Lecture 8: Organization of Epithelial Tissue

Transcription

1 Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Organization of Epithelial Tissue Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16

2 9. November / 103 Contents 1 Organization of Epithelial Tissue 2 Euler s Theorem - Average Neighbour Number 3 Driving Forces behind the Polygon Distribution Topological Models: Impact of Cell Division Tissue Mechanics 4 Inference of cellular properties from images 5 Cell-based Simulation Frameworks

3 Organization of Epithelial Tissue Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

4 Fletcher, A. G., et al. (2014). Biophys J 106(11): Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103 Organization of Epithelial Tissue In an epithelium, each cell is tightly coupled to its neighbours by a ring of intercellular junctions which, in addition to maintaining the cohesion between cells, also forms a tight permeability barrier between cells. The membrane boundaries of individual cells within an epithelium are, to a good approximation, polygonal in shape.

5 9. November / 103 Apical Surface of the Drosophila Wing Disc Epithelium Drosophila Wing Disc: Heller, D., et al. (2016) Dev Cell 36,

6 9. November / 103 Polygon Distribution in the Drosophila Wing Disc Heller, D., et al. (2016) Dev Cell 36,

7 9. November / 103 Edge Number increases during Cell Cycle Heller, D., et al. (2016) Dev Cell 36,

8 Epithelial cells have on average 6 neighbours Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

9 Average Cell Area related to Polygon class Lewis Law (1928) A n = A 0 (n 2) 4N A n average apical area n number of its edges N total number of cells A 0 total tissue area. Either one or both of N and A 0 can be time dependent, to account for growth and cellular division; yet the relationship is valid throughout the development of the tissue. Heller, D., et al. (2016) Dev Cell 36, Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

10 9. November / 103 Aboav-Weaire s Law: Properties of Neighbours Aboav-Weaire s Law (1970) m n = n m n refers to the average number of edges of a randomly chosen neighbouring cell of a cell with with n neighbours. Aboav. The arrangement of grains in polycrystals. Metallography 3, (1970)

11 Euler s Theorem - Average Neighbour Number Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

12 9. November / 103 Relation of the number of faces, edges & vertices Every face, F n, of a polygon with n sides has n edges. Every edge, E, separates two faces, F. In total, nfn = 2E. (1) Every edge joins two vertices and every vertex of connectivity α has α incident edges: αvα = 2E. (2) If on average α edges per vertex, V = 2 E. (3) α

13 9. November / 103 Euler s Theorem The topological constraint follows from Euler s theorem relating the number of faces F = N, edges E and vertices V covering a two-dimensional manifold of Euler-Poincare characteristics χ, F E + V = χ. χ describes the global character of the manifold. For a sphere, χ = 2. Wikipedia

14 9. November / 103 Constraint on Number of Faces Using we obtain E = 1 2 nfn ; V = 2 α E. F E + V = χ F E + 2 α E = F + 2 α α E = χ F + 2 α nfn = χ 2α

15 9. November / 103 Average Neighbour Number We further write for the fraction of cells with n neighbours, p n = F n F = N n N (4) and can the write nfn = F np n = F < n >. (5) Here, < n > is the average number of neighbours.

16 9. November / 103 Average Neighbour Number F + 2 α nfn 2α = χ F + 2 α 2α < n > = χ F + 2 α 2α F < n > = χ ( F α ) 2α < n > = χ 2α 2 α + < n > = 2αχ (2 αf ) 2α < n > = α 2 + 2αχ (2 αf )

17 9. November / 103 Limit of Large Tissues < n > = 2α α 2 + 2αχ (2 αf ) Unlike F, E and V, χ does not scale with the number of faces. For a large tissue, χ = O(1) is negligible in comparison with F such that lim F 2αχ (2 αf ) = 0. In the limit of large tissues we thus have < n > 2α α 2.

18 Average Number of Neighbours < n > 2α α 2 In a cellular mosaic, most vertices have three edges, i.e. α = 3. Thus, < n > 2α α 2 = 6. Some tissues, have a low number of vertices with 4 edges. In the unrealistic case that α = 4 for the entire tissue, we would have < n > 2α α 2 = 4. We thus expect that the average number of neighbours in tissues is close to 6, and certainly greater than 4. Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

19 9. November / 103 Epithelial cells have on average 6 neighbours To minimize surface tension, cells tend to assume a spherical shape. However, because of the tight cell-cell interactions, epithelial cells take on polygonal shapes. Each cell would still minimize its individual surface tension by assuming a quasi-circular shape and thus a polygonal shape with very high edge number, n. However, because of topological constraints < n >= 6. each epithelial cell has on average six neighbours.

20 Driving Forces behind the Polygon Distribution Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

21 9. November / 103 So what defines the distribution of the polygon types? 1 Impact of Cell Division (Gibson and co-workers) 2 Impact of Tissue Mechanics Heller, D., et al. (2016) Dev Cell 36,

22 Cell Topology, Geometry, and Morphogenesis in Proliferating Epithelia Cellular geometry: specifies cell shape Cellular topology: connectivity among cells in a tissue. Manipulations that change cell shape, but maintain topology: stretching or deforming a sheet of cells in such a way that the cells respective shapes change, but all neighbor relationships are preserved. Manipulations that change cell topology: processes such as perforation or tearing, in which cell contacts are broken, or convergent extension, in which cell contacts are both made and broken. In epithelia, various elementary processes, such as cell division, cell rearrangement, and cell disappearance, modify cell sheet topology in stereotyped ways. Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

23 1. Impact of Cell Division Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

24 9. November / 103 Impact of Cell Division (Gibson et al, 2006) Model Assumptions: 1 cells are polygons with a minimum of four sides 2 cells do not re-sort 3 mitotic siblings retain a common junctional interface 4 cells have asynchronous but roughly uniform cell cycle times 5 cleavage planes always cut a side rather than a vertex of the mother polygon 6 mitotic cleavage orientation randomly distributes existing tricellular junctions to both daughter cells. Gibson et al. Nature (2006).

25 9. November / 103 Topological Model In topological terms, each tricellular junction is a vertex, each cell side is an undirected edge, and each apical cell surface is a polygonal face. v t : number of vertices after t divisions e t : number of edges after t divisions f t : number of faces after t divisions. If we assume that cells divide at a uniform rate, then the number of faces (cells) will double after each round of division: f t = 2f t 1 v t = v t 1 + 2f t 1 each cell division results in +2 vertices e t = e t 1 + 3f t 1 each cell division results in +3 edges

26 9. November / 103 Topological Model Boundary effects become negligible for large t so that mean number of sites per cell can be determined as s t = 2e t = 2(e t 1 + 3f t 1 ) = s t 1 f t 2f t (6) This recurrence system leads to such that for s t = t (s 0 6) (7) t s t 6 (8)

27 9. November / 103 Topological Model t s t 6 (9) This implies that even in epithelia devoid of minimal packing, the system will assume a predominantly hexagonal topology as a consequence of cell division. This behaviour is independent of cleavage plane orientation, and is instead a result of the formation of tricellular junctions. Importantly, however, an average of six does not necessitate a prevalence (or even existence) of hexagons.

28 Markov chain model for the proliferation dynamics for polygonal cells. s: number of sides with s > 3 p s : relative frequency of s-sided cells in the population p (t) = [p 4, p 5, p 6, p 7, p 8, p 9,...]: system state at time t The state dynamics is described by p (t+1) = p (t) PS, (10) where P and S are probabilistic transition matrices. Gibson et al. Nature (2006). Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

29 9. November / 103 Markov chain model p (t+1) = p (t) PS The entries P ij represent the probability that an i-sided cell will become j-sided after mitosis. Matrix S: Topological arguments indicate that a cell will gain an average of one new side per cycle due to neighbour divisions.

30 9. November / 103 Probability that an i-sided cell will become j-sided after mitosis s t 1 sides (or junctions) at generation t 1. K t : random variable for the number of junctions distributed to one daughter cell on division at generation t, leaving s t 1 K t for the other. No triangular cells are allowed / observed empirically Assumptions: junctions are distributed uniformly at random around the mitotic cell the cleavage plane orientation is chosen uniformly at random (to bisect the rounded mitotic cell s area) The (unnormalized) entries of P are the coefficients of Pascal s triangle.

31 9. November / 103 Markov chain model for the proliferation dynamics for polygonal cells. In equlibrium, p (t+1) = p (t) From this, one can obtain the equilibrium distribution to be approximately 28.9% pentagons, 46.4% hexagons, 20.8% heptagons and lesser frequencies of other polygon types. Gibson et al. Nature (2006).

32 9. November / 103 Markov chain model for the proliferation dynamics for polygonal cells. The model also predicts that the population of cells should approach this distribution at an exponential rate. Consequently, global topology converges in less than eight generations, even for initial conditions where every cell is quadrilateral, hexagonal, or nonagonal. Gibson et al. Nature (2006).

33 9. November / 103 Predicted and Observed polygon distributions. Gibson et al. Curr Top Dev (2009)

34 9. November / 103 Cleavage Plane NOT random. Hertwig s rule: division perpendicular to longest axis.

35 9. November / 103 Neighbours determine longest axis. Gibson et al Cell (2011).

36 9. November / 103 Bisection of smallest polygon. Gibson et al Cell (2011).

37 Better match of polygon distribution with division perpendicular to longest axis. Gibson et al Cell (2011). Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

38 9. November / 103 Different polygon distributions. The fact that also other polygon distributions exist in nature then those predicted by the cell division-induced topological changes suggests that some other important aspect is still missing.

39 2. Tissue Mechanics Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

40 9. November / 103 Modelling an epithelial sheet We will model an epithelium as a two-dimensional sheet of contiguous, coupled polygonal cells (i.e. as a collection of nodes connected by line segments). We begin by describing a mechanical model for a single cell. To model an epithelium, it will then only be necessary to mechanically couple a suitable number of cells into a continuous two-dimensional sheet. Further details in Weliky, M. and G. Oster (1990). Development 109:

41 9. November / 103 Modelling an epithelial sheet Ishihara & Sugimura (2012) Theor Biol

42 9. November / 103 Vertex Model Representation of cells: Fletcher et al., 2013 cells represented by polygons neighboring cells share edges, intersection point = vertex

43 9. November / 103 Elastic tension forces The elastic tension forces at cell node, n, are represented by a pair of two-dimensional vectors, T (n,n 1) and T (n,n+1), applied to the node. These vectors are directed along the line segments connecting this node to each of the two adjacent nodes, where T (n,n 1) denotes the vector directed towards the node in the counterclockwise direction and T (n,n+1) towards the clockwise node.

44 9. November / 103 Hooke s Law According to Hooke s law, the force F needed to extend or compress a spring by some distance x is proportional to that distance. That is, F = k x.

45 Here P is the cell perimeter, and k elas is the elastic modulus of the cell cortex (i.e. the circumferential microfilament bundle and the cortical actin gel matrix). Larger values of k elas reflect increasing contraction and/or density of microfilament bundles, while lower values reflect solation, which reduces the elastic modulus of the cortical actin gel matrix. Increasing the cell perimeter, P, stretches cortical fibers thus increasing their elastic restoring forces. Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103 Elastic tension forces We will use the two-dimensional unit vectors t (n,n 1) and t (n,n+1) to describe the tension vectors direction. We can describe the elastic vector forces applied to each cell node as follows T (n,n+j) = t (n,n+j) k elas P

46 9. November / 103 Pressure Intracellular pressure is assumed to be uniform within the cell. The pressure force at each node is represented by the two-dimensional vector P directed outward from the cell boundary and bisecting the angle between the two line segments joined at a node; that is, the pressure acts normal to the polygonal surface at each node.

47 9. November / 103 Pressure The two-dimensional unit surface normal vector n describes the direction of the nodal pressure. The standard assumption is that a change in cell volume (surface area) results in a change in the osmotic pressure difference such that the pressure difference is inversely proportional to the volume (surface area, A n ): P n = const n A n

48 9. November / 103 The net force at a node The net force acting at a cell node, n, is the vector sum of the nodal tensions and pressure: F n = P n + T (n,n 1) + T (n,n+1) The total force acting at an epithelial node is the sum of all net forces from all cells sharing this node plus the external forces: F = N F n + F ext n=1 where N is the number of cells sharing the common junctional node.

49 9. November / 103 Equations of Motion The configuration of the model epithelium is specified by giving the location of each node. Therefore, the equations of motion describing the changing configuration have the form µ d x dt = F (x) where µ is a viscous coefficient, x is the position vector of the nodes, and F (x) is the vector of forces acting on the nodes. Inertial forces are assumed to be negligible.

50 9. November / 103 Equations of Motion µ d x dt = F (x) is a set of coupled differential equations where the number of equations is equal to the number of nodes in our simulation, which can be many hundreds. For the position vector of each node, x n, we have µ d x n dt N = P n + T (n,n 1) + T (n,n+1) + F ext n=1 Since the equations are nonlinear, an analytical solution to this system is not possible, so we must apply numerical methods.

51 9. November / 103 Energy Functional Alternatively, we can use an energy functional and determine the steady state distribution: E = λ i,j l i,j <i,j> }{{} surface tension γ α Π 2 α }{{} cortical tension + α + κ α (A A 0 ) 2 i }{{} area elasticity

52 9. November / 103 Relation between Forces and Energy-based View Evolution in time: vertex i moves according to: µ dx i dt = F i forces derived from energy potentials: F i = E R i E(R i ) = K α α 2 (A α A 0 ) 2 + <i,j> Γ ij 2 (r ij l ij ) 2 term 1: area elasticity, K α = elasticity coefficient term 2: perimeter contractility, Γ ij = contractility coefficient

53 9. November / 103 Junctional Rearrangement Brodland, G. W., et al. (2010) PNAS 107: Fletcher, A. G., et al. (2014). Biophys J 106:

54 9. November / 103 Vertex Model of epithelial sheet Membrane Tension versus Cell Junction Strength Farhadifar, R., et al. (2007). Curr Biol 17(24):

55 9. November / 103 Comparison to Wing Disc Data same polygon distribution same area distribution same response to laser ablation

56 9. November / 103 FEM Tissue Simulations & longest axis division Better match of polygon distribution with division perpendicular to longest axis. Gibson et al Cell (2011)..

57 9. November / 103 Conclusion Both tissue mechanics and the cell division angle determine the polygon distribution in epithelia.

58 9. November / 103 Summary Properties of Epithelia: < n >= 6 (Euler s Theorem - Topological Argument) Polygon Distribution: Cell Division Angle & Tissue Mechanics Lewis Law: A n = A 0 4N (n 2) (??) Aboav-Weaire s Law: m n = n (??)

59 Inference of cellular properties from images Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

60 9. November / 103 Inference of cellular properties from images Ishihara, S. & Sugimura, K. (2012) Bayesian inference of force dynamics during morphogenesis. J Theor Biol 313,

61 9. November / 103 Inference of cellular properties from images Ishihara, S. & Sugimura, K. (2012) Bayesian inference of force dynamics during morphogenesis. J Theor Biol 313,

62 9. November / 103 Batchelor Stress Tensor σ = 1 A i P i A i I + [ij] T ij r ij rij r ij (11) where I is the two-dimensional identity matrix and A = i A i is the total tissue area. Note that the scale of σ is undetermined. But quantities like the maximal stress direction can be obtained.

63 9. November / 103 Forces acting at a given node r i = (x i, y i ): positions of vertices T ij : tension of the vertex that connects the i-th and j-th vertices P i : pressure of the i-th cell r i = ( y i, x i ): orthogonal vector F = r 2 r 1 r 2 r 1 T 1 + r 6 r 1 r 6 r 1 T ( [ r2 r 1 ] ) 2 r 2 r 1 r 2 r 1 (P 1 P 0 ) + 1 ( [ r1 r 6 ] ) 2 r 1 r 6 r 1 r 6 (P 1 P 0 )

64 Forces: component notation r i = (x i, y i ): positions of vertices T ij : tension of the vertex that connects the i-th and j-th vertices P i : pressure of the i-th cell Let us consider the force acting on the 0-th vertex at the origin r 0 = (0, 0). The forces in the x and y directions are given by F x 0 = F y 0 = 3 i=1 3 i=1 x i r i T i y i r i T i + 3 i=1 3 i=1 y i 2 (P i P i+1 ) x i 2 (P i P i+1 ) Ishihara, S. et al. (2013) The European physical journal. E, Soft matter 36, 9859 Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

65 Forces: component notation F x 0 = F y 0 = 3 i=1 3 i=1 x i r i T i y i r i T i + 3 i=1 3 i=1 y i 2 (P i P i+1 ) x i 2 (P i P i+1 ) Orientation of the edge: θ ij = tan 1 ( y ij x ij ) x i y i Ishihara, S. et al. (2013) The European physical journal. E, Soft matter 36, 9859 r i = cos (θ ij); r i = sin (θ ij) Pressures act along the sides of the cells in their normal direction. Projection to the x- and y-axes gives the pre-factors y i and x i, respectively, of the normal force. Half of the force acts on each end-point. Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

66 9. November / 103 Full system of Equations Repeating the same derivation of force-balance equations for every vertex, we obtain a vector F = ( F x, F y ) T (12) that represents the forces acting on vertices in the x and y directions as F = A T T + AP P = A X. (13) Here, T and P are column vectors composed of T ij and P i, respectively. X = ( T, P) T represents the unknown variables to be inferred. A T and A P represent matrices with the coefficients of T ij and P i.

67 9. November / 103 Dimension of Full system of Equations F = A T T + AP P = A X. Suppose we have an image, in which N cells are surrounded by R cells: E: numbers of cell contact surfaces (edges) V : number of vertices Dimensions T : [E 1], P: [N + R 1], X: [E + N + R 1] F : [2V 1] A T : 2V E, A P : [2V (N + R)], A: [2V (E + N + R)]

68 9. November / 103 Quasi-steady State Assumption Under the assumption of quasi-static cell shape changes, the force-balance equation becomes F = A T T + AP P = A X = 0 (14) This gives us a relationship between the observable geometry (angles and lengths) of cells and the unknown tensions T and pressures P to be determined. Note that we can only determine relative forces and pressure differences.

69 9. November / 103 Overdetermined System F = A T T + AP P = A X = 0 (15) Number of unknowns = length of X == number of cells (N + R) plus cell contact surfaces (E). Number of conditions = 2 number of vertices Number of unknowns is smaller than the number of the conditions for a sufficient number of cells. To get plausible and unique estimates, the inverse problem must be formulated to handle this indefiniteness.

70 The Inverse Problem Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

71 9. November / 103 Estimated relative pressures and tensions against their true values Ishihara, S. & Sugimura, K. Bayesian inference of force dynamics during morphogenesis. J Theor Biol 313, , doi: /j.jtbi (2012).

72 9. November / 103 In vivo Validation Ishihara, S. & Sugimura, K. Bayesian inference of force dynamics during morphogenesis. J Theor Biol 313, , doi: /j.jtbi (2012).

73 Inferred Pressure & Tension in the Wing Disc Ishihara, S. & Sugimura, K. Bayesian inference of force dynamics during morphogenesis. J Theor Biol 313, , doi: /j.jtbi (2012). Computational Biology Group (CoBi), D-BSSE, ETHZ 9. November / 103 Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16

74 Inferred Pressure & Tension in the Wing Disc Ishihara, S. & Sugimura, K. Bayesian inference of force dynamics during morphogenesis. J Theor Biol 313, , doi: /j.jtbi (2012). Computational Biology Group (CoBi), D-BSSE, ETHZ 9. November / 103 Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16

75 9. November / 103 CELLFit: Segment also Curvature of Membranes Laplace Equation: p ij = γ ij ρ ij (16) ρ ij : radius of curvature γ ij : tension Brodland, G. W. et al (2014) Plos One

76 9. November / 103 Summary Inference of Epithelial Properties: Images of epithelia can be used to infer hydrostatic pressure and membrane tension Challenge to deal with overdetermined system and low quality data What is the correct model?

77 9. November / 103 Literature Rivier, N. and A. Lissowski (1982). On the Correlation between Sizes and Shapes of Cells in Epithelial Mosaics. J. Phys A: Math. Gen. 15(3): L143-L148. Weliky, M. and G. Oster (1990). The mechanical basis of cell rearrangement. I. Epithelial morphogenesis during Fundulus epiboly. Development 109: Gibson, M. C., et al (2006) The emergence of geometric order in proliferating metazoan epithelia. Nature 442, Farhadifar, R., et al. (2007). The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing. Curr Biol 17(24): Gibson, W. T., et al. (2011). Control of the mitotic cleavage plane by local epithelial topology. Cell 144(3): Ishihara, S. & Sugimura, K. (2012) Bayesian inference of force dynamics during morphogenesis. J Theor Biol 313, Brodland, G. W. et al. (2014) CellFIT: A Cellular Force-Inference Toolkit Using Curvilinear Cell Boundaries. Plos One 9

78 4. Cell-based Simulation Frameworks Computational Biology Group (CoBi), D-BSSE, ETHZ Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16 9. November / 103

79 Different Levels of Details... Tanaka et al., 2015 Computational Biology Group (CoBi), D-BSSE, ETHZ 9. November / 103 Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16

80 9. November / 103 Why cell-based modeling? Representation of: cell division (directed and undirected) cell migration cell adhesion cell differentiation cell polarity cell proliferation/cell death

81 9. November / 103 Cell-based Simulation Frameworks Tanaka et al., 2015

82 9. November / Spheroid Model Representation of cells: particle-like cells spherical, isotropic shape represented by a soft spehre interaction potentials: Johnson-Kendall-Roberts (adhesive spheres), Hertz, harmonic potentials,... I J I#J 6 IJ & 4 % 0A S % 0A S D IJ C L D IJ L Drasdo et al., 2007

83 9. November / Spheroid Model Evolution in time: 1 solving eq of motion deterministically: η x t = f with η being the mobility coefficient. 2 solving stochastic eqs of motion (Langevin): ( ) γ + Γ f u i + ( Γ f u i u j) = is ij j i where γ is an effective friction coefficient. N F ij + η i (t) j=0

84 3.1 Spheroid Model: Applications (a) N=1 (c) N=100 (A ) N=5000 N= (sagital cut) (B) (C) N= (sagital slice) (b) (d ) Drasdo et al., 2007 Drasdo et al., 2005 Computational Biology Group (CoBi), D-BSSE, ETHZ 9. November / 103 Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16

85 9. November / Spheroid Model: Discussion simple representation: cheap large number of cells 3D no cellular details coupling to signaling restricted software: CellSys

86 9. November / 103 Subcellular Model Representation of cells: V inter β j α i V intra i j Newman et al., 2005 one cell = many subcellular elements similar to spheroid model: forces derived from interaction potentials between elements (often Morse potential used)

87 9. November / 103 Subcellular Model Equation of motion for position y αi of a subcellular element α i of cell i: η y α i t where ζ αi ( = ζ αi αi V yαi intra y βi ) ( αi V yαi inter y βi ) β i α i is Gaussian noise. term 1: intra-cellular interactions, summation runs over all remaining elements β i of cell i term 2: inter-cellular interaction, pair-interactions between the elements β j of other cells j and the elements α i of cell i Morse Potential: V (r) = U 0 exp ( r/ζ 1 ) V 0 exp ( r/ζ 2 ) j i β j

88 Subcellular Model: Applications 2D simulation of one migrating cell initially located to the right (a: white, bc: orange) interacting with a second cell via cell-cell adhesion compared to experiments (d). The cell to the left is quiescent (a: red) or actively remodelling its cytoskeleton form regions of low to regions of high density as defined by dai (b: blue) or daint (c: blue). i The last row shows an experimental observation of two living cells circling around each other (d). Milde et al., 2014 Computational Biology Group (CoBi), D-BSSE, ETHZ 9. November / 103 Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16

89 9. November / 103 Subcellular Model: Discussion explicit, detailed resolution of shape 3D implementation straightforward however, computationally expensive Exercise: subcellular model

90 9. November / Cellular Potts Model Glazier et al., 2007 originates from the Ising model generalization: Potts model spin: σ (x) Z +,0 spin value = cell identity

91 9. November / Cellular Potts Model Hamiltonian: ) 2 ( ( ( + J τ (σ (x)), τ σ x ))) (1 ( ( δ σ (x), σ x ))) H = σ λ v ( Vσ V T σ (x,x ) term 1: volume constriction, term 2: cell-cell adhesion Metropolis algorithm: probability of converting the spin σ (x ) : p ( σ (x) x ) = Problems: What is the time step? What is the temperature T? { 1 if H (σ (x) x ) < 0 exp ( H (σ (x) x ) /T ) otherwise

92 3.2 Cellular Potts Model: Applications Glazier et al., 1993 Computational Biology Group (CoBi), D-BSSE, ETHZ 9. November / 103 Hester et al., 2011 Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16

93 9. November / Cellular Potts Model: Discussion widely used, well-studied and mature simple mechanism computationally intensive: lattice, time step level of abstraction relatively high software: CompuCell3D

94 9. November / Immersed Boundary Cell Model Tanaka, 2015 membrane is discretized polygons elastic cell membranes fluid inside and outside of the cells fluid-structure interaction cell-cell junctions cell division, growth, differentiation

95 9. November / Immersed Boundary Cell Model Immersed Boundary method: force on vertex distributed to local fluid neighborhood: f (x, t) = F (q, r, s, t) δ (x X (q, r, s, t)) Tanaka, 2015 fluid equation solved velocity of vertex interpolated from local fluid neighborhood: X t (q, r, s, t) = u (X (q, r, s, t), t) = u (x, t) δ (x X (q, r, s, t)) dx iteration

96 9. November / Immersed Boundary Cell Model: Applications Rejniak, 2007 Dillon et al., 2008

97 9. November / Immersed Boundary Cell Model: Discussion currently only in 2D very high level of detail, down to individual cell-cell junctions physical representation of tissue mechanics computationally expensive software: LBIBCell

98 9. November / LBIBCell LBIBCell: A Cell-Based Simulation Environment for Morphogenetic Problems, Tanaka et al., Bioinformatics, 2015 LBIB: Lattice Boltzmann - Immersed Boundary coupled modeling of signaling and tissue growth

99 9. November / LBIBCell Tanaka et al., 2015

100 b Clonal cell population c Negative Mean curvature Growth direction a Positive Vertex Model: 3D Application Viscous friction weight ( ) t t = 8 [cell cycle] Time (t [cell cycle]) Viscous friction weight ( ) Time (t [cell cycle]) Viscous friction weight ( ) Computational Biology Group (CoBi), D-BSSE, ETHZ 9. November / f Viscous friction weight ( ) e 10-4 Gauss Local Gaussian curvature ([ Mean 101 ]) 10-2 Tubular length in growth dir. ([ ]) 10-3 t = 8 [cell cycle] Local mean curvature ([ d Compressive stress in growth dir. ([ ]) 10-4 ]) Viscous friction weight ( ) t = 8 [cell cycle] Okuda et al., 2014 Prof Dagmar Iber, PhD DPhil Lecture 8 MSc 2015/16

101 9. November / 103 Summary: Discrete tissue mechanics tissue consists of cells cell dynamics considered simulation of small tissues/ cell groups various models to represent the cells: (sub-) cellular models coarse grained discrete models Immersed Boundary models

102 9. November / 103 Outlook cell-based modelling techniques is an ongoing field of research models have to be implemented efficiently models have to be compared to each other and validated against experimental data more and more: (live) (3D) data with (sub-) cellular resolution

103 9. November / 103 Thanks!! Thanks for your attention! Slides for this talk will be available at: