Tracking the Plasma membrane in order to locate the developing actin cortex
during Blebbing
Introduction
In this project, cortical reassembly in blebbing cells was studied. Blebs are
initially devoid of an actin cortex and so provide a good model system to study it’s
reassembly. Advances have been made in determining the causes of blebbing, the
order in which proteins are recruited to blebs and finding the biochemical key
players responsible for reassembling the cortex. However, a lot is still unknown
about the assembly and dynamics of actin filaments in the cortex. The aim of the
current project as a whole is to track actin monomers as they move through the
reassembling cortex in blebs to try to learn more about the kinetics of
polymerisation in the construction of the cortex.
In order to track actin monomers, speckle microscopy is used. This involves
injecting cells with very low amounts of actin binding green fluorescence protein
(GFP) so that only a few actin monomers become tagged and fluoresce. Since each
monomer is only tagged by one GFP molecule, the speckles that are seen are very
faint and, therefore, very difficult to track. The aim of this case presentation within
this larger project was to provide extra information about the probable location of
the actin monomers by tracking and studying the motion of the plasma membrane.
The actin monomers in the reassembling cortex tend to move parallel to the plasma
membrane and remain close to it. By knowing the location of the membrane as the
bleb retracts, more information about the likely location of the actin filaments can
be introduced to the speckle-tracking program and so they should become easier to
follow more accurately. Furthermore, when analysing the actin motion data, this
will allow de-convolution of the motion of the actin from the motion of the bleb as a
whole. From the motion of the actin, and the nucleator which attaches to the barbed
end of the filament, we can learn about the kinetics of polymerisation.

The plasma membrane
Figure 1: The plasma membrane, comprised of a lipid bilayer with embedded proteins and
protein complexes. (taken from ‘http://www.freewebs.com/ltaing/homogenization.htm’)
The plasma membrane separates the cell interior from the external environment1. It
consists of a lipid bilayer embedded with proteins and protein complexes that,
together, can be considered to be a 2-dimensional fluid2. As well as isolating the
interior of the cell, it is involved in signalling, transport of necessary proteins and
plays a role in attaching the cell to the external matrix in cell movement and
grouping1. Alone the membrane provides little rigidity and cannot maintain cell
shape. Mechanical support is provided by an underlying cytoskeleton to which the
membrane is anchored1,3.

Actin filaments and the cytoskeleton
Figure 2: Fluorescence images of mouse fibroblasts showing microtubule and actin
cytoskeletons by Jan Schmoranzer (taken from
‘http://www.maths.bris.ac.uk/~matbl/research/biophys.html’)
The cytoskeleton is a mesh of actin microfilaments, intermediate filaments and
microtubules contained in the cytoplasm that gives the cell shape and structure1.
Actin microfilaments, which are composed of linear polymers of actin, are the
thinnest of the filaments. They generate force by elongating their leading edge
whilst depolymerising at the rear resulting in net movement. The Rho family of
GTPases control actin structures, for example during contraction which is controlled
by Rho itself (see section on blebbing) or when creating protrusions (e.g. Cdc43 and
Rae)1.

The actin cortex
Figure 3: f-actin fibers in the cortex (Taken from
‘http://web.uvic.ca/~rburke/burkelab/integrins.htm’)
A thin gel known as the cell cortex lies beneath the membrane. It is 100nm to
500nm thick and primarily composed of cross-linked actin filaments, myosin and
actin binding proteins1,4. It is the main cellular component affecting cell shape and
providing rigidity in animal cells1. Myosin molecular motors can exert forces on the
cortex that allow the cortical cytoskeleton to contract and reshape when required
for various cellular processes1. It is thought that spectrin is responsible for the
cross-linking of the actin filaments, which increases the rigidity of the cortex1,5.
Attachment proteins connect the actin network to transmembrane proteins keeping
the membrane bound1. The actin cortex increases the rigidity of the membrane 5fold, protecting the bleb against external forces6. Protrusions of the plasma
membrane can occur under the control of the underlying cytoskeleton where the
membrane is pushed outwards by microtubules (e.g. cilia) or actin filaments (e.g.
filapodia, lamellipodia). Furthermore contraction of the cortex can lead to the
protrusions studied in this project, known as blebs.
Contractions of the cortex are required in lamellipodial motility in order to detach
the cell rear from the substrate and pull the cell forwards14. Furthermore, as well as
this conventional form of motility, many types of cell, including metastatic cancer
cells, can migrate through two alternative methods that rely on cortical contraction
for initiation of blebbing, as well as retraction of the cell body12,52. These methods of
directed motility are important since they allow metastatic cancer cells to escape
drugs. This is because these forms of motility don’t require proteolytic degradation
of the surrounding matrix9-13. The cortex is also essential for several other cellular
processes including cytokinesis and morphogenesis as well as any process requiring
the cell to respond to or create forces7,8. Understanding its structure, assembly and
behaviour is therefore important. A brief review of the role of cortical contractility in
cellular processes is given in appendix A.

Blebbing
Blebs are spherical protrusions of the plasma membrane from a cell that last of the
order of minutes and extend to around 2μm long. They differ from other types of
protrusion (e.g. filopodia, lamellipodia and podosomes) in that they are not formed
as a result of actin polymerisation pushing the protrusion outwards15. Instead they
result from the plasma membrane locally separating from the actin cortex, causing
cytosol to rush into the gap which in turn pushes the membrane outwards.
Polymerisation of actin does, however, occur in order to stop expansion and an actomyosin cortex is generated to retract the bleb16-18. Blebs have historically been
associated with apoptosis but are also seen in healthy cells, for example during
cytokinesis19, cell spreading20,21 and movement in tumour and embryonic cells22.
During apoptosis, two types of blebbing are seen. The first are similar to those seen
in healthy cells23 whilst the second do not result from acto-myosin contractions and
are larger and do not retract24. Since blebs initially do not contain actin and
reassemble a cortex throughout the blebbing cycle, they are a useful tool for
studying cortical actin nucleation and the mechanics and assembly of the cortex.
Furthermore, blebbing is often used as a reporter of high cortical contractility since
this is a major factor in their formation52.
Figure 4: Blebbing (Taken from ref. 9) The membrane separates from the cortex due to either
a rupture or an increase in hydrostatic pressure. Cytosol rushes into the gap causing the bleb
to expand outwards. This expansion is halted as an actin cortex begins to reform. Finally the
newly assembled cortex contracts until the bleb re-joins the bulk of the cell.
The Bleb Cycle
1. Initiation
Blebs can be initialised through two different mechanisms, both of
which result from myosin driven contraction of the actin cortex. In the first of
these, the cortex is ruptured as a result of the increased tension25. The
second involves separation of the plasma membrane from the cortex as the
cortex contracts inwards, increasing the pressure in the cell17. This pressure
causes cytosol to push through the porous cytoskeleton to detach the
membrane from the cytoskeleton. Blebs have been shown to occur more
frequently where the membrane is weaker16, 26 and so it is possible that, as
well as the pressure increase, a change in the level of proteins related to
cortex-membrane attachment may be required in order to detach the
membrane from the cortex. PIP2 has been suggested since a decrease in
cortex-membrane adhesion results from loss of PIP227.
If the Rho-ROCK-myosin cascade is inhibited at any level in a cell,
blebbing does not occur12, 17, 18 28, 29 demonstrating that the cortical
contraction required for blebbing acts under control of this cascade with
myosin as the motor protein driving contractility. Myosin light chain is
directly phosphorylated by ROCK (Rho-associated kinase), which in turn is
activated by Rho. ROCK also inhibits Myosin light chain dephosphorylation
and so also prolongs Myosin II activation30.
It has been shown that the hydrostatic pressure differences generated
as a result of the cortical contraction wouldn’t have time to redistribute
throughout the cell before blebbing occurs. Furthermore, localised addition
of drugs leads to changes in blebbing only in the treated area. Therefore,
initiation of blebbing is a local effect depending on local hydrostatic
pressures and cortical tensions17, 31.
2. Growth
In the next step, cytosol pushes against the membrane causing it to
inflate outwards to form a spherical protrusion by flowing through the pores
and holes in the cytoskeleton. This inflation also causes further tearing of the
membrane from the cytoskeleton, increasing the size of the bleb’s rim32. This
process of expansion lasts roughly 30 seconds.
The increase in the size of the membrane as it balloons outwards
from the cell cannot be explained through unfolding of the membrane
wrinkles and tearing at the rim alone. This is because the perimeter to base
ratio of the bleb increases with time whereas a roughly linear relationship
between the two would be expected if the extra membrane was entirely due
to unfolding32. The additional membrane could result from the flow of lipids
in the membrane through the bleb neck. In other contexts the flow of lipids in
membranes has been observed so this is possible33, 34. A lipid flow of 2
micrometres per second would be required to account for the discrepancy
and there is no evidence that this happens32. Another possible explanation is
the fusion of vesicles to the bleb membrane, but there is no evidence that this
occurs and no fusion with vesicles could be observed when imaged with lipid
markers35.
It is possible that, while blebs appear to initially have no actin
cytoskeleton, they may have an erythrocytic cytoskeleton throughout
growth. The mechanical properties of the membrane are similar to that of
red blood cells, which have an erythrocytic cytoskeleton6, 36. Furthermore,
protein 4.1 and ankrin (components of the erythrocytic cytoskeleton) are
both localised to the membrane throughout growth37.
3. Slowing and reassembly of the cortex
As inflation slows, a new cortex begins to form beneath the bleb
plasma membrane, formed from a mesh of actin and myosin, which, in turn
can begin to contract17. It has been shown that regrowth of the actin cortex is
triggered by recruitment of nucleators rather than alternatives such as
growth of the cortex into blebs from the sides or regrowth stemming from
actin templates stuck to the membrane38. The cortex at the base of the bleb
also disassembles at roughly the same time and it is unclear both what
causes this disassembly as well as whether actin from this is recycled in the
new cortex.
In the paper ‘Reassembly of the contractile actin cortex in cell blebs’
(Charras et al, 2006)37, the sequence of proteins recruited to the forming
bleb cortex is studied. It is shown that the retraction of a bleb results from a
sequential assembly of actin-membrane linker proteins (such as ezrin), actin,
actin bundling proteins, regulatory proteins and lastly myosin motor
proteins. In ‘The cellular cortex is a composite of two independently nucleated
actin networks’ (Bovellan et al, 2012) it is shown that that two independently
acting proteins, Arp2/3 and Diaph1, are responsible for nucleating the bulk
of the actin cytoskeleton38. These contribute roughly equal amounts of actin
to the cortex, but Diaph1 works 5 times more quickly38. The regrowth of the
cortex, the recruitment of proteins and the role of the 2 nucleators are
discussed in more detail in Appendix B.
Secondary blebs can occasionally protrude out from an existing bleb
during cortex creation and retraction. These could be caused by the
increased pressure in the cytosol as a result of contraction or the increased
tension causing rupture in the bleb32.
4. Retraction
The bleb shrinks as the newly formed cortex undergoes myosin
driven contraction until the bleb membrane re-joins the bulk cell membrane
and all of the cytosol is pushed back into the cell body. The process of
retraction takes around 2 minutes. It is unknown whether the bleb cortex
joins the existing cell cortex or whether it is depolymerised and a new cortex
is assembled which links more strongly with the rest of the cell.
Speckle microscopy for studying the assembly of an actin cortex
Speckle microscopy is useful for studying the motion of individual proteins
or molecules in a cell. In conventional fluorescence microscopy, used for studying
the overall motion and behaviour of groups of molecules, as much fluorescing
protein as possible is injected into the cell. However, speckle microscopy involves
using very little fluorescing protein so that only a few of the molecules of interest
show in an image. Each spot of light in an image then corresponds to only one of the
molecules being studied and so the behaviour of individual molecules can be
determined. Tracking these speckles can be challenging and so an appearance model
telling the tracking program what the speckle should look like is combined with a
motion model telling the program where the speckle is most likely to be.
The aim of this case presentation was to analyse conventional fluorescence
microscopy images of the plasma membrane in order to provide extra information
for the motion model when speckle microscopy is used to follow individual actin
filaments in the developing cortex. Snaking was used to track the membrane
through images. As described in the introduction, this would also allow us to
separate the motion of the actin and the actin nucleator sitting at the barbed end of
the filament from the motion of the bleb as a whole, allowing us to learn about the
kinetics of polymerisation.
Single molecule fluorescence imaging has been used to further knowledge in
many areas of biology, including tagging single actin molecules in filaments which
has helped, for example, in studying lamellipodia43-47. In ‘Actin polymerisation-driven
molecular movement of mDia1 in living cells’ (Higashida et al, 2004)78 a similar
technique was used to study the formin mDia1. By studying the motion of the formin
and of actin in the cell, it was discovered that formins have the ability to attach to
the extending end of actin filaments and so can move without the use of molecular
motors. A review of the role of formins in actin polymerisation is given in ‘Formins:
processive cappers of growing actin filaments’ (Watanabe and Higashida, 2004)79
Snaking48, 49
The first step of any feature detection is to produce a feature map. This
shows how likely it is that each pixel is placed on the feature of interest. To
construct this, a convolution is usually performed between a small prototype image
of the feature (e.g. a strong gradient or a v-shaped intensity function) and the image
in which the feature is being sought, in order to give a pixel-by-pixel map of the
likelihood that each pixel is placed on the feature. There must then be some process
of deciding the location of the feature from these probabilities and in low-level
image analysis a cut off is usually used.
Active contours or “snakes” offer an improvement over low-level image
analysis in that the properties that the feature boundary must be continuous and
smooth are imposed onto the image rather than hoping they will come from the data
alone. These properties are important to ensure the tracked outline of the
membrane is realistic and complete.
A snake is an elastic continuous flexible curve or rod, which tries to move and
bend in order to fit the feature of interest, in our case the blebbing membrane. As
well as finding a feature in one image, snakes can be used to track a moving object in
a series of images, which will allow the membrane to be followed as it retracts.
These snakes are referred to as dynamic contours.
Snakes are deformable curves r(s) which move and bend over a feature map
F in order to maximise the response of F(r(s)) over the length of the curve (0≤s≤1)
without breaking the elastic properties imposed upon them. The problem is
formalised as if an elastic rod were moving within some external potential energy
field in which this external field is defined by the feature map. The external forces
are countered by those stemming from the internal potential energy of the rod
which aim to preserve smoothness.
In equilibrium the system is characterised by the equation:
𝑑(𝑤1 𝑟) 𝑑 2 (𝑤2 𝑟)
(
−
) + ∇𝐹 = 0
𝑑𝑠
𝑑𝑠 2
When this system is solved iteratively, the snake will move towards ridges on
the feature map. The constants w1 and w2 control the elasticity and stiffness
respectively and can be adjusted to include prior knowledge of the system. They can
also be allowed to vary with s (along the snake) although, since the mechanical
properties of the membrane do not greatly change around a bleb, this was not
necessary in this project.
In practice, computations are performed in discrete space. This involves
approximating the continuous line with a series of points s=si, i=1…N, spread at
equal intervals Δs along its length. The first and second order spatial derivatives can
then be calculated using finite difference equations:
𝑑𝑟(𝑠𝑖 ) 𝑟𝑠 (𝑠𝑖+1 ) − 𝑟𝑠 (𝑠𝑖−1 )
=
𝑑𝑠
2∆𝑠
𝑑2 𝑟 𝑟𝑠 (𝑠𝑖+1 ) − 2𝑟𝑠 (𝑠𝑖 ) + 𝑟𝑠 (𝑠𝑖−1 )
=
𝑑𝑠 2
∆𝑠 2
The more complex finite element method can also be used. This treats the
points r(si) as nodes from which the full continuous curve can be reconstructed. For
example, spline curves can be used which are especially computationally efficient
since they automatically maintain smoothness. Spline curves are reviewed in
Appendix C.
For dynamic contours, such as the retracting bleb membrane, the feature
map varies with time, and therefore, so must the positioning of the snake. The snake
must follow the peak response as it moves between images. The equation of motion
for the snake in this case becomes:
𝑑𝑟 𝑑(𝑤1 𝑟) 𝑑 2 (𝑤2 𝑟)
𝑑2𝑟
(𝛾
−
+
) + ∇𝐹 = 𝜌 2
𝑑𝑡
𝑑𝑠
𝑑𝑠 2
𝑑𝑡
In which 𝜌 is the mass density and the term containing this corresponds to
the inertia in the system whilst 𝛾 is the viscous resistance and this term dictates the
viscosity. The new coefficients can also be used to include prior knowledge into the
system. Through this equation, the position and motion of the snake in previous
frames can be used to predict the position of the feature in the next frame. This
prediction can then be combined with the actual image data to best locate the
feature of interest.
Using Snaking to track the plasma membrane in blebbing cells

The snaking program
An existing snaking program named ‘Snake: Active Contour’ (Dirk Jan Kroon,
2010)50 was used in order to track the membrane in the images, although this was
modified extensively throughout the project. This roughly follows the procedure
described above. The program requires the user to input a feature map, some initial
points giving the rough location of the snake and various parameter values. It then
outputs 100 points defining the location of the membrane in the image.
Unfortunately it requires that the snake is a closed loop while the bleb
membrane is a curved arch. This is a general feature of snaking rather than a fault
of the program itself since the elasticity term in the snake equation favours smaller
gaps between each point. If the loop were not closed, all of the snake points would
come together. The only alternative would be to find a method that calculates
exactly where the bleb membrane joins the bulk cell membrane in every image and
then fix the ends of the snake at these points. This could be done in future work if
greater accuracy is required but, in the timeframe of this 6-week project, a closed
loop snake was used.

Finding a feature map
The snake program required a feature map as an input and so the first task was
to find a suitable way of creating the feature map for each image. A feature map of
an image is a second image that defines how likely it is that the feature of interest
(the membrane) is in each pixel (see snaking section).
Since the images used in this project were of cells in which the membrane had
been tagged with fluorescent protein, the brightness of a pixel already
corresponded, to some extent, to the likelihood of that pixel showing membrane and
therefore, as an initial method, intensity alone was used. A lot of smoothing was
applied to the images in order to remove the noise. It was hoped that this would also
smooth out the membrane so that, instead of just have a thin line of bright pixels
surrounded by dark, there would instead be a gradient of brightness for the snake to
follow towards a maximum brightness at the membrane.
However, when a feature map created in this way was used in the snake
program, the membrane was not well tracked. This was tried for several different
values of the snake parameters, but a force field that moved the snake towards the
membrane could not be found.
Since smoothing of the original images had failed to work sufficiently well, a
more complex method of calculating the feature map had to be found. A program
named ‘An Unbiased Detector of Curvilinear Structures’ (Steger, 1996)51 was used. A
full description of this program can be found in the paper51. However it essentially
works by, firstly, calculating the Hessian (the matrix of second spatial derivatives) at
every point. At each point, it then finds the eigenvector corresponding to the largest
eigenvalue of the Hessian. This points in the direction in which the intensity change
is greatest. If the change in intensity is very large, this point may mark the edge of a
membrane. The program then checks if the neighbouring pixels also show a large
change in intensity since this would be expected if the point did lie on a membrane.
The second eigenvalue points perpendicular to the largest change in intensity,
which can be used to follow around the edge of the membrane giving an outline
(fig5a). A line drawn though the center of this outline should, therefore, follow the
center of the membrane (fig5b).
Figure 5: a - left) The outline of the plasma membranes in a blebbing image as detected using
ref. 51, b - right) A line following the membrane taken from the same program.
This method doesn’t enforce the properties of elasticity and continuity and so,
zigzagged and sometimes broken lines are given. This method could therefore not
be used alone to track the membranes. It does however give a good outline for the
snake to follow. The distance transforms of the outputs from this program were
used as the feature map. The distance transform replaces the value of each nonmembrane pixel in a binary image with a value corresponding to its proximity to the
nearest membrane.
Since the distance transform provides a simple continuous gradient towards
the position of the membrane, the Gaussian derivative methods used in the snake
program were no longer needed and, in fact, made results worse. A more simple
0 1
0
method was written which simply convolved the image with the [1 0 −1]
0 −1 0
matrix. This ensured the snake was always moving towards the maximum likelihood
curve.
This method located the membrane well, and figure 6 shows a bleb
membrane with the snake overlaid. A range of values were used for the various
snake parameters and the values which best fitted the membrane whilst giving a
smooth curve were chosen (when judged by eye, see later for rigorous error
analysis). When tracking the membrane in an image, the snake program needed to
be initialised by the user clicking roughly around the membrane. The snake could
find the membrane well, even if initialised relatively far from the membrane.
Figure 6: The membrane of a bleb that has been located by the snaking program.

Tracking the membrane through time
Since the program could now locate the membrane of a bleb in a single frame
when given rough coordinates by the user, the next step was to get the snake to
track the membrane through several frames. To do this, a script was written which
uses the output from the previous frame as the initialisation of the next frame. In its
most automated form, the program follows the snake until the area of the snake
becomes smaller than some specified cut-off (default 100 pixels). However, an
option can be used in which, instead of calculating if the bleb has retracted from the
area, the script instead asks the user every 5 frames if the bleb has fully retracted. A
second option can also be used which asks the user every 5 frames if the snake is
successfully following the bleb. If the user says the tracking has gone wrong, they
can reinitialise the snake. Once the snake has been reinitialised, the program also
goes back and corrects the last few frames. It does this by assuming a linear
transition of the snake points between the last frame that the user had stated to be
well tracked and the current reinitialised frame.
Movie 1: A video showing the snake track a bleb as it retracts. The pictures are at 25second intervals. The frame rate of the original movies was 5s and every 5th frame is shown.

Calculating the Dynamics of individual points on the snake
The next step was to try to characterise how each small segment of membrane
moved throughout the blebbing cycle. To do this, the program was modified so that
it could follow individual points on the snake through frames.
This was more complicated than simply presuming the nth point on the snake
in frame 1 moves to the nth point on the snake in frame 2 and so on. This is because
the points may rotate around the snake as it contracts. Furthermore, points are not
evenly spaced around the snake but instead localise more to the regions of the snake
where the feature map values are larger.
In order to overcome the latter of these problems, the points themselves
were not directly used, but instead were used as control points for fitting spline
curves to the snakes. New points were then sampled at evenly spaced intervals from
these curves. These intervals could be made very small by sampling many points.
Spline curves are described in Appendix C.
This still left the problem that the points may rotate throughout the frames. A
method was written to calculate which point in each frame corresponded to the
same point in the first frame. This involved rotating each set of re-sampled snake
points to find the configuration in which they most closely matched the snake points
in the first frame. The similarity was calculated by summing the squared distance
between corresponding points on the two snakes.
Movie 2: Following the motion of individual points on the snake in movie 1.

Calculation of the error in the tracking
Now that a program had been developed that allowed the membrane to be
tracked when judged by eye, a more vigorous method of calculating the error in
tracking was written. This is described in detail in Appedix D. For 5 movies, 3 repeat
trackings were compared with manually drawn curves to get an average difference
in pixels. For 4 of the 5 curves the tracked curve points were, on average, less than 1
pixel from the manually clicked curve and therefore within the error bars of the
manually clicked curve. For the final curve, for the majority of frames the membrane
was well tracked. However for a roughly 20-frame period an adjacent, brighter
membrane was tracked instead of the correct membrane leading to a higher error.
This suggests that, whilst the code works well in most situations, a limitation of the
code is that it may become inaccurate if a second bright membrane comes close to
the one it should be tracking.
The area of blebs through the blebbing cycle
Figures 12 a and b show the area of the bleb cross sections from movies 3
and 7 (Appendix D) plotted against time. Rapid expansion is seen for the first 30
seconds as cytosol rushes into the gap between the cortex and the membrane. As the
cortex begins to reassemble, expansions slows and a peak is reached at around 100s
for the first larger (16μm2) bleb cross-section and at around 50s for the second
smaller (7 μm2) bleb cross-section. Next, the slow retraction begins as the newly
assembled cortex contracts. After around 3μm2 the bleb areas fell below the
minimum number of pixels and were no longer tracked. Through this phase the
larger bleb cross section shrank at a rate of around 0.05μm2/s while the smaller
shrank at a rate of roughly 0.03μm2/s. It is worth noting that these are crosssections and different rates and sizes may be seen if the blebs were studied in a
different plane, but the general behaviour should be the same.
Figure 12 a b: The area of the bleb cross sections shown in a) movie 3 and b) movie 7 plotted
against time.
In ‘coordination of Rho GTPase activities curing cell protrusions’ (Machacek et
al, 2009)80 the start of expansion and the start of retractions of lamellipodia
were used as reference points to align the expression patterns of 3 different
GTPases (RhoA, Rac1 and Cdc42) imaged separately. Expression levels were
compared with edge velocities of the expression area. This allowed them to
study the spatial and temporal behaviour of the 3 proteins at the same time,
overcoming the difficulties involved with simultaneous imaging of different
proteins. They found the location of the maximum expression of the proteins
relative to the membrane as well as the timing of their expression relative to
protrusion events. This could be another use of the code. The program has been
used in this project to track the line marking the maximum expression of
proteins, rather than the leading edge of an expression area. However, the
program defining the feature map already has an option to give an area of
expression, rather than a ridge defining maximum expression and so this could
be used instead.
This paper also used the level-set method as a particularly mathematically
rigorous method of determining which section of the leading edge in one frame
corresponded to each section in the previous frame. An original secondary aim
of my project was to use level set in a similar way to track points on the
membrane with time. The level set method is particularly useful for biological
uses since it deals well with edges that change in a complex manner with time.
However, due to time constraints, this was left for future work. As described
earlier, a simpler method was used instead based on simply sampling the
membrane curves with a large number of points. The points on each frame were
then rotated until they gave the shortest summed distance to the points on the
first frame. Each point could then be tracked through images and used to define
how each particular section of membrane moved in time (movie 2). This method
should work well for membranes whose shape remains qualitatively similar, i.e.
if the area of bleb remains one smooth ‘blob’. Since the blebs membranes were
tracked using snaking, which enforces smoothness and continuity, they in
general evolved in a fairly simple manner. This method may have limitations if
tracking unusually shaped blebs, or if the cut off area is set very low so that the
bleb is tracked as it collapses back into the cell.
A thorough description of the level set method along with it’s application to
tracking points on curves representing GTPases in lamellipodia is given in
‘Morphodynamic profiling of protrusion phenotypes’ (Machacek and Danuser,
2006)81. A description of it’s application to modelling cellular deformations as
well as its use in a more predictive setting is given in the paper ‘Modeling cellular
deformations using the level set formalism’ (Yang et al, 2008)82. A review of the
level set method is given in Appendix E.
APPENDIX -A
Examples of actin cortex contraction and instability in cellular processes

Actin cortex contraction in lamellipodial motility
Lamellipodial motility is the most studied type of cell motion. It involves a
three-step cycle53 in which firstly, a protrusion forms at the front of the cell. Unlike
blebs, this type of protrusion, the lamellipodia, requires actin polymerisation
induced by Rac or Cdc42 localised at the leading edge of the cell54, 55. Next, the cell
adheres to the surrounding substrate before, finally, myosin motors contract the
cortex at the rear of the cell propelling the nucleus and cytoplasm forwards into the
protrusion56-58. Due to the substrate adhesions, this leads to net movement of the
cell. The cortex contractility is induced by Rho GTPases as described in the blebbing
section. Understanding actomyosin assembly and contraction is therefore not only
important for understanding amoeboid motility but also for understanding
lamellipodial motility. It is thought that the microtubule system could be
responsible for ensuring that the proteins inducing polymerisation localise to the
leading edge of the cell whilst the Rho remains localised to the rear59, 60.

Actin cortex contraction in amoeboid motility
The two alternative methods of motility (studied by Sahai and Marshall,
are based on the bleb cycle and depend upon cortical contraction both to
create the protrusion and to pull the cell body forwards25. As described in the
‘blebbing cycle’ section, enhanced acto-myosin contractility can lead to either cortex
rupture as a result of cortical instability or the separation of the membrane of the
cortex. Both of these lead to the membrane expanding outwards as cytosol flows
into the gap.
The first method of motility arising from this involves one large bleb
expanding out, followed by cortical contraction at the rear of the cell and
reassembly of a cortex in the bleb, similar to lamellipodial motility52, 12. Under
normal conditions the cortical contractility is not strong enough to push the nucleus
through the bleb rim52. However, in conditions of enhanced contractility (either
natural12, 61 or artificially induced62, 63, 64) the nucleus no longer limits motion and
200312)
contraction waves are seen13, 52, 65, 66. Due to adhesions of the membrane to the
surrounding matrix this leads to net propulsion in the cell.
The alternative form of motility that arises as a consequence of enhanced
contractility, involves the formation of many blebs at the leading edge of the cell.
This method of motility, which allows the cells to migrate through 3D matrices, was
studied in A375m melanoma cells12. In human neutrophils, a method of motility has
been observed in which one large bleb at the front of the cell drives motility but
several small blebs appear at the rear of the cell as it slows suggesting that multiple
blebs can also be used to slow movement67. This type of motility relies on
Rho/ROCK signalling and does not involve proteolytic breakdown of the
surrounding matrix12.
The spontaneous cortex behaviours leading to one or many ruptures forming
these two types of blebbing are studied in to papers ‘Dynamic modes of the cortical
actomyosin gel during cell locomotion and division’ (Paluch et al, 2006)52 and
‘Cracking up: symmetry breaking in cellular systems’ (Paluch et al, 2006)68. In the
later of these, actin is grown around beads in order to create a simplified model for
spontaneous cortex rupture under tension. The conditions leading to single or
multiple ruptures are studied in the bead system. It is suggested that, as with
nucleation and growth of polymerisation reactions77, many blebs are seen when
‘nucleation’ of rupture is fast relative to growth of the hole, whilst a larger single
bleb is seen in the opposite case. They suggest if cortical tension is well below the
instability threshold, it is unlikely that nucleation will be seen in more than one
place. Especially if the tension is so low that rupture cannot occur spontaneously
and must be induced.

Actin cortex contraction in cytokinesis
The behaviour of the cortex is also important for cytokinesis, both for
determining the position of the cleavage furrow as well as in the segregation of
proteins. Before the onset of division, Rho activity increases, leading to a more
contractile cortex and hence, rounder cells. As cytokinesis begins, a gradient of
cortical contractility leads to cortical flows directed towards the equator69-71 and
there is experimental evidence to suggest that it is these flows that lead to the
creation of a cleavage furrow8, 72. The mechanism leading to these flows is
unclear, but in ‘Site selection for the cleavage furrow at cytokinesis’ (Burgess and
Chang, 2005)76 it is shown that the mitotic spindle plays an important role in
determining the position of the cleavage furrow.
As well as this role in establishing the cleavage furrow, cortical flows have
also been associated with the segregation of proteins. For example, before
division in C. elegans embryos, increased cortical contraction and flows are
seen73, 74. In ‘Cortical flows powered by asymmetrical contraction transport Par
proteins to establish and maintain anterior-posterior polarity in the early C.
elegans embryo’ (Munro et al, 2004)75 it is shown that these flows in turn lead to
the transport of several cortex-associated proteins to specific parts of the cell.
Similar effects have also been seen in later divisions in c. elegans75 as well as in
Drosophila embryos63.
APPENDIX - B
The recruitment of proteins and the assembly of an actin cortex
Some of the proteins, which regulate the actin cytoskeleton are present at the
membrane throughout the blebbing cycle. These include RhoA37, it’s downstream
affecter ROCK10, 39, 40, and Rho GEFs which are activators of RhoA37. However their
role in cortical reassembly is unclear.

Ezrin is the first protein to be recruited
Proteins related to the assembly of the actin cortex are recruited to the bleb
in sequence. The first protein in this sequence is ezrin 37. Ezrin is an ERM protein
that links the actin cytoskeleton to the cell membrane. In cells expressing the
dominant FERM domain of ezrin, the cortex still regrows but the blebs do not fully
retract 37. This is consistent with ezrin’s role in tethering the cortex to the
membrane. The ERM protein Moesin is also recruited to the bleb membrane37.

Actin recruitment and nucleation
Actin is recruited soon after ezrin forming a cortex beneath the bleb
membrane 37. In order to find the actin nucleators responsible for assembling this
structure, Charras et al combined proteomic analysis of isolated cortex-rich blebs
with a localisation/shRNA screen for phenotypes in which the cortex was weaker or
less able to contract38. From this, 2 proteins were found to produce the bulk of the
actin in the cortex. These were the formin Diaph1 and the Arp2/3 complex. When
one of these was depleted, the cell still contained a cortex but when both were
depleted together, the cortex was almost completely lost. This suggests that the two
nucleators act independently. This is similar to the situation seen at the front of
migrating cells. However, in this case the two nucleators are spatially separated,
whilst in blebs they are not. It was shown that either protein contributed roughly
equal amounts of actin, but that Diaph1-mediated assembly happened four times
faster than Arp2/3-mediated assembly.
Depletion of the two nucleators gave opposite phenotypes relative to wild
type cells with Diaph1 depletion leading to larger blebs, whilst Arp2/3 depletion led
to smaller blebs. Therefore, cells may be able to adjust the contribution of each
nucleator in order to fine-tune their cortical mechanical properties.
Two possible causes of the opposite phenotypes are suggested in the paper.
The first is based on the polymerisation speed of the two nucleators. Diaph1 is
faster, and so the blebs may become larger when this is depleted since all of the
actin then has to be nucleated by the slower Arp2/3. When Arp2/3 is depleted, the
blebs would then be smaller due to an increase in the amount of Diaph1 nucleated
actin. Secondly, the difference could be because the depletion of the two nucleators
has opposite effects on the cortical tension since a more contractile cortex leads to
larger blebs.

Bundling proteins and finally motor proteins are recruited
The next proteins to be recruited to the bleb are actin bundling proteins αactinin, coronin and later fibrin. Trypomyosin and myosin, the contractile motor
proteins were the final proteins to be recruited. Once recruited, the myosin localised
to distinct foci whilst the other proteins were distributed evenly throughout the
cortex37.

The actin network doesn’t act statically during myosin driven contraction
Around 50 myosins are initially recruited to start retraction and this number
increases roughly 5-fold throughout retraction. It has been shown that the cortex
cannot simply act statically throughout contraction since not enough myosins are
recruited to buckle it6. This has been calculated by comparing the force exerted by
myosin with the force required to buckle the cortex, which is estimated from the
wavelength of the wrinkles and the cortex bending rigidity41. It has been shown that
actin depolymerisation doesn’t occur during retraction and so this cannot account
for the difference6, 37. One possible explanation is that the stochastic nature of actin
bundling proteins allows actin filaments to occasionally detach, increasing their
effective length and so allowing buckling. A second is that actin is severed
throughout retraction. This severing could be mediated through aip1-coronin-cofilin
since coronin is present in blebs37, 42.

Actin turnover is low once the cortex is established
Once a continuous cortex has been assembled in the bleb, actin turnover is very low.
This has been demonstrated by imaging with fluorescently tagged actin as well as by
treating blebs with cytochalasin D, a protein which caps the barbed ends of actin
filaments inhibiting polymerisation37.
APPENDIX C
Spline Curves49
Spline curves are a useful tool for fitting a curve to a series of data points.
They are convenient for a number of computational reasons. Also, they represent
data well because they treat the data points as ‘control points’, which they must go
close to, but not necessarily pass directly through and therefore a smoother curve is
given.
A spline curve is a curve, (x(s), y(s)), where x and y are spline functions of a
coordinate s. s is a coordinate marking how far along the curve a point is. A spline
function consists of a series of concatenated polynomial segments joined together to
form one continuous function. These polynomials are usually order 3 or 4. Higher
orders would be able to represent more complex curves but would also hugely
increase computation. Therefore, instead of increasing the order of the polynomial
to represent a more complex curve, it is common to increase the number of
segments.
If a function x(s) consists of NB concatenated weighted segments (basis
functions), it is referred to as a B-spline. Each of these basis functions, Bn(s), is
defined over a unit length span of the s-axis and are joined at ‘knots’.
The spline function can be written in the form:
𝑁𝐵−1
𝑥(𝑠) = ∑ 𝑥𝑛 𝐵𝑛 (𝑠)
𝑛=0
where xn are the weights corresponding to each basis function.
𝐵−1
∑𝑁
𝑛=0 𝐵𝑛 (𝑠) = 1 for all s because Bn(s) are basis functions.
The equation can be written in matrix form:
𝑥(𝑠) = 𝐵(𝑠)𝑇 𝑄 𝑥
𝐵(𝑠)𝑇 = [𝐵0 (𝑠) …
𝐵𝑁𝐵−1 (𝑠)]
𝑥0
𝑄𝑥 = [ … ]
𝑥𝑁𝐵−1
There are methods for calculating the basis functions depending on the order
of polynomial used and whether or not the knots are evenly spaced, see the
appendix of reference 49 for more details.
The current equations apply to a bi-infinite spline function and therefore
require and infinite number of basis functions which is, of course, impossible to
program. Real data exists over a finite range and, therefore, finite basis can be
used. These basis sets can either be periodic (as in the case of the membrane)
where the first basis function meets with the final one, or aperiodic. The
mathematics for aperiodic spline functions is slightly more complicated.
For the case of a periodic, regular (evenly spaced knots), quadratic spline
function there is a simple method of finding the weights that can be found in the
appendix of reference 49. There is also a convenient way of storing the basis
functions as a matrix of coefficients for this case. The mathematics surrounding
calculating norms, inner products and other useful properties of spline functions
are also found in chapter 3 of this book.
This shows how a one-dimensional set of points could be represented by a
spline function. However, the membrane in this project spans over 2 spatial
dimensions and therefore requires a 2-dimensional spline function to represent
it. In this case, the data are represented by a curve 𝑟(𝑠) = (𝑥(𝑠), 𝑦(𝑠)) where x
and y are spline functions. In this case the weights xn are replaced by vector
‘control points’, 𝒒𝒏 = (𝑞𝑛 𝑥 , 𝑞𝑛 𝑦 ).
The curve can then be represented as:
𝑁𝐵−1
𝒓(𝒔) = ∑ 𝒒𝒏 𝐵𝑛 (𝑠)
𝑛=0
Once again, the mathematics for calculating properties such as the norm or
inner product in the two dimensional case are given in chapter 3 and the
appendix of reference 49.
APPENDIX D
Error Analysis
Since there was no reference for the ‘correct’ location of the membrane, it was
decided that the best way of getting a curve for comparison was to manually click
several points on the membrane in every 5th frame. Curves were fitted to these
points and then, for each point outputted by the snaking program, the shortest
distance to the reference curve was calculated and the average over all points was
taken. This gives an average error on points for every fifth frame throughout the
bleb cycle. This is of course not perfect since there is error in the manually clicked
curve. When two adjacent membranes were very close or when the bleb was small it
was often difficult to identify exactly where the membrane was. Furthermore, the
brightest part of the membrane is often up to 3-4 pixels thick so both the manual
curve and the snake could be identifying equally likely section of the curve as
membrane whilst giving different coordinates. Another problem was that the snake
curve was a closed loop whilst the bleb membrane was not, so the snake curve may
differ from the reference curve at the bleb rim where the curves do not follow
membrane. As a result of this, an average error of less than 1 pixel was effectively
considered to be tracking the curve as well as can be judged with a manually drawn
reference curve.
The errors were calculated for 5 blebs that I had not previously looked at and so
had not been used in writing and configuring the code (Movies 3-6). Blebs were
chosen which looked quite different in order to test the program in different
situations and to try to find the limitations of the program. Three trackings were
compared to the reference curve in each case to get an average error.

Test bleb 1
Movie 3: The first test tracking. Every 5th frame is shown (25 second intervals). It can be
seen from the images that the membrane is least visible at the start and end of the blebbing
cycle when the bleb is smallest and this is when the largest errors were seen.
The average error on points from the three repeats for movie 3 for all of the
frames was 0.699 pixels suggesting that the curve was tracked very well. As
described above, an average error of less than 1 is considered to be accurate when
comparing to a manually drawn reference curve.
Figure 7 shows the average error for each frame, plotted against frame number
for the bleb. The error is largest at the start and end of the blebbing cycle when the
bleb was smallest. This is expected since the membrane is less bright and sharply
defined in these smaller blebs. For the larger, better-defined curve, an average error
of 0.5-0.6 pixels is seen.
Figure 7: The average error for each frame plotted against frame number for the bleb shown
in movie 3.

Test bleb 2
Movie 4: The second test tracking. Every 5th frame is shown (25 second intervals). A large
error is seen for frame numbers 25-45 (the6-10th images shown) where the snake was tracking
an adjacent membrane to the left of the bleb.
The average error from the three repeats for movie 4 was 0.898. This is larger
than in the previous case since, as can be seen from the movie, for frames 25-45 (the
6-10th images shown) one side of the snake is tracking an adjacent, brighter
membrane, rather than the membrane it should be tracking. This suggests that a
possible limitation of the program is that the snake may jump to nearby, brighter
membranes.
Figure 8 shows the average error for each frame plotted against the frame
number for this bleb. Although the average error for all frames was less than 1, large
peaks of 1.3-1.4 pixels were seen for the frames in which the snake was tracking the
adjacent membrane, as well as at the end when the bleb was small and the
membrane less visible. At other times an average error of 05-0.7 pixels is seen.
Figure 8: The average error for each frame plotted against frame number for the bleb shown
in movie 4.

Test bleb 3
Movie 5: The third test tracking. Every 5th frame is shown (25 second intervals). As can be
seen in the images, the membrane separating the tracked bleb with the smaller bleb to the
right of it becomes very feint between the 5th and 7th images but the snake continues tracking
properly.
The average error on the bleb shown in movie 5 was 0.648 pixels suggesting that
this was well tracked. In this movie, the boundary between the bleb being tracked,
and the bleb to the right of it becomes less well defined (frames 16-31), and so this
tests the code in a situation where the boundary between two blebs is feint. Figure 9
shows the average errors for each frame plotted against frame number for this bleb.
The errors were highest for this bleb at the start when the bleb is small, before
falling to between 0.5 and 0.7 for the remaining frames.
Figure 9: The average error for each frame plotted against frame number for the bleb shown
in movie 5.

Test bleb 4
Movie 6: The fourth test tracking. Every 5th frame is shown (25 second intervals). The
racked bleb forms in the gap between two other blebs and then retracts back to the cell.
The average error for the bleb shown in movie 6 was 0.718, and so the snake
was close enough to the reference to be considered accurate. The bleb in this movie
forms between two existing blebs and then retracts back to the cell, and so tests the
code in unusual circumstances. Figure 10 shows the average errors for each frame
plotted against frame number and, as with the previous bleb, it can be seen that the
error starts relatively high whilst the bleb is small and the membrane is less well
defined, before falling to between 0.5 and 0.8 for the remaining frames.
Figure 10: The average error for each frame plotted against frame number for the bleb shown
in movie 6.

Test bleb 5
Movie 7: The fifth test tracking. Every 5th frame is shown (25 second intervals).
The average error for the bleb shown in movie 7 was 0.699 and so this bleb was
also well tracked. The bleb in this movie moves slightly as a whole while going
through the blebbing cycle since it is pushed by the bleb beneath it. This therefore
suggests that the code can track blebs that move whilst going through the blebbing
cycle.
Figure 11 shows the average error for each frame plotted against the frame
number for this bleb. It can be seen that the error starts high where the bleb is
small. There is a second peak around frame 20.
Figure 11: The average error for each frame plotted against frame number for the bleb shown
in movie 7.
APPENDIX E
The level set method81
The level set method is used to quantify the evolution of complex boundaries,
ensuring that they continuously propagate in their normal direction. It was
suggested by Osher and Sethian (1988)83. Following the derivation in
‘Morphodynamic profiling of protrusion phenotypes’ (Machacek and Danuser,
2006)81, if we have two cell boundaries (αT, αT+1) in frame numbers T and T+1, the
level set method allows the boundary to continuously propagate from the former to
the latter and allow the user to follow local regions of the boundary thoughout. In
level set, the 2D cell boundary αT is converted into a 3D surface ΦT, the level set,
which is the signed distance function from the boundary. I.e. that value of ΦT at each
point is given by its proximity to the nearest boundary
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