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Title: Generation of Diverse Biological Forms through Combinatorial Interactions between Tissue Polarity and Growth}
Article:
Abstract:
A major problem in biology is to understand how complex tissue shapes may arise through growth. In many cases this process involves preferential growth along particular orientations raising the question of how these orientations are specified. One view is that orientations are specified through stresses in the tissue( axiality-based system). Another possibility is that orientations can be specified independently of stresses through molecular signalling( polarity-based system). The axiality-based system has recently been explored through computational modelling. Here we develop and apply a polarity-based system which we call the Growing Polarised Tissue( GPT) framework. Tissue is treated as a continuous material within which regionally expressed factors under genetic control may interact and propagate. Polarity is established by signals that propagate through the tissue and is anchored in regions termed tissue polarity organisers that are also under genetic control. Rates of growth parallel or perpendicular to the local polarity may then be specified through a regulatory network. The resulting growth depends on how specified growth patterns interact within the constraints of mechanically connected tissue. This constraint leads to the emergence of features such as curvature that were not directly specified by the regulatory networks. Resultant growth feeds back to influence spatial arrangements and local orientations of tissue, allowing complex shapes to emerge from simple rules. Moreover, asymmetries may emerge through interactions between polarity fields. We illustrate the value of the GPT-framework for understanding morphogenesis by applying it to a growing Snapdragon flower and indicate how the underlying hypotheses may be tested by computational simulation. We propose that combinatorial intractions between orientations and rates of growth, which are a key feature of polarity-based systems, have been exploited during evolution to generate a range of observed biological shapes.
Introduction:
Although there have been many experimental and theoretical studies on patterns of gene activities and their establishment in animals and plants[1]–[6] much less is known about how patterns of activity are linked to tissue growth and deformation. Addressing this problem represents a challenge because final form is usually not a direct readout of locally specified properties, but depends on mechanical constraints from neighbouring regions. For example, if the margin of a leaf has a higher specified growth rate than the centre, a wavy edge will emerge. The wavy edge is not directly specified but is a feature that emerges through the interaction between patterns of specified growth and the mechanical constraints of tissue continuity[7]. In such cases we may distinguish between specified growth, which is the growth that would be attained if each region grew independently of its neighbours( i. e. in mechanical isolation), and resultant growth, which is the growth observed when mechanical constraints of neighbours are taken into account( i. e. mechanically connected tissue). In animal systems a similar distinction is made between an imposed active deformation, and an elastic passive deformation[8]. Resultant growth can be measured experimentally by tracking tissue deformations over time[9]–[13]. However, to understand the mechanisms by which resultant growth arises we need to know how genes influence specified growth. Where specified growth is isotropic, genes need to control a single parameter, the local rate of growth. However, in many cases specified growth may be anisotropic requiring orientations as well as rates of growth to be under genetic control. Controlling orientations of growth requires a local axis to be defined( i. e. axiality, represented as a field of lines). In this respect growth is similar to stress which also has axiality. This similarity has led to the suggestion that stresses provide the primary cues for orienting growth. According to such a stress-based axiality mechanism, gene activity influences stresses in the tissue, the orientations of which are transduced to influence molecular properties of cells such as the cytoskeleton. These in turn modulate growth orientations which may further feed back to influence the pattern of stresses[14]–[18]. Recent support for such mechanisms in plants have come from studies of the effect of stresses on microtubule patterns[19]. A different way of specifying orientations of growth is through differential concentrations of signalling molecules. The varying concentrations define a local( cellular) polarity which includes both axiality and directional components( represented by a field of arrows). The axiality component is then used to orient growth. In this polarity-based axiality system, genes influence the distribution of signalling molecules which define a coordinated field of polarities. Incorporation of mechanical constraints then leads to resultant growth, which may feed back to influence, for example, tissue polarity orientations. In support of this system, there is considerable evidence that polarity is prevalent in biological tissues and may modulate growth[20]–[24]. For example, planar cell polarity( PCP) systems have been described in animals and implicated in processes such as growth of wings in Drosophila and convergent-extension in vertebrates[25]–[27]. Similarly, the polarised distribution of auxin transporters( PIN molecules) has been shown to be important for outgrowths of primordia in plants[28]. Elements of the stress-based axiality system have recently been modelled[29],[30]. Here we describe a framework and software implementation for the alternative polarity-based axiality approach, which we call GPT-framework. ur software, called GFtbox, is a MATLAB application available from http://www. uea. ac. uk/cmp/research/cmpbio/GFtbox. This framework was developed with plant growth in mind, although it may also be useful for modelling animal systems where cell movement is limited. In accompanying papers we show how a biologically relevant model can be derived using the GPT-framework[31],[32]. This model of Snapdragon flower development is constrained by a range of experimental data including gene expression patterns, mutant phenotypes, clonal analysis, growth dynamics and changes in geometry. It provides a working hypothesis for how growth is specified and shows how reorientation of growth can account for key observations. In this paper we explore a series of simplified models which illustrate how growth and polarity may interact combinatorially during morphogenesis to generate a wide range of forms. The results highlight the value of being able to specify orientation independently of stresses in the generation of complex tissue shapes. In addition, we provide the theoretical foundations on which our modelling depends.
Discussion:
We model growth through the accumulation of a series of small deformations of the tissue( canvas). Stresses are generated during the process as the canvas is mechanically interconnected. This may lead to anisotropic resultant growth even when growth is specified to be isotropic( e. g. Case R). In principle, such resultant stresses could be used, through stress-based axiality, to orient all forms of anisotropic growth. However, this would mean that specified orientations of growth would be dependent on differential rates of growth, precluding the possibility of independent control. By contrast, we show how a polarity-based axiality system allows diverse forms to be generated through combinatorial interactions between specified orientations and rates of growth. In this system, a key element in controlling growth orientations is the distribution of polarity organisers. These are of two types, or, allowing polarity fields to be anchored at both ends. Even when specified anisotropic growth is uniform over the canvas, a range of forms can be generated by varying the pattern of organisers. For example, starting from an initial square canvas it is possible to generate rectangles( Case A), concavities( Case F), small spikes( Case J) and ripples( Case O). In these Cases polarity was fixed after a setup period. Biologically, this would correspond to an initial period when polarity propagates across the tissue( when the tissue is small), followed by polarised cells maintaining their polarity and passing it on to their daughters. Another possibility is that polarity continues to propagate during growth leading to slight modifications of the resulting shape( compare Cases F and G). The range of shapes may be greatly extended by combining polarity fields with differential growth rates. For example, tissue polarity may reinforce or antagonise curvature arising through differential growth( Cases C and I). Both aspects are incorporated into the growing Snapdragon tube- reinforcement of curvature during the early phase leading to arching over( Case S), followed by antagonism of curvature leading to straightening( Cases T). It is also straightforward to generate extended outgrowths and apices by combining a single organiser with enhanced growth( Cases L and M). A further feature of polarity-based axiality systems is the emergence of asymmetries through interactions between polarity fields. For example, asymmetric spurs may arise because of counterflowing polarity on one side( Cases P and Q). The asymmetry of the outgrowths in these Cases results from the underlying polarity interactions and would not have arisen from a simple system with only stress-based axiality. In these examples only a few organisers are needed to achieve major shape transformations. To test whether the same simplicity might underly more complex biological transformations, we modelled growth of the Snapdragon flower[31]. This model is constrained by a range of experimental data. The expression pattern of the genes DIV, CYC, DICH and RAD are set according to experimental observations. The model has to not only account for the wild-type phenotype but also double( cyc, dich) and triple( cyc, dich, div) mutants. The model is also constrained by the observed changes in 3D shape determined by optical projection tomography at several developmental stages. In addition the pattern of growth rates and directions over each model petal need to be similar to those observed by clonal analysis. The model starts with an initial cylindrical canvas with five lobes and a proximodistal pattern of polarities established through two polarity organisers( and)( Figure 7 A). During the early phase of growth the ventral region of the tube arches over through differential anisotropic growth. To account for the observed pattern of clones a third organiser() is introduced( Figure 7 C). In the absence of this organiser the tube bulges out( Figure 7 F) similar to what happens in the simplified corolla with no reorientation of growth( Case S). However, with the introduction of the organiser the tube automatically straightens out during later stages, consistent with experimental observations. Thus, this biologically relevant case provides evidence for three organisers underlying major shape transformations and growth dynamics. In the Snapdragon model, the reorientation of growth is under the control of DIV, a gene that encodes a Myb-like transcription factor that affects flower shape and symmetry[31]. As well as its effect on organiser activity, DIV also influences growth rates. Thus, although rates and orientations of growth are specified separately in the model they can be regulated by a common gene. The polarity-based axiality system has the flexibility to account for global shape changes, observed growth patterns and clones without invoking large numbers of polarity organisers. This alone does not demonstrate the validity of invoking tissue polarity for the control of growth orientations. Nevertheless, tissue polarity is commonly observed in animals, for example, polarised cell movements[27] and in plants where the polar distribution of molecules within cells, such as PIN auxin transporters, suggests that cell polarity is also common[23],[24]. It has also been proposed that an auxin concentration maximum at a vascular boundary in the root tip establishes a distal polarity organiser in the root[20]. The GPT-framework allows hypotheses on polarity-based axiality growth to be established that can be subjected to further tests such as mechanical or genetic perturbations. The Snapdragon model, for example, was evaluated against predictions of shapes of multiple mutants not used to build the model[31],[32]. The results showed a good, quantitative, fit between predicted and observed shapes. The model also makes important predictions about the location of polarity organisers. Polarity markers are predicted to show reversals( i. e. arrows pointing away or towards each other) at these locations. In all our Cases we make the simplifying assumption that the tissue is linear over small deformations and has isotropic material properties. An elaboration of the GPT-framework would be to incorporate non-uniform properties, although this would also require these properties to be measured across the tissue during growth. The GPT-framework is consistent with current hypotheses regarding the mechanisms in which plant tissue grows under turgor pressure through the loosening and formation of bonds( Theorems 1 and 2, Methods). Loosening bonds in the cell wall allows the tissue to grow. If new material is inserted that restores the properties of the cell wall then the residual strain returns to zero( ‘snip and fill’[31]). Biologically this would require some form of feedback between resultant stresses( or strains) and cellular properties[34]. Feedback from stresses to microtubule patterns has been proposed[19], and this can be interpreted as reflecting the need to dissipate residual stresses rather than being the primary way of orienting specified growth. Cutting provides a convenient experimental way to evaluate the extent to which residual stresses accumulate or dissipate in a given biological system. Often they accumulate in certain regions in later developmental stages. For example, the dorsal and ventral petals of the adult Snapdragon flower press against each other holding the flower shut( not a part of the model in Green et al[31]). The observation that the accumulation of residuals varies systematically from region to region suggests that the process of dissipating or accumulating residuals is under genetic control. Stresses that are accumulated can be modelled with the GPT-framework and, to enable direct comparison with experimental results, the resulting shapes can be cut allowing the structure to spring into a new shape. The GPT-framework assumes that regions( e. g. cells) in a tissue do not slide or move past each other. This is valid for plants[35], making them particularly appropriate for this approach. The GPT-framework may also be applicable to some aspects of animal development. For example, finite element models have been used to capture deformations during Drosophila ventral furrow formation driven by apical constriction and apicobasal elongation of cells[8]. Comparable deformations can also be generated using GPT-framework by using a posterior-anterior polarity field[36] and incorporating negative growth( contraction) on one side of the canvas( Figure 8). Although this model does not incorporate all biologically relevant features such as constraints of the external vitelline membrane, it illustrates the flexibility of the approach. Clones generated in early wing development of Drosophila often stay as contiguous patches, indicating that connectivity is broadly maintained and extensive mixing of cells does not occur[37]. Greater cell mixing is observed for clones in developing mammalian tissues such as the heart or limb, although even in these cases cell movements are not sufficient to disrupt formation of clonal clusters or patches[38]. At the tissue scale it may therefore be reasonable to model many animal structures with the framework described here, particularly as orientated cell behaviours are thought to play a critical role[39],[40]. As well as multicellular tissues the canvas could represent a region of a plant or bacterial cell wall. By extension of the GPT-framework it may also be possible to capture the growth of compartments enclosed by a canvas( e. g. cells with their walls) or growth of a bulk solid. Thus, the GPT-framework provides a general approach that can be applied to growing tissues at many scales. The GPT-framework with its assumption of tissue polarity as a key component of growth specification provides an economical way of generating diverse shapes and forms. We hypothesise that this combinatorial richness is not only computationally attractive but has also been exploited during evolution to generate a range of observed biological shapes.
How do genes control the growth of cells into complex tissue shapes such as flowers , wings or hearts ? A key requirement is that genes must be able to modulate growth along particular directions . Two mechanisms have been proposed for how this may work; one based on the directions of mechanical stresses in the tissue and the other on molecular signals that propagate and provide local polarities . Here we show how a polarity-based system has the advantage of being able to act in combination with growth rates to generate a wide range of shapes . By applying this system to the development of the Snapdragon flower , we show , by comparison of computational simulations with actual flower development , how a simple set of polarity controls may underlie the formation of complex biological structures .