Ordingly the cell membrane can be represented by @O0 . Thereby, the substrate domain can be defined as O ?fx x ?2 L; x ? O0 g 2 ?1??0?During cell migration, both domains O0 and O vary such that O0 [ O = and O0 \ O = ;. To correctly incorporate adhesivity, z, of cell in the cell front and rear, it is essential to define the cell anterior and posterior during cell motility. Assuming is a plane passing by the cell centroid, O, with unit normal vector n, parallel to epol, and s(X0) is a position vector of an arbitrary node located on @O0 (Fig 3), projection of s on n can be defined as d ?2?Consequently, nodes with positive are located on the cell membrane at the front, @O0 +, while nodes with negative belong to the cell membrane at the cell rear, @O0 -, where @O0 = @O0 + [ @O0 – should be satisfied. We assume that the cell extends the TSA biological activity protrusion from the membrane vertex whose position vector is approximately in the direction of cell polarisation, on the contrary, it retracts the trailing end from the membrane vertex whose position vector is totally in the opposite direction of cell polarisation. Thus, the maximum value of delivers the membrane node located on @O0 + from which the cell must be extended while the minimum value of represents the membrane node located on @O0- from which the cell must be retracted. Assume eex 2 O is the finite element that the membrane node with the maximum value of belongs to its space and ere 2 O0 is the finite element that the membrane node with the minimum value of belongs to its space. To integrate cell shape changes and cell migration, simply, ere is moved from the O0 domain to the O domain, in contrast, eex is eliminated from the O domain and is included in the O0 domain [68]. In the present model the cell is not allowed to obtain infinitely thin shape during migration. Therefore, consistent with the experimental observation of Wessels et al. [93, 94], it is considered that the cell can extend approximately 10 of its whole volume as pseudopodia.PLOS ONE | DOI:10.1371/journal.pone.0122094 March 30,10 /3D Num. Model of Cell Morphology during Mig. in Multi-Signaling Sub.Fig 3. Definition of extension and retraction points as well as anterior and posterior parts of the cell at each time step. R3, and 0 represent the 3D working space, matrix and cell domains, respectively. X stands for the global coordinates and X0 represents the local cell coordinates located in the cell centroid, O. is a plane passing by the cell centroid with unit normal vector n parallel to the cell polarisation direction, epol. P denotes a finite element node located on the cell membrane, @. @0 + and @0 – are the finite element nodes located on the front and rear of the cell membrane, respectively. doi:10.1371/journal.pone.0122094.gFinite element implementationThe present model is implemented through the commercial finite element (FE) software ABAQUS [95] using a TSA msds coupled user element subroutine. The corresponding algorithm is presented in Fig 4. The model is applied in several numerical examples to investigate cell behavior in the presence of different stimuli. It is assumed that the cell is located within a 400?00?00 m matrix without any external forces. The matrix is meshed by 128,000 regular hexahedral elements andPLOS ONE | DOI:10.1371/journal.pone.0122094 March 30,11 /3D Num. Model of Cell Morphology during Mig. in Multi-Signaling Sub.Fig 4. Computational algorithm of migration and cell morphology changes in a multi-.Ordingly the cell membrane can be represented by @O0 . Thereby, the substrate domain can be defined as O ?fx x ?2 L; x ? O0 g 2 ?1??0?During cell migration, both domains O0 and O vary such that O0 [ O = and O0 \ O = ;. To correctly incorporate adhesivity, z, of cell in the cell front and rear, it is essential to define the cell anterior and posterior during cell motility. Assuming is a plane passing by the cell centroid, O, with unit normal vector n, parallel to epol, and s(X0) is a position vector of an arbitrary node located on @O0 (Fig 3), projection of s on n can be defined as d ?2?Consequently, nodes with positive are located on the cell membrane at the front, @O0 +, while nodes with negative belong to the cell membrane at the cell rear, @O0 -, where @O0 = @O0 + [ @O0 – should be satisfied. We assume that the cell extends the protrusion from the membrane vertex whose position vector is approximately in the direction of cell polarisation, on the contrary, it retracts the trailing end from the membrane vertex whose position vector is totally in the opposite direction of cell polarisation. Thus, the maximum value of delivers the membrane node located on @O0 + from which the cell must be extended while the minimum value of represents the membrane node located on @O0- from which the cell must be retracted. Assume eex 2 O is the finite element that the membrane node with the maximum value of belongs to its space and ere 2 O0 is the finite element that the membrane node with the minimum value of belongs to its space. To integrate cell shape changes and cell migration, simply, ere is moved from the O0 domain to the O domain, in contrast, eex is eliminated from the O domain and is included in the O0 domain [68]. In the present model the cell is not allowed to obtain infinitely thin shape during migration. Therefore, consistent with the experimental observation of Wessels et al. [93, 94], it is considered that the cell can extend approximately 10 of its whole volume as pseudopodia.PLOS ONE | DOI:10.1371/journal.pone.0122094 March 30,10 /3D Num. Model of Cell Morphology during Mig. in Multi-Signaling Sub.Fig 3. Definition of extension and retraction points as well as anterior and posterior parts of the cell at each time step. R3, and 0 represent the 3D working space, matrix and cell domains, respectively. X stands for the global coordinates and X0 represents the local cell coordinates located in the cell centroid, O. is a plane passing by the cell centroid with unit normal vector n parallel to the cell polarisation direction, epol. P denotes a finite element node located on the cell membrane, @. @0 + and @0 – are the finite element nodes located on the front and rear of the cell membrane, respectively. doi:10.1371/journal.pone.0122094.gFinite element implementationThe present model is implemented through the commercial finite element (FE) software ABAQUS [95] using a coupled user element subroutine. The corresponding algorithm is presented in Fig 4. The model is applied in several numerical examples to investigate cell behavior in the presence of different stimuli. It is assumed that the cell is located within a 400?00?00 m matrix without any external forces. The matrix is meshed by 128,000 regular hexahedral elements andPLOS ONE | DOI:10.1371/journal.pone.0122094 March 30,11 /3D Num. Model of Cell Morphology during Mig. in Multi-Signaling Sub.Fig 4. Computational algorithm of migration and cell morphology changes in a multi-.