TY - JOUR AU - Dumais, Jacques AB - Abstract The bewildering morphological diversity found in cells is one of the starkest illustrations of life’s ability to self-organize. Yet the morphogenetic mechanisms that produce the multifarious shapes of cells are still poorly understood. The shared similarities between the walled cells of prokaryotes, many protists, fungi, and plants make these groups particularly appealing to begin investigating how morphological diversity is generated at the cell level. In this review, I attempt a first classification of the different modes of surface deformation used by walled cells. Five modes of deformation were identified: inextensional bending, equi-area shear, elastic stretching, processive intussusception, and chemorheological growth. The two most restrictive modes—inextensional and equi-area deformations—are embodied in the exine of pollen grains and the wall-like pellicle of euglenoids, respectively. For these modes, it is possible to express the deformed geometry of the cell explicitly in terms of the undeformed geometry and other easily observable geometrical parameters. The greatest morphogenetic power is reached with the processive intussusception and chemorheological growth mechanisms that underlie the expansive growth of walled cells. A comparison of these two growth mechanisms suggests a possible way to tackle the complexity behind wall growth. Cell mechanics, cell wall, chemorheology, equi-area deformation, Euglena, inextensional deformations, intussusception, morphogenesis, pollen grains, prokaryotes, S-layer, tip growth, turgor pressure. Introduction Of cellular morphogenesis it can justly be said that we know much but understand little. Franklin Harold (1990) If something can be said of the cells that populate even the smallest drop of water, it is that they are morphologically diverse. Cells are also the smallest bits of matter that can be appropriately described as living—thus the multifarious forms of cells is a unique window into the power of organization of life in its purest and simplest form. As we elucidate how cells build themselves from the non-living molecular components within them, we are getting a bit closer to understanding how life can spring and develop from the inorganic world. Cells occupy a unique length scale that lies just above the reaches of molecular self-assembly; so, although the much smaller (and non-living) viruses can use proteins as literal building blocks to create shapes (Crick and Watson, 1956; Caspar and Klug, 1962), cells have had to make use of other strategies for shape generation where the relationship between the genetic information contained in DNA and the shape of the cell is not so literal as in protein self-assembly. With the exception of perhaps the biconcave geometry of erythrocytes (Sheetz and Singer, 1974; Elgsaeter et al., 1986), none of the morphogenetic strategies used by cells is very well understood. There is certainly abundant information on the molecular controls of morphogenesis, yet the painstaking work of putting the pieces together into compelling morphogenetic models is only beginning. Unfortunately, Franklin Harold’s statement is as valid today as it was more than two decades ago. The broadest class of cells encountered in nature comprises those endowed with a stiff extracellular matrix—the so-called walled cells. Extracellular polymeric walls are the norm in prokaryotes, many protists, fungi and plants. Walled cells distinguish themselves from other cells in that their shape is maintained by their surface. Although internal structures such as the various components of the cytoskeleton play a critical role in controlling the morphogenesis of walled cells, they do not directly maintain cell shape. Accordingly, most walled cells keep their shape even when the entire cytoplasmic content has been removed, leaving behind a perfectly formed wall ghost. Besides the most common polysaccharide-based cell walls, special cases are represented by the protein-based pellicle of euglenoid algae (Suzaki and Williamson, 1985, 1986) and the crystalline surface layers found in prokaryotes (Houwink, 1956; Pum et al., 1991) and certain Volvocales algae (Roberts et al., 1985; Woessner and Goodenough, 1994). Although structurally distinct from the polysaccharide walls, the cortical pellicle and surface layers are mechanically similar to the classical walls in that cell shape is defined by the properties of a stiff layer located directly below or above the plasma membrane. The shared similarities between the walled cells broadly defined make this group particularly appealing to begin investigating how morphological diversity is generated at the cell level. A few investigators have attempted to identify different strategies of walled-cell morphogenesis based on kinematics (Green et al., 1970; Geitmann and Ortega, 2009). These classifications distinguish, for example, between the well-known cases of diffuse growth and tip growth. However, the limits of a classification based on geometrical attributes have also been noted. At the level of the cell, Green (1969) emphasized that the same final morphology can be achieved via many surface deformation pathways (Fig. 1A). Therefore, an observed shape change is not sufficient to specify the mechanism of wall deformation. Even the deformation of a small wall element may have alternative explanations that cannot be inferred from geometry alone. For example, anisotropic wall expansion can emerge from isotropic stresses acting on a mechanically anisotropic cell wall or from anisotropic stresses acting on a mechanically isotropic wall (Fig. 1B) (Dumais et al., 2004). In terms of cellular control, these two alternatives have very distinct implications that would be missed if attention were paid only to the kinematics rather than the details of the deformation mechanism at the microstructural level. In other words, a strict focus on wall kinematics is often problematic if our goal is to arrive at an understanding of the mechanism of wall deformation. Fig. 1. View largeDownload slide Limits on the geometrical characterization of wall deformation. (A) Two contrasting ways to deform a cylindrical cell into a flaring trumpet (based on Green, 1969). Three wall patches illustrate the local pattern of surface deformation. For inhomogeneous expansion, shape change results from a gradient in the degree of area expansion. The three patches enlarge to varying degrees but remain approximately square. For anisotropic expansion, there is no gradient in area expansion (final wall patches have all the same area), but there is a gradient in the anisotropy or direction of expansion. (B) Two ways to deform a wall element anisotropically: using anisotropic stresses on an isotropic cell wall (top) or using isotropic stresses on an anisotropic cell wall (bottom). Macroscopically, the deformations are identical but at the microstructural level, the deformations proceed from distinct mechanisms. Fig. 1. View largeDownload slide Limits on the geometrical characterization of wall deformation. (A) Two contrasting ways to deform a cylindrical cell into a flaring trumpet (based on Green, 1969). Three wall patches illustrate the local pattern of surface deformation. For inhomogeneous expansion, shape change results from a gradient in the degree of area expansion. The three patches enlarge to varying degrees but remain approximately square. For anisotropic expansion, there is no gradient in area expansion (final wall patches have all the same area), but there is a gradient in the anisotropy or direction of expansion. (B) Two ways to deform a wall element anisotropically: using anisotropic stresses on an isotropic cell wall (top) or using isotropic stresses on an anisotropic cell wall (bottom). Macroscopically, the deformations are identical but at the microstructural level, the deformations proceed from distinct mechanisms. Here, I adopt a classification of cell morphogenesis using the mechanism of wall deformation as the central criterion. In this context, diffuse growth and tip growth emerge as two related morphogenetic strategies within the rich universe of wall-deformation mechanisms. The diversity of walled cells is such that the list provided is likely to be incomplete. Yet, it seems that even a tentative classification can play an important role in organizing the field and highlighting important areas of research. The walled cell: a diagram of forces in equilibrium D’Arcy W. Thompson (1942) famously said that organic form is a ‘diagram of forces in equilibrium’. This description is particularly accurate for the shape of walled cells. The cell wall evolved to bear the force of turgor pressure and is among the stiffest structures present within cells. Like any other material, the cell wall deforms only to the extent that forces are acting on it. The nature and extent of these forces play a major role in how the cell surface deforms and grows. However, as implied by Thompson’s statement, the forces applied on the cell surface are balanced by counteracting forces in the wall, i.e. they are in equilibrium. In the simple case of a turgid and non-growing cell, turgor pressure is balanced by tensions in the polymeric network constituting the wall. For a given pressure, the cell can respond in many different ways depending on the constitution of its wall. The so-called constitutive properties of the wall material or the mechanism of wall deformation are the basis of the classification presented here. The idea of focusing on the constitutive response of the wall material is not new, as it is, implicitly or explicitly, the starting point of all mechanical analyses. However, in cell biology, this approach has not been used systematically to identify different modes of morphogenesis. The modes of deformation considered here are those that emerge from different constitutive behaviours of the wall rather than their geometrical attributes (Fig. 2). The geometrical features that are sometimes used in defining a mode of deformation (e.g. inextensibility) are those that arise from a definite deformation mechanism and leave little doubt about the constitutive properties of the wall. Finally, the presentation sequence is meant to follow increasing degrees of freedom in deforming the cell wall (Fig. 2). Accordingly, I have adopted as a starting point (mode 0), a rigid wall that excludes any type of deformation. Fig. 2. View largeDownload slide Modes of deformation of cell walls. The modes are ordered roughly from the least to the most deformable wall. Mode 0 is a rigid wall element that will not undergo any deformation. All motions can be described as the sum of a solid body displacement and a rotation. Mode 1 is an inextensional bending that preserves all the length within the plane of the wall. Mode 2 is an equi-area shear. The length and width of the strips are conserved, but the strips are allowed to slide with respect to each other. Mode 3 is an elastic stretching, here illustrated in the special case where a family of inextensible cords is present (i.e. l is constant during the deformation). Mode 4 is a processive intussusception of new wall material by enzyme complexes (stars). Mode 5 is a chemorheological growth process where turgor stresses and the rate of wall deposition by secretion and wall synthesis (stars) contribute to the deformation. Fig. 2. View largeDownload slide Modes of deformation of cell walls. The modes are ordered roughly from the least to the most deformable wall. Mode 0 is a rigid wall element that will not undergo any deformation. All motions can be described as the sum of a solid body displacement and a rotation. Mode 1 is an inextensional bending that preserves all the length within the plane of the wall. Mode 2 is an equi-area shear. The length and width of the strips are conserved, but the strips are allowed to slide with respect to each other. Mode 3 is an elastic stretching, here illustrated in the special case where a family of inextensible cords is present (i.e. l is constant during the deformation). Mode 4 is a processive intussusception of new wall material by enzyme complexes (stars). Mode 5 is a chemorheological growth process where turgor stresses and the rate of wall deposition by secretion and wall synthesis (stars) contribute to the deformation. Mode 0: rigid walls Imagine a wall element that is not growing and cannot stretch or bend in any way. Such a wall element is undeformable and can solely undergo rigid body rotations and translations (Fig. 2, mode 0). A cell confined by a continuous layer of this rigid material would be prisoner of its own wall. This explains why most rigid walls are articulated in some way. They come either in the form of overlapping plates (e.g. the scales of coccolithophores) or of imbricating elements (e.g. the frustules of diatoms). Moreover, these rigid walls, unable to deform, must be ‘assembled’ using an organic template as a mould. In diatoms, assembly of the frustule is performed within the silica deposition vesicle where adjacent areolar vesicles help control the deposition of the silica to form the frustule (Pickett-Heaps et al., 1990). For coccolithophore scales, a cellulosic base plate helps define the size and overall symmetry of the future calcium carbonate scale, while many of the finer details of the inorganic scale come from accretion at the surface of CaCO3 crystals (Marsh, 2003; Henriksen et al., 2004; Henriksen and Stipp 2009). The inability of a cell to remodel its wall leads to unusual strategies to accommodate growth and mitosis. In the most extreme cases, the cell must escape the protection of its own wall in order to grow, as in foraminifers (Spero, 1988). In diatoms, the cell’s inability to remodel its wall leads to a reduction in size at each cell division; these cells would vanish if it were not for periodic rounds of sexual reproduction that restore their original size (Pickett-Heaps et al., 1990). Mode 1: inextensional bending I next consider a wall that cannot stretch. More specifically, I consider a thin wall whose mid-plane does not allow any length changes (Fig. 2, mode 1). If it is deformed, the deformation must be length preserving. For such a wall, the only possible deformation is bending, i.e. a change in the curvature of the mid-plane. A sheet of paper is a good analogue of an inextensible wall. Because it is thin, the sheet can bend or fold easily; however, most sheets of paper are sufficiently stiff to prevent any significant stretching. If we were to draw a series of lines on the sheet, no amount of folding and bending would alter the length of these lines. Under what conditions would the constitutive behaviour of the wall approximate inextensibility? Certainly, a convex cell cannot increase its volume without at the same time stretching its wall. Inextensibility is not a mode of deformation that is compatible with cell expansion. In contrast, when a walled cell deforms because it loses volume, say, as a result of desiccation, its wall may comply with the decreasing volume by inextensional bending. Some examples of inextensional deformation with functional significance include the spores of some eusporangiate ferns (Hovenkamp et al., 2009) and the annular cells of fern leptosporangia (Noblin et al., 2012). However, the best illustration of this mode of deformation is the angiosperm pollen grain whose entire suite of structural adaptations seems to be tailored to favour large-scale inextensional bending of its wall (Halbritter and Hesse, 2004; Katifori et al., 2010; Couturier et al., 2013). Pollen grains are live cells that must leave the protective environment of the anther during the pollination process. As with all living cells, their small size makes them vulnerable to rapid desiccation when exposed to air. The danger of dehydration remains until the pollen grain lands on the stigma where it can absorb water before germinating to complete fertilization. Without any protective mechanism against desiccation, most pollen grains arriving at the stigma would be dead or would be in a state of deep dormancy that would preclude quick germination and passing on their genes to the next generation. The near-universal solution to the desiccation problem is harmomegathy (Vesque, 1883; Wodehouse, 1935; Bolick, 1981). Harmomegathy is the characteristic infolding of the pollen grain’s apertures in response to a decreasing cellular volume (Fig. 3A–C). The apertures on the pollen surface provide the main routes for water exchange because, unlike the exine present elsewhere on the pollen surface, their cellulosic wall is not impregnated with sporopollenin and thus remains permeable to water (Heslop-Harrison, 1979a,b). The structure of the pollen wall is designed to allow the apertures to fold inwardly during harmomegathy, thus reducing the rate of water loss (Fig. 3C). Specifically, the apertures function as local soft spots that guide various pollen grains along specific folding pathways (Katifori et al., 2010). For thin shells, bending modes of deformation (i.e. the modes that change the local curvatures of the surface) are known to be energetically less expensive than stretching modes (Box 1). Considering inextensional (bending-only) deformations of the wall, we can arrive at simple solutions for the folding of pollen grains. For a closed convex surface such as a sphere, at least two distinct types of inextensional deformation exist: a mirror reflection of a segment of the surface about a dissecting plane, and folding of the surface past a meridional slit (Fig. 3F). These two types of inextensional deformation provide a good approximation of the deformation observed in inaperturate and aperturate pollen grains, respectively (Fig. 3). Box 1. Thin shells favour bending over stretching Surface elements within an elastic shell can respond to loads in two contrasting ways. They can stretch in the plane of the shell or they can bend (e.g. change their curvature). The question is, what is the best way to deform if the shell is thin, as most walled cells are. It is a well-known result that thin shells favour bending over stretching whenever the loading conditions allow it (Pogorelov, 1988). A simple calculation of the bending and stretching strains illustrates why it is so. Imagine a pollen grain where the edges of the aperture are being pulled toward each other (A) and consider how a narrow equatorial strip ought to deform. The strip can respond to this load by: (i) stretching over the aperture while keeping its radius constant, or (ii) bending into a tighter roll to close the gap (B). We can easily compute the strains associated with these two alternatives. The stretching strains are and the bending strains are , where is the change in curvature and all other variables are illustrated in the figure. The ratio of bending to stretching strains is simply . It is clear from this relation that, for a thin shell (i.e. h <