The complexity metrics

Morphological metrics

The metrics discussed below focus broadly on morphological aspects of an organ- ism; that is, in terms of how a cell fate distribution is described.

Number of cells: One of the simplest proposed indicators of the complexity of an organism is its size. Bonner (1988) argues that as as organisms grow larger they must by necessity become more complex as the internal requirements for sup- porting their larger size become more specialised. The advantage of this metric is its simplicity. However, interpreting this metric requires caution: applied strictly it would imply that larger organisms are always more complex than smaller organ- isms, which is clearly not always the case (despite being larger, a blue whale is not necessarily more complex than a dolphin). The number of cells of a cell lineage was defined as its number of terminal nodes.

Number of cell types: Bonner (1988) also proposed that an increase in the complexity of an organism would be reflected by an increase in the number of specialised cell types it contained. As with the number of cells, this metric has considerable intuitive appeal. A potential problem with applying this metric to real organisms is the difficulty in classifying different types of cells and the potential for bias in favour of more well-studied organisms (McShea, 1996). The number of cell types of a cell lineage was defined as the number of different types of

96 The Structure and Composition of Ontogenetic Space

terminal nodes. Given the definition of the DRGN-lineage model, an upper limit was imposed on this metric by the structure of the underlying control network. Therefore, while there was some scope for variation due to a DRGN not employing all possible cell types during development, its range was limited by the number of output nodes in the DRGN.

Number of hierarchical levels: Between two organisms containing an equal number and type of components, there may be a difference in how those compo- nents are arranged. Hierarchical, in this context, refers to the number of levels of nestedness of a morphology (McShea, 2001). For example, in the sequence “or- ganelle, cell, organ, organism”, each component contains and partially constrains the behaviour of the earlier components. Given that a cell lineage captures an ontogenetic, rather than a physical, relationship between cells, it is not possible to define a formal measure of hierarchy in this context. However, if we accept a relationship between the ontogeny of a cell and its morphological context, the algorithmic measures of complexity described below may be taken as a proxy for levels of hierarchical organisation.

Developmental metrics

An alternative approach to defining complexity metrics in terms of describing an object is to describe a process. The following metrics focus broadly on the development of an organism; in terms of how a cell fate distribution is generated.

Algorithmic complexity (deterministic): One approach to measuring the complexity of a system is by considering the length of its shortest algorithmic description—an approach formalised as Kolmogorov complexity (Badii and Politi, 1997). A measure of cell lineage complexity based on Kolmogorov complexity was introduced by Braun et al. (2003) and further refined by Azevedo et al. (2005). This measure is calculated by transforming a cell lineage into a series of unique production rules of the form X → {Y, Z}, indicating that a cell of type X divides to form cells of type Y and Z. X is necessarily an undifferentiated (non-terminal) cell, while Y and Z may be differentiated (terminal) or undifferentiated. The plane of cell divisions is lost (X → {Y, Z} is equivalent to X → {Z, Y}), but otherwise these rules provide a complete description of a lineage. This initial set of rules

5.1 Study 4: Characteristics of cell lineage complexity 97

Figure 5.2: An example application of the deterministic algorithmic complexity metric. First, a cell lineage is transformed into a set of production rules by creating a rule for each division. Redundant rules (highlighted) from this set are then removed. Algorithmic complexity is measured as the proportion of unique cell divisions (Alg. Cx. = 4÷6'0.67).

is then reduced by removing equivalent rules until a minimal set is arrived at. Deterministic algorithmic complexity is then defined as the size of this minimal set as a proportion of the total number of divisions (Figure 5.2).

Algorithmic complexity (non-deterministic): One of the implications of the algorithmic complexity measure used by Azevedo et al. (2005) is that each of the rules corresponds to an intermediate cell state that will always produce an identical sublineage in a deterministic fashion. An alternative view is that an intermediate cell state could define the subset of terminal cell fates possible in that sublineage, but not necessarily the exact structure of that sublineage. To investigate the effect of this definition, we defined a second algorithmic complexity metric in which the production rules were non-deterministic. Rules now took the form of {A, B, C} → {{A, B}, C}, where {A, B, C} and {A, B} are undifferentiated cells that will eventually give rise to differentiated cells of types A, B and C, or A and B, respectively, and C is either a differentiated cell that may or may not continue

98 The Structure and Composition of Ontogenetic Space

Table 5.1: Complexity values for sample lineages

Lineage Number of Number of Algorithmic Algorithmic Weighted (Figure cells cell types complexity complexity complexity

5.3) (det.) (non-det.) A 32 1 0.1613 0.0645 2.064 B 32 4 0.1935 0.1290 4.128 C 18 4 0.7059 0.6471 11.65 D 4 4 1.0 1.0 4.0 E 2 2 1.0 1.0 2.0

to divide in a proliferative fashion (i.e., to give rise to more cells of type C). As with deterministic algorithmic complexity, redundant rules are removed from this set, and non-deterministic algorithmic complexity is defined in terms of the size of this minimal set as a proportion of the total number of divisions.

Weighted complexity: As discussed further below, each complexity metrics displayed certain limitations when applied to cell lineages. A final complexity met- ric was designed to address these limitations, that combined both morphological and developmental aspects of complexity. Weighted complexity was defined as the product of the number of cells in a lineage and the non-deterministic algorithmic complexity of a lineage.