Case Presentation Essay
“A model for passive cell rounding at the onset of
mitosis – an energy minimisation approach.”
Rosanna Smith
Supervisors:
Dr Buzz Baum
Dr Mark Mikodownik
Word Count:
4163
1
A model for passive cell rounding at the onset of mitosis
– an energy minimisation approach.
Introduction
Recent studies by Kunda et al. have suggested that the active form of the membranebinding protein moesin may be a major cause of the morphological changes observed
in Drosophila cells at the onset of mitosis. As moesin binds the bilayer membrane to
the underlying cortex and it has been found to be both necessary and sufficient for cell
rounding, then this may be not be an active process. Instead, it could be due to the
passive process of energy minimisation, driven by changes to the mechanical
properties of the cell membrane. The biological constraints of the system are initially
reviewed before looking at possible cell models and their application to the feasibility
of cell rounding due to energy minimisation.
Moesin involvement in Mitosis
Cell division by mitosis is a well studied phenomenon, but there are still aspects of
the process which are not understood. The eucaryotic cell cycle can be divided into
four phases as shown below:
Fig 1. The four phases of the cell cycle. Mitosis
(chromosome condensation and segregation) and
cytokinesis (physical splitting of the cell in two) occurs
during M phase. G1 and G2 are checkpoint phases and
DNA replication is carried out during S phase. Together
the S, G1 and G2 phases are known as the interphase.
During M phase the DNA replicated during S phase condenses into chromosomes,
these are then are aligned by the mitotic spindle before the chromatids are separated
to opposite sides of the cell and cytokinesis is carried out (physical splitting of the cell
into two). When cells cultured on a substrate undergo cell division, it is observed that
the cells change shape from being flat and spread out on the substrate during
interphase, to being almost spherical for the onset of mitosis (prophase) [1]. The cells
2
then stay in a rounded shape through mitosis until after cytokinesis, when the two
daughter cells regain the flat shape. This process of cell rounding has not been fully
characterised, nor has it been determined why this shape change is important for high
fidelity in the division process.
In general, cell dynamics is dependent upon the behaviour of the protein polymer
actin, which makes up a significant proportion (5%) of the cell cytoplasm.[2] Actin can
form bundles of filaments or a cross-linked mesh such as that which is found at the
cell cortex (the outer layer of the cytoplasm adjacent to the cell membrane). It has
been shown that at least part of the cell rounding mechanism is based on actin, but
that unlike the contraction of muscle cells or the contractile ring during cytokinesis,
the mechanism is unlikely to be based on a myosin-like molecular motor [1]. This
leaves the possibility that a novel molecular motor is at work during cell rounding
(although one has yet to be found) or that the process does not involve a molecular
motor at all and is instead a passive process.
It has recently been found by Kunda et al.[3] that another protein (moesin) is highly
involved in Drosophila cell rounding. This membrane-associated protein is part of
the ERM (ezrin-radixin-moesin) family of proteins that regulate the structure of the
cell cortex through their ability to link the actin filaments of the cross-linked cell
cortex mesh to membrane proteins. ERM proteins contain a FERM (Four-point one,
ezrin, radixin, moesin) domain at the amino end (N-terminus), which binds to
membrane-bound proteins and a C-ERMAD domain (carboxy or C-terminus ERM
associated domain) that can bind to an actin filament. The FERM and C-ERMAD
domains have a high binding affinity for each other leading to the majority of ERM
proteins in the cytoplasm existing as globular monomers with the C-ERMAD acting
as a ligand for the binding site for the FERM domain. Phosphorylation of the ERM
causes a conformational change, lowering the FERM/C-ERMAD binding affinity and
unmasking the FERM binding site. This phosphorylated form is the active of the
ERM and is able to link the actin cortex to the cell membrane[4].
Fig 2. The N and C terminals of an ERM protein are shown in the normal (autoinhibitory) state and in
two different membrane binding states after phosphorylation. Taken from Bretscher et al. [4]
3
The only member of the ERM family produced by Drosophila is moesin, making it a
simple system in which to study moesin function. Mammalian systems are
complicated by the overlapping functions of the different ERM proteins. The recent
investigation of moesin involvement in cell rounding has shown that moesin in
Drosophila becomes activated and localised to the cell cortex at the onset of mitosis
and that active moesin is necessary for rounding. During rounding an associated
stiffening of the cortex has been observed by measuring the Young’s modulus of the
cell by atomic force microscopy (AFM). If active moesin is absent from the cortex
then there is a delay in mitosis at the metaphase due to defects in spindle growth and
orientation. This suggests that the behavioural properties of the cortex are important
in spindle morphogenesis. As these spindles control chromosome alignment and
segregation there is a knock-on effect on accurate cell division.
It seems feasible that as active moesin is necessary for rounding then the rounding
mechanism may not be due to a novel molecular motor, but instead due to the changes
in the mechanical cell properties that occur when moesin binds the actin cortex to the
membrane. I shall investigate this possibility using an appropriate cell model.
Models of the Cell
Mechanical modelling of the cell is not a trivial matter as it involves the interaction of
many components on many scales. For example, the cytoskeleton of an animal cell is
known to consist of the following protein polymers: Actin filaments, microtubules
and intermediate filaments. The structures, growth mechanisms and mechanical
properties of these individual polymers are well studied. However, the polymers
interact not only with each other, but with many different proteins which means that it
becomes difficult to elucidate the function of a particular component in a mechanical
process that affects the whole cell. As a result, models are required to use what may
seem like gross oversimplifications of the biological system to make the problem
tractable.
There are two broad categories of single cell models: continuum and non-continuum
(microstructural). Continuum models do not consider the individual components of a
cell on the molecular scale, but instead assume that the cell is composed of either a
single homogeneous material or of a few different homogeneous materials confined to
specific regions. Non-continuum models recognise the inhomogeneity of the actual
intracellular environment by taking one or more components and considering how
extended networks of these would behave. The aim of either type model is to try and
predict known behaviour that has been determined from experiment whilst
maintaining as much realism as possible. Below, I briefly discuss continuum and
non-continuum approaches, the experimental evidence in support of each and whether
they are applicable to the problem of cell rounding.
Continuum Models
The cell can be described, with varying levels of complexity, as either a liquid drop or
a solid. The liquid drop model assumes that the bulk of the cell has the same
characteristics as a Newtonian fluid (constant viscosity) whilst the cell cortex is an
anisotropic fluid layer that provides static tension but does not provide any resistance
4
to bending. This model was developed by Yeung and Evans[5] as an explanation of
leukocyte cell behaviour during aspiration into narrow pipets [Fig. 3]. The basic
model liquid drop model can be modified to increase realism. For example, adding a
nucleus as an internal, more viscous liquid drop can explain discrepancies in the
experimentally determined apparent viscosity. Also, modelling the fluid as
Maxwellian (elastic and viscous components) instead of Newtonian for small
deformations can help explain the observed elastic behaviour of leucocytes as they
rapidly enter the pipet[6]. The Newtonian liquid model is the simplest of all continuum
models and considering it’s simplicity, displays remarkable similarity to the
behaviour of real cells for small deformations. A solid model on the other hand does
not maintain a separate cortical layer and instead the whole cell acts as a viscoelastic
solid i.e. the elastic modulus (ratio of stress to strain) is both time and loading history
dependent. There is good agreement between the expected deformation from pipette
aspiration and flat-punch indentation as determined by the viscoelastic solid model
and that observed from experiment[6].
One particular property of cells that can not be predicted from either solid or liquid
drop models is the dependency of the storage modulus of a cell on the frequency of a
perturbing force, as measured by magnetic twisting cytometry (MTC). Conversely,
the soft glassy rheology model (SGR) which can explain this phenomenon does not
adequately predict aspiration deformations[6]. This situation highlights the
unfortunate specificity of models to a particular environment or probing mechanism.
Experimental techniques
Mechanical models developed
• cortical shell–liquid core model
• solid model
• cortical shell–liquid core model
• solid model
• power-law structural damping model
• power-law structural damping model
• solid model
• solid model
• biphasic model
• solid model
Fig 3. Some common techniques used to probe the mechanical behaviour of cells and the
corresponding which are compatible with the experimental results (not all of these models have been
mentioned. Taken from the review by Lim et al. [6]
5
Non-continuum Models
One particular model that takes account of microstructures present inside the cell is
the tensegrity model. The definition of this framework (or architecture) for cell
structure has been much debated,[7] but basically postulates that certain components of
the cytoskeleton bear only stress, whilst other components are purely tensile. The
structure as a whole is stabilised through the continuous tension that exists in the
system, whereas stress is discontinuous and localised. The tensile components of the
network are considered to be actin and microfilaments but there is not yet a consensus
concerning the identification of the stress components. One controversial aspect of
the tensegrity model concerns the prediction of “action at a distance”, where a local
stress can cause spatial rearrangement of the entire network. Experiments that
examined this idea by pulling on cell surface receptors did not show the global
movement predicted,[8] however, the basis of the experimental procedure has been
questioned and the tensegrity model can provide convincing simulations of cell
behaviour under other experimental conditions. The tensegrity model gives a
framework into which new information concerning cytoskeletal components and
organisation can be integrated, but there remains controversy surrounding the premise
of assuming the cell maintains a tensegrity architecture.
Other non-continuum models focus on understanding the characteristics of actin
networks. Due to its abundance in the cell in different forms, actin networks have
been modelled both as single component systems and in conjunction with a single,
rigid actin binding protein[9]. Typically the elastic characteristics of the network are
theoretically determined for variations in parameters such as filament length, binding
protein concentration and applied stress. The resulting state diagrams differ
substantially from those of non-biological polymer networks. This approach is still in
its infancy and has yet to be extended to more realistic systems with multiple, nonrigid actin linkers which could lead towards understanding the elastic properties of the
whole cell.
The brief examination of current single cell models given above shows that different
approximations are needed when examining cell structure on varying scales and when
looking and the cells response to varying stimuli. In relation to the problem of cell
rounding discussed earlier, the change in cell shape is not associated with an external
force, but a change of membrane structure. For this reason, a model is needed in
which the membrane and cortex region is the main focus. None of the
aforementioned models can provide this focus when taken on their own, so instead I
shall consider the interior of the solid as fluid but use a membrane model (as
described in the following section) that does not assume zero rigidity.
Membrane models
Models of the cell membrane in animal cells mainly focus on the phosoplipid bilayer
that acts as a barrier between the outside environment and the internal workings of the
cell. As well as controlling what is allowed to enter and leave the cell, this bilayer
contributes towards the cell mechanics as it can resist bending and expansion. The
exact composition of each monolayer influences the mechanical properties of the
bilayer, for example, the addition of a small amount of cholesterol into a single
component bilayer can dramatically increase the elastic modulus.[10] The mechanical
6
properties are not necessarily spatially constant at different points on the surface, but
due to the complexity of the actual cell bilayer I shall consider a parameter such as the
elastic modulus or bending rigidity as being an average for the whole cell. Bilayer
properties have been extensively researched using lipid vesicles and the resulting
information has being applied to the red blood cell. In this case, models of the bilayer
and thin underlying cortex have successfully simulated the distinct shape
transformations that occur as red blood cells enter narrow pipettes (similar to blood
capillaries)[11]. The basis of the model is the determination of the lowest energy
spatial configuration for a certain environment and this is also the approach I shall
take for modelling the cell membrane at the onset of mitosis.
The bending energy of a membrane can be determined using the spontaneous
curvature model. In this case the membrane is considered as a thin shell with a
characteristic bending resistance. The following expression for energy density ()
was derived by Helfrich et al.[12] and assumes that for small deformations the local
energy can be expanded in terms of H (mean curvature) and K (Gaussian curvature).
See Appendix A.

B
2
(H  C0 ) 2  GK
Where C0 is the spontaneous curvature, B the bending rigidity and G the Gaussian
bending rigidity. The spontaneous curvature is a measure of the curvature that the
 possess due to differences in composition of the inner and
unstrained bilayer would
outer layers. The bending rigidity B and the Gaussian bending rigidity G are
measures of the energy required to deform unit area membrane to have unit mean
curvature and unit Gaussian curvature respectively. For membrane shapes that have a
vertical axis of rotation, such as those shown in [Fig. 4] then it is possible to
demonstrate (Appendix A) that the energy of the surface is given by the expression
below, where s is the length along the rotated curve and  is the angle between the
interior normal to the curve s and the vertical.
E[s]   B
S

x(
S
sin  d
d

 C0 ) 2 ds  2G  sin  ds
x
ds
ds
Another possible analysis for the bending energy includes the idea that the two layers
of the membrane are not mechanically coupled, so that they can slide past one another
 stress. This is the area difference elasticity (ADE) model[13] and it depends
to relieve
on the discrepancy between the area difference of the inner and outer layers (A) and
what this value would be for an unstressed cell (A0). The ADE model is more
rigorous than the spontaneous curvature model, however, as a first approximation it is
simpler to include the change in membrane properties due to moesin binding as a
change in spontaneous curvature of the membrane.
7
Energy Minimisation Model for Cell Rounding
It shall be assumed that when the cell is flat then this is the minimum energy
configuration for the membrane properties before mitosis (low levels of moesin
binding at the membrane). I shall also assume that the rounded cell is the lowest
energy configuration for the membrane properties that occur when the moesin binds
the cortex and the bilayer. The total energy for each configuration must take into
account bending energy, adhesion energy and gravitational potential energy. There is
no volume or density change of the interior of the cell so the energy associated with
the fluid is purely gravitational. By examining the energy of these configurations it
should then be possible to determine how the membrane properties would need to
change in order for rounded configuration to be of a lower energy, endorsing the
theory of passive rounding due to moesin binding.
Bending Energy
A rounded cell is idealised as a sphere and a flat cell is approximated as a donutshaped body (without a hole in the middle) where  = R/r >> 1. i.e. A thin donut.
Axis of rotation
Axis of rotation
2r


S
S
x
2r
2R
Fig 4. Idealised cell shapes. (Not to the same scale)
The resulting bending energy for the sphere is exact, but the donut shape requires the
approximation  >> 1 to simplify integration. This approximation seems valid for flat
cells.
flat
E bending
 (1 C0normalr flat )(8 Bnormal   2 Bnormal  4Gnormal)
round
E bending
 (1 C0moesinr round )(8 Bmoesin  4Gmoesin )

The spontaneous curvatures and bending rigidities are labelled by “normal” for the
cell before rounding and “moesin” for the cell after rounding in order to reflect the
 are structural properties of the membrane and not dependent on the cell
fact that these
shape. It can be seen that the energy associated with the spontaneous curvature is
coupled to the smallest radii of curvature (R and r), whereas the energy associated
with the mean and Gaussian curvatures are scale invariant (do not depend on size –
only shape). I shall assume that the cell does not change in volume during cell
rounding, so the volume of the sphere 43  (r round )3 and the volume of the donut
2 (r flat )3 ( 2  23 ) must be equal. This leads to the following restraint on r round,
assuming that r flat and R flat can be determined from observation


8
r round  r flat ( 3 
 1)
4
1
3
B can be determined by experiment for both flat cells with normal moesin and
rounded cells with bound moesin. Methods include using micropipet aspiration [14]
and shear flow induceddeformation.[15] Typical values for pure bilayers and red
blood cells are of the order 1x10-19 J but an order of magnitude higher for the cell
membrane of Dictyostelium discodeum (a slime mould) [15]. Values of G are harder
to determine experimentally as simple shape changes will not change the Gaussian
contribution to the bending energy (it is topologically invariant), and thus the
behaviour of the cell if deformed. However, some values have been obtained for pure
bilayers[16] which indicate that BG. This means that the bending energy equations
can be simplified to contain only measurable bending rigidities.
flat
E bending
  Bnormal(12  )(1 C0normalr flat )
round
E bending
 12 Bmoe sin (1 C0moesinr flat ( 3
) 3)
4
1

Unfortunately I do not know of any current methods by which the spontaneous
curvatures may be determined analytically or experimentally so this parameter for the
flat and roundcell will remain unquantified. The order of magnitude of these bending
energies will be similar to those where there is no spontaneous curvature, which for a
sphere is approximately 12B  3x10-17 J.
Adhesion Energy
Another important contribution to the energy of a certain configuration is due to any
adhesion of the cell surface to the substrate. In general, the cell is adhered to the
substrate non-continuously via protein linkers (integrins) at points known as focal
contacts. Kunda et al.[3] have shown that the cell is still attached to the substrate at
focal contacts whilst round so the adhesion energy at this point can not be assumed to
be negligible. There are several different experimental approaches to finding the
adhesion energy of a cell, including the use of a shear flow to distort the cell in the
direction of flow. Reflection interference contrast microscopy can then be used to
image the cell and determine the contact angle, which in conjunction with the external
stress due to the shear flow can be used to calculate the adhesion energy[15]. This
method leads to a measured value of adhesion energy density of the order of 1x 10-5
Jm-2 and adhesion energy of the order 1 x 10-16 J for a cell with a contact area of
10m2. This gives an indication that the adhesion energy could be an order of
magnitude greater than the bending energy. However, it is changes in energy that are
important for the passive rounding theory and hence it is important to consider the
changes in adhesion energy that occur rather than absolute values.
A more direct measurement that does not assume the interface between the cell and
the substrate is at least quasi-continuous, could perhaps be found by utilizing a force
sensor array. Previous studies[17] have used small cantilevered pads to determine the
local force exerted in a vertical direction by a motile cell but the cells only adhered to
a few pads at a time as well as to the intervening substrate, so the forces exerted by
the cell could not be determined over the whole cell. Another method uses a micro9
fabricated sheet of pillars, whose deflections from rest position can be tracked for
horizontal directions[18]. In this situation the cell adhered only to the many pillars it
covered, allowing the applied forces to be measured over the whole cell-substrate
interface. A combination of these two approaches could perhaps use fabricated an
array similar to that sketched in Fig. 5. The idea would be to track the vertical
displacement of the levers in the same manner as for the horizontal displacement of
pillars. During rounding the centre of mass of the cell changes its displacement from
the substrate. If this distance change can be determined for different stages during
cell rounding then a curve could be fitted to a plot of the vertical component of total
force versus cell displacement. The area underneath this graph would be an estimate
of the change of adherence energy as the cell rounds.
d
Vertical displacement of microlever = d
Fig 5. Images of the cantilevered beams [17] and micropillars [18] used in previous studies of forces
applied to the substrate by cells. The sketch shows a possible array of microlevers to measure vertical
displacement.
Gravitational Potential Energy
Another energy change that may need to be taken into account is the gravitational
potential energy that the centre of mass of the cell must overcome during rounding.
Estimating the radius of the cell to be in the order 10m, the averaged density of the
cell to be similar to that of water (which will be a lower bound) and the vertical
displacement of the centre of mass also to be roughly 10m, the change in
gravitational potential energy is of the order 1x10-16 J. This has the same magnitude
as the bending energy and adherence energy so must be taken into account.
Change in gravitational potential energy (this is a negative value):
flat
round
E gravitational
 E gravitational
 G  (

10
4 (r round ) 3
g(r flat  r round )
3
Condition for passive rounding
For passive rounding the total energy of the rounded cell must be lower than that of
the flat cell, leading to the following inequality:
round
round
round
flat
flat
flat
(E bending
 E adhesion
 E gravitational
)  (E bending
 E adhesion
 E gravitational
)
Including all the parameters that have been mentioned, this can be expressed as:

D

moesin
C
round 3 
0
4
(
) D
1
3
normal
C
flat
0
A  G  (Dnormal  Dmoesin )

r flat
flat
round
 E adhesion
 A , Dmoesin  12 Bmoesin , Dnormal   Bnormal(12  ) and
Where E adhesion
  1. If the above condition holds, then within the restrictions of this model, cell

rounding at the onset of mitosis can be attributed to the passive mechanism of energy
minimization through shape change. The restrictions of the model and parameter



estimates are summarized in the following section.
Discussion
Several simplifications have been made to the biological situations in the derivation
shown above. In particular, the shape of the membrane when flat on the substrate does
not closely mimic that of the thin donut shape used. It would be possible to go some
way towards correcting the resultant bending energy aberration by approximating the
rotation curve as the top half of an ellipse lying on top of two quarter-circles (bottom
half of the donut shape). Other shapes can also be used, although integration
constrains which shapes have bending energies that can be solved analytically. In any
case the gradient of the curve must be continuous to avoid infinite bending energies.
The radius of the circle and lengths of the ellipse axes could be varied to most closely
resemble that the actual cells. Some other biological features which have not been
included in this model include the contribution of blebs and folded bilayer sections
towards the bending energy. However, these local membrane deformations may be
caused by local changes in the membrane properties. As parameters such as bending
rigidity are the average value for the whole cell, then local variations in configuration
are roughly accounted for. As shown in the preceding section, it is possible to
construct a condition to describe the circumstances in which passive rounding due to
energy minimization is a feasible mechanism. However, there are a large number of
parameters in the model that need to be determined by experiment specifically for
these Drosophila cells. Current or possible methods for measuring these parameters
have been mentioned, except for spontaneous curvature. Unfortunately this means
that until there is a way to uncover the spontaneous curvature, it will not be possible
to evaluate the passive rounding condition. Perhaps in the future it will be possible to
calculate C0 using more complex models of actin and binding protein interactions;
building on the studies by Gardel et al.[9] Alternatively there may be a novel method
to determine C0 experimentally. For the time being, the passive rounding condition
can not be tested but its derivation has shown which approaches are useful in
modelling the cell rounding process.
11
Appendix A
The curvature (C) at a given point on a curve is given by the reciprocal of the radius
(r) of the osculating circle, which is a circle whose circumference approximates the
curve at that point. The centre of the circle lies on the normal to the curve.
C
1
r
Every plane that intersects a surface has an associated osculating circle and curvature
for every point on the curve defined by the intersection of the surface and intersecting

plane. The maximum and minimum
of the curvatures are defined as the principle
curvatures C1 and C2. The arithmetic mean of these is the mean curvature (H) and the
product is the Gaussian mean (K). An interesting property of the Gaussian mean is
that it is topologically invariant i.e. it has the same value for spheres and ellipsoids
(g=1 surfaces) but a different value for planes and cylinders (g=0 surfaces).
H  12 (C1  C2 )
K  C1C2  g
 , surfaces that have an axis of rotation take the following
As demonstrated by Boal[19]
parameters:

sin(  )
d
C1 
C2 
x
ds
Where x is the distance of the rotated curve from the axis of rotation, s is the path
along the curve and  is the angle between the inward normal of the curve and the


rotation axis.
This leads to the expression given in the main text for the bending energy as the
energy density () is integrated over the surface with surface element dA  (2x)ds.
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