On Braitenberg's Vehicles, Compound Polygons, and Evolutionary Developmental Structural Complexity

  1. 1.  Orthogonal Research
  2. 2.  OpenWorm Foundation

Abstract

The discrete evolution of phenotypes and nervous systems can provide much information about the structures that carry forward the evolutionary process. There are also relationships at the intersection of branching processes, semiotics, the geometry of form, and biological function that remain unexplored. One path to understanding involves establishing a link between representations of phenotype and computational structures that provide generalizable quantitative information regarding biocomplexity and geometric coherence. Two proof-of-concept analysis are conducted to demonstrate a structural characterization of developmental and evolutionary processes. The first is a network analysis of Valentino Braitenberg’s Vehicles to represent discrete traits in functional phenotypes. The second employs a network of polygons (composite polygons) to represent the process and geometry of embryogenesis. For each approach, an appropriate analytical technique is used. The vehicle phenotypes are analyzed using a syllabic network, while the composite polygon phenotypes are analyzed using an Euler walk. The implications of each approach and corresponding analysis are discussed in the context of phylogeny and developmental processes.

Introduction

To better understand the structure and integration (Pigliucci, 2003) of functional phenotypes, we can approach the emergence of complex phenotypes through the synthesis of syntax, geometry, and the major transitions of living matter. While this is a rather unconventional approach, it allows us to speculate with respect to the complexity of life before major body plans are established. It also allows us to approximate the role of functional constraint prior to the emergence of evolutionary constraint. In approximating the evolutionary features of early phenotypes, we can use two models that capture the first and second major transitions of life, respectively (Trifonov, 2008). The first, replication, can be operationalized as the replication of parts as required by the environment. The second, best described as coding between independent domains, is operationalized as the integration of multicellular phenotype into a unified system.

 

In both of these transitions, syntax-like mechanisms play an important role. In a biological context (Noth, 2007), the rules of syntax provide an ontology (Emmeche, 1999) through which to demonstrate how innovations and shared derived traits both emerge and diverge in early multicellular diversity. By configuring these syntactic elements in a semi-directed network, we can calculate statistics that approximate the relative degree of biocomplexity at the phenotypic level. Meanwhile, geometric representations of simple multicellular forms also allows us to understand the organizational and evolvable aspects of multicellular phenotypes as they integrate (Marshall, 2011; Libby et.al, 2014). Such systems can be fully integrated, partially disordered, or inherently modular.

 

Aside from the representation of phenotype as a discrete set of syntactical nodes linked together by their full description, we can also use compound polygons to examine the integrability of multicellular systems. While this is immediately useful in terms of investigating what phenotypic configurations are possible, it is also valuable in examining the topological relationship between cells in different configurations. Specifically, as cells exchange paracrine and metabolic information, we expect to see an efficiency of exchange, with every edge between enclosed geometric objects being traversed only in a single location.

 

Technical Approach

We can approach the stated problem in a representational manner by using geometric representations of phenotype. This work is motivated in part by the thought experiments posed by Valentino Braitenberg in his book “Vehicles” (Braitenberg, 1984). These pre-constructed synthetic vehicle-like organisms consist of a body, limbs, sensors, and effectors. Vehicles provide us with a diversity of organisms that adapt to their environment by adding or replicating parts to their phenotype. While Braitenberg vehicles are presented sequentially in order of complexity, no explicit set of phylogenetic relationships (origin of shared, derived characteristics) are given. Thus, vehicles can be considered static in that there is no explicit phylogenetic history.

 

While Braitenberg uses vehicles to focuses on the evolution of nervous systems, he does not directly address the issues of integration of phenotypes and vehicle complexity. Indeed, the analyses done on Braitenberg vehicles has focused on functional aspects of the phenotype (Travers, 1987; Salomon, 1999; Rano, 2012). The approach introduced here is to treat vehicles and associated structures as outcomes of a branching process (Kimmel and Axelrod, 2015), particularly the shared derived components of this process (Lyons-Weiler and Milinkovitch, 1997; Lyons-Weiler, 1996) that can be analyzed in terms of syntactic content and traversability. These two components (Table 1) give us some understanding of relational information, particularly with respect to how different phenotypic configurations (deep homology) might exhibit constraints (Shubin et.al, 2009).

 

Table 1. Comparison of the two alternate approaches taken in this paper.

 

Static Phenotype

Dynamic Phenotype

Representation Used

Braitenberg Vehicle

Composite Polygon

Abstraction Used

Syllabic Networks

Network of Geometric Shapes

Evaluation

Measure

Kolmogorov (K-) Complexity

Euler Path Score

Evaluation Type

Complexity Metric

Coherence Metric

 

Word transformation as basis for syntactic network

The use of syllabic networks is related to an associated problem in symbolic logic called the word transformation problem. This was first formalized in the form of Thue’s problem (see Methods). In the world of contemporary Computer Science (dynamic programming), this has been operationalized as the edit distance for approximating the relationship between two or more strings (Post, 1947). As syllables are the fundamental components of syntax, we can use ensembles of common word fragments joined together by transformational steps. As these syllables can be independent of meaning (semantic content), these fragments provide a link between common ancestry, functional equivalence, and symbolic logic.

 

One English language example is the transformation of the word “green” to the word “pretty”. The transformation process proceeds by reducing each word to a set of common syllables, which allows us to approximate word transformation as a discrete parsimonious process (Hirshberg, 1975). While there are a number of potential evolutionary paths, limiting this to a finite set of symbols (syllables) results in two transformational steps. “Green” is broken up into “gre” and “en”: the syllable “gre” is mutated to “pre”, while the resulting syllable “pre” is added to a new syllable “tty”. While this appears to be somewhat arbitrary, the component syllables can be constrained by both semantic relevance (functional meaning) and the existence of larger word sets. The addition of words to our original list allows us to find syllables that are more generalizable. For example, the word list “deceit”, “receipt”, and “recover” have both direct (“cei” in “deceit” and “receipt”) and indirect (“deceit” and “recover” are related through “receipt”) relationships. In addition, increasing the number of words in a syllabic network increases the odds that they might share a common prefix or suffix. Words may also share a common motif related to function (combination of letters such as “qu”).

 

Syntactic content

Syntactic content will be defined by decomposing vehicles sensu Braitenberg into a series of linguistic syllables in symbolic language. We will show that each component of the vehicle can be mapped to an alphabetical representation. If we consider multiple vehicles in terms of a single representational structure, we can derive classes which are defined by their similarities of components and form. This representation also serves to define syllables in larger words that provide a description length for individual vehicle morphologies (Rissanen, 1978; Zenil et.al, 2016). Furthermore, these syllables can be related to each other in the form of a semi-directed network. Taken together, the network topology and description length can be used to assess the Kolmogorov complexity (Zenil et.al, 2016) of a given phenotype.

 

A syllabic network is constructed in the following manner: develop a syntax using a collection of phenotypes, find the common (hub nodes) and unique (external nodes) syllables, and then construct generalizable networks based on specific word description lists. Once the syllabary is established, individual syllables are connected given their relationships within and between word length descriptions of vehicles. While syllabic networks are arbitrary with respect to context, they may be computationally universal. One problem that arises in the construction of syllabic networks is that their nodes (syllables) can change along with their connectivity (set of transformations). As a model for the analysis of Braitenberg vehicles, this is not problematic, since this conception of vehicle morphology is static. However, evolution presents us with a dynamic context, particularly as it relates to phenotypic geometry.

 

Complex phenotypes as composite polygons

We can step back from the specificity of vehicle morphologies and use a more abstract model of biological shape and form to understand the collective structure of cell assemblies and other simplistic phenotypes (Martens et.al, 2009). While classifying structurally heterogeneous phenotypes allows us to use a grammar, investigating simple multicellular organisms requires a different approach. Such phenotypes may be considered simplistic because they may or may not possess hierarchical structure.

 

One example of this is in the transformation of a simple rectangle. We can choose a specific compound polygon as a target end point, in this case divisions along the horizontal axis into five equally wide strips. The differentiation process begins by dividing the original rectangle into two parts, and then proceeding from left to right. We can map this differentiation process to a so-called differentiation tree, which captures the shape and location of each newly-born polygon. The next step is to determine whether or not each cell belongs to the same community. To do this, an Euler path (Skiena, 1990) is drawn according to a variant of Fleury’s algorithm (Thorup, 2000) for the compound polygon. This results in an Euler path, an analysis of which can evaluate the composite polygons after every internal differentiation event. In general, if every edge can be crossed only once by a continuous path trajectory, then the coherence metric value is 1.0. On the other hand, if every edge can only be crossed by a discontinuous path trajectory, then the coherence metric value will be less than 1.0. In the case of highly-integrated topologies, where the boundaries between shapes are continuous, incomplete Euler walks tend to cluster around 1.0. In cases where the shapes are spatially dispersed, the coherence metric value should range between 0.0 and 1.0.

 

Traversability of a composite polygon

Traversability will be defined by the existence of a complete Eulerian walk (McKay and Robinson, 1995) across a complex phenotype. In this case, we will move away from Braitenberg-style vehicles and focus on composite polygons (e.g. sets of squares, rectangles, and triangles). A series of Euler walks are made for a limited set of composite polygons, and associated topological data will provide us with an indicator of relative plausibility. A complete Eulerian walk across a compound polygon also indicates whether a given phenotype exists as a modular system (Niklas, 2014) or as a single connected component (Jena, 2016).

 

To make an assessment of traversibility, a series of compound polygons will be generated and analyzed using the Eulerian walk method. This will be demonstrated using a dataset consisting of 26 shape networks represent a variety of geometries, from triangular to rhomboid. Each shape network can be characterized further by extracting several topological parameters. Eulerian walks will be generated on each shape, resulting in a coherence metric that is related to fitness and modularity. The results of this walk will be summarized by a coherence metric, which measures the completeness of the walk. If a complete Euler path can be drawn (one that traverses each edge only once), then the cellular system is considered a coherent functional unit. If the Euler path is incomplete as defined by a coherence metric value less than 1.0, the compound polygon is said to exhibit a bifurcation point in its topology. This could lead to the establishment of developmental modularity or discrete organs and tissue types.

 

Methods

 

Thue’s problem

Thue’s problem can be posed in the following manner: a sequence (PQ to PQ’) can be transformed from PQ to MN a given a finite set of symbols (a1, a2,…….an). In this formulation, arbitrary strings PQ and MN are similar if they contain identical substrings (A) that can be regarded as interchangeable.

 

Mapping vehicle elements to syllables

Each vehicle has a series of components that can be described using a symbolic language. In this case, we have decided to use alphabetic code to represent each component. The components can be the type and location of an element (e.g. RW is used for “rear wheel” and FS is used for “front sensor”) or the functional relationship between elements (CC is used for “cross-connectivity”). The description of a single vehicle is a sequence of descriptions (e.g. RWCCFSFS) forming a word. This sequence can be partitioned into syllables, particularly syllables common to multiple vehicles, will serve as the nodes of our network.

 

Kolmogorov complexity

In the context of syllabic networks, Kolmogorov Complexity is the minimum description length (MDL) of a single word using syllables. For example, if the word “strawberry” is described using the syllables “st”, “raw”, “be”, and “rry”, the MDL would be 4. A word in which every letter constituted its own syllable would have a MDL identical with respect to word the character length. This scenario also demonstrate maximum Kolmogorov Information, calculated by finding the ratio of the MDL in syllables to the character length of the entire word.

 

Euler trail analysis

We can define an Euler trail as a path that crosses edges of the network sequentially and without repeated crossings. An Euler trail is complete if it crosses every edge in the graph. We can also evaluate each composite polygon network in terms of its representative topological properties by analyzing its connectivity and shape features. The resulting topological information can be used to reveal the connections between incomplete Euler trails and specific geometric configurations.

 

Coherence metric

To measure the coherence of certain configurational topologies, a network comprised of composite polygons can be used. A composite polygon network provides us with a series of complex topologies that represent a certain level of complexity. Each network’s structure is analyzed using an Euler trail analysis (McKay and Robinson, 1995). A complete Euler trail (no missing edges) indicates high coherence, while a decreasing score resulting from an incomplete path indicates a lower coherence. The relative level of completeness of an Euler trail is consistent with the coherence of a composite polygon network. In general, the Euler trail approach can provide a measure of configurational coherence that maps directly back to the importance of form in the emergence of life.

 

Results

Figure 1 and Table 2 demonstrate a simple syllabic network. Syllabic networks are collections of words (descriptions in the form of single strings) broken into syllables and configured into a semi-directed network. A description is followed by moving from node to node in the network, and two or more descriptions can share the same node or set of nodes. Figure 1 demonstrates how a syllabic network can describe a system of word descriptions by breaking the words down into syllables and finding the similarities between words. In this case, or network topology reduces a list of 9 words (between 4 and 9 characters long) to 11 nodes and 14 edges. The potential compactness of this representation can be observed in how similar words can be added to the network without a corresponding increase in complexity.

 

Figure 1. Example of a syllabic network. Word list (w) = 9 and Network size (n) = 11. INSET (left): word list. Network can be characterized by average chunk size (2.0), standard deviation of average chunk size (1.10), and the ratio of word list length to number of nodes (0.82).

 

Encoding a series of word descriptions in this way also allows us to quantify the complexity for each word. This can be done by calculating the Kolmogorov Complexity and the Kolmogorov Information of each word description. The K-complexity is a summary of the number of syllables needed to describe a single word, and the K-information is a relation between K-complexity and the description length in characters. In this network, the average value for each is 2.67 and 0.58, respectively. While this is somewhat trivial for loosely associated English words, this becomes important when alphabetic descriptions are used to characterize geometric and other features of phenotypic diversity.

 

Table 2. Kolmogorov Complexity (minimum description length) and Kolmogorov Information (minimum description length relative to word length) for each word in Figure 1.

 

Words

 

Kolmogorov Complexity

 

 

Kolmogorov Information

 

Real

2

0.5

Money

4

0.8

Repossess

3

0.33

Pony

3

0.75

Deal

2

0.50

Demon

3

0.60

Dead

2

0.5

Deny

3

0.75

Pool

2

0.5

 

Figure 2: Syllabic network for descriptions of five unique vehicle designs (2a: RWLCFSFS, 2b: RWCCFSFS, 2c: RWFCFSFS, 3: RWLCCCLCCCCCLCCCLCFSFSFSFSFSFS, and 8: RWFWHCFS) as defined in Braitenberg [1]. Word list (w) = 5 and Network size (n) = 9. Network can be characterized by average chunk size (4.2), standard deviation of average chunk size (3.53), the average ratio of word list length to number of nodes (0.56), the average Kolmogorov Information (3.4), and the average Kolmogorov Complexity (3.6).

 

Figure 2 demonstrates a syllabic network constructed from descriptions of five Braitenberg vehicles: 2a, 2b, 2c, 3, and 8. Each component was described by using the first-letter initials of their English name (e.g. “rear wheels” is encoded as RW). Creating a description in this way ignores any potential semantic content; indeed, we can assume that no such semantic content exist. Unlike in the case of an English word list with very loose syntactic and semantic associations, the average K-Complexity is 3.6, while the average K-Information is 3.4. On the other hand, the syntactic organization of natural languages (in particular syntactic chunking) might reduce its complexity (Lu et.al, 2016).

 

Figure 3. A comparison of the phylogenetic (left) and syllabic network (right) approach using Vehicles 2a, 2b, and 2c. TOP: phylogeny annotated with the same syllables as used in the syllabic network. BOTTOM: phylogeny annotated with descriptions of the differences between vehicles.

 

We can also compare the syllabic network representing the analysis of Vehicles 2a, 2b and 2c with a comparable phylogeny. Figure 3 shows a comparison of the syllabic network approach to a phylogeny built using the same vehicle features and is constructed using a parsimonious argumentation approach (Hennig, 1965). While we can see some correspondence in topology between the phylogenetic tree and syllabic network, information in the syllabic network tends to be much more compact. This is due to each edge in the syllabic network being representative of a single transformation. However, there is a lack of correspondence between the presence or absence of specific traits and their common ancestor. From this example, we can say that while syllabic networks are unrootable, they can provide insights into the structural connectivity between traits in a complex phenotype.

 

Figure 4 demonstrates this method on three DNA sequences generated for purposes of comparison with the phenotypic descriptions. In the case of both the Vehicles and DNA toy examples, there is a high degree of redundancy due to the existence of repetitive elements. Yet in the DNA case, syllable size is much shorter than in the vehicles example. This leads to a higher amount of K-information but a lower amount of K-complexity. Interesting, there may be semantic content in DNA sequences (Crofts, 2007) that does not exist in the vehicles, as carriers of DNA have made the second transition in the origin of life (Niklas, 2014). Nevertheless, this very small-scale analysis demonstrates the flexibility of syllabic networks and their generalizability beyond a purely linguistic context.

 

Figure 4. Example of a syllabic network for three DNA sequences (CGCCTTCGAATA, GGCTTAATTGCC, CCCCCTTACGGT). Word list (w) = 3 and Network size (n) = 10. The only syllabic network rule is the addition of syllables to the sequence. Numbers adjacent to arcs represent an addition relation for a specific DNA sequence (where “’1” is CGCCTTCGAATA, “2” is GGCTTAATTGCC, and “3” is CCCCTTACGGT). Network can be characterized by average chunk size (2.0), standard deviation of average chunk size (0.67), the ratio of word list length to number of nodes (0.3), the average Kolmogorov Information (6.0), and the average Kolmogorov Complexity (2.0).

 

Now we turn to the second technique for analyzing multicellular complexity. In this case, we generate a small dataset of compound polygons and attempt to parse the resulting networks using Euler trails. Figure 5 and Table 3 show the results of this analysis. Figure 5 shows six examples of these structures and the existence of an Euler path for each edge (red lines with arrowheads pointing the direction of the path). Amongst these examples, four exhibit complete Euler paths (Rows 1-3, left; Row 4, right), while the other four exhibit incomplete Euler paths (Rows 1-3, right; Row 4, left). In cases in which the Euler paths are incomplete, it was found that out of 26 sample topologies, a maximum of three edges could not be crossed (see Table 3). This suggests that all of the sampled compound polygons are representative of either fully integrated or partially bifurcated phenotypes.

 

Figure 5. Six examples (A-F) of potential Euler paths crossing the edges of a composite polygon network. Of these examples, only A is a complete Euler walk, while three (C, E, and F) exhibit two bifurcation points. Euler path creation algorithm used in this example requires path to begin and end outside of the composite polygon bounds.

 

Figure 6 shows two examples (3-cell phenotype and 5-cell phenotype) of a developing compound polygon with their corresponding differentiation trees (A and B) and Euler path analysis (C and D). As mentioned in the original example, the compound polygon begins as a simple shape primitive (e.g. a rectangle). In both examples A and B, this creates a differentiation tree that sorts the smaller partition from the larger partition at each step. These are analogous to expansion and contraction waves (Gordon, 1999), which in biological embryos denote the emergence of structure and related asymmetries. We can use the geometry formed by these networks to represent larger-scale processes in a dividing embryo, or to compare different shapes exhibited by multicellular community. In such a case, the higher-dimensional aspects of phenotypic geometry (Wolf, 2002) become important to understanding functional diversity.

 

Figure 6. Two examples of how 3-cell (A and C) and a 5-cell (B and D) composite polygon networks can be represented as a differentiation tree (A and B), and how a composite polygon network can be analyzed using an Euler path analysis (C and D). An example of an inaccessible edge is shown in D.

 

In Figure 6D, we see a region of inaccessibility in the compound polygon network. This edge represents a bifurcation point. In the next round of division (not shown), this edge would be partitioned first, with subsequent Euler trails being drawn through this area. This will provide structure to the emerging complexity of cell shapes and sizes. This region of inaccessibility results from a number of divisions, but also a shift in the orientation of division. This might also result in two coherent subsystems inside of a coherent system. Once a region of inaccessibility is produced through partitioning of the original shape, the foundation of a new subsystem (e.g. network module) is initiated.

 

Conclusions

In this paper, we have introduced two complementary analytical approaches to understanding developmental phenotypes: decomposition of simple static phenotypes, and a dynamic geometric analysis of phenotypic form. The former approach, instantiated in the form of Braitenberg vehicles, provides us with a series of discrete complex traits. The latter approach provides a generalized instantiation of cell colonies or simple multicellular forms emerging from a single progenitor.

 

Table 3. Sample dataset of topological analysis for n = 26 composite polygons. Coherence is measured by dividing the number of edges crossed by the Euler walk into the number of edges in a single composite polygon network.

 

Edges

Coherence

Triangle

Square

Rectangle

Rhombus

Polygon

Corner

1

8

1

1

0

0

2

6

2

2

8

1

1

0

0

2

6

2

3

8

0.88

2

0

1

0

6

2

4

8

0.75

4

0

0

0

5

0

5

3

1

3

0

0

0

3

3

6

4

1

0

1

0

0

4

4

7

6

0.83

1

0

0

1

5

3

8

6

1

1

0

0

1

5

3

9

7

0.86

3

0

0

0

5

1

10

11

0.91

1

1

0

2

8

2

11

16

0.94

0

2

3

0

12

4

12

14

1

1

1

2

1

9

3

13

10

1

0

0

3

0

8

4

14

14

1

2

0

2

0

10

3

15

7

1

0

0

2

0

6

4

16

15

1

3

0

3

0

11

2

17

16

1

0

0

5

0

12

4

18

9

0.78

4

0

0

0

6

2

19

7

0.86

3

0

0

0

5

2

20

9

0.78

4

0

0

0

6

3

21

13

0.77

6

0

0

0

8

4

22

9

0.89

4

0

0

0

6

1

23

5

0.8

2

0

0

0

4

2

24

8

0.88

3

0

0

0

6

3

25

14

0.93

2

0

3

0

10

2

26

17

1

4

2

0

0

12

5

 

Extension to a “trees within trees” approach

To make the dynamic aspect of these models explicit, we can adopt a “trees within trees” approach to development and evolution. While this approach may enable the analysis of two or more conflicting lineage relationships across evolutionary history (Page and Charleston, 1998), we propose a version of this approach for understanding developmental and evolutionary processes in the same context. In Gordon (1999), it is proposed that differentiation trees (or trees that capture the differentiation process in embryogenesis) can be incorporated into each species-level node of a phylogeny.

 

In the case the abstractions presented here, the relationships between developmental trees and phylogenetic trees is demonstrated in Figure 6A and B. In the case of Braitenberg vehicles, the developmental aspects are less clear. Nevertheless, a series of embryonic vehicles could be constructed and treated in the same manner. Future work might focus on the relationship between developmental and evolutionary bifurcations, particularly as they relate to cell lineage trees, differentiation trees, and species phylogenies.

 

Further Considerations

In this paper, a proof-of-concept based on toy models of evolutionary and developmental processes has been introduced that may have broad applicability to the structural features of organismal-level biocomplexity. Both models introduced here can also be used to understand the role of regulation in the formation and maintenance of complex phenotypes. When combining these structural models with either genetic algorithms or cybernetic models of genetic regulation, the deep structural features of the relationship between genotype and phenotype might become discoverable.

 

More specific to phenotypic integration, the introduced methods might be able to show how phenotypic modules and other groups of traits are constrained together by virtue of functionality. This is particularly true in the case of Braitenberg vehicles, for which the chosen phenotypic traits are functionally relevant to an incipient nervous system. However, compound polygons also feature common constraints, as common sets of shapes can scale up in network size without adversely affecting the coherence metric value. These structural features of generative complexity can also be utilized in the analysis of cultural evolution. The analysis of Braitenberg’s vehicles might be particularly informative in this regard.

 

Acknowledgements

Thanks go to the DevoWorm research group (Drs. Richard Gordon, George Mihailovsky, and Tom Portegys) for their feedback on a presentation of this work. Thanks also go to Robert Stone, who encouraged this paper to be developed as a self-contained open system.

 

 

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