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# Source: arxiv 1806.04138 # Title: Simulation of morphogen and tissue dynamics # Sections: all # Downloaded: 2026-03-03T05:17:23.817566+00:00 |
Simulation of morphogen and tissue dynamics Abstract Morphogenesis, the process by which an adult organism emerges from a single cell, has fascinated humans for a long time. Modelling this process can provide novel insights into development and the principles that orchestrate the developmental processes. This chapter f... |
Introduction During morphogenesis, the coordination of the processes that control size, shape, and pattern is essential to achieve stereotypic outcomes and comprehensive functionality of the developing organism. There are two main components contributing to the precisely orchestrated process of morphogenesis: morphogen... |
Simulating morphogenesis is challenging because of the multi-scale nature of the process. The smallest regulatory agents, proteins, measure only a few nanometers in diameter, while animal cell diameters are typically at least a 1000-fold larger cp. alberts2014molecular VanDenHurk2005 , and developing organs start as a ... |
Given the multiscale nature of morphogenesis, combining signalling dynamics with tissue mechanics in the same computational framework is a challenging task. Where justified, models of morphogenesis approximate tissue as a continuous domain. In this case, patterning dynamics can be described by reaction-advection-diffus... |
In this review, we provide an overview of approaches to describe, couple and solve dynamical models that represent tissue mechanics and signalling networks. Section deals with the mathematical representation of morphogen dynamics, tissue growth and tissue mechanics. Section covers cell-based simulation frameworks. Fina... |
Mathematical representation of morphogen and tissue dynamics 2.1 Morphogen dynamics A fundamental question in biology is that of self-organisation, or how the symmetry in a seemingly homogeneous system can be broken to give rise to stereotypical patterning and form. In 1952, Alan Turing first introduced the concept of ... |
in his seminal paper “The Chemical Basis of Morphogenesis”, cf. Lucas:Turing1990-li , in the context of self-organisation and patterning. He hypothesized that a system of chemical substances “reacting together and diffusing through a tissue” was sufficient to explain the main phenomena of morphogenesis. |
Morphogens can be transported from their source to target tissue in different ways. Transport mechanisms can be roughly divided into two categories: extracellular diffusion-based mechanisms and cell-based mechanisms muller2013morphogen In the first case, morphogens diffuse throughout the extracellular domain. Their mov... |
and cytonemes Michael:RK99 According to the transcytosis model, morphogens are taken up into the cell by endocytosis and are then released by exocytosis, facilitating their entry into a neighbouring cell rodman1990endocytosis In this way, morphogens can move through the tissue. The morphogen Decapentaplegic (Dpp) was p... |
According to the cytoneme model, cytonemes, i.e. filapodia-like cellular projections, emanate from target cells to contact morphogen producing cells and |
vice versa , cf. Michael:RK99 Morphogens are then transported along the cytonemes to the target cell. Several experimental studies support a role of cytonemes in morphogen transport across species, see Michael:KS14 Michael:BSAR+13 sanders2013specialized However, so far, the transport kinetics and the mechanistic detail... |
A key concept for morphogen-based patterning is Lewis Wolpert’s French Flag model , cf. wolpert1969positional According to the French Flag model, morphogens diffuse from a source and form a gradient across a tissue such that cells close to the source experience the highest morphogen concentration, while cells further a... |
In the original publication, the source was included as a fixed boundary condition. No reactions were included in the domain, but morphogen removal was included implicitly by including an absorbing (zero concentration) boundary condition on the other side. The resulting steady state gradient is linear and scales with t... |
wartlick2011dynamics A threshold-based read-out as postulated by the French Flag model is nonetheless possible because the amplitude increase and the imperfect scaling of the pre-steady state gradient compensate such that the Dpp concentration remains constant in the region of the domain where the Dpp-dependent pattern... |
2.2 Mathematical Description of Diffusing Morphogens Morphogen behaviour can be modelled mathematically using the reaction diffusion equation, which we derive here. We assume, for the moment, that there is no tissue growth and the movement of the morphogen is a consequence of random motion. We denote the concentration ... |
[MATH] is [MATH] and the rate of change of the total concentration is [EQUATION] The rate of change of the total concentration in [MATH] is a result of interactions between the morphogens that impact their concentration and random movement of the morphogens. The driving force of diffusion is a decrease in Gibbs free en... |
[EQUATION] where [MATH] is the diffusion coefficient or diffusivity of the morphogen. This is a measure of how quickly the morphogen moves from a region of high concentration to a region of low concentration. The total flux out of [MATH] is then |
[EQUATION] where [MATH] is the boundary of [MATH] and [MATH] is the normal vector to the boundary. Reactions between the morphogens also affect the rate of change of [MATH] We denote the reaction rate [MATH] . The rate of change of the concentration in the domain [MATH] due to morphogen interactions is |
[EQUATION] As the rate of change of the total concentration in [MATH] is the sum of the rate of change caused by morphogen interactions and the rate of change caused by random movement, we have |
[EQUATION] Now, the Divergence Theorem yields [EQUATION] Substituting ( ) and ( ) into ( ) and exchanging the order of integration and differentiation using Leibniz’s theorem gives |
[EQUATION] Taking into account that this equilibrium holds for any control volume [MATH] , we obtain the classical reaction-diffusion equation |
[EQUATION] This partial differential equation (PDE) can be solved on a continuous domain to study the behaviour of morphogens in a fixed domain over time. If there is more than one morphogen then their respective concentrations can be labelled [MATH] for [MATH] where [MATH] is the number of morphogens. The reaction ter... |
It is important to keep in mind that reaction-diffusion equations only describe the average behaviour of a diffusing substance. This approach is therefore not suitable if the number of molecules is small. In that case stochastic effects dominate, and stochastic, rather than deterministic, techniques should be applied, ... |
2.3 Morphogen Dynamics on Growing Domains In the previous paragraph we introduced morphogen dynamics on a fixed domain. However, tissue growth plays a key role in morphogenesis and can play a crucial part in the patterning process of the organism |
kondo1995reaction henderson2002mechanical . Growth can affect the distribution of the morphogens, transporting them via advection and impacting the concentration via dilution, see Figure In turn, morphogens can influence tissue shape change and growth, for example, by initiating cell death and cell proliferation respec... |
It is then necessary to modify equation ( ) to account for growth. Applying Reynolds transport theorem to the left-hand side of equation ( ) we get |
[EQUATION] For a more detailed derivation, we refer to Michael:ITFG+14 This results in the reaction-diffusion equation on a growing domain: |
[EQUATION] By the Leibniz rule, there holds [MATH] These terms can be interpreted as the dilution , i.e. the reduction in concentration of a solute in a solution, usually by adding more solvent, and advection i.e. movement of a substance in a fluid caused by the movement of the fluid, respectively. |
2.4 Modelling tissue growth The details of the process of tissue growth still remain to be elucidated. It is therefore an open question of how best to incorporate it into a model. One approach considers the velocity field to be dependent on morphogen concentration, i.e. [MATH] , where [MATH] is again the morphogen conc... |
iber_control_2013 menshykau_kidney_2013 Another is “prescribed growth”, in which the velocity field [MATH] of the tissue is specified and the initial domain is moved according to this velocity field. A detailed measurement of the velocity field can be obtained from experimental data. To this end, the tissue of interest... |
There are also other techniques to model tissue growth. If the local growth rate of the tissue is known, the Navier-Stokes equation can be used. Tissue is assumed to be an incompressible fluid and tissue growth can then be described with the Navier-Stokes equation for incompressible flow of Newtonian fluids, which read... |
[EQUATION] where [MATH] is fluid density, [MATH] dynamic viscosity, [MATH] internal pressure, [MATH] external force density and [MATH] the fluid velocity field. The term [MATH] is the local mass production rate. The parameter [MATH] |
is the molecular mass of the cells. The impact of cell signalling on growth can be modelled by having the source term [MATH] dependent on the morphogen concentration, i.e. [MATH] . Note that the source term results in isotropic growth. External forces as implemented in the [MATH] term can induce anisotropic growth. Bas... |
The Navier-Stokes description, with a source term dependent on signalling, has been used in simulations of early vertebrate limb development, see dillon2003short An extended anistropic formulation has been applied to Drosophila |
imaginal disc development in bittig2008dynamics It has also been used to model bone development, cf. tanaka2013inter and coupled with a travelling wave to simulate the developing |
Drosophila eye disc, see Michael:FSAL+16 In the case of the developing limb, the proliferation rates were later determined, see boehm2010role They were then used as source terms in the isotropic Navier-Stokes tissue model. There was, however, a significant discrepancy between the predicted and actual growth. The shapes... |
2.5 Tissue mechanics Tissue expands and deforms during growth. Given its elastic properties, stresses must emerge in an expanding and deforming tissue. Cell rearrangements are able to dissipate these stresses and numerous experiments confirm the viscoelastic properties of tissues |
Michael:For98 Michael:FFSS98 Michael:FFPS94 Michael:MMKA+01 Over long time scales, as characteristic for many developmental processes, tissue is therefore typically represented as a liquid, viscous material and is then described by the Stokes equation Michael:DO99 Michael:FSAL+16 Michael:ITFG+14 Over short time scales,... |
The mathematical representation of kinematics, i.e. the description of motion of points and bodies, is usually performed with respect to two different frameworks. They are called Lagrangian |
(or material ) and Eulerian (or spatial ) coordinates. The Lagrangian framework adopts a particle point of view, for example the perspective of a single cell, and tracks its movement over time. In contrast to this, the Eulerian framework adopts the perspective of an entire body, for example a tissue, and describes its ... |
[EQUATION] deformation field . The deformed body at a given time [MATH] is then denoted by [MATH] , see Figure for a visualisation. The position of the particle [MATH] at time [MATH] is therefore given by |
[MATH] , which is the description in Eulerian coordinates. On the other hand, we can also consider [MATH] which is the description in Lagrangian coordinates. More important than the deformation is the displacement |
[EQUATION] respectively. Based on the displacement, one can consider balance principles of the form [EQUATION] which is Newton’s second law and is also known as Cauchy’s first equation of motion, see e.g. Michael:Hol00 Herein, the tensor field [MATH] characterises the stresses inside the body, the vector field [MATH] s... |
[EQUATION] Note that the steady state in morphogen concentrations is reached very fast compared to the time scale on which growth happens. |
Several models exist to describe material behaviour. A material is called elastic if there exists a response function with [EQUATION] |
where [EQUATION] is the deformation gradient Linearly elastic materials are described by Hooke’s law In this case, the function [MATH] is linear. For the description of tissues, non-linear material responses are better suited. To that end, hyperelastic material models are used. They are characterised by the response fu... |
[EQUATION] where [MATH] and [MATH] is a scalar strain energy density function For the modelling of soft tissues, Fung-elastic materials might be employed, see e.g. Michael:Fun93 Michael:Tab04 . Here the strain energy density function is for example given by |
[EQUATION] In Michael:Tab04 , this model is suggested to simulate the blastula stage of the sea urchin. Figure .A shows the corresponding computational model. By considering only a cross section, the model can be reduced to two spatial dimensions. In Figure .B, pressure versus radius curves for different values of the ... |
[MATH] and we chose [MATH] , see also Michael:Tab04 Michael:IP17 Cell-based simulation frameworks Cell-based simulations complement continuum models, and are important when cellular processes need to be included explicitly, i.e. cell adhesion, cell migration, cell polarity, cell division, and cell differentiation. Cell... |
A wide range of cell-based models has meanwhile been developed. The approaches differ greatly in their resolution of the underlying physical processes and of the cell geometries, and have been realised both as lattice-based and lattice-free models. In lattice-based models the spatial domain is represented by a one, two... |
In the following, we will provide a brief overview of the most widely used cell-based models for morphogenetic simulations, i.e. the Cellular Potts model , the spheroid model the Subcellular Element model , the vertex model and the Immersed Boundary Cell model , see Figure |
for their arrangement with respect to physical detail and spatial resolution. In the following, we will focus on the main ideas behind each model and name common software frameworks that implement the aforementioned methods. A more detailed description of the models and their applications in biology can be found in Mer... |
The Cellular Potts Model (CPM) is a typical on-lattice approach as it originates from the Ising model, see Ising1925 It represents the tissue as a lattice where each lattice site carries a spin value representing the cell identity. The update algorithm of the CPM is the Metropolis algorithm, cf. Metropolis1953 which ai... |
[EQUATION] The first term describes the volume constriction with [MATH] being the coefficient controlling the energy penalisation, |
[MATH] being the actual volume and [MATH] the target volume of cell [MATH] The second term represents the cell-cell adhesion, where [MATH] |
denotes the surface energy term between two cell types and [MATH] the Dirac [MATH] -function. The CPM has been used to simulate various processes in morphogenesis, including kidney branching morphogenesis Hirashima:2009er somitogenesis Hester:2011cc , and chicken limb development Manuscript2009 As for many lattice-base... |
The spheroid model is an example for off-lattice agent-based models and represents cells as particle-like objects being a typical off-lattice approach. The cells are assumed to have a spherical shape being represented by a soft sphere interaction potential like the Johnson-Kendall-Roberts potential or the Hertz potenti... |
[EQUATION] where [MATH] is the mobility coefficient and [MATH] represent the forces acting on each particle, or by solving the stochastic Langevin equation. The simple representation of cells enables the simulation of a large number of cells with the spheroid model and further, to simulate tissues in 3D. However, as al... |
The Subcellular Element Model (SEM) represents a cell by many subcellular elements assuming that the inner of a cell, i.e. the cytoskeleton, can be subdivided. The elements are represented by point particles which interact via forces that are derived from interaction potentials such as the Morse potential. A typical eq... |
[MATH] of cell [MATH] reads Tanaka2015 [EQUATION] with [MATH] being the viscous damping coefficient and [MATH] Gaussian noise. The first term describes intra-cellular interactions between the subcellular element [MATH] and all other subcellular elements [MATH] of cell [MATH] The second term represents inter-cellular in... |
for example the Morse potential can be used: [EQUATION] where [MATH] is the distance between two subcellular elements, [MATH] are the energy scale parameters and |
[MATH] are the length scale parameters defining the shape of the potential. Therefore, the SEM is very similar to agent-based models with the difference that each point particle represents parts of and not an entire cell. The SEM offers an explicit, detailed resolution of the cell shapes, further, a 3D implementation i... |
In the vertex model , cells are represented by polygons, where neighboring cells share edges and an intersection point of edges is a vertex. It was first used in 1980 to study epithelial sheet deformations Honda1980 The movement of the vertices is determined by forces acting on them, which can either be defined explici... |
[EQUATION] with [MATH] being the junctions direction of vertex [MATH] The first term describes elastic deformations of a cell with [MATH] being the area elasticity coefficient, |
[MATH] the current area of a cell and [MATH] the resting area. The second term represents cell movements due to cell-cell adhesion via the line tension between neighboring vertices [MATH] and [MATH] with [MATH] being the line tension coefficient and [MATH] the edge length. The third term describes volumetric changes of... |
In the Immersed Boundary Cell Model (IBCell model) the cell boundaries are discretized resulting in a representation of cells as finely resolved polygons. These polygons are immersed in a fluid and, in contrast to the vertex model, each cell has its own edge, cf. Rejniak2007 Therefore, there are two different fluids: f... |
Overview of numerical approaches As we have seen so far, the mathematical description of biological processes leads to complex systems of reaction diffusion equations, which might even be defined with respect to growing domains. In this section, we give an overview of methods to solve these equations numerically. For t... |
4.1 Finite element method The finite element method is a versatile tool to treat partial differential equations in one to three spatial dimensions numerically. The method is heavily used in practice to solve engineering, physical and biological problems. Finite elements were invented in the 1940s, see the pioneering wo... |
dune-fem and FEniCS The pivotal idea, the Ritz-Galerkin method, dates back to the beginning of the 20th century, see Michael:GW12 for a historical overview. The underlying principle is the fundamental lemma of calculus of variations: Let [MATH] be a continuous function. If |
[EQUATION] i.e. for all compactly supported and smooth functions [MATH] on (0,1), then there holds [MATH] , cf. Michael:GF63 We can apply this principle to solve the second order boundary value problem |
[EQUATION] for a continuous function [MATH] , numerically. The fundamental lemma of calculus of variations yields [EQUATION] Rearranging the second equation and integrating by parts then leads to |
[EQUATION] Dependent on the right hand side, the above equation does not necessarily have a solution [MATH] However, solvability is guaranteed in the more general function space [MATH] which consists of all weakly differentiable functions with square integrable derivatives. Then, introducing the bilinear form |
[EQUATION] and the linear form [EQUATION] the variational formulation of ( ) reads [EQUATION] The idea of the Ritz-Galerkin method is now, to look for the solution to the boundary value problem only in a finite dimensional subspace |
[MATH] [EQUATION] Let [MATH] be a basis of [MATH] Then, there holds [MATH] Moreover, due to linearity, it is sufficient to consider only the basis functions [MATH] as test functions |
[MATH] Thus, we obtain [EQUATION] and consequently, by setting [MATH] [MATH] and [MATH] , we end up with the linear system of equations |
[EQUATION] The latter can now be solved by standard techniques from linear algebra. A suitable basis in one spatial dimension is, for instance, given by the linear hat functions, see Figure .A for a visualisation. Figure .B shows how the function [MATH] |
is represented in this basis on the grid [MATH] for [MATH] . For the particular choice of the hat functions, the coefficients are given by the evaluations of [MATH] at the nodes [MATH] , i.e. |
[EQUATION] Note that using the hat functions as a basis results in a tridiagonal matrix [MATH] The presented approach can be transferred one-to-one to two and three spatial dimensions and also to more complex (partial) differential equations. In practice, the ansatz space |
[MATH] is obtained by introducing a triangular mesh for the domain in two spatial dimensions or a tetrahedral mesh in three spatial dimensions, and then considering piecewise polynomial functions with respect to this mesh. To reduce the computational effort, the basis functions are usually locally supported, since this... |
4.2 Arbitrary Lagrangian-Eulerian (ALE) The underlying idea of the Arbitrary Lagrangian-Eulerian (ALE) description of motion is to decouple the movement of a given body |
[MATH] from the motion of the underlying mesh that is used for the numerical discretisation. We refer to Michael:DHPR99 and the references therein for a comprehensive introduction into ALE. As motivated by the paragraph on continuum mechanics, we start from a deformation field |
[EQUATION] which describes how the body [MATH] evolves and moves over time. Remember that the position of a particle [MATH] at time [MATH] |
in Eulerian coordinates is given by [MATH] , whereas its Lagrangian coordinates read [MATH] Analogously, the corresponding velocity fields for [MATH] are then given by |
[EQUATION] in Lagrangian coordinates and by [EQUATION] in spatial coordinates, respectively. Usually, the representation of quantities of interest changes with their description in either spatial or material coordinates. Consider the scalar field [MATH] . We set |
[EQUATION] i.e. [MATH] is the time derivative of [MATH] where we keep the material point [MATH] fixed. Therefore, [MATH] is referred to as material derivative of [MATH] The chain rule of differentiation now yields |
[EQUATION] Thus, the material derivative [MATH] is comprised of the spatial derivative [MATH] and the advection term [MATH] see e.g. Michael:Hol00 for further details. |
As a consequence, given the fixed computational mesh in the Eulerian framework, the domain [MATH] moves over time. The Eulerian description is well suited to capture large distortions. However, the resolution of interfaces and details becomes rather costly. On the other hand, in the Lagrangian framework, it is easy to ... |
[MATH] . In order to bypass the drawbacks of both frameworks, the ALE method has been introduced, see Michael:DHPR99 Michael:HAC74 . Here, the movement of the mesh is decoupled from the movement of the particles [MATH] . The mesh might, for example, be kept fixed as in the Eulerian framework or be moved as in the Lagra... |
Within the ALE framework, the reaction diffusion equation on growing domains can be written as [EQUATION] Herein, [MATH] the relative velocity between the material velocity |
[MATH] and the mesh velocity [MATH] If the velocity of the mesh and the material coincide, i.e. [MATH] the Lagrangian formulation is recovered. On the other hand, setting [MATH] , we retrieve the Eulerian formulation, i.e. equation ( ), cf. Michael:ITFG+14 Michael:MMNI16 In practice, the mesh velocity is chosen such th... |
4.3 Diffuse-Domain method The ALE method facilitates the modelling of moving and growing domains. Due to the underlying discretisation of the simulation domain, the possible deformation is still limited and topological changes cannot be handled. The Diffuse-Domain method, introduced in Lucas:Kockelkoren2003-rf decouple... |
The general idea is appealingly simple. We extend the integration domain to a larger computational bounding box and introduce an auxiliary field variable |
[MATH] to represent the simulation domain. A level-set of [MATH] describes the implicit surface of the domain, see Figure for a visualisation. To restrict the partial differential equations to the bulk and/or surface, we multiply those equations in the weak form by the characteristic functions of the corresponding doma... |
[EQUATION] First, we extend the Poisson equation into the bounding box [MATH] Thus, equation ( 10 ) becomes [EQUATION] Now, the Neumann boundary conditions can be enforced by replacing the B.C. -term by |
[MATH] or [MATH] where [MATH] and [MATH] approximate the Dirac [MATH] -function. Dirichlet and Robin boundary conditions can be dealt with similarly, see Lucas:Li2009-sa It is possible to show that in the limit [MATH] the explicit formulation ( 10 ) is recovered, cf. Lucas:Lervag2014-pg Lucas:Li2009-sa Depending on the... |
Several methods exist to track the diffuse interface of the computation domain. In the level set method, the interface [MATH] is represented by the zero isosurface of the signed-distance function to the surface [MATH] In contrast to this artificial level set formulation, phase fields are constructed by a physical descr... |
[EQUATION] where [MATH] is the domain, [MATH] controls the thickness of the interface, and [MATH] is the phase field, cf. Lucas:Aland2012-xe Moreover, the function [MATH] is a double-well potential having its two minima in the values representing the bulk surfaces, e.g. |
[EQUATION] having its two minima at [MATH] and [MATH] The second term of the Ginzburg-Landau energy penalises gradients in the concentration field and thus can be interpreted as the free energy of the phase transition, see Lucas:Eck2011-cu With [MATH] defined as above, [MATH] is also referred to as surface energy or Ca... |
[EQUATION] The solution of this equation has two homogenous bulks describing the two phases and a [MATH] -profile in between, see e.g. Lucas:Aland2012-xe The Allen-Cahn equation then reads as |
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