Bulletin of Mathematical Biology manuscript No.
(will be inserted by the editor)
Cell Physician: Reading cell motion
A mathematical diagnostic technique through analysis of single cell motion
Hasan Coskun · Huseyin Coskun
Received: August 22, 2010/ Accepted: 2010
Abstract Cell motility is an essential phenomena in almost all living organisms. It is natural to think that behavioral or
shape changes of a cell bear information about the underlying mechanisms that generate these changes. Reading cell motion,
namely, understanding the underlying biophysical and mechanochemical processes is of paramount importance. The mathematical model developed in this paper determines some physical features and material properties of the cells locally through
analysis of live cell image sequences and uses this information for making further inferences about the molecular structures,
dynamics, and processes within the cells, such as, the actin network, microdomains, and chemotaxis. The generality of the
principals used in formation of the model ensures its wide applicability to different phenomena at various levels. Based on
the model outcomes, we hypothesize a novel biological model for collective biomechanical and molecular mechanism of
cell motion.
Keywords cell motility · inverse problem · material parameters · actin network · chemotaxis · lateral signal diffusion ·
microdomains · membrane ruffling · adhesion formation · retrograde flow
1 Introduction
Cell motility is an essential phenomena in almost all living organisms. Organ and tissue formation during development of
an embryo, maintenance of tissues, wound healing, white blood cell movement in immune response, and generation of new
blood vessels, etc., are some examples in which cell motility is essential [1, 4]. On the other hand, pathology of motion
causes many diseases. To name a few, arthritis and neurological birth defects have their roots in cell motility dysfunction.
Increase in the motility of tumor cells may be associated with cancer metastasis. Aside from motion, differences in material
properties of cancer and normal cells are also observed [24, 41, 42].
Recent biological theories explain that the important steps of cell motion are protrusion, adhesion, and contraction of
the cell through the action of the cytoskeleton [4, 11, 28]. Briefly, it is thought that polymerization and bundling of the
actin filaments produce directed force to create protrusion at the cell membrane. The protruding site contacts and adheres
to the substratum. Activation of the actomyosin complex, with depolymerization and unbundling in some cases, causes
contraction at the rear (or near the nucleus) of the cell. In terms of mechanical terminology, the material of the cell is known
to be viscoelastic and heterogeneous according to experimental measurements [5, 26, 43]. It is proposed in [9] that, the
models for single cell motility in the literature can be divided into ‘discrete’ and ‘continuous’ models in physical terms
[2, 3, 8, 9, 12, 13, 21, 29, 30, 37, 44]. The model, called the Anchor Model, introduced in this paper provides an example of
the combination of both; it uses continuum mechanical means on discrete subunits.
Hasan Coskun
Department of Mathematics
Texas A&M University-Commerce
Commerce, TX
E-mail: hasan coskun@tamu-commerce.edu
Huseyin Coskun
Mathematical Bioscience Institute
Ohio State University
Columbus, OH
Corresponding author E-mail: hcoskun@mbi.osu.edu
2
Hasan Coskun, Huseyin Coskun
Cells are living organisms, and it is natural to think that behavioral or shape changes of a cell bear information about the
underlying mechanisms that generate these changes. Reading cell motion, namely, understanding these underlying biophysical and mechanochemical mechanisms of behavioral and shape changes is of paramount importance. This is analogous to the
examination of patients, analysis of their symptoms and behavioral changes by a physician for diagnosis. The mathematical
model developed here plays the role of a physician or the physical exam in diagnosis of cellular pathologies instead of human
subjects. Thus, the technique translates the most obvious physical information, i.e. cell position, into practical and clinically
usable form.
The first attempt in analysis of motion in the literature to derive such useful information is found in [7, 9]. Dynamic material properties such as elasticity and viscosity, quantitative relations between molecular dynamics and continuum mechanics,
and the definition of force generation as a scalar potential field are formulated coherently in [7]. The technique developed in
that work is classified as model based inverse problem to emphasize the difference from the experimental measurements of
the material properties. While forward problem formulates the governing equations of cell motion where a set of material parameters is given, inverse problem formulates equations to extract information from the motion and determines the material
parameters. These parameters in turn can be used as input for the forward problem to mimic the given cell motion. Forward
problem is formulated by a system of ordinary differential equations and inverse problem by linear algebraic systems. The
model based inverse problem approach allows observing and analyzing the motion as it naturally takes place. It seems to be
advantageous over some experimental methods which use physical constraints or, in some cases, external agents that may
possibly interfere with the motion. Another advantage of the technique is its time and cost efficiency in comparison to the
experiments. In line with a series of papers on systematic treatment and analysis of the phenomena, this paper introduces a
novel quantitative technique for extracting local information from the motion as opposed to the global analysis of the Ring
Model (Fig. 1(a)) [9].
Although the Anchor Model provides local information, namely the relevant quantities at discrete points in space at
specific times, it uses continuum mechanical means, more specifically, linear viscoelastic solid material relations to derive
the model formulation. Local analysis keeps the model equations simple with a very efficient computing time regardless of
the dimension, number of subunits used, or the duration of the experiment. Extension of the model to three dimensional
analysis, for example, is immediate after some minor modifications. In the Anchor Model, determination of the evolution
of the points for local analysis is a nontrivial task. In [9] positions of the points are found manually from cell and nuclear
morphology. Later, the process of point tracking is automated through the development of an open access software which
includes extended active snake techniques for better segmentation of cell boundary [36] (see appendix).
This paper further addresses important molecular components of cell motion such as microdomain dynamics, actin
polymerization and network dynamics, myosin–induced cell boundary retraction and cell body contraction, adhesion site
formation, and chemotaxis based on the model outcomes. Polymerization and active myosin concentration, in mechanical
terms protrusion and retraction, are quantified with a linear function of positive and negative normal displacement, respectively, at the cell boundary and these relations are justified through model application to and analysis of the experimental
data. The strain function, then, provides a quantitative tool to track polymerization history and consequently to estimate the
network topology. Motile microdomains/microclusters/lipid rafts have been under investigation as scaffolds for signal transduction and as molecular structures which organize membrane proteins in the literature [6, 27, 38]. We propose that strain
rate indicates spatio-temporal organization of microdomains on the cell membrane and so determines signaling dynamics
and cell’s local chemotactic response.
Besides the Ring Model [9], there have been recent studies that investigate evolution of single cell boundary and regular
pattern formation associated with the analysis of displacement and normal velocities [10, 21, 25]. As similar patterns appear
in all model applications to different cell types, our results agree with the findings that local membrane waves constitute
a universal organizational principle or dynamic pattern of motile cells [10, 25]. The Anchor Model, on the other hand,
is based on novel forward and inverse problems which incorporates strain and strain rates into the analysis, determines
characteristic physical parameters of the cell, and develops a mathematical diagnostic tool for cell profiling, characterization,
and classification based on the model outcomes. Moreover, we hypothesized a novel biological model to address the universal
principle in formation of regular patterns by relating them to microdomain dynamics and focal adhesion site formation. This
model incorporates other important biomechanical and molecular mechanisms of cell motion, such as retrograde flow and
membrane ruffling and addresses their collective behavior. Although transversal wave propagation is observed in our work as
in other studies, the waves are not continuous. This difference maybe due to the non-physical mechanical model or the level
set technique used in which the predetermined speed formulation for evolution of the level sets subsequently determines the
velocity of the zero level set (cell boundary) [25].
The generality of the underlying principles of the model ensures its applicability to many organismal phenomena at
various levels for different interpretations guided by the biology of the phenomena at which the nonrigid morphological
changes are important. Similar to the crawling cell motion which is used for the model applications in this paper, it can also
be applied, for example, to the contractile motion of a beating myocyte at cellular level and that of the heart and the bladder
Cell Physician: Reading cell motion
3
at organ level. Locality of the technique allows use of the model for the dynamics of subcellular structures as well. It can also
be applied to tumor tissue for similar analysis, derivation of the material properties of the cancer tissue and its growth rate,
etc. Through intervention studies, evolution of the cell boundary and underlying molecular dynamics can be investigated for
identification of cancer biomarkers. The model may also prove useful in novel drug development as the effect of the drugs
on single cells can be quantified by this technique.
Since all computations are made based on the analysis of live cell image sequences which are 2D projections of a 3D
motion, and since pixel and frame numbers are used instead of actual space dimensions and time, the quantities determined
and discussed in this paper are mainly approximations. The results are compared with actual experimental measurements
in some cases after necessary unit conversions. Qualitative analysis of each application studied in the paper also justifies
the use and wide applicability of the model to various cell types. The conclusive results, however, can be obtained through
application of the method systematically to a population of specific cell types for comparison. As an example of a systematic
application of the model, the morphological changes and motility of single cancer cells are being investigated and their
comparison to normal cells are being studied for characterization and classification of cancer cells in an ongoing work. The
results will appear in a separate paper.
Other than physical parameters, two other quantitative tools are developed for analysis of model outcomes and interpretation of cell motion; motility index and deformation index. These quantities are introduced and used in the following
sections.
2 The Anchor Model
Let a space-wise parametrization
be given as
St
of the smooth, closed cell boundary, ψ , in counterclockwise direction at a fixed time t
St
: τ ∈ [0, τ̄ ] ⊂ R −→ (x(τ ,t), y(τ ,t)) ∈ R2 ,
(2.1)
and time parametrization of the trajectory of a point on the boundary for a fixed τ , T τ , is given by
Tτ
: t ∈ R −→ (x(τ ,t), y(τ ,t)) ∈ R2 .
(2.2)
For x = (x, y), the boundary condition for a closed boundary and the initial condition becomes
x(τ , 0) = x0 (τ ),
x(0,t) = x(τ̄ ,t).
(2.3)
If a set of discrete points on the boundary at τ = τi are chosen to be
Ti
: t ∈ R −→ xi = (xi (t), yi (t)) = (x(τi ,t), y(τi,t)) ∈ R2 ,
i = 1, . . . , n,
(2.4)
then the local, moving coordinate vectors uti and uni can be approximated as
uti =
xi+1 − xi−1
|xi+1 − xi−1 |
and
uni = C uti
where
C=
cos(π /2) sin(π /2)
0 1
=
-sin(π /2) cos(π /2)
−1 0
(2.5)
is the clockwise rotation matrix. The first point x1 = St (0) is called the pivot point, the superscript t stands for tangential
direction, and subscript n for normal direction with respect to the boundary (Fig. 1(b)).
The velocity, vi = (vti , vni ) := ẋi (t), displacement, di = (d ti , d ni ) := xi (t) − xi (t0 ), and force, σ i = (σ ti , σ ni ), vectors can
be written in terms of local moving coordinates at each point xi as
fi = f ti uti + f ni uni .
(2.6)
The coordinates, then, can be written in terms of the wedge product of fi and coordinate vectors,
1
f ti
fi ∧ uni
=
.
f ni
uti ∧ uni uti ∧ fi
(2.7)
4
Hasan Coskun, Huseyin Coskun
Forward Problem The equation for vi becomes the governing equation of the forward problem:
ẋi = vti uti + vni uni .
(2.8)
Force components, σ α , can be interpreted as principal stress at the point where the force applied. The strain values can be
written, using finite difference formulas, in the form of
γ ti ≈
d ti+1 − d ti
∂ d ti
≈
∂s
△s
γ ni ≈
and
d ni+1 − d ni
∂ d ni
≈
∂s
△s
(2.9)
where
the displacement is discretized as di (t j ) = xi (t j ) − xi (t j−1 ), that is, t0 = t j−1 at each time step t = t j , s(τ ,t) =
Rτ
0 |d St /d τ | d τ is the arc length at time t, and △s = |xi+1 (t j ) − xi (t j )|.
It is known that an actin monomer has a diameter of about 4 − 5 nm. Due to the helical structure of the filaments,
a monomer binding elongates the filament tip about 2.5 nm. Since each step in elongation generates a displacement in
the direction of the filament tip according to the configuration before the polymerization takes place, linear relations are
proposed between the positive part of normal displacement, d ni ,+ , and the average number of polymerized actin, pi , as well
as the negative part of normal displacement, d ti ,− , and the average number of active myosin, mi , in this model. That is,
d ni ,+ (t) = k p pi (t),
d ni ,− (t) = km mi (t)
at the point xi where k p and km are global material proportionality constants and
d ni
(2.10)
=
d ni ,+ − d ni ,− .
Inverse Problem A simple linear viscoelastic constitutive equation can be stated as
σ = Ḡ γ + η̄ γt ,
(2.11)
where σ is stress, γ is strain, γt is strain rate, and Ḡ and η̄ are elasticity and viscosity coefficients respectively. Since the
Anchor Model formulates local analysis in time and space, this choice of linear constitutive equation provides a reasonable
approximation.
All points are assumed to be masses with negligible weigh work against drag forces during the motion due to cellsubstrate interactions. A force balance equation can be stated at each point as σ α − µ α vα = 0 where µ is the drag coefficient
with units of N · s/m3 . The force balance equation then reads, after employing the equation Eq. (2.11) and scaling, as
Gα γ α + η α γtα = vα ,
α = ti , ni ,
(2.12)
where Gα = Ḡα /µ α and η̄ α = η α /µ α with the units of m/s and m respectively. Under homogeneous substrate assumption,
i.e. µ α ≡ const., Gα and η α approximate physical elasticity and viscosity parameters linearly, only by a constant factor. The
parameters quantify material characteristics of cells at the boundary in normal (α = n) and tangential directions (α = t). The
relaxation time is defined in terms of scaled (Gα , η α ) or original parameters (Ḡα , η̄ α ) as
λα =
ηα
η̄ α
= α.
α
G
Ḡ
(2.13)
The relaxation time, with unit of s, is a quantity that measures how close the material is to be an elastic solid (small values)
or a viscous fluid (large values). The mean relaxation time in normal direction λmn of the motion is then defined to be
λmn = mean(mean(λ ni (t j ))),
i
j
j = 1, . . . , m,
(2.14)
The mean relaxation time can similarly be defined in tangential direction; λmt .
Assuming that the material properties are the same for any two neighboring subunits, the parameters can be obtained by
solving
(2.15)
V α , j = Gα Γ α , j + η α Γt α , j , j = 1, . . ., m,
where, for α = ni , for example,
V
ni , j
n , j
v i−1
=
,
vni , j
Γ
ni , j
n , j
γ i−1
=
,
γ ni , j
for each time t = t j . The solution to Eq. (2.15) then becomes
α
α, j
α, j
1
G
V ∧ Γt
=
α, j ∧ V α, j .
ηα
Γ α , j ∧ Γt α , j Γ
(2.16)
(2.17)
Time derivatives of strain are obtained using spline interpolation techniques. When the solutions to the system Eq. (2.17)
have negative values, the built-in optimization routine, lsqlin of Matlab, is used for positive, optimized solutions.
Cell Physician: Reading cell motion
5
3 Results
The model is tested first with arbitrary data sets. A set of data whose visualization resembles motion of a live cell is numerically generated. The given motion of this computational cell, after the parameters (vti , vni ) of the dynamical system Eq. (2.8)
are computed by Eq. (2.7), was accurately reproduced through the solution of the system (results not shown). This indicates
that the model and results are reliable for any set of arbitrary data.
The model is then applied to data from various cell types including neutrophil, keratocyte, epithelial cell, and glioblastoma cell line (Fig. 8 and movies S1-S7). In all applications, the evolution of the pivot point, x1 , is first determined on the
membrane at each time step using the morphology (curvature) of the cell boundary segmented by CellTrack software [36].
Other points are determined uniformly in equal number at each time step with respect to the pivot point (Fig. 1(b)). The number n is taken to be 100 for the applications in this paper. In all cases, for the data extracted from live cell image sequences,
pixel is used as the space unit and frame number in the image sequence is used as the time unit, i.e. t j = j, instead of actual
time course of the motion (see appendix).
In the annotated live cell movies three markers are used; the asterisk (∗) is used for the normal displacement at the initial
point (the points on the cell membrane) and the triangle (△) for the normal velocity at the terminal point of the vectors. The
colors of the markers are determined based on the sign of the quantities they represent; red is used for positive values and
green for the negative values. The cyan circles (◦) represent zero values of either quantities. The size of the vectors indicates
the magnitude of the normal strain, i.e. they actually are γ ni uni (magnified by a factor of 30 for clarity of visualization). The
vectors, thus, compactly summarize information about signs of normal displacement and normal velocity and magnitude of
normal strain at a point on the membrane. The point indices in multiples of 10 are annotated on the live cell images. The
image sequences is included in the supporting online materials in the movie format (Movies S1-S7). In the title of the contour
graphs for displacement, strain, and strain rate, the index, i, is dropped.
3.1 Strain rate indicates spatio-temporal microdomain dynamics and cell’s chemotactic response
The normal strain, γ ni , indicates actin network topology and polymerization history and the normal strain rate, γtni , provides
information about actin network dynamics as discussed below. The strain rate gives insight also on the signaling and/or
microdomain dynamics on the membrane and on chemotaxis.
Strain and strain rates in all applications show branch-like regular patterns formed by the positive (negative) values of
the functions which represent locally synchronous motion. Actin network is a branched structure [31]. The positive strips
(green) of strain contour maps mark polymerizing and actively pushing filament ends at the leading edge, and as a result
provide information on the polymerization history and actin network topology. The positive strips (green) of strain rate
maps mark location of the driving force of the motion or chemotactic agent and indicates signalling and actin network
dynamics, and cell’s local chemotactic response. We propose that formation of the regular patterns on strain and strain
rate maps maybe due to the microdomains which have roles in signal transduction ([22, 38]) or other microclusters of
membrane receptors [6]. All domains with similar functionality will be referred to as microdomains hereafter. Microdomains
are motile and signal may propagate along with their motion through the membrane and activate different filament tips as
they propagate which results in these regular patterns. This is natural in the sense that propagating microdomains spread the
force generated due to the polymerization along the cell boundary instead of concentrating the force at a stationary point
(Fig. 11). In compliance with this interpretation, different filament orientation to the cell front is observed experimentally
[19, 40]. Another experimental justification is the observation in recent studies that lateral signal propagation behavior is
associated with endothelial growth factor receptor (EGFR) activation [34]. Merging (∧) and splitting (∨) stripes (Fig. 2(f))
may indicate inter-talk and physical contact among microdomains [27]. There maybe some other interior molecular dynamics
that regulate spatio-temporal organization of polymerization at the leading edge parallel to the membrane as well [25, 38].
In line with Anchor Model formulation, we use the term of “lateral signal propagation” induced by motile microdomain
dynamics instead of “membrane protrusion or retraction wave propagation” used in some other works [10, 25].
For quantification of the contour graphs for the strain rates, the stripes for positive values are represented by line segments
that completely resides in the strips. The average length of the stripes, ml , which shows the strength of signal, the average
size of the angles between the stripes and horizontal, ma = mean(a, a′ ), which indicates the lateral spread of signal on the
membrane, and the average size of the angles between two merging/splitting stripes, mb = mean(b, b′ ), which represents the
interaction among microdomains are defined accordingly (Fig. 2(f)). The angles a (a′ ) are formed by the strips and positive
(negative) x−axis and the angles b (b′ ) are upward (downward) angles on the graphs. The smaller ma is, the more spread the
signal on the membrane and the smaller mb is, the less the interaction among microdomains. Using these quantities, lateral,
local signal propagation speed, ν , is defined to be ν = cot(ma ), signal persistence, l, is defined to be l = ml sin(ma ), and
region of influence of the signal, r, is defined to be r = ml cos(ma ). Point index, [i], is used as space unit and frame index,
6
Hasan Coskun, Huseyin Coskun
[j], is used as time unit for all these quantities defined on the strain rate maps. As an example, ν has the unit of [i/j] and
quantifies the propagation speed of signal for each cell relative to its boundary arc length. A matrix called motility index,
m, is then defined to be m = [ml , ma , mb ; r, l, ν ]. It should be noted that average size of the angles a and a′ as well as b and
b′ , separately, are very close to each other in each application which indicates a globally uniform lateral signal propagation
speed in both trasnversal and inverse transversal directions.
Based on experimental justifications, it is concluded in [32] that lamellipodium and lamella are kinematically, kinetically, molecularly, and functionally two distinct but spatially colocalized network of actin filaments. Through application of the Anchor model to the cell boundary and the transition line, separately, the motility index found to be mb =
[9.91 i, 32.70◦ , 112.12◦ ; 8.05 i, 5.65 j, 1.47 i/j] and mn = [12.98 i, 14.46◦ , 143.71◦ ; 12.33 i, 3.11 j, 9.80 i/j], respectively. These
results show that, while the strain rate patterns are branch-like at the lamellipodium (cell) boundary (Fig. 4(b)), they are
step-like at the lamella boundary (transition line) (Fig. 4(d)). This may indicate that the signal fully develops and influences
a wider region at the transition line as it diffuses into the cell (Fig. 11, blue line at time step j3 ). The mean relaxation time
is λmt = 1.29 j at the cell boundary and is λmt = 1.19 j at the transition line. This increase indicates an increase in fluidity
at the cell boundary which is consistent with the experimental measurements of [43]. These differences imply that lamella
and lamellipodium are mechanically and structurally different as well and so, they might not be spatially colocolized as suggested in different experimental reports [40] (Movies S3, S4). The mean elasticity as defined in Eq. (2.14) in both normal and
tangential directions, (Gnm , Gtm ), has almost a two fold increase in lamella with the values of (7.6 i/j, 19.3 i/j) in comparison
to the results for lamellipodium (4.1 i/j, 13.6 i/j). These results are also in parallel with the experimental measurements of a
recent study with an even higher, a five fold, increase in the lamella stiffness [17].
For a better visualization of the strain rate patterns and their correlations with corresponding normal displacement functions, auto/cross correlation maps are presented in Fig 5, only for this application. The maps further confirm the existence
of the regular patterns and show the cross correlations of the functions. Space-time correlation map for the functions ϕ (s,t)
and φ (s,t) is defined to be:
C ϕ ,φ (∆ s, ∆ t) = q
R tm R s(τ̄ ,t)
0
0
R tm R s(τ̄ ,t)
0
0
(ϕ (s + ∆ s,t + ∆ t) − ϕm ) (φ (s,t) − φm ) ds dt
q
R R s(τ̄ ,t)
(ϕ (s,t) − ϕm)2 ds dt 0tm 0
(φ (s,t) − φm )2 ds dt
(3.1)
where the mean values ϕm and φm are defined as in Eq. (2.14), s = s(τ ,t) is the arc length at time t ∈ [0,tm ], and τ ∈ [0, τ̄ ] ⊂ R.
3.2 Displacement quantifies actin polymerization and myosin-induced retraction
The results discussed below indicate that the relation proposed, Eq. (2.10), between normal displacement, d ni , and polymerization or active myosin concentration are also consistent. That is, normal displacement can efficiently predict polymerization
or myosin–induced retraction at the cell boundary.
In [32], actin dynamics in potoroo kidney (PtK1) epithelial cells is presented. It is shown in Fig. 2(a) that the results
from the experimental measurements of polymerization and depolymerization in [32] seem, in general, to be consistent with
computational results of the Anchor Model. The pivot points (green marker) for the epithelial cell (PtK1) applications are
chosen to be the first point on the membrane on the right side of the image (Fig. S2). A local view of the membrane is given
in Figs. 2(b)-2(d) for three consecutive time steps, j = 25, 26, 27, for the mid-points i = 47, . . . , 53. Since polymerization is
represented by red and depolymerization by green florescent speckles in the original movie, color analysis of the markers
indicates that the experimental data and model predictions match reasonably well, even though the data extracted from the
original image sequence is an averaged representation of the polymerization whereas the Anchor Model predictions are local
in time and space and the cell boundary for this application was not clearly detectable for tracking due to the coloring.
As an overall look, positive and negative regions of the the contour graph Fig. 2(e) for normal displacement (accumulated
red/blue small islands for i > 50, j < 50, for example) is consistent with the polymerization, depolymerization regime
(protruding and retracting boundary segments) of PtK1 cells (Movies S1, S2). The motility index for this application is
m p = [9.66 i, 30.38◦ , 117.09◦ ; 8.0 i, 5.15 j, 1.67 i/j] and the mean relaxation times are λmn = 1.42 j and λmt = 1.25 j.
PtK1 cells are also treated with cytochalasin D (CytD) to inhibit polymerization of free barbed ends in [32]. The contour
graph of the normal displacement (Fig. 3(a)) indicates where the polymerization take place before and after CytD treatment.
While the positive values of normal displacement form flat horizontal stripes on the graph before the treatment ( j < 55), they
form vertical stripes instead after the treatment ( j > 55). The vertical stripes and active, polymerizing sharp corners formed
after the treatment coincide (Figs. 3(a), 3(b), Movie S5). Quantitatively, the motility indices before and after the treatment,
mc, j<55 = [10.09 i, 41.83◦ , 97.36◦ ; 8.81 i, 4.68 j, 1.98 i/j] and mc, j>55 = [9.33 i, 55.36◦ , 73.93◦ ; 7.08 i, 5.86 j, 1.27 i/j], respectively, indicate that although the signal strength does not change much, the merging angle decreases after the treatment,
which maybe an indication of less inter-talk between the microdomains. The signal speed decreases almost by half from
Cell Physician: Reading cell motion
7
1.98 i/j to 1.27 i/j. The result comply with the experimental report that microcluster formation is abrogated by actin polymerization inhibitor latrunculin A [6]. Converting to actual measurement units (1 pixel ≈ 106 nm, and i = 474 nm, j = 10 s,
i/j = 47.4 nm/s), the difference in the signal speed before and after treatment becomes 94 nm/s − 60 nm/s = 34 nm/s, approximately. It is reported in [43] that cytoskeletal disruption with cytochalasin D made body and/or trailing regions 50%
less elastic and less viscous in locomoting neutrophils. Our computations agree with this experimental result at the leading
edge: similar to the signal speed, mean values for elasticity and viscosity defined as in Eq. (2.14), (Gnm , ηmn ), dropped from
(4.23 i/j, .29 i) to (2.88 i/j, .19 i) almost by half after the treatment. Interestingly, while λmn decreases from 1.46 j to 1.28 j after
the treatment likely due to the domination of stationary filament network after polymerization stops, λmt increases from 1.44 j
to 2.128 j, which indicates an increase in fluidity in tangential direction. It should also be noted that, the experimental result
as reported in [43] for relaxation time (.3 s) at the leading edge of a locomoting neutrophil and our computational result
(λmn = 1.46 j = .15 s) for this PtK1 cell are comparable in magnitude.
3.3 Model outcomes allow cell profiling through analysis of their motion
The correspondence set forth between normal displacement and actin polymerization/contraction and normal strain rate
and signaling dynamics are further justified with three different applications: neutrophil chasing bacteria, keratocyte, and
glioblastoma cell motion. The results outline similarities and differences among different cell types which allow cell profiling
based on the physical parameters obtained from the model and motility and deformation indices.
In Fig. 6(a), an annotated live cell image of the neutrophil and in Fig. 6(d) part of the data extracted from the movie are
presented. The tip of the neutrophil tail (the point of highest curvature) is taken to be the pivot point (green marker). The
annotation is done as before, except the closest points to the two bacteria chased by the neutrophil on the membrane are
marked by black and red markers in Fig. 6. There are some artificial lines and coloring at and around the times t = 33, 56, 72,
in all contour plots. This is due to the camera motion during the movement of the cell, that is likely because of the low
technology of the time the movie is recorded. These lines and accompanying coloring are disregarded in what follows. The
results show that, the normal displacement, d ni , and the polymerization dynamics match as expected (Fig. 6(b)). The strip
of polymerization mimics the bacteria trajectories (black and red markers on the contour plots). It is interesting to see that
the cell is more aggressive in catching the bacteria it started to chase first (red markers). As a result, the strip around the
first bacteria is wider than the other (Fig. 6(b)). The analysis of the strain rate (Fig. 6(e)) shows that the regular branched
patterns are reorganized around the bacteria trajectories, are mostly pointing towards them, as expected, and the highest
values (small, red islands) are located mostly around the trajectories. This indicates that strain rate bear information about
signal for the directed motion and chemotaxis. The location of the chemotactic signal is further marked by both normal
and membrane strains. It is clearly visible that the curved stripes mimics the bacteria trajectories in both cases. Positive
stripes for normal strain (Fig. 6(c)) indicate a forward push, whereas the negative stripes for membrane/tangential strain
(Fig. 6(f)) indicate lateral contraction at these locations likely due to the initiation of phagocytosis: when cell has physical
contact with the bacteria or get very close, it stops and locally contracts in normal direction to get the bacteria inside
(Fig. 6(a)), which generates a strong spatial strain gradient at the location (Movie S6). The motility index for this application,
ml = [9.84 i, 28.94◦ , 117.84◦ ; 8.25 i, 5.16 j, 1.71 i/j], is closest to m p . Among all applications, the relaxation times are the
highest in both normal and tangential directions (λmn = 1.77 j, λmt = 2.89 j) which is a reflection of fluidity and likely a highly
deformable behavior of the cell (assuming a frame rate closer to other applications, see appendix).
Similarly, the model is applied to a keratocyte cell motion (Fig. 7), with the pivot point is chosen to be tip of the
lower (right) wing as being one of the two point of highest curvature (Fig. 8(a)). The model effectively predict where the
polymerization and contraction takes place based on the analysis of normal displacement, which clearly identifies spatiotemporal dynamics of actin and myosin in the cell (Figs. 7(a), 7(c)). There are other computational studies demonstrate,
consistent with our findings, that in the rapidly crawling keratocytes, myosin concentrates at the rear boundary [35]. In spite
of the simple, compartmentalized displacement patterns (Fig. 7(b)), strain rate shows regular, branched patterns as before,
but the strips are larger in this case, likely due to the existence of larger microdomains (Fig. 7(b)). The motility index reads
mk = [8.60 i, 61.40◦ , 53.52◦ ; 7.08 i, 4.72 j, 1.57 i/j] with the smallest merging angle (mb = 53.52◦ ) among all applications.
The angle is even smaller for the rear of the cell (1 < i < 50) with the value of mb = 45.50◦ . This minimal inter-talk among
the microdomains and small lateral signal speed are reflected at the minimal deformations or steady shape of the cell body.
Similarly, the relaxation time (λmn = 1.25 j) is the smallest among all applications and this indicates highly elastic nature
of keratocyte cell boundary likely due to the presence of a dense actin network. Both strain rate and strain shows that the
extreme values (small red/blue islands) takes places at or around the tips of the wings, namely i = 1 or i = 50, which
generates maximum normal stress at those locations (Figs. 7(b), 7(d), Movie S7). These high traction forces are actually
observed experimentally in other studies [14, 23]. It should be noted that, extreme strain/strain rate values also colocalize
with retrograde flow in keratocytes (see Sec. 4.1 and Fig. 11) [17, 39].
8
Hasan Coskun, Huseyin Coskun
Finally, application of the model to glioblastoma cell line (Figs. 9(a)) gives similar results. Normal displacement can
precisely estimate membrane motion (Figs. 9(c)) as it can be seen by comparing the color patterns of the normal displacement
(accumulation of small, red/blue islands for i = 40, . . . , 100 and j = 1, . . . , 35) to movement of the points on the trajectories
(boundary segments with maximum extension/retrection) (Figs. 9(a), 9(c))). Strain and strain rate with similar branched
patterns give the information around where the cell moves more aggressively (red small islands) (Figs. 9(b), 9(d)). In various
experimental studies ([24, 41, 42]) reduced viscosity and elastic strength and consequently 50% more deformability in static
cancerous cells are determined. Cytoskeletal disruption with CytD also made the cell 50% less elastic and less viscous
in locomoting neutrophils as discussed above [43]. This resemblance imply that cancerous cells may have deformation
in signaling mechanisms or actin network function. Computational results of the Anchor Model outlines this parallelism.
The motility index for brain tumor cell, mg = [8.51 i, 54.76◦ , 73.21◦ ; 6.56 i, 5.28 j, 1.31 i/j], is the closest one among all
applications to that of PtK1 cell after CytD treatment, mc, j>55 . The mean relaxation time in both normal and tangential
directions for brain tumor cell, however, are smaller than that for all other applications (λmn = 1.31 j and λmt = 1.15 j). This
indicates more elasticity (small rigidity) of the tumor cell boundary. The difference between experimental observation of
increased cancer cell deformability which indicates more fluidity and our result maybe caused by the measurements at
different stages of cancer, such as precancerous or metastatic stages. It may also be caused by having the experimental
measurements done with cells in static conditions as opposed to our analysis of motile cell. Nevertheless, considering high
standard deviation in the results reported for relaxation time of normal and cancerous cells (2.5 ± 0.7 s and 3.1 ± 1.2 s,
respectively), it is safe to say that the Anchor Model results indicating increasing elasticity for this brain tumor cell matches
some experimental measurements [42].
Another tool developed in this paper for cell profiling through quantification of cell motion is called deformation index,
n. It is a vector quantity with three components: deformation area, na , deformation arc length, nc , and deformation radius,
nr , that is, n j = [na j , nc j , nr j ] for each time step j. If the area of cell is a j at time t j and the non–overlapping area between
the configurations at jth and j + 1st time steps is o j , the deformation area is then defined to be na j = o j /a j . Similarly, if
the total arc length of cell is c j := s(τ̄ ,t j ) at time t j , the deformation arc length is defined to be nc j = (c j+1 − c j )/c j . Let
the radii of the circles which are tangent to the cell boundary from inside and outside be ri and ro , respectively. Finally,
the deformation radius at time t j is then defined as nr j = (rij+1 /roj+1 )/(rij /roj ). Smoothed graphs of the deformation indices
for three applications; neutrophil, keratocyte, and brain tumor cell movements (nl, j , nk, j , and ng, j , respectively) are given in
Fig. 10 for j = 1, . . . , 30. Deformation area for keratocyte seems to be almost constant as expected considering the steady
shape of the cell during its motion (Fig. 10(a)). Deformation arc lengths for all three cells look like sinusoidal. This may be
an indication of a pulsative motion in third direction (z-axis) during cell movement (Fig. 10(b)). Although there are large
fluctuations on the deformation radius graphs for neutrophil and keratocyte, the deformation radius of brain tumor cell looks
like sinusoidal (Fig. 10(c)).
4 Conclusions
We developed a model based inverse problem approach that formulates forward and inverse problems to analyze single cell
movement and deformation. The model determines physical parameters such as elasticity and viscosity using continuum
mechanical tools and kinematic functions such as displacement, strain, and strain rate. These quantities then analyzed for
further inference about the molecular mechanisms that generate the motion. In other words, the molecular mechanisms
are addressed through qualitative reasoning based on the mechanical and physiological cues. We concluded that normal
displacement quantifies actin polymerization and myosin-induced retraction and normal strain rate indicates spatio-temporal
signaling/microdomain dynamics and cell’s chemotactic response. A locally uniform retrograde flow is assumed in the
interpretation of the model outcomes. The boundary motion together with retrograde flow is analyzed in a recent work
through a continuum model formulation [21]. We observed regular patterns in strain/strain rate graphs in all applications
which is an indication of globally uniform, local signal propagation speed and implies a universal principal behind this
behavior. We hypothesized a biological model to address this principle by relating microdomain dynamics to formation of
regular patterns, below. The model incorporates other important molecular mechanisms of cell motion such as adhesion,
membrane ruffling, and retrograde flow and address their collective behavior.
These computational result in compliance with the experimental findings show that the model outcomes provide quantitative tools for mechanical and molecular analysis of single cells through their motion and consequently for diagnostic
prediction of cell abnormalities including malignancy. The motility index, m, deformation index, n, and the mean relaxation
time, λmα , are defined in a way to profile cells through their motion. Although the quantitative results presented in the paper
are not conclusive, similar values within the same cell type (epithelial cells, for example) and differences between the different types show that these quantities can be used for cell profiling, characterization and classification. The model may provide
Cell Physician: Reading cell motion
9
useful information through systematic analysis of cell populations of the same type under different physical and chemical
conditions.
4.1 Collective biomechanical and molecular mechanism of cell motion: Microdomains weave cytoskeleton and their
interactions mark the location for formation of new adhesion sites
Based on the Anchor Model outcomes, we hypothesize a biological model for collective molecular mechanism of cell
motion. We propose that microdomain signaling dynamics organizes cytoskeleton and its interaction with substratum. As
microdomains trigger and maintain active polymerization of actin filaments, their propagation and zigzagging motion on
the membrane generate a highly interlinked network of curved or linear filaments oriented at a wide spectrum of angles to
the cell boundary. Microdomain interaction may also mark the formation of new focal adhesion sites at the cell periphery.
Myosin interaction with actin network then generate membrane retraction/ruffling, retrograde flow, and contractile forces for
forward motion. Finally, continuous application of stress on the old focal adhesion sites could result in the calcium–induced
calpain activation and consequently the detachment of focal adhesions ([16]) which completes the cycle (Fig. 11).
More specifically, we propose that microdomains (Fig. 11, green curves on the cell boundary) may interact each other
as indicated at the merging (∧) and splitting (∨) angles of the strain rate graphs (Fig. 2(f)). This interaction could results
in down–regulation of signaling ([6]) which is manifested as discontinued ∧ formation in the strain rate graphs (Fig. 11,
time step j2 ). After physical interaction, which may include a highly complex set of biochemical reactions that may result in
gaining different functionality ([27]), microdomains may propagate in the opposite directions as manifested in ∨ formation as
well. Co-occurrence of both forms × on the graphs (Fig. 11, time step j3 , Fig. 2(f)). A splitting angle without merging history
(∨ alone) may represent signal initiation by microdomains [38]. As the microdomains move, filament tips are activated
along with, as sunflowers face towards the sun (Fig. 11, lines marked with red end points). This might be the mechanism
responsible for the filaments orientation at various angles to the cell edge and curved filament trajectories as reported in
recent experimental studies [19]. This mechanism could also explain why the proportion of filaments at lower angles to the
cell edge is higher in slowing and pausing lamellipodia when compared with continuously protruding sites [19]. As signal
propagates away from its origin, filaments following the signal become parallel to the cell edge and some of them may
be part of the cell cortex while others, which can survive depolymerization, retrograde flow to the lamellipodium–lamella
interface (Fig. 11, time step j2 ).
Since the positive strips represent active forward push generated by actin polymerization and since the filaments need a
mechanical support for forward push instead of a backward transmission of the force generated, we propose that × shapes
on the strain rate contour graphs indicate location of newly formed adhesion sites (Fig. 11, cyan points at the × intersection). This interpretation and experimental results indicating that some filaments originate from foci at the lamellipodium
front ([19]), high-avidity adhesion could be induced by clustering the membrane microdomains ([20]), lipid microdomain
clustering induces a redistribution adhesion molecules on human T lymphocytes ([27]), central cluster of T cell receptors is
surrounded by a ring of adhesion molecules ([6]), and similar results referenced in these studies seem to be consistent.
Myosin is considered to be a marker for lamellipodium and lamella interface ([19]) and a characteristic of lamella
[17, 32]. Myosin interaction with actin network generates the retraction state, a third stage after protrusion and pause. We
propose that while myosin activation in the inner region encompassed by the lower arms of × cause contraction of cell
body forward, recruitment of myosin at the outer region cause experimentally observed myosin–induced retrograde flow or
membrane ruffling [15, 17, 19] (Fig. 11, magenta arrows at time step j3 ). This might be a reason behind different strain rate
patterns at the cell boundary and the transition line (Fig. 4, Fig. 11, blue line segment at time step j3 ). While condensing
lamella, this mechanism may also sparse cell front for newly growing filament tips.
5 Appendix
A Image Quantification & Data Analysis
A.1 Edge Detection and Point Tracking
Tracking the points on the cell and nuclear membranes is a nontrivial task. In the paper [9] where the Ring Model (Fig. 1(a)) is introduced, point
coordinates are determined manually from cell and nucleus morphology. The process of tracking the points is automated through the development
of an open access software called CellTrack in [36]. The figures Fig. 3 in both [9] and [36] show similar point tracking results, done manually
and automatically, respectively. The software uses a combination of known imaging techniques for a better approximation. In particular the active
snakes algorithm of [18] is extended by adding a new term, Ematch , for a finer match of the cell shape.
10
Hasan Coskun, Huseyin Coskun
A.2 Filtering and Smoothing the Parameters
The results for the parameter values are filtered to exclude outliers from subsequent computations. The values that are more than three standard
deviation away from the mean of the parameter in question is replaced by the average value of the function within a 3 × 3 space-time window at
each time step. This filtering process followed by convolution with the Gaussian of the support of the same window. The contour graphs for the
displacement, strain, and strain rate are presented without a Gaussian convolution after the filtering.
A.3 Live Cell Applications
The model is applied to data from various cell types including neutrophil, keratocyte, epithelial cells, and glioblastoma cell line. In all cases pixel
is used as the space unit and frame number in the image sequence is used as time unit instead of the actual time course. Each movie has a different
frame rate: 5 min for the glioblastoma cell lines [9] , 10 s for epithelial cells [32], and 15 s for keratocyte [33]. The frame rate for neutrophil motion
is not recorded in the original movie source. The sources of the movies are as follows:
1. Brain tumor cell motion: The movie for brain tumor, glioblastoma cell line, is taken from the paper [9].
2. Epithelial cell motion: The movies for potoroo kidney (PtK1 ) epithelial cells (S6, S7, S9) are taken from the paper [32].
3. Keratocyte motion: The movie for keratocyte motion (Keratocyte actin-based motility) is taken from the Theriot Lab [33]:
http://cmgm.stanford.edu/theriot/movies.htm#Current
4. Neutrophil chasing bacteria: The movie for a crawling neutrophil chasing two bacteria is taken from the Fenteany Lab:
http://www.biochemweb.org/fenteany/research/cell migration/neutrophil.html
The graphs in Figs. 8(a), 6(d), 8(b), and 9(a) are constructed using the data obtained from the movies S5, S6, S7, and the movie in the paper [9],
respectively, as listed below. Each curve on a graph represents the position of the plasma membrane of the cell at a particular time step. In each case
100 points are chosen on the membranes at each time step. The point indices in multiples of 10 are marked blue and annotated on the figures for
only the initial configuration. The pivot points are marked green in each case.
B Legends to Supporting Movies
Three markers are used in the annotated live cell movies. The asterisk (∗) is used for the normal displacement at the initial point (the points on the
cell membrane) and the triangle (△) for the normal velocity at the terminal point of the vectors. The colors of the markers are determined based
on the sign of the quantities they represent; red is used for positive values and green for the negative values. The cyan circles (◦) represent zero
values of either quantities. The size of the vectors indicates the magnitude of the normal strain, i.e. they actually are γ ni uni (magnified by a factor
of 30 for clarity of visualization). The vectors, thus, compactly summarize information about signs of normal displacement and normal velocity
and magnitude of normal strain at a point on the membrane. The point indices in multiples of 10 are annotated on the live cell images. The image
sequences are included as supporting online materials in the movie format (Movies S1-S7).
Movie S1 Annotated PtK1 epithelial cell membrane motion shows the relation between displacement and polymerization or depolymerization.
See the Results section of the paper for further explanation. The original movie is S6 in [32].
Movie S2 Annotated PtK1 epithelial cell membrane local motion (i = 47,... ,53) shows the relation between displacement and polymerization
or depolymerization. See the Results section of the paper for further explanation. The original movie is S6 in [32].
Movie S3 Annotated PtK1 epithelial cell (lamellipodium) membrane motion. See the Results section of the paper for further explanation. The
original movie is S9 in [32].
Movie S4 Annotated PtK1 epithelial cell lamella (transition) line motion. See the Results section of the paper for further explanation. The original
movie is S9 in [32].
Movie S5 Annotated PtK1 epithelial cell treated with CytD. See the Results section of the paper for further explanation. The original movie is
S7 in [32].
Movie S6 Annotated movie of a neutrophil chasing bacteria. See the Results section of the paper for further explanation. The source of the movie
is listed in the previous section.
Movie S7 Annotated movie of a keratocyte. See the Results section of the paper for further explanation. The source of the movie is listed in the
previous section.
Cell Physician: Reading cell motion
11
xi
i−1
x
i+1
x
u
u
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s
i
f
i−1
i
uni
uti
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ui
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un
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x
i
x
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x
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i−1
(a) Ring Model
Fig. 1 Schematic representation of the Ring and Anchor Models.
time t = 26 for the points i = 1 to 100
60
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50
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80
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99
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1
(a) annotated live cell image of PtK1 epithelial cell
time t = 25 for the points i = 47 to 53
time t = 26 for the points i = 47 to 53
time t = 27 for the points i = 47 to 53
60
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d
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(d) local view of (a)
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Fig. 2 Application to epithelial cell membrane motion for protrusion/retraction analysis (see the Results section for details)
12
Hasan Coskun, Huseyin Coskun
dn
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Fig. 3 Application to epithelial cell membrane motion before and after CytD treatment (see the Results section for details)
References
1. Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walter, P.: Molecular Biology of the Cell. Garland, New York
(2002)
2. Alt, W., Dembo, M.: Cytoplasm dynamics and cell motion: two-phase flow models. Math. Biosci. 156, 207–228 (1999)
3. Bottino, D., Mogilner, A., Roberts, T., Stewart, M., Oster, G.: How nematode sperm crawl. J. Cell Sci. 115, 367–384
(2002)
4. Bray, D.: Cell Movements. Garland, New York (2001)
5. Caille, N., Thoumine, O., Tardy, Y., Meister, J.J.: Contribution of the nucleus to the mechanical properties of endothelial
cells. J. Biomech. 35(2), C177–C188 (2002)
6. Campi, G., Varma, R., Dustin, M.L.: Actin and agonist mhcpeptide complexdependent t cell receptor microclusters as
scaffolds for signaling. The Journal of Experimental Medicine 202(8), 1031–1036 (2005)
7. Coskun, H.: Mathematical models for cell movements and model based inverse problems. Ph.D. thesis, University of
Iowa (2006)
8. Coskun, H.: A continuum model with free boundary formulation and the inverse problem for ameboid cell motility
(2009). Preprint
Cell Physician: Reading cell motion
13
time t = 2 for the points i = 1 to 100
30
40
50
60
time t = 2 for the points i = 1 to 100
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70
20
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99
70
10
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50
20
30
40
10
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1
90
99
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(c) lamella (transition) interface
γtn
γn
t
2
6
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1.5
90
80
1
80
70
0.5
70
5
4
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0
−0.5
50
60
2
time
time
60
50
1
−1
40
40
−1.5
30
0
30
−2
20
−1
20
−2.5
10
−2
10
−3
10
20
30
40
50
60
70
80
point indices
(b) normal strain rate at cell boundary
90
10
20
30
40
50
60
70
80
90
point indices
(d) normal strain rate at transition interface
Fig. 4 Application to epithelial cell boundary and transition interface between the lamellipodium and lamella for the analysis of differences between
lamellipodium and lamella (see the Results section for details)
9. Coskun, H., Li, Y., Mackey, M.A.: Ameboid cell motility: A model and inverse problem, with an application to live cell
imaging data. Journal of Theoretical Biology 244(2), 169–179 (2007)
10. Dbereiner, H.G., Dubin-Thaler, B.J., Hofman, J.M., Xenias, H.S., Sims, T.N., Giannone, G., Dustin, M.L., Wiggins,
C.H., Sheetz, M.P.: Lateral membrane waves constitute a universal dynamic pattern of motile cells. Phys. Rev. Lett.
97(3), 038,102– (2006)
11. Defilippi, P., Olivo, C., Venturino, M., Dolce, L., Silengo, L., Tarone, G.: Actin cytoskeleton organization in response to
integrin-mediated adhesion. Microsc. Res. Tech. 47, 67–78 (1999)
12. DiMilla, P.A., Barbee, K., Lauffenburger, D.A.: Mathematical model for the effects of adhesion and mechanics on cell
migration speed. Biophys. J. 60, 15–37 (1991)
13. Dong, C., Skalak, R.: Leukocyte deformability: finite-element modeling of large viscoelastic deformation. J. Theor.
Biol. 158, 173–193 (1992)
14. Fournier, M.F., Sauser, R., Ambrosi, D., Meister, J.J., Verkhovsky, A.B.: Force transmission in migrating cells. The
Journal of Cell Biology 188(2), 287–297 (2010)
15. Giannone, G., Dubin-Thaler, B.J., Rossier, O., Cai, Y., Chaga, O., Jiang, G., Beaver, W., Dbereiner, H.G., Freund, Y.,
Borisy, G., Sheetz, M.P.: Lamellipodial actin mechanically links myosin activity with adhesion-site formation. Cell
128(3), 561–575 (2007)
16. Glading, A., Lauffenburger, D.A., Wells, A.: Cutting to the chase: calpain proteases in cell motility. Trends in Cell
Biology 12(1), 46–54 (2002)
17. Gupton, S.L., Anderson, K.L., Kole, T.P., Fischer, R.S., Ponti, A., Hitchcock-DeGregori, S.E., Danuser, G., Fowler,
V.M., Wirtz, D., Hanein, D., Waterman-Storer, C.M.: Cell migration without a lamellipodium. The Journal of Cell
Biology 168(4), 619–631 (2005)
18. Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. International Journal of Computer Vision pp.
321–331 (1987)
19. Koestler, S.A., Auinger, S., Vinzenz, M., Rottner, K., Small, J.V.: Differentially oriented populations of actin filaments
generated in lamellipodia collaborate in pushing and pausing at the cell front. Nat Cell Biol 10(3), 306–313 (2008)
20. Krauss, K., Altevogt, P.: Integrin leukocyte function-associated antigen-1-mediated cell binding can be activated by
clustering of membrane rafts. Journal of Biological Chemistry 274(52), 36,921–36,927 (1999)
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Fig. 5 Auto correlations of strain rate functions (Figs. 4(b), 4(d)) and their cross correlations with corresponding displacement functions:
The second rows in each graph represent equal space, C(∆ s,0), and time, C(0, ∆ t), correlation maps, where ∆ s and ∆ t are space and time lags,
respectively. (a) Auto correlation of strain rate γtn presented in Fig. 4(b). (b) Cross correlation of strain rate γtn presented in Fig. 4(b) and corresponding normal displacement function d n (displacement function is not shown separately). (c) Auto correlation of strain rate γtn presented in
Fig. 4(d). (d) Cross correlation of strain rate γtn presented in Fig. 4(d) and corresponding displacement function d n (displacement function is not
shown separately) (see the Results section for details).
21. Kuusela, E., Alt, W.: Continuum model of cell adhesion and migration. Journal of Mathematical Biology 58(1), 135–161
(2009)
22. Laude, A.J., Prior, I.A.: Plasma membrane microdomains: Organization, function and trafficking (review). Molecular
Membrane Biology 21(3), 193–205 (2004)
23. Lee, J., Leonard, M., Oliver, T., Ishihara, A., Jacobson, K.: Traction forces generated by locomoting keratocytes. The
Journal of Cell Biology 127(6), 1957–1964 (1994)
24. Lekka, M., Laidler, P., Gil, D., Lekki, J., Stachura, Z., Hrynkiewicz, A.Z.: Elasticity of normal and cancerous human
bladder cells studied by scanning force microscopy. European Biophysics Journal 28(4), 312–316 (1999)
25. Machacek, M., Danuser, G.: Morphodynamic profiling of protrusion phenotypes. Biophysical Journal 90(4), 1439–1452
(2006)
26. Marella, S.V., Udaykumar, H.S.: Computational analysis of the deformability of leukocytes modeled with viscous and
elastic structural components. Phys. Fluids 16(2), 244–264 (2004)
27. Mitchell, J.S., Kanca, O., McIntyre, B.W.: Lipid microdomain clustering induces a redistribution of antigen recognition
and adhesion molecules on human t lymphocytes. J Immunol 168(6), 2737–2744 (2002)
28. Mogilner, A., Edelstein-Keshet, L.: Regulation of actin dynamics in rapidly moving cells: a quantitative analysis. Biophys. J. 83, 1237–1258 (2002)
29. Mogilner, A., Marland, E., Bottino, D.: A minimal model of locomotion applied to the steady gliding movement of fish
keratocyte cells. In: Maini, P.K., Othmer, H.G. (eds.) Mathematical Models for Biological Pattern Formation, IMA Vol.
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Fig. 6 Application to neutrophil motion (see the Results section for details)
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Fig. 7 Application to keratocyte motion (see the Results section for details)
Keratocyte Motion
Epithellial Cell Membrane Motion
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Cell Physician: Reading cell motion
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Fig. 9 Application to brain tumor cell motion. (see the Results section for details)
Math. Appl., vol. 121, pp. 269–294. Frontiers in Application of Mathematics, Springer, New York (2000)
30. Mogilner, A., Verzi, D.: A simple 1-D physical model for the crawling nematode sperm cell. J. Stat. Phys. 110, 1169–
1189 (2003)
31. Pollard, T., Blanchoin, L., R.D., M.: Biophysics of actin filament dynamics in nonmuscle cells. Annu. Rev. Biophys.
Biomol. Struct. 29, 545576 (2000)
32. Ponti, A., Machacek, M., Gupton, S.L., Waterman-Storer, C.M., Danuser, G.: Two distinct actin networks drive the
protrusion of migrating cells. Science 305(5691), 1782–1786 (2004)
33. Ream, R.A., Theriot, J.A., Somero, G.N.: Influences of thermal acclimation and acute temperature change on the motility
of epithelial wound-healing cells (keratocytes) of tropical, temperate and antarctic fish. J Exp Biol 206(24), 4539–4551
(2003)
34. Reynolds, A.R., Tischer, C., Verveer, P.J., Rocks, O., Bastiaens, P.I.H.: Egfr activation coupled to inhibition of tyrosine
phosphatases causes lateral signal propagation. Nat Cell Biol 5(5), 447–453 (2003)
35. Rubinstein, B., Fournier, M.F., Jacobson, K., Verkhovsky, A.B., Mogilner, A.: Actin-myosin viscoelastic flow in the
keratocyte lamellipod. Biophysical Journal 97(7), 1853–1863 (2009)
36. Sacan, A., Ferhatosmanoglu, H., Coskun, H.: Celltrack: an open-source software for cell tracking and motility analysis.
Bioinformatics 24(14), 1647–1649 (2008)
37. Schmid-Schönbein, G., Kosawada, T., Skalak, R., Chien, S.: Membrane model of endothelial cells and leukocytes. A
proposal for the origin of a cortical stress. J. Biomech. Eng. 117, 171–178 (1995)
38. Simons, K., Toomre, D.: Lipid rafts and signal transduction. Nat Rev Mol Cell Biol 1(1), 31–39 (2000)
39. Vallotton, P., Danuser, G., Bohnet, S., Meister, J.J., Verkhovsky, A.B.: Tracking retrograde flow in keratocytes: News
from the front. Mol. Biol. Cell 16(3), 1223–1231 (2005)
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Fig. 10 Components of the deformation index for comparison of three model applications; neutrophil, keratocyte, and brain tumor (glioblastoma)
cell line motion. (see the Results section for details)
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Fig. 11 Schematic representation of the collective biomechanical and molecular mechanism of cell motion: Actin network is represented
by vertical, thin, black lines for clear visualization of the idea presented. Filaments marked with red ends represent actively pushing, polymerizing filaments following signal by the microdomains. Green curves on the membrane represent microdomains and left/right arrows represent
microdomain/signal propagation speed, ν . Cyan dots represent integrins and yellow dumbbells represent myosin family proteins. Blue line segment
represents the region pulled forward by the actomyosin system (Fig. 4(d)). Up and down thick, magenta arrows represent cell contraction and
myosin induced retrograde flow, respectively (see the Conclusions section for details).
40. Vallotton, P., Small, J.V.: Shifting views on the leading role of the lamellipodium in cell migration: speckle tracking
revisited. J Cell Sci 122(12), 1955–1958 (2009)
41. Ward, K., Li, W., Zimmer, S., Davis, T.: Viscoelastic properties of transformed cells: role in tumor cell progression and
metastasis formation. Biorheology 28(3-4), 301–13 (1991)
42. Wottawah, F., Schinkinger, S., Lincoln, B., Ebert, S., Mller, K., Sauer, F., Travis, K., Guck, J.: Characterizing single
suspended cells by optorheology. Acta Biomaterialia 1(3), 263–271 (2005)
43. Yanai, M., Butler, J.P., Suzuki, T., Sasaki, H., Higuchi, H.: Regional rheological differences in locomoting neutrophils.
Am. J. Physiol. Cell Physiol. 287, C603–C611 (2004)
44. Yeung, A., Evans, E.: Cortical shell-liquid core model for passive flow of liquid-like spherical cells into micropipets.
Biophys. J. 56, 139–149 (1989)