A model study on biomorphological description

Abstract

-Description of the morphology of organisms is mostly done in terms of features assuming certain states. Biological patterns are subjected to certain constraints imposed on them by the fact that they all are developed from a single cell. Using a model in which some of the constraints of biological development are incorporated we investigate the relation between different modes of description of such patterns. Pattern generation Pattern recognition Numerical taxonomy Parallel rewriting systems L-systems I. I N T R O D U C T I O N Organisms show for each species characteristic, patterns in which cells and regions of different types occur. Much biological work is done on the description of these patterns: in taxonomy and morphology the patterns are described and classified, in developmental biology sequences of patterns occurring during the development are described, while in evolutionary biology observed patterns are ordered into sequences in such a way that the transitions seem small. In all these areas of research, patterns are described in terms of ~'features" (subpatterns) assuming certain ~'states". Given this description, classification or ordering may be done either intuitively, selecting a few important features (classical taxonomy), or by cluster analysis or multidimensional scaling techniques where selection of features is less severe (numerical taxonomy). Nevertheless, in all cases the selection of features remains crucial. Moreover, the recognition of these features is sometimes difficult, and it is not clear which subpatterns should be compared (homology problem). Patterns of multicellular organisms are subjected to some obvious constraints. All are produced by repeated cell division, via a sequence of patterns, from a single cell, All cells contain the same genetic material. Obviously the behaviour of the cells (division or differentiation) can be controlled only by local conditions working on the genetic material, not by the current global form of the organism. In this paper we try to gain insight in the relations between global patterns and their morphological description in terms of subpatterns and the description of the local transformations [controlled by local (non morphological) conditions], by which the patterns are formed. These relations seem important for the methodology of pattern recognition in biology: often morphological similarities are interpreted as representing similarities on the genetic level. Moreover. we try to obtain some feeling about "small" transitions between morphological patterns under the stated constraints; small transitions between successive stages in development and those caused by small changes in the rules specifying the development (studied by developmental biology and evolutionary theory respectively). We do this by studying a set of generative systems, defined on the basis of the above named characteristics of biological systems and the sequences of patterns generated by these systems. Thus we obtain the following modes of description of the set of patterns: (1t The generative system (a set of rules specifying local transformations). (2) String representations of the sequences of patterns generated by these rules. (3) Pictorial representations of the strings (as threedimensional branching patterns). (4) Description of the pictorial representations in terms of morphological features and the classification on the basis of these features. (5) Description of the sequences of the strings and the pictorial representations in terms of recurrence relations. 165 166 P. HOGEWEG and B. HESPER It should be stressed that we do not consider these generative systems as realistic models for biological development: they only serve to investigate pattern description and pattern recognition of patterns subjected to some of the most obvious constraints of biological development. Biological development may be subjected to more constraints and the simplified two-state system we used is obviously not realized in biological systems. It. THE GENERATIVE SYSTEMS We used for our experiments the generative systems proposed by Lindenmayer a'z~ (and later called L-systems by van Dalen ~3)) for the description of the sequence of developmental stages of growing filamentous organisms. The systems may be described in terms of arrays of simultaneously operating, interacting cellular automata or in terms of growing strings of symbols. The description in terms of cellular automata suggests strongly the analogy with cellular organisms. In this paper we will use the linguistic terminology. The main characteristics of these generative systems, as compared to Chomsky's generative grammars, are: (1) The production rules are applied to a:ll symbols simultaneously. (2) No terminal alphabet is defined, since the emphasis lies on the sequence of strings formed and not on alternative strings developed by the same grammar. In these systems we denote by the word "language" all strings formed from a given axiom, by applying the production rules any number of times and not only those strings which contain terminal symbols only. (3) The starting string (axiom) is contained in the alphabet (Rozenberg and Doucet)/4~ Lindenmayer ¢~m introduced three different types of L-systems, characterized by the influence of the context on the productions. In 0-L-systems the production rules are such that each symbol is to be transformed independently of the neighbouring symbols (context); in 1L and 2-L systems the production rules are such that each symbol is to be transformed according to its one or two-sided context respectively. The context is locally defined by the neighbouring symbol(s) or, when there is no neighbouring alphabetic symbol, it is defined by end symbols representing the environment. The simultaneity of the productions introduces a non localness in the systems. Formal definitions of L-systems are given by van Dalen, ~3) Herman ~'6) and Rozenberg and Doucet/~ * For a formal definition of bracketed i-L systems, see Lindenmayer/-'~ The systems are called propagating if the empty string is not generated by any of the production rules. For the description of branching structures Lindenmayer ~-'~ introduced brackets in the L-systems. In the bracketed deterministic L-systems which we have used, all the production rules have the forms: (1) ~(x,a,y) = bit] ; (2) 6(x,a,y) = bc; or (3) 6(x,a,y) = b. where a, b, c are symbols of the alphabet, x is the left and y is the right context (possibly unspecified as in 0-L and I-L systems). The brackets are terminal symbols and once introduced are included in all subsequent strings. The brackets are defined as changing the context of the symbols. In 0-L systems the presence of brackets has no influence on the subsequent development of a string of alphabetic symbols. The rules that define the context in bracketed, propagating deterministic I-L and 2-L systems (bracketed P D i L systems; i = 1,2) are as follows: Isee Fig. 1). (1) When the left or right neighbouring symbol is an alphabetic symbol, this symbol defines the context (as is the case in non-bracketed iL systems). (2) When the left neighbouring symbol is an opening bracket, the leftsided context is defined by the first alphabetic symbol in the string towards the left, which is separated from this opening bracket by an equal number (possibly 0) of opening and closing brackets. (3) When the left neighbouring symbol is a closing bracket, the leftsided context is defined by the first alphabetic symbol towards the left which is separated from the symbol by an equal number of opening and closing brackets (including the neighbouring closing bracket). Moreover. in case of a 2-L system: (4) When the right neighbouring symbol is a closing bracket, the rightsided context is defined by the symbol at the end of the string, which represents the environment. (5) When the right-sided symbol is an opening bracket, the rightsided context is defined bv the first alphabetic symbol towards the right, which is separated from the symbol by an equal number of opening and closing brackets (including the neighbouring closing bracket).* (N.b. "left" and "right" are introduced by convention and could be interchanged. The form of some production rules then becomes: 6(x.a.y) = [b]c.)

DOI: 10.1016/0031-3203(74)90019-3

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Cite this paper

@article{Hogeweg1974AMS, title={A model study on biomorphological description}, author={Paulien Hogeweg and Ben Hesper}, journal={Pattern Recognition}, year={1974}, volume={6}, pages={165-179} }