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, 11 (10), e0165570
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Modelling Chemotactic Motion of Cells in Biological Tissues

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Modelling Chemotactic Motion of Cells in Biological Tissues

Bakhtier Vasiev. PLoS One.

Abstract

Developmental processes in biology are underlined by proliferation, differentiation and migration of cells. The latter two are interlinked since cellular differentiation is governed by the dynamics of morphogens which, in turn, is affected by the movement of cells. Mutual effects of morphogenetic and cell movement patterns are enhanced when the movement is due to chemotactic response of cells to the morphogens. In this study we introduce a mathematical model to analyse how this interplay can result in a steady movement of cells in a tissue and associated formation of travelling waves in a concentration field of morphogen. Using the model we have identified four chemotactic scenarios for migration of single cell or homogeneous group of cells in a tissue. Such a migration can take place if moving cells are (1) repelled by a chemical produced by themselves or (2) attracted by a chemical produced by the surrounding cells in a tissue. Furthermore, the group of cells can also move if cells in surrounding tissue are (3) repelled by a chemical produced by moving cells or (4) attracted by a chemical produced by surrounding cells themselves. The proposed mechanisms can underlie migration of cells during embryonic development as well as spread of metastatic cells.

Conflict of interest statement

The authors have declared that no competing interests exist.

Figures

Fig 1
Fig 1. Migration of cells during gastrulation in the chick embryo.
A: Schematic illustration of cellular flows in the chick embryo at early gastrulation [15]. Yellow spot represents a morphological structure called Hensen’s node which consists of roughly 80 cells (size is about 100μm) and moves with a speed of about 2μm/min. B: Process shown in panel A as modelled by Cellular Potts Model [8]. Translational motion of yellow cells can be explained by four chemotactic mechanisms: (i) yellow cells are attracted by chemical produced by red cells, (ii) yellow cells are repelled by chemical produced by blue cells, (iii) red cells are repelled by a chemical produced by yellow cells, and (iv) blue cells are attracted by a chemical produced by yellow cells. C: Schematic illustration of stem zone migration at late gastrulation in the chick embryo [16]. Red contoured white spot represents stem zone which consists of about 1000 cells (size is 0.3mm) and moves with a speed of about 2μm/min. D: Process shown in panel C as modelled by Cellular Potts Model [9]. Translational motion of the stem zone (represented by red cells which are proliferating and differentiating into green cells) is explained by chemo-repulsion by FGF8 [10] which is produced in the stem zone [17].
Fig 2
Fig 2. Illustrations of 2D- (A) and 1D- (B, C) models.
A: Group of cells (green) forms a circular domain in a tissue (composed by red cells) in two-dimensional Cellular Potts Model [18]. B-C: One-dimensional model represented by a cross-section of two-dimensional model tissue (panel A) through the centre of the green domain. Group of cells now is represented by a segment of a line (of length a) along the ξ-axis. Symmetric u-profile for stationary (c = 0, panel B) and asymmetric u-profile for travelling wave (c≠0, panel C) solutions given by Eq (3) are shown. On the asymmetric profile (panel C) the maximum concentration lags behind the mid-point of the segment and the concentration on the back of the segment is above the concentration on its front (the difference is denoted by Δu on panel C).
Fig 3
Fig 3. Travelling wave solutions in the system (2, 4).
A: Plots of y = f(c) for four positive (dotted lines) and four negative (solid lines) values of c0 and the plot of y = c (dashed line) are shown. Abscissa of the points of intersection of solid/dotted lines with dashed one gives the speed, c, (for the value of c0 corresponding to the solid/dotted line) satisfying Eqs (3) and (5). Points of intersections in the first quadrant are associated with the segment moving to the right (c>0), while the points of intersection in the third quadrant (c<0)–to the left. Model parameters used for plotting solid lines: k1 = 2.5*10−4, k2 = 2*k1, D = 0.5; a = 50, c0 = -12,-8,-4,-1—for (highest to lowest in the II quadrant) dotted lines and c0 = 1, 4, 8 and 12—for (lowest to highest in the I quadrant) solid lines. B: u-profile for the segment moving to the right (given by the Eq (3) where c = 0.02). C: u-profile for the segment moving to the left (given by the Eq (3) where c = -0.02).
Fig 4
Fig 4. Plots illustrating dependence of the velocity of traveling wave solutions, c, in the system (2, 4), on the strength of chemotaxis, c0, and indicating the presence of a supercritical pitchfork bifurcation in Eq (8).
A: Plots of c(c0) satisfying the system (3, 5) for analytical solution (solid black line) and its cubic approximation (dashed line) are shown (for the values of model parameters taken from Fig 3). Plots illustrate that non trivial travelling wave solutions exist below a critical value, c0*, of the bifurcation parameter c0. Travelling solutions emerge in pairs, moving with an equal speed in the opposite directions. B: Our stability analysis indicates that the pitchfork bifurcation is supercritical and solutions corresponding to travelling waves are stable.
Fig 5
Fig 5. Travelling wave solutions and their stability in the systems (2, 16, 4).
A-B: u-profiles for non-moving (A) and moving to the right (B) segments in the system (17, 18). C: Supercritical pitchfork bifurcation in Eq (8) for the system (2, 16, 4). Plots of analytical solutions c = c(c0) satisfying the system (17, 18) are shown. Model parameters are the same as in Fig 2, however now the production term in Eq (2) is given by Eq (16). When the chemoattraction gets strong enough, c0>c0*, the non-moving segment becomes unstable and two new (stable) solutions corresponding to moving segments emerge.
Fig 6
Fig 6. Speed of the MD versus its size in two modifications of the model.
A-B: Solution of the Eq (5) for small- (A) and large-sized (B) segments. C-D: Solution of the Eq (22) for small- (C) and large-sized (D) segments. Plots are produced using MAPLE with the following values of model parameters: k1 = 7.5*10−4, k2 = 2*k1, D = 0.5, c0 = 10.
Fig 7
Fig 7. The MD in four chemotactic scenarios simulated using the Cellular Potts Model.
A: Cells (green) forming the MD are repelled by the chemotactic agent produced by themselves. B: Cells (green) forming the MD are attracted by the chemotactic agent produced by surrounding (red) cells. C: Cells (red) surrounding the MD are attracted by the chemotactic agent produced by cells (green) forming the MD. D: Cells (red) surrounding the MD are repelled by the chemotactic agent produced by themselves. Locations of the MD at three consequent instances of time are shown for each scenario. Concentration of chemotactic agent is represented by shadows of grey from black (highest concentration) to white (lowest concentration). To maintain the direction of motion (to the right) the initial asymmetry was set in the following manner: green cells were initially set close to the left border; during initial 4000 time units the chemotaxis was turned off giving time for the concentration field to establish; then the concentration field was shifted 10 grid points left and chemotaxis was turned on. From this moment (T = 0) the domain was moving due to chemotaxis. The medium contains 31x55 cells, green domain– 9 cells, average cell contains 49 grids. Model parameters describing the dynamics of morphogen: D = 0.5, k1 = 2.5*10−4, k2 = 2*k1; β = -8700, both, time and space, steps are set to one. Other model parameters in CPM (for the explanation of these parameters see [9]) elasticity of cells in CPM, λ = 0.6; Boltzmann temperature, T = 5.0; adhesiveness matrix (cell_type1/cell_type2):J=[red/redred/greengreen/redgreen/green]=[3772].
Fig 8
Fig 8. Movement patterns of cells in the vicinity of the MD in Cellular Potts Model.
Panel A illustrates cell movement pattern for the two active mechanisms while panel B—for passive. Panel A shows the area framed in the snapshot for T = 4000 in Fig 7A, while panel B—framed in Fig 7C, T = 16000. Both panels demonstrate locations of green and red cells and their traces, which are shown in blue and taken for 500 time units in panel A and 2000 time units in panel B (four-fold difference in time intervals compensates four-fold difference in speeds of the active and passive MDs). Conclusions on movement directions for (green and red) cells in different locations are illustrated by (green and red) arrows.

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References

    1. Wolpert L (1969) POSITIONAL INFORMATION AND SPATIAL PATTERN OF CELLULAR DIFFERENTIATION. Journal of Theoretical Biology 25: 1-&. - PubMed
    1. Grimm O, Coppey M, Wieschaus E (2010) Modelling the Bicoid gradient. Development 137: 2253–2264. 10.1242/dev.032409 - DOI - PMC - PubMed
    1. Vasieva O, Rasolonjanahary Mi, Vasiev B (2013) Mathematical modelling in developmental biology. Reproduction (Cambridge, England) 145: R175–184. - PubMed
    1. Baker RE, Maini PK (2007) Travelling gradients in interacting morphogen systems. Mathematical Biosciences 209: 30–50. 10.1016/j.mbs.2007.01.006 - DOI - PubMed
    1. Stern CD (2004) Gastrulation in the chick In: Stern CD, editor. Gastrulation: from cells to embryo. New York: Cold Spring Harbor Laboratory Press; pp. 219–232.

Grant support

This work was funded by the BBSRC; URL: http://www.bbsrc.ac.uk/. Grant number: BB/K002430/1. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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