1. Generic Modelling Framework

2. Modelling chemical morphogenesis

3. Modelling mechanical morphogenesis
     Lattice-based Models    Centre-based Models    Deformable Cell Models  

4. References

Modelling biological development refers to using mathematical/computational models to simulate episodes of morphogenesis, organogenesis or even pathological processes as the growth of cancerous tumors.

Computational modelling of biological development has taken ground due to the widespread use of mathematical modelling coupled in recent years with ever increasing computing power. Computer simulations present over wet lab experiments the advantages of simplicity, reduced risk and more control over experimental conditions and parameters.^[1].

Generic Modelling Framework

Vertebrate development is home to spectacular and species-consistent morphological events that gradually transform the living creature from a single egg into a fully functional adult ^[2]. Morphological changes in embryonic tissues are driven by coordinated dynamics of individual cells. During development, vertebrate cells exhibit a wide range of non-trivial behaviours including cell growth and division, cell death, cell migration, polarisation, ECM secretion, cell-cell adhesion, cell-cell signalling and epithelial to mesenchymal transition ^[3]. These emerge as the result of intense activity within and between cells dominated by tissue mechanics, gene expression and molecular signalling.

By their movements or change in form, cells generate the physical forces that shape the embryo. Also, differences in genetic activity lead to the synthesis of various proteins which explain differentiation into distinct cell types ^[4]. Development is therefore the result of the interplay between two mutually influencing processes: "Mechanical Morphogenesis", "Chemical Morphogenesis" ^[5]. Metaphorically, development can be seen as a mechanism with double dynamics featuring "shaping", by which the embryo "sculpts" itself (mechanical morphogenesis), and "atlassing", by which the embryo "paints" itself (chemical morphogenesis).

Feeding on this description, modelling development becomes a question of how much of each of these processes can be integrated into a unified mathematical/computational formulation to account for what is being observed experimentally. The several modeling techniques currently employed can be viewed as different compromises to this problem. Multiple approaches based solely on either of these mechanisms have been proposed. In recent years, models that couple both mechanical and chemical variables have been on the rise ^[6], ^[3], ^[7], ^[8].

Modelling chemical morphogenesis

It is generally accepted that chemical structures called morphogens which react together and diffuse through the tissue are responsible of pattern formation. Alan Turing pioneered this idea by proposing a mathematical formulation, the reaction diffusion system, to artificially reproduce this phenomenon ^[9]. Another approach used to explain differences in cell behaviours is Positional Information. Lewis Wolpert coined the terms while addressing the issue of how complex patterns could emerge in biological tissue. This approach claims that further asymmetries appear in tissues as a result of existing discriminations (hidden colors) which in turn were built upon earlier heterogeneities. ^[10], ^[11]. It has been shown that combining these two approaches, by making one act upstream or downstream of the other, could reproduce interesting phenomena observed in living beings ^[12].

The above methods assume either static concentrations of morphogens (in the case of positional information) or a limited number of acting substances (an activator and an inhibator in the case of reaction-diffusion systems). However, gene expression within the cell is the output of a highly dynamic and complex graph of interaction between genes, proteins, and other substances within the cell. Gene Regulatory Networks (GRN) can graps these behaviours in a remarkable way. Common computational approaches for Gene Regulatory Networks models include Bayesian networks, dynamic Bayesian networks, Boolean networks, probabilistic Boolean networks, ordinary differential equations and neural networks ^[13]

Modelling mechanical morphogenesis

Lattice-based Models

It might appear that investigating the physics of cells and tissues biomechanics is impossible without a deep exploration of sub-cellular mechanisms. Surprisingly, models assuming simple rules at cell level have successfully simulated complex tissue-level behaviours among which cell sorting, proliferation, cell death, differentiation and polarisation ^[14]. In these models, cells reside on a fixed lattice and the dynamics derives from swapping lattice site states under certain conditions. This allows the simulation of large populations of cells at a computationally friendly cost. Macroscopic phenomena like Cell Sorting have been reproduced with these models using the Differential Adhesion Hypothesis ^[15]. However, interesting single cell behaviours can generally not be observed in this context. Additionally the simple rules that cells follow here do not represent the physics involved in a meaningful way.

Centre-based Models

Another class of models, Centred Based Model (CBM), represent cells as single particles immersed in a 3D highly viscous environment. These models assume, in formal analogy to physical particles, that cell trajectories in space can be described by an equation of motion ^[16]. The gained physical friendliness means that it is possible to use established theories of contact mechanics to model precise cell-cell adhesion. Interesting experimentally observed phenomena have been simulated using CBM going from proliferation within monolayers to complex cell rearrangements induced by intercalation observed during zebrafish gastrulation ^[17]. However, appreciation of mesoscale properties such as cell shapes remains an issue: cells generally have a constant geometric shape.

Deformable Cell Models

At the higher end of biological realism integrated within the cell stand Deformable Cell Models (DCM). In DCM the cell body is discretised by a number of nodes which are connected by viscoelastic elements interacting via pairwise potential functions, creating a flexible scaffolding structure with arbitrary degrees of freedom per cell ^[18]. Nodes and their interactions mimic the activity of the cytoskeleton, the nucleus and other cytoplasmic organelles. In some cases, a distinction is made between internal nodes (representing subcellular elements within the cell) and external nodes (representing the cell membrane). In addition to allowing arbitrary cell shapes, complex cellular mechanisms like mechanotransduction can be simulated with this approach.

References

• K-1 World MAX 2005 Champions Challenge
• K-1 World MAX 2005 Championship Final
• K-1 World MAX 2005 Japan Tournament
• K-1 World MAX 2006 Champions Challenge
• K-1 World MAX 2006 Champions' Challenge
• K-1 World MAX 2006 Final
• K-1 World MAX 2006 Final Elimination
• K-1 World MAX 2006 World Championship Final
• K-1 World MAX 2006 World Tournament Final
• K-1 World MAX 2006 World Tournament Open
• K-1 World MAX 2008 Final
• K-1 World MAX 2008 Final 8
• K-1 World MAX 2008 World Championship Tournament Final
• K-1 World MAX 2008 World Championship Tournament Final 8
• K-1 World MAX 2009 Final
• K-1 World MAX 2009 Final 16
• K-1 World MAX 2009 Final 8
• K-1 World MAX 2009 World Championship Tournament Final
• K-1 World MAX 2009 World Championship Tournament Final 16
• K-1 World MAX 2009 World Championship Tournament Final 8
• K-1 World MAX 2010 -70kg World Championship Tournament Final
• K-1 World MAX 2010 FInal
• K-1 World MAX 2010 Final 16
• K-1 World MAX 2010 World Championship Tournament Final
• K.2