Cellular “bauplans”: Evolving unicellular forms by means of Julia sets and Pickover biomorphs
Introduction
In biological thought, the concept of a body plan has derived from two distinct intellectual paths (Raff, 1996). On the one side, Owen had put forward in the 1840s the notion of archetypes to characterize the essential design that underlies the diversity of species within a group. And on the other side, a little bit earlier, von Baer's developmental studies were concerned on how the body plan (bauplan) is deeply immanent in the process of ontogeny for each species. The possibility that this discussion, also connected with the great controversies Cuvier–Geoffroy of that time, could apply not only to the structures of multicellular organisms but also to individual cells, was not considered except by the pioneering approach of D’Arcy Thompson, a few decades later. His discussion on surface tension, viscosity, “rolling curves”, and stability as applying to the determination of cellular form was seminal; later on authors such as Waddington, Goodwin and others were inspired by it, as we are going to argue.
The extension of the idea of a “body plan” to the organization of the cell itself was due to Lynn and Tucker (1976), about thirty years ago, who first considered the possibility of two different mechanisms as responsible for defining organelles’ position inside cells. Structural positioning and chemical signaling were instructing organelles to follow a well-ordered and characteristic pattern in many unicellular organisms, such as ciliates. The fact that differentiated structures within a cell, such as a mitochondrion, vacuole, or chloroplast, that perform a specific function, were occupying specific positions inside different classes of cells was the observational and experimental support of the “morphogenetic field” notion in the cellular realm. Further, these authors proposed that the spatial complexity of the organelle arrays is comparable with the multicellular arrangement of different cell types, thus tissues (i.e. muscle, nerve, etc.) and organs (structures made up of several tissues capable of performing some special biological function) in animals. Frankel (1975) argued that some of the mechanisms underlying the spatial specification of pattern are the same both in ciliates (a class of protozoan) and multicellular animals. However, whereas the genes dedicated to specifying shape seem important in higher organisms (for example the Drosophila fly), there appear to be very few of such genes in microorganisms (Harold, 1990). In consequence, at least in microorganisms or unicellular organisms, morphogenesis would be a matter of large-scale spatial organization arising epigenetically, not being encoded directly in the genome.
In agreement with Harold (1990), this morphogenetic idea had been considered before by Waddington, who introduced the notion of epigenetic landscape—the fact that a morphogenetic surface which specific topography depends on all the interacting genes or their mutations and not on a particular gene or mutation. In consequence, the form of unicellular organisms could not be a direct expression of a genetic program. Explanations of this kind have led to contemporary evo-devo biologists to reconsider the notion of morphogenetic field (Raff, 1996), assuming under a theoretical point of view that living organisms either unicellular or pluricellular are organized by fields; in point of fact revisiting molecularly the classical notion of bauplan, nowadays also extended within the cellular level after the pioneering work of Lynn and Tucker (1976).
Properly in developmental studies, the notion of morphogenetic field had been introduced in the 1920s by Spemann, 1921, Spemann, 1938, Gurwitsch (1922), and Weiss, 1923, Weiss, 1939 which also used the terms of “developmental” or “embryonic” fields. Formally, as we will discuss later on, a morphogenetic field may be defined as a plane A with points representing the locations where cells are differentiated or sub-cellular components (or organelles) organized, thus being the plane A either a tissue sheet or a whole cell, respectively. According with Harold (1990), such field has special spatial and temporal features with some ‘agency’ acting in a coordinated way over the area A. For instance, the agency may correspond to signaling molecules defining the morphogen gradients introduced by Turing (1952), or to endogenous electric fields (Nuccitelli, 1984, Nuccitelli, 1988), to ionic currents (Limozin et al., 1997), visco-elastic field tensions (Briere and Goodwin, 1990), osmotic fields (O'Shea, 1988) or even to electromagnetic fields (Chovnjuk et al., 2001). No matter which one of the aforementioned agencies is the plausible one, morphogenetic fields always consider some physical rule that, in the last term, is responsible for the genesis and control of such field.
Nowadays, two main different points of view are considered, which are not totally incompatible between them (even more, for some authors they are elegantly intertwined: Basler and Struhl, 1994, Marijuán, 1996). One paradigm considers that A locations ‘hold’ positional information that controls the morphogenetic destiny or the self-organization events experimented by cells or their sub-cellular components. Under this view, originally based on the morphogen theory proposed by Turing (1952), the local interactions between points (i.e. the activator and inhibitor morphogens) govern the final emergent pattern. At present, this view is supported by the notion of ‘positional information’ or ‘embryonic organizer’ proposed by Wolpert (1969), Spemann and Mangold (1924), as well as by some genetic mechanisms proposed to explain the formation of morphogen gradients (Gierer, 1981, Gierer and Meinhardt, 1972, Meinhardt, 2003). Alternatively, the other developmental view considers that cells located at points of A into the morphogenetic field follow a ‘master plan’ resulting the final whole organism as a consequence of some ‘attractors’ in the morphospace. This latter perspective, mainly proposed by Waddington, 1940, Waddington, 1942, Waddington, 1957, Goodwin (1994) and D’Arcy Thompson and Bonner (1992), considers that organisms are dynamical systems. In a mathematical sense, the organism may be defined as a manifold M called the ‘organism state’, and a smooth evolution function ϕ(t) that for any element of t, the time, maps a point of the phase space back into the phase space. The organism evolves through transformations which are mainly explained by the molecular organization and the mechanical properties of the cytoplasm (i.e. via cytoskeleton and networks of filamentous protein polymers, see Hotani et al., 1992, Goodwin and Trainor, 1985). That cytoskeletons may create a ‘mechanical’ environment guiding cellular development and differentiation is receiving strong empirical support. In a very challenging paper, Ainsworth (2008) brings forward two striking experiments of how a mechanical field of physical forces (i.e. governed by cytoskeleton) could be affecting the cells fate. One of these experiments describes how stem cells growing on soft, medium or rigid matrices resembling their natural environments start developing into neurons, muscle and bone cells. Another experiment is related with the mechanics of the cancerous cell: the extracellular matrix or microenvironment surrounding cells with mutated oncogenes becomes stiffer even before those cells form tumours. We will discuss these challenging views, and how mechanical forces, together with chemical morphogens, could organize the developmental and cell differentiation processes of an organism.
The evolution of the morphogenetic field is a relevant topic in Biology since the ways in which morphogenetic fields can change over time may yield insights into the evolutionary possibilities of unicellular and pluricellular organisms. For instance, Perales-Graván and Lahoz-Beltra (2004) simulated the evolution of the morphogenetic field in the zebra skin pattern. In the aforementioned paper we used a genetic algorithm with chromosomes coding the diffusion distance and the morphogenetic constant field values of two different morphogen molecules, the activator and inhibitor, codifying such values the transition rules of a cellular automata population. This cellular automata model of the morphogenetic field of the zebra skin pattern (Young, 1984) was the discrete version of the reaction–diffusion equations proposed by Turing (1952). Surprisingly we found that during evolution spot patterns appear more often than stripe patterns on the simulated skin of zebras, concluding that the stripe pattern of zebras may be a result of other biological features (i.e. genetic interactions, the Kipling hypothesis, etc.) not included in this model—see Perales-Graván and Lahoz-Beltra (2004).
Assuming the ‘positional paradigm’, where A locations contain information governing the morphogenetic events, two principal mechanisms have been proposed to convey positional information in the mathematical models (Kavlock and Setzer, 1996). The most common model is the reaction–diffusion approach based on Turing's morphogen theory, where pairs of diffusing chemicals interact with each other to form stable patterns (i.e. zebra strip patterns). One example of this model is the aforementioned paper (Perales-Graván and Lahoz-Beltra, 2004) where we simulated the evolution of the morphogenetic field in the zebra skin pattern. In the other approach, “genetic addresses” and gradients of chemicals provide the positional information (Basler and Struhl, 1994, Marijuán, 1996). For instance, based on this approach Levin (1994a) supposes that the chemicals of the genes (similar to ‘proteins’) interact intracellularly producing complicate developing patterns, modeling such patterns with fractal and chaos theory. The author also considers that functions in complex variables can be used to simulate genetic interactions, resulting in position-dependent differentiation. Further, this is shown to be equivalent to computing modified Julia sets, and may produce a very rich set of morphologies.
Following the assumptions and restrictions introduced by Levin (1994a), in the present paper we study the morphogenetic field evolution yielding from an initial population of cells to different unicellular and microbial organisms (Perry et al., 2002) as well as specialized eukaryotic cells (Lodish et al., 2000). Both types of cells are represented as Julia sets and Pickover biomorphs (Pickover, 1986, Pickover, 1990) using to simulate Darwinian natural selection a simple genetic algorithm. The rest of this paper has been organized as follows. After this introductory section, Section 2 presents the model and includes a brief description of the main theoretical notions and definitions related with cellular–biological forms, describing the chromosome coding of complex mathematical functions, plus the evolving biomorphs (unicellular organisms as well as specialized eukaryotic cells) obtained by a simple genetic algorithm with a fitness function which evaluates biomorphs based on thirteen selected features related with the form and physical properties of a biomorph. Section 3 presents the computer simulation experiments, and Section 4 makes further comments on the whole results of the simulation experiments. Finally, Section 5 discusses the possible impact of this work and its future directions.
Section snippets
Model
Let P(z) denote a function in a complex plane and z0 () a starting value. Given Pn(z) as the function resulting of the composition of P with itself n-times, then the sequence of points P(z), P2(z), …, Pn(z), etc., given by the recursive family zn+1 = Pn(z) with n varying over the set of natural numbers, defines an orbit O(z0) of its starting point z0 (Devaney, 1994). For a given function P(z), the behavior of the orbit O(z0) defined by the set or sequence of points Pn(z0) depends on the
Genetic Algorithm Simulation Experiments
Several simulation experiments were carried out assuming in each one that an initial population of cells (or biomorphs), the form of which was plotted from chromosomes, could be evolved in different environments (or under different fitness functions, see Lahoz-Beltra et al., 2008) to one of the following cellular classes:
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Class I. This class is defined by ancient cells without cell nucleus and mitochondrion, having a thin membrane. These cells live in a watery environment.
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Class II. Is a kind of
Simulation Results
Computer simulation experiments were carried out using the population size, gene values as well as the recombination and mutation probabilities referred in Section 2.3. In Fig. 4 we show two representative performance graphs of the simulation experiments. Performance was measured as the average fitness per generation as is usual in experiments based on genetic algorithms. The evolutionary dynamics of class IV (Fig. 4a) is also displayed by classes III and V. In a similar fashion the dynamics
Discussion
In this paper we have examined how unicellular organisms, as well as some specialized eukaryotic cells, evolve through a master plan of transformations under Darwinian natural selection from an initial population of cells simulated as Pickover biomorphs (Fig. 1, Fig. 2). An interesting finding of our study is that the Pickover criterion promotes the evolutionary appearance of cell populations with a higher diversity of size and form than those populations of cells evolved under the Euclidean
Acknowledgments
We would like to thank I. Barradas Bribiesca and A. Ceseña Quiñones for useful comments and suggestions. The first author was supported by the University of Guanajuato (Mexico), the University of Guerrero (Mexico), CONACYT (Consejo Nacional de Ciencia y Tecnología, Mexico) as well as the Complutense University of Madrid (Spain). The second and third authors acknowledge support of the Instituto Aragonés de Ciencias de la Salud (Gobierno de Aragón, Spain); and the forth author was supported by
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