Development

As with gene regulation, there are many different ways of modelling the process of development (Stanley and Miikkulainen, 2003, provides an extensive review). Unlike gene regulation however, there are relatively few standardised approaches. This section therefore considers how various models have addressed two specific issues: the control of development and representation of developing entities.

Developmental control

Broadly speaking, two approaches to modelling the control of a developmental process have been considered, termed grammatical and cell chemistry approaches by Stanley and Miikkulainen (2003). Grammatical approaches describe develop- ment using a set of production rules, which are applied iteratively to transform an initial state into a final phenotype. Each production rule consists of a non- terminal symbol on the left, which is replaced by some combination of terminal and non-terminal symbols on the right. The use of grammars to model biological systems was first introduced by Lindenmayer (1968), who developed L-systems as a means of describing the complex fractal patterns observed in nature. L- systems remain in wide use today, particularly for describing the architecture of plants (Prusinkiewicz, 2004). Various extensions have been proposed to extend the descriptive capabilities of L-systems, including parameterised rules, environmental interactions and stochasticity.

Grammatical approaches have also been used for the evolutionary design of neural networks (Kitano, 1990, Gruau, 1995), and robot morphologies and con- trollers (Hornby and Pollack, 2002). The motivation for using grammatical (or

44 A Computational Model of Developmental Cell Lineages

‘generative’) encodings in these contexts was to increase evolvability through the use of a representation that was scalable and inherently modular. The issue of representation is reviewed further below (§3.2.3).

The cell chemistry approach to developmental control is a bottom-up approach to simulating a growth process. Rather than explicitly specifying phenotypic change using rewriting rules, developmental events are derived from the dynamics of an underlying cell chemistry system. One of the earliest such models was the reaction-diffusion system described by Turing (1952), which was capable of pro- ducing a variety of natural-looking spatial patterns. The interacting components of Turing’s system were abstract chemicals known as morphogens; more recent cell chemistry approaches have used metabolic or gene regulatory networks as devel- opmental controllers. Many of the network modelling formalisms reviewed above (§3.2.1) were designed to address issues in biological development.

In a cell chemistry model of development, the dynamics of the underlying net- work are used to control cellular events such as division and differentiation. In many cases, specific components of the network (i.e., genes or metabolites) are assigned a particular function, such as cell division or cell death. When this com- ponent becomes active, its assigned event takes place (Fleischer and Barr, 1994, Fleischer, 1996, Eggenberger, 1997). In other models, cell division and cell death depend on a cell’s volume: a cell whose volume exceeds some upper bound will divide and form two cells, while falling below a lower bound initiates apopto- sis (Hogeweg, 2000a,b). Cell differentiation is typically defined in terms of pat- terns of gene activity, with each stable pattern indicating a distinct type (Kaneko and Yomo, 1994, Furusawa and Kaneko, 1998, Hogeweg, 2000b, Sol´e et al., 2003, Ker¨anen, 2004). Cell behaviour is usually modelled as a product of both internal dynamics and external signals. The external signals may originate from other cells, either via direct contact or diffusion; or a pre-specified field, such as a morphogen gradient.

Phenotypic representation

The second issue arising in the design of a developmental model is how a devel- oping entity is represented. Existing models may be classified according to their treatment of the spatial field in which development occurs:

3.2 Existing computational models 45

which chemicals diffuse and react (Turing, 1952, Meinhardt, 1982). In such models, there are no discrete cells, and often no notion of organism growth (i.e., the size of the field is fixed).

• Most developmental models take individual cells as the fundamental building blocks. In some models, all cells are of an identical shape and size, and the arrangement of cells is fixed to a regular square (Eggenberger, 1997, Ker¨anen, 2004) or hexagonal (e.g., Marnellos and Mjolsness, 1998) grid. In other mod- els, cells may be located freely within space, and even adopt irregular shapes and sizes (Hogeweg, 2000b, Kumar and Bentley, 2003). The dimensionality of the space in which development occurs ranges from one (Salazar-Ciudad et al., 2000) to three (Eggenberger, 1997, 2003, Kumar and Bentley, 2003). In general, most models use two dimensions, which allows for a reasonably diverse range of morphological forms, without the computational expense involved in scaling up to three dimensions (Fleischer, 1996). Discrete cell models may exist in a continuous substrate, through which signals can diffuse between cells or across which morphogen gradients may be defined (Rudge and Geard, 2005).

• A further class of models utilises building blocks at a level above that of an individual cell. Many of the generative grammar models of developmental control described above fall into this category: each developmental unit rep- resents a morphological module. A single module may be a branch segment or leaf in a plant model, or a body or limb segment in an animal model. Cell chemistry models may also adopt this approach. Bongard and Pfeifer (2003) uses macrocellular units as components of a robot morphology: each unit is a complex morphological component involving sensors, actuators and neural control elements.

• One final possibility is that models may not include an explicit spatial com- ponent. The grammatical approaches to modelling neural networks (Kitano, 1990, Gruau, 1995) are one example: the important feature of the network phenotypes is the relationship between individual neurons, rather than their location in space. Another form of phenotypic representation that is an organisational rather than a spatial description is a cell lineage (described in a biological context in §2.1.1). Yoshida et al. (2005) use a cell lineage

46 A Computational Model of Developmental Cell Lineages

representation of a developing phenotype in order to detect the occurrence of recursive patterns of cell differentiation. A cell lineage representation of development may be depicted and represented as a one dimensional array of cells whose length increases over time. Alternatively, individual division events may be labelled with their orientation (e.g., left-right, dorsal-ventral or anterior-posterior), making it possible to interpret a cell lineage in multi- ple dimensions. The cell lineage model used in this thesis is described further in§3.3.2.