Analysis of protrusion dynamics in amoeboid cell motility by means of regularized contour flows
Daniel Schindler, Ted Moldenhawer, Maike Stange, Carsten Beta, Matthias Holschneider, Wilhelm Huisinga
AAnalysis of protrusion dynamics in amoeboid cellmotility by means of regularized contour flows
Daniel Schindler , Ted Moldenhawer , Maike Stange , Carsten Beta ,Matthias Holschneider , Wilhelm Huisinga Institute of Mathematics, University of Potsdam, 14476 Potsdam, Germany, Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany* [email protected]
Abstract
Amoeboid cell motility is essential for a wide range of biological processes includingwound healing, embryonic morphogenesis, and cancer metastasis. It relies on complexdynamical patterns of cell shape changes that pose long-standing challenges tomathematical modeling and raise a need for automated and reproducible approaches toextract quantitative morphological features from image sequences. Here, we introduce atheoretical framework for obtaining smooth representations of the spatiotemporalcontour dynamics from stacks of segmented microscopy images. Based on a Gaussianprocess regression we propose a one-parameter family of regularized contour flows thatallows us to continuously track reference points (virtual markers) between successive cellcontours. We use this approach to define a coordinate system on the moving cellboundary and to represent different local geometric quantities in this frame of reference.In particular, we introduce the local marker dispersion as a measure to identify localizedmembrane protrusion and provide a fully automated way to extract the properties ofsuch protrusions, including their area and growth time. We use time-lapse microscopydata of the social amoeba
Dictyostelium discoideum to illustrate our approach.May 27, 2020 1/22 a r X i v : . [ q - b i o . CB ] M a y ntroduction Amoeboid motion is one of the most wide-spread forms of cell motility in the livingworld [1]. It plays a key role in many essential functions of the human body, such asresponses of the immune system [2] or the healing of injured tissue [3]. Its medicalrelevance also extends to the field of cancer research, as metastatic tumor cells rely onamoeboid motility to invade the surrounding tissue [4]. Amoeboid locomotion is basedon dynamical changes of the cell shape. Specifically, localized protrusions of the cellmembrane, often called pseudopodia, that are extended in the direction of motion, aregenerally seen as the basic morphological entities that drive amoeboid movement [5].Together with membrane retraction at the back of the cell body, their extension resultsin a displacement of the center of mass of the cell. This requires, besides thecoordinated pattern of protrusion and retraction, also the formation and rupture ofadhesive contacts to a substrate or to a surrounding extracellular matrix [6].The mechanical forces that drive the shape changes of amoeboid cells are generatedby the actin cytoskeleton, a dynamic filament meshwork at the inner face of the cellmembrane [7]. The main building blocks of this network are actin filaments that aresubject to a constant turnover by polymerization and depolymerization, resulting in acontinuous rapid reorganization of the network structure. This process is assisted by ahost of auxiliary cytoskeletal proteins that initiate the nucleation, capping, or severingof actin filaments as well as filament bundling, branching, and membrane cross-linking.The cytoskeletal machinery is orchestrated by biochemical signaling pathways thatcoordinate the spatiotemporal patterns of activity in the actin system across the cellcortex [8]. These upstream signaling pathways also provide a link to membrane receptorsignals, so that cells may react to gradients of extracellular cues by moving directionallytowards a chemical source—a process commonly referred to as chemotaxis [9].The mathematical modeling of amoeboid motion is a long-standing challenge thathas been addressed at different levels of complexity, ranging from random walk modelsfor the center of mass of the cell [10–12] up to detailed high-dimensional models for theintracellular signaling activity [13]. Typically, current models focus on selectedmechanistic aspects of amoeboid motility and describe, for example, actin dynamics,cell-to-cell variability, or the switching between different migratory modes in moredetail, see for example [14–16]. Also, first attempts have been made to incorporateseveral key components, such as dynamic signaling patterns, polarity formation, andcytoskeletal activity, in a modular approach [17]. These models, however, remainqualitative and their comparison with experimental data oftentimes relies on visualinspection. As the entire biological process involves many hundreds of interactingproteins and signalling molecules, with many mechanistic details yet unknown, aquantitative model that includes the full molecular details remains out of reach.To advance the mathematical description of amoeboid motility, we envision thatcurrent efforts of mechanism-based modeling are complemented by a more systematic,data-driven approach. This requires a mathematical framework that allows us tosystematically develop a quantitative model based on experimental data. Such aframework should rely on observables that encode the key characteristics of amoeboidmotility and, at the same time, are readily accessible experimentally. Trajectories of thecenter-of-mass of the cell can be easily recorded in large amounts from low-resolutionbright-field microscopy data but reflect only a very limited, integral information on theentire process. The intracellular signaling pathways and the cytoskeletal mechanisms,on the other hand, are difficult to access and knowledge on this part of the systemremains highly incomplete. We therefore concentrate on the cell shape as the centralreference quantity. The cell shape is fully accessible by standard microscopy techniquesand can be easily recorded with sufficient spatial and temporal resolution. Moreover, itsdynamic evolution implicitly reflects the intracellular processes and determines theMay 27, 2020 2/22enter-of-mass trajectory of the cell.In our long-term quest for a quantitative, data-driven model of amoeboid motility,several steps are required: First, the development of a mathematical framework for thedescription of experimentally observed shape dynamics; second, the design of a model ofthe contour dynamics that predicts realistic shape evolutions; and finally, theincorporation of mechanistic information on key intracellular processes as the drivingdeterminants of the contour dynamics. These become accessible by imaging offluorescently tagged fusion proteins and by more advanced methods, such asknock-sideways and optogenetic approaches. Here, existing mechanistic models mayprovide a useful basis and might merge into a joint modelling concept.In this article, we concentrate on the first aspects of the above agenda. We provide amathematically well-defined approach that allows for a detailed analysis of the complex,multifaceted contour dynamics of amoeboid cells. A key ingredient is the concept ofregularized flows between contours that define an evolution of virtual markers in time.Using contour flows, we define a coordinate system on the evolving contours (stronglyregularized case) and approximate local quantities of interest (weakly regularized case).In the spirit of previous approaches, we rely on the widely used concept of kymographsto graphically represent the space-time dynamics of different geometric quantities, suchas the speed of membrane displacement or the local curvature, along the cell contour.Based on these geometric quantities, we propose a novel criterion based on a singledynamic quantity for defining membrane protrusions/retractions. Previous approachesto identify protrusions relied on the simultaneous matching of multiple criteria, such ascritical values of curvature, protrusion speed, and pseudopod lifetime [18] or identifiedprotrusion events as points in time only [19], without providing major protrusionproperties, such as area growth rate, shape and others. Moreover, all existingapproaches have in common an undesirable blending of the protrusion-defining criteriawith their numerical implementation. This makes it difficult to discern biological effectsfrom numerical artefacts. We instead first define our criterion in mathematical termsand only subsequently implement it numerically, allowing to control numerical errorsand avoid artefacts.The article is organized as follows. We first develop the mathematical frameworkand introduce the concept of regularized contour flows. We then illustrate our approachin applications to experimental recordings of the social amoeba
Dictyosteliumdiscoideum , a widely used model organism for the study of amoeboid motility. Finally,based on the results of our analysis, we illustrate in the Discussion section how weenvision the next steps towards a quantitative, data-driven model of amoeboid motility,based on a point process of protrusive activity.
Methods
Data acquisition and image segmentation
Experiments were performed with AX2 cells of the social amoeba
Dictyosteliumdiscoideum . As a marker for filamentous actin, cells expressed fluorescently taggedLifeact (C-terminally fused with mRFP, plasmid kindly provided by Igor Weber, Zagreb,HR). They were grown at 20 ◦ C in liquid culture flasks containing HL5 mediumincluding glucose (Formedium, Hunstanton, GB) and 10 µ g/ml G-418 sulfate (CaymanChemical Company, US) as a selection marker. Before each experiment, cells wereharvested from culture flasks and grown in 25 ml overnight shaking cultures at 180 rpmunder otherwise identical conditions. Afterwards, nutrients were removed bycentrifugation and washing of the cell pellet with Sørensen phosphate buffer at pH 6(14.7 mM KH PO , Merck, Darmstadt, DE; 2 mM Na HPO × H O, Merck, Darmstadt,May 27, 2020 3/22E). Then, cells were resuspended in fresh buffer and droplets were formed in a Petridish to initiate the streaming processes.For image acquisition cells were transferred after 5 hours to a glass bottom dish(Fluorodish, ibidi GmbH, Martinsried, DE). During imaging, they were kept in Sørensenphosphate buffer at 20 ◦ C. Images were taken with a Zeiss LSM780 laser scanningconfocal microscope (Carl Zeiss AG, Oberkochen, DE) at a frame rate of one image persecond, using a 63 × or 40 × oil immersion objective. Fluorophores were excited at651 nm and emission was recorded between 562 nm and 704 nm. For details seealso [15], where the same data set was used in a different context.Fluorescence image (8-bit gray scale) were segmented using a modified version of theactive contour (snake) algorithm described in [19, 20]. Based on this algorithm, weparameterized the cell boundary in each frame by a closed string of M = 400equidistant nodes. As frames were taken at discrete time points t , . . . , t K − with equaltime difference δt = t k +1 − t k = 1 sec, we denoted the discrete representation of the cellcontour in a given frame at time t k as( x k, , y k, ) , . . . , ( x k,M − , y k,M − ) ∈ R . (1)In the result section, the datasets consists of K = 500 to 1000 time frames. For laterreference, we set t = 0 and t K − = T . Estimate of contour dynamics
We used a real valued smoothing spline for the x and y coordinates based on Gaussianprocess regression using a Poisson kernel, for details see File S1. This yielded aparametrization of the contour Γ k at time frame t k = k · δt Φ k : [0 , π ) (cid:51) θ (cid:55)→ ( x ( θ ) , y ( θ )) ∈ R (2)in terms of a finite sum of smooth kernels (see e.g. [21, 22] for details) x ( θ ) = M − (cid:88) m =0 c m P ( θ m − θ ) , y ( θ ) = M − (cid:88) m =0 d m P ( θ m − θ )where P is a suitably scaled Poisson kernel. Support points were chosen to correspondto normalized secant length along the contour θ k,m = 2 π (cid:80) mi =0 d k,i (cid:80) M − i =0 d k,i , d k, = 0 , d k,i = (cid:0) ( x k,i − x k,i − ) + ( y k,i − y k,i − ) (cid:1) / for m = 0 , . . . , M − k = 0 , . . . , K −
1, see File S1 for details. We parametrized thecontour in the mathematical positive sense, i.e., the interior of the cell is on the leftwhen going around the contour with increasing θ . In the numerical implementation, weused the rescaled arclength coordinates, which we denoted again by θ . This gives (cid:107) ∂ θ Φ k ( θ ) (cid:107) ≡ L k π , with L k = (cid:90) π (cid:107) ∂ θ Φ k ( θ ) (cid:107) d θ, (3)denoting the length of contour Γ k . Note that Φ k is only uniquely determined up to aphase shift, i.e., for every Φ k and τ , also Φ k,τ ( · ) = Φ k ( · + τ ) is a valid parametrizationof Γ k . The phase shift was chosen by an additional requirement in the next section.The smooth parametrization Φ k allowed us compute local quantities along thecontour, e.g., its curvature κ = R π/ ∂ θ Φ k ( θ ) · ∂ θ Φ k ( θ ) (cid:107) ∂ θ Φ k ( θ ) (cid:107) , (4)May 27, 2020 4/22here R π/ is an anti-clockwise rotation by π/
2. As global quantity, we determined thecenter of mass C k of contour Φ k as C k = 12 π (cid:90) π Φ k ( θ ) dθ. (5)Since the segmentation points were rather densely spaced over the contour, they wellconstrained the smooth contours. Based on the kernel representation, all geometricquantities, such as arclength and curvature were defined intrinsically for each contour,and may be easily computed numerically and with high precision.Connecting the contours along the time axis, however, is not intrinsically welldefined, and is bound to choices. In a first step, we constructed a global coordinate flow,which served as a reference frame for further local flows to be defined subsequently. Inthe limit of infinitely densely sampled contours, this global coordinate system results ina mapping Φ : [0 , T ] × [0 , π ) → R , ( t, θ ) (cid:55)→ Φ( t, θ ) . Vice versa, any coordinate system defines a coordinate flow over the tube of contours,i.e., the cell contours in 2d mapped into a 3d space-time coordinate system. If p = ( x , y ) is a point on the first contour at t = 0 and if θ is its arclength coordinate,then t (cid:55)→ Φ( t, θ ) corresponds the movement of p over the space–time tube of contours,see Fig S1. Of note, this coordinate flow is a theoretical construct that allows us toanalyze amoeboid contour dynamics and should not be misinterpreted as a flow ofspecific membrane lipids or proteins. Nevertheless, such a global coordinate system isuseful and allows for graphical visualization of the contours and any locally definedquantity in form of a kymograph. maintaining maximal stiffness along the contour (i.e.arclength along the contour). The maximal correlation coordinate system (MCCS).
The starting point arethe parameterized contours Φ k in Eq. (2) for k = 0 , . . . , K −
1. To make the influence ofthe sampling rate more prominent, we also used the notationΦ( kδt, θ ) = Φ( t k , θ ) = Φ k ( θ ) . As stated above, the parametrization of Φ k is only determined up to a phase shift byEq. (3). We finally chose the phase shift and therefore the parametrization of Γ k +1 byminimizing the distance to the previous contour Γ k in a mean squared sense, i.e., τ k +1 = argmin τ (cid:90) π (cid:107) Φ k +1 ( θ − τ ) − Φ k ( θ ) (cid:107) d θ. (6)We may alternatively interpret Eq. (6) as optimizing the cross covariance between thetwo contours when interpreted as vector-valued functions τ k +1 = argmin τ (cid:90) π Φ k +1 ( θ − τ ) · Φ k ( θ ) d θ. In the sequel, we used (cid:101) Φ k +1 ( · ) = Φ k +1 ( · + τ k +1 ) and omitted the tilde for ease ofnotation. Effectively, choosing the phase shift amounts to fixing the ’zero point’ Φ k +1 (0)on Γ k +1 .This is the coordinate system that from now on was used to represent the contourgeometry, i.e., we represented a scalar quantity q defined on the contour Γ k , i.e., q = q ( t k , ( x, y )) with ( x, y ) ∈ Γ k as function ( k, θ ) (cid:55)→ q ( kδt, θ ) of discrete time andcontinuous space w.r.t. the chosen coordinate system Φ.May 27, 2020 5/22 he Eulerian and Lagrangian points of view. Any flow that maps the contourΓ k into Γ k +1 is determined by a mapping which describes the translation along thecontour φ k . To ensure that θ (cid:55)→ φ k ( θ ) is a one-to-one map, we required in addition ∂ θ φ k ( θ ) > . (7)The iteration θ k +1 = φ k ( θ k )describes the trajectory ( θ k ) k =0 ,...,K − of the starting point at coordinate θ on the firstcontour in our coordinate system. This approach to visualize the flow shall be called theEulerian point of view, since it describes the translation vector field from Γ k to Γ k +1 inthe coordinate system of Γ k :Φ k +1 ( φ k ( θ )) = Φ k ( θ ) + δtV k ( θ ) . (8)The Lagrangian point of view instead describes the flow in the coordinate system itgenerates, which is different from our MCCS. Denote the coordinate of a point on thefirst contour by its angle coordinate θ , and let χ k be the mapping of Γ to Γ k recursively defined by χ k +1 ( θ ) = φ k ( χ k ( θ )) , χ ( θ ) = θ . This gives χ k ( θ ) = θ k for k = 0 , . . . , K −
1. The Lagrangian description Ξ( t k , θ ) ∈ R is linked to the Eulerian description viaΞ( t k , θ ) = Φ( t k , χ k ( θ )) . Both points of view are useful to understand and describe a flow over the contour. Thetranslation vector W k in Lagrangian coordinates is simplyΞ( t k +1 , θ ) = Ξ( t k , θ ) + δtW k ( θ )and is linked to the Eulerian description via V k ( χ k ( θ )) = W k ( θ ) . We used the functions φ k , V k and W k interchangeably to specify the flow from Γ k toΓ k +1 . Transport along the flow.
For any flow, we may define the instantaneous dilationrate LD of the flow. Considering two infinitesimally nearby points on contour Γ k , wesee that the local relative dilation/contraction factor is obtained from φ k viaLD k ( θ ) δt = log( ∂ θ φ k ( θ )) . (9)Note that our global coordinate system MCCS above induces a flow with uniformdilation rate. To describe the transport of a density under the flow, consider points onthe contour Γ k that are distributed according to a density µ k ( θ )d θ . Under the flowinduced by φ k this density changes according to µ k +1 ( φ k ( θ ))d θ = µ k ( θ )d θ∂ θ φ k ( θ ) (10)Starting from µ ( · ) ≡
1, this defines the transported density on all contours. In theLagrangian picture this transport preserves the density µ ( · ), which follows from thefact that by definition the starting angle does not change under the flow. The density µ k can be written in Lagrangian coordinates as µ k ( χ k ( θ )) = 1 ∂ θ χ k ( θ ) . (11)May 27, 2020 6/22 regularizing family of flows. We next defined a family of local mappings φ k between successive contours that yields a compromise between the reverse normal flowand the uniform dilation coordinate flow.The mean squared velocity of the flow (with respect to a density µ k ) is given as F k [ φ k ] = (cid:90) π (cid:13)(cid:13)(cid:13)(cid:13) Φ k +1 ( φ k ( θ )) − Φ k ( θ ) δt (cid:13)(cid:13)(cid:13)(cid:13) µ k ( θ )d θ = (cid:90) π (cid:107) V k ( θ ) (cid:107) µ k ( θ )d θ. (12)The normal flow from contour Γ k to Γ k +1 is the flow that departs from the first contourin the normal direction until it intersects with the second contour. The normal flowfrom Γ k +1 to Γ k shall be called the reverse normal flow from Γ k to Γ k +1 . This is theunconstrained minimizer of F k . If there are no intersections of flow lines, it defines aone-to-one mapping between the two contours. In general, however, direct minimizationof the functional F k does not yield a valid flow because of self intersections, and as aconsequence the induced map is multiple valued. We therefore need to regularize theflow. A natural requirement is that the flow has the tendency to enforce non-uniformlydistributed points on contour Γ k towards more uniformly distributed points on contourΓ k +1 . We proposed to quantify the degree of non-uniformity of a distribution µ ( θ )d θ bymeans of U [ µ ] = (cid:90) π d θµ ( θ ) . (13)Since any distribution satisfies (cid:82) π µ ( θ ) = 1 and µ ( θ ) >
0, the minimizer actuallycorresponds to the uniform distribution. Other measures of non-uniformity are possible,for instance the entropy S [ µ ] = (cid:90) π µ ( θ ) log( µ ( θ )) dθ. In this paper, we used the characterization in Eq. (13), since it leads to a more readilytractable numeric quadratic minimization problem. In terms of the defining mapping φ k , the functional U reads U k [ φ k ] = (cid:90) π ∂ θ φ k ( θ ) µ k ( θ ) d θ, (14)as follows from Eq. (10). The regularized flow is defined as the flow that minimizes acompromise between both cost functions φ k,λ = argmin φ k F k [ φ k ] + λU k [ φ k ] , λ > . (15)Note that H k [ φ k ] = F k [ φ k ] + λU k [ φ k ] depends on both, φ k and the measure µ k . Wheniterating over all contours, one needs to update the measure before optimizing the flowfor the next time step. There are two end member cases for the regularized flow: • For large λ the optimal flow essentially immediately uniformizes the density of theinitial contour. Thereafter, it is the uniform stretching flow that minimizes themean square distances between the contours. This is precisely the coordinate flowdefined before. • For small λ the optimal flow allows for arbitrary local stretching rates to minimizethe point-wise distance. Here the limit defines the regularized reverse normal flow.Note that straight forward pointwise minimization of the flow distance from Γ k to Γ k +1 may lead to overlapping connections and hence singular mappings between the contours.If instead we regularize with small λ , such overlaps are avoided.May 27, 2020 7/22 he virtual marker picture. For numerical implementation, we discretized thecost functionals using the concept of virtual markers on the contours. The virtualmarkers are a discretized version of the Lagrangian coordinates. Since µ k is thetransported density, the first cost functional for φ k in the Lagrangian point of viewusing Eq. (11) is given by F k [ φ k ] = (cid:90) π (cid:107) V k ( χ k ( θ )) (cid:107) d θ = (cid:90) π (cid:13)(cid:13)(cid:13)(cid:13) Φ k +1 ( φ k ( χ k ( θ ))) − Φ k ( χ k ( θ )) δt (cid:13)(cid:13)(cid:13)(cid:13) d θ , while the second functional using Eq. (11) is given by U k [ φ k ] = (cid:90) π | ∂ θ χ k +1 ( θ ) | d θ . (16)See File S1 for details of the derivation. Both equations are well suited for a discretenumerical approximation for a given function f and initially equidistant θ ,i = 2 πi/N with i = 0 , . . . , N − N − (cid:88) i =0 f ( θ ,i )( θ ,i +1 − θ ,i ) (cid:39) (cid:90) π f ( θ )d θ . If we now approximate the continuous mapping φ k by its discrete values on the virtualmarker points θ k = ( θ k, , . . . , θ k,N − ) with θ k +1 ,i = φ k ( θ k,i ) , (17)then the first cost function may be approximated as F k [ φ k ] (cid:39) F k (cid:2) θ k +1 | θ k (cid:3) = 2 πN δt N − (cid:88) i =0 (cid:13)(cid:13) Φ k +1 ( θ k +1 ,i ) − Φ k ( θ k,i ) (cid:13)(cid:13) (18)and the second cost function as U k [ φ k ] (cid:39) U k (cid:2) θ k +1 | θ k (cid:3) = N π N − (cid:88) i =0 (cid:12)(cid:12) θ k +1 ,i +1 − θ k +1 ,i (cid:12)(cid:12) . (19)For the entropy based measure of uniformity, consider a collection of points θ k, , . . . , θ k,N − ∼ µ k on Γ k that are distributed according to the density µ k ( · ) (notnecessarily uniform). Then for any function f , it is1 N N − (cid:88) i =0 f ( θ k,i ) (cid:39) (cid:90) π f ( θ ) µ k ( θ )d θ, yielding S k [ φ k ] (cid:39) S k (cid:2) θ k +1 | θ k (cid:3) = 1 N N − (cid:88) i =0 log( θ k +1 ,i +1 − θ k +1 ,i ) . In this discrete virtual marker approximation, the local dilation rate at θ i,k , also calledthe local dispersion, reads LD k,i = 1 δt log θ k +1 ,i +1 − θ k +1 ,i θ k,i +1 − θ k,i . (20)Finally, the discrete optimization problem is given by φ k,λ = argmin φ k F k (cid:2) φ k ( θ k ) | θ k (cid:3) + λU k (cid:2) φ k ( θ k ) | θ k (cid:3) , λ > . (21)Note that in the discrete optimization problem we do not pose any requirements on thespace of transformations φ k ensuring condition Eq. (7). If φ k ( θ k,i +1 ) − φ k ( θ k,i )= θ k +1 ,i +1 − θ k +1 ,i ≤ k , we say that φ k exhibits mapping violations.May 27, 2020 8/22 arker re-initialization for weakly-regularized contour flows. In general, thedistribution of virtual markers θ k ∼ µ k on Γ k depends on the initial distribution ofvirtual markers θ ∼ µ on Γ , since the density µ k results from the transport of µ bythe flow. As a consequence, functions derived from the local contour flow like, e.g., thelocal dilation rate along Γ k , may depend on the initial distribution on Γ . Byre-initializing the distribution of virtual markers on Γ k for any k = 1 , . . . , K −
1, thisdependence may be removed. A natural choice is to re-initialize with the uniformdistribution, i.e., using θ k +1 ,i = φ k ( ξ i ) (22)with ξ i = 2 πi for i = 0 , . . . , N − φ = ( φ k ) k =0 ,...,K − between contours, we thus solved the re-initialized optimizationproblem φ k,λ = argmin φ k F k (cid:2) φ k ( ξ ) | ξ (cid:3) + λU k (cid:2) φ k ( ξ ) | ξ (cid:3) , λ > . (23)with ξ = ( ξ i ) i =0 ,...,N − . For the local dispersion, e.g., this resulted inLD k,i = 1 δt log φ k,λ ( ξ i +1 ) − φ k,λ ( ξ i ) ξ i +1 − ξ i (24)with no dependence on the initial distribution of markers on Γ . Algorithmic workflow.
We summarize the proposed numerical workflow:1. Given the segmented contours in Eq. (1), determine the continuousrepresentations Φ k defined in Eq. (2) satisfying the conditions (3)2. Determine contour based quantities like the curvature in Eq. (4) or the center ofmass in Eq. (5)3. To determine the (strongly-regularized) coordinate flow, consider N equallyspaced markers θ = ( θ ,i ) i =0 ,...,N − θ ,i = ξ i = 2 πiN on the initial contour and choose a large λ value. Iteratively solve theregularization problem (23) to determine the coordinate markers θ k +1 for Γ k +1 from the coordinate markers θ k for Γ k . Since both θ k +1 and θ k are approximatelyequally spaces, solving the minimization problem amount to choosing θ k +1 , .4. To determine the (weakly regularized) local flow φ = ( φ k ) between successivecontours, choose a small λ value and solve the regularization problems (23) todetermine the coordinates θ k +1 ,i = φ k,λ ( ξ i ) on Γ k +1 based on N equally spacedmarkers ξ , . . . , ξ N − on Γ k .5. Determine contour flow based quantities like the local dispersion in Eq. (24) orthe local motion:LM k,i = (cid:107) V k ( ξ i ) (cid:107) = (cid:107) Φ k +1 ( φ k,λ ( ξ i )) − Φ k ( ξ i ) (cid:107) δt , (25)see also Eq. (8).May 27, 2020 9/22 m )60708090100110120 y ( m ) (A)
20 40 60x ( m ) (B) m )50 y ( m ) (C) m ) (D) Fig 1. Cell trajectory of a persistently moving amoebae. (A)
Fluorescenceimage with closed string of M = 400 equidistant nodes resulting from the segmentationprocess; shown is only every fifth segmentation point (blue points). (B) Smoothrepresentation Φ k of the cell contour (orange line) obtained by spatial Gaussianregression on the segmentation points. Every fifth cell contour is displayed as dashedgrey line. (C) Entire cell track of K = 500 cell contours (only every fifth shown). (D) Global trace of the cell track (grey area) and the trajectory of the center of mass ofthe contour (solid line, color coded as in panel (C)). The initial contour is shown asdashed black line and the final contour as dashed grey line.
Results
Degree of regularisation controls distribution of virtual markers
We applied our approach to time-lapse microscopy data of the social amoebae
D.discoideum . Fig 1 illustrates the data aquisition process. For each time point t k , weobtained a sequence of segmentation points from a fluorescence image (see panel (A)and Eq. (1)) and their corresponding continuous representation Φ k (see panel (B) andEq. (2)). The entire sequence of continuous contours is shown in panel (C), while thetrace of the cell track and the center of mass trajectory are shown in panel (D).The continuous representations Φ , . . . , Φ K − of all contours are the input to theoptimization problem Eq. (15) to determine the regularized contour flow. Fig 2 (A)-(C)shows the impact of the regularization parameter λ on the virtual marker trajectories(shown for two illustrative contours). In the absence of any regularization (panel (A), λ = 0), virtual markers are thinned out in some regions (linked to protrusive areas),while they are clustered in others (in particular at the back of the cell). The case λ = 0corresponds to minimizing the translation from the first contour to the next, i.e., eachvirtual marker on the first contour is linked to its nearest neighbor on the secondcontour. This may result in mapping violations θ k +1 ,i +1 − θ k +1 ,i ≤ λ = 0 is closely related to the reversed normalMay 27, 2020 10/22 m )110120130140150160170 y ( m ) (A)
10 20 30 40 50x ( m ) (B)
10 20 30 40 50x ( m ) (C) (D) ( ms ) (E) ( m ) Fig 2. Impact of regularization on the distribution of virtual markers for (A) no regularization ( λ = 0), (B) weak regularisation with λ = 1; and (C) strongregularisation with λ = 1000, illustrated on two frames (roughly 20s apart forillustration purpose). Using the strongly regularised so-called coordinate markers as ameans to map local characteristics into a kymograph, the lower panel shows the localmotion (D) and curvature (E) . The local motion is defined by the magnitude of eachmapping vector, which are determined based on the weakly regularised marker flow. Allpanels correspond to the persistently motile cell of Fig 1.flow. Thus, instead of taking the nearest neighbors on the next contour, one could alsochoose the shortest normal vector from the second contour to the first one as anoptimization criterion.In the weakly regularized case, virtual marker thinning and clustering is sillprominent (see panel (B) with λ = 1), but to a lesser extent. Moreover, in the presenceof regularization, virtual marker trajectories are interdependent, which results intrajectories without mapping violations. In the limit of strong regularization, themarker points remain uniformly distributed on every contour, while minimizing theoverall distance between contours (see panel (C) with λ = 1000). This makes thestrongly regularized virtual marker trajectories an ideal candidate for a time-evolvingreference frame and corresponds to the previously defined MCCS coordinate system.By choosing a strong regularization for the coordinate system, however, informationon local contour changes is largely lost. Therefore, we used the strongly regularized caseonly to determine the coordinate system (set of N = 400 virtual marker trajectories),while we determined local contour characteristics (e.g. local motion or local dispersion)based on a re-initialized weakly regularized flow. The local characteristics weresubsequently represented in the coordinate system obtained from a strongly regularizedflow. In panel (D), the local motion obtained from the re-initialized weakly regularizedflow with λ = 1 is shown for the same cell track as in Fig 1 and the entire time-lapsemicroscopy recording (500 frames, δt = 1s). The local motion clearly shows regions offast moving membrane parts at the leading edge (red areas) and at the back of the cell(blue areas). The curvature kymograph in panel (E) shows characteristic lines ofstrongly convex (orange) and concave (green) membrane parts. Inclined lines ofMay 27, 2020 11/22
50 100 150 200 250x ( m )50100150200250 y ( m )
50 100 150 200 250x ( m ) 50 0 50 100 150x ( m ) 0100200300400Time (s)0 ( s ) ( ms ) ( m ) Fig 3. Comparison of different cell tracks of
Dictyostelium discoideum : persistently motile (left), weakly motile (middle) and almost stationary (right). Thecorresponding kymographs contain information on the local dispersion (left), localmotion (middle) and curvature (right). For details see text.curvature (and local motion) may result from adherent parts of the cell moving alongthe cell contour as well as shifting effects of virtual markers due to arc length changes.It is important to notice that a kymograph depends on the underlying time-evolvingcoordinate system.In contrast to the procedure mentioned above, one may also compute localcharacteristics along the global flow without re-initializing from a uniform distributionof markers after every time step. However, this is not recommended, because eitherlocal information gets lost (strongly regularized flow) or clustering and thinning effectsof markers become too prominent (weakly regularized flow); see Fig. S2 for global flowsbased on different regularization parameters and the resulting kymographs. Kymographs of local properties show characteristic patterns ofamoeboid motility
The kymographs of local dispersion (LD), local motion, and curvature can be used tovisualize, analyze, and quantify the protrusive activity of a cell track. Note that thethree quantities are closely related. Fig 3 shows cell tracks and correspondingkymographs for three different motility patterns: (A) persistently motile cell; (B)weakly motile cell; and (C) an almost stationary cell. All kymographs are smoothed bya Gaussian filter with a standard deviation of three markers in space and a standarddeviation of one contour in time. See Fig S4 for the same kymographs but withoutsmoothing. Moreover, videos of all three cell tracks and their corresponding kymographscan be found in Video S1–S3.For the persistently motile cell, the LD kymograph shows strong (positive) activityin a band-like structure along roughly half of the cell contour (width π ), while theMay 27, 2020 12/22
100 200 300 400t (s)0 (A) ( s ) (B) -0.16-0.107-0.0530.00.0530.1070.16Local dispersion ( s )
50 100 150 200x ( m )75100125150175200225250 y ( m ) (C)
50 100 150 200x ( m )75100125150175200225250 y ( m ) (D) Fig 4. From the local dispersion kymograph to protrusive areas andprotrusion events . (A) Local dispersion of a persistently motile cell as in Fig 1 andFig 3 (left-hand side). (B)
Discretised local dispersion with different areas of activity:high (dark red), medium (light red), low (white) protrusive activity, and low (white),medium (light blue), high (dark blue) retractive activity. Local maxima of positive localdispersion are depicted as black dots. Areas of medium and high protrusive (C) andretractive (D) activity mapped back on the trace of the cell track.activity of local dispersion is less localised for the weakly motile cell and much lesspronounced for the stationary cell. A similar scenario is seen in the local motionkymographs of the three cells. In broad terms, the local motion kymographs show moreactivity, e.g., areas of red and dark red color, than the LD kymographs. This is alsoapparent from a correlation plot of the two quantities (see Fig S3).In the following section, we chose the local dispersion as a basis to identifyprotrusions being a product of local velocity and curvature. Another reason to chooseLD as the quantity to define protrusion is that many patches of high activity in thelocal motion kymograph contain multiple LD areas. The LD allowed us to divide thesepatches of high local motion into single separated protrusions with high LD rate.
Virtual marker dispersion allows to identify and characterizeprotrusions
Using the persistently motile cell track in Fig 1, we describe next, how to use the LD todefine protrusion areas and protrusive events. Based on the local dispersion kymographin Fig 4 (A), we defined areas of medium (light red) and high (dark red) protrusiveactivity as well as medium and high retractive activity (light and dark blue,May 27, 2020 13/22 m )120140160180200 y ( m ) (A)
60 80 100 120 140x ( m )190200210220230240250260 y ( m ) (B) Fig 5. Protrusive and retractive areas with corresponding protrusionevents.
Illustrative sequence of the contour dynamics for 96 s ≤ t ≤ s (left), and316 s ≤ t ≤ s (right) based on the cell track shown in Fig. 4. Features with high andmedium protrusive activities are shown in dark and light red, respectively. Features withhigh and medium retractive activities are shown in dark and light blue, respectively. Allpatterns shown possess a minimal growth time of 3 s . The black dots show local maximaof the local dispersion in areas of medium and high protrusive activity.respectively), see Panel (B). In this context, we determine thresholds for protrusiveactivity by dividing the 90th percentile of all positive LD values of the kymograph intothree intervals of equal length. For retractive activities (LD <
0) the opposite thresholdswere taken. By leaving out the largest and smallest values of the LD kymograph, theclassification becomes less depending on outliers. Additionally, we performed a priorsmoothing of the kymograph as mentioned in the first paragraph of the previous sectionto reduce noise and, therefore, to reduce the number of small and separated patterns.To highlight the events of highest local protrusive activity, we included positive localmaxima of the LD kymograph (black dots) in panel (B). Local maxima falling insideregions with high or middle protrusive activity were depicted as bold dots. Using thetime-evolving coordinate system obtained from strongly regularized flow, theprotrusion/retraction areas shown in panel (B) is mapped back into the 2d plane ofamoeboid motion, see panels (C) and (D), respectively. As a result, we obtain anautomated visualization of the protrusive activity during amoeboid locomotion.The kymograph in panel (B) clearly shows the trace of protrusive activity at theleading edge, located initially at around 3 π/ π . At the sametime, retractive activity occurs mainly at a distance of π from the leading edge. Inpanels (C) and (D), one can nicely see the explorative dynamics of the protrusions atthe cell front and the stably retracting uropod at the back of the cell, where dark blueareas indicate faster retractions. Fig S5 and S6 show the analogous graphics for theweakly motile and almost stationary cell track.In Fig 5, we present a close-up of two sequences of cell contours. The core protrusiveareas that shape the evolving cell contour can be seen clearly, e.g., in panel (A). Theevents of highest protrusive activity (black dots) seem to drive the protrusion in manycases. Panel (B) illustrates strong retractive activity at the uropod, and the retractionof the membrane between two nearby protrusions (blue area sandwiched between redareas). The opposite effect can be seen in panel (A) at the back of the cell, where aconcave region between two convex retractions is identified as protrusion. Thesepatterns nicely illustrate the concept of local dispersion, being a product of the localvelocity of virtual markers and the curvature along the contour segment.May 27, 2020 14/22 (A) -0.16-0.107-0.0530.00.0530.1070.16Local dispersion ( s ) A r e a g r o w t h ( m s ) (B) s )6420246 N u m b e r o f f e a t u r e s (C) Fig 6. Statistical analysis of example cell track. (A)
Local dispersionkymograph with thresholds as in Fig. 4. (B)
Area growth of cell segments which arepart of identified protrusions (red) and retractions (blue) of high and middle intensity. (C)
Number of protrusive and retractive areas with high intensity with respect to time.The time with > t ≥ Statistical analysis of motility patterns
In this section we illustrate the ability to statistically analyse a cell track based on ourregularized contour flow approach. We used the persistently motile cell track forillustration.Fig 6 (A) shows the local dispersion kymograph of the persistently motile celldivided into four different phases (note that here the y axis starts at π /4). Until t = 200 s , the cell moves upward with well observable protrusions roughly between 1 . π to 0 . π , while the uropod is slightly above π/
2. Then, the cell begins a phase ofreorientation that lasts until t ≈ s , where larger protrusions occur also at the formerback of the cell. Subsequently, the cell changes its direction downward toward 3 / π to π (in the video it moves rightwards). In the last phase, the cell moves to the right-handside by creating protrusions at the front left, front right and again front left of the cell.See Video S1 for a better understanding of the cell track.We analyzed the protrusive and retractive areas shown in Fig 4 (C)+(D) withrespect to the activity level (medium/high), duration and position along the cellcontour. In addition, we investigated the differences between protrusions andretractions. Fig 6 (B) displays the area growth of protrusios and retractions for highactivity. The identified protrusive and retractive areas are naturally partitioning by thesequence of contours into smaller ’slices’ (see, e.g., Fig 5). We defined the area growthof a protrusion/retraction as the area of the slice divided by the frame rate δt . Weobserved that the overall change of cell area attributed to protrusions of high activity issubstantially larger than for retractions of high activity. This illustrates that the cellmotility of this cell track is driven by a higher number fast (and potentially explorative)May 27, 2020 15/22 .6 0.4 0.2 0.0 0.2 0.4 0.6Local dispersion ( s ) D e n s i t y (A) ( ms ) D e n s i t y (B) ( m ) D e n s i t y (C)
0% 2% 4%
FrontBack RightLeft (D) s )020406080100Area ( m ) (E) R e l a t i v e f r e q u e n c y ( % ) (F) s )0510152025 R e l a t i v e f r e q u e n c y ( % ) Fig 7. Statistical analysis of example cell track.
Distributions of local dispersion (A) , local motion (B) and curvature (C) inside protrusive and retractive patterns withhigh intensity. (D)
Circular histogram displaying the angle, where high protrusiveactivity appears along the cell contour. (E)
Correlation between area and growth timeof identified patterns. (F)
Histograms of growth times of protrusions and retractionswith high intensity.protrusions, and a small number of fast retractions. Since in broad terms, the total areagain balances the total area loss, it further illustrates that area loss is to a larger extentattributed to slower and steadier retractions than it is for protrusions. After the secondphase ( t > t > . .
2) . Inaddition, we determined the fraction of time f T protr > and f T retrac > with more thantwo simultaneous protrusions and retractions, respectively. In the first phase, it is f T protr > = 0 .
28 vs. f T retrac > = 0 .
06. In the following phase, these fractions increasefor protrusions: f T protr > = 0 .
42; 0 .
39; 0 .
54 (for phase 2; 3; 4), but more so forretractions: f T retrac > = 0 .
16; 0 .
20; 0 .
43 (for phase 2; 3; 4). This illustrated that theincrease activity, as seen in panel (B), goes along with an increased number ofprotrusions and retractions. This indicates a more explorative character of the cellmotion at this time, while the cell seemed to be more stabilized during the first phase.Fig 7 gives further insight into the protrusive activity. Panel (A) shows thedistribution of LD values of all virtual markers within protrusive and retractive areaswith high activity. In other words, the local dispersion distribution shows only valuesthat are larger than the thresholds for high protrusive activity or smaller than thethreshold for high retractive activity (the thresholds are ± . . − . ± µm/s forprotrusions and retractions. Minor peaks correspond to inward protrusion and outwardretractions. These features have discussed in relation to Fig 5 (see red protrusive areaswith blue retractive region in panel (A), and blue retractive area within protrusiveregion in panel (B)). For the curvature, the distributions are bimodal with almost noweight for zero curvature for retractions.Fig 7 (D) shows the distribution of high activity protrusions in direction of themoving cell. We identified two peaks in direction of the cell movement (front-left andfront-right). Another peak is located at the back of the cell. A similar behaviour waspresented for pseudopods in [23, 24], where two different types of pseudopods weredistinguished: (left/right) splitting pseudopods and de-novo pseudopods. By comparingthe correlation between the growth time and the area of protrusions and retractions inpanel (E), we observed that protrusions possess an area often twice as large asretractions with similar growth times. This indicates the difference between the fasterand more explorative character of protrusions at the cell front and the slower and stablyretracting character of the uropod. This is also in line with the corresponding activitiesshown in Fig. 6 (B).Finally, in panel (F) the distribution of growth times is shown for both, high activityprotrusions and retractions. Note that we used the term ’growth time’ for both,protrusions as well as retractions, owing to the fact that also retractive areas can beexpanding, as discussed earlier. The majority of growth times felt inside a range of 0 s to 10 s . Nevertheless, there are patterns, especially long persistent uropods, with growthtimes much larger than the range presented in these histograms. The average growthtime of pseudopods observed in [23] is much higher (12 . s ) than the growth times ofprotrusions presented in this work (4 . s ). This is not surprising, since our definition ofprotrusions also takes short-lived objects into account. For example, the averagenumber of protrusions per minute for the persistently motile cell track ( ≈ .
8) is muchhigher than the average frequency of pseudopods per minute (2 . ± .
2) as observedin [23]. Additionally, concave regions between two retractions as in Fig 5 (A) weredetected as protrusive areas as well, which raises the total number of protrusions. SeeFig S8 and Fig S9 for similar graphics of the other two example cell tracks from Fig 3.
Discussion
With the ever increasing amount of live cell imaging data and the continuously growingcomputational power, computer-automated techniques to analyze the morphology ofcells have steadily developed over the past two decades. In particular, in cell motilityresearch, morphological characteristics are commonly used to pinpoint phenotypicdifferences between mutant cell lines thus highlighting the mechanistic role of individualcomponents of the underlying signaling pathways. While many static measures of cellshape have already been introduced early on [25, 26], dynamic measures that quantifythe temporal evolution of the cell shape proved to be more difficult to implement. Firstattempts focused on temporal changes of the projected cell area to deduce overallprotrusion rates, for an example see [27]. These approaches were later refined by localmeasures of cell boundary motion along selected line segments perpendicular to the cellborder [28]. Also local space-time plots have been defined in this way [29, 30]. However,in all these cases, the direction of interest or the local placement of the kymograph hadto be chosen manually, which severely limits a reliable long-time tracking of morecomplex cell shapes and introduces an arbitrariness related to the manual processing.The most promising approach to overcome this limitation relies on an active contour(snake), a closed chain of connected nodes (virtual markers) that is placed along the cellMay 27, 2020 17/22erimeter [31]. Different rules have been proposed to propagate the markers from onecontour to the next. In some cases, the distance between virtual markers is keptconstant and markers are added or removed as the contour evolves [32]. Otherapproaches that are inspired by mechanical spring models or concepts fromelectrostatics keep the number of markers constant and allow for local variations in themarker spacing [33]. Here, we can distinguish two limiting cases. On the one hand,equidistance is enforced and markers on adjacent contours are connected in a one-to-onemapping by minimizing the sum of square distances between pairs of connectedmarkers [20]. On the other hand, markers are propagated from one contour to the nextin normal direction (normal flow) while the marker spacing evolves without constraints.The latter approach has been implemented using a level set method to cope withproblems related to finite time sampling [34]. However, high computational costs andthe rapid buildup of highly uneven marker distributions limit the use of the level setmethod in practical applications. In this article, we have introduced a family of markerflows that incorporates these different cases into a general framework. In particular, theregularization that we introduced in Eq. (15) includes the two extreme scenariosdescribed above as limiting cases. In the limit of large λ , we obtain an equidistantmapping, whereas in the limit of small λ , we approach the (reverse) normal flow.Tuning λ allows us to systematically shift between these two limits.Once a flow of virtual markers is computed on the evolving cell contour by any ofthese methods, it defines a coordinate system, in which different local quantities can bedisplayed, such as curvature, membrane displacement, or the intensity of a fluorescentlytagged membrane-associated protein. Note that for all of these coordinate choices it isgenerally acknowledged that the dynamics of the virtual membrane markers do notreflect the motion of specific membrane lipids or proteins, as the membrane itself is avery complex and dynamic structure [20, 34]. In particular, lateral flows may occur dueto membrane recycling, so that the dynamics of individual molecules or domains in themembrane do not necessarily correspond to morphological changes and will prevent aone-to-one mapping of molecular markers on adjacent contours.Amoeboid motion is primarily driven by localized membrane protrusions, so-calledpseudopodia. Identifying and tracking pseudopodia has thus been an important focus ofthe morphodynamic analysis of amoeboid cells. The first substantial pseudopodstatistics were generated by computer-assisted manual image processing, relying on theexpert judgement of the investigator [23, 35]. From this, a first automated softwareroutine for pseudopod tracking was developed [18] and successfully applied also toanalyze the chemotactic navigation of amoeboid cells [24]. It relies on a complexdecision tree that defines pseudopodia based on a sequence of threshold criteria appliedto the local curvature, the virtual marker movement and the local area change. In thisway, frequency and direction of pseudopod formation, their sizes, lifetimes and otherquantitative measures were extracted. While successfully providing a first quantitativedata base of pseudopod characteristics, this approach has the drawback that it requiresthe choice of several parameters that are tuned to the characteristic properties ofpseudopods in starvation-developed D. discoideum cells. If cells display protrusions witha more diverse range of shapes and time scales, a reliable tracking is difficult to achievewith this approach.Later, a more compact criterion for the detection of localized protrusions wasproposed [19, 20]. It relies on thresholding a distance measure between the currentposition of virtual markers and the cell boundary at a later time point. The calculationof this distance measure, however, lacks a clear underlying definition and is computed inan ad hoc fashion for the specific data set (see supplementary material of Ref. [19]):First, each virtual marker is mapped from its current position onto the closest point onthe future cell contour. As this generates a highly non-uniform distribution of targetMay 27, 2020 18/22 % 1% 2% 3%
FrontBack RightLeft (A)
Local motion D e n s i t y (B) t ( s ) (C) t ( s ) (D) Local motion ( ms ) Fig 8. Future outlook.
Underlying distributions are based on averaging over severalcell tracks. (A)
Circular histogram displaying the position, where protrusions andretractions events appear along the cell contour. (B)
Distribution of local local motionof protrusion and retractions events above a threshold ± / (C) Simulated dataobtained from self-excited Poisson point processes (so-called Hawkes processes) on theunit circle. Afterwards, we obtained regions of high (red) and low (blue) intensity dueto some cluster algorithm. (D)
Continuous kymograph obtained from a regressionmodel (e.g. Gaussian process regression) based on sampled magnitudes at eventlocations shown in (C) from the distribution in (B).markers, with protrusive areas particularly poorly covered, the target markers on thenew contour are then redistributed by two successive smoothing steps, using averagingwindows of specific sizes. The time interval between the successive contours for thedistance projection, as well as the smoothing parameters for redistribution of the targetmarkers were hand-picked by the investigator. Note, however, that this could beenvisioned as one step in an iterative procedure to minimize our cost functional.In the present work, we introduce a novel approach to define, identify and analyzelocalized protrusions on dynamically evolving contours of amoeboid cells. Protrusiveareas are defined via a single threshold value. In contrast to previous approaches, wechose the virtual marker dispersion as the underlying quantity, since it combinesinformation on both, the marker displacement and the local curvature. The markerdispersion is mathematically well defined by Eq. (24) and does not require additionalempirical smoothing steps. Based on this criterion, we not only detect individualprotrusion events but we capture the entire shape and the complete temporal evolutionof a protrusion in a fully automated fashion, see Fig 5 for examples.As outlined in the Introduction, the overall aim is a quantitative, data-driven modelof amoeboid motility. The presented theoretical framework is a first step in thisdirection. We envision that point events of high protrusive activity may be used todefine a point process in the space-time coordinate system. To reflect the often observedpersistence in motility, so-called self-excited Poisson cluster processes or Hawkesprocesses may be favourable choices. The point process can serve as a skeleton forprotrusion activity that is ’completed’ to a random realisation of a kymograph, based onthe statistics of a local quantity. We illustrated this idea in Fig 8, where we used thelocal motion statistics (B) to reconstruct a local motion kymograph (D). The idea is touse such realisations of kymographs to reconstruct a cell track and eventually toassimilate the time-lapse microscopy data into a mathematical model of amoeboidmotility.May 27, 2020 19/22 upporting information
S1 File. Supplementary methods and supporting computations. (PDF)
S1 Fig. Three-dimensional space–time tube of contours , where consecutivecontours are stacked onto each other. (PDF)
S2 Fig. Comparison of global flows in cases of no, weak and strongregularization. (PDF)
S3 Fig. Correlation between local dispersion and local motion.
Thecorrelation is shown for the cell track in Fig. 3: Persistently motile, weakly motile andalmost stationary cell. (PDF)
S4 Fig. Kymographs as in Fig 3 without prior smoothing. (PDF)
S5 Fig. Feature identification of weakly motile cell. (PDF)
S6 Fig. Feature identification of almost stationary cell. (PDF)
S7 Fig. Collection of identified protrusive areas and events of high intensityfor the persistently motile cell. Only features with minimal persistence length ∆ t ≥ S8 Fig. Statistical analysis of weakly motile cell. (PDF)
S9 Fig. Statistical analysis of stationary cell. (PDF)
S1 Video. Persistently motile cell. (MP4)
S2 Video. Weakly motile cell. (MP4)
S3 Video. Stationary cell. (MP4)
Acknowledgments
The research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) -SFB1294/1 - 318763901. We thank Gregor Pasemann (Technische Universit¨at Berlin),and Till Brettschneider & Piotr Baniukiewicz (University of Warwick/UK) for fruitfuland stimulating discussions.