Análisis filogenéticos de las HEs presentes en intrones L2263 y L2512

The previous section described five formalised complexity metrics: two morpho- logical, two developmental and one that combines both morphological and devel- opmental aspects. To provide a comparison of how each of these metrics worked in practice, they were applied to a set of five sample lineages (Figure 5.3). The sample lineages were chosen to exhibit a range of lineage types: a large, homo- geneous lineage (A); a large, heterogeneous lineage with regular structure (B); a medium-sized heterogeneous lineage with irregular structure (C); and two smaller lineages (D and E). The results of applying each of the metrics to this sample set are summarised in Table 5.1. For each metric, the most complex lineage(s) are indicated in bold.

5.1 Study 4: Characteristics of cell lineage complexity 99

Figure 5.3: The five sample lineages used to compare complexity metrics: a large, homogeneous lineage (A); a large, heterogeneous lineage with regular structure (B), a medium-sized heterogeneous lineage with irregular structure (C) and two smaller lineages (D and E).

Number of cells: Using number of cells as a measure of complexity clearly satisfies our first intuition about complexity: that lineages containing more cells are more complex. However, one significant disadvantage is apparent: counter to our second intuition, the large, homogeneous lineage (A) is ranked as being of equal complexity to the large, heterogeneous lineage (B) and of greater complexity than all of the smaller, heterogeneous lineages (C, D and E).

100 The Structure and Composition of Ontogenetic Space

Table 5.2: Deterministic and non-deterministic rule sets for sample lineages Deterministic Non-deterministic

Lineage Rule Set Rule Set

A 1 → {2,2} {R} → {R},{R} 2 → {3,3} 3 → {4,4} 4 → {5,5} 5 → {R, R} Y 1 → {2,2} {R, G, B, Y} → {R, G, B, Y},{R, G, B, Y} 2 → {3,3} {R, G, B, Y} → {R, G},{B, Y} 3 → {4,4} {R, G} → {R},{G} 4 → {5,6} {B, Y} → {B},{Y} 5 → {Y, B} 6 → {G, R} C 1 → {2,3} {R, G, B, Y} → {R, G, B, Y},{R, G, B, Y} 2 → {4,5} {R, G, B, Y} → {R, G, B, Y},{R, G, B} 3 → {6,7} {R, G, B, Y} → {R, G, B},{B, Y} 4 → {8,5} {R, G, B, Y} → {R, G,},{B, Y} 5 → {9, R} {R, G, B, Y} → {R},{G, B, Y} 6 → {8,10} {R, G, B} → {R},{G, B} 7 → {11, R} {B, Y} → {B, Y},{B} 8 → {12, B} {R, G} → {R},{G} 9 → {G, B} {G, B, Y} → {G},{B, Y} 10 → {G, R} {G, B} → {G},{B} 11 → {G,12} {B, Y} → {B},{Y} 12 → {Y, B} D 1 → {2,3} {R, G, B, Y} → {R, G},{B, Y} 2 → {Y, B} {R, G} → {R},{G} 3 → {G, R} {B, Y} → {B},{Y} E 1 → {G, R} {G, R} → {G},{R}

Number of cell types: Using number of cell types as a measure of complexity mitigates the problem with large homogeneous lineage (A); however, it suffers from two disadvantages. At the scale of the lineages considered here, it is a very coarse measure: lineages B, C, and E are all classified as being equally complex despite the differences in the size and regularity of the lineage. Furthermore, as mentioned above, in the DRGN-lineage model, cell type number is constrained at the point of model definition: a lineage can never contain more than NO cell types.

5.1 Study 4: Characteristics of cell lineage complexity 101

Algorithmic complexity (deterministic): The first of the developmental com- plexity metrics had several advantages over the morphological metrics. The large, heterogeneous lineage (B) was ranked as having greater complexity than the large, homogeneous lineage (A). The smaller but less regular lineage (C) was ranked as having greater complexity than both lineages A and B. However, problems emerge with the remaining small lineages (D and E). Lineage D contains only a single cell division event, which is by definition unique, and therefore obtains a maxi- mal complexity value of 1.0, as does lineage E, with three unique division events. A further limitation is less obvious but may be detected by considering the rule set that describes lineage A. Intuition suggests that this lineage is a product of a single rule (X → X, X) applied a fixed number of times. However, the procedure described generates unique rules for each level of non-terminal cells, resulting in a larger rule set than anticipated (Table 5.2).

Algorithmic complexity (non-deterministic): The ordering of the complex- ity values according to the non-deterministic algorithmic complexity measure is identical to that of the deterministic complexity measure, although the actual val- ues differ. Lower complexity values assigned to lineages A (60% lower), B (33% lower) and C (9% lower) reflect a loss of information about the structure of the lin- eage. Equal complexity values are assigned to lineages D and E indicating that the first problem identified with deterministic algorithmic complexity persists: lineages D and E are assigned disproportionately high complexity values. Considering the rule sets indicates that the problem of repeated proliferative divisions producing new rules at each level has been addressed by the introduction of non-deterministic rules (Table 5.2). One possible disadvantage of this metric is that there is no longer a unique mapping between a rule set and a lineage. The rule set describing lineage A can be used to describe homogeneous lineages containing any number of levels of cell division.

Weighted complexity : The final complexity metric addresses several of the limitations of the previous metrics. By incorporating lineage size into the measure, the bias of the algorithmic complexity measures towards small lineages has been balanced. Specifically, the small lineages (D and E) are no longer assigned dis- proportionately high complexity values as with the other algorithmic definitions. Conversely, the large, homogeneous and regular lineages (A and B) no longer have

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excessively large complexity values due to their size alone.

5.1.3

Discussion

Any measure of complexity will have strengths and limitations and be, to some extent, specific to a particular task or observer. In the absence of a single accepted definition of complexity, the decision to use any one metric over another was guided by pragmatic requirements. Given the focus of this research on the control of de- velopmental processes, the most suitable complexity metric was deemed to be one which reflected the number and diversity of control decisions required to produce a given lineage.

This study illustrated the strengths and limitations of several different measures of complexity from the perspective of control decisions. The results suggest that a size-weighted algorithmic complexity measure, based on a modified version of the lineage complexity metric introduced by Azevedo et al. (2005), accords with our intuitions about which lineages should be considered more or less complex.

Study 5: Visualisation of ontogenetic space used the metrics described in this section to explore how lineages, as quantified by complexity, varied over parame- terised regions of space.