dynamics of the radius of curvature $R$, the spanning angle $\theta$
and the width $w$ are given by

$\displaystyle\frac{1}{R}\frac{dR}{dt}=-\Phi_{R}\frac{\partial E}{\partial
R}\;,$ (S.11a)
$\displaystyle\frac{1}{\bar{w}}\frac{d\bar{w}}{dt}=-\Phi_{w}\frac{\partial
E}{\partial\bar{w}}\;,$ (S.11b)
$\displaystyle\frac{1}{\theta}\frac{d\theta}{dt}=-\Phi_{\theta}\frac{\partial
E}{\partial\theta}\;,$ (S.11c)

where $\Phi_{R}$, $\Phi_{\theta}$ and $\Phi_{w}$ are the rate constants. The
steady-state values for the radius of curvature of the centerline and the
cross-sectional width is given by the solutions to $\partial_{R}U=0$ and
$\partial_{w}U=0$. As shown in Supplementary Fig. 4a the energy density $U$
admits a stable absolute minimum in width, controlled by the stiffness
parameter $k_{m}$ such that for lower values of $k_{m}$ (red curve in
Supplementary Fig. 4a), $U$ does not have a minimum and the width grows in
time rendering the rod-like shape unstable. Similarly, the parameter $k_{c}$
controls cell curvature, such that the energy density has a stable absolute
minimum in $R$ beyond a critical stiffness $k_{c}$ (Supplementary Fig. 4b). At
lower values of $k_{c}$ the cell does not have a stable radius of curvature
and the energy is minimized by increasing $R$. This leads to straightening of
the cell’s medial axis. In the parameter range where the cell maintains a
stable value for $\bar{w}$ and $R$, the energy density $U$ becomes a constant
during cell elongation. From the growth law introduced in the main text, the
spanning angle $\theta$ evolves in time according to the equation

$\frac{d\theta}{dt}=-(\Phi_{\theta}U)\theta(t)\;,$ (S.12)

such that the condition for exponential growth becomes $U<0$.

Supplementary Figure 4: Dependence of the mechanical energy on cell width and
radius of curvature. (a) Energy density $U$ as a function of mean cell width
$\bar{w}$ at various values of bending stiffness (in units of $nN\mu$m):
$k_{m}=30$ (red), $k_{m}=40$ (green) and $k_{m}=50$ (blue), for a fixed radius
of curvature $R=4.4\ \mu$m and $k_{c}=2$ nN$\mu$m2. The absolute minima (if it
exists) are indicated by solid circles. (b) $U$ as a function of radius of
curvature $R$ at various values of crescentin stiffness (in units of
$nN\mu$m2): $k_{c}=1.9$ (red), $k_{c}=2.0$ (green) and $k_{c}=2.1$ (blue), for
a fixed width $\bar{w}=0.74\ \mu$m and $k_{m}=40$ nN$\mu$m. Other parameters:
$P=0.3$ MPa, $\gamma=50$ nN/$\mu$m, $R_{c}=0.5$ $\mu$m, $R_{m}=0.31$ $\mu$m.

Constriction dynamics. As described in the main text, we assume that the shape
of the constriction zone is given by two intersecting hemishperical segments
with diameter $w_{\text{max}}$. The total surface area of the septum is given
by $S(t)=\pi w_{\text{max}}\ell_{s}(t)$, where $\ell_{s}(t)$ is the total
length of the spherical segments (see Supplementary Fig. 5a). Using elementary
geometry, we obtain the following relation between $\ell_{s}(t)$ and the
minimum width, $w_{\text{min}}(t)$:

$w_{\text{min}}(t)=w_{\text{max}}\sqrt{1-(\ell_{s}(t)/w_{\text{max}})^{2}}\;.$
(S.13)

If septal growth occurs by addition of new peptidoglycan strands while
maintaining the curvature of the spherical segments, then $\ell_{s}(t)$ is the
shape variable controlling the growth of septal surface area $S$ as well as
the constriction dynamics of $w_{\text{min}}(t)$. To describe the dynamics of
$\ell_{s}(t)$, we assume that initial phase of elongation occurs with a
velocity $v_{0}$, and thereafter growth follows an exponential law with rate
$\kappa_{d}$,

$\frac{d\ell_{s}}{dt}=v_{0}+\kappa_{d}\ell_{s}(t)\;.$ (S.14)

Multiplying both sides of the equation by $\pi w_{\text{max}}$, we derive the
growth dynamics of the surface area S(t),

$\frac{dS}{dt}=\kappa_{0}+\kappa_{d}S(t)\;,$ (S.15)

where $\kappa_{0}=\pi w_{\text{max}}v_{0}$. Having determined the dynamics of
$w_{\text{min}}$, one can evaluate the time-dependence of the average width
$\bar{w}$ defined as,
$\bar{w}(t)=\frac{1}{\ell(t)}\int_{0}^{\ell(t)}w(u,t)du$, where $u$ is the
coordinate along the centerline. Using the simplified geometry of the
constricting cell, given by two toroidal segments (peripheral regions)
connected by two intersecting hemispheres (septal region), one gets,

$\bar{w}(t)=\frac{1}{l(t)}\left[\bar{w}(\ell(t)-\ell_{s}(t))+2\int_{0}^{\ell_{s}(t)/2}w_{s}(u,t)du\right]\;,$
(S.16)

where $w_{s}(u)=w_{\text{max}}\sqrt{1-(2u/w_{\text{max}})^{2}}$, is the width
in the septal region and $\bar{w}$ is the average width of the stalked and
swarmer components as determined by Eq. (S.11b). As shown in Fig. 2c in the
main text (blue solid line), Eq. (S.16) is in excellent agreement with the
experimental data and captures the dip in $\bar{w}(t)$ seen for $\phi>0.5$.
Fig. 2b in the main text also shows that the radius of curvature of the
centerline ($R(t)$) drops for $\phi>0.5$ when cell constriction is prominent.
Therefore we examine the role of constriction dynamics on the time-dependence
of the centerline radius of curvature.

The centerline spans a distance $\ell_{s}(t)$ and an angle $\theta_{s}(t)$ in
the septal region, as shown in Supplementary Fig. 5a. Consider a small segment
$\delta\ell_{s}(t)$ along the centerline with a local radius of curvature $R$
and spanning angle $\delta\theta_{s}$. We then have the relation
$\delta\ell_{s}(t)=R(t)\delta\theta_{s}(t)$. This leads to the following
identity,

$\frac{1}{\delta l_{s}}\frac{d(\delta
l_{s})}{dt}=\frac{1}{R}\frac{dR}{dt}+\frac{1}{\delta\theta_{s}}\frac{d(\delta\theta_{s})}{dt}\;.$
(S.17)

Now the geometry of the septal region (Supplementary Fig. 5a) directly relates
the rate of increase in the spanning angle $\delta\theta_{s}(t)$ to the rate
of drop in the tangent angle ($\psi(t)$) at the constriction site,
$\frac{1}{\delta\theta_{s}}\frac{d(\delta\theta_{s})}{dt}=-\frac{1}{\delta\psi}\frac{d(\delta\psi)}{dt}$.
Furthermore, using the relations,
$w_{\text{max}}\sin{\psi(t)}=w_{\text{min}}(t)$ and
$w_{\text{max}}\cos{\psi(t)}=l_{s}(t)$, we get $\delta\psi(t)=\delta
w_{\text{min}}(t)/l_{s}(t)=-\delta l_{s}(t)/w_{\text{min}}(t)$. The last
equality follows after variations of Eq. (S.13). As a result we have the
following kinetic relation,

$\frac{d(\delta\psi)/dt}{\delta\psi}=\frac{dw_{\text{min}}/dt}{w_{\text{min}}}-\frac{d(\delta
l_{s})}{\delta l_{s}}=-\frac{d(\delta\theta_{s})/dt}{\delta\theta_{s}}\;.$
(S.18)

Combining the identities in Eq. (S.17) and (S.18) we get,
$R^{-1}dR/dt=w_{\text{min}}^{-1}dw_{\text{min}}/dt$, showing that the radius
of curvature drops at the same rate as $w_{\text{min}}(t)$ shrinks.

Supplementary Figure 5: Geometry of a constricting cell contour. (a) Schematic
of the constriction zone of a dividing cell where the dashed line indicates
the centerline with length $l_{s}$, radius of curvature $R$ and spanning angle
$\theta_{s}$. S (blue) denotes the septal cell wall, $\psi$ is the tangent
angle at the constriction site and $\delta$ is the width of the division ring.
(b) Compartmentalizing the cell contour into the (1) stalked pole $p_{st}$
(red), (2) swarmer pole $p_{sw}$ (purple) (3) upper wall $1_{st/sw}$ (with
crescentin bundle attached, 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 in cell shape we study an effective two-
dimensional contour model for the cell shape. This approach facilitates closer
comparison with the two-dimensional splined contours obtained from our
experimental data. The contour description of the rod-like cell shape
negelects circumferential variations in cell geometry. The model incorporates
non-uniform materials properties and mechanical constraints across the cell
wall, with the poles and the septal region being mechanically stiffer than the
rest of the cell wall. By exploiting the rod-like geometry, one can use the
centerline curve to divide the contour into two parts, the upper and the lower
cell wall. As shown in Supplementary Fig. 5b we further 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 ${\bf R}(u,t)$, the length of the centerline
$l(t)$ and the width $w(u,t)$, where $u$ is the absolute distance along the
centerline from the stalked pole such that $u\in[0,l(t)]$. If $\hat{{\bf n}}$
denotes the outward unit normal vector on the centerline, the curves defining
the upper and lower parts of the cell contour, ${\bf r}_{\pm}(u,t)$, are given
by the relation, ${\bf r}_{\pm}={\bf R}\pm w_{\pm}\hat{{\bf n}}$, where
$w_{\pm}(u)$ represent the perpendicular distances of the top and bottom
curves from the centerline. The total cell width is then given by
$w(u,t)=w_{+}(u,t)+w_{-}(u,t)$.

It is convenient to switch to polar coordinates, where the shape of the cell
contour is given by the re-parametrized curve ${\bf r}_{\pm}(\varphi,t)$,
where $0<\varphi<\theta(t)$ is the angular coordinate spanning the centerline,
which can be approximated as the arc of a circle with radius $R(t)$. Since the
ratio $w_{\pm}/R\ (\simeq 0.1)$ is small at all times, one can approximate the
local curvature as

$c_{\pm}(\varphi,t)=\frac{1}{R(t)}\left(1-\frac{w_{\pm}^{\prime}(\varphi,t)+w_{\pm}^{\prime\prime}(\varphi,t)}{R(t)}\right)+\mathscr{O}(w_{\pm}^{2},w_{\pm}^{\prime
2},w_{\pm}^{\prime\prime 2})\;,$ (S.19)

where prime denotes derivative with respect to $\varphi$ and the subscripts
$\pm$ represent the upper and the lower part of the cell contour respectively.
Furthermore, in the linear regime, the differential arc length can be
approximated as $du\simeq Rd\varphi$. The dynamics of the shape parameters
$l(t)$, $\theta(t)$, 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 the following shape equation,

$P-\gamma(u)c_{\pm}(u,t)+f_{\text{width}}(u,t)+f_{\text{cres}}(u,t)=0\;,$
(S.20)

where, $\gamma(u)$ is the tension on the cell contour and
$f_{\text{width}}(u,t)$ and $f_{\text{cres}}(u,t)$ are the linear force
densities on the contour due to maintenance of width and the crescent shape,
respectively. In the contour model we simplify the energetic contribution due
to maintenance of width (acting on the sections $1_{st/sw}$ and $2_{st/sw}$)
as

$E_{\text{width}}=\frac{K_{m}}{2}\int du\left(w_{\pm}-R_{m}\right)^{2}\;,$
(S.21)

where the elastic constant $K_{m}$ depends on the bending rigidity $k_{m}$
introduced in the main text as $K_{m}=k_{m}/R_{m}^{4}$. This linear
approximation holds if $(w_{\pm}-R_{m})/R_{m}\ll 1$. From our data and mean
field model fits we get $(w_{\pm}-R_{m})/R_{m}\simeq 0.15$. The resultant
force density is given by
$f_{\text{width}}(u,t)=-K_{m}\left(w_{\pm}(u,t)-R_{m}\right)$.

The bending energy induced by crescentin protein bundles anchored onto the
cell wall (regions $1_{st/sw}$) is given by

$E_{\text{cres}}=\frac{k_{c}}{2}\int ds\left[c_{+}(u)-c_{0}\right]^{2}\;,$
(S.22)

where $c_{0}=1/R_{c}$ is the spontaneous curvature of the crescentin bundle,
and s is the arc-length parameter along the upper part of the cell contour
($1_{st/sw}$) which is related to $u$ as $ds=du-w_{+}d\varphi$. To obtain the
force density we consider an infinitesimal deformation, $\delta r$, of the
upper contour as ${\bf r}_{+}\rightarrow{\bf r}_{+}+{\bf n}_{+}\delta r$,
where ${\bf n}_{+}$ is the outward unit normal. Accordingly the curvature and
the differential arc-length changes, $c_{+}\rightarrow c_{+}+\delta c_{+}$ and
$ds\rightarrow ds+\delta(ds)$, where $\delta c_{+}=c_{+}^{2}\delta
r+d^{2}(\delta r)/ds^{2}$ and $\delta(ds)=-c_{+}ds$ [Smumford1994, ]. The
resultant force density is obtained after variations of the energy functional,
$\delta E_{\text{cres}}=\int f_{\text{cres}}\delta rds$, where
$f_{\text{cres}}\delta rds=\delta[(c_{+}-c_{0})^{2}ds]$. This leads to the
following non-linear force contribution:

$f_{\text{cres}}=\frac{k_{c}}{2}\left(2\frac{d^{2}c_{+}}{ds^{2}}+c_{+}^{3}-c_{+}c_{0}^{2}\right)\;.$
(S.23)

Using Eq. (S.19), $f_{\text{cres}}$ can be linearized and expressed using the
angular coordinate $\varphi$ as

$f_{\text{cres}}(u,t)=\frac{k_{c}}{R^{4}}\left[-w_{+}^{\prime\prime\prime\prime}(u,t)+\frac{1}{2}(c_{0}^{2}R^{2}-5)w_{+}^{\prime\prime}(u,t)+\frac{1}{2}(c_{0}^{2}R^{2}-3)w_{+}(u,t)+R(1-c_{0}^{2}R^{2})\right]\;.$
(S.24)

Compartmentalizing the cell contour. (1) Pole caps: The cell poles are assumed
to be mechanically inert in the sense that they do not interact with the
active cytoskeletal proteins such that $f_{\text{width}}=f_{\text{cres}}=0$.
The mechanical forces acting on the cell poles come from turgor pressure $P$,
and the tension $\gamma_{p}$. The shape of the cell poles are then described
by the two-dimensional Laplace’s law,

$\gamma_{p}^{st,sw}c_{\pm}(u,t)=P\;,$ (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:

$P-\gamma c_{+}(u,t)+f_{\text{width}}(u,t)+f_{\text{cres}}(u,t)=0\;.$ (S.26)

At the endpoints of the segments $1_{st/sw}$ we impose the boundary condition
that the curvature $c$ must equal the longitudinal 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 obeys the force-balance
equation:

$P-\gamma c_{-}(u,t)+f_{\text{width}}(u,t)=0\;.$ (S.27)

(4) Septal region: We incorporate the effect of constriction in the cell shape
equation by imposing the constraint (boundary condition) that
$w(\ell^{st}(t),t)=w_{\text{min}}(t)$, where $w_{\text{min}}(t)$ 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 the curvature of the newly formed poles. Subject to these boundary
conditions, the width profile in the upper septal region
($\ell^{st}-\ell_{s}/2<u<\ell^{st}+\ell_{s}/2$) is given by,

$P-\gamma_{s}c_{+}(u,t)+f_{\text{width}}(u,t)+f_{\text{cres}}(u,t)=0\;,$
(S.28)

with a surface tension ($\gamma_{s}$ ) of the newly formed poles chosen to be
much higher than the peripheral region, $\gamma_{s}\simeq 4.5\gamma$. The
contour below the centerline obeys the equation,

$P-\gamma_{s}c_{-}(u,t)+f_{\text{width}}(u,t)=0\;.$ (S.29)

The governing cell shape equation given in Eq. (S.20), is then solved
numerically in each part of the cell contour with matching boundary conditions
in $w_{\pm}(u,t)$ and its derivatives.

## Supplementary References

## References

  * (1) Brandt, J. W.  Convergence and continuity criteria for discrete approximations of the continuous planar skeleton.  _CVGIP-Imag Understan_ 59, 116–124 (1994).
  * (2) Goriely, A., Robertson-Tessi, M., Tabor, M. & Vandiver, R.  Elastic growth models.  In Mondaini, R. P. & Pardalos, P. (eds.) _Mathematical Modelling of Biosystems_ , 1–44 (Springer-Verlag, Berlin, 2008).
  * (3) Cosgrove, D. J.  Cell wall yield properties of growing tissue evaluation by in vivo stress relaxation.  _Plant Physiol_ 78, 347–356 (1985).
  * (4) Koch, A. L.  _Bacterial Growth and Form_ (Springer Science+Business Media, Dordrecht, 2001).
  * (5) Boudaoud, A.  Growth of walled cells: from shells to vesicles.  _Phys Rev Lett_ 91, 018104 (2003).
  * (6) Landau, L. D. & Lifshitz, E.  _Course of Theoretical Physics Vol 7: Theory and Elasticity_ (Pergamon Press, London, 1959).
  * (7) Boulbitch, A., Quinn, B. & Pink, D.  Elasticity of the rod-shaped gram-negative eubacteria.  _Phys Rev Lett_ 85, 5246 (2000).
  * (8) Garner, E. C. _et al._ Coupled, circumferential motions of the cell wall synthesis machinery and MreB filaments in B. subtilis.  _Science_ 333, 222–225 (2011).
  * (9) Gahlmann, A. & Moerner, W.  Exploring bacterial cell biology with single-molecule tracking and super-resolution imaging.  _Nat Rev Microbiol_ 12, 9–22 (2014).
  * (10) Colavin, A., Hsin, J. & Huang, K. C.  Effects of polymerization and nucleotide identity on the conformational dynamics of the bacterial actin homolog MreB.  _Proc Natl Acad Sci USA_ 111, 3585–3590 (2014).
  * (11) Bathe, M., Heussinger, C., Claessens, M. M., Bausch, A. R. & Frey, E.  Cytoskeletal bundle mechanics.  _Biophys J_ 94, 2955–2964 (2008).
  * (12) van den Ent, F., Amos, L. A. & Löwe, J.  Prokaryotic origin of the actin cytoskeleton.  _Nature_ 413, 39–44 (2001).
  * (13) Reimold, C., Soufo, H. J. D., Dempwolff, F. & Graumann, P. L.  Motion of variable-length mreb filaments at the bacterial cell membrane influences cell morphology.  _Mol Biol Cell_ 24, 2340–2349 (2013).
  * (14) Gittes, F., Mickey, B., Nettleton, J. & Howard, J.  Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape.  _J Cell Biol_ 120, 923–934 (1993).
  * (15) Guzman, C. _et al._ Exploring the mechanical properties of single vimentin intermediate filaments by atomic force microscopy.  _J Mol Biol_ 360, 623–630 (2006).
  * (16) Mumford, D.  Elastica and computer vision.  In Bajaj, C. L. (ed.) _Algebraic Geometry and its Applications_ , 491–506 (Springer-Verlag, New York, 1994).