TY - JOUR
AU - Asada, H., Harry
AB - Abstract An integrative cell migration model incorporating focal adhesion (FA) dynamics, cytoskeleton and nucleus remodeling and actin motor activity is developed for predicting cell migration behaviors on 3-dimensional curved surfaces, such as cylindrical lumens in the 3-D extracellular matrix (ECM). The work is motivated by 3-D microfluidic migration experiments suggesting that the migration speed and direction may vary depending on the cross sectional shape of the lumen along which the cell migrates. In this paper, the mechanical structure of the cell is modeled as double elastic membranes of cell and nucleus. The two elastic membranes are connected by stress fibers, which are extended from focal adhesions on the cell surface to the nuclear membrane. The cell deforms and gains traction as transmembrane integrins distributed over the outer cell membrane bind to ligands on the ECM, form focal adhesions, and activate stress fibers. Probabilities at which integrin ligand–receptor bonds are formed as well as ruptures are affected by the surface geometry, resulting in diverse migration behaviors that depend on the curvature of the surface. Monte Carlo simulations of the integrative model reveal that (a) the cell migration speed is dependent on the cross sectional area of the lumen with a maximum speed at a particular diameter or width, (b) as the lumen diameter increases, the cell tends to spread and migrate around the circumference of the lumen, while it moves in the longitudinal direction as the lumen diameter narrows, (c) once the cell moves in one direction, it tends to stay migrating in the same direction despite the stochastic nature of migration. The relationship between the cell migration speed and the lumen width agrees with microfluidic experimental data for cancer cell migration. Insight, innovation, integration To understand three-dimensional crawling behaviour of a single stalk cell on the surface of the angiogenic lumen, we developed an integrated cell migration model incorporating three key mechanisms of cell biology, consisting of remodelling of the cell and nuclear membranes, focal adhesion dynamics, and actin motor activity. After we successfully compared our model with an existing experimental work of spontaneous cancer cell migration in a confined microfluidic device, we predicted stalk cell migration in various circular tubes. When the cell migrates on the wall of a wide lumen, the cell tends to stretch out along the circumference, where the radius of curvature is smaller than that of the longitudinal direction, resulting in a high probability for transverse migration. The new cell migration model can be further developed toward a more complex model with the inclusion of cell–cell interactions to predict emergent behaviours of collective cell migrations in various geometries. Introduction Cells generate traction forces at focal adhesion (FA) sites, triggered by chemotaxis and haptotactic responses from the extracellular environment,1 and contractile forces through myosin II motor activity in actin stress fibers by an intracellular signalling cascade involving the RhoA small GTPase.2 Studies indicate that actomyosin activity in cell migration is critical for many biophysical phenomena, including angiogenesis, tumor growth, metastasis, and wound healing in multicellular phenomena.3–5 Additionally, myosin II motor activity is essential for cell adhesion to the substratum and for changes in cell morphology through mechanical force balances between the generation of traction force at the leading edge and the release of FAs in the rear of the cell.6 A study reports that cells contain at least three different types of contractile actin-based stress fibers: transverse arcs, dorsal stress fibers, and ventral stress fibers;7 transverse arcs are curved actomyosin bundles that are not directly associated with focal adhesion on either end, dorsal stress fibers consist of one end bound to a focal adhesion site and the other end attached to the nucleus or transverse arcs, whereas ventral stress fibers bridge two focal adhesions on both ends. Among the different types of actin based stress fibers, actin stress fibers connected to the nuclear membrane, in particular, appear to play an important role in cell migration and contraction. For example, the recent experiments by Chancellor et al.8 demonstrate that the actomyosin tension is balanced in part by the nucleus through mechanical links mediated by nesprin-1 (nuclear membrane proteins that bind to F-actin). Interestingly, when its connection to the nucleus is inhibited, the number of ventral SFs is rapidly increased, which results in the reduction of cell migration speed. In the recent literature,9 the reduction of cell migration speed due to the disconnection of actin stress fibers to the nuclear membrane was also verified using a computational cell migration modeling approach. Previous experimental works mostly involve 2-dimensional cell migration on planar surfaces. However, 2-D cell migration on the 3-D extracellular matrix (ECM) surface, e.g. stalk cell migration in angiogenesis, remains poorly understood. When migrating on a 3-D curved surface, e.g. the inside surface of a tunnel produced by matrix degradation of a lead or tip cell, interactions between the transmembrane integrins and the surrounding ECM create complex spatiotemporal dynamics in forming focal adhesions and stress fibers, leading to complex migratory behaviors strikingly different from the 2-D migration we observe in traditional gel surface experiments.10 The objective of the present work is to build a computational model to predict cell migration on 3-D curved surfaces of ECM by integrating multifaceted mechanisms. Recently, the development of a three-dimensional cell culturing technique has provided a method to investigate a novel mechanism of cell migrations in the ECM which has been rarely observed using two-dimensional substrates.11–14 In the 3D ECM environment, cells can interact with ligands of ECM proteins with many microscopic properties including fiber density, fiber strength, degree of cross-linking, filament length and constitutive deformability of the scaffold through integrin receptors, which lead to activation of signalling networks and cytoskeleton remodeling.15 Several studies have emphasized the differences between 3-D cell migration and 2-D cell migration.16–18 Interestingly, it has been recently reported that the higher activation level of Rac GTPase is observed in 2D than 3D, which promotes cell spreading and inhibits uniaxial migration phenotype observed in 3D.16 Another recent study addresses, unlike in 2D, the number of focal adhesions in cells fully embedded in collagen 3D matrices is smaller than that in cells on curved matrix surfaces (2.5D) because cells tend to migrate toward stiffer 3D gel regions.18 On the other hand, current studies on angiogenic sprouting using 3-D microfluidic assays present clear observations of focal adhesions along the surface of the lumen, which differ from such embedded 3-D cell experiments since the cells migrate on curved surfaces. This study is motivated by two experimental works; one on cancer cell migration through conduits with diverse cross-sections,19 and the other on angiogenic sprouting using 3-D microfluidic assays.14 Both experiments have indicated that 3-D interactions between a cell's cytoskeleton and a curved surface directly affect the migration speed and direction. Furthermore, when proceeding through a narrow conduit, the cell exhibits a unique deformation pattern; not only does its cytoskeleton conform to the geometric constraint, but its nucleus deforms elastically and changes its location relative to the rest of the cell.20,21 The specific goal of the present work is to build a computational model to elucidate and predict these experimentally observed migratory behaviors in relation to the geometry of migration surfaces. This entails (a) deformation mechanics of both cytoskeleton and nucleus, (b) 3-D interactions between transmembrane integrins and ECM ligands, leading to focal adhesion formation, and (c) stress fiber formation and traction generation. Integration of these key mechanisms is pivotal for elucidating the aforementioned migration behaviors. Here we describe 3-dimensional spatiotemporal dynamics of cell migration by incorporating focal adhesion dynamics, cytoskeleton and nuclear remodeling and actin motor activity, all interfaced with a curved ECM surface. In the following, experimental observations of cell migration on cylindrical surfaces of the angiogenic lumen are first discussed, an integrated migration model on such geometries is then presented, and numerical simulation experiments demonstrate the diverse migration behaviors in relation to the geometry of migrating surfaces. Experimental observations To gain insights into cell migration on a curved surface, experiments using 3-D microfluidic assays were conducted (Fig. 1D(1)). The results showed cell behaviors significantly different from traditional on-the-gel experiments.10,Fig. 1A shows the schematic of a 3-D microfluidic assay for angiogenic sprouting experiments of human Micro Vascular Endothelial Cells (hMVECs). The cells seeded on one side of the collagen gel are exposed to vascular endothelial growth factor (VEGF) while a concentration gradient is created across the gel matrix. Sprouts are formed from the monolayer of the seeded cells that extend towards the higher concentration of VEGF. Holes are created in the gel matrix by tip cells that cleave the gel with matrix-degrading enzymes (matrix metalloproteinases, or MMPs)22–24 (Fig. 1D(1)). Stalk cells14,25 migrate along the hole created by the tip cell, crawling on the curved surface of the conduit. Fig. 1B shows 3-D images of a stalk cell migrating along a narrow conduit (15 μm). The cell is stretched out parallel to the axis of the conduit, while the nucleus (shown in blue) deforms to fit the narrow conduit. Focal adhesions (red spots) are distributed 3-dimensionally across the curved conduit surface. Typically, focal adhesions are more highly concentrated on the front of the migrating cell, as shown in Fig. 1B(2), the top slice of the 3-D image. The middle slice (Fig. 1B(3)) indicates that focal adhesions are also created along the side wall of the conduit, while Fig. 1B(4) shows the bottom slice where the nucleus is located (Movie S1, ESI†).‡ Significant deformations of both the cytoskeleton and the nucleus in the longitudinal direction are observed along with the 3-D distribution of FAs. Fig. 1 Open in new tabDownload slide Experimental observations of focal adhesion sites and actin stress fibers on the lumen in a 3-D collagen matrix: (A) 3-D microfluidic assay with hMVECs seeded on one side of the collagen gel.14 Higher concentration of VEGF is supplied to channel A and lower concentration of VEGF to channel B so that a gradient of VGEF is created across the gel. (B) Stalk cells migrating into the gel are observed; (1) collapsed confocal 3-D image (120×) showing a stalk cell migration along a narrow lumen, and slices at selected heights of (2) z = 6.3 μm (top), (3) 10.08 μm (middle), and (4) 16.38 μm (bottom); nucleus and focal adhesion sites are stained with Hoechst (blue) and vinculin (red), respectively. (C) Actin stress fibers in a larger lumen with a magnification of 120×; sectional slices showing stress fibers at selected heights; (1) z = 0 μm, (B) 0.76 μm, (3) 5.32 μm and (4) 6.84 μm; nucleus and actin stress fibers are stained with Hoechst (blue) and Rhodamine phalloidin (red), respectively. (D(1)) A reflectance microscopy image showing the creation of a hole by a migrating tip cell in the 3-D collagen matrix; nucleus and vascular endothelial (VE)-cadherins are stained with Hoechst (blue) and anti-VE cadherin (green), (D(2)) collapsed confocal 3D image (120×); nucleus and actin stress fibers are stained with Hoechst (blue) and Rhodamine phalloidin (red), respectively, (D(3)) the longitudinal cross sectional view across line aa′ shown in D(2); short actin stress fibers are seen beneath the nucleus, (D(4)) the longitudinal cross sectional view across line bb′ shown in D(2); long stress fibers (yellow arrows) connected to the nucleus extend towards the leading edge of the migrating cell. (E) Quantification of the aspect ratio of the nucleus under two conditions when the cell migrates into the lumen with a diameter less than 25 μm and larger than 25 μm. Data are means ± SD. In each case 35 cells were analysed. Fig. 1 Open in new tabDownload slide Experimental observations of focal adhesion sites and actin stress fibers on the lumen in a 3-D collagen matrix: (A) 3-D microfluidic assay with hMVECs seeded on one side of the collagen gel.14 Higher concentration of VEGF is supplied to channel A and lower concentration of VEGF to channel B so that a gradient of VGEF is created across the gel. (B) Stalk cells migrating into the gel are observed; (1) collapsed confocal 3-D image (120×) showing a stalk cell migration along a narrow lumen, and slices at selected heights of (2) z = 6.3 μm (top), (3) 10.08 μm (middle), and (4) 16.38 μm (bottom); nucleus and focal adhesion sites are stained with Hoechst (blue) and vinculin (red), respectively. (C) Actin stress fibers in a larger lumen with a magnification of 120×; sectional slices showing stress fibers at selected heights; (1) z = 0 μm, (B) 0.76 μm, (3) 5.32 μm and (4) 6.84 μm; nucleus and actin stress fibers are stained with Hoechst (blue) and Rhodamine phalloidin (red), respectively. (D(1)) A reflectance microscopy image showing the creation of a hole by a migrating tip cell in the 3-D collagen matrix; nucleus and vascular endothelial (VE)-cadherins are stained with Hoechst (blue) and anti-VE cadherin (green), (D(2)) collapsed confocal 3D image (120×); nucleus and actin stress fibers are stained with Hoechst (blue) and Rhodamine phalloidin (red), respectively, (D(3)) the longitudinal cross sectional view across line aa′ shown in D(2); short actin stress fibers are seen beneath the nucleus, (D(4)) the longitudinal cross sectional view across line bb′ shown in D(2); long stress fibers (yellow arrows) connected to the nucleus extend towards the leading edge of the migrating cell. (E) Quantification of the aspect ratio of the nucleus under two conditions when the cell migrates into the lumen with a diameter less than 25 μm and larger than 25 μm. Data are means ± SD. In each case 35 cells were analysed. In a larger diameter conduit (∼30 μm) the migrating cell stretches more along the circumferential direction (Fig. 1C). The migration speed in the axial direction was substantially lower than that for the narrower conduit (15 μm). Actin stress fibers stained in red are distributed across the wall of the conduit. The nucleus appears to be pulled down by many short stress fibers (approximately 2 μm) (Movie S2, ESI†). On the other hand, in a narrow conduit (∼15 μm) the migrating cell elongates more along the longitudinal direction (Fig. 1D). Actin stress fibers are distributed parallel to the longitudinal direction. Longitudinal cross-sectional views of lines aa′ and bb′ in Fig. 1D(2) are shown in Fig. 1D(3) and D(4), respectively. The white arrows in Fig. 1D(3) indicate short stress fibers beneath the nucleus, while yellow arrows in Fig. 1D(3) and D(4) indicate long stress fibers connected to the nucleus. These long stress fibers extend towards the front side of the migrating cell. The nucleus also appears to be elongated along the longitudinal direction; the aspect ratio of semi-major and semi-minor is ∼2. The aspect ratio of nuclear size was quantified and contrasted between cells in lumen <25 μm (1.91 ± 0.26), and in lumen >25 μm (1.37 ± 0.19) (Fig. 1E). The aspect ratio of nuclear size is progressively increased as the diameter of the lumen becomes narrower. In summary, the experimental observations suggest that: The nucleus of a migrating cell deforms depending on the geometry of a contacting surface; The cell cytoskeleton stretches out when passing a narrow conduit; The cell stretches out in the circumferential direction when the conduit diameter is large; and Short stress fibers are formed beneath the nucleus, pulling it down towards the cytoskeleton, and long stress fibers can be observed between the nucleus and focal adhesions at the front side of the cytoskeleton. These experiments support the hypothesis that migrating cell behaviors are different depending on the surface geometry. The results are aligned with the in vitro cancer cell migration experiments by Irimia and Toner.19 They showed that the speed of migrating cancer cells differs significantly depending on the cross-sectional area of the microfluidic channel through which the cell spontaneously migrates.§The direction of migration and its persistency, too, differ depending on the width of the channel.19 Finally, the recent experiments by Chancellor et al.8 demonstrate that the actomyosin tension is balanced in part by the nucleus through mechanical links mediated by nesprin-1 (nuclear membrane proteins that bind to F-actin). We have also observed SFs connected to the nucleus, as shown in Fig. 1C and D. At the zero optical height, short SFs (2–3 μm) are formed and connected to the nuclear membrane on the bottom lumen that appear to tether the nucleus to the cell membrane.26 Long SFs are also formed that point towards the nucleus. These SFs connected to the nucleus play an important role in cell migration. In ref. 8 the authors also demonstrated that nesprin-1 depleted endothelial cells showed decreased migration speed with no SFs connected to the nuclear membrane. Furthermore, Khatau et al.26 highlighted the interplay between cell shape, nuclear shape, and cell adhesion mediated by the perinuclear actin cap. Based on these experimental observations, we constructed a 3-dimensional cell migration model incorporating mechanical links between FAs and the nucleus through contractile actomyosin motor activity, and formation of FAs through interactions between transmembrane integrins and ligands on 3D curved ECM surfaces. Our motivation is to investigate how these mechanisms are coordinated to create migratory motion on a curved surface, how a migrating cell deforms and spreads out over a curved substrate, and how the migration speed and direction are affected by the geometry of the curved surface. Simulated migration behaviors will be compared to the experimental results in terms of migration speed. Methods We aim to build a computational model to investigate cell migration behaviors on a curved surface, as described in the Experimental section. Specifically, we simulate haptotactic migration due to a gradient in ligand density on a curved surface. We simulate binding kinetics between integrin receptors and extracellular matrix protein ligands (e.g. collagen, fibronectin and laminin), model the formation of stress fibers, and predict how the forces acting on the cell deform the nucleus and the cytoskeleton, resulting in diverse patterns of the cell profile and migratory motion. Integrated cell migration model in the curved surface We have built an integrated cell migration model incorporating focal adhesion (FA) dynamics, cytoskeleton and nucleus remodeling and actin motor activity, and detailed modeling of cell migration in the 2-D planar surface and fibronectin coated patterns.9 The integrated cell migration model was further extended to simulate cell migration on 3-D curved surfaces. The simulation utilizes a Lagrangian approach with the time dependent motions of nodes in double membranes of cell and nucleus. The geometry of this double mesh structure is shown in Fig. 2A; the outer mesh representing the cell membrane and the inner mesh representing the nuclear membrane. Each mesh consists of N nodes connected elastically to adjacent nodes, forming a double elastic membrane. The inner and outer mesh nodes may be connected when Stress Fibers (SFs) that are formed between membranes of the nucleus and the cell.8,27 Multiple transmembrane integrins are bundled together and placed at each node on the outer mesh. They can bind to ligands on the matrix substrate, forming focal adhesions, to which stress fibers are connected (Fig. 3A). Fig. 2 Open in new tabDownload slide (A) Integrated cell migration model consisting of the cytoskeleton, the nucleus, N integrin nodes on the surface of cytoskeleton, N nucleus nodes on the surface of nucleus, and actin stress fibers which connect the integrin node to the nucleus node; (A) a top view of the model showing triangular mesh network of double membranes of cytoskeleton and nucleus. (B) The free body diagram of the i-th integrin node in the circle marked in (A) where five external forces are acting. Note that, while showing in 2-D, the force balance exists in 3-D. Fig. 2 Open in new tabDownload slide (A) Integrated cell migration model consisting of the cytoskeleton, the nucleus, N integrin nodes on the surface of cytoskeleton, N nucleus nodes on the surface of nucleus, and actin stress fibers which connect the integrin node to the nucleus node; (A) a top view of the model showing triangular mesh network of double membranes of cytoskeleton and nucleus. (B) The free body diagram of the i-th integrin node in the circle marked in (A) where five external forces are acting. Note that, while showing in 2-D, the force balance exists in 3-D. Fig. 3 Open in new tabDownload slide 3-D integrated cell migration model: (A) 2-D schematic representation of a cell migration model on the curved substrate, showing deformable cell and nuclear membranes, and actin stress fibers, (B) a magnified view in A showing the structure of focal adhesion including the attachment of the end of stress fibers through an integrin node to the underlying extracellular matrix, (C) a magnified view in B illustrating a stochastic ligand–receptor bonding process at the focal adhesion site, and (D) a magnified view in A showing the structure of actin stress fibers connected to a nucleus node. Fig. 3 Open in new tabDownload slide 3-D integrated cell migration model: (A) 2-D schematic representation of a cell migration model on the curved substrate, showing deformable cell and nuclear membranes, and actin stress fibers, (B) a magnified view in A showing the structure of focal adhesion including the attachment of the end of stress fibers through an integrin node to the underlying extracellular matrix, (C) a magnified view in B illustrating a stochastic ligand–receptor bonding process at the focal adhesion site, and (D) a magnified view in A showing the structure of actin stress fibers connected to a nucleus node. Fig. 2B shows the free body diagram of the i-th node of the cytoskeleton, called the i-th integrin node, where a bundle of integrins is formed. Acting on this node are force vectors due to frictional dissipative FD,ic(= −Ccvic) where Cc is a friction coefficient (0.001 N s m−1)28–30 associated with the energy dissipation at the integrin node and vic is the velocity of the i-th integrin node, focal adhesion force FFA,ic, elastic energy force FE,ic, and stress fiber force FSF,ic. Similarly, acting on the i-th node of the nucleus are force vectors due to frictional dissipative force FD,in(= −Cnvin) where Cn is a friction coefficient (0.001 N s m−1)28–30 associated with the energy dissipation at the nuclear node and vin is the velocity of the i-th nuclear node, elastic energy force FE,in and stress fiber force FSF,in. The equation of motion for each integrin node is given by micdvicdt=FD,ic+FFA,ic+FE,ic+FSF,ic,  i=1,...,N.1 where vic is the velocity vector of the i-th integrin node. Similarly, the equation of motion for each node of the nucleus is given by mindvindt=FD,in+FE,in+FSF,in,  i=1,...,N.2 where FD,in, FE,in and FSF,in are frictional dissipative force, elastic energy force and SF force at the i-th nuclear node, respectively, and vin is the velocity of the i-th nuclear node. It should be noted that sum of forces at any node is zero because their motions are very slow and in quasi-equilibrium in time, thus motion equations in terms of the velocities vic and vin (dxicdt=vic,dxindt=vin) can be simplified to the following six ordinary differential equations: Ccdxicdt=FFA,i+FE,ic+FSF,ic,  i=1,...,N3 Cndxindt=FE,in+FSF,in,  i=1,...,N4 where xic and xin represent coordinates of the i-th integrin node and the i-th nuclear node (Fig. 3A), respectively. Most of the frictional dissipative term FD,ic arises from the rupture of stretched ligand–receptor bonds (Fig. 3B) in the focal adhesion dynamics; when they rupture, the stored strain energy is released and dissipated. Similarly, FD,in also arises from the energy stored in SFs that, when F-actins are depolymerized, the stored strain energy is released and dissipated. The focal adhesion force FFA,i acts between the i-th integrin node and the point on the curved ECM surface where the extension of the unit normal vector, n̂R,i, intersects with the curved ECM surface.9 From Fig. 3B this intersection position, that is, the root location of receptor and ligand bonds (xL,i), is given by xL,i=xic+Lbn^R,i=xic−hpn^R,in^w⋅n^R,i5 where Lb is the bond length, n̂w is the unit normal vector of the ECM surface, and hp is the gap between the i-th integrin node and the curved ECM surface, as shown in Fig. 3B. These expressions are valid only when n̂w·n̂R,i < 0 and the gap hp is less than a critical height (hc) of 300 nm: hp < hc. The latter condition is to restrict the formation of receptor–ligand bonds within the upper limit hc. The focal adhesion force of the i-th integrin node FFA,i is computed as FFA,i=nb,i KLR(Lb−λ)n^R,i6 where nb,i is the number of ligand–receptor bonds at the i-th integrin node, κLR is an effective spring constant for a single ligand–receptor bond (∼1 pN nm−1),31 and (Lb − λ) represents averaged stretched distance from the equilibrium length (λ; 30 nm32) (see Fig. 3C). The detailed explanations of focal adhesion dynamics using Monte Carlo simulation methods can be found in the literature.9 Forces due to actin SFs' motor activity at the i-th integrin and j-th nuclear nodes are given by FSF,ic=−KSFNSF,i(dSF,i−NSF,iLSF,i1)∂dSF,i∂xic7a FSF,jn=−FSF,ic7b where KSF(=ESFASFLSF,i1) is stiffness of a SF which is variable depending on Young's modulus of SFs (ESF; 230 kPa33), average cross-sectional area of SFs (ASF; 250 nm in radius34) and a length of a single compartment of the i-th SF (LSF,i1), NSF is the number of compartments of the i-th SF, and dSF,i is full length of SFs under tension (see Fig. 3D). In particular, considering myosin II's slide on F-actin filaments with a sliding rate of vm at both ends,35–37LSF,i1 can be calculated with following discretized form of equation at every time step (Δt): dLSF,i1dt=−2vm8a LSF,i1=LSF,i0−2vmΔt8b where LSF,i0 indicates the length of a single unit of the i-th SF at the previous time (t − Δt). These forces are generated when focal adhesions have been formed and F-actin filaments are fully polymerized. It has been known that SF assembly occurs over several minutes,38–40 but SFs disassemble rapidly within seconds.41–44 In addition, it takes several minutes to form FAs from focal complexes (FCs). These observations suggest that myosin motor activities in SFs are switched off during the remodelling of the actin cytoskeleton (polymerization) and SFs turnover. In our simulations, time for full formation of F-actin is set to be Tp = 180 s, and time for the complete disassembly of F-actin is set to 1 s, based on the above reference information. The elastic forces, FE,ic and FE,in, are obtained by using the virtual work theory in structural mechanics.9 It should be noted that the elastic forces FE,ic and FE,in at the i-th node represent the resultant force acting on the i-th node that is calculated by vectorial addition of elastic forces from neighbouring nodes. To compute this, first the coordinates of each node are updated in each time cycle, and distances from each node to neighbouring nodes are computed along with the areas of the surrounding rectangles. The elastic forces are derived from these distances and areas for individual nodes. Computation of an “integrated cell migration model” Cell migration simulations were carried out using a fourth order Rosenbrock method45 based on an adaptive time-stepping technique for integrating ordinary differential equations with the convergence criterion <10−4. The ordinary differential equations were solved for the 6 × N (N = 98) unknown variables associated with the mesh node position vectors for both cell membrane and nuclear membrane: xic, xin, i = 1 – N. For cell migration simulation the Rosenbrock method outperforms the standard Runge–Kutta method which requires a relatively large number of iterations.45 Furthermore, the Rosenbrock method consumes less computing time by using adaptive time-step control that ranges from 10−3 s to 10−2 s in the present work. Thus, it is suitable for simulating transient cell migratory behaviors over 10 hours. The methods for building geometrical models for the simulation of cell migration have been well documented in the literature.46,47 See also Fig. S1 (ESI†). One practical issue in computing finite mesh geometric models is to check geometrical compatibility. As the coordinates of cell membrane and nuclear nodes are updated based on the equations of motion, geometrically incompatible situations occur occasionally in the configurations of the cell membrane mesh and that of the nucleus in relation to the curved ECM surface. For example, some cell membrane nodes intersect with the conduit, and the nucleus intersects with the cell membrane. These incompatible situations must be checked in every computational cycle, and necessary corrections must be made. Cell migration in the collagen gel matrix hMVECs, cell culture media and supplements were purchased from Lonza. All cell culture work was carried out in sterile tissue culture hoods and cell culture was carried out in a 5% CO2 humidified incubator at 37 °C. hMVECs were cultured in EGM-2MV (Lonza) culture medium, grown in a T75 tissue culture flask (Corning), and trypsinized when the culture flask becomes confluent (0.25% with EDTA, Gibco). Afterwards, 75 μL of cell suspension with a cell density of 2 × 106 cells mL−1 were loaded into the Micro-fluidic channel B (Fig. 1A) of the in vitro microfluidic device for observing the angiogenesis.14 The pressure difference between channels A and B (Fig. 1A) enables interstitial flow and seeded cells to adhere on the side surface of the collagen gel matrix, followed by putting the device into the 5% CO2 humidified incubator for several hours to form a monolayer on the side of the collagen gel matrix. Once the confluent monolayer was formed on the surface of channel B, fresh media with 20 and 40 ng mL−1 of vascular endothelial growth factor (VEGF; R&D systems) were, respectively, replenished into channels A and B to generate a gradient.11,14 Stalk cell migration was monitored by confocal microscopy with an incubator keeping 5% CO2 at 37 °C with a time-interval of 30 minutes for 12 hours. Immediately after the live cell imaging, immunofluorescence staining was performed to visualize the intracellular structure of the stalk cell at the end-point of the experiment. Cells were fixed with 4% paraformaldehyde (Sigma-Aldrich) for 15 minutes at room temperature, and permealized with 0.1% Triton X-100 (Sigma-Aldrich) for 5 minutes. Actin stress fibers, vinculin and nuclei were stained with Rhodamine-Phalloidin (Sigma-Aldrich), monoclonal anti-vinculin antibody produced in mouse (Sigma-Aldrich) and Hoechst (Sigma-Aldrich), respectively. Results and discussion Simulations of cell migration were performed for various conduit geometries and ligand density gradients. Type I collagen was considered for the ligands on the conduit surface. Two cases were examined: (a) uniform ligand concentration, and (b) graded ligand concentration. For case (a), the ligand concentration was uniformly set to be 0.8 mg mL−1 over a longitudinal conduit length of 100 μm. Since the molecular mass of Type I collagen is 350 kDa, the corresponding ligand surface density was 750 molecules μm−2 using the relationship between plating concentration and ligand surface density of type I collagen48 (Fig. 4). For case (b), the high ligand concentration was varied continuously from 2.60 mg mL−1 to 3.35 mg mL−1 over a longitudinal conduit length of 100 μm. This created a density gradient of 1.2 ng mm−3, whose ligand surface density was curve-fitted as 1.25 × 103 molecules μm−2 at the lowest end and 1.55 × 103 molecules μm−2 at the highest end using the relationship between plating concentration and ligand surface density of type I collagen48 (Fig. S2, ESI† and Fig. 5). Fig. 4 Open in new tabDownload slide (A) Simulated trajectories of cell migrations along seven rectangular conduits with the identical height of 3 μm, and different widths of 6 μm, 10 μm, 15 μm, 20 μm, 30 μm, 50 μm and 70 μm. Cells are initially spherical. The ligand surface density is 750 molecules μm−2 and constant over a longitudinal conduit length of 100 μm. The black lines indicate trajectories of nuclei for the first three hours, (B) comparison of average cell migration speeds: the simulation model vs. experimental data by Irimia and Toner.19 Average speed and standard error of mean (N = 5) are shown for the seven different channels, and (C). Linear regression (R2 = 0.771) of simulated migration speed vs. experimental migration speed. Fig. 4 Open in new tabDownload slide (A) Simulated trajectories of cell migrations along seven rectangular conduits with the identical height of 3 μm, and different widths of 6 μm, 10 μm, 15 μm, 20 μm, 30 μm, 50 μm and 70 μm. Cells are initially spherical. The ligand surface density is 750 molecules μm−2 and constant over a longitudinal conduit length of 100 μm. The black lines indicate trajectories of nuclei for the first three hours, (B) comparison of average cell migration speeds: the simulation model vs. experimental data by Irimia and Toner.19 Average speed and standard error of mean (N = 5) are shown for the seven different channels, and (C). Linear regression (R2 = 0.771) of simulated migration speed vs. experimental migration speed. Fig. 5 Open in new tabDownload slide (A) Average cell migration speeds and standard error of mean (N = 5) are shown for cylindrical luminal diameters of 6, 8.8, 12, 15, 20 and 30 μm, 5 hour-trajectories of cell migrations in corresponding lumens in (B) cross-sectional views, (C) top views, and (D) and (E) longitudinal and transverse cross sectional views of migrating cells along a narrow (8.8 μm) and wider (20 μm) cylindrical conduits, and (F) an example of backward cell migration in the lumen despite a positive ligand gradient on its surface (diameter of 12 μm). Fig. 5 Open in new tabDownload slide (A) Average cell migration speeds and standard error of mean (N = 5) are shown for cylindrical luminal diameters of 6, 8.8, 12, 15, 20 and 30 μm, 5 hour-trajectories of cell migrations in corresponding lumens in (B) cross-sectional views, (C) top views, and (D) and (E) longitudinal and transverse cross sectional views of migrating cells along a narrow (8.8 μm) and wider (20 μm) cylindrical conduits, and (F) an example of backward cell migration in the lumen despite a positive ligand gradient on its surface (diameter of 12 μm). At the initial state of each simulation, both cell and nuclear membranes were assumed to be round. Since the migration model is stochastic, simulations were repeated multiple times under the same initial conditions. Table 1 lists all the parameters used for the simulations with numerical values and their sources. Table 1 List of simulation parameters Parameter . Definition . Value, (equation) . Sources . ASF Averaged SFs' sectional area/μm2 0.196, (7a) 34 Cc Friction coefficients associated with the energy dissipation at the integrin node/N s m−1 0.001, (3) 28–30 Cn Friction coefficients associated with the energy dissipation at the nuclear node/N s m−1 0.001, (4) 28–30 F Force/N (1)–(4) and (6)–(7) L Length (5), (6) and (7a) Lb Stretched length of bonds between receptors and ligands Variable, (5) LSF,i1 Length of the i-th single unit of SFs at the present time/nm Variable, (7a) and (8) Current work LSF,i0 Length of the i-th single unit of SFs at the previous time/nm Variable, (8b) Current work N Number of nodes at each membrane 98 Current work NSF,i Number of contractile compartments in the i-th SFs Variable, (7a) Current work dSF,i Distance between i-th integrin and j-th nuclear nodes/m Variable, (7a) Current work hc Critical height/nm 300, (5) Current work hp Height from the surface to the i-th integrin node/nm Variable, (5) Current work κLR Effective spring constant of ligand-receptor bond/pN nm−1 1.0, (6) 31 κSF Effective stiffness of the i-th single unit of SFs/N m−1 Variable, (7a) Current work nb,I Number of bonds between receptors and ligands Variable, (6) Current work n̂R,i Unit normal vector at the i-th integrin node (5) and (6) Current work n̂w Unit normal vector at the local surface of the lumen (5) Current work t Time/s (1)–(4), and (8) v Velocity vector/nm s−1 (3) and (4) vm Sliding rate of non-muscle myosin II on the actin filaments/nm s−1 10, (8) 35–37 x Location vector/μm (3)–(5), and (7a) xL,i Location vector of the root of ligand–receptor bonds on the local surface of the lumen/nm (5) λ Equilibrium distance of an integrin/nm 30, (6) 32 Superscripts D Drag or friction (1) and (2) E Elastic (1)–(4) FA Focal adhesion (1), (3) and (6) SF Stress fiber (1)–(4) and (7) c Cytoskeleton (1), (3), (5) and (7) n Nucleus (2), (4) and (7b) i i-th node (1)–(8) 0 Previous time or initial state (8b) 1 Present time (7a) and (8a) Subscripts b Bond (5) and (6) Parameter . Definition . Value, (equation) . Sources . ASF Averaged SFs' sectional area/μm2 0.196, (7a) 34 Cc Friction coefficients associated with the energy dissipation at the integrin node/N s m−1 0.001, (3) 28–30 Cn Friction coefficients associated with the energy dissipation at the nuclear node/N s m−1 0.001, (4) 28–30 F Force/N (1)–(4) and (6)–(7) L Length (5), (6) and (7a) Lb Stretched length of bonds between receptors and ligands Variable, (5) LSF,i1 Length of the i-th single unit of SFs at the present time/nm Variable, (7a) and (8) Current work LSF,i0 Length of the i-th single unit of SFs at the previous time/nm Variable, (8b) Current work N Number of nodes at each membrane 98 Current work NSF,i Number of contractile compartments in the i-th SFs Variable, (7a) Current work dSF,i Distance between i-th integrin and j-th nuclear nodes/m Variable, (7a) Current work hc Critical height/nm 300, (5) Current work hp Height from the surface to the i-th integrin node/nm Variable, (5) Current work κLR Effective spring constant of ligand-receptor bond/pN nm−1 1.0, (6) 31 κSF Effective stiffness of the i-th single unit of SFs/N m−1 Variable, (7a) Current work nb,I Number of bonds between receptors and ligands Variable, (6) Current work n̂R,i Unit normal vector at the i-th integrin node (5) and (6) Current work n̂w Unit normal vector at the local surface of the lumen (5) Current work t Time/s (1)–(4), and (8) v Velocity vector/nm s−1 (3) and (4) vm Sliding rate of non-muscle myosin II on the actin filaments/nm s−1 10, (8) 35–37 x Location vector/μm (3)–(5), and (7a) xL,i Location vector of the root of ligand–receptor bonds on the local surface of the lumen/nm (5) λ Equilibrium distance of an integrin/nm 30, (6) 32 Superscripts D Drag or friction (1) and (2) E Elastic (1)–(4) FA Focal adhesion (1), (3) and (6) SF Stress fiber (1)–(4) and (7) c Cytoskeleton (1), (3), (5) and (7) n Nucleus (2), (4) and (7b) i i-th node (1)–(8) 0 Previous time or initial state (8b) 1 Present time (7a) and (8a) Subscripts b Bond (5) and (6) Open in new tab Table 1 List of simulation parameters Parameter . Definition . Value, (equation) . Sources . ASF Averaged SFs' sectional area/μm2 0.196, (7a) 34 Cc Friction coefficients associated with the energy dissipation at the integrin node/N s m−1 0.001, (3) 28–30 Cn Friction coefficients associated with the energy dissipation at the nuclear node/N s m−1 0.001, (4) 28–30 F Force/N (1)–(4) and (6)–(7) L Length (5), (6) and (7a) Lb Stretched length of bonds between receptors and ligands Variable, (5) LSF,i1 Length of the i-th single unit of SFs at the present time/nm Variable, (7a) and (8) Current work LSF,i0 Length of the i-th single unit of SFs at the previous time/nm Variable, (8b) Current work N Number of nodes at each membrane 98 Current work NSF,i Number of contractile compartments in the i-th SFs Variable, (7a) Current work dSF,i Distance between i-th integrin and j-th nuclear nodes/m Variable, (7a) Current work hc Critical height/nm 300, (5) Current work hp Height from the surface to the i-th integrin node/nm Variable, (5) Current work κLR Effective spring constant of ligand-receptor bond/pN nm−1 1.0, (6) 31 κSF Effective stiffness of the i-th single unit of SFs/N m−1 Variable, (7a) Current work nb,I Number of bonds between receptors and ligands Variable, (6) Current work n̂R,i Unit normal vector at the i-th integrin node (5) and (6) Current work n̂w Unit normal vector at the local surface of the lumen (5) Current work t Time/s (1)–(4), and (8) v Velocity vector/nm s−1 (3) and (4) vm Sliding rate of non-muscle myosin II on the actin filaments/nm s−1 10, (8) 35–37 x Location vector/μm (3)–(5), and (7a) xL,i Location vector of the root of ligand–receptor bonds on the local surface of the lumen/nm (5) λ Equilibrium distance of an integrin/nm 30, (6) 32 Superscripts D Drag or friction (1) and (2) E Elastic (1)–(4) FA Focal adhesion (1), (3) and (6) SF Stress fiber (1)–(4) and (7) c Cytoskeleton (1), (3), (5) and (7) n Nucleus (2), (4) and (7b) i i-th node (1)–(8) 0 Previous time or initial state (8b) 1 Present time (7a) and (8a) Subscripts b Bond (5) and (6) Parameter . Definition . Value, (equation) . Sources . ASF Averaged SFs' sectional area/μm2 0.196, (7a) 34 Cc Friction coefficients associated with the energy dissipation at the integrin node/N s m−1 0.001, (3) 28–30 Cn Friction coefficients associated with the energy dissipation at the nuclear node/N s m−1 0.001, (4) 28–30 F Force/N (1)–(4) and (6)–(7) L Length (5), (6) and (7a) Lb Stretched length of bonds between receptors and ligands Variable, (5) LSF,i1 Length of the i-th single unit of SFs at the present time/nm Variable, (7a) and (8) Current work LSF,i0 Length of the i-th single unit of SFs at the previous time/nm Variable, (8b) Current work N Number of nodes at each membrane 98 Current work NSF,i Number of contractile compartments in the i-th SFs Variable, (7a) Current work dSF,i Distance between i-th integrin and j-th nuclear nodes/m Variable, (7a) Current work hc Critical height/nm 300, (5) Current work hp Height from the surface to the i-th integrin node/nm Variable, (5) Current work κLR Effective spring constant of ligand-receptor bond/pN nm−1 1.0, (6) 31 κSF Effective stiffness of the i-th single unit of SFs/N m−1 Variable, (7a) Current work nb,I Number of bonds between receptors and ligands Variable, (6) Current work n̂R,i Unit normal vector at the i-th integrin node (5) and (6) Current work n̂w Unit normal vector at the local surface of the lumen (5) Current work t Time/s (1)–(4), and (8) v Velocity vector/nm s−1 (3) and (4) vm Sliding rate of non-muscle myosin II on the actin filaments/nm s−1 10, (8) 35–37 x Location vector/μm (3)–(5), and (7a) xL,i Location vector of the root of ligand–receptor bonds on the local surface of the lumen/nm (5) λ Equilibrium distance of an integrin/nm 30, (6) 32 Superscripts D Drag or friction (1) and (2) E Elastic (1)–(4) FA Focal adhesion (1), (3) and (6) SF Stress fiber (1)–(4) and (7) c Cytoskeleton (1), (3), (5) and (7) n Nucleus (2), (4) and (7b) i i-th node (1)–(8) 0 Previous time or initial state (8b) 1 Present time (7a) and (8a) Subscripts b Bond (5) and (6) Open in new tab Comparison to experimental data The first set of cell migration simulations was aimed to compare the integrated model against the experimental data published previously. Irimia and Toner19 performed cancer cell migration experiments in confined microfluidic channels having a uniform ligand concentration. They have reported that the observed spontaneous cell migration along conduits of uniform ligand concentration differs significantly depending on the conduit cross-section. The simulation results, too, showed similar behaviours. Fig. 4A and Fig. S2A (ESI†) show sample trajectories of simulated cell migrations along rectangular conduits of various widths. The conduit sizes used for the simulations matched those of the available experimental data; the depth of the conduit was fixed to 3 μm, while the width was varied to 7 different values: 6 μm, 10 μm, 15 μm, 20 μm, 30 μm, 50 μm and 70 μm. First the total path length of each trajectory was obtained and was divided by the travelling time, 3 hours, to obtain the average migrating speed. In the experiments, the speed of cancer cell migration was monitored in every 6 minutes, and was time averaged over the entire migration period for each of the channel geometries. Fig. 4B compares the average migration speed between the experiment and simulations. The experimental data show that the cell velocity is the lowest when migrating in the narrowest conduit, increases with the increasing cross-sectional width, reaches a maximum value at the width of 20 μm, and then decreases as the conduit becomes wider (Fig. 4B and Fig. S2B, ESI†). The simulated cell migration speed, too, shows a trend similar to the experiments: slow for a very small cross section, the fastest in the midrange, then slower again. Both experiments and simulations attain the fastest speed around 20–30 μm of width, or 60–90 × 10−12 m2 of cross-sectional area. As the width became even wider, the simulated speed decreased modestly while the experiment speed decreased more rapidly. Overall both show a good agreement over the range of 6–30 μm of conduit width. Statistical analysis of linear regression was performed by comparing the experiment and the simulation in terms of the “mean values” of time-averaged cell migration speed for the same conduit cross section. As shown in Fig. 4C and Fig. S2C (ESI†), significant correlations were found between the two with R2 = 0.771 and R2 = 0.719, respectively. Thereby, cell migration speeds are strongly dependent on channel's widths. Interestingly, as the width becomes narrower than 15 μm, cell trajectories are almost straight lines since the cell contacts all four walls of the channel. On the other hand, as the width becomes wider, the cell tends to wander, following a curved trajectory until it contacts a side wall of the conduit. Since the cell's motility mechanism is stochastic due to the binding and rupture kinetics of integrin-receptors and ligands, the cell does not necessarily move in the direction of higher ligand density. To further examine migration direction, the same simulations were repeated for conduits having the ligand concentration gradient, 1.2 ng mm−3, as described previously. Despite the gradient the cells migrated laterally or backward as shown in Fig. S2A and Movie S3a (ESI†). Since the ligand density has a gradient, on average the cell moves towards the higher ligand density (Fig. S2A, ESI†). But, it may move in the opposite direction with a certain probability. Occasional backward migration was observed in our experiments as well (Movie S3b, ESI†). It is interesting to note that the cell, once committed to moving in one direction, tends to keep moving in the same direction. It persists to move in the same direction until it stochastically switches direction. The duration of the persistent migration varies depending on the ligand gradient as well as on the geometry of the conduit. The conduit width affects this persistency of migration direction. As shown in Fig. S2A (ESI†), the trajectories for the conduits of widths 6, 10 and 15 μm are almost straight, while the ones for 50 and 60 μm of conduit width are winding. It is noticeable that the cell membrane profile for the 6 μm conduit is so narrow that the cell membrane is in contact with the four walls surrounding the cell, creating a smaller difference in the FA numbers between the leading and tailing edges. This results in a smaller traction force and slower migration speed than others. As the width gets wider, the cell has more room for spreading the shell-shaped cell membrane on the leading edge, resulting in higher speeds, as shown in 20 and 30 μm conduits (Fig. S3 and Movie S3c, ESI†). Cell migration along cylindrical lumens Migration simulations under the same ligand gradient along cylindrical conduits with diverse diameters: 6 μm, 8.8 μm, 12 μm, 15 μm, 20 μm and 30 μm, produced results similar to the rectangular conduit cross sections. The results show that migration speed is maximized at a specific diameter of conduit and that the migration direction was influenced by the conduit diameter (Fig. 5A–C). The maximum speed was found at a diameter around 10 μm, which is interestingly equivalent to the capillary diameter (Fig. S4B, ESI†). As the diameter was reduced to 6 μm, the migratory speed became very slow because most of integrin nodes (97%) contacted the cylindrical wall. This caused the density of FAs to be almost the same between the leading and tailing edges. Furthermore, the magnitude of actin SF forces at the tailing edge is relatively strong because the actin SFs are aligned with the longitudinal direction of the lumen (Fig. S4C, ESI†). This makes the backward force larger, resulting in a small net traction (Movie S4a, ESI†). As the conduit diameter became larger, 8.8 and 12 μm, the density of tailing edge FAs reduced, and SFs became short and not aligned with the longitudinal direction, while the leading edge had many long SFs well aligned with the longitudinal direction, resulting in a large traction force and fast migration (Movie S4b, ESI†). Fig. 5D shows longitudinal and transverse cross-sectional views of the cell. In this range of conduit diameter the cell membrane contacted the entire circumference of the conduit, thereby plugging the conduit. This prevented the cell from moving sideways, and directed it only towards the longitudinal direction; the trajectories of the nucleus are almost straight lines, as seen in Fig. 5C for diameters of 6, 8.8, and 12 μm. The cell body stretched out in the longitudinal direction, and its length is approximately inversely proportional to the conduit cross-sectional area: longitudinal body lengths of 23 μm, 15 μm, and 10 μm were observed for conduit diameters of 6 μm, 8.8 μm, and 12 μm, respectively (Fig. S4, ESI†). Furthermore, the nucleus is most elongated when the lumen is narrowest. Similar experimental observations were reported that cells were restricted and nuclei were stretched at the widths 10–50 μm of the underlying FN patterns.26 As the conduit diameter further increases, the cell no longer plugged the conduit, but tended to move in the transverse directions. See the conduit diameter of 15, 20 and 30 μm, in Fig. 5C. The conduit curvature affects cell movement significantly; the probability of integrin–ligand bond formation for a curved surface (transverse direction) is significantly higher than that of a planar surface (longitudinal direction), since the distance between an integrin and a ligand becomes shorter (see Fig. 3B). As a result, the cell stretched out in the circumferential direction, and tended to move transversely. See the cross-sectional views in Fig. 5E (Movies S4d and S4e, ESI†). Spiral movements were also observed for the conduit diameters of 15 μm and 20 μm. When the diameter of the lumen was 30 μm, the cell wandered in the transverse direction, and stretched out along the conduit circumference (Movie S4f, ESI†). However, the probability of the lateral cell migration will increase with a shorter distance between integrin–ligand formation as the diameter increases. However, note that there is an exceptional case when the radius of curvature increases largely, the lumen becomes flatter locally. In this case, the distance between integrin–ligand formation is uniform such that cell migration becomes increasingly similar to randomised 2-D cell migration on a planar surface. It is of interest to identify the critical radius of curvature for the transition from directed, lateral cell migration to randomised cell migration. However, this critical radius of curvature for the transition may be different for different cell types due, for example, to the size of the cell relative to the diameter of the lumen; as the size of the stretched cell increases, the critical radius of curvature for the transition may be relatively increased more. We presume the critical radius of curvature for the transition in current simulations is at the value larger than our maximum radius of 30 μm. The stochastic persistence phenomenon was also observed. Once the cell began to move in one direction, it continued to move in that direction for some time, but it occasionally switched the direction of migration (Movie S4c, ESI†). Computation of dissipative friction coefficients The frictional dissipation term, FD,ic = −Cc[|vic|]vic, in the equation of motion is a state-dependent, nonlinear term. In the simulations, a fixed value was used for the coefficient Cc based on the literature information: Cc = 0.001 N s m−1.28–30 This coefficient value is now examined below. The dominant effect of this frictional dissipation comes from the energy release due to the rupture of integrin receptor–ligand bonds. The dissipated energy arises from the tension built up at each focal adhesion, which is released when the bond ruptures. The probability of bond rupture as well as bond formation depends on the local geometry of each integrin and surrounding ligands, the relationship of which varies depending on the velocity of the integrin node relative to the substrate, vic. By simulating this process we can obtain the relationship between the frictional dissipation force and the velocity. As shown in the inset of Fig. 6, the cell membrane is moved at a constant speed |vic| along the bed of ligands, and the formation and rupture of ligand–receptor bonds are simulated based on the stochastic computation algorithm described previously. Since the transmembrane integrins are at nodes of the elastic mesh structure, the integrins are suspended elastically, as illustrated in the inset of Fig. 6. Once a bond is formed, the integrin pulls the ligand as the cell membrane moves, the tension increases as it elongates, and finally the bond ruptures. Fig. 6 shows stochastic simulations delineating the relationship between the cell membrane speed and the rupture-induced dissipative energy release per unit time, i.e. power loss. (See ESI† for mathematical derivation.) The figure also shows how long individual bonds last, i.e. bonding time, indicated with color-coded dots. There are two strikingly different groups of data points; one in a lower speed range and the other at higher speeds. At low speeds, the bonds tend to last for a longer time, generating a larger force and a larger amount of energy release, while at higher speeds the bonds rupture immediately. Each data group can be approximated by a curved solid line, the slope of which gives the coefficient Cc (see ESI† for details). These two different values of Cc are resulted from different speeds of |vic|. For example, as the power loss is lower and |vic| is higher, the value of Cc becomes lower (equation (A10), ESI†). In the migration simulation, the fixed coefficient value Cc = 0.001 N s m−1 was used, which gives the linear relationship shown by the red straight line in Fig. 6. Note that this value taken from the literature can approximate the data very accurately. The caveat is that at higher speeds the coefficient Cc reduces significantly, as shown by the blue line in Fig. 6. The power loss due to the quick ruptures is negligibly small compared to that of the red line, and the occurrence probability of quick ruptures is much lower than that of the red line. Therefore, it is justifiable to use the fixed value Cc = 0.001 N s m−1 found in the literature for migration simulations. Fig. 6 Open in new tabDownload slide Dissipative friction due to bond ruptures; a graph of power released by ruptures of integrin receptor–ligand bonds versus |vic|2 a color map indicates receptor–ligand bonding time, and a schematic in the inset of the i-th integrin node representing frictional dissipations due to ruptures of integrin receptor–ligand bonds. Fig. 6 Open in new tabDownload slide Dissipative friction due to bond ruptures; a graph of power released by ruptures of integrin receptor–ligand bonds versus |vic|2 a color map indicates receptor–ligand bonding time, and a schematic in the inset of the i-th integrin node representing frictional dissipations due to ruptures of integrin receptor–ligand bonds. Integrated cell migration model The current integrated cell migration model has been developed for the general cell migration on the surface of ECM molecule or intercellular adhesion molecule (i.e. VE-cadherin) coated substrates independent of cell type. In order to mimic a specific cell type simply requires changing the size of the cell, or the numbers of adhesion molecules per node, or per sectional area of actin stress fibers etc. For example, in the case of fibroblasts which have a higher number of focal adhesions, the density of integrin nodes on the cell membrane (Fig. 2) can be increased. Additionally, although our model as currently constructed is limited to migration along a surface, it has the potential to incorporate effects that would permit the simulation of 3-D migration. This would require the addition of MMP activity which is ongoing work. Finally, the model can be further extended to simulate homogeneous cell–cell interactions as well as heterogeneous cell–cell interactions to simulate paracellular or transcellular migration of immune cells across endothelial monolayers.49 Cell migration is a complex multifaceted process, triggered by chemotaxis and haptotactic responses from the extracellular environment.1 The motion of cell migration model is initially triggered by strong traction force from the ECM molecules like a thin lamellipodium protrusion at the leading edge, followed by the assembly of a number of focal adhesions between the lamellipodium base and the ECM. Afterwards, actin stress fibers extend from nascent focal adhesions and some of which connect to the nucleus. The corresponding motor activity exerts force on the FAs fore and aft, enabling the generation of a traction force and the release of FAs in the rear of the cell, creating the cell body's forward movement. The current integrative cell migration can provide new biological insight into designing a better experiment. For example, as an extension of cell migration on the curved surface, it is of interest to predict cell migration on a wavy surface. The model can predict whether cells migrate perpendicular to the grooves or how the migration direction differs if the cells are on a concave or convex surface. The model we proposed will not only provide new insight into better building experiments, but also such an experiment will allow us to better validate the model. Thus, as a selected application, we confirm how this cell migration model may be applied to the designs of efficient experiment for cell migration and further experiment for spontaneous cancer cell migration for a diagnostic assay. Conclusion An integrated computational model for predicting cell migration on 3-D curved surfaces has been presented. The equations of motion based on the elastic double mesh structure allowed both cell membrane and nucleus to deform flexibly in accordance with forces acting on each mesh node, including focal adhesion (FA) force, elastic membrane forces, and frictional dissipative force. Integrins distributed over the cell membrane interact stochastically with ligands on the ECM lumen surface, form FAs, anchor them to the surface, create stress fibers (SFs), and collectively generate a traction force for migration. Probabilities for formation and rupture of ligand–receptor bonds are affected by the surface curvature and lumen geometry, resulting in unique migration behaviors, which cannot be explained by migration models for 2-D flat surfaces. Specifically, we have found. (1) The migration speed takes a maximum at a particular diameter or width of the conduit along which the cell migrates. The computational model successfully produced the speed vs. conduit width relationship consistent with the existing microfluidic experimental data of cancer cell migration. (2) For a narrow lumen, the cell is confined and stretched along the longitudinal direction, contacting all the circumference of the lumen and resulting in a straight movement directed towards the longitudinal direction. For a wide lumen, the cell tends to stretch out along the circumference, where the radius of curvature is smaller than that of the longitudinal direction, resulting in a high probability for transverse migration. The model can be extended to a more complex model including more details about mechanisms of cell–cell interactions. Acknowledgements The authors thank the Singapore-MIT Alliance of Research and Technology for financial support of this work. This material is based upon work supported by the National Science Foundation under Grant No. EFRI-0735997 and Grant No. STC-0902396. References 1 L. 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See DOI: 10.1039/c2ib20159c ‡ In the recent literature18 the formation of focal adhesions in cells embedded in collagen 3-D matrices has been addressed. The present experiments, however, differ from such embedded 3-D cell experiments since the cells migrate on curved surfaces. Focal adhesions were clearly observed along the surfaces. § This will be discussed in detail later in the results and discussion section. See Fig. 4A. The journal is © The Royal Society of Chemistry 2012
TI - Integrating focal adhesion dynamics, cytoskeleton remodeling, and actin motor activity for predicting cell migration on 3D curved surfaces of the extracellular matrix
JF - Integrative Biology
DO - 10.1039/c2ib20159c
DA - 2012-10-22
UR - https://www.deepdyve.com/lp/oxford-university-press/integrating-focal-adhesion-dynamics-cytoskeleton-remodeling-and-actin-uHdxYPuzOx
SP - 1386
VL - 4
IS - 11
DP - DeepDyve
ER -